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Pavel Akimov · Nikolai Vatin
Editors

Proceedings of FORM 2021

Construction The Formation of Living
Environment
Editors
Pavel Akimov Nikolai Vatin
Moscow State University of Civil Peter the Great St. Petersburg Polytechnic
Engineering University
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Contents

Modern Building Materials

Viscoelastic Vibrations of a Layered Composite with Internal
Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Tatiana Bobyleva and Alexey Shamaev
Vibration Damping Problems for Some Models of Viscous Fluids . . . . . . 13
Tatiana Bobyleva and Alexey Shamaev
Calculation of Retention Profiles in Porous Medium . . . . . . . . . . . . . . . . . . 21
Galina Safina
Mechanical and Durability Properties of High-Performance
Concrete in Corrosive Medium of Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Nguyen Duc Vinh Quang, Olga Aleksandrova, and Boris Bulgakov
Modifying Heracleum sosnowskyi Stems
with Monoethanolamine(N→B)-trihydroxyborate
for Manufacturing Biopositive Building Materials . . . . . . . . . . . . . . . . . . . . 45
Irina Stepina, Marc Sodomon, Vyacheslav Semenov,
Elizaveta Dorzhieva, and Irina Titova
Phase Content of Plasticized Cement Systems in the Early Stages
of Heat-Moisture Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Andrei Leshkanov, Lev Dobshits, and Sergey Anisimov
Plasticizer Type Influence on HCP Radiation Resistance . . . . . . . . . . . . . . 65
Vyacheslav Medvedev
Properties of Epoxy Composites with Halloysite Nanotubes
Subjected to Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Evgeniya Tkach and Maxim Bichaev
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice
Husk Ash Content on the Properties of Geopolymer Concrete . . . . . . . . . 85
Tang Van Lam, Pham Van Ngan, and Nguyen Dac Binh Minh

v
vi Contents

Composite Forming by the Method of Prestressing of Carbon

Unidirectional Tape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Vasilii Plevkov, Artem Ustinov, Andrei Plyaskin, Victor Bunkov,
and Yulia Silman
Influence of Superplasticizer and Silica Fume on the Structure
Formation and Properties of Cement Stone . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Aleksandr Smirnov, Lev Dobshits, and Sergey Anisimov
Cellular Structure Formation of Composite Materials . . . . . . . . . . . . . . . . 123
Olga Miryuk
Mechanical Properties of Butyl Rubber Composites
with Microspheres Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Yuriy Yurkin, Amadeo Benavent-Climent, Pavel Kovtonyuk,
Darya Varankina, and Irina Voloskova

Reliability of Buildings and Constructions and Safety in

Construction
Experimental and Theoretical Studies of the Concrete
Static-Dynamic Stress–Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Natalia Fedorova, Michael Medyankin, Sergey Fedorov, and Sergey Savin
Simulation of Effects the Degree of Water-Saturation on Stress–
Strain State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Armen Ter-Martirosyan and Ahmad Othman

Modelling and Mechanics of Building Structures

Elastic–Plastic Equilibrium of a Hollow Ball Made
of Inhomogeneous Ideal-Plastic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Vladimir Andreev and Mikhail Maksimov
Assessments of Solutions of the Uniform Elastic Boundary Value
Problem in the Tip Area of a Boundary Wedge-Shape Notch . . . . . . . . . . . 189
Lyudmila Frishter
Computer Programs Developing for Solving Problems
of Cylindrical Shells Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Stepan Cheremnykh
Estimation of the Vibration Waves Level at Different Distances . . . . . . . . 207
Mirziyod Mirsaidov, Muhammadbobir Boytemirov, and Faxriddin Yuldashev
Mode Shapes of Transverse Vibrations of Rod Protected
from Vibrations in Kinematic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Mirziyod Mirsaidov, Olimjon Dusmatov, and Muradjon Khodjabekov
Contents vii

Pile and Elastic–Plastic Soil Mass Interaction . . . . . . . . . . . . . . . . . . . . . . . . 229

Evgeny Sobolev and Vitalii Sidorov
Regularities of Formation of Residual Stresses in the Fatigue
Crack Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Oleg Emel’yanov
Nonlinear Dynamic Analysis of Truss with Initial Member Length
Imperfection Subjected to Impulsive Load Using Mixed Finite
Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Vu Thi Bich Quyen and Dao Ngo.c Tien
Optimal Scale Modeling of Surf Zone Waves . . . . . . . . . . . . . . . . . . . . . . . . . 259
Izmail Kantarzhi and Alexander Gogin
Features of Numerical Modeling of CFRP Steel Bars . . . . . . . . . . . . . . . . . . 271
Evgeniy Shchurov and Alexander Tusnin
The Concept of Bearing Capacity Distribution in the Supports
of Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Alexandre Danilov and Ivan Kalugin
Force Combinations Inducing Lateral Torsional Buckling . . . . . . . . . . . . . 289
Yana Makzhanova
Dynamic Analysis of Reinforced Concrete Beams on Yielding
Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Oleg Kumpyak, Zaur Galyautdinov, Daud Galyautdinov,
and Tatiana Rakhimova
Modeling of High-Speed Interaction of Composite Barrier
and Steel Striker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Andrei Plyaskin, Nikolai Belov, Nikolai Yugov, Emil Useinov,
and Marina Savintceva
Information Modeling of Wind Flows for Object of Parametrical
Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Galina Kravchenko, Elena Trufanova, and Lyubov Pudanova
Iterative Refinement of the Boundary Condition in the Numerical
Solution of the Thermoelasticity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Filipp Sergeyev and Fyodor Kiselyov
Pochhammer—Chree Wave Dispersion in Hollow Cylinders . . . . . . . . . . . 339
Tagibek Gadzhibekov
Prefabricated Steel Structures with a Corrugated Web (Part 2.
Load-Bearing Capacity of a Steel Beam with a Profiled Sheet Web) . . . . 347
Alexander Ibragimov, Ekaterina Zinoveva, Stanislav Rosinsky,
and Lyubov Gnedina
viii Contents

Evaluation of the Dynamic Behavior of Multi-connected Shell

Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Tulkin Mavlanov, Sherzod Khudainazarov, and Feruza Umarova
Impact of Construction Seams on the Bearing Capacity
of a CVC-RCC Combined Dam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Viktor Tolstikov and Yara Waheeb Youssef
Assessment of Embankment Dam Slope Stability
with Consideration of Its Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Mikhail Sainov
Interaction of Long Piles with a Multilayer Soil Mass, Taking
into Account the Elastic and Rheological Properties . . . . . . . . . . . . . . . . . . 393
Zaven Ter-Martirosyan and Aleksander Akuletskii
Propagation of a Spherical Wave in Elastoplastic Medium
with Complex Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Sherzod Khudainazarov and Burkhon Donayev
Identification Vibration Characteristics of Structures
by Operational Modal Analysis (OMA) Technique . . . . . . . . . . . . . . . . . . . . 421
Trung Duc Tran, Anh Tuan Le, and Dinh Huong Vu
Pile Rows for Protection from Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . 433
Aleksandr Dudchenko, Daniel Dias, and Sergey Kuznetsov
Longitudinal-Transverse Bending of Physically Nonlinear Rods
by Quasi-Static Loads and Mass Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Yury Nemirovskii and Sergey Tikhonov
The Stress–Strain State of Reinforced Concrete Arches with a View
of Concrete Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Serdar Yazyev, Vladimir Andreev, and Leysan Akhtyamova
Differential Equations with Fractional Derivatives for Studying
an Oscillator with Viscoelastic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Alexander Andreev, Temirkhan Aleroev, Mohammad Khasambiev,
and Hedi Aleroeva

Engineering and Smart Systems in Construction

Macrokinetic Model of Biochemical Oxidation . . . . . . . . . . . . . . . . . . . . . . . 487
Vladislava Nikolaevna Volkova and Viktor Leontievich Golovin
Buildings Enclosures Coupling by Its Energy Efficiency, Seismic
Resistance and Microclimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Erkin Boronbaev, Berikbay Unaspekov, Aigul Abdyldaeva,
Kamoliddin Holmatov, and Nurbubu Zhyrgalbaeva
Contents ix

Pre-ammonization in the Preparation of Chromaticity Water

for Drinking Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
Zhanna Govorova, Uliana Rudich, and Oleg Govorov
Verification of Heat Supply System Telemetry Data . . . . . . . . . . . . . . . . . . . 513
Elena Kitaytseva
Groundwater Treatment Plants as a Sustainable Source
of Iron-Containing Nanopowders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Lev Maksimov, Rowan Baker, Ruslan Safargaliev,
Svetlana Maksimova, and Viktor Mironov

Global Environmental Challenges

Particles Transport with Deposit Release in Porous Media . . . . . . . . . . . . . 539
Liudmila Kuzmina and Yuri Osipov
Physically-Based Streamflow Predictions in Ungauged Basin
with Semi-Arid Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Dmitry Kozlov and Anghesom Ghebrehiwot

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Modern Building Materials
Viscoelastic Vibrations of a Layered
Composite with Internal Friction

Tatiana Bobyleva and Alexey Shamaev

Abstract The article considers the problem of viscoelastic vibrations of a layered

composite material of two pairwise alternating layers. Three types of layer materials
with different properties are considered. The first case is two elastic materials with
internal dissipation of mechanical energy, which is described by an integral term
of the convolution type with an exponential kernel. The second case is a layered
composite of elastic materials with Kelvin-Voigt friction, and finally, in the third
case, the internal dissipation in the layers of the material is described by the fractional
Kelvin–Voigt friction. We consider the transverse vibrations of this layered composite
and give a qualitative picture of the behavior of the spectra. The purpose of this work
is to reveal the influence of internal friction for its various models on the vibration
spectrum. This technique can be used to create building materials with predetermined
properties, for example, for sound insulation.

Keywords Homogenization · Viscoelasticity · Eigenvalues · Spectral problem ·

Abelian kernel

1 Introduction

Our goal is to study the vibration spectra of a layered composite with various models
of internal friction, in particular, with the Kelvin-Voigt fractional friction model. By
fractional Kelvin-Voigt friction, we mean such internal friction, the defining relation
for which contains not the usual time derivative of the strain tensor, but the time
derivative of a non-integer positive order less than unity.
The development of a method for calculating the vibration spectra of composites
which consist of materials with internal dissipation can be used to design composite

T. Bobyleva (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
A. Shamaev
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Pr.
Vernadskogo, 101-1, Moscow 119526, Russia
Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 3

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_1
4 T. Bobyleva and A. Shamaev

materials with specified properties. In particular, the proposed technique can be used
to create building materials with extreme parameters of vibration protection and noise
insulation for given strength properties of the composite material being created. As
is well known, such an operator of non-integer differentiation is an integral oper-
ator with an Abelian kernel. Fractional derivatives are used to study materials with
power-law nonlocality, power-law long-term memory or fractal properties [1–4]. The
homogenization theory of inhomogeneous bodies and issues of averaging resonant
frequencies of elastic composites were considered in [5–8]. The book [9] describes
the mechanics and thermodynamics of solid deformable bodies in a viscoelastic
solid state. Applications of this theory are given in [10]. Natural frequencies and
damping coefficients of multilayer composites were studied in [11, 12]. The prob-
lems in which the averaging theory was applied for elastic materials with memory
were considered in [13–15]. Theoretical calculations can be verified by experiment,
as in [16] the EMAR (electro-magnetic acoustic resonance) method is given, which
allows to measure the resonant frequencies of free vibrations of a material, and the
non-contact nature of this method also allows to accurately measure the internal
friction of materials such as composites, ceramics, porous materials. Multilayered
models are commonly used for behaviour structures on soil foundations and for
high-frequency behaviour of railway tracks [17]. The article [18] provides a compre-
hensive overview on various theoretical models of elastic and viscoelastic founda-
tions in oscillatory systems with applications in structural mechanics, nanosystems,
composite structures.
The presence of internal friction makes it necessary to consider a math problem
that is the spectral problem for a non-self-adjoint operator. Its spectrum will contain
both real and complex eigenvalues. We establish qualitative differences in the
behaviour of the spectra for three different models of internal friction and propose
a constructive method for calculating both complex and real values of the elements
(points) of these spectra by solving algebraic equations in the complex plane.

2 Problem Specification and Decision

We consider the problem of finding the spectrum of natural vibrations of a layered

composite for three cases of materials from which the layers are made (Fig. 1),
namely, a visco-elastic material, a material with Kelvin-Voigt internal friction and a
material with Kelvin-Voigt fractional internal friction.
The case of a layered composite of two elastic materials without taking into
account internal friction was considered in detail earlier [5]; in this case, the vibration
spectrum is purely imaginary.
The aim of this paper is a comparative analysis of the vibration spectra of a
viscoelastic layered composite for three different cases of internal energy dissipation
in the process of vibrations.
The first one is a model of viscoelastic oscillations with a time-distributed memory,
which is described by convolution with a damped exponential kernel.
Viscoelastic Vibrations of a Layered Composite … 5

Fig. 1 Layered structure of

the model, ε is the
characteristic size of the
periodicity cell

The second case is the classical Kelvin–Voigt model of internal friction, which
is given by the time derivative of a tensor describing internal scattering during
motion. This tensor contains components of the deformation tensor with factors
that determine dissipation. Only local operators are present in this model.
The third model is the Kelvin–Voigt fractional friction model, when, instead of
the integer time derivative, the operator of fractional time differentiation with a
positive order less than one is used for vibrations of a layered composite sample,
strictly perpendicular to the layers and with the conditions for fixing the upper and
lower layers, it is possible to reduce the problem of constructing the spectrum of
such vibrations to the algebraic problem of nalysing the roots of polynomials in the
complex domain. Consider these three cases.

2.1 Viscoelastic Layered Material

The equations of motion in this case are as follows:

ρ(x3 )ü i = (σij )xj , (1)

where

σi j = cjj kl ekl + djj kl (t) ∗ ekl , (2)

t
djj kl (t) ∗ ekl = djj kl (t − τ ) ekl (τ ) d τ, (3)
0

1 
ekl = (u + u xl ) (4)
2 xk
6 T. Bobyleva and A. Shamaev

In formulas (1–4) in the Cartesian coordinate system (x1 , x2 , x3 ), the following

designations are introduced: σi j (i, j = 1 ÷ 3) are components of the stress tensor, u iε
are components of the displacement vector, cjj kl (i, j, k, l = 1 ÷ 3) are components
of the elastic modulus tensor, djj kl (t) (i, j, k, l = 1−3) are Volterra integral operators
(3), ρ is a material density, two dots represent the second time derivative, t is a time
variable.
Components of the elastic tensors and Volterra integral operators in (2 and 3) have
the form (5 and 6) since isotropic materials are considered below [9]:

cjj kl = λδij δkl + μ(δik δjl + δil δjk ), (5)

1 1
djjkl = −(Dv (t) − Dsh (t))δij δkl − Dsh (t)(δik δjl + δil δjk ). (6)
3 2
We denote here by Dsh and Dv the regular part of the shear and the bulk
viscosity respectively, by δij Kronecker symbol. Suppose that the amplitude of a
bulk viscosity kernel is proportional to the amplitude of the shear viscosity kernel
with a proportionality coefficient ks for each layer, that is: (Dv )s = ks (Dsh )s , ks =
const, ks > 0, (s = 1, 2).
Further, Dsh is denoted by D.
In this problem, all elastic and viscoelastic modules are periodic functions of
the coordinate y = xε3 , (ε is the characteristic size of the periodicity cell) and are
piecewise constant functions of this variable, i.e., elastic modulus, material density
and viscosity kernel have the form [13]:
 
λ1 , y ∈ [0; h] μ1 , y ∈ [0; h]
λ(y) = , μ(y) = ,
λ2 , y ∈ [1 − h; 1] μ2 , y ∈ [1 − h; 1]
 
ρ1 , y ∈ [0; h] D1 (t), y ∈ [0; h]
ρ(y) = , D(y, t) =
ρ2 , y ∈ [1 − h; 1] D2 (t), y ∈ [1 − h; 1].

We denote here by λs , μs , (s = 1, 2) Lame parameters for each layer and choose

Volterra integral operators kernels of exponential type for each layer: Ds = gs e−αs t ,
where αs , gs are constants, αs > 0,gs > 0.
Let us consider the case of one-dimensional motion in a direction strictly
perpendicular to the plane of the layers.

∂ 2 (u 3 )s ∂ (σ33 )s
ρs = , (7)
∂t 2 ∂ x3
t
(σ33 )s = L s (e33 )s + G s e−αs (t−τ ) (e33 )s dτ . (8)
o

 
Here are L s = λs + 2μs , G s = gs ks + 23 , (s = 1, 2).
Viscoelastic Vibrations of a Layered Composite … 7

We apply to (7), (8) the Laplace transform in time:

∞
f˜( p) = f (t)e− pt dt. (9)
0

The result is the equation of elasticity theory with a complex parameter p. Since
we are considering a layered material, we apply to this equation homogenization
method described in [5, 6]:

1
ϕ̂ = ϕ = ϕ (y) dy. (10)
0

As a result, we get
 −1 −1
G ∂ 2u0
p ρu 0 =
2
L− (11)
p+α ∂ x32

Here, for brevity, u 0 is the displacement, that is the solution to the averaged
problem
 of the theory of viscoelasticity in Laplace images, and L = λ + 2μ, G =
g k + 23
After performing the averaging operation, we have

p 2 [ρ1 h + ρ2 (1 − h)]u 0

[L 1 ( p + α1 ) − G 1 ][L 2 ( p + α2 ) − G 2 ]
=
( p + α1 )h[L 2 ( p + α2 ) − G 2 ] + ( p + α2 )(1 − h)[L 1 ( p + α1 ) − G 1 ]
∂ 2u0
× ,
∂ x32
 
Here are L s = λs + 2μs ,G s = gs ks + 23 , (s = 1, 2).
The previous equation can be written like this

∂ 2u0
− au 0 = 0, (12)
∂ x32

where the variable a depending on the complex parameter p is defined by the next
formula

p 2 [ρ1 h + ρ2 (1 − h)]{( p + α1 )h[L 2 ( p + α2 ) − G 2 ] + ( p + α2 )(1 − h)[L 1 ( p + α1 ) − G 1 ]}

a=
[L 1 ( p + α1 ) − G 1 ][L 2 ( p + α2 ) − G 2 ]
(13)
8 T. Bobyleva and A. Shamaev

We need to add Dirichlet boundary conditions on the horizontal planes of the

sample:

u 0|x3 =0 = 0, u 0|x3 =H = 0, (14)

where H is the thickness of the layered sample.

We have reduced the original problem to the classical Sturm–Liouville problem.
Its eigenvalues can be obtained by equating the value a, which depends on the complex
parameter p, to the values ak :

π 2k2
ak = − , k = 1, 2, . . . (15)
H2
For each k = 1,2, … it is necessary to determine the roots of the algebraic equation
a = ak . The whole spectrum is obtained by combining the set of these roots by k =
1,2, …

2.2 Viscoelastic Layered Material with Internal Kelvin-Voigt

Friction

In this case, the relationship between stress and strain tensors is as follows:

σi j = cjj kl ekl + vjj kl (t)ėkl (16)

In (16), the tensor cjj kl is determined by formula (5), and the viscous stress tensor
vjj kl has a similar form:

vjjkl = βδij δkl + η(δik δjl + δil δjk ), (17)

where β, η are viscosity coefficients [19, 20].

Similarly to the previous subsection, we apply the Laplace transform (9–16) and
perform the averaging procedure (10). As a result, we get

−1 ∂
2
u0
p 2 ρu 0 = (L + pN )−1 (18)
∂ x32

There are L = λ + 2μ, N = β + 2η in (18). After transformations, we have

∂ 2 ũ
− a ũ = 0 (19)
∂ y2

where we denoted
Viscoelastic Vibrations of a Layered Composite … 9

p 2 [ρ1 h + ρ2 (1 − h)][h(L 2 + pN2 ) + (1 − h)(L 1 + pN1 )]

a= (20)
(L 1 + pN1 )(L 2 + pN2 )

The spectrum in this case is obtained by combining the solutions of the algebraic
equations

π 2k2
a = ak = − , k = 1, 2, . . . (21)
H2

2.3 Visco-elastic Layered Material with Fractional Internal

Kelvin-Voigt Friction

In this case, the relationship between stress and strain tensors is as follows:
γ
σi j = cjj kl ekl + wjj kl d t ekl (22)

In (22), the tensor cjj kl is determined by formula (5), and the viscous stress tensor
γ
wjj kl has a similar form wjjkl = ξ δij δkl + ς (δik δjl + δil δjk ), operator d t is the operator
of fractional order γ differentiation in time t, which is given by the following formula

t
γ 1
d t ekl = (t − τ )γ −1 ėkl dτ (23)
(γ )
0

(γ ) is the Euler gamma function in (23). Performing the Laplace transform and
averaging similarly to the previous subsections, we obtain

−1 ∂
2
u0
p 2 ρu 0 = (L + M p γ )−1 (24)
∂ x32

There are L = λ + 2μ, M = ξ + 2ς .

1
Next, we will consider the case γ = 21 and introduce the variable p 2 = q. In this
case, the equation for determining the eigenvalues takes the form:

q 4 [ρ1 h + ρ2 (1 − h)][h(L 2 + q M2 ) + (1 − h)(L 1 + q M1 )] π 2k2

= − 2 , k = 1, 2, . . .
(L 1 + q M1 )(L 2 + q M2 ) H
(25)

In this case, when combining by k = 1, 2,…. One should take into account only
those roots of this equation lying in the left half-plane, since the roots lying in the
right half-plane have no mechanical interpretation. As a result, the spectrum will be
obtained as a union of roots for all natural k.
10 T. Bobyleva and A. Shamaev

3 Conclusions

The algebraic expressions we found for the elements (points) of the spectra of trans-
verse vibrations make it easy to calculate their values using the standard software for
calculating the roots of polynomials in the complex plane. The results obtained for
various models can be compared with each other and with experimental data. In addi-
tion, a qualitative picture of the spectra of the problems under consideration can be
constructed using asymptotic analysis, since the algebraic equations for calculating
the spectra contain a large integer parameter k.
This analysis shows that in the first case, in comparison with the spectrum of a
purely elastic composite, the spectrum shifts to the left half-plane, and it has a vertical
asymptote.
In the second case, the spectrum contains only a finite number of points describing
damped oscillations. Other points of the spectrum are real and describe motions that
asymptotically lead to rest without oscillation. The third case can be considered as
something intermediate between the first and the second, here there are an infinite
number of elements (points) of the spectrum describing the oscillatory motion, but
with an increase in the number of the eigenvalue, not only the frequency increases
indefinitely, but also the damping decrement, there can be no vertical asymptote.
Our results can serve as a basis for the analysis of more complex problems in the
analysis of vibrations of composite materials in the presence of dissipation, and also
help in choosing a model describing the dissipation of mechanical energy during
vibration of samples made of composite materials. The formulas obtained can form
the basis for the formulation and construction of solutions for the optimization of
vibration and noise protection properties of the building composite materials being
created. It is obvious that the solution of such problems is important for the design
of building materials with desired properties.

Acknowledgement The work was done on the subject of the state assignment of the IPMech RAS
AAAA-A20-120011690138-6.

References

1. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential
equations. Elsevier, Amsterdam
2. Tarasov VE (2013) International Journal of Modern Physycs B 27:9
3. Carpintery A, Mainardi F (1997) Fractals and fractional calculus in continuum mechanics.
Springer, Wien
4. Evans RM, Katugampola UN, Edwards DA (2017) Comp Math Appl 73
5. Oleynik OA, Shamaev AS, Yosifian GA (1992) Mathematical problems in elasticity and
homogenization. Elsevier, North-Holland
6. Bardzokas DI, Zobnin AI (2005) Mathematical modeling of physical processes in composite
materials of periodic structure. URSS, Moscow
7. Pobedrya BE (1984) Mechanics of composite materials. MSU, Moscow
Viscoelastic Vibrations of a Layered Composite … 11

8. Christensen RM (2005) Mechanics of composite materials. Dover, New York

9. Ilyushin AA, Pobedrya BE (1970) Foundations of the mathematical theory of thermovisco-
elasticity. Nauka, Moscow
10. Pobedrya BE (1979) Mech Compos Mater 15
11. Shamaev AS, Shumilova VV (2018) IFAC PapersOnLine 51:2
12. Shamaev AS, Shumilova VV (2020) Doklady Phys 65:4
13. Bobyleva TN, Shamaev AS (2017) Soil Mech Found Eng 54:4
14. Bobyleva TN, Shamaev AS (2018) MATEC Web of Conf 251:04039
15. Bobyleva TN, Shamaev AS (2018) IFAC PapersOnLine 51:2
16. Hirano M, Ogi H (2017) Elastic constants and internal friction of advanced materials.
Electromagnetic acoustic transducers, Springer, Tokyo
17. Connolly D, Kouroussis G, Laghrouche O, Ho C, Forde M (2015) Constr Build Mater 92
18. Younesian D, Hosseinkhani A, Askari H, Esmailzadeh E (2019) Nonlinear Dyn 97
19. Petrov N, Brankov Y (1986) Modern problems of thermodynamics. Mir, Moscow
20. Zarubin VS, Kuvyrkin GN (2002) Mathematical models of thermodynamics. Fizmatlit,
Moscow
Vibration Damping Problems for Some
Models of Viscous Fluids

Tatiana Bobyleva and Alexey Shamaev

Abstract This article deals with control problems for dynamical systems with non-
local convolution type terms. A method is proposed to get conditions under which
the moving system will go into complete rest. The force acting on the system is
distributed over the entire moving domain. Domains of one, two and three dimensions
are considered. For these three cases of dimension and two types of fluids (Oldroyd
fluid and Kelvin-Voigt fluid), the initial conditions are formulated for the problem
posed, with the help of which the proposed method can bring these systems to
complete rest in a finite time. Sufficient conditions are given that must be satisfied by
the initial oscillations of the systems, under which the spectral method we use can
bring these initial oscillations to complete rest. This is a condition on the smoothness
of the initial functions and some additional boundary conditions for them. The article
presents a new technique for damping unwanted vibrations in visco-elastic building
materials.

Keywords Control functhion · Viscoelasticity · Damping · Eigenvalues · Spectral

problem

1 Introduction

The design and construction of high-rise buildings requires ensuring their safety
under wind and seismic loads, as well as the ability to control the basic dynamics
and characteristics of building structures. One way of solving these problems is the
use of vibration control systems. With the help of such control systems, it is possible to
limit unwanted deformations, displacements and stresses, and to control the dynamic
characteristics. Controlling forces will resist external influences. This paper proposes

T. Bobyleva (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
A. Shamaev
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Pr.
Vernadskogo, 101-1, Moscow 119526, Russia
Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 13

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_2
14 T. Bobyleva and A. Shamaev

an algorithm for damping vibrations in Oldroyd and Kelvin-Voigt fluids. Such visco-
elastic compositions can be used as backfill materials for repair and insulation works
in especially difficult conditions. These methods can also be used to damp vibrations
of viscoelastic fragments of building structures, as well as fragments of machines and
mechanisms using active elements (actuators). Such modern methods of stabilizing
structures have now begun to find practical application. Another example is geosyn-
thetic materials, which are finding ever new areas of application in construction;
therefore, the question of the influence of their viscoelastic properties on dynamic
effects is relevant.
This article presents a method for bringing viscoelastic mechanical systems to rest
in a finite period of time. This method can also be used to bring to the rest vibrations
of rods, plates, shells and other elastic bodies. Spectral theory is the basis of this
method. As viscoelastic mechanical systems, the Oldroyd fluid and the Kelvin-Voigt
fluid are considered for the cases of various domains differing in both dimension and
shape.
Many Russian and foreign works are devoted to the problems of control of mechan-
ical systems with integral aftereffect. It is proved by controlling one end of the string
one can stop the vibrations of the string in a finite time [1–3]. Control problems in
the case when the behavior of the system is described by the Gurtin-Pipkin equation
were considered in [4–6], it was found that it is possible to control such a system
using a limited external force, which is distributed over the entire domain under
consideration. In [7], it was proved that this cannot be done in the case when the
force is applied only to some part of the domain. Evolution equations with memory
were studied in [8]. This work shows that although it is impossible to control the
entire domain using a force on a part of it, but if a force is applied to a moving
subdomain, then the control problem is solvable under certain conditions. This paper
discusses integro-differential models that are often used in the study of viscoelastic
systems [9, 10]. The correctness of models (1)–(2) was investigated in [11]. It can
be shown that in these problems controllability takes place with the help of a force
applied to the entire area (to a segment), and the absolute value of the force can be
of little value. It was shown in [12] that for a number of systems without an inte-
gral aftereffect, the motion can be completely stopped by a force limited in absolute
value and applied to the entire region, under certain conditions on the initial data. In
[13], for a mechanical system defined by a linear integro-differential equation with
a non-local term of the convolution type, the possibility of damping oscillations in
a finite period of time for any initial conditions was proved. In the works [14–16]
it is proved that even with the tightening of the requirements for the control force,
the controllability of the system is not lost, and on the whole the qualitative picture
corresponds to [1]. The methods used in these works were developed in [17–19].
Dynamic problems for systems with integral time delay are presented in [20].
The spectral method is used in this article to solve the problem of damping fluid
vibrations in two-dimensional and three-dimensional regions.
Vibration Damping Problems for Some Models of Viscous Fluids 15

2 Problem Specification and Decision

Consider the following vibration control problems for simplified Oldroyd and Kelvin-
Voigt fluid models. The simplification consists in considering the scalar case and in
the absence of pressure. The domain Q can be a one-dimensional interval, a square, a
cube, a two-dimensional or three-dimensional bounded domain, that is, a total of five
cases. The equations of fluid dynamics for the Oldroyd and Kelvin-Voigt models,
the boundary and initial conditions have, respectively, the following form:
⎧
⎪ t
⎪
⎨ u̇ − c2 u + K e−λ(t−τ ) u(τ, x)dτ = f (t, x) in Q × [0, T ],
0 (1)
⎪
⎪ u(t, x) = ϕ(x), u(t, x) = 0 on ∂ Q × [0, T ],
⎩
| f (t, x)| ≤ ε, f (t, x) is a contr ol f unction,
⎧
⎪ t
⎪
⎨ u̇ − c2 ∂t∂ u + K e−λ(t−τ ) u(τ, x)dτ = f (t, x) in Q × [0, T ],
0 (2)
⎪
⎪ u(t, x) = ϕ(x), u(t, x) = 0 on ∂ Q × [0, T ],
⎩
| f (t, x)| ≤ ε, f (t, x) is a contr ol f unction.

The problem is to build a distributed control f (t, x) over the entire area Q. Func-
tion f (t, x) satisfies the next constraint | f (t, x)| < ε, and it is such that u(t, x) ≡ 0
for T > t ≥ T ∗ > 0. Thus, our problem is to stop fluid oscillations using control
function f (t, x) in a finite time.
To construct this control function, we will apply the spectral method, which
consists in finding f (t, x) in the form of a decomposition
∞

f (t, x) = cn (t)vn (x), (3)
n=1

where cn (t) are time functions to be determined t, and {vn (x)} is the system of
eigenfunctions of the Dirichlet problem for the Laplace equation in the domain
Q. The search for a solution in this form will lead to the problem of stopping the
oscillations of the counting system of pendulums, the specific form of which will
be given below. To implement the spectral method, we need some estimates for the
eigenfunctions and Fourier coefficients.
Let {vn (x)} and {λn } be sets of eigenfunctions and eigenvalues of the Laplace
operator in a domain Q with a smooth boundary, that is, v = λn vn in Q, moreover
vn|∂ Q = 0 and  vn  L 2 (Q) = 1 for n = 1, 2, . . . .
We estimate the value  vn  C(Q) = max| vn (x)|.
x∈Q
According to S. L. Sobolev’s lemma [21]  vn  C(Q) ≤ C0 (d) vn  [ d2 ]+1 ,
H (Q)
where C0 (d) > 0 is a constant, independent of vn (x), d is the dimension of space.
We consider the next values d = 1,2, 3. Then the following is true
16 T. Bobyleva and A. Shamaev

 vn  C(Q) ≤ C0  vn  H 2 (Q) ≤ C1 |λn | ·  vn  L 2 (Q) ≤ C1 |λn |, (4)

based on the estimate for the solution of the elliptic boundary value problem u =
f (x) in Q, u | ∂ Q = 0, having the form

 u H 2 (Q) ≤ C̃ f  L 2 (Q) (5)

The multiplier on the right side does not depend on f . Now let u ∈ H (2k) (Q)
and function u(x) also satisfy the boundary conditions.
s−1
u | ∂ Q = 0, . . . , [ 2 ] u | ∂ Q = 0, s = 0, 1, . . . , k, k is some integer.

∞
Let u(x) = cn u n (x) be an expansion of a function u(x) in a Fourier series in
n=1 
terms of eigenfunctions {vn (x)}, cn = u(x) · vn (x)d x.
Q

∞
According to well-known results [21] we have cn2 |λ n |k < ∞. Hence it follows
n=1
cn2 |λ n |k ≤ c̄, c̄ > 0 does not depend on n = 1, 2, . . . .
Also it’s known behavior of the eigenvalues of elliptic boundary value problems
2
in a bounded domain | λ n | ∼ K · n d , consequently,
2
| c n| ≤ K · n d (6)

The last estimate shows how the Fourier coefficients of the expansion of the
function u(x) ∈ H 2k (Q) decrease as the smoothness of the function (parameter k)
increases, and k boundary conditions are satisfied on the boundary of the region ∂ Q.
Without the specified boundary conditions, it is impossible to assert an increase the
rate of decrease |cn | at n → ∞.
The maximum modulus of eigenfunctions satisfies the estimate
2
 vn  C(Q) ≤ K · n d (7)

Using the example of a circle-shaped domain Q on the plane (d = 2), it can be

shown that at n → ∞ the maximum of the modulus of the normalized in L 2 (Q)
eigenfunctions does indeed increase. In this case, the real growth rate of the modulus
of eigenfunctions will be less than the estimate we have obtained. Getting an accurate
estimate for a circle is a separate problem related to the theory of Bessel functions,
since for the eigenfunctions of Laplace operator in a circle of radius r0 , normalized
in L 2 (K r 0 ), (where K r 0 is a circle of radius r0 ) there is a representation Jn (μ(n)
m ·
−1
r ) sin(n θ ) · r0 Jn (μ(n)
m , where (r, θ ) are polar coordinates, μ (n)
m is m-th root
of the Bessel function Jn (x). For a closed line segment, a square, and a cube, the
maxima of the normalized eigenfunctions are limited in aggregate due to the explicit
representation {(sin nx) sin(my) sin(l z)}, m, n, l = 1, 2, . . . dl d = 3.
The following statement holds.
Vibration Damping Problems for Some Models of Viscous Fluids 17

Statement 1
Consider the next sequence of systems
⎧
⎨ ü n + An u̇ n + An u n = gn (t), n = 1, 2, . . .
2 2
2
u n (0) = an , u̇ n (0) = bn n d , (8)
⎩
|gn (t)| ≤ cεn −( d +1+γ ) .
2

Here gn (t) is a control function, ε is an arbitrary positive number. Let be, An ∼

c1 n, Bn ∼ c2 n, an ≤ c3 , bn ≤ c4 , c1 , c2 , c3 , c4 , c, γ are some constants. Then
there are a constant T > 0 independent of n and a control function gn (t), such that
u n (t) ≡ 0 at t ≥ T and gn (t) ≡ 0 if t ≥ T, n = 1, 2, . . ..
Proof.
The general solution of (8) has the form.

bn − αn an αn t
u n (t) =an eαn t cos βn t + e sin βn t
βn
t
1
+ eαn (t−τ ) sin βn (t − τ )gn (τ ) dτ , (9)
βn
0

where αn ± βn are the roots of the characteristic equation.

We will look for a control function gn (t) in the form

gn (t) = C1n (T )e−αn t cos βn t + C2n (T )e−αn t sin βn t, (10)

where C1n (T ), C2n (T ) are to be determined from the conditions u n (T ) = u̇ n (T ) =

0. Calculations show that the following inequality holds: C1n (T ) + C2n (T ) ≤
|γn |
C exp(γn T ), γn < 0, |α n|
> 1 + δ, n = 1, 2, . . . , δ > 0 is some constant.
For the two real roots of the characteristic equation, control function gn (t) should
also be sought in the form
(1) (2)
gn (t) = K 1n (T )e−λn t + K 2n (T )e−λn t , (11)

where λ(1) (2)

n ≤ λn < 0 are two negative real roots of the characteristic equation.
In this case, it is also easy to show that K 1n (T ) + K 2n (T ) ≤ C exp(γn T ), γn <
|γn |
0, |α n|
> 1 + δ, n = 1, 2, . . ., δ > 0 is some constant. Hence it can be seen that for
a sufficiently large T there will be gn (t) ≤ ε, if t < T, ε > 0 is given small number,
and moreover T > 0 can be chosen independent of n = 1, 2, . . ..
Let us reduce systems (1), (2) to systems for oscillators. To do this, we differentiate
Eq. (1) with respect to t, then we multiply the result by λ and add it to (1). For the
Oldroyd fluid model, we obtain a system that does not contain integral terms:
18 T. Bobyleva and A. Shamaev
⎧
⎨ ü − c2 u̇ + λu̇ + (K − λc2 )u = g(t, x),
u |t=0 = ϕ(x), (12)
⎩
u̇ |t=0 = ψ(x), u | ∂ Q×[0,T ] = 0,

here g(x, t) = f˙(x, t) + λ f (x, t), ψ(x) = c2  ϕ(x).

Let us now differentiate Eq. (12) and add it with (2). For the Kelvin-Voigt fluid
model, we also obtain a system without integral terms:
⎧
⎨ ü + λu̇ − c2 ü + K u − c2 u̇ = g(t, x)
u |t=0 = ϕ(x), (13)
⎩
(u − c2 u )˙ |t=0 = 0, u | ∂ Q×[0,T ] = 0

The second of the initial conditions in (13) can be easily obtained from (2) if we
assume f (0, x) = 0. Without this assumption, the second initial condition would be
inhomogeneous.
Our task is to bring to rest the oscillations of systems (12)–(13) using the control
function g(t, x), limited in modulus. Then, as it is easy to see, it is possible to
construct a limited, small in absolute value control f (t, x) for systems (1), (2) with
nonlocal terms of the convolution type, which bring the oscillations of these systems
to complete rest in a finite time. For this, it suffices to use the equality:

f (t, x) = e−λ(t−τ ) g(τ, x)dτ

0

Let us apply the method of expansion in a Fourier series to reduce the problems
posed to the problems of controlling a countable number of the simplest oscillatory
systems.
For the Oldroyd fluid model, the indicated system of oscillators will take the form:
⎧
⎨ ü n − (c2 λn − λ) u̇ n + (K − λc2 )λn u n = gn (t),
u n (0) = ϕn , (14)
⎩
u̇ n (0) = c2 λn ϕk ,


∞ 
∞
where u(t, x) = u n (t)v n (x), g(t, x) = g n (t)v n (x).
n=1 n=1
2 
∞ 2
Because |v n (x)| ≤ Cn d , |g(t, x)| < ε, if n d |gn (t)| ≤ ε, for the last
n=1
inequality to be satisfied, it is sufficient that |g n (t)| ≤ 2
C
, gde δ = const > 0.
n d +δ+1
It was previously found that |ϕ n | ≤ K n − , λn ∼ n . An integer k = 1, 2, . . .
k 2
d d

determines the smoothness of the initial condition ϕ(x) and the number of additional
conditions for ϕ(x) on ∂ Q, sufficient for the specified estimate of |ϕ n |. Using State-
ment 1, it is easy to prove that in the case of the Oldroyd fluid model it is sufficient to
Vibration Damping Problems for Some Models of Viscous Fluids 19

require the existence of second derivatives, and no additional conditions are required
on the boundary of the domain Q.
For the Kelvin-Voigt fluid model, the corresponding system of oscillators is:
⎧ λ−c2 λn K ·λn gn (t)
⎨ ü n + 1−c2 λn
u̇ n + 1−c 2λ un =
n 1−c2 λn
,
u n (0) = ϕn , (15)
⎩
u̇ n (0) = 0.

The difference between the system of oscillators corresponding to the Kelvin-

Voigt model is that as the oscillator number tends to infinity, the oscillation frequency
and damping decrement tend to finite values, while for Oldroyd’s fluid they increase
indefinitely. The proof of an analogue of Statement 1 for this case and further simple
estimates show that in this case, too, the condition k > 4+d is sufficient for reducing
the entire countable system of oscillators in a finite time at rest by means of a control

∞
{ g n (t)}, such that g n (t)v n (x) < ε, namely |g(t, x)| < ε.
n=1

3 Conclusions

In both cases, for the Oldroyd and Kelvin-Voight fluid models, it is possible to bring
the oscillations to a state of complete rest in a finite time using a force distributed
over the entire region of a limited absolute value. However, the Kelvin-Voight model
requires additional conditions on the smoothness of the initial function, as well as
additional boundary conditions for it. Of course, these conditions are only sufficient,
we do not present any necessary conditions in this work. In our opinion, controlla-
bility for the subdomain is impossible here. An interesting question is whether it is
possible to damp fluid oscillations when a control force is applied to a moving subdo-
main. Similar methods of distributed control can be easily transferred to the model of
a viscoelastic deformable rigid body. Recently, such methods are finding increasing
practical application for stabilizing the elements of machines and mechanisms, as
well as visco-elastic fragments of building structures.
The work was supported by a Grant from the Russian Science Foundation №
21-11-00151.

References

1. Butkovskiy AG (1965) Theory of optimal control of distributed parameter systems. Nauka,

Moscow
2. Lions JL (1972) Optimal control of systems governed by partial differential equations. Mir,
Moscow
3. Lions JL (1988) Soc Ind Appl Math Rev 30:1
4. Pandofi L (2005) Appl Math Optim 52
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5. Ivanov S, Pandofi L (2009) Math Analysis and Appl 355

6. Romanov IV, Shamaev AS (2018) J Math Sci 234:4
7. Shamaev AS, Romanov IV (2016) J Optim Theory App 170:3
8. Chaves-Silvia F, Zhang X, Zuazua E (2017) arXiv:1705.07683 [math.OC]
9. Ilyshin AA, Pobedrya BE (1970) Foundations of the mathematical theory of thermovisco-
elasticity. Nauka, Moscow
10. Christensen RM (2010) Theory of viscoelasticity. Dover, New York
11. Oskolkov AP (1989) Proc Steklov Inst Math 179
12. Chernousko FL (1992) Appl Math Mech 56:5
13. Bobyleva T, Shamaev A (2020) Inst Phys Puplishing Conf Ser Mater Sci Eng 869, 022011
14. Romanov IV (2011) Moscow University Bulletin Ser 1, Math Mech 2
15. Romanov IV Shamaev AS (2016) Proc Russ Acad Sci 94
16. Romanov IV, Shamaev AS (2018) arXiv:1603.01212
17. Russel DL (1978) Soc Ind Appl Math Rev 20
18. Lagnese JE (1989) Boundary stabilization of thin plates. SIAM, Philadelphia
19. Kapitonov BV (1993) Russian Acad Sci. Sb Math 76:2
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21. Mikhailov VP (1978) Partial differential equations. Mir Publishers, Moscow
Calculation of Retention Profiles
in Porous Medium

Galina Safina

Abstract To construct the foundations, tunnels and underground structures in loose

rocks, the soil has to be strengthened and protected from formation and flood waters.
Liquid grout pumped into the loose soil under pressure, filters in the pores and
strengthens the porous rock. A porous rock with a hardened hardener forms a
new building material. The use of reinforcements with dissimilar particles makes
it possible to obtain a variety of building materials with desired properties. The
paper purpose is modeling of filtration of a bidisperse suspension or colloid in a
porous media with linear filtration and concentration functions and a size-exclusion
particle capture mechanism. The problem is solved numerically by the finite differ-
ence method. The retention profiles of large and small particles and the total retention
are constructed for different times. The main result of the work is the uneven distri-
bution of the sediment depending on the particle size. The profiles of large particles
always decrease monotonically, the profiles of small particles decrease monotoni-
cally at a short time and monotonically increase at a long time. The profiles of the
total retention retain or change their monotonicity depending on the parameters of
the problem. At some time, a maximum point appears on the plots of non-monotonic
profiles, moving from the entrance to the exit of the porous media with increasing
time. The limit velocity of maximum points movement of non-monotonic profiles
depends on the model coefficients. This makes it possible to describe the properties
of materials obtained by filtration of bidisperse suspensions in a porous medium
theoretically.

Keywords Deep bed filtration · Suspension · One-dimensional filtration problem ·

Retention profile · Large and small particles

1 Introduction

In order to construct tunnels and underground structures, it’s necessary to strengthen

the loose rock and make a waterproof wall in the ground. Liquid grout pumped into

G. Safina (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 21

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_3
22 G. Safina

the loose soil under pressure, filters in the pores, strengthens the porous rock and
prevents the flow of formation water [1, 2].
When filtering suspensions and colloids in a porous media, some particles pass
freely from entrance to exit, others stick in the pores and form sediment. With deep
bed filtration, the deposit forms throughout the porous media, not just at the entrance.
Particles are retained in the porous media either in the throats of narrow pores or in
wide pores under the influence of electric, hydrodynamic, gravitational forces, diffu-
sion into dead-end pores, etc. [3]. When filtering a 2-part suspension, the yyvxyte
is unevenly distributed. The aim of this work is to study the distribution of deposited
particles on the framework of a porous media, depending on the type of particles.
To study the suspension filtration in a porous media, mathematical modeling
methods are used [4, 5]. The classical model of deep bed filtration of a monodis-
perse suspension in a porous media with a size-exclusion particle capture mechanism
includes the kinetic equation of sediment growth and the mass balance equation of
suspended and retained solids [6]. For some one-dimensional models, exact analyt-
ical solutions are obtained, for a wide class of models it is possible to construct
asymptotics [7–9]. We can use numerical methodsfor unknown analytical solutions
[10, 11].
The most important characteristic of the filtration process is the retention profile—
the dependence of retained particles concentration on the spatial coordinate at a fixed
time. The profiles of a monodisperse suspension always decrease monotonically. If
suspensions or colloids contain particles of different sizes, the profiles lose their
monotony. In [12], the profiles of 2-particle colloids formed during filtration in
a porous media were calculated. The profiles of large particles keep a monotonic
decrease, while the profiles of small particles change monotonicity. The behavior of
the total retention profile depends on the parameters of the problem.
The filtration problem of a 2-particle suspension is considered in the paper. The
retention profiles of large and small particles and the total retention are plotted.
The process of transition of the profiles of small particles and total retention from
a monotonic decrease at a short time to a monotonic increase at a long time is
investigated. At some time, a maximum point appears on the profiles, which moves
from the entrance to the exit of the porous media. The velocity of maximum point
movement and its asymptotics for a large time are calculated.

2 Mathematical Models

The mathematical model assumes that the filtration function is linear and blocking
(Langmuir coefficient), and the suspended solids concentration is small. In this case,
the concentration function is linear [13].
In the domain  = {x ≥ 0, t ≥ 0}, consider the quasilinear hyperbolic system

∂ci ∂ci ∂si

+ + = 0, (1)
∂t ∂x ∂t
Calculation of Retention Profiles in Porous Medium 23

∂si
= (1 − b)λi ci , i = 1, 2 (2)
∂t
for unknown concentrations of suspended ci and retained si solids, where

b = B1 c10 s1 + B2 c20 s2 (3)

and λ1 , λ2 , B1 , B2 , c10 , c20 are positive values, c10 + c20 = 1.

The uniqueness of the solution to system (1), (2) is ensured by the boundary and
initial conditions

x = 0 : c1 = c10 , c2 = c20 , (4)

t = 0 : c1 = 0, c2 = 0, s1 = 0, s2 = 0 (5)

The suspended and retained solidsconcentration front t = x divides the domain

 into two zones. In zone 0 = {x ≥ 0, 0 < t < x} there are no suspended and
retained solids and there is zerosolution. In zone 1 = {x ≥ 0, t > x} there are
suspended and retained solids, in 1 there is positivesolution. The conditions (4) and
(5) are not compatible at the origin, the solution ci is discontinuous at the front of
concentration; in the  the solution si is continuous.
On the front of concentration t = x the solution s1 = 0, s2 = 0. Substitute
Eq. (2) into (1)

∂ci ∂ci
+ + λi ci = 0, i = 1, 2. (6)
∂t ∂x
Solution of the Eqs. (6) with the conditions (4)

ci− = e−λi x , i = 1, 2. (7)

Formula (7) determine the solution on the concentration front from the side of
zone 1 (behind the front), ahead of the front the solution is zero.
Below we study the solution in zone 1 , where the solution to the problem is
continuous and positive.
The exact solution for a monodisperse suspension was obtained in [14]. For a
bidisperse suspension, the analytical solution in zone 1 is unknown, the problem
is calculated.
24 G. Safina

3 Numerical Calculation

The calculation of the problem was performed by the finite difference method
according to an explicit difference scheme [15–17]. The grid steps in time and coor-
dinate were chosen in accordance with the Courant convergence condition [18]. In
Fig. 1 the profiles of total deposit (green line) and partial deposits of large (blue line)
and small (red line) particles are presented at different times. The curves are obtained
for parameters c10 = c20 = 0.5, B1 = 0.125, B2 = 0.025, λ1 = 5, λ2 = 25.
According to Fig. 1 the profiles of large particles decrease monotonically at any
fixed time. The retention profiles of small particles decrease monotonically at a short
time and increase monotonically at a long time. The profiles of the total deposit
retain a monotonic decrease or change the monotonicity depending on the problem
parameters. The variationof the monotonicity of the profiles is associated with the
appearance of a maximum point separating the monotonically increasing part of the
plot from the monotonically decreasing one. With increasing time, the maximum
point moves from the entrance to the exit of the porous media, gradually increasing
the increasing part of the plot. The maximum point on the profile of the total retention
lags behind the maximum point of the small particle retention.
The limit velocities of the profile maximum point are shown in Table 1. The
velocities v2 of small particles retention profile and v0 of total retention profile are
calculated as the slope of the asymptote at large times.
Table 1 shows that the limit maximum points velocities of small particles reten-
tion profile and total retention profile coincide. When changing any parameter, the
velocity of maximum points changes disproportionately. With a proportional change

Fig. 1 Plots of s1 , s2 and s at a t = 0.3; b t = 1; c t = 5; d t = 10; e t = 25; f t = 50

Calculation of Retention Profiles in Porous Medium 25

Table 1 Parameters and limit maximum points velocities

No. λ1 λ2 B1 B2 c10 c20 v2 v0
a 25 5 0.125 0.025 0.5 0.5 0.028 0.029
b 12.5 2.5 0.125 0.025 0.5 0.5 0.028 0.029
c 50 10 0.125 0.025 0.5 0.5 0.029 0.029
d 50 10 0.0625 0.0125 0.5 0.5 0.0148 0.0149
e 25 5 0.0625 0.0125 0.5 0.5 0.0144 0.0148
f 100 20 0.0625 0.0125 0.5 0.5 0.0149 0.015
g 12.5 2.5 0.25 0.05 0.5 0.5 0.054 0.057
h 6.25 1.25 0.25 0.05 0.5 0.5 0.056 0.059
i 25 5 0.25 0.05 0.5 0.5 0.055 0.057

of the coefficients λ1 , λ2 , the limit velocity does not change. Thus, the limit velocity
of maximum points is a nonlinear function of its parameters, a uniform degree of
zero in terms of parameters λ1 , λ2 .
In Fig. 2 the dependence on time of maximum points of the profiles is shown for
parameter from Table 1 (Fig. 2a corresponds to the first line of the Table, Fig. 2b
corresponds to the second line of the Table, etc.). Green line corresponds to the
maximum points of total deposit profiles and red line corresponds to the maximum
points of partial deposit of small particles.
Figure 2 show that the plots of the maximum points have an asymptote. Conse-
quently, the velocity of maximum points tends to a constant with an unlimited increase
in time.

4 Discussion

In the paper a numerical solution is received for a one-dimensional filtration problem

of a suspension with 2-size particles in a porous media. The retention profiles of large
and small solids, as well as of total retention profile, were constructed.
The retention profiles of large particles decrease monotonically at any time. The
retention profile of small particles is not monotonic. The total retention profile
either always decreases monotonically or changes monotonicity depending on the
parameters.
For a short time, all profiles decrease monotonically. With increasing time, a
maximum point appears on the profiles, moving from the entrance to the exit of the
porous media. The maximum point of total retention profile lags behind the maximum
point of the profile of small particles. When the maximum point reaches the exit of
the porous media, the profile becomes monotonically increasing.
26 G. Safina

Fig. 2 Maximum points of the profiles for parameters λ1 , λ2 , B1 , B1 , c10 , c20 given in Table 1

With a long time, the limit velocities of the maximum points of the total and partial
retention profiles coincide. The velocity depends nonlinearly on the coefficients of
the problem and is homogeneous in parameters λ1 , λ2 .

5 Conclusions

The profiles of sediments—the distribution of trapped particles in a porous media

depending on the distance to the entrance at a fixed time—have been investigated. It
is shown that the monotonicity of the profiles depends on the given time and on the
type of particles. The profiles of large particles always decrease monotonically, while
the profiles of small particles decrease monotonically at a short time and increase
monotonically at a long time.
Intermediate profiles of small particle and full sediment profiles have a maximum
point that moves from entrance to exit with increasing time. For suspension with 2-
size particles in a porous mediathe limit maximum points velocities of small particles
Calculation of Retention Profiles in Porous Medium 27

retention profile and total retention profile coincide. The velocity of maximum points
tends to a constant with an unlimited increase in time.
This work makes it possible to describe the properties of inhomogeneous materials
obtained by filtration of bidisperse suspensions in a porous media theoretically. It
is shown that the properties of materials depend on both the particle size of the
suspension and the filtration time.
The study of retention profiles becomes more difficult if the filtration and concen-
tration functions are non-linear [19]. Behavior of profiles of a polydisperse suspen-
sion containing particles of 3 or more different sizes is also of interest. Separate
studies will be devoted to these problems.
Calculation of retention profiles allows fine tuning of experiments and decrease
the cost and amount of laboratory experiments [20, 21].

References

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3. Tsuji M, Kobayashi S, Mikake S, Sato T, Matsui H (2017) Post-grouting experiences for
reducing groundwater inflow at 500 m depth of the mizunami underground research laboratory.
Procedia Eng 191:543–550
4. Tien C (2012) Principles of filtration. Elsevier, Oxford
5. Herzig JP, Leclerc DM, Legoff P (1970) Flow of suspensions through porous media—applica-
tion to deep filtration. Ind Eng Chem 62(5):8–35
6. Vyazmina EA, Bedrikovetskii PG, Polyanin AD (2007) New classes of exact solutions to
nonlinear sets of equations in the theory of filtration and convective mass transfer. Theor
Found Chem Eng 41(5):556–564
7. Kuzmina LI, Osipov YV, Zheglova YG (2018) Analytical model for deep bed filtration with
multiple mechanisms of particle capture. Int J Non-Linear Mech 105:242–248
8. Zhang H, Malgaresi GV, Bedrikovetsky P (2018) Exact solutions for suspension-colloidal
transport with multiple capture mechanisms. Int J Non-Linear Mech 105:27–42
9. Fogler HS (2006) Elements of chemical reaction engineering. Prentice Hall, Upper Saddle
River, NJ
10. Kuzmina L, Osipov Y (2018) Deep bed filtration with multiple pore-blocking mechanisms.
MATEC Web Conf 196:04003
11. Wang S (2018) An improved high order finite difference method for non-conforming grid
interfaces for the wave equation. J Sci Comput 77:775–792
12. Osipov Y, Safina G, Galaguz Y (2018) Calculation of the filtration problem by finite differences
methods. MATEC Web Conf 251:04021
13. Galaguz YuP, Safina GL (2016) Modeling of particle filtration in a porous media with changing
flow direction. Procedia Eng 153:157–161
14. Safina GL (2019) Numerical solution of filtration in porous rock. E3S Web Conf 97, 05016
15. Crist JT, Zevi Y, McCarthy JF, Throop JA, Steenhuis TS (2005) Transport and retention
mechanisms of colloids in partially saturated porous media. Vadose Zone J 4(1):184–195
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brates. Mar Ecol Prog Ser 418:255–293
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17. You Z, Bedrikovetsky P, Badalyan A, Hand M (2015) Particle mobilization in porous media:
temperature effects on competing electrostatic and drag forces.Geophy Res Lett 42(8), 2852–
2860
18. Borazjani S, Bedrikovetsky P (2017) Exact solutions for two-phase colloidal-suspension
transport in porous media. Appl Math Model 44:296–320
19. Kuzmina LI, Nazaikinskii VE, Osipov YuV (2019) On a deep bed filtration problem with finite
blocking timerussian. J Math Phys 26(1):130–134
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tion of fines migration system using laboratory pressure measurements. J Nat Gas Sci Eng 65,
108–124
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during reactive flows in porous media. J Nat Gas Sci Eng 34:1422–1433
Mechanical and Durability Properties
of High-Performance Concrete
in Corrosive Medium of Vietnam

Nguyen Duc Vinh Quang, Olga Aleksandrova, and Boris Bulgakov

Abstract Nowadays, High-performance concrete (HPC) has been used to gradually

alternative ordinary conventional concrete for structures exposed to harsh environ-
ments to improve the long-term durability of concrete. The principal aim of this
research was to utilize industrial waste by-products like fly ash, silica fume as an
addition to cement at different rates incorporating quartz powder in HPC mixes for
application in concrete structures subjected to severe environmental conditions. The
corrosion resistance of HPC was examined by the mass and compressive strength
loss of 15 × 15 × 15 cm3 cubes immersed in seawater and saline-alkaline environ-
ments. The result experimental indicates that an optimum mixture could be achieved
with the use of 12.5% SF, 20% FA for cement substitution, and 20% Qp substitute
sand. The compressive strengths of all mixes curing in potable-water obtained varies
between 70.76 and 91.15 MPa at 28th days, these values gained 78.1 to 101.6 MPa at
540th days. Compared to cubes immersed in freshwater, similar compressive strength
values have been received reduced from 0.13 to 2.78% strength for seawater environ-
ment, whereas exposed to saline-alkaline medium has decreased from 0.13 to 2.78%
and mass loss ranges from 0.11 to 2.1% of specimens, after curing period 540 days.
It is observed that the HPC produced with locally available materials in Vietnam
proved its excellent mechanical and durability properties. In the future, HPC will be
used vastly in the construction industry, as well as for concrete structures exposed to
synthetic corrosive medium similar to Vietnam.

Keywords High-performance concrete · Mechanical · Durability · Corrosion ·

Corrosive medium

N. D. V. Quang (B)
Hue Industrial College, Hue City, Vietnam
e-mail: ndvquang@hueic.edu.vn
N. D. V. Quang · O. Aleksandrova · B. Bulgakov
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 29

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_4
30 N. D. V. Quang et al.

1 Introduction

In Vietnam, the research on HPC is a relatively new research topic, not much research
has been published. Meanwhile, this type of concrete has been strong developed in
the world since the 80 s. Vietnam is a developing country, in recent years, with the
help of countries like Japan, Korea, Australia, China by ODA loans and construction
technology, Vietnam has begun to apply high-strength concrete with aggregate and
conventional cement, using superplasticizer, with water to cement ratio about 0.35–
0.40, Slump reaches 15–20 cm, keep at least 60 min, compression strength 50–
70 MPa and early strength R7 = 0.85R28. However, Vietnam is located in the wet-
hot dry tropics, contains high chloride ion content, with a temperature varying from
12 to 41 °C. In the north, the climate is monsoonal with four distinct seasons while
in the south, the climate is tropical monsoon with two seasons (rainy and dry). The
long seacoast more than 3200 km and this is the area immediately affected climate
change, rising seawater, seawater intrusion incorporation with chemical wastewater
from industrial areas is one of the basic causes of the deterioration of the construction
as well as the impact of the “sour” and “salty” environments, which leads to corrosion
and destruction of the construction due to the quality of concrete that does not meet the
requirements of corrosion resistance in corrosive environment aggressive. Structures
that are exposed to corrosive medium will be destroyed quickly after the short-
term, it was also a major challenge facing the concrete industry for sustainable
infrastructure development purposes while ensuring economic efficiency for coastal
region structures Vietnam. In recently nowadays, HPC was designed with properties
far superior to conventional concrete both on strength and durability, to help extend
the service life of concrete structures. In order to solve this problem, HPC was
designed with properties far superior to conventional concrete both on strength and
durability, help extend the service life of concrete structures and for this reason, it
is a perfect choice for constructions in harsh climatic and environmental conditions,
such as in Vietnam. Concrete structures work in environments with high temperature
and humidity as well as in hot and humid climates with continuous changes in
climatic conditions according to the seasons in the year, concrete will be continuously
shrinking. According to the change of the surrounding environment, resulting in
rapid degradation and reduced service life, this problem was also presented in Shah’s
research [1]. Besides, environmental factors affect the strength of HPC such as humid-
wet environment, hot–cold temperature…, destroying concrete structures from the
outside [2–4].
Mechanical and Durability Properties of High-Performance … 31

2 Materials and Methods

2.1 Materials

HPC constituent materials used in research is locally available materials in Vietnam,

complies with the requirements for each type of material listed according to TCVN
10306-2014 [5], consists of sulfate-resistant Portland cement PCSR 40 (type V) of
Luks Cement (Vietnam) Limited. Fly ash (FA) used is a by-product of the thermal
power plants Pha Lai (Vietnam). Silica Fume (SF) used is Sikacrete® PP1. Crushed
aggregate have maximum size 19 mm (60%) and 9.5 mm (40%) and average strength
105.593 MPa. Fine aggregates is nature sand exploited from Bo river with fineness
modulus as 3.0. Quartz powder (Qp) is milled from local silica white sand in the
Central region of Vietnam, specific gravity 2.56 g/cm3 and it is in white color powder
form and has particles size ranging from 5 to 20 μm, which replaces partial fine aggre-
gate. The chemical admixture used is Sika® ViscoCrete® -8100. Water used to mix
and curing concrete is the potable water. These materials also had been used to design
HPC mixes in previous investigations [6–8]. The physicochemical composition of
cementitious materials presented in Table 1.

2.2 Mix Proportions and Casting of Specimens

2.2.1 Preliminary Investigations

The material types like cement, aggregates, mineral materials… etc., before used in
the mix, they have been preliminarily examination to physical and chemical analyses
to determine whether they are in compliance with the standard used. A total of
15 different mixes are designed with concrete grades higher or equal to M80, fits
TCVN 10306-2014 [5], in which the properties workability, as well as strength and
durability, were tested. In there, the total dosage of cementitious materials kept fixed
at 550 kg/m3 . Similarly, the dose of aggregates, chemical admixture, and w/cm ratio
was fixed in all mixes. Change mineral materials like silica fume with content selected
as (0, 5, 7.5, 10, 12.5%) combined with class F fly ash has content at levels (0, 20,
30, 40%) to part replace dosage of sulfate-resistant cement, by weight of cement in
mixes.

2.2.2 Mix Proportioning

The HPC was produced by mixing the designed mix proportions of cement, FA, SF,
Qp, SP, and water together follow the existing mix design methods using a standard
mixer complying with TCVN 10306:2014 [5]. To start with the experimentation, a
mix design for the control mix targeted for a strength of 80 MPa, with a slump of
 32

Table 1 Chemical composition of cementitious materials used in this study

Cementitious materials Chemical composition (% by mass)
SiO2 Al2 O3 Fe2 O3 CaO MgO MnO Na2 O TiO2 K2 O SO3 L.O.I
Cement 20.6 3.82 5.1 62.1 1.7 1.25 0.135 0.12 0.63 2.05 1.1
Fly ash 57.4 24.07 6.1 0.7 0.95 2.62 0.27 0.7 3.61 – 5.8
Silica fume 92.5 0.87 1.9 0.31 0.84 0.14 0.38 – 1.23 – 1.68
Quartz powder 99.6 0.044 0.04 0.052 0.036 – 0.02 0.05 0.007 – 0.04
N. D. V. Quang et al.
Mechanical and Durability Properties of High-Performance … 33

(50–100) mm and were obtained without considering the addition of any mineral
admixtures and also no containing Qp. Mix proportions for 1m3 HPC mix, consist of
total dosage binder 550 kg/m3 + 1088 kg/m3 coarse aggregate + 621.7 kg/m3 fine
aggregate + 156 l/m3 water + 9.9 l/m3 superplasticizer (similar to 1.8% by mass
of binder), entrapped air is 2%, and w/cm ratio fixed 0.3, meanwhile part of sand
replaced by 20% quartz powder, all them kept fixed for different 15 mixes. The total
dose of mineral substitutes for cement is represented by the (SF)α (FA)β formula,
where α and β are the percentages of SF and FA, respectively, to replace cement in
the mixture. The mix combinations were categorized into four SF replacement levels
are 5, 7.5, 10, and 12.5%; and three FA replacement levels are 20, 30, and 40% were
proposed. In each of the SF series, the FA content was set at 20, 30 and 40%, as
shown in Table 2.

2.2.3 Casting and Curing of Test Specimens

The first, the dry materials (fine and coarse aggregates, quartz powder, cement, SF,
FA) were first blended for 2–3 min inside a plastic bag in dry conditions. Second,
water containing the plasticizer was added and blended for another 4–5 min. After
6–8 min in the mixer or until concrete mix achieved the homogeneous and uniform
consistency, during mixing, temperature variation ranging follow ambient tempera-
ture from 30–41 °C. After that, the concrete mixes will be put into different casting
mold like cubes/cylinders/prisms specimens and preserved in the room at tempera-
tures ranged from 30 ± 2 °C until 24 h. Then the specimens were stripped molds
and transferred to the fresh-water tank to curing samples until the time of testing.

2.3 Experimental Methods and Instrumentation

The 15 × 15 × 15 cm3 cube specimens were used to test compressive strength (f c )

after time 3, 7, 28, and 540-day curing conforms to the standard TCVN 10303-2014
[9]. The 10 × 20 cm cylinder specimens used for tensile splitting strength (f ct,28 )
tests as per the code TCVN 3119-1993 [10] at age 28 days. The prism specimens
of 10 × 10 × 40 cm were used to test flexural strength (f t,28 ) of concrete after
28th days of curing and three specimens were prepared for each batch, according to
TCVN 3119–1993 [10] standards. The prism specimens of 10 × 10 × 40cm were
used to test static modulus of elasticity (Ecm ) of concrete after 28th days of curing is
described in TCVN 5276-1993 [11]. Water absorption was carried out according to
TCVN 3113-1993 [12] standards with 15 cm cubic specimens. The standard cylinder
specimens have a diameter of 150 mm and 150 mm height were used to test water
permeability of concrete after 28th days.
 34

Table 2 Mix proportions of 1 m3 concrete

Mix No. Cementitious materials, kg/m3 Aggregates, kg/m3 Water Super (W + SP)Cm Slump
Cement Silica fume Fly ash Crushed agg Sand river Quartz powder plasticizer
kg % kg % kg kg kg kg kg ltr – mm
S1 550 – – – – 1088 622 – 156 9.9 0.3 105
S2 550 – – – – 1088 498 124 156 9.9 0.3 185
S3 412.5 5.0 27.5 20 110 1088 622 – 156 9.9 0.3 245
S4 412.5 5.0 27.5 20 110 1088 498 124 152 9.9 0.3 220
S5 357.5 5.0 27.5 30 165 1088 498 124 156 9.9 0.3 260
S6 302.5 5.0 27.5 40 220 1088 498 124 156 9.9 0.3 265
S7 398.8 7.5 41.3 20 110 1088 498 124 156 9.9 0.3 260
S8 343.8 7.5 41.3 30 165 1088 498 124 156 9.9 0.3 260
S9 288.8 7.5 41.3 40 220 1088 498 124 156 9.9 0.3 265
S10 385 10 55 20 110 1088 498 124 156 9.9 0.3 245
S11 330 10 55 30 165 1088 498 124 156 9.9 0.3 265
S12 275 10 55 40 220 1088 498 124 156 9.9 0.3 270
S13 371.3 12.5 68.8 20 110 1088 498 124 156 9.9 0.3 240
S14 316.3 12.5 68.8 30 165 1088 498 124 156 9.9 0.3 230
S15 261.3 12.5 68.8 40 220 1088 498 124 156 9.9 0.3 265
N. D. V. Quang et al.
Mechanical and Durability Properties of High-Performance … 35

2.4 Durability Test

Vietnam, with a coastline of 3260 km stretching from 23°23 to 8°27 North latitude,
has high temperature and humidity year-round. Under the direct impacts of climate
change, the saline intrusion has occurred continuously, this equivalent to increasing
the saline-alkaline environment, especially in the South of Vietnam [13]. In Vietnam,
annual ocean floods are projected to particularly affect the densely populated Mekong
Delta and the northern coast around Vietnam’s capital, Hanoi, including the port
city of Haiphong [14]. The effects of climate change and sea-level rise make many
areas frequently affected by seawater intrusion through flood tides. Especially, the
Mekong Delta with the area is about 3, 9 × 10.5 km2 . In which, about 60% area
of this area is an alkaline medium with pH ranges from 3 to 6, 5 and the main
corrosive agent is sulfuric acid. Plus, the salinity intrusion area accounts for about
27%, the main corrosive agents are sulfate salts and chloride salts. This is reason
leads to caused corrodes reinforced concrete structures. The content sulfate average
about 1320 mg/L, the amount of NaCl from 4.5 to 37 g/L, structure concrete in the
alkaline water environment stronger corrosive than the salinity zone about 1.3 to
1.6 times, but corrosion of reinforced steel in concrete less than 2 to 5 times. In the
environment hot-humid climates, most of the construction work in saline-alkaline
medium such as the Mekong Delta region after a period of service showing signs
of corrosion, for example, protective concrete layer peeling, the concrete surface
showed signs inert crushed aggregate, corrosion of steel… etc. leading to structural
damage does not guarantee to reduce durability and longevity of construction works
[15]. To have the basis for finding solutions to enhancing the anti-corrosion-resistant
concrete structures working in the above corrosive environment aggressive. In this
study, the physico-mechanical properties of concrete are determined through the
change compressive strength and loss of weight of the concrete samples immersed
in three environments after a period of 540 days. (1) The saline-alkaline medium
were chosen to soak experimental samples in position Hung Thanh–Dong Thap,
it characterizes for the environment in the Mekong River Delta region (Southern
Vietnam); (2) Seawater environment used to curing of concrete samples is Chan
May Port–Thua Thien Hue Province (South Central Coast–Central Vietnam); (3) and
potable water of Thua Thien Hue Water Supply Joint Stock Company (Vietnam) used
to curing of controlled concrete is the control environment. The chemical composition
of the sample immersion environments shown in Tables 3 and 4.
In order to solve this problem, it is necessary to have solutions to enhancing
the anti-corrosive resistance due to chemical effects for concrete, and the ability to
resist both chemical and mechanical abrasion of the flow containing the seeds, solid
and domestic. The research orientation of this topic is to use the locally available
materials in conformity with the practical conditions in Vietnam.
36 N. D. V. Quang et al.

Table 3 The composition of the main ions in source of water used for sample concrete curing
Code Corrosive water pH SO24 - (mg/l) Cl - (mg/l) Mg2 + (mg/l) Ca2 + (mg/l)
environment
1 Potable water 6.53 103 182 – –
2 Saline-alkaline 3.52 490 97.2 696 560
environment
3 Seawater 7.8–8.4 (1.4–2.5) × (6.5–18) × 103 (0.2–1.2) × –
103 103

Table 4 The pH value and the salt content (g/l) of saline-alkaline medium soak
Months 1 2 3 4 5 6 7 8 9 10 11 12 Aver.
value
pH 5.08 5.02 4.56 4.10 3.52 3.12 3.13 3.46 3.96 5.22 6.20 6.05 4.43
Salt 25.9 25.5 24.3 23.3 22.3 22.0 22.2 23.7 25.2 25.6 26.1 27.0 24.43
content

3 Results and Discussion

There are numerous benefits to incorporate SCMs into HPC mix design, include
improving the workability of fresh concrete, reducing/eliminating the free lime
content, decreasing the C/S ratio of C–S–H in hardened cement pastes, miti-
gating alkali-aggregate reactions, etc. The products resulting from the reactions
between lime and SCMs refine the pore structure of hardened pastes and reduce
the permeability of hardened pastes.

3.1 Impact of Mineral Material on Properties of Concrete

The slump of the fresh concrete mixes was determined by Abrams standard cone with
a chopped shape d × D × H = 100 × 200 × 300 mm, according to standards TCVN
3016-1993 [16]. Table 2 represents the slump values required range of 105–270 mm.
In theory, the addition mineral admixtures (MAs) finer than cement to fill into the
voids between the cement grains and the addition of another even finer MAs to fill into
the voids between the larger particles should be able to reduce the voids to a greater
extent than possible with the addition of just one MA. Basically, concrete mixes
consist of cement, FA, SF, and sand flour …etc. can improve the packing density of
binding materials. This is due to when fillers added finer than cement particles, can be
inert form as filler, it will fill the pore of cement paste or may participate in pozzolanic
reactions generate more form CHS gel to complement for concrete structure. Because
of, the shape of FA particles typically spherical particles, silt-sized, it finer than
Portland cement, due to it can fill the voids between cement particles whereas SF
Mechanical and Durability Properties of High-Performance … 37

particles size < 0.1 μm, hence the voids between FA and cement will be filled into
by SF particles to successively fill up this voids to enhance efficiency as well as
generate the optimization on packing density. The mineral admixtures, particularly,
fly ash can provide plasticizing effects enhancing the paste and concrete workability
[6–8]. Because of, the spherical shape of fly ash particles creates good conditions
the movement of neighboring particles special in case it used at high dosage will
create a ball bearing effect. Moreover, Portland cement can be replaced by a part
dosage of the fly ash will bring highly cost-effective that offers lower environmental
impact than chemical superplasticizers in enhancing the flowability. Added to that,
quartz sand powder particles can fill into the voids between aggregates particles will
create the maximizing optimal packing density for matrix microstructure of Portland
cement–SF–FA–quartz powder–aggregates.

3.2 Mechanical Properties of HPC

The role of incorporating FA, SF, and quartz powder in mixes was exerted their effects
on reacting with calcium hydroxide Ca(OH)2 to produce C–S–H that contribute in
strengthen the cement paste itself and the interfacial transition zone as well with,
in turn, improve the strength and modulus of elasticity. On the other hand, it acts
as a fine-filler to fill empty spaces in cement paste and the interface between it
with the aggregates which improve the compactness of concrete leads to reducing
porosity and cause pore blockage in the microstructure brings to decrease concrete
permeability, which is an evidence to the efficiency of incorporating FA, SF, and Qp
in producing impermeable concrete. Describe the information presented in Table 5
shows mechanical properties gradually increases according to curing time, the lowest
value was achieved at age test 3-day, followed by from 7th to 28th day augmented
slowly and it starts to increase fastly from the date 28th of all samples, and then
continue to increase gradually at later ages. The compressive strength results at age
28 days range between 70.759–91.8100 MPa and the 540 days strength of all mixes
were in the range of 78.102–101.597 MPa. At the age of 3 days, the specimen S4
has the lowest value of 52.44 MPa, meanwhile samples S13 reached the highest
strength at 101.597 MPa after 540th curing, increased by 48.4% from 3th day. This
data indicates that the strength improved gradually follow time curing, which is
due to forms C–H–S gel continuously supplemented through pozzolanic reactions,
that is why the strength of concrete be improved and increases gradually by time.
The reason is that the pozzolan reaction in concrete mixtures occurs slowly, fly-ash
concrete achieves significant improvement in its mechanical properties at later ages,
example reacts to makes more than denser form C-S–H gel, as follows: 3Ca(OH)2 +
2SiO2 → 3CaO · SiO2 · 3H2 O or mCa(OH)2 + nSiO2 + pH2 O → mCaO · nSiO2 ·
(m + p)H2 O…etc., similar to the previous findings [17, 18].
The diagram in Table 6 shows the average flexural strength ( f t28 ) of different
HPC mixes after 28-day curing. Control mix has a flexural strength of 8.8 MPa,
while the industrial by-product waste materials modified groups had the flexural
 38

Table 5 Mechanical properties test result for varying mineral admixtures replacement levels in HPC
Mix Compressive strength of concrete cured in Compressive strength and weight of concrete cured in corrosive medium, MPa
No. freshwater environment, MPa Seawater environment Saline-alkaline medium
3 days 7 days 28 days 540 days 540 days Stre. loss 540 days Stre. loss M128 . (g) M2540 (g) Mass loss
(%) (%) (%)
S1 67.581 71.111 83.643 86.353 86.353 0.39 82.157 8.10 8320 8145 2.10
S2 72.615 82.558 89.246 95.05 94.269 0.82 93.822 1.29 8310 8299 0.13
S3 41.333 62.222 65.283 67.27 65.689 2.35 60.975 9.36 8330 8159 2.05
S4 52.44 75.967 81.812 92.774 92.053 0.78 91.331 1.56 8100 8085 0.19
S5 67.333 72.889 80.209 86.476 85.795 0.79 84.433 2.36 8238 8225 0.16
S6 56.667 60.889 70.759 78.102 75.931 2.78 74.431 4.70 8120 8096 0.30
S7 60.000 81.039 89.451 94.195 93.026 1.24 92.442 1.86 8125 8115 0.12
S8 69.323 74.957 84.04 88.442 88.115 0.37 87.966 0.54 8105 8096 0.11
S9 59.111 66.667 74.776 84.146 83.34 0.96 82.534 1.92 8050 8028 0.27
S10 62.148 81.914 87.921 95.426 94.926 0.52 94.020 1.47 8150 8125 0.31
S11 63.861 67.54 88.827 92.582 91.610 1.05 91.400 1.28 8105 8078 0.33
S12 65.333 72.444 80.624 85.129 83.924 12 81.677 4.06 8090 8050 0.49
S13 58.974 83.819 91.810 101.597 101.08 0.51 100.994 0.59 8143 8112 0.38
S14 72.178 76.089 84.161 92.55 87.511 0.85 86.133 2.41 8075 8059 0.20
S15 56.444 57.333 76.096 88.011 84.919 0.13 83.106 2.27 8030 7990 0.50
M 28
1 (g) is weight of sample concrete at age 28 days, 24 h after removing from the mold and cured in freshwater tank at temperature variation ranging follow
ambient temperature from 28–39 °C; and M 540
2 (g) is weight of sample concrete at age 540 days since after sample removing from the freshwater tank and was
soak in saline-alkaline medium at temperature variation ranging follow ambient temperature from 22–41 °C
N. D. V. Quang et al.
Table 6 Flexural, splitting tensile strength, water absorption and permeability coefficients (Kt * 10–11 , cm/sec) test results of concrete at 28th day
Mix No S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15
f t,28 (MPa) 8.8 9.16 7.84 8.77 8.69 8.16 9.18 8.76 8.39 9.1 9.24 9.16 9.26 8.9 8.46
f ct,28 (MPa) 5.68 5.99 4.86 5.65 5.58 5.13 6.01 5.64 5.33 5.93 6.06 6.02 6.08 5.76 5.39
m 0 (g) 8290 8275 8335 8060 8125 8049 8125 8100 8035 8140 8060 8050 8091 8049 7975
m 1 (g) 8413 8321 8480 8088 8155 8082 8151 8123 8065 8168 8087 8079 8113 8077 8006
Wa (%) 1.48 0.56 1.74 0.35 0.37 0.41 0.32 0.28 0.37 0.34 0.33 0.36 0.27 0.35 0.39
Mechanical and Durability Properties of High-Performance …

Kt * 10–11 , (cm/s) 3.60 2.18 4.96 1.21 1.33 2.60 1.13 1.21 1.98 1.10 1.03 1.09 1.01 1.08 1.63
39
40 N. D. V. Quang et al.

strength of 9.18, 9.1, 9.24, 9.19, and 9.26 MPa for the samples S7, S10, S11, S12,
and S13, all them taller than the control mixture up to 4.32, 3.41, 5.0, 4.43, and 5.23%,
respectively. Besides, remaining specimens also received the flexural strength value
is approximately equal to control mix. The addition of same as above materials
improved the interface characteristics between the bonding layer and aggregate,
leads to raising the interface adhesion, and hence ameliorate the bending resistance
of the concrete. On the other hand, Table 6 shows that the splitting tensile strength
values obtained greater than the controlled mix up to 7.4% when the mixture using
combination of 10% SF plus 30–40% FA or 20% FA plus 7.5/12.5% SF. Conversely,
some mixture showed fall in tensile strength of about 6.1% at the age of 28-day
curing period to the controlled mix, as in mixtures used silica fume at levels of 5,
7.5 and 12.5% combined with 40% content amount of fly ash to part replace cement
content in the mixture.
The result from the tests for modulus of elasticity of reference concrete (S1
specimen) is 40.302 GPa, with relative deformation ranged (ε2 − ε1 ) from 52.4
× 10–5 to 52.6 × 10–5 . In comparison with reference concrete, the HPC mixes
from S4 to S15 specimen with incorporating FA, SF and Qp showed a significant
increase in E δ with the values obtained considerably ranged between 38.165 GPa
(correspondence S6 specimen—5%SF40%FA) and 43.12 GPa (correspondence S13
specimen—12.5%SF20%FA).

3.3 Durability of Hardened Concrete in Corrosive

Environment Aggressive

The results of mass loss and compressive strength for concrete samples during the
immersion period of 18 months in marine and saline-alkaline medium are summa-
rized in Table 5. Based on the data obtained, on the other hand, simultaneously
observe the surface of the experimental samples as shown in Fig. 1 indicated that the
samples have begun to show signs of corrosion, e. g., sample surfaces appeared the
fine aggregate particles; discolored concrete surfaces, salt agglomeration, or marine

Fig. 1 Effect of curing methods on corrosion resistance of concrete a, b seawater exposed specimens
and c, d saline-alkaline immersed specimens
Mechanical and Durability Properties of High-Performance … 41

creatures stick on the surface … etc., also in the seawater environment concrete
surfaces still flat smooth after a period of 540 days immersed in corrosive environ-
ment aggressive, due to corrosion of hardened cement paste., this demonstrates that
the strength and weight of concrete diminished gradually by curing time.
The sample of cement PCSR 40 concrete (S1 sample) in the saline-alkaline medium
after 540 days exhibited strong damage followed by high mass loss of about 2.1% and
compressive strength loss at 5.12% is higher than in seawater environment (strength
loss 0.39%). Especially S3 sample also occurs similarly with compressive strength
loss at 9.36% shown dropped degradation of physical–mechanical properties, so
further investigation was stopped. On the other hand, cement content of (45–52.5)%
replaced by fly ash at 40% and of SF from 5 to 12.5% for the samples S6, S11,
S12, and S15 show compressive strength loss at 4.7, 1.92, 6.4 2.27% and mass loss
of about 0.3%, 0.27%, 0.54%, 0.5%, respectively. Contrary to cement content of
(25–32.5)% replaced by fly ash at 20% and of SF from 5 to 12.5% for the samples
S4, S7, S10, and S13 show compressive strength loss at 1.56, 1.86, 1.47, 0.59 and
mass loss of about 0.19%, 0.12%, 0.31%, 0.38%, respectively. Nevertheless, cement
content of (35–42.5)% replaced by fly ash at 30% and of SF from 5 to 12.5% for
the samples S5, S8, S11, and S14 show compressive strength loss at 2.36, 0.54,
1.28, 2.41 and mass loss of about 0.16%, 0.11%, 0.33%, 0.2%, respectively. All
these mixtures show that weight loss, as well as strength, is much higher than when
immersed in the seawater environment. We could conclude that during this period,
different corrosion resistance in seawater and saline-alkaline medium is obviously
the consequence of the applied fillers as FA, SF, and Qp. The samples corroded
in varying degrees depending on different percentages of active mineral admixture
ratios used to replace for cement content in concrete.

3.4 Water Permeability and Water Absorption of Concrete

Water absorption Wa (%) test results obtained in Table 6 an only approximate value.
This problem was due to the influences of casting and preparing specimens leads to
specimens have the surface of porous concrete will be more water absorption than
compared to smooth surfaces. m 1 value is determined in the water-saturated state, i.e.
specimens immersed in tap water till the time of testing at age 28 days. In addition,
during the procedure specimens are oven-dried for a long time at high temperature
(≈ 105 °C) then part of the aggregate on the sample surface will be detached leads to
inaccurate m 0 valuation. The water absorption was calculated to follow equation: Wa
(%) = 100(m1 −m2 )/m0 . The variations on water permeability coefficients of HPC
mixes are summarized in Table 6. In general, the concrete grades for watertightness
fell within the range ≥ W16 (correspondence Kt ≤ 5 × 10–11 cm/s), only the S3
mixture received water-tightness value with W = 14 is of the least value. The length
of water permeability under pressure within the range 15.5 to 7.5 mm. From the
results, it can be seen that the addition of Qp can improve the water impermeability
of concrete composite containing SF and FA evidently.
42 N. D. V. Quang et al.

4 Conclusions

Based on the observations and the experimental studies through the results achieved
in this study, the following conclusions may be drawn:
High-performance concrete with respect to strength and durability can be
produced from mixtures with water-binder ratios at 0.35 and having cementitious
material content at or below (least) 550 kg/m3, including 20–40 percent fly ash and
5–12.5 percent silica fume, at the age of 56 days almost concrete mixes designed in the
above research attained strength ranging from 85 to 105 MPa, and physics properties
it’s will continue to grow at older ages. The results experiment showed that in almost
mixes containing SF content at 10% reached the highest physics properties at all
ages. Hence dosage 10% is optimum replacement content of SF in concrete mixtures
when incorporating with of FA content from 20 to 40% achieve good overall results
on properties of concrete like strength, and durability for the construction structures
under hot and humid climate conditions in the tropics as well as in Vietnam. Besides,
In terms of economics, reduce industrial waste, namely, fly ash to protect the envi-
ronment, as well as corrosion resistance of concrete when they work in corrosive
environments, the figures above show that the optimum high-performance concrete
mixes content used for construction works in corrosive environment aggressive with
hot-humidity climate conditions of Vietnam, consists of, S8(7.5%SF + 30%FA),
S10(10%SF + 20%FA), S11(10%SF + 30%FA), S13(12.5%SF + 20%FA) mixtures
respectively.
The utilization of industrial by-products materials like fly ash, silica fume to
replace and decline cement content in concrete mix lead to reducing vehicle emissions
CO2 contribute to protecting the environment (by minimizing the volume of waste
disposed on the wasteland and minimizing the emission of greenhouse gases that are
released during cement production). On the other hand, due to its beneficial chemical-
physical properties leads to reduce water demand in concrete by using low w/b ratio,
improving properties of HPC conduct increases durability, low creep, and shrinkage,
and reduces the cost maintenance and repairs, and smaller depression as a fixed cost,
help increase the service life of structures constructions. High-performance concrete
is vast applications in the construction industry. Especially, it is used in the bridge
structures, tall buildings, dams, and water retaining structures and structures located
in the worst or extreme weather region.

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Modifying Heracleum sosnowskyi Stems
with Monoethanolamine(N→B)-
trihydroxyborate for Manufacturing
Biopositive Building Materials

Irina Stepina, Marc Sodomon, Vyacheslav Semenov, Elizaveta Dorzhieva,

and Irina Titova

Abstract One of the modern trends in construction is the orientation towards biopos-
itive building materials and technologies. In this regard, the use of plant raw materials
for the production of building materials is of relevance. The aim of the research was
to study the possibility of chemical modification of the ligno-carbohydrate complex
of Heracleum sosnovsky to obtain biostable plant raw materials. The hypothesis was
formulated, according to which a significant increase in the biostability of plant
raw materials based on Heracleum can be achieved by modifying it with a four-
coordination boron compound–monoethanolamine(N→B)-trihydroxyborate. This
goal was achieved through laboratory studies of the sorption of the modifier, studying
the microstructure of the samples and carrying out IR spectroscopy. As a result of the
research, it was found that the degree of modification of Heracleum stems multiply
depends on the modification temperature and drying temperature, the corresponding
dependences have been established. It was found that under the action of the modi-
fier, depolymerization of lignin and hemicellulose occurs, as well as the chemical
interaction of the reagent with the substrate. Analysis of the microstructure indi-
cates a uniform distribution of the modifier in the intercellular space. The novelty
of the conducted research consists in the substantiation and experimental confirma-
tion of the possibility of increasing the biostability of plant raw materials—Hera-
cleum. The significance of the result lies in determining the optimal parameters of
the modification process.

Keywords Biopositive building materials · Heracleum sosnowskyi ·

Monoethanolamine(NB)-trihydroxyborate

I. Stepina (B) · M. Sodomon · V. Semenov

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
E. Dorzhieva
Department of Building Materials, Roads and Woodworking, East Siberia State University of
Technology and Management, Klyuchevskaya, 40V, Ulan-Ude 670013, Russia
I. Titova
Core Facility Centre «Progress», East Siberia State University of Technology and Management,
Klyuchevskaya, 40V, Ulan-Ude 670013, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 45

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_5
46 I. Stepina et al.

1 Introduction

Heracleum sosnovsky was imported to the Russian Midlands from Georgia in the
middle of twentieth century, later domesticated and throughout few decades has been
grown for feeding livestock [1–4]. This was due to the biological specific features
of the plant species, namely low maintenance, rapid growth and development of
green matter, frost tolerance, high content of vitamins, proteins and microelement in
the composition of cell walls [5–8]. However, the presence of furocoumarins in the
plant green matter has exacerbated the quality of agricultural products. Moreover,
there was discovered the potential hazard of dermatitis caused in the case of contact
with Heracleum [9]. Thus, its cultivation was halted as early as the 1980s, at first in
Europe, than on the territory of the Commonwealth Independent States. Nevertheless,
throughout the period of its cultivation, this species has become wide spread in
Eastern Europe, in particular on the territories of the Former Soviet Republics.
In the Central Part of Russia, the first specimen of wilding Heracleum sosnovsky
was first detected in 1948 in Moscow Region. Over the subsequent years, Heracleum
have been found, mostly, near the cultivation areas; at the very least in Moscow
Region this species had not exhibited the tendency to diffusion into natural vege-
tation until the 1970s when Heracleum naturalizing became wide spread [1, 10].
Nowadays, the expansion of this invasive plant is becoming environmental catas-
trophe. If the green matter of Heracleum was used more actively in the national
economy, this would facilitate its removal from agricultural ecosystem as well as
invasive offloading. Presently, the scientists are searching the possible applications of
Heracleum biomass in engineering, pulp-and-paper production, medicine and phar-
maceutical industry [11–14]. Porous stem structure of Heracleum has quickened
scientists’ interest in this material in the function of concrete admixture lowering its
thermal conductivity [15]. Taking into account the current trend for the development
of biopositive building technologies [16], the use of plant raw materials, in partic-
ular Heracleum, for the production of various building materials is of relevance and
practical interest.
However, this plant raw material requires pre-treatment with antiseptics, because
despite the high content of extractives in the composition of cell walls of Hera-
cleum, there was observed the active growth of fungi on the stem surface when
stored in contact with moist environment. The previous studies have shown that
boracium-nitrogen compounds are rather effective, from the point of bio stability
enhancement of plant raw materials used in construction. For example, the papers
[17, 18] have demonstrated that application of four-coordination compounds of
boracium for surface modification allows ensuring 100% bio stability of timber
structures for the period of at least 20 years. It was suggested that application of four-
coordination boracium compound–monoethanolamine(N→B)-trihydroxyborate for
modifying cell walls of Heracleum stems will allow to enhance its biostability. Thus,
the purpose of the present research was the study of possible chemical modifica-
tion of ingo-carbohydrate complex of Heracleum sosnovsky with four-coordination
boracium compound in order to increase the biostability of plant raw materials for the
Modifying Heracleum sosnowskyi Stems with Monoethanolamine … 47

production of building materials. In order to realize the set objective, the following
tasks were to be solved: determining optimal temperatures for modifying and drying
of source raw material on the base of monoethanolamine(N→B)-trihydroxyborate;
identifying the pattern of interaction of monoethanolamine(N→B)-trihydroxyborate
with ingo-carbohydrate complex of Heracleum sosnovsky cell walls.

2 Methodology

As source raw materials of plant origin, there were used grinded stems of Heracleum
sosnovsky gathered in September 2020 in the Pushkinsky district of Moscow Region.
The stems were preliminary cleaned from the extraneous bodies and air-dried until
they reach the constant weight. Then they were mechanically grinded to the particle
size ranging 1–5 mm; the particles’ shape being irregular. Modifying was carried
out by means of dipping into 50% aqua solution of monoethanolamine(N→B)-
trihydroxyborate, pH = 9, for 3 h. Modifying temperature accounted for 25, 50,
75 and 100 °C correspondingly. The samples were further filtered off on the filter
paper and dried until they reach the constant weight by two methods: (a) air-drying
at room temperature and (b) in the drying cupboard at 105 °C. In order to remove
the excess of unreacted modifier and hydrolysis products, threefold extraction of
modified and reference samples was done by distilled water at room temperature with
further air-drying at room temperature until they reach a constant weight. Then, the
surface layer of samples was investigated by the methods of infrared spectroscopy
and electronic microscopy. The measurements were taken by the infrared Fourier
spectrometer VERTEX 70v made by BRUKER (Germany) with the application of
the auxiliary device FTIR (frustrated total internal reflection) GladyATR made by
PIKE (USA) with diamond operating element. Spectra were obtained directly from
finely grinded samples, non-treated additionally. The spectra taken by the method
of frustrated total internal reflection (FTIR) were converted into absorption spectra
by factoring in the relationship between penetration depth of infrared radiation into
the sample and wave length, done by the OPUS application incorporated into the
device software. Microphotographs of modified samples were received by means of
the raster electron microscope JSM-6510LV JEOL.

3 Results and Discussion

The results of samples’ weighing before and after modifying at various temperatures
as well as after extraction and drying are outlined in the Table 1.
From the data presented in the Table 1 it is apparent that light porous base has
high sorption capacity, it absorbs from 1.4 g up to 3.18 g of modifier per 1 g of
the base as a function of temperature. However, less modifier is bonded chemically.
This was stated by means of samples weighing after extraction. Extraction enables
48 I. Stepina et al.

Table 1 Mass change of the samples occurred in the process of modifying, extraction, and drying
NN Tmodifying , °C Tdrying, °C Initial mass of Samples’ mass after Samples’ mass after
samples, g modifying and extraction and
drying**, g/% of drying**, g/% of
increment increment
1 25 25 5 12.0/140 7.3/46
2 25 105 5 12.6/152 8.8/76
3 50 25 5 13.7/174 7.6/52
4 50 105 5 14.0/180 7.8/56
5 75 25 5 13.7/174 8.1/62
6 75 105 5 14.3/186 10.3 / 106
7 100 25 5 18.3/266 11.8/136
8 100 105 5 20.9/318 12.5/150
9 Reference* Reference 5 5 4.4/ − 12
* Reference samples were not modified by monoethanolamine(N→B)-trihydroxyborate but
undergone the extraction under the same conditions as modified samples
**mean values of three series of weighing

to eliminate the excess of unreacted monoethanolamine(N→B)-trihydroxyborate

from the composition of support pores. According to the table data, the level of
modification has multiply increased as the temperature grows. Along with that, not
only the temperature of modification has the meaning but the temperature of drying.
The highest degree of modifier’s sorption is observed at the temperature of 100 °C
and drying in the drying cupboard at 105 °C. Under such conditions, there has
been observed threefold exceeding of the modifier’s content in the base composition
if compared with the process of modifying at room temperature. But also at room
temperature of modifying and drying, the content of modifier in the base composition
is noticeable—46%. This could ensure good security against bio corrosion and further
researches will be aimed at the lowering of modifier’s concentration and the search
of the optimal concentration.
Lingo-carbohydrate complex (LCC) of cell walls of Heracleum has the highest
reaction capacity in relation to monoethanolamine(N→B)-trihydroxyborate if
compared with LCC of timber [19]. This phenomenon could be explained by the
following. Firstly, in the contact with modifier LCC of cell walls of Heracleum
experiences lower steric hindrances if compared with LCC of timber due to less
dense structure of tissues. Secondly, there are essential differences in the component
ration in the LCC-composition. In the LCC-composition of timber, less chemically
reactive cellulose dominates over chemically reactive lignin twofold, while in the
LCC-composition of Heracleum the ration of cellulose and lignin is equal to 1:1
approx. Along with that, the attention should be paid on the negative mass incre-
ment of reference samples after extraction. This gives an evidence that hydrolysis
of extractive substance from the initial substance composition takes place. Taking
into account that modifying solution is characterized by alkaline medium (pH = 9),
Modifying Heracleum sosnowskyi Stems with Monoethanolamine … 49

then during the modifying process, the percent of extractive substance outwashing
must be significantly higher. Moreover, the hydrolysis process for low molecular
polysaccharose (hemicellulose) and lignin in alkaline medium is inevitable. Never-
theless, the data of the Table 1 demonstrate that the mass loss is not observed (the
last column of the table). This could be explained by the fact that significant mass
increment arising due to modification overlaps by far the mass loss occurring during
hydrolysis.
The Fig. 1 represents the fragments of infrared spectra in the area undergone
significant changes during modification process. The Fig. 1 shows that all the spectra
of modified samples 1.1.–1.8 are similar and have relatively insignificant differences.
The variation from the reference sample spectrum (1.9) is much more prominent.
The data presented at the Fig. 1 confirm that the most part of changes in the dipped
base is observed at the curves peaks related to hemicellulose (1737, 1268, 1100,
1056 cm−1 ) and lignin (1601, 1268 cm−1 ). These changes are the result of lignin and
hemicellulose depolymerization caused by a modifier, whereas cellulose has not been
depolymerized that ensures secure fixation of modifier in the composite formulation
[20].
After modifying and drying the spectral band of 1735 cm−1 has almost disap-
peared, the spectral band of 1580 cm−1 is significantly reduced (has nearly
disappeared).
This goes to prove chemical interaction between lignin hydroxyls and modifier’s
molecules as well as partial hydrolysis of aromatic constituent of Heracleum LCC.
The structure of the wide absorption band in the range of 1315–1470 cm−1 has
also undergone a change. The changes prove that for all the samples, regardless the
temperature of modifying and drying, chemical interaction of reactive chemical and
the base takes place. Modified wide absorption band in the range of 1315–1470 cm−1
appeared after modification (Fig. 1) testifies the presence of the coordinate bond
N→B in the composition of modified base [21]. Once can also observe the appearance
of band in the range of 1630 cm−1 , which corresponds to the effect of bonding NH2
in the spectra of the samples 1.1–1.8, Fig. 1 [21].
Microphotographs of modified and reference samples (magnification 500×)
are presented in the Fig. 2. As the presented photographs show, modifier wraps
around cell walls. At the microphotographs 2.1–2.3, even distribution of modifiers
in the intercellular space is observed. In addition to this, the higher modification
temperature, the more saturated near-wall layers by the modifier’s molecules.
This is explained by the fact that elevated temperature in alkaline medium
leads to more constitutive hydrolysis of hemicellulose and lignin; and their place
in the composite structure are taken by the molecules of monoethanolamine
(N→B)-trihydroxyborate.
50 I. Stepina et al.

1.1 1.2

1.3 1.4

1.5 1.5

1.7 1.8

1.9

Fig. 1 Infrared spectrum of modified and references samples: 1.1 Tmod = 25 °C, Tdrying = 25 °C;
1.2 Tmod = 25 °C, Tdrying = 105 °C; 1.3 Tmod = 50 °C, Tdrying = 25 °C; 1.4 Tmod = 50 °C, Tdrying
= 105 °C; 1.5 Tmod = 75 °C °C, Tdrying = 25 °C; 1.6 Tmod = 75 °C, Tdrying = 105 °C; 1.7 Tmod =
100 °C, Tdrying = 25 °C; 1.8 Tmod = 100 °C, Tdrying = 105 °C; 1.9 Reference sample (non-modified
sample)
Modifying Heracleum sosnowskyi Stems with Monoethanolamine … 51

2.1 2.2

2.3 2.4

Fig. 2 Microphotographs of modified and reference samples (magnification 500×): 2.1 Tmod =
25 °C, Tdrying = 25 °C; 2.2 Tmod = 50 °C, Tdrying = 25 °C; 2.3 Tmod = 100 °C, Tdrying = 25 °C;
2.4 Reference sample (non-modified sample)

4 Conclusions

Thus, the obtained experimental data allow making the following conclusions. When
modifying plant raw materials represented by grinded stems of Heracleum sosnovsky
by the composition on the base of monoethanolamine(N→B)-trihydroxyborate,
depolymerization of lignin and hemicellulose occurs, and the modifiers molecules
replace them in the composition of lignin-carbohydrate complex of cell walls. This
process is reinforced as the temperature of modifying and drying grows that leads to
the increased content of boracium-nitrogen compound in the base composition. The
process of dipping plant raw material in monoethanolamine(N→B)-trihydroxyborate
is accompanied by chemo sorption. This is evidenced by the presence of modifier’s
molecules confirmed by the data of infrared spectroscopy and microphotography, in
the intercellular space after continuous water extraction. The novelty of the conducted
research consists in the substantiation and experimental confirmation of the possi-
bility of increasing the biostability of plant raw materials—Heracleum–by modifying
with monoethanolamine(N→B)-trihydroxyborate, as well as establishing the mech-
anism of this process. When applied as preservatives of plant raw materials, it is
recommended to carry out modifying and drying at room temperature as the high
level of modification is reached at these conditions.

Acknowledgements The team of authors would like to thank B.V. Lokshin [A.N.Nesmeyanov
Institute of Organoelement Compounds of Russian Academy of Sciences (INEOS RAS)] and
52 I. Stepina et al.

E.G. Leonova, T.B. Tumurova (CCU “Progress”, East Siberia State University of Technology and
Management) for help in research.

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4:27–33
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[MGSU Bulletin] 12(14):1555–1571
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structures: monograph. MGSU, Moscow
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19. Lesar B, Podlesnik B, Pohleven F, Humar M, Kralj P, Veber M (2008) J Membr Sci 53(3):17–26
20. Haider S, Park S-Y (2009) Carbohyd Polym 328:90–96
21. El Oudiani A et al (2017) Vestnik MGSU [MGSU Bulletin] 164:242–248
Phase Content of Plasticized Cement
Systems in the Early Stages
of Heat-Moisture Treatment

Andrei Leshkanov, Lev Dobshits, and Sergey Anisimov

Abstract The use of superplasticizers (SPs) in concrete makes it possible to obtain

high-strength concretes at the design age. However, their application can reduce the
early concrete strength. The use of the latest SPs based on polycarboxylate esters
(PCE) allows you to adjust the heat-moisture treatment (HMT) regime. A compar-
ative analysis of the cement stone structure formation was carried out depending
on the modification of PCE and sulfonated naphthalene formaldehyde (SNF) under
HMT conditions at 60 °C. For the first 2 h of HMT, the amount of C3 A and C4 AF
is greater in the stone with PCE by 12 and 8.5% than in the compositions without
additives and 6 and 2% higher than in the compositions with SNF. The introduction
of SNF in cement paste leads to an increased content of AFt by 32% compared to a
stone with PCE. The content of CaSO4 · 2H2 O in the compositions with PCE was
found to be twice as high as in the samples with SNF and without additives for 2 h
of HMT. The number of clinker minerals after HMT according to the mode (0 + 6
+ 2) h in the stone with PCE is higher than in the compositions without additives
and with SNF, but the differences are lower. In general, the results indicate the relax-
ation of temperature stresses in the cement stone body when PCE is added. Thus, the
study indicates the possibility of regulating the HMT modes during the hardening of
plasticized cement systems.

Keywords Cement · Heat-moisture treatment · Superplasticizer · Polycarboxylate

ester · Sulfonated naphthalene formaldehyde · Phase content · Gypsum · Ettringite

A. Leshkanov (B) · S. Anisimov

Department of Construction Technologies and Highways, Volga State University of Technology, 3
Lenin square, Yoshkar-Ola 424000, Russia
L. Dobshits
Department of Construction Materials and Technologies, Russian University of Transport (MIIT),
9, b 9 Obrazcova street, Moscow 127994, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 53

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_6
54 A. Leshkanov et al.

1 Introduction

In modern construction practice, concrete is increasingly used with various addi-

tives. According to the Russian State Standard (GOST) 24211–2008, additives are
divided into the regulatory properties of concrete and mortar mixtures (f. ex., super-
plasticizers), regulating properties of concretes and mortars (f. ex., accelerators and
retarders, curing), giving concrete and solutions with special properties (f. ex., frost
and water-repellent) and mineral supplements (f. ex., slag, metakaolin) [1–3].
The most common additives in concrete technology are plasticizers and super-
plasticizers. In such additives based on naphthalenes (SNF) and melamine sulfonates
(SMF), and modified lignosulfonates (LS), the polymer molecule, due to its nega-
tive electric charge, is adsorbed on the surface of the entire cement grain. At the
same time, sulfonates are rapidly and almost completely absorbed. As a result of this
phenomenon, water loses access to the cement components, and the setting rate of
the cement is significantly reduced up to the rupture of the plasticizer film [1, 4, 5].
In the last decade, the market for concrete modifiers has been increasingly
replenished with superplasticizers based on PCE. Their adsorption properties can
be purposefully controlled by changing the number of carboxylate groups and the
charge density. For example, shorter side chains with a constant carboxylate-to-ether
ratio (C/E) and an increase in the C/E ratio with a constant side chain length lead to
an increase in the retarding effect. This effect is explained by the fact that in both
cases the density of the carboxylate groups increases, which leads to more efficient
adsorption and, probably, to a longer delay in the cement hydration [6, 7]. Due to the
steric repulsion in PCE plasticizers’ presence, the cement grains are dispersed and
kept at a distance. This principle of operation is called “spatial dispersion” [2, 8].
Depending on the specific tasks caused by the concrete manufacturing technology,
it becomes possible to regulate the properties of concrete mixtures by optimizing the
PCE molecules’ chemical structure. Polymers with long side chains make it possible
to obtain concretes with high early strength. This is explained by the following
mechanism of action of the additives: the molecules of such PCEs quickly adsorbed
on cement particles, are oriented by their side chains perpendicular to the surface.
Creating a spatial repulsion effect, they do not cover the cement particle’s entire
surface but provide access to water molecules for hydration. At the same time, there
is an intensive increase in the strength of the cement stone [9–11].
Heat-moisture treatment (HMT) is the most common method for producing
concretes with high tempering strength. However, the concrete subjected to HMT
has lower physical and mechanical properties (strength, frost resistance, crack resis-
tance) than a similar concrete stone of natural hardening. That is due to a temperature
gradient in the solidifying concrete body, the formation of directed capillary porosity.
Elevated temperatures (up to 90–95 °C), prolonged steaming (12–16 h) significantly
increase energy costs, and as a result, harmful CO2 emissions into the atmosphere.
Besides, the cost of reinforced concrete products increases, the formwork’s turnover
and the production rate of finished products and structures significantly decrease
[12, 13].
Phase Content of Plasticized Cement Systems … 55

Consumers and manufacturers are always interested in obtaining high-quality

concrete in the shortest possible time with the lowest energy costs [14, 15]. Since the
most important component of the concrete matrix, which determines all its properties,
is cement stone, the study of its structure formation, under the combined action of
plasticizing additives and temperature, is of undoubted interest and will allow you
to regulate the concrete properties directly.
It is known that PCE adsorption at the initial stage mainly occurs on the surface
of C3 A and C4 AF, also monosulfates and ettringite. In contrast, adsorption of LS,
SNF, and SMF is less selective for the cement phases [11, 16–18]. It is important to
understand what changes occur in cement stone composition in the early stages at
elevated temperatures. Particularly, the hydration rate of cement increases with the
rise of temperature, leading to higher early strength.
Morphology and microstructure of hydration products are directly linked to the
curing temperature. As the temperature increases, there is an increase in the C–S–H
density and a more heterogeneous hydration product structure [19, 20]. With the
addition of PCE, there is a certain slowdown in hydration, which depends on the
molecular structure and dosage of the polymer [9, 21].
It is also noted that when modifying cement pastes at an early stage, dispersed
short bars of ettringite are formed. The main PCE effect on ettringite precipitation is
related to a strong increase of ettringite specific surface area. After 5 min of hydration,
the surface area increase related to ettringite precipitation can be six times higher
in the presence of PCE. At the same time, when adding PCE, the total amount of
ettringite by weight decreases [22].
The microstructure of SNF plasticized cement pastes hydrated for 28 days,
composed closely packed structure consisting of ill crystalline and nearly amorphous
calcium silicate hydrates representing the main hydration product, also hexagonal
crystals of calcium aluminate hydrates and calcium hydroxide. This is a highly dense
structure [4].
At the same time, the microstructure of cement stone with PCE is represented
by fine needle-like ettringite minerals and thin Ca(OH)2 layers with a large specific
surface area. Consequently, the formed neoplasms that crystallize in the presence
of a complex additive in a finely dispersed form fill the pores and capillaries of the
cement stone, strengthening and compacting its structure [23].
The authors conducted a study of cement stone phase composition with PCE,
depending on the duration of presteaming to the temperature rise under conditions
with a 6-h HMT at 60 °C and 2-h cooling. It is proved that there is a tendency to
reduce the amount of ettringite and portlandite minerals in cement stone in the PCE
presence. The proportion of unreacted clinker minerals C3 A and C4 AF is higher in
compositions with PCE after HMT. No significant differences were recorded in the
mineralogical composition of cement stone samples that harden during HMT with
presteaming for 2 h and without it [24, 25].
However, the change in the phase composition of cement stone at the early stages
of hydration, particularly during the first hours of HMT, has not been sufficiently
studied. To further develop production concrete technology and establishing the
factors that aimed to regulate the structure of the formed cement stone is necessary
56 A. Leshkanov et al.

to determine which growths arise in the cement when it is modifying with superplas-
ticizers of different action types in different temperature conditions. In this regard,
we have conducted studies of the phase composition of cement stone with PCE and
SNF additives in the early stages of HMT.

2 Methods

The main objective of this investigation is to study the phase composition of

neoplasms of plasticized cement systems at the early stages of hardening.
Portland cement CEM I 42.5N produced by “Gornozavodskcement” company
was used as a binder. The physical and mechanical properties of cement and the
mineralogical composition of cement clinker are shown in Table 1.
The reduction of the water-cement ratio was achieved due to the polycarboxylate
superplasticizer Sika ViscoCrete 24 HE of the Swiss concern “Sika”. For comparison,
cement stone structure formation with the introduction of a superplasticizer based on
SNF compounds S-3 from “Polyplast” is considered. The dosage was taken following
the technical data sheets of additives (0.4% by cement mass). The amount of mixing
water was selected based on the condition of equal mobility of cement pastes. The
mobility was controlled by lowering the Vic device’s pestle according to the Russian

Table 1 Some properties of the cement used

No. Description of Unit of measurement Actual readings
characteristics
Physical and mechanical properties of cement
1 Normal consistency of % 27.0
cement paste
2 Compressive strength after MPa 36.0
HMT according to the
Russian State Standard
(GOST) 10178-85
3 Efficiency group after I
steaming according to the
Russian State Standard
(GOST) 10178-85
4 Surface area cm2 /g 3370
Mineral composition
5 C3 S (3CaO · SiO2 ) % 60.7
6 C2 S (2CaO · SiO2 ) % 13.4
7 C3A (3CaO · Al2 O3 ) % 7.5
8 C4 AF (4CaO · Al2 O3 % 12.8
· Fe2 O3 )
Phase Content of Plasticized Cement Systems … 57

Table 2 Compositions of the studied samples of cement stone

No. HMT mode Admixture Dosage of the admixture, in % w/c
by cement mass
1 2 h of HMT Without admixtures – 0.27
2 2 h of HMT Sika ViscoCrete 24HE (PCE) 0.4 0.234
3 2 h of HMT S-3 (SNF) 0.4 0.264
4 (0 + 6 + 2) h Without admixtures – 0.27
5 (0 + 6 + 2) h Sika ViscoCrete 24HE (PCE) 0.4 0.234
6 (0 + 6 + 2) h S-3 (SNF) 0.4 0.264

standard (GOST) 310.3-76 until the normal density was reached. The compositions
of the test samples are presented in Table 2.
The samples were tested for 2 h of hardening at 20 °C and 8 h after HMT according
to the mode without presteaming, 6-h steaming at 60 °C, and 2-h cooling. After
reaching the required time (2 and 8 h), the cement stone samples were crushed in a
press and ground to powder. The resulting material was sifted through a sieve with
a cell size of 0.27 mm and placed in a 50 ml container, and filled with a 99.99%
solution of isopropyl alcohol to stop hydration.
X-Ray Diffraction (XRD) analysis is chosen to determine the mineralogical
composition, changes, and identification of neoplasm products during cement stone
hardening. CuKa radiation is used, monochromatized (λ(Cu–K) = 1.54060 Å) by a
germanium curved Johanson monochromator; the operating mode of the X-ray tube
is 30–40 kV, 20–30 mA, the scanning step is 0.05°. The experiments were performed
at a temperature of 20 °C in the Bragg-Brentano geometry with a flat sample.

3 Results and Discussion

The water-cement (w/c) ratio of equal-moving cement pastes decreased due to

the water-reducing action of PCE (Sika ViscoCrete 24 HE) by 13.3%, while the
introduction of an additive based on SNF (S-3) reduced it by only 2.3%.
Figure 1 shows the XRD pattern of a cement stone sample without additives after
2 h of HMT (sample No. 1).
Figures 2 and 3 show X-ray images of a cement stone sample with 0.4% PCE
Sika ViscoCrete 24HE and 0.4% additive S-3 (SNF) after 2 h of HMT (samples No.
2 and No. 3, respectively).
Based on the conducted studies (Figs. 1, 2 and 3), it was found that the modification
of cement pastes with PCE leads to the blocking of the positive-potential minerals
of trialcium aluminate with d = [2.68 Å] and brownmillerite with d = [7.29; 2.65;
2.05 Å], as evidenced by an increase in the intensity of the corresponding diffraction
maxima in comparison with samples No. 1 (without additives) and No. 3 (with the
addition of SP based on SNF). Besides, in sample No. 2, the amount of ettringite
58 A. Leshkanov et al.

Fig. 1 XRD pattern of cement stone sample without additives after 2 h of HMT at 60 °C (sample
No. 1)

Fig. 2 XRD pattern of a cement stone sample with Sika ViscoCrete 24HE after 2 h of HMT at
60 °C (sample No. 2)

with d = [9.73; 5.61; 4.70; 2.20 Å] decreased, resulting from PCE adsorption on this
mineral and the lack of water tolerance to the mineral C3 A.
The amount of clinker minerals alite and belite in the sample with polycarboxylate
SP is higher compared to the non-additive sample No. 1 and with the SP based on
SNF No. 3, as evidenced by an increase in the diffraction maxima with interplane
distances d = [5.93; 5.48; 3.03; 2.96; 2.77; 2.74; 2.60; 2.32; 1.98; 1.77 Å] and d =
[2.88; 2.77; 2.60; 2.29 Å], respectively.
Modification of cement pastes by SP based on SNF leads to a decrease in port-
landite content with d = [4.91; 3.11; 2.63; 1.93; 1.80 Å] compared to samples No.
1 and No. 2. At the same time, the amount of ettringite formed (d = [9.73; 5.61;
4.70; 2.20 Å]) is significantly higher than in sample No. 1 (without additives) and
Phase Content of Plasticized Cement Systems … 59

Fig. 3 XRD pattern of a S-3 modified cement stone sample after 2 h of HMT at 60 °C (sample No.
3)

No. 2 (with PCE), which indicates the formation of a loose aluminate structure for
the first 2 h of hardening in the sample with S-3. This phenomenon is explained by
the different actions of PCE (mainly steric repulsion effect) and the products of SNF
polycondensation (electrostatic effect).
Distinct peaks with d = [7.62; 4.26 Å] corresponding to the mineral CaSO4 ·2H2 O
added to the cement to regulate the setting time were recorded in the samples with
polycarboxylate SP at 2 h of HMT. In samples No. 1 and No. 3, the relative intensity
of the peaks is lower by 29 and 41%, respectively, compared to the gypsum peak of
sample No. 2, which allows us to conclude that tricalcium aluminate is blocked, as
a result of which unbound CaSO4 ·2H2 O minerals are detected.
The above-described conclusions about the structure formation of the phase
composition for the first 2 h of the HMT indicate the relaxation of the resulting
stresses from thermal action in the cement stone body modified with PCE.
Figures 4, 5, and 6 show X-ray images of cement stone samples without additives
and with various SPs after HMT in the mode (0–6–2) h (samples No. 4, 5, and 6,
respectively).
Similarly, as in the first 2 h of HMT at 60 °C in sample No. 5 with a PCE after
the HMT mode according to the scheme (0–6–2) h, the amount of unreacted clinker
minerals of alite d = [5.93; 3.03; 2.96; 2.77; 2.74; 2.60; 2.32; 1.98; 1.77 A], C3 A
with d = [2.68 Å], and brownmillerite with d = [7.29; 2.65; 2.05 Å] are lower than
in the samples without additives and with SP based on SNF, which indicates a slight
slowdown in hydration.
X-ray phase analysis of cement stone samples was performed. The semi-
quantitative calculation is performed, taking into account 19–29% of the amorphous
phase. The mineral content in the cement stone samples is shown in Table 3.
According to the study results, it was found that the use of PCE and SNF leads to
a slowdown in the hydration processes of Portland cement for the first 2 h of HMT.
60 A. Leshkanov et al.

Fig. 4 XRD pattern of a sample of cement stone without additives after HMT according to the
mode (0–6–2) h at 60 °C (sample No. 4)

Fig. 5 XRD pattern of a cement stone sample with Sika ViscoCrete 24HE after HMT in the mode
(0–6–2) h at 60 °C (sample No. 5)

However, the slowing of hydration is more pronounced in the composition

with polycarboxylate SP, which is reflected in a significant amount of unreacted
minerals of tricalcium aluminate and brownmillerite, as well as alite and belite. This
phenomena is explained by the selective adsorption of PCE-based SP on clinker
minerals C3 A and C4 AF, and on ettringite. As a result, the CaSO4 ·2H2 O content was
twice as high in compositions with PCE.
A noticeable decrease in the concentration of portlandite in a sample with SNF
after 2 h of HMT and a significant increase in ettringite content at the early stage
of Portland cement hardening was found, which explains the low strength of the
composition.
With an increase in HMT time to 8 h (at the mode of (0–6–2) h), the amount
of clinker minerals in the composition with PCE is also higher than in the sample
Phase Content of Plasticized Cement Systems … 61

Fig. 6 XRD pattern of a cement stone sample modified with S-3, after HMT in the mode (0–6–2)
h at 60 °C (sample No. 6)

Table 3 Results of quantitative calculation of cement stone

Name of the 2 h of HMT HMT by regime of (0 + 6 + 2) h
phase No. 1 No. 2 (with No. 3 No. 4 (without No. 5 (with No. 6
(without SV 24HE) (with additive) SV 24HE) (with
additive) S-3) S-3)
C3 S 36.3 39 40.5 24.9 25.7 25.1
β-C2 S 13.5 14.0 15.1 9.2 9.6 12.6
CaSO4 · 2H2 O 3.4 6.0 3.1 0.7 1.4 0.7
C4 AF 5.3 5.8 5.7 5.0 5.5 5.3
C3 A 6.1 6.9 6.5 5.7 6.7 6.4
Ca(OH)2 4.6 3.6 3.1 14.2 13.5 14.7
AFt (ettringite) 6.8 4.8 7.1 11.4 8.5 11.2

without additives and with SP based on SNF. Still, the differences are significantly
lower than after 2 h of HMT.

4 Conclusions

1. It was found that the amount of unreacted clinker minerals C3 A and C4 AF in

compositions with PCE is 12 and 8.5% higher than in compositions without
additives after 2 h of HMT, which is explained by the selective adsorption of
PCE on cement grains in contrast to SNF.
2. The amount of ettringite formed in the first 2 h of HMT is higher in cement stone
with SNF by 4% and 32% in comparison with compositions without additives
62 A. Leshkanov et al.

and with PCE, respectively. The concentration of portlandite in the sample with
SNF is noticeably lower, which leads to the formation of a loose cement stone
structure at this stage of cement hardening.
3. In cement stone with PCE by the end of the first 2 h of HMT, the gypsum content
is two times higher than samples without additives and SNF, which allows us
to conclude that tricalcium aluminate is blocked, as a result of which unbound
minerals CaSO4 ·2H2 O are found.
4. The conducted studies have shown that the amount of clinker minerals in the
composition with PCE is also higher than in the sample without additives and
with SNF after TVE according to the regime (0–6–2) h, but these changes are
significantly less than after 2 h of HMT.
5. 5. There is a reduced content of ettringite and portlandite minerals in the cement
stone with PCE at the 8-h HMT mode by 3% and 8% compared to the sample
with SNF and without additives, respectively.
6. The above-described conclusions on the structure formation of the phase compo-
sition for the first 2 h of HMT indicate the relaxation of the resulting stresses
from thermal exposure when modifying compositions with SP based on PCE.
It becomes possible to regulate the HMT modes in terms of accelerating
construction’s pace without significantly reducing the physical and mechanical
properties.

References

1. Gelardi G, Mantellato S, Marchon D, Palacios M, Eberhardt AB, Flatt RJ (2016) Science and
technology concrete admixtures. Elsevier Inc., pp 149–218
2. Dobshits LM, Kononova OV, Anisimov SN, Leshkanov AY (2014) Fundam Res 5:945
3. Khudhair MHR, El Hilal B, Elharfi A (2018) J Mater Environ Sci 9:1722
4. El-Gamal SMA, Al-Nowaiser FM, Al-Baity AO (2012) J Adv Res 3:119
5. Robeyst N, De Schutter G, Grosse C, De Belie N (2011) Mag Concr Res 63:707
6. Winnefeld F, Becker S, Pakusch J (2007) Cem Concr Compos 29:251
7. Yamada K, Ogawa S, Hanehara S (2000) Spec Publ 351
8. Flatt RJ, Houst YF (2001) Cem Concr Res 31:1169
9. Schmidt W, Brouwers HJH, Kühne HC, Meng B (2014) Cem Concr Compos 49:111
10. Falikman VR (2009) Pop Concr Sci 2:88
11. Plank J, Hirsch C (2007) Cem Concr Res 37:537
12. Mironov SA (1966) Symposium on structure of portland cement paste and concrete.
Washington, pp 465–474
13. Neville AM (2012) Properties of concrete
14. Smirnova OM (2016) Mag Civ Eng 6:12
15. Leshkanov AY, Anisimov SN, Kononova OV, Minakov YA, Smirnov AO (2015) Mod Probl
Sci Educ 2:6
16. Schmidt W (2014) Design concepts for the robustness improvement of self-compacting
concrete: effects of admixtures and mixture components on the rheology and early hydration
at varying temperatures. Ph.D. Thesis, Eindhoven University of Technology, The Netherlands
17. Yoshioka K, Tazawa EI, Kawai K, Enohata T (2002) Cem Concr Res 32:1507
Phase Content of Plasticized Cement Systems … 63

18. Regnaud L, Nonat A, Pourche S, Pellerin B (2006) Proceeding of the 8th CANMET/ACI inter-
national conference on superplasticizers and other chemical admixtures in concrete. Sorrento,
pp 389–408
19. Gallucci E, Zhang X, Scrivener KL (2013) Cem Concr Res 53:185
20. Lothenbach B, Matschei T, Möschner G, Glasser FP (2008) Cem Concr Res 38:1
21. Zingg A, Winnefeld F, Holzer L, Pakusch J, Becker S, Figi R, Gauckler L (2009) Cem Concr
Compos 31:153
22. Dalas F, Pourchet S, Rinaldi D, Nonat A, Sabio S, Mosquet M (2015) Cem Concr Res 69:105
23. Bogdanov RR, Ibragimov RA (2017) Mag Civ Eng 5:14
24. Leshkanov AY, Dobshits LM, Anisimov SN (2020) Conf IOP. Ser Mater Sci Eng 869:032038.
https://doi.org/10.1088/1757-899X/869/3/032038
25. Leshkanov AY, Dobshits LM, Anisimov SN (2020) Conf IOP. Ser Mater Sci Eng 896:012094.
https://doi.org/10.1088/1757-899X/896/1/012094
Plasticizer Type Influence on HCP
Radiation Resistance

Vyacheslav Medvedev

Abstract In recent years, the variety of such additives with different functional
effects has reached an unprecedented amount. Of greatest interest is the use of func-
tional additives in the construction of modern nuclear power plants, since this directly
affects the quality and timing of construction. However, exposure to radiation is a
unique environment, and radiation-induced changes in concrete have only been little
studied. Each type of plasticizing additives has its own chemical structure and mech-
anism of action, therefore their influence on concrete composition will be different.
Judging by the existing experimental data on various additive research, it can be
concluded that the study of the effect of the chemical structure or type of the plasticizer
on radiation resistance is of significant practical interest for modern construction. In
the current work, the analysis method using DSC is adopted as a research technique.
Since the study of plasticizing additives based on polycarboxylate ethers was carried
out in the recent work, to analyze the effect of the type of the plasticizing additive,
in the current work the plasticizing additives based on naphthalene formaldehyde,
melamine formaldehyde and lignosulfonates were chosen as the object of study.

Keywords Plasticizers · Chemical additives · Concrete · HCP · Radiation

resistance

1 Introduction

The use of various mineral and chemical additives in the construction of modern
buildings and structures is not uncommon. In recent years, the variety of such addi-
tives with different functional effects has reached an unprecedented amount. Of
greatest interest is the use of functional additives in the construction of modern
nuclear power plants, since this directly affects the quality and timing of construc-
tion. However, exposure to radiation is a unique environment, and radiation-induced
changes in concrete have only been little studied. The use of various additives directly

V. Medvedev (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
e-mail: MedvedevVV@mgsu.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 65

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_7
66 V. Medvedev

affects the chemical and mineral composition of concrete, which creates an even
greater variety of data for study.
Plasticizing additives are among the most common and in demand in the industry.
One of the most developing directions in the field of improving the technology of
concrete work is the use of concrete mixtures with additives for concreting mono-
lithic structures, as well as highly mobile and cast concrete mixtures obtained by
introducing plasticizing additives. The use of such additives makes it possible to
simplify the technology of molding products and structures, to abandon energy-
consuming equipment for vibration compaction of concrete mixtures and to achieve
the required quality of concrete structures. However, plasticizing additives slow down
the hydration of alite, which reduces the durability of hardened cement paste (HCP)
and concrete in the early stages and is a serious drawback of this line of research,
because a decrease in the time for placing the concrete mixture in the formwork will
be accompanied by an increase in the time for the concrete to reach the estimated
durability [1]. At the same time, the positive effect of the use of this type of addi-
tives is expressed in a significant decrease in the water-cement ratio, an increase
in the mobility of the concrete mixture, an increase in the physical-mechanical and
operational properties of concrete, which leads to an acceleration of the construction
process and a decrease in construction costs.
In terms of technical effects, plasticizers are dispersants—stabilizers of the cement
system, forming a structured film as a result of adsorption on the interface between
the solid and liquid phases [2, 3].
The experience in the production of plasticized concrete is closely related to the
preliminary assessment of the following main indicators: the rheological efficiency
of the plasticizing additive in cement concretes, the early hydraulic activity of the
cements in the presence of the plasticizing additive, the mineralogical composition
and fineness of the selected cement [4–6].
Ensuring the workability of the mixture is one of the key factors for accelerating
the production of concrete works and creating a homogeneous structure of concrete
structures, providing the design characteristics and the estimated life of the final
product [6–8]. For the construction of critical structures of nuclear power plants
(such as, for example, dry containment of the nuclear reactor), the workability of
the mixture and the uniformity of the structure directly affect the efficiency and
safety of the entire project. The use of plasticizers for concrete of such structures
can significantly improve the quality of work. However, it is not known how such
concretes will behave when irradiated. At the same time, the composition or the type
of the plasticizer itself plays a significant role in the formation of the structure of the
HCP, which requires special attention when conducting experiments in the study of
the radiation resistance of concrete.
Plasticizer Type Influence on HCP Radiation Resistance 67

2 State of the Issue and Research Methods

To date, there is a lot of data on the study of the effect of various additives on
the radiation resistance of concrete. However, most of the experimental data were
obtained in the 80s and 90s of the last century. At the same time, today the variety
of additives used cannot be compared with what was at that time [9–11].
In works [12–15], a study of the influence of most of the most popular mineral
and chemical additives used in the construction of existing nuclear power plants
was carried out: fly ash, finely ground quartz, a number of anti-freeze additives,
granulometric slag, high and low calcium ash, tripoli, amorphous silica, a number
of set retarders, a number of plasticizing additives, etc. All these data were obtained
during the actual irradiation of cement stones and concretes with additives with
various neutron fluences and are of great interest to the scientific community.
At the same time, the authors of works [3, 7, 9, 12, 16] note that in the case of
using most of the studied chemical additives in the manufacture of HCP, the radiation
resistance of the HCP practically does not change or increases, since the values of
radiation-thermal shrinkage and decrease in mechanical properties practically do not
change or decrease.
It should be noted that conducting such a full-scale test is fraught with enormous
difficulties and costs in terms of time and resources.
In continuation of these works, methods were developed for conducting accel-
erated tests to determine the radiation resistance of cement stones and concrete. In
works [1, 16, 17] one of such techniques is described and its efficiency is proved.
In this case, the main emphasis is placed on the similarity of thermal and radiation-
thermal changes in concretes, which makes it possible to reduce tests to determine
the radiation resistance of concrete to the study of the thermal behavior of samples
[16]. This significantly reduces the cost of testing and allows you to obtain data that
correlate with the actual tests of concrete under irradiation as shown in [1, 16, 17].
According to the classification of the British Association for Concrete Additives
[18], depending on the chemical composition, plasticizing additives are divided into
the following types:
1. Based on sulfonated melamine-formaldehyde polycondensates
2. Based on sulfonated naphthalene-formaldehyde polycondensates
3. Based on lignosulfonates refined from sugars
4. Based on polyacrylates and polycarboxylates.
Each type of plasticizing additives has its own chemical structure and mecha-
nism of action, therefore their influence on concrete composition will be different.
The first three types of plasticizers are also called “traditional”. Plasticizers based
on polyacrylates and polycarboxylates, which have become widespread in the last
decade, are more effective than other types of plasticizing additives. Many experts
associate the advantage of the latter with the structure of molecules: other types of
plasticizing additives are characterized by a linear form of the polymer chain; for
additives based on polyacrylates and polycarboxylates, the spatial form of polymer
value with cross links is characteristic [19, 20].
68 V. Medvedev

In [1, 12, 16, 17], the effect of polycarboxylates on the radiation resistance of
HCP is considered, but the authors of [12] say that the most significant increase in
radiation resistance is provided by plasticizing additives based on formaldehyde. At
the same time, all authors agree that it is the plasticizers that have the greatest impact
on the radiation resistance of concrete.
Based on these data, it can be concluded that the study of the effect of the chemical
structure or type of the plasticizer on radiation resistance is of significant practical
interest for modern construction. In this work, the method described in [17] is adopted
as a research technique. Since the study of plasticizing additives based on polycar-
boxylate ethers was carried out in [17], in this work, to analyze the effect of the type of
the plasticizing additive, plasticizing additives based on naphthalene formaldehyde,
melamine formaldehyde and lignosulfonates were chosen as the object of study.

3 Results

As a part of the experiment the plasticizing additives, which are currently used in the
construction of the structures of the Voronezh, Leningrad and Rostov nuclear power
plants were selected. At the same time, the key requirement for those additives was
that the selected additives should not delay the hardening of concrete and have a
stable effect for cements of different phase composition.
As it was shown in [17], the current studies were carried out on HCP made
in accordance with the method described in mentioned work. For the subsequent
correlation of the obtained data with the results of work [17], a similar Portland
cement with the following content of the main minerals was chosen: C3S—64.8%;
C2S—11.1%; C3A—4.4%; C4AF—15.5%; Bassanite—2.3%; Gypsum—1.9%. For
each experimental composition, the dosage of the additive, which made it possible
to obtain the cement pastes closest in rheology at the same W/C was selected (Table
1).

Table 1 Compositions of HCP accepted for the current research

No. Additive Type of plasticizing additive Recommended Dosage W/C,
marking dosage, % used, % l/kg
0 Control – – – 0.26
1 MF Based on sulfonated 0.3–1.0 0.41 0.24
melamine-formaldehyde
polycondensates
2 NF Based on sulfonated 0.4–0.8 0.44 0.24
naphthalene-formaldehyde
polycondensates
3 LS Based on lignosulfonates No data 0.42 0.24
refined from sugars
Plasticizer Type Influence on HCP Radiation Resistance 69

Fig. 1 DSC graphs obtained for samples of HCP: a sample 0 (Control sample); b sample 1 (MF);
c sample 2 (NF); d sample 3 (LS)

It should be noted that the selected additives can be used both in the form of a
powder and in the form of a suspension, therefore, the dosage data are indicated in
terms of the dry residue.
DSC was carried out using a specialized thermal analysis apparatus, using open
corundum crucibles. The test conditions were also chosen corresponding to the work
[17]: constant air flow, preheating to 50 °C to stabilize the heat flow, heating rate
10 °C/min, the samples were preliminarily milled to a homogeneous powdery state,
the mass of each test sample was 50 ± 3 mg.
The results of the experiments performed are presented in the Fig. 1.

4 Discussion

In general, all studied samples of HCP exhibit a similar thermal behavior. Throughout
the experiment, a stable weight loss is observed, which may be associated with the
decomposition of aluminate phases and calcium hydrosilicate compounds.
At the early stage of the experiment (up to 130–140 °C), the process of removing
chemically bound water is observed. Significant endothermal effects at 500–530 °C
and 780–810 °C, accompanied by weight loss, are characteristical for cement
70 V. Medvedev

Table 2 Estimated content of base components

No. Additive marking Estimated content of base components, %
Ca(OH)2 CaCO3
0 Control 9.04 7.80
1 MF 12.74 6.27
2 NF 14.55 5.77
3 LS 6.87 9.64

systems and indicate the decomposition of portlandite and calcite, respectively. The
exothermal effect at 810–840 °C indicates the formation of wollastonite as a result
of the phase transition of the dehydrated C–S–H gel under thermal exposure.
In general, the differences in the thermal behavior of samples 0, 1, and 2 are of
a similar nature. The effect of the plasticizing additive is expressed in an increased
content of portlandite and a decrease in the amount of calcite in the HCP sample in
comparison with the control sample (Table 2).
At the same time, in the case of a sample with a plasticizing additive based on
lignosulfonate, a significant decrease in the initial amount of portlandite and an
increased content of calcite are observed. Also, the graph of this sample shows the
presence of an exotermal effect at 300–310 °C without a sharp change in the dynamics
of weight loss, which may indicate the formation of new phases upon the additive
usage.

5 Conclusions

The results of the analysis and their comparison with the data obtained in [17]
showed that the use of plasticizing additives based on naphthalene formaldehyde
and melamine formaldehyde can potentially lead to a slight decrease in the thermal
stability of the HCP, since their usage increases the amount of chemically bound water
in the samples, but at a slight decrease in strength is predicted due to a decrease in the
amount of calcite. This may be due to the greater dispersion effect of these additives.
It should be noted that the increased content of portlandite and chemically bound
water in the HCP as a whole has a positive effect on the radiation resistance of
concrete, as it slows the irradiation effect.
The additive based on lignosulfonate, in general, provided an increase in the
thermal stability of the test sample, as evidenced by the lower overall weight loss
compared to the rest of the samples and the presence of a larger amount of calcite,
which also indicates the potentially higher strength of such a cement stone.
According to the technique indicated in [1, 16, 17], the investigated additives can
be used in the manufacture of concrete structures exposed to neutron fluences up to
3 × 1023 neutron/m2 .
Plasticizer Type Influence on HCP Radiation Resistance 71

However, a comparison of the data with those obtained in [17] testifies to the
greater efficiency of plasticizing additives based on polycarboxylate esters in compar-
ison with other groups of additives, presumably due to the steric effect of action and
the special effect of binding cement paste molecules using side polymer chains.

All tests were carried out using research equipment of The Head Regional Shared Research
Facilities of the Moscow State University of Civil Engineering.

References

1. Ershov VYu (1992) Radiation resistance of Portland cement stone with chemical and mineral
additives. The dissertation for the degree of candidate of technical sciences, Moscow
2. Batrakov VG (1998) Constr Mater Prod, 768
3. Batrakov VG, Shanov FM, Silina BS, Falikman VR (1988) Constr Mater Prod 1.2(7):59
4. Ramachandran VS, Feldman RF, Beaudoin JJ (1981) Concr Sci 427
5. Nithya R, Barathan S, Govindarajan D, Raghu K, Anandhan N (2010) Int J Chem 2(1):121–127
6. Hilsdorf HK, Kropp J, Koch HJ (2014) International Committee on Irradiated Concrete (ICIC)
Information Exchange Framework Meeting, vol 1, pp 223–251
7. Thomas DR (1965) Nucl Struct Eng 1:368–384
8. Nowak-Michta A (2015) Procedia Eng 108:262–269
9. Denisov A (2017) Civ Eng J 5:70–87
10. Nkinamubanzi PC, Mantellato S, Flatt RJ (2016) Science and technology of concrete
admixtures, pp 353–377
11. Field KG, Remec I, Le Pape Y (2015) Nucl Eng Des 282:126–143
12. Dubrovsky VB, Korenevsky VV, Muzalevsky LL, Pergametsik BK, Sugak EB (1980) Radiation
safety and protection of nuclear power plants, 4:240
13. Dubrovsky VB, Korenevsky VV, Pospelov VP, Sugak EB (1985) Investigation of a binder
for protective concretes with increased plasticizing properties (in Russian). In: Proceedings
of the third all-union scientific conference on protection against ionizing radiation of nuclear
engineering installations, vol 5, 27–29 October, 1985, Tbilisi, Georgia, pp 62–69
14. Sugak EB, Denisov AV, Korenevsky VV, Muzalevsky LL, Pergametsik BK (1978) Energy
Constr 9:11–13
15. Ershov VYu, Dubrovsky VB, Muzalevsky LL, Kolesnikov NA (1988) Questions of atomic
science and technology (Design and construction series), vol 2, pp 120–129
16. Medvedev V, Pustovgar A (2015) Appl Mech Mater 725–726:337–382
17. Medvedev V (2020) IOP Conf Ser Mater Sci Eng 869:032033
18. Vavrzhin F, Krchma R (1964) Stroyizdat 18
19. Jeknavorian A, Roberts J, Jardine L et al (1997) Condensed polyacrilic acid-aminated poly
ether polymers as supciplasticizers for concrete. In: Proceedings of the fifth CANMET ACI
international conference, Rome, Italy, vol 173, p 52
20. Ohta A, Sugiyama T, Tanaka Y (1997) Fluidizing mechanism and application of
polycarboxylate-based superplasticizers. In: Proceedings of the fifth CANMET ACI inter-
national conference, Rome, Italy, vol 173, p 359
Properties of Epoxy Composites
with Halloysite Nanotubes Subjected
to Tensile Testing

Evgeniya Tkach and Maxim Bichaev

Abstract For the rational consumption of resources, humanity needs to invest in

the creation of new composite materials. The main properties of new composites
should exceed traditional ones by 4 times, and the energy consumption during their
production should be reduced by 2 times. The relevance of this work lies in the search
for technological solutions that can lead to the creation of new composite materials.
Objective of the study: obtaining an epoxy composite with increased mechanical
properties under tension using a biocompatible nanoscale filler-halloysite nanotubes
(HNTs). The samples were made and tested according to the national standard of the
Russian Federation GOST R 56800-2015. The current study provides an analysis
of the tensile properties of the resulting composites in comparison with the litera-
ture. For the epoxy base, an epoxy composition similar to the literature was adopted.
The following results were obtained: uniaxial tensile strength was improved by +
18.19%, elastic modulus—by +6.54%, elongation decreased by −15.57% in samples
containing 10 wt% halloysite. HNTs are also effective as a filler for epoxy compo-
sition. The research results presented in this article can serve as early support and
reference materials for the creation of new epoxy-based composites using a tubular
nanosized halloysite clay filler.

Keywords Halloysite · Epoxy resin · Composite · Mechanical properties

1 Introduction

According to the UN Population Division, in 1950 there were about 2.5 billion people
living on the planet. Now, in 2020 there are 7.7 billion, and in 2050 there will be
9.8 billion. By the end of the century, the UN expects the world’s population to be
11.2 billion people [1]. It is obvious that population growth will lead to a significant
consumption elevation of our planet’s resources. A rational solution to this problem
can be the improvement of existing technological developments in modern materials
science.

E. Tkach · M. Bichaev (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 73

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_8
74 E. Tkach and M. Bichaev

Therefore, is important to reform the material basis of human life in such a way
that, for example, by 2050, it would be possible to meet supply demand of 9.8 billion
people on the entire planet with all the necessary benefits of modern civilization.
However, this requires cutting the production of materials and energy by about half.
At the same time, it is necessary to increase the properties of basic materials by more
than 4 times. The solution to this problem may be the creation of new composites
whose mechanical properties are superior to traditional materials.
One area of interest in modern materials science is the physicochemical modifi-
cation of epoxy resins with various fillers. Epoxy resins, are widely used in various
fields of technology due to their unique characteristics: high adhesion strength to
the material surface, relatively high mechanical strength, hardness, and excellent
chemical and heat resistance. However, most cured epoxy systems have poor frac-
ture toughness, low impact resistance and a tendency to delaminate. This article
discusses a promising natural filler that can improve the mechanical properties of
epoxy compositions.
It is generally accepted that the level of development of technology is largely
determined by the availability of the necessary materials. This can be seen most
clearly in the development of ancient civilizations, when the invention or creation of
a new progressive material became the impetus for the development of technology
and civilization. No wonder the technical level of development of civilization is
characterized by the type of material.
In the last century, polymer composite materials such as rubber and phenol-
formaldehyde resin-based press material were invented. Since then, many attempts
have been made to modify polymers with various additives [2–4]. The best results
were achieved due to the synergistic effect of using different fillers in polymers.
However, it is known that the introduction of various fillers can lead to both an
increase and a decrease in the mechanical properties of modified polymer materials.
In recent years, a great quantity of research has been done on polymer modi-
fication using rigid nanoparticles or nanofibers. It was found that inorganic addi-
tives such as particles of silica, alumina, and glass are promising modifiers for
enhancing the properties of epoxy resins [5–9]. The resulting epoxy nanocomposites,
as expected, had improved mechanical characteristics in comparison with conven-
tional polymer composites [8, 10]. In particular, rubber particles with a core-shell
structure (Core Shell Rubber—CSR) are excellent materials for strengthening epoxy
materials [11–14].
To improve the properties of epoxy resins, particles of montmorillonite (MMT)
were analyzed [15] as a potential modifier. Nevertheless, no significant positive
changes in the mechanical properties of epoxy nanocomposites modified by MMT
have yet been found. In study [16] has shown that potassium titanate whiskers have a
significant effect on the mechanical properties of cured epoxy composites. Samples
filled with potassium titanate whiskers showed improvements in properties. Hard-
ness, density, wear resistance have increased. However, the strength characteristics
decrease with increasing filler content.
In [17], the compressive and tensile properties of epoxies (usually as matrices
of fiber composites) modified by different types of nanoparticles were investigated
Properties of Epoxy Composites with Halloysite Nanotubes … 75

for the following study on the mechanical properties of epoxy composites. Epoxy
composites were fabricated by vacuum assisted resin infusion molding (VARIM)
with the matrices modified with nanosilica and liquid rubber (CTBN).
The joint effect of rubber nanoparticles (with an average particle size of 100 nm)
with SiO2 (with an average particle size of 16 nm) on the mechanical properties of
the cured epoxy resin was studied in [18].
Currently, the most successful fillers for epoxy systems are carbon nanotubes
(CNTs) [19–21].
However, CNTs have a significant drawback—its cost and complexity in the
synthesis. In particular, graphene nanotubes (single-walled carbon nanotubes—
SWCNTs) manufactured by OCSIAL (Russian Federation) cost $ 870 per 100 g
of nanomaterial [22]. World production at the level of 15–20 tons of these nanotubes
is quite enough for the manufacture of super composites for mass consumption. Still,
at the moment, this high cost is a limiting factor for the industrial introduction of
CNTs [23]. For this reason, the development of technological production for polymer
materials with enhanced performance of CNTs is still nowadays an economically
demanding task. Furthermore, there is a need to find a more affordable universal
filler in use—an analog of CNTs with similar physical and mechanical properties,
but at the same time affordable, biocompatible, non-toxic, and foremost it most have
the ability to increase the mechanical properties of cured polymers.
The ideal filler described above may be of halloysite clay, a material, mainly
consisting of tubular particles. Halloysite nanotubes are a two-layer aluminosilicate
of naturally occurring origin with a spiral conformation (Fig. 1) and the structural
formula (Al2 Si2 O5 (OH)) 4 * nH2 O). The distance between the layers depends on
the degree of hydration: at n = 0, the interlayer distance will be 0.7 and 1 nm at n
= 2. It is known that halloysite has a layered structure: the inner—Al–OH and the
outer—Si–O–Si surfaces have positive and negative charges, respectively [24]. The
cost of this nanomaterial is significantly lower than CNTs and ranges from $ 4–5 per
kilogram [25].
In recent years, nanoclay has become a subject of special interest for many scien-
tists and researchers in the field of chemistry, physics, technology and biology due to
its excellent mechanical properties and sustainability [26–29]. It was with them that

Fig. 1 TEM image of halloysite (a). Model of the spiral configuration of the halloysite tube (b) [35]
76 E. Tkach and M. Bichaev

the development of new “smart” materials began: halloysite nanotubes were used as
a container with controlled desorption for various substances [30–32]. The creation
of a polymer composite with enhanced performance is based on the unique structure
of aluminosilicate filler; the presence of a hydroxyl group coating on the surface of
a HNT [33–37], makes possible the chemical interaction between the surface OH
groups of the aluminosilicate and the polymer, the result of which is the formation
of strong non-hydrolyzable bonds [2].
The purpose of the study was to demonstrate the application of the effective use
HNT as a filler for epoxy systems. The tasks that must be solved to achieve this goal
is to conduct a series of tests, which will determine mechanical characteristics of
filled samples under tension.

2 Methods

First, epoxy mixtures were prepared, from which epoxy composites were then made.
Composites were prepared by mixing the polymer composition AE-1 (Inter Aqua,
Moscow, Russian Federation, Table 1) with halloysite nanotubes (Imerys Ceramics,
New Zealand, Matauri Bay, Table 2) in an amount of 5%, 10% by weight from epoxy
resin.

Table 1 Technical characteristics of the epoxy composition AE-1

Component appearance Homogeneous thixotropic system—paste
Color Component A—light yellow Component B—dark gray
Ratio of components 100 50
The density of the epoxy system A 1.65
+ B at a temperature of (20 ±
2) °C, g/cm3 , not more than
Pot life at temperature (20 ± 2) °C, 60
min, not less than
Adhesion strength (adhesion), 2.7 (concrete demolition)
MPa, not less than
Modulus of elasticity, GPa 2.4–2.8

Table 2 Morphological characteristics of HNTs

HNTs Length External Inner Wall Surface Porosity Ratio
(nm) diameter diameter thickness area (cm3 /g)
(nm) (nm) (nm) (m2 /g)
Matauri 100–3000 50–200 15–70 20–100 22.1 0.06 12,4
Bay, New
Zealand
Properties of Epoxy Composites with Halloysite Nanotubes … 77

AE-1 is a two-component cold-cured compound made on the basis of epoxy-

dianic components-soluble, fusible and reactive oligomers based on diphenylpropane
epichlorohydrin (component “A”—epoxy resin) and modified aliphatic polyamine
(component «B»—epoxy hardener).
Samples were made in the following order: firstly, HNTs were added to the
container with component A, then mixed at 20 °C during 1 h. After this, the hard-
ener (component B) was added to the system (HNTs + «A») in the proportion «A»:
«B» = 100: 50 by weight and mixed during 5 min until a homogeneous mass was
obtained. Then, the resulting mixtures were poured into molds for the manufacture of
flat samples with subsequent curing under normal conditions during the day. Finally,
control samples without alumino-silicate additives were also prepared to compare
mechanical characteristics.
Liquid epoxies were put into flat specimen molds and cured for 24 h at room
temperature. After they were removed from the molds, the tensile specimens were
prepared in accordance with National Standard of the Russian Federation GOST R
56800-2015 «Polymer composites. Determination of mechanical tensile properties of
unreinforced and reinforced materials» (Fig. 2) [39]. The parallelism of the surfaces
of the tensile specimens was obtained using a grinder.
Samples were prepared in the amount of 6 pieces for each test mode. The first test
mode is without the use of halloysite nanotubes. The second and third test modes
were carried out with the addition of nanotubes at 5% and 10% by weight of epoxy
resin, respectively. After processing, the prepared samples were sent to a universal
test machine (Fig. 3, Instron 150 LX).
The standard size of the specimens was assigned on the basis of a previously known
or assumed modulus of elasticity, material and loading speed mm/min indicated in
Table 4 of this standard [39].
Thus, we know the following data for tensile testing of specimens:
• Test speed 1 mm/min;
• Type of specimen: flat with a rectangular cross-section;
• Specimen size: Type 5;
• Minimum thickness: h = 4 mm;
• Cross-sectional area: S = 12.72 mm2 .

Fig. 2 Specimen for uniaxial tensile testing [39]

78 E. Tkach and M. Bichaev

Fig. 3 Equipment for tensile testing. a Instron 150 LX universal testing machine, b grabs of
a sample-blade for uniaxial tension with a sensor for determining the relative transverse and
longitudinal deformations

Mechanical tests were carried out in order to determine the elastic modulus,
elongation at break and tensile strength at break according to national standard of
the Russian Federation GOST R 56800-2015 «Polymer composites. Determination
of mechanical tensile properties of unreinforced and reinforced materials».
The study of the microstructure of the cured epoxy specimens was carried out
using a scanning electron microscope (SEM, Philips XL30).

3 Results and Discussion

According to the research carried above, it was found that in unfilled samples, failure
begins with a defect on the surface or inside of it, experiencing a constant growing
deformation up to the final critical deformation. However, for epoxy resins modified
with halloysite, the modulus of elasticity is higher due to the reinforcing effect of
halloysite nanotubes. Cracks are caused by defects developing on the surface of the
sample or inside it. It was also found that the surfaces of the modified samples in the
maximum tensile zone are rougher than that of cured pure epoxy resin, which shows
the effect of halloysite particles.
Moreover, in recent years, impressive results have been published on hardening
of epoxy resins with fillers such as carbon nanotubes or fibers, and montmorillonite
particles. Changes in the sample’s mechanical properties depending on the selected
filler are shown in Table 3. As can be seen from Table 3, the strength and modulus
Properties of Epoxy Composites with Halloysite Nanotubes … 79

Table 3 Comparison of improvements in mechanical properties for epoxy-based nanocomposites

with various fillers
Filler Filler content, % Tensile strength, Young’s modulus, Elongation at
MPa GPa break
Improvement, % Improvement, % Improvement,
%
Potassium titanate 0 0.0 0.0 0.0
whiskers [16] 2.5 −4.1 9.4 −7.6
5 −7.9 17.1 −10.8
7.5 −11.1 26.3 −15.7
Silica [17] 0 0.0 0.0 −
10 4.86 14.3 −
CTBN liquid 0 0.0 0.0 −
rubber [17] 5 1.5 −28.9 −
10 3.0 11.2 −
Nano-SiO2 with 0 0.0 0.0 0.0
CSR 10 wt% [18] 1 6.1 3.2 −5.5
2 8.5 12.9 −10.9
3 −2.4 16.1 −18.2
5 −11.0 19.4 −29.1
Single walled 0 0.0 0.0 −
carbon nanotubes 0.05 −48.2 −51.4 −
[38]
0.1 −33.3 −34.3 −
0.2 −24.1 −25.7 −
0.4 −20.4 −20.0 −
0.6 −53.7 −54.3 −
1 −63.0 −60.0 −
Multi-walled 0 0.00 0.00 −
carbon nanotubes 0.05 −44.4 −42.9 −
[38]
0.1 −25.9 −25.7 −
0.2 −18.5 −17.1 −
0.4 −13.0 −14.3 −
0.6 −11.1 −11.4 −
1 1.9 0.6 −
Halloysite 0 0.0 0.0 0.0
nanotubes (current 5 14.7 3.1 −11.4
study)
10 18.2 6.5 −15.6
80 E. Tkach and M. Bichaev

70
60
50
Stress [MPa]

40
30
20
10
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Longitudinal strain [mm/mm]
0 wt% halloysite 5 wt% halloysite 10 wt% halloysite

Fig. 4 Effect of halloysite clusters on tensile strength characteristics

of cured composites increase with loading of rigid particles (HNTs, nano-SiO2 ),

whereas, they decrease with loading of soft particles (liquid rubber CTBN).
When analyzing various studies, it was found that epoxy resins filled with
halloysite nanotubes have mechanical properties no worse than those of a hybrid
filler consisting of nanosilica and rubber particles with a core-shell structure [18].
In the course of the experiment, it was found that halloysite particles have a
tendency to form different types of clusters. A detailed study of the microstructure
of epoxy samples with nanotubes was carried out using the scanning electron micro-
scope. Halloysite-rich layer with a thickness of about 200 µm was observed at the
bottom of the samples. Also, it was noted that oversized halloysite clusters in the
lower surface layer could cause premature tensile failure of specimens with signifi-
cant reduction in strength and fracture strain, as shown in Fig. 4. In this case, the use
of halloysite filler is not desirable.
However, if the lower surface layer with halloysite clusters is removed and the
test is repeated, we will see, that the premature failure of the specimens tested before
under tension was significantly reduced.
The process of removing the halloysite-rich layer from the underside of the spec-
imens changes the overall percentage of halloysite in epoxies. Resulting in a content
of approximately 4.5 wt% in samples with 5 wt% of epoxy resin, and 8.5 wt% for
10 wt%.
Figure 5 shows typical strain versus stress curves of tensile plotted from the
average test results of six specimens with different contents of halloysite nanotubes.
It is seen that composites containing halloysite particles have increased strength and
elastic modulus compared to unmodified samples.
Thus, samples filled with halloysite have a tendency to cluster formation of
nanoparticles. This phenomenon is most likely associated with the poor quality of
dispersion of the filler in the epoxy matrix. A solution can be found if we apply
a more advanced method of dispersing nanosized particles [40]. This method is a
multi-step approach involving filler modification, mechanical mixing of the filler in
Properties of Epoxy Composites with Halloysite Nanotubes … 81

80
70
60
Stress [MPa]

50
40
30
20
10
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Longitudinal strain [mm/mm]

0 wt% halloysite 5 wt% halloysite 10 wt% halloysite

Fig. 5 Average tensile strength values

an epoxy matrix, and ultrasonic treatment. The proposed multistage method differs
from ultrasonic treatment in that it does not depend on various parameters: mixture
temperature, frequency, filler cross-section. During dispersion, these variables remain
constant, however, each stage of the process has a different duration of action. As
shown in [40], due to the application of this multistage approach, the authors managed
to significantly improve the quality of dispersion of nanosized filler particles in an
epoxy matrix, which was accompanied by a further improvement in the mechanical
characteristics of the cured samples.

4 Conclusions

The following conclusions were made:

1. The maximum values of mechanical properties were obtained on specimens
with 10% by weight filler content: the uniaxial tensile strength was improved
by +18.19%, the elastic modulus by +6.54%, but the elongation decreased by
–15.57%.
2. It was found that halloysite nanotubes, without any surface treatment, create a
deposition as oversized particles in the bottom of the samples.
3. Removal of the 200 µm thick halloysite bottom layer resulted in an improvement
of mechanical properties and a decrease in the overall filler content from 0.5 to
1.5% by weight.
It is obvious that the improvement of filler dispersion methods will make possible
to obtain more uniform samples with improved properties. This study showed that
halloysite nanotubes are an effective filler for epoxy compositions.

Acknowledgements The authors thank the staff of SKB-ENGINEERING LLC for the opportunity
to conduct this research.
82 E. Tkach and M. Bichaev

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Effect of Liquid-to-Alumino-Silicate
Material Ratio and Rice Husk Ash
Content on the Properties of Geopolymer
Concrete

Tang Van Lam, Pham Van Ngan, and Nguyen Dac Binh Minh

Abstract This paper combines various contents of Vietnamese rice husk ash (RHA)
and fly ash (FA) with an alkali-activator solution to produce geopolymer concrete
(GPC). In which FA and RHA are used as alumino-silicate material in GPC mixtures.
The effects of the liquid-to-alumino-silicate material (L/ASM) ratio (0.35–0.50) and
the RHA content (0–60%) on the properties of the GPC were then investigated.
The theoretical calculation combined with the experiment was used to determine
the compositions of these GPC. The workability of fresh GPC was tested by the
slump flow test. Further, the strength of specimens was performed in accordance with
Russian standards. Results found that both the L/ASM ratio and RHA content greatly
affected the workability of mixtures and compressive strength of the specimens. The
compressive strength of GPC-specimens prepared with a L/ASM ratio of 0.45 and an
RHA content of 40% exhibited higher than the control GPC-specimens. Moreover,
the relationship between 28-day compressive strength and the L/ASM ratio of GPC
was also determined. These results refer to the use of RHA and FA in mixes GPC
is not only environmental but also cost-effective for concrete producers, as well as
improved properties of the green concrete in the future.

Keywords Liquid-to-alumino-silicate material ratio · Rice-husk ash · Fly ash ·

Geopolymer concrete · Compressive strength

1 Introduction

Portland cement (PC) clinker is made by heating a mixture of raw materials, including
limestone and clay, to a calcining temperature of above 600 °C and then a fusion
temperature, which is about 1450 °C to sinter the materials into clinker. Ordinary

T. Van Lam (B)

Hanoi University of Mining and Geology, 18 Pho Vien, Duc Thang, Bac Tu Liem, Ha Noi,
Vietnam
P. Van Ngan · N. D. B. Minh
Institute of Regional Research and Development, 70 Tran Hung Dao, Hoan Kiem, Ha Noi,
Vietnam

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 85

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_9
86 T. Van Lam et al.

Portland/pozzolanic cements manufacture are not only an energy-intensive process

but also releases significant quantities of toxic fumes, and carbon dioxide into the
atmosphere [1, 2]. In recent years, many investigations were focused on increasing
attention on the potential for natural materials, as fly ash, bottom ash, slag, rice
husk ash… to be used as a replacement for PC not only to reduce the cement
content in the mixture concrete but also on the development of modern cementi-
tious materials in construction materials manufacturing. In 1978, Davidovits [3, 4]
proposed that a new binder could be produced by a polymerisation process involving
a reaction between alkali-liquid solutions and compounds containing aluminium
and silicon. These binders created were termed “geopolymers”. The geopolymer,
including alumino-silicate material with alkaline liquids, is currently one of the most
promising of these environmentally-friendly binders for green concretes in building
constructions [5].
Material types containing silicon (Si) and aluminium (Al) atoms in amorphous
form, which come from natural mineral additions or by-product materials, and indus-
trial wastes, could be used as source materials for geopolymer concretes and mortar.
On the other hand, some pozzalanic materials are by-product materials such as fly
ash (FA), bottom ash, rice husk ash (RHA), and palm oil fuel ash etc. [6, 7]. For the
manufacture of geopolymer products, the choice of raw materials depends mainly
on their availability and cost, the type of application, and the specific demand of the
producers [8, 9].
In the synthesis of geopolymer materials, fly ash and Vietnamese rice husk ash-
based GPC specimens are provided excellent properties in both the fresh and hard-
ened state these made them suitable materials for structural applications in green
buildings [10–12].
For the geopolymer concrete and mortar, the type of alkali-liquid used plays an
important role in the polymerisation process as a binder of concrete. The alkali-
activator solutions (AAS) of NaOH and Na2 SiO3 or KOH and K2 SiO3 with different
concentrations are the most common alkali-liquid solutions used in geopolymeri-
sation [13, 14]. The previous research [9–11] showed that the engineering prop-
erties of concrete/mortar increases when waterglass (Na2 SiO3 ) is added to the
sodium hydroxide, compared with using only NaOH. Further, the addition of water-
glass increases the ratios of Si/Al and Na/Al, resulting in increased formation
of sodium aluminosilicate gel (Na2 O–Al2 O3 –SiO2 –H2 O) which indicates higher
strength behavior of tested samples.
Industrial waste material, in particular FA obtained from thermal power plants
(TPP), was used widely as mineral additive materials for concrete because of its
pozzolanic characteristics. At present, already over a million tons of FA TPP are
generated each year globally [15–17]. The class-F FA TPP, containing a high content
of amorphous SiO2 and Al2 O3 , was used as a raw material to produce geopolymer
concrete and mortar.
Moreover, rice husk ash-RHA is an agricultural waste that contains a high silica
content in the form of non-crystalline or amorphous silica—SiO2 . In Vietnam, a
source of serious environmental pollution is agricultural waste, as the rice husks and
straws, which are most often dumped into ponds, lakes, and rivers [18, 19]. Hence,
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice … 87

the reduction of agricultural waste, especially rice husks, is very necessary today.
The Vietnamese rice husk ash, which has been obtained in burning rice husks and
used as pozzolanic material for concrete and mortar. While rice husks or hulls are
generated during the first stage of rice milling. Depending on the burning conditions
of rice husks, the SiO2 content in the RHA varies from 85 to 95 wt%, which exists
predominantly in an amorphous phase and to the very large surface area. This material
is reactive with the alkali-activator solution to produce the aluminosilicate gel that
binds the aggregate types and provides the mechanical properties of GPC [20, 21].
Because both Vietnamese RHA and FA TPP “Vung Ang” are contained high
levels of Si and Al, these alumino-silicate materials are effective for producing GPC
and mortar. Furthermore, the AAS plays an important role in the dissolving process
of Si–Al from amorphous silica-alumina to form gel geopolymer precursors and
alumino-silicate materials. In addition, the liquid-to-alumino-silicate material ratio
significantly affects the workability of mixtures and compressive strength of the
tested GPC-specimens. Therefore, the overarching purpose of the current research is
to investigate the effect of liquid-to-alumino-silicate material ratio and RHA content
on the properties of fly ash and rice husk ash-based geopolymer concrete.

2 Materials and Experimental Methods

2.1 Material Properties

(a) A coarse aggregate wasn’t used for making geopolymer concrete in this test.
The fine aggregate in this geopolymer concrete of the mixtures was used Red
River quartz sand (QS) (Vietnam) with fineness modulus Mk = 3.0 and a
specific gravity of 2.65 g/cm3 .
(b) The alumino-silicate materials (ASM) used include rice husk ash -RHA and
fly ash -FA from TPP “Vung Ang” (Vietnam) class-F in this study. The exper-
imental results of chemical compositions and physical properties of RHA and
FA TPP “Vung Ang”, determined by X-ray fluorescence analysis of raw mate-
rials, are given in Table 1. The class-F FA TPP “Vung Ang” was composed
mainly of SiO2 (54.62%) and Al2 O3 (25.17%) and that the main constituent
of RHA was SiO2 (88.2%) (Fig. 1).
Particle size distributions of FA and RHA, determined by XRD analysis, are given
in Figs. 2 and 3, respectively. Moreover, these results showed that the particles of
RHA were significantly smaller than the particles of FA.
(c) Alkaline liquid: In the current experimental research, a combination of
Na2 SiO3 and NaOH solutions was used as the alkali-activator solution. The
molarity of this NaOH solution was 14.
1. The sodium hydroxide solution was obtained by dissolving the NaOH
flakes in the water. In this case of the study, taking 14 Molar of the NaOH
88 T. Van Lam et al.

Table 1 Physical properties and average chemical compositions of Fly ash TPP “Vung Ang” and
Rice husk ash
Materials Fly ash TPP “Vung Ang” Rice husk ash
Physical properties Specific weight (g/cm3 ) 2.35 2.25
The volume of natural 765 572
porous state (kg/m3 )
Specific surface area 0.755 0.850
(m2 /g)
Mean particle size (mm) 17.6 14.8
Average chemical SiO2 54.62 88.2
composition (%) Al2 O3 25.17 1.25
Fe2 O3 7.11 1.75
SO3 0.25 0.5
K2 O 1.28 1.14
Na2 O 0.2 2.67
MgO 1.57 0.8
CaO 1.45 0.52
TiO2 2.35 0.15
P2 O5 765 0.25
Loss on ignition 0.755 2.77

Fig. 1 Particle size distribution of Fly ash TPP “Vung Ang”

solution, this can be obtained by mixing 42.4% of a NaOH solid with

57.6% of water. This solution’s specific gravity is 1.45 g/cm3 .
2. The Na2 SiO3 liquid used was procured from Viet-Tri Co., Ltd., with a ratio
SiO2 /Na2 O = 2.5, containing 29.5% SiO2 , 11.8% Na2 O, 58.7% H2 O, and
its specific gravity of 1.55 g/cm3 .
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice … 89

Fig. 2 Particle size distribution of Vietnamese RHA

Fig. 3 The development of GPC compressive strength at L

AS M = 0.35

(d) The super-plasticizer admixture SR-5000F “SilkRoad” (SP) was procured from
Hanoi-Korea Co., Ltd., with a specific gravity of 1.12 g/cm3 . This admixture
was used to increase the workability of the fresh concrete and to reduce the
ratios of water-cement, while increasing the strength of GPC-samples.

2.2 Methods of Test

To date, there have been very limited studies on the mixture design of geopolymer
concrete raw materials, let alone the combined effects of alumino-silicate raw mate-
rials on the GPC properties. In 2008, the previous research [22–24] proposed the
90 T. Van Lam et al.

method for a composition of fly ash-based GPC but this method did not discuss
how to deal with the effects of a super-plasticiser content or the air content in the
concrete mixture. In this study, the calculation method of the compositions of GPC
mixture was applied in accordance with the absolute volume method combined with
the experimental results.
• The workability of concrete mixture is determined by the standard slump cone
with dimensions of 40 × 70 × 80 mm by Vietnam standard TCVN 3106:2007.
• The compressive strength of GPC was conducted to evaluate the compressive
strength development of the tested GPC-specimens for different time periods.
This test was performed on 100 × 100 × 100 mm cubic these specimens at 3, 7,
14, 28, and 90 days of curing age using a 500 T computer-controlled compression
tester machine “Controls Advantest 9” with a constant loading rate of 1000 N/s in
order to keep the loading rate to a minimum rate in the processing test of concrete
patterns. The compressive strength test was performed in accordance with GOST
10180-2012 (Russian standard).

2.3 Sample Preparation

The sodium hydroxide solution was initially prepared by dissolving the NaOH flakes
in water in concentrations of 14 Molarity. This solution was then mixed with the
sodium silicate solution and allowed to cool to room temperature, while, the alkali-
activator solutions had Na2 SiO3 /NaOH ratios of 2.5 and liquid-to-alumino-silicate
materials (L/ASM) ratios of 0.33, 0.40, 0.45, and 0.50, respectively. Data of Table
2 details shows the mixture composition used in 1 m3 of the concrete by weight.
Furthermore, a quartz sand-to-alumino-silicate materials ratio of 1.30 was maintained
for all of the mixtures in this study. The super-plasticizer “SR-5000F SilkRoad” is
equally to 1.0% by mass of ASM (ASM = RHA + FA) [5]. In addition, using the
RHA to replace from 0, 20, 40, to 60% of mass the FA TPP “Vung Ang” in the
geopolymer concrete mixtures. Besides, relative volume of entrapped air is 3.0% in
1 m3 of the tested concretes.
In this work, all the tested patterns of geopolymer concrete series were demoded
24 h after casting. Next, the cubic geopolymer samples were heated at 100 °C for 6 h
in the oven and subjected to standard maintenance Treatment (temperature 20–25 °C
and 90–95% relative humidity) until the required testing ages.

3 Results and Discussion

The all compositions of GPC-mixture used in this investigation are calculated in

Table 2. Therefore, density, and slump of fresh geopolymer concrete, as shown in
Table 2.
Table 2 Mix compositions, density, and slump of GPC-mixture
L
No. Compositions AS M Mix proportions of ingredients (kg/m3 ) Density (kg/m3 ) Slump (cm)
of GPC-mixture FA RHA NaOH Na2 SiO3 QS SP
Mix-01 100% FA 0.35 840 0 84 210 1092 8.4 2233 15.5
Mix-02 100% FA 0.40 816 0 93 233 1061 8.2 2212 15.5
Mix-03 100% FA 0.45 794 0 102 255 1033 7.9 2193 16.5
Mix-04 100% FA 0.50 774 0 111 276 1006 7.7 2174 17.5
Mix-05 80% FA + 20% RHA 0.35 670 167 84 209 1088 8.4 2226 14.5
Mix-06 80% FA + 20% RHA 0.40 651 163 93 233 1058 8.1 2205 15
Mix-07 80% FA + 20% RHA 0.45 634 158 102 255 1030 7.9 2186 16.5
Mix-08 80% FA + 20% RHA 0.50 617 154 110 275 1003 7.7 2167 17.5
Mix-09 60% FA + 40% RHA 0.35 500 334 83 209 1084 8.3 2219 14.5
Mix-10 60% FA + 40% RHA 0.40 487 324 93 232 1055 8.1 2198 14.5
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice …

Mix-11 60% FA + 40% RHA 0.45 474 316 102 254 1026 7.9 2179 16
Mix-12 60% FA + 40% RHA 0.50 461 308 110 275 1000 7.7 2161 17
Mix-13 40% FA + 60% RHA 0.35 333 499 83 208 1081 8.3 2212 13
Mix-14 40% FA + 60% RHA 0.40 323 485 92 231 1051 8.1 2191 13
Mix-15 40% FA + 60% RHA 0.45 315 472 101 253 1023 7.9 2172 15
Mix-16 40% FA + 60% RHA 0.50 307 460 110 274 997 7.7 2154 15.5
91
92 T. Van Lam et al.

Data in Table 2 was shown that the addition of RHA with different levels only
slightly decreased the workability of GPC-mixtures, their slump was in the range
of 13–17.5 cm when the molarity of the NaOH solution and the ratio of the sodium
silicate-to-sodium hydroxide solutions remain the same in all mixes. This could be
explained by (1)—the specific surface area of RHA was larger, and (2)—the RHA
particles were significantly smaller than the FA TPP “Vung Ang” particles, which
have tended to increase water requirement in the mixing for these concrete mixtures.
These results also are shown that with effects combined of liquid-to-alumino-
silicate material ratio and SR-5000F super-plasticizer on the microstructure of
mixtures concrete, different values for slump mixes concrete were obtained from
different L/ASM. In the present study, the workability increase in the slump was
obtained that corresponded to the increase in liquid-to-alumino-silicate material ratio
from 0.35 to 0.50. The results of this investigation, similar to the results found in
previous studies [5, 13].
Effect of liquid-to-alumino-silicate material ratio and rice husk ash content
on compressive strength of fly-ash and rice-husk ash based geopolymer concrete
samples used in this study are presented in Table 3.
Data in Table 3 presented that the compressive strength values at 3, 7, 14, 28, and
90 days curing time of the tested geopolymer concrete samples were, respectively, in
the range of 20.3 ÷ 36.0, 32.2 ÷ 44.1, 46.1 ÷ 59.1, 48.7 ÷ 62.9, and 55.4 ÷ 73.1 MPa.
The experimental results also were shown the compressive strength development
of the GPC-specimens prepared with not only different Liquid-to-alumino-silicate
material ratio, but also different levels of RHA content.
Figures 3, 4, 5, and 6 are presented in detail in the development of the compressive
strength of GPC-samples at the different of RHA contents.
The relations between the 28-day compressive strength—fcs and the liquid-to-
alumino-silicate material ratio of the GPC-samples are shown in Fig. 7.
Figure 7 shows the relationship between the liquid-to-alumino-silicate mate-
rial ratio and 28-day compressive strength of GPC-samples. Similar findings were
reported by Ferdous et al. [5]. The data required to plot these relationships for GPC-
samples were obtained from laboratory conditions in which the cubic samples were
heated at 100 °C for 6 h in the oven. The relationship is useful at the start of the
calculation of the compositions in different geopolymer concrete types when the
liquid-to-alumino-silicate material ratio has still not been clearly determined by the
researchers.
(a) Effect of liquid-to-alumino-silicate material ratio on strength development of
fly ash and rice husk ashcbased geopolymer concrete
The compressive strength of the GPC-specimens was expected to increase
with increased liquid-to-alumino-silicate material ratio due to the fact that
more Si and Al atoms in an amorphous phase of FA and RAH are dissolved
in the alkaline solution. It can be seen in Table 4, an increase in the liquid-to-
alumino-silicate material ratio from 0.35 to 0.45 clearly increased the compres-
sive strength of the GPC. However, when the ASLM was 0.50, the compressive
strength started to slightly decline. Based on the final results of this study, it
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice … 93

Table 3 Average value of compressive strength of GPC-samples at different liquid to alumino-

silicate material ratio and different levels of RHA content
L
No. Compositions of AS M Compressive strength fcs at different curing ages (MPa)
GPC-mixture 3-day 7-day 14-day 28-day 90-day
Mix-01 100% FA 0.35 24.9 36.6 46.5 49.9 56.7
Mix-02 100% FA 0.40 25 37.9 49.2 53 57.9
Mix-03 100% FA 0.45 25.8 38.7 51.1 55.7 61.7
Mix-04 100% FA 0.50 22.9 36.2 47.5 51.8 57.9
Mix-05 80% FA + 20% 0.35 34 43.4 55.1 58.8 63
RHA
Mix-06 80% FA + 20% 0.40 33.2 44.1 56.7 60 65.9
RHA
Mix-07 80% FA + 20% 0.45 32.3 43.8 58.3 62.0 68.4
RHA
Mix-08 80% FA + 20% 0.50 31.5 42.9 56.6 60.2 67.7
RHA
Mix-09 60% FA + 40% 0.35 35.2 42.6 57.6 60.8 70.8
RHA
Mix-10 60% FA + 40% 0.40 36.0 43.4 58.8 62.4 72.8
RHA
Mix-11 60% FA + 40% 0.45 34.7 42.9 59.1 62.9 73.1
RHA
Mix-12 60% FA + 40% 0.50 33.4 41.2 57.4 61.6 70.9
RHA
Mix-13 40% FA + 60% 0.35 23.6 33.9 47.5 48.8 55.4
RHA
Mix-14 40% FA + 60% 0.40 24.1 34.7 48.3 50.8 58.1
RHA
Mix-15 40% FA + 60% 0.45 22.9 35.5 48.9 51.6 60
RHA
Mix-16 40% FA + 60% 0.50 20.3 32.2 46.1 49.1 56.1
RHA

was observed that the GPC samples activated with ASLM = 0.45 had the most
ideal alkali-activator solution and had the highest strength average values of all
tested GPC. The tested GPC-samples compressive strength determined in the
current work confirms the trends observed in similar studies [5, 10, 11, 13].
(b) Effect of rice-husk-ash content on strength development of fly ash and rice husk
ash—based geopolymer concrete
Next, data presented in Table 3 indicates that these GPC compressive strengths
were increased not only with curing periods (from 3 to 90 days) but also with
the increase in content of Vietnamese rice husk ash for all of tested patterns.
According to the study by Hwang et al. [6], it has reported that silica atoms
increase with the increase in levels of RHA content. Besides, it was believed
94 T. Van Lam et al.

Fig. 4 The development of GPC compressive strength at L

AS M = 0.40

Fig. 5 The development of GPC compressive strength at L

AS M = 0.45

that Si–O–Si bonds can be stronger than either Si–O–Al bonds or Al–O–Al
bonds [6, 7, 21]. And finally, this strength of aluminosilicate network in GPC
structures should increase significantly with the increase in amount of rice
husk ash. It is evident, in the range of this investigation, the GPC compressive
strength increased with the RHA content from 0 to 40% by mass of FA TPP
“Vung Ang”, then descends. The results of the current research, average value
of compressive strength at the 28-day curing age of GPC-specimens these
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice … 95

Fig. 6 The development of GPC compressive strength at L

AS M = 0.50

Fig. 7 Correlation between 28-day compressive strength of GPC-samples and the liquid-to-
alumino-silicate material ratio

contained 0%, 20%, 40%, and 60% RHA were, respectively, in the range of
49.9 ÷ 55.7 MPa; 58.8 ÷ 62.0 MPa; 60.8 ÷ 62.9 MPa; and 48.8 ÷ 51.6 MPa.
In particular, in Figs. 3, 4, 5, and 6 were shown that the addition of Vietnamese
RHA in the concrete mixes modifies significantly the GPC properties, especially
with the RHA contents of 20 and 40%. These are cause by: on the one hand—the
increased the levels of reactive silica (Si in SiO2 ) from the Vietnamese RHA resulted
96 T. Van Lam et al.

in a higher density of Si–O–Si bonds, leading to higher compressive strength of

the GPC-samples, on the other hand—with a higher levels of rice husk ash content
containing a high specific surface area for modifying and improving the compressive
strength of the GPC-specimens. Similar to the results presented in published studies
by Hwang et al. [6] and He et al. [21].
Moreover, contrary to these principles, while the current work found that the
strength behavior increased as Vietnamese rice husk ash increased to 40%, that
strength levels start declined after the content of RHA content is 60% (Table 3 and
Figs. 3, 4, 5 and 6). This could be explained by the effect of other parameters in the
geopolymerization process on the engineering properties of tested GPC-samples [6,
26]. Factors possibly responsible for these phenomenon include: (1) relatively larger
Vietnamese rice husk ash solid particles negatively affect the rate and extent of the
geopolymerization process when this rice husk ash comprise a higher content of the
GPC mixture [21], leading in weaker GPC-samples; and (2) a higher concentration
of soluble silica, but lower soluble alumina, which hinders the reorganization of Si
and Al atoms in the material structures and weakens tested GPC-samples [6, 11,
24]. According to experimental results of the strength behavior, the current research
found a level of 40% rice husk ash amount, delivering the highest compressive
strength value for the GPC-samples.

4 Conclusions and Future Work

1. Both the liquid-to-alumino-silicate material ratio and Vietnamese rice husk ash
content greatly affected the workability mixtures concrete and strength behavior
of GPC-specimens.
2. The compressive strength of all of GPC-specimens increased with curing times
of the test from 3 to 90 days. These results of this study are supported that
curing periods is promoted the development of compressive strength of GPC-
specimens and was similar to the basic principles of PC concrete.
3. With regard to the FA and RHA-based geopolymer concrete, the optimum of
both liquid-to-alumino-silicate material ratio and rice husk ash content was
consistently associated with a higher average value of strength. The compressive
strength of GPC-specimens maximized at the optimum value and then decreased
gradually as liquid-to-alumino-silicate material ratio and Vietnamese rice husk
ash content increased. Based on the results in the current research showed that the
geopolymer samples prepared with a liquid-to-alumino-silicate material ratio
of 0.45 and an RHA content of 40% exhibited high strength behaviors that were
comparable with or even much higher than the control specimens of GPC.
4. The development of compressive strength in the GPC- specimens was dependent
on not only the curing age of this test but also the liquid-to-alumino-silicate
material ratio (0.35–0.50) and RHA content (0–60%).
Effect of Liquid-to-Alumino-Silicate Material Ratio and Rice … 97

5. Producing geopolymer concrete using FA TPP “Vung Ang” and RHA of

Vietnam is feasible. The use of fly ash and Vietnamese rice husk ash as alumino-
silicate material in mixes GPC is not only environmental but also cost-effective
for concrete producers, as well as improved properties of the green concrete in
the future.

Acknowledgements This study was supported by the Ministry of Education and Training of
Vietnam with No. B2021-MDA-11.

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Composite Forming by the Method
of Prestressing of Carbon Unidirectional
Tape

Vasilii Plevkov, Artem Ustinov, Andrei Plyaskin, Victor Bunkov,

and Yulia Silman

Abstract The performed analysis of a large number of scientific studies in the field
of building structures reinforcement with prestressed carbon composites showed
this subject matter to be relevant, and the authors’ proposed solution allows for more
efficient use of prestressed composites. The paper presents a new method of forming
a composite where the pre-tension of a carbon unidirectional tape is applied The
deformed surface state is determined by the correlation method of digital images. The
stress-strain state of a carbon composite under a carbon tape prestressing and gradual
removing of tensile stresses is studied. The results of the conducted research proved
that the carbon tape pre-tensioning during the molding process can be successfully
applied to increase the strength and deformation characteristics of carbon composites.
The limited effectiveness of carbon composites use in the reinforcement of building
structures associated with the perception of only temporary loads and deformation
parameters of the material, can be expanded by using pre-tension of a carbon tape
while creating a composite on the structure surface.

Keywords Carbon composite · CFRP (Carbon Fiber Reinforced Plastic) ·

Prestress · Composite reinforcement · Tensile strength · Fiber tensile ·
Stress-strain state · Therma-threading

1 Introduction

Currently, the use of carbon composite materials in the reinforcement of building

structures is widespread. Unidirectional carbon tapes with various surface densities
are widely used for the composite reinforcement [1–9].
The traditional method of reinforcement consists of the reinforced element surface
preparing and pasting the loaded sections with carbon composite. Cavities, chips,
ulcers are filled with epoxy spackle and the surface is smoothed down. The tape is
pre-wetted on both sides in a binder, placed on the structure and rolled out with a

V. Plevkov · A. Ustinov · A. Plyaskin (B) · V. Bunkov · Y. Silman

Tomsk State University of Architecture and Building, 634003 Tomsk, Russia
e-mail: PVS@Tomsksep.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 99

P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_10
100 V. Plevkov et al.

roller to remove excess glue and air. Due to this method of composite reinforcement
the carbon tape retains its weaving shape at all stages of reinforcement: wetting,
forming, rolling out and strength accumulating of the epoxy binder.
The test results comparison of carbon lamellas and carbon composite formed by
the traditional method [10] showed that the decrease in strength and deformation
characteristics depends on the structure of the composite. The plain weave type used
in the production of the carbon tape bends strands of carbon fiber in the form of
waves. In turn, carbon lamellas are produced by forming a composite by reinforce-
ment with straight carbon strands extended immediately before forming. The given
difference in the composite structure can reduce the modulus of elasticity by 2–3
times, and the strength by two times. The strength and deformation characteristics
of carbon composites can be increased, the carbon tape pre-stressing being applied.
The influence of pre-tensioning of carbon lamellas using various anchor systems is
presented in [11–14].
The purpose of the work is to increase the strength and deformation characteristics
of carbon composites by applying carbon tape pre-stress.
To achieve this goal, the main tasks of the study were set:
• to determine the maximum level of the carbon tape pre-stress in order to perform
the technological process of creating a composite;
• to determine the residual stress in the carbon fiber after the strength of the binder
is attained;
• to investigate the stress-strain state of a carbon composite under carbon tape
pre-stress and stepwise removal of tensile stresses.

2 Materials and Equipment

The authors used a Carbon Wrap® Tape 530/150 as reinforcement of composite.

An innovative product is specially designed for reinforcement of ribbed floor slabs,
T-beams of bridge spans with a small rib width, beam elements of frame structures,
trusses and small-sized structures (Table 1).
The matrix of a composite is a two-component Epoxy binder CarbonWrap® Resin
530+ for the impregnation of external reinforcement systems with increased surface
density of carbon filler (Table 2).
The use of the VIC-3D digital optical measurement system made it possible
to obtain data on the sample surface displacements in three coordinate axes while
loading [15].
The test bench for the carbon tape prestressing (Fig. 1) is made of two shafts for
winding the carbon tape ends and a lever system for hitching weights, that does not
change the prestressing value at the tape deformation (relaxation, straightening of the
threads, slipping on the support or tightening). The choice of the loading system in
the form of a lever and loads is based on the experience of experimenters describing
the manual hydraulic system of the tape prestressing [14]. The described system did
Composite Forming by the Method of Prestressing of Carbon … 101

Table 1 Technical characteristics of the Carbon Wrap® Tape 530/150

Name Value
Surface density, g/m2 530 ± 15
Width, mm 150
Type of warp thread Carbon fiber 12/24 K
Type of weft thread Adhesive therma-threading
The density of the warp threads, threads per 10 cm2 64 ± 1/32 ± 1
The density of weft threads, threads per 10 cm 10 ± 1
The tensile strength of the fiber, HPa At least 4.9
Elasticity modulus at fiber tensile, HPa At least 245
The elongation at break of the fiber 1.8%

Table 2 Technical characteristics of the binder Carbon Wrap® Resin 530+

Name Value
The external appearance of the components Homogeneous transparent system without
foreign inclusions
Material color Component A—colorless; component
B—pale yellow
Brookfield dynamic viscosity, at (25 ± 0.5) ° Component A N = 4, 4000–10,000
C—at 20 rpm Component B N = 2, 15–50
The mixture density of components A + B at the ~1.20
temperature of (20 ± 2) °C, g/cm3 , not more than
Viability time at the temperature of (20 ± 2) °C, 50
min, not less
Bond strength (adhesion to concrete), >2.5 (destruction to concrete)
MPa, not less than
Displacement strength of the adhesive samples 12
(7 days at 23 °C), MPa, not less
Experimental modulus of elasticity, MPa 638

not maintain the required pressure during the composite forming process due to the
carbon tape creeping.

3 The Carbon Tape Structure in the Composite

Images from an optical microscope of the carbon composite reinforcement among

the weft threads allowed the tape structure evaluation. When the composite is made
by the traditional method, the tape has curved carbon fibers (Fig. 2a). The method
of tape prestressing allows us to straighten the carbon fibers in a strand (Fig. 2b), as
102 V. Plevkov et al.

Fig. 1 Test stand

well as to straighten the warp thread—the carbon strand i.e. therma-threading, the
weft thread being bent.

3.1 Preparation and Testing

A “load” moment perceived by the tensile force in the tape on the shoulder of the
gripper shaft radius was transmitted to the movable gripper with the help of a lever
system. The prestressing was performed by deadweights, that in turn prevented stress
decreasing in the composite during relaxation (scutching, sliding on the canvas or
tightening).
The strength of the epoxy binder being gained, stress relief, i.e. prestress removal
from the carbon reinforcement of the composite was carried out in six stages, equal
in value to the load.
Stereoscopic images of the carbon composite working zone were recorded in the
process of relief. A speckle image is applied to the recording zone to determine the
deformed state. Five stereoscopic images were taken at each stage of prestressing for
Composite Forming by the Method of Prestressing of Carbon … 103

Fig. 2 The images from the

carbon composite
microscope among the weft
therma-threadings: a without
prestressing, b with
prestressing

data statistical processing. Image processing was carried out in the VIC-3D software
product.

3.2 Results of Prestress Relief from Carbon Composite

The diagram of gradual prestress relief in the composite is shown in Fig. 3. Three
broken lines are combined on the diagram: actual stresses—stresses in the composite
reduced to a thickness of 0.6 mm, planned stresses—stresses at the stages of linear
relief, stress difference—the broken line showing the percentage difference between
the planned and actual stress reduction.
In case of using the loads with a mass of 2 kg, there is a deviation of actual
stresses by 50.8% from the planned stresses at the 5th stage and 29.3% one at the
4th stage. The given deviation is compensated when constructing the deformation
diagram “σ–ε”.
The total length of the prestressed tape between the axes of the tensioning rolls
was 1256 mm. The registration zone of deformations was 316 × 150 mm (Fig. 4)
and it was located in the middle of the pre-tensioned tape. Therefore, the defined
displacements along the X-axis are a half of the prestressed tape total displacements
(Fig. 5).
104 V. Plevkov et al.

250 60.0
actual stresses
50.0
200
planned stresses
Stresses, MPa 40.0

Difference, %
150 stress difference
30.0

100 20.0
10.0
50
-
- -10.0
0 1 2 3 4 5 6
Stage of prestress relief

Fig. 3 The diagram of prestress relief

5 см

Fig. 4 Isofields of the X-axis values of carbon composite working zone with tape prestressing
The difference in the displacements

5.0% 5.00
Difference, %
4.5% 4.50
of the outermost strands, %

X-axis displacements, mm

4.0% Displacement, mm 4.00

3.5% 3.50
Difference, mm
3.0% 3.00
2.5% 2.50
2.0% 2.00
1.5% 1.50
1.0% 1.00
0.5% 0.50
0.0% 0.00
0 1 2 3 4 5 6
Stages of prestress relief

Fig. 5 Graphs: displacements along X-axis, difference of outermost strands displacements in

absolute and relative values
Composite Forming by the Method of Prestressing of Carbon … 105

The analysis of the obtained data on the absolute displacement along the X-
axis showed their unevenness between the outermost strands of the carbon tape,
i.e. between the top and the bottom strands, the maximum value of 4.55% being
determined at the 5th stage of relief (Fig. 3).
General Displacements
Figure 6 shows the patterns of longitudinal relative deformations in a carbon
composite at all stages of prestress reduction.

0 1

2 3

5 см

5 6

Fig. 6 Patterns of longitudinal relative deformations at prestress reduction stages. Line L0-L0—
cross-section along the middle strand of carbon tape
106 V. Plevkov et al.

Deformations ε, % 0.0%

-0.1%

-0.1%

-0.2%

-0.2%

-0.3%
0 50 100 150 200 250 300
Registration zone length, mm

Fig. 7 The graph chart of the distribution of longitudinal relative deformations along the L0-L0
cross-section at the 4th stage of prestress reduction

Cyclic repetition of zones with maximum and minimum values of relative defor-
mations with an approximately 115 mm pitch is observed in the images of longi-
tudinal relative deformations (Fig. 6, stage 4) and the graph chart of longitudinal
relative deformations distribution along the L0-L0 cross-section (Fig. 7) at the 4th
stage of relief. The zones of about 50 mm wide run at 60° to the longitudinal axis of
the carbon tape due to uneven reduction.
Figure 8 shows the distribution of longitudinal relative deformations along the
L0-L0 cross-section at all stages of prestress reduction.
To increase frequency of zones with the decrease in their width is observed in the
images of longitudinal relative deformations at the 5th and the 6th stages of relief.
Statistical processing of longitudinal relative deformation data resulted in
constructing a graph chart of the reduction in total carbon composite deformations
(Fig. 9). The broken line of actual stresses is also duplicated in the graph chart. The
bending shape of the broken stresses and deformations coincides. The stress graph

0.1%

0.0%

-0.1%

-0.1%
ε

-0.2%

-0.2%

-0.3% 0 этап 1 этап 2 этап 3 этап

-0.3% 4 этап 5 этап 6 этап
0 50 100 150 200 250 300
Registration zone length, mm

Fig. 8 The graph chart of distribution of longitudinal relative deformations along L0-L0 cross-
section at all stages of prestress reduction
Composite Forming by the Method of Prestressing of Carbon … 107

Fig. 9 The graph chart of 250 0.20%

total deformations
Stresses
200
0.15%

Deformations, %
Deformations

Stresses, MPa
150
0.10%
100

0.05%
50

- 0.00%
0 1 2 3 4 5 6
Stages of prestress relief

is shown in a symmetrical form for clarity and logic of reducing both stresses and
deformations during relieving.
The stress-deformation “σ–ε” diagram while relieving (Fig. 10) is linear that
reflects the elastic behavior of carbon reinforcement.
Forming a carbon composite at pre-tensioning of the reinforcing material in the
form of a unidirectional carbon tape allows straightening the carbon strands that wrap
the adhesive therma-threadings. The shape of the carbon strand was not changing
still being straight since the prestress relief was carried out after gaining the epoxy
binder strength.
Carbon composite materials have long been used in the construction industry
while designing new structures, as well as existing structures strengthening. As a
result, the effectiveness of the composites in case of reinforcement was limited by
the perceived stresses from the structure additional loading after reinforcement, i.e.,
the area of temporary loads. The use of carbon tape pre-stress when creating a

Fig. 10 The diagram of 250

“σ–ε” deformation while 0
relieving in stages 200 1
2
σ, MPa

150
3
100
4
50
5
6
-
0.00% 0.10% 0.20%
ε,%
108 V. Plevkov et al.

composite on the structure surface allows creating the effective layered structures
using the entire strength resource of carbon fiber.

4 Conclusions

1. Being 17% of the ultimate stresses of the composite, the prestressing of the
carbon tape at the level of 205.54 MPa technologically allows for the process
of prestressing the dry tape saturating the tape with a binder and holding to the
full strength of the matrix, without prestress reducing associated with ruptures
of overstressed carbon fibers.
2. The composite matrix from the “Resin 530” binder with an elasticity modulus
of 638 MPa allows maintaining about 0.4% of the stress in the carbon tape from
the prestressing level.
3. The developed carbon tape prestressing method showed the displacements
unevenness of the extreme strands of less than 4.55% of the total longitudinal
displacements.
4. Cyclicity and width of the zones of maximum and minimum relative deforma-
tions being revealed in the images of longitudinal deformations during relief
are dependent on the level of prestress in the composite.

References

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2310:020149
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strain-stress state of vertical tank reinforced by carbon tyre based on numerical researches in
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plastic deformation of the steel/steel adhesive joint using digital image correlation method.
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unbonded and bonded prestressed CFRP-strengthened steel plates. In: Proceedings of the eighth
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Arch 2(61):79–97
Influence of Superplasticizer and Silica
Fume on the Structure Formation
and Properties of Cement Stone

Aleksandr Smirnov, Lev Dobshits, and Sergey Anisimov

Abstract The influence of superplasticizer and silica fume on cement hydration,

structure formation, phase composition, and cement stone properties is investigated.
It was found that using a polycarboxylate superplasticizer reduces the normal consis-
tency of the cement paste and increases the cement stone strength at 28 days by 22%.
Replacing cement with silica fume in the absence of a superplasticizer does not
increase the cement stone strength, but it reduces the open capillary porosity of the
cement stone by 7%. The combined use of silica fume and superplasticizer increases
the cement stone strength by 27% and reduces open capillary pores volume by 17%
compared to the control sample without admixtures. According to the X-ray phase
analysis results, it was established that the use of polycarboxylate superplasticizer
leads to a slowdown in the hydration processes of clinker minerals at 1 day. The use
of silica fume accelerates the cement hydration in the early hardening stages and
compensates for the plasticizing admixture slowing effect. At 28 days, in the cement
stone with silica fume, a decrease in the portlandite content by 39% is observed.
The combined use of superplasticizer and silica fume leads to the formation of a
cement stone structure with increased content of amorphized low-basic hydrated
calcium silicates by 22%, which significantly densifies and strengthens the cement
stone structure.

Keywords Cement stone · Silica fume · Superplasticizer · Normal consistency ·

Compressive strength · Porosity · Phase composition · Pozzolanic reaction

1 Introduction

Significant progress in modern concrete technology has been achieved due to the
widespread introduction of various highly effective chemical and mineral additives
into its production [1].

A. Smirnov (B) · S. Anisimov

Volga State University of Technology, Yoshkar-Ola, Russia
L. Dobshits
Russian University of Transport (MIIT), Moscow, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 111
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_11
112 A. Smirnov et al.

The use of mineral fillers as a component of concrete mixtures is one of the

priority areas for reducing cement consumption, improving their technological and
performance characteristics [2, 3].
In high-performance concretes, much attention is paid to the ferroalloy industry
by-product—silica fume [4, 5]. Silica fume is a dust-like material consisting of
ultrafine spherical particles obtained in gas purification of technological furnaces
in the silicon and ferrosilicon production. The main component of silica fume is
amorphous silicon dioxide, which is 85–98%. The average particle size of silica
fume is 0.1–0.2 μm, and the specific surface area is 15–25 m2 /g [6].
Due to its large specific surface area and amorphous structure, silica fume has
a high pozzolanic activity and is an effective microfiller. The cement stone struc-
ture formation in the presence of silica fume is based on the interaction of calcium
hydroxide formed during the cement hydration with the active silicon dioxide. The
formation of low-basic hydrated calcium silicates instead of portlandite crystals leads
to the cement stone structure densification, increasing its strength and resistance to
aggressive environments [7, 8].
However, the use of silica fume, due to its high specific surface area, leads to a
significant increase in water demand. In conditions of equal workability of concrete
mixtures, the use of silica fume is ineffective and gives a negligible strength effect
[9]. To reduce the water demand of concrete mixes with silica fume requires the
use of a water-reducing admixture. The most effective among such admixtures are
polycarboxylate-based superplasticizers [10–12].
In contrast to superplasticizers based on naphthalene and melamine sulfonates,
polycarboxylate superplasticizers are characterized by the spatial structure of
molecules with branched side chains, contributing to more efficient dispersion of
cement floccules due to the steric effect [10–12].
These superplasticizers have a significant water-reducing and plasticizing ability
and provide self-compacting concrete mixtures with high retention of rheological
properties [13, 14].
The goal of research was to study the separate and joint effect of polycarboxylate
superplasticizer and silica fume on the processes of cement hydration, structure
formation, phase composition, and properties of cement stone.

2 Methods

For the preparation of cement stone samples, Portland cement CEM I 52.5N
according to GOST 31108-2016 produced by LLC «South Ural Mining Processing
Company» was used as a binder. The specific surface area of this cement was
404 m2 /kg. The clinker had the following mineralogical composition: C3 S—62.1%;
C2 S—15.8%; C3 A—5.0%; C4 AF—13.2%. Condensed silica fume produced by
PJSC «Novolipetsk Metallurgical Plant» (TS 14-106-709-2004) with a bulk density
of 175 kg/m3 and mass content of silicon oxide SiO2 of 92% was used as an active
mineral additive. Polycarboxylate superplasticizer Sika ViscoCrete 25 HE-C was
Influence of Superplasticizer and Silica Fume on the Structure … 113

Table 1 Mix proportions of cement stone samples

No. Mix ID Components content, wt%
Cement Silica fume Superplasticizer Water
1 Control 100 – – 28.5
2 10% SF 90 10 – 28.5
3 0.6% SP 100 – 0.6 28.5
4 0.6% SP + 10% SF 90 10 0.6 28.5

used as a water-reducing admixture. Dosages of silica fume and superplasticizer

were taken based on previous studies and amounted to 10 and 0.6% of the total mass
of binder [9].
The study of the effect of admixtures on the normal consistency and setting time
of cement paste was carried out using the Vicat apparatus according to the Russian
standard GOST 310.3-76.
The study of the physical and mechanical characteristics of cement stone was
carried out on samples-cubes with a size of 20 × 20 × 20 mm. The amount of
mixing water in the compositions was constant and equal to the normal consistency
of the cement paste without admixtures. The compositions of the studied samples of
cement stone are shown in Table 1.
The compressive strength of the cement stone samples was monitored at 1, 3,
and 28 days of hardening under normal conditions. The density of cement stone
was determined by hydrostatic weighing at the age of 28 days according to GOST
12730.1-78. The open capillary porosity of cement stone was determined by the
volume water absorption of samples according to GOST 12730.3-78.
To identify the pattern of changes in the phase composition during hydration and
structure formation of cement systems with superplasticizer and silica fume, an X-ray
phase analysis of cement stone samples at 1 and 28 days was performed.
During the research, a Bruker D2 Phaser diffractometer was used to measure
powder samples in the Bragg-Brentano geometry using monochromatic CuKα radi-
ation with a wavelength of λ = 1.5406 Å, in a step scan mode. Measurement and regis-
tration modes: X-ray tube voltage—30 kV, current—10 mA, scanning step—0.02°,
speed—1°/min.

3 Results and Discussion

3.1 Normal Consistency and Setting Time of Cement Paste

Table 2 shows the superplasticizer and silica fume effect on the normal consistency
and setting time of cement paste.
114 A. Smirnov et al.

Table 2 Normal consistency and setting time of cement paste

No. Mix ID Normal consistency, Initial setting time, Final setting time,
% min min
1 Control 28.5 160 200
2 10% SF 30.5 190 225
3 0.6% SP 23.0 260 340
4 0.6% SP + 10% SF 20.0 220 270

It was found that the cement replacement with silica fume due to the high specific
surface area of its particles leads to an increase in the normal consistency of the
cement paste by 7%. Due to the increase in the water content in the composition with
silica fume, the initial setting time of cement paste slows down by 30 min compared
to the sample without additives. At the same time, the use of silica fume reduces the
interval between the initial and final setting time of cement paste by 5 min compared
to the control sample.
The use of a polycarboxylate superplasticizer, due to its plasticizing ability,
reduces the normal consistency of the cement paste by 19%. At the same time,
due to the adsorption of the admixture on the cement particles, there is a significant
slowdown in the initial and final setting time for 100 min and 140 min, respectively.
There is also an increase in the setting time of the cement paste by 40 min compared
to the sample without additives.
When silica fume is added to the composition with the superplasticizer, an addi-
tional decrease in the normal consistency of the cement paste by 11% is observed. This
effect can be explained by the fact that in the presence of a superplasticizer, smooth
spherical silica fume particles improve the particle packing density and reduce the
friction between the cement particles, acting as «ball bearings» [15, 16]. Also, the
addition of silica fume to the composition with the superplasticizer accelerates the
initial setting time of cement paste by 40 min and the final setting time by 70 min
compared to sample No. 3.

3.2 Physical and Mechanical Properties of Cement Stone

Table 3 shows the effect of superplasticizer and silica fume on the physical and
mechanical properties of cement stone.
It was established that the cement replacement with silica fume leads to a decrease
in the early strength of the cement stone at 1 day by 23%. At 28 days, the decrease
in the cement stone strength with silica fume is only 4%, which is explained by the
acceleration of the additive’s pozzolanic activity. Also, the use of silica fume reduces
the open capillary porosity of cement stone by 7% due to the additive’s microfilling
effect.
Influence of Superplasticizer and Silica Fume on the Structure … 115

Table 3 Physical and mechanical properties of cement stone

No. Mix ID Compressive strength, MPa Density, kg/m3 Open capillary
1 day 3 days 28 days porosity, %

1 Control 37.8 56.4 83.7 1935 29.8

2 10% SF 29.2 45.1 80.3 1865 27.8
3 0.6% SP 30.0 65.9 102.0 1953 29.1
4 0.6% SP + 10% SF 38.2 68.3 106.3 1942 24.8

It was found that the use of a polycarboxylate superplasticizer at a constant W/C

leads to a decrease in the early strength of cement stone at 1 day by 21%. At the
same time, at 28 days, there is already an increase in the cement stone strength by
22% compared to the sample without additives. Also, using a superplasticizer leads
to a slight increase in density and a decrease in the cement stone porosity.
The use of silica fume together with a superplasticizer makes it possible to
compensate for the slowing effect of the plasticizing admixture on the growth of
the cement stone’s early strength. At 1 day, the cement stone strength with addi-
tives is 1% higher than the strength of the sample without additives. At 28 days,
the combined use of admixtures can increase the cement stone strength by 27%.
At the same time, the combined use of a superplasticizer and silica fume leads to
a compaction of the structure of the cement stone and a decrease in its capillary
porosity by 17% compared to the sample without admixtures.

3.3 Phase Composition of Cement Stone

To identify the pattern of changes in the phase composition during hydration and
structure formation of cement systems with superplasticizer and silica fume, an X-ray
phase analysis of cement stone samples was performed.
Figures 1 and 2 show X-ray diffraction patterns of cement stone samples at 1 and
28 days.
In the course of a qualitative analysis of X-ray diffraction patterns, it was found
that the following minerals are present in the studied samples of cement stone:
• alite (3CaO·SiO2 ) with interplanar distances d = [5.90; 3.03; 2.97; 2.77; 2.74;
2.61; 2.45; 2.32; 2.18; 1.98 Å];
• belite (β-2CaO·SiO2 ) with d = [3.24; 2.88; 2.78; 2.74; 2.61; 2.45; 2.41; 2.28;
2.19; 1.98 Å];
• brownmillerite (4CaO·Al2 O3 ·Fe2 O3 ) with d = [7,24; 3,63; 2,77; 2,67; 2,63; 2,04;
1,92 Å];
• tricalcium aluminate (3CaO·Al2 O3 ) with d = [2.70; 1.91 Å];
• anhydrite (CaSO4 ) with d = [3.49; 2.85 Å];
 116

(c)
(a)

(d)
(b)

SP + 10% SF
9.73 9.73 9.73 9.73

10
7.24 7.24 7.24 7.24

5.90 5.90

15
5.90 5.90
5.61 5.61 5.61 5.61

4.93 4.93 4.93 4.93

4.69 4.69 4.69 4.69

20
3.88 3.88 3.88 3.88
3.63 3.63 3.63 3.63

25
3.49 3.49 3.49 3.49

3.24 3.24 3.24 3.24

3.11 3.11 3.11 3.11
3.03 3.03 3.03 3.03

30
2.97 2.97 2.97 2.97
2.88 2.88 2.88 2.88
2.77 2.77 2.77 2.77
2.74 2.74 2.74 2.74
2.67 2.67 2.67 2.67
2.63 2.63 2.63 2.63
2.61 2.61 2.61 2.61

35
2.57 2.57 2.57 2.57
2.45 2.45 2.45 2.45

2Theta (Coupled TwoTheta/Theta) WL=1.54060

2.41 2.41 2.41 2.41
2.32 2.32 2.32 2.32
2.28 2.28 2.28 2.28

40
2.18 2.18 2.18 2.18

2.10 2.10 2.10 2.10

2.04 2.04 2.04 2.04

45
1.98 1.98 1.98 1.98
1.93 1.93 1.93 1.93

50

Fig. 1 X-ray diffraction patterns of cement stone at 1 day: a Control, b 10% SF, c 0.6% SP, d 0.6%
A. Smirnov et al.
 (c)

(d)
(b)
(a)

5
9.73 9.73 9.73 9.73

10
7.24 7.24 7.24 7.24

d 0.6% SP + 10% SF
5.90 5.90

15
5.90 5.90
5.61 5.61 5.61 5.61

4.93 4.93 4.93 4.93

4.69 4.69 4.69 4.69

20
3.88 3.88 3.88 3.88
3.63 3.63 3.63 3.63

25
3.48 3.48 3.48 3.48

3.24 3.24 3.24 3.24

3.11 3.11 3.11 3.11
3.03 3,03 3.03 3.03

30
2.97 2.97 2.97 2.97
2.88 2.88 2.88 2.88
2.77 2.77 2.77 2.77
2.74 2.74 2.74 2.74
2.67 2.67 2.67 2.67
2.63 2.63 2.63 2.63
Influence of Superplasticizer and Silica Fume on the Structure …

2.61 2.61 2.61 2.61

35
2.57 2.57 2.57 2.57
2.45

2Theta (Coupled TwoTheta/Theta) WL=1.54060

2.45 2.45 2.45
2.41 2.41 2.41 2.41
2.32 2.32 2.32 2.32
2.28 2.28 2.28 2.28

40
2.18 2.18 2.18 2.18
2.10 2.10 2.10 2.10
2.04 2.04 2.04 2.04

45
1.98 1.98 1.98 1.98
1.93 1.93 1.93 1.93

50
117

Fig. 2 X-ray diffraction patterns of cement stone at 28 days: a Control, b 10% SF, c 0.6% SP,
118 A. Smirnov et al.

• ettringite (3CaO·Al2 O3 ·3CaSO4 ·32H2 O) with d = [9.73; 5.61; 4.69; 3.88; 3.48;
2.57; 2.21 Å];
• portlandite (Ca(OH)2 ) with d = [4.93; 3.11; 2.63; 1.93 Å].
Table 4 shows the results of quantitative X-ray phase analysis of cement stone by
the Rietveld method.
The obtained results of quantitative X-ray phase analysis of cement stone show
that the use of silica fume leads to an acceleration of hydration of clinker minerals
at 1 day. In the composition with silica fume, there is a decrease in the content
of unreacted minerals: alite—by 12%, belite—by 10%, brownmillerite—by 11%,
tricalcium aluminate—by 13%. Acceleration of cement hydration in the early stages
of hardening is a consequence of negatively charged silica fume particles to adsorb
calcium ions on their surface [17]. The adsorption of Ca2+ ions on the surface of
silica fume leads to a decrease in their concentration in the pore solution, which
accelerates the dissolution of clinker minerals [18–20]. Despite the increase in the
degree of hydration of cement minerals in the presence of silica fume, the amount of
formed portlandite in the cement stone is 5% lower compared to the sample without
additives.
Using a polycarboxylate superplasticizer leads to a slowdown in the hydration
of clinker minerals at 1 day. In the composition with a superplasticizer, there is an
increase in the content of unreacted alite—by 6%, belite—by 1%, brownmillerite—
by 12%, tricalcium aluminate—by 27% compared to the control sample without
admixtures. At the same time, slowing down the hydration of cement leads to a
decrease in the content of hydrate phases in the composition of cement stone: port-
landite—by 13%, ettringite—by 10%. There is also a slowdown in the formation of
the amorphous phase by 11% compared to the control sample without additives.
Replacing cement with silica fume in the presence of a superplasticizer leads to the
acceleration of cement hydration processes at 1 day. There is a decrease in the amount
of unreacted minerals: alite—by 13%, belite—by 10%, brownmillerite—by 10%,
tricalcium aluminate—by 24%. At the same time, the complex use of admixtures
leads to an increase in the content of portlandite in the cement stone by 5% and
ettringite—by 14% compared to the cement stone sample using a superplasticizer.
At 28 days, there are no peaks of tricalcium aluminate on the X-ray diffraction
patterns of all cement stone samples, which indicates the complete hydration of this
mineral.
In the cement stone with silica fume, a significant decrease in the Ca(OH)2 content
by 39% is observed compared to the sample without additives. This indicates accel-
erating the pozzolanic reaction of silica fume at the later stages of hardening of
cement systems. In this case, an increase in the amorphous phase content by 18% is
observed, which indicates the formation of a structure with an increased content of
poorly crystallized low-basic hydrated calcium silicates C–S–H.
When using a superplasticizer at 28 days, a decrease in the content of alite by
19% and belite by 2% is observed compared to the sample without additives, which
indicates an acceleration of their hydration later stages of hardening in the presence
of a superplasticizer. Also, in the presence of a superplasticizer, there is an increase in
Table 4 Phase composition of cement stone
No. Mix ID Phase composition, %
Alite Belite Brownmillerite Tricalcium Anhydrite Portlandite Ettringite Amorphous
3CaO·SiO2 β-2CaO·SiO2 4CaO·Al2 O3 ·Fe2 O3 aluminate CaSO4 Ca(OH)2 3CaO·Al2 O3 · phase
3CaO·Al2 O3 3CaSO4 ·32H2 O C–S–H
1 day
1 Control 33.6 15.6 7.6 3.0 1.0 10.5 7.2 21.5
2 10% SF 29.5 14.1 6.8 2.6 1.3 10.0 7.7 28.0
3 0.6% SP 35.6 15.8 8.5 3.8 1.6 9.1 6.5 19.1
4 0.6% SP 31.0 14.2 7.7 2.9 1.4 9.6 7.4 25.8
+ 10% SF
28 days
1 Control 17.5 12.5 6.8 – – 13.6 8.3 41.3
Influence of Superplasticizer and Silica Fume on the Structure …

2 10% SF 16.9 12.5 6.1 – – 8.3 7.5 48.7

3 0.6% SP 14.1 12.3 7.4 – – 14.9 8.9 42.4
4 0.6% SP 14.4 12.2 6.4 – – 8.9 7.9 50.2
+ 10% SF
119
120 A. Smirnov et al.

the content of portlandite by 10%, ettringite by 7%, and an increase in the amorphous
phase by 3% compared to the control sample without additives.
The use of silica fume together with a superplasticizer at 28 days leads to the
formation of a structure with a reduced content of portlandite by 40% and ettringite
by 11%. At the same time, amorphized hydrated calcium silicates prevail in the
structure of cement stone with the complex use of additives, the content of which is
22% higher than that of the sample without additives. Also, in the composition with
silica fume at 28 days, a gradual slowdown in the hydration processes of C3 S and
C2 S is observed. This is due to an increase in the density of C–S–H gel around the
cement grains due to the pozzolan reaction of silica fume, compaction of the cement
stone structure, and a decrease in its permeability [18–20].
As a result of the research, it was found that the combined use of a polycarboxylate
superplasticizer and silica fume as a complex admixture in cement systems leads
to a synergistic effect and can significantly improve the structure of cement stone
and increase its physical and mechanical properties, which cannot be achieved with
their separate addition. Moreover, each component of the complex admixture not
only retains its positive influence but also enhances the positive effect of the other
component.

4 Conclusions

1. The influence of superplasticizer and silica fume on the properties of cement

compositions is investigated. It was found that the cement replacement with
silica fume due to the high specific surface area of its particles leads to an
increase in the normal consistency of the cement paste by 7%. A polycarboxylate
superplasticizer reduces the normal consistency by 19%, but its use leads to a
significant increase in the cement paste setting time by 100…140 min. When
silica fume is added to the composition with a superplasticizer, an additional
decrease in the cement paste normal consistency by 11% and an acceleration of
its setting time by 40…70 min are observed.
2. It was found that using a polycarboxylate superplasticizer at constant W/C leads
to a decrease in the cement stone strength at 1 day by 21% and its increase by
22% at 28 days. The use of silica fume makes it possible to compensate for the
plasticizing admixture slowing effect on the cement stone strength growth at
1 day. At 28 days, the complex use of admixtures can increase the cement stone
strength by 27% compared to the control composition without additives.
3. The positive effect of silica fume on reducing the cement stone porosity due
to its microfilling and pozzolanic action is noted. It was found that the cement
replacement with silica fume leads to a decrease in the open capillary porosity
of cement stone by 7%. The combined use of silica fume and superplasticizer
reduces the open capillary pores volume of cement stone by 17% compared to
the control composition without admixtures.
Influence of Superplasticizer and Silica Fume on the Structure … 121

4. According to the X-ray phase analysis results, it was found that the use of
polycarboxylate superplasticizer leads to a slowdown in the hydration processes
of clinker minerals at 1 day. The use of silica fume accelerates the cement
hydration in the early hardening stages and compensates for the plasticizing
admixture slowing effect.
5. At 28 days, in the cement stone with silica fume, a significant decrease in the
portlandite content by 39% is observed compared to the composition without
additives, which indicates accelerating the pozzolanic reaction of silica fume at
the later stages of hardening of cement systems. The combined use of super-
plasticizer and silica fume leads to the formation of a cement stone structure
with increased content of amorphized low-basic hydrated calcium silicates by
22%, which densifies and strengthens the cement stone structure.
6. As a result of the research, it was found that the combined use of a polycar-
boxylate superplasticizer and silica fume as a complex admixture in cement
systems leads to a synergistic effect and can significantly improve the struc-
ture of cement stone and increase its physical and mechanical properties, which
cannot be achieved with their separate addition. Moreover, each component of
the complex admixture not only retains its positive influence but also enhances
the positive effect of the other component.

References

1. Aïtcin P-C, Flatt RJ (2016) Science and technology of concrete admixtures. Elsevier
2. Juenger MCG, Siddique R (2015) Cem Concr Res 78:71
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Cellular Structure Formation
of Composite Materials

Olga Miryuk

Abstract The article is devoted to magnesia thermal insulation materials. The aim of
the work is to study porization and hardening of magnesia composites with a cellular
structure. Technogenic fillers were used for synthesis of composite materials. Mate-
rials’ cellular structure was formed by foaming. Properties of the foam mass were
evaluated by multiplicity, density, and durability. The materials were studied using
X-ray phase analysis, electron microscopy, physical and mechanical methods. Prop-
erties of foam obtained from solutions of various salts have been investigated. Protein
foam concentrates are preferable for obtaining stable, fine-pored foam from magne-
sium chloride solution. Foam control is achieved by combining magnesium chloride
and zinc chloride solutions. It has been determined that hardening of composite mate-
rials slows down in the presence of foam concentrates. There has been developed a
method to stimulate hardening processes of porous composites. It was found out that
mechanical activation of a composite magnesia binder and molding mixture’s sepa-
rate preparation accelerate hydration and increase porous material’s strength by 60%.
It was revealed that strengthening of a cellular structure of composites is achieved
by fibrous magnesium hydroxychlorides formation. A resource-saving technology
has been developed for effective heat-insulating magnesia materials with a density
of less than 500 kg/m3 .

Keywords Composite magnesia binders · Porization · Hydration · Hardening ·

Cellular structure

1 Introduction

Efficiency of magnesia materials is determined by the low energy consumption of

production, intensive hardening and high strength.
Activating effect of magnesia hydroxychloride cement on natural and man-
made materials served as the basis for development of composite binders made of
caustic magnesite and mineral filler [1–12]. There are disadvantages of composite

O. Miryuk (B)
Rudny Industrial Institute, Kostanay Region, 111500 Rudny, Kazakhstan

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 123
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_12
124 O. Miryuk

magnesia binders in improving the properties of materials while saving scarce caustic
magnesite; rational use of man-made raw materials [13–20].
Cellular concrete occupies a leading position in the global production of thermal
insulation wall materials. Increase in efficiency of cellular concrete is associated with
the expansion of the range of the binder component of molding mixtures. Binders are
preferred, which provide strengthening of cellular structure framework, increasing
porosity of the stone [21, 22].
The use of caustic magnesite for cellular concretes production is known [4].
Information about magnesia composite materials of a porous structure is scarce.
There is no information about influence of saline solutions’ composition on
formation and stability of the foam mass. Processes of porous structure formation
and hardening of magnesia materials with mineral additives have not been studied.
Solving these problems will contribute to development of efficient thermal insulation
materials’ technology.
The aim of the work is to study the processes of porization and hardening of
cellular materials based on composite magnesia binders.
To achieve this goal, it is necessary to solve the following research tasks:
• study the process of foam formation obtained from saline solutions of various
compositions;
• determination of mineral filler’s influence on magnesia materials porization;
• development of a method for producing foam mass from composite magnesia
materials.

2 Materials and Methods of Research

Caustic magnesite of the PMK—75 brand with MgO content of 75–90% was used
in the experiments. Binder grinding fineness was 3–5% of the residue on the sieve
No. 008. Standard consistency is 40%. Setting time was as follows: start—20 min;
end—2 h 40 min. Composite binders were obtained by adding a technogenic filler to
caustic magnesite. Magnesium compositions were mixed with solutions containing
MgCl2 , MgSO4 , ZnCl2 , FeSO4 . To obtain cellular molding mixtures, surfactants of
various origins were introduced: protein foam concentrate «Polymir», «Zelly» and
synthetic foam concentrate «Fairy» in the amount of 3%.
The foams were prepared according to a single-stage method: a suspension
obtained by mixing the components was foamed in a mixer. Properties of the foam
mass were evaluated by multiplicity, density, and durability. To test the strength of
composite materials, samples with dimensions of 40 × 40 × 40 mm were formed.
The microstructure was studied by electron microscopy.
Ability of salt solutions to form foam was characterized by multiplicity and
stability of the foam. Foam was obtained from aqueous solutions with various salt
concentrations.
Cellular Structure Formation of Composite Materials 125

3 Results and Discussion

3.1 Formation of Foam from Salt Solutions

Foams were prepared on the basis of saline solutions of various compositions.

Characteristics of foam obtained from water were used for comparison.
Introduction of protein foam concentrate «Polymir» into solutions of magnesium
chloride of various densities (1100–1300 kg/m3 ) provide stable foam. The expansion
rate of foam decreases with an increase in concentration of MgCl2 in the solution
(Fig. 1).
Multiplicity of foam from protein foam concentrate «Polymir», formed on the
basis of magnesium sulfate solutions of various densities (1100–1300 kg/m3 ) is
comparable to multiplicity of foam from magnesium chloride solutions (Fig. 1).
However, foams based on MgSO4 solutions degrade quickly.
Foams obtained from solutions of complex composition have been investigated.
Adding zinc chloride solution to magnesium chloride solution (solution density is
1250 kg/m3 ) increases the foam rate. As content of ZnCl2 solution increases, the
foam stability decreases. Therefore, the content of ZnCl2 solution in the foam liquid
should be limited to 10% (Table 1). Combination of solutions of ferrous sulfate and
magnesium sulfate (solution density is 1200 kg/m3 ) is accompanied by increase in
foam stability and decrease in foam expansion (Table 1). To obtain foam of satisfac-
tory quality, liquid composed of 15% ferrous sulfate solution and 85% magnesium
sulfate solution is preferred.
Foam formation from magnesium chloride solutions based on «Fairy» synthetic
foam concentrate is difficult. The foam has a low expansion rate and quickly degrades.
Adding 20% ZnCl2 solution to MgCl2 solution (solution density is 1250 kg/m3 )

12
Polymir/MC
Foam expansion ratio

10
Fairy/MC
8 Polymir/М
6 Fairy/М

0
1100 1150 1200 1250 1300 water
Density of the salt solution, kg/m3

Fig. 1 Effect of the salt solution composition on the foam multiplicity (MC—magnesium chloride,
M—magnesium sulfate)
126 O. Miryuk

Table 1 Influence of liquid’s composition on properties of the foam from «Polymir» foam
concentrate
Solution content, % Foam expansion Foam stability during 80 min
MgCl2 ZnCl2 MgSO4 FeSO4 ratio Liquid outflow, Foam
% shrinkage, %
100 – – – 4.9 20 4
95 5 – – 5.6 18 3.5
90 10 – – 6.3 21 3.8
85 15 – – 6.5 22 4.2
80 20 – – 6.6 26 4.6
– – 100 – 4.8 25 5.2
– – 95 5 4.8 25 5.1
– – 90 10 4.6 21 4.6
– – 85 15 4.5 18 4.2
– – 80 20 4.3 17 4.1

increases the expansion rate of the foam and increases stability of the foam (Table
2).
The process of foam formation from magnesium sulfate solutions based on
«Fairy» foam concentrate depends on the density of salt solution.
Solutions with a density of 1100–1150 kg/m3 form foam with multiplicity of 5.0–
5.5 (Fig. 1). When density of MgSO4 solution is higher than 1150 kg/m3 , the foam
expansion decreases. Foam from liquid containing solutions of magnesium sulfate

Table 2 Influence of liquid’s composition on properties of the foam from «Fairy» foam concentrate
Solution content, % Foam expansion Foam stability during 80 min
MgCl2 ZnCl2 MgSO4 FeSO4 ratio Liquid outflow, Foam
% shrinkage, %
100 – – – 1.1 40 9.2
95 5 – – 1.2 37 9.0
90 10 – – 1.5 31 8.5
85 15 – – 3.4 28 5.4
80 20 – – 4.1 26 4.6
75 25 – – 4.2 25 4.4
– – 100 – 2.0 38 9.2
– – 95 5 2.8 33 8.3
– – 90 10 3.1 27 6.6
– – 85 15 3.5 23 4.3
– – 80 20 3.3 22 3.8
– – 75 25 3.2 22 3.7
Cellular Structure Formation of Composite Materials 127

and ferrous sulfate (solution density is 1150 kg/m3 ) is characterized by the highest
expansion rate when FeSO4 solution content is up to 15% (Table 2).
It can be assumed that improvement in properties of the foam upon transition to
solutions of complex composition is due to a change in viscosity and surface tension
of the liquid.

3.2 Porization of Composite Magnesia Materials

To prepare foam mass on the basis of magnesia binders of different compositions

there was used a magnesium chloride solution with the density of 1220 kg/m3 and
foam concentrate «Polymir» (Table 3). The increased porosity of compositions is
achieved with the introduction of silica fillers (waste from processing of magnetite
ores, metallurgical slag), content of which for low-density materials should be limited
to 30%.
When using a combined solution (90% magnesium chloride solution and 10% zinc
chloride solution), the expansion rate of foam mass increases. Porous materials have
reduced density (Table 4). Preparation of a mixture with HPP ash is accompanied
by rapid desolation of the molding mass. Comparison of strength characteristics of
porous composites with their analogues revealed the following. Composite binders
containing 30–70% of the filler are not inferior in strength to caustic magnesite,
differing in slow hardening in the period up to 3 days. The strength of cellular
materials with fillers is significantly lower than the strength of foam-magnetite.
Study of effect of foam concentrates on hardening of magnesia binders revealed
a decrease in strength of the stone by 1.5–3.0 times (Table 5).
Increased sensitivity of composite binders to the presence of foam concentrate is
due to a decrease in activating ability of magnesium oxide; the shielding effect of
the surfactant on the filler.

Table 3 Influence of the filler type on properties of porous materials

Filler Foam mass Composite Compressive
Type Content in the multiplicity density, kg/m3 strength, MPa
binder, %
No 0 3.1 490 3.6
Waste from 30 3.3 470 2.4
processing of 50 3.4 455 1.5
magnetite ores
Metallurgical slag 30 2.5 485 1.9
50 2.6 470 1.2
Ash from power 30 1.8 495 2.0
plants 50 1.9 485 1.3
128 O. Miryuk

Table 4 Influence of filler type on the properties of porous materials while using a combined
solution of chloride salts
Filler Foam mass Composite’s Compressive
type Content in the multiplicity density, kg/m3 strength, MPa
binder, %
No 0 4.5 450 3.1
Waste from 30 4.8 425 1.9
processing of 50 4.7 410 1.3
magnetite ores
Metallurgical slag 30 4.5 435 1.8
50 3.9 460 1.1
Ash from power 30 2.3 485 2.1
plants 50 2.5 470 1.2

Table 5 Effect of foam concentrate on the strength characteristics of binders

Content of waste from Foam concentrate Compressive strength, MPa, age,
processing of magnetite Type Content, % day (samples 2 × 2 × 2, cm)
ores in the binder, %
1 14
0 no 0 40 82
«Polymir» 2 25 73
3 21 68
«Zelly» 2 26 65
3 19 63
30 no 0 27 84
«Polymir» 2 14 67
3 12 63
«Zelly» 2 15 65
3 10 63

3.3 Stimulating Porization and Hardening of Composite

Materials

To reduce negative impact of the foam concentrate on hardening of magnesia mate-

rials, a complex technological method was used, which provides for mechanical
activation of the binder and improvement of the method for preparing the foam
mass.
To increase the hydration capacity, components of the composite magnesia binder
were separately and jointly subjected to additional grinding in a laboratory mill-
activator «Emax». The effect of mechanical activation is transition of the passive
Cellular Structure Formation of Composite Materials 129

Table 6 Effect of activation on hardening of magnesia binders

Composition of the binder, % Compressive strength,
MPa, age, day
(samples 2 × 2 × 2,
cm)
Caustic magnesite Waste from the processing of 1 28
magnetite ores
Initial After activation Initial After activation
50 – 50 – 22 90
50 – – 50 27 97
– 50 50 – 31 101
– 50 – 50 43 115

surface of the substance to a chemically active state, which is expressed in increased

ability to react during subsequent technological processes.
At the same time of powder processing, the specific surface area of caustic magne-
site increased from 350 to 570 m2 /kg; of filler—from 360 to 680 m2 /kg. The highest
strength values were achieved with mechanical activation of both components of the
mixed binder (Table 6).
Activation of the binders intensified the structure formation and allowed to reduce
duration of hardening of cellular compositions by 30–35% till acquisition of decaying
strength.
The ambiguous role of foam concentrate in magnesia compositions led to the
study of methods for preparing foam mass, which differ in sequence of introducing
components in preparation of a suspension subjected to foaming (Table 7).
When comparing the structure and properties of foam concrete of various prepara-
tions, method 4, which provides the greatest strength of the composite, is considered
preferable. Primary contact of the binder with a highly concentrated solution of
magnesium chloride initiates the active hydration of magnesium oxide, ahead of the
impact of the foam concentrate.
This is confirmed by the results of diffractometric analysis (for example, reflec-
tion of 0.197 nm), indicating an increased content of magnesium pentahydrooxy-
chloride 5 Mg(OH)2 ·MgCl2 ·8H2 O in the composite material (Table 7). Addition of
an aqueous solution of foam concentrate reduces the concentration of MgCl2 in the
raw suspension to the set value.
Method’s efficiency increases when using an activated binder: strength of the
composite increases by 18%. Combined effect of composite binder’s activation and
a method of preparing the foam mass is expressed in an increase of 60% in the strength
of the composite compared to the material obtained by conventional technology
(Table 7).
Magnesia composites are characterized by uniform porosity with cell sizes not
exceeding 1 mm (Fig. 2).
 130

Table 7 Effect of the foam mass preparation method on the properties of the composite
Foam mass preparation method* Foam mass multiplicity Composite density, Compressive strength, Intensity of the
kg/m3 MPa diffraction reflections
0.197 nm, %
1. (MC + FM + CMB) – mixing – foaming 3.2 475 2.7 100
2. (MC + FM) – mixing + CMB – mixing – foaming 3.2 480 2.1 87
3. (MC + CMB) –mixing + FM– mixing – foaming 3.3 460 3.3 107
4. (MC of increased density + CMB) –mixing + 3.4 450 3.8 112
(FM + W) – mixing – foaming
4a. (MC of increased density + CMB activation) 3.4 445 4.5 120
–mixing + (FM + W) – mixing – foaming
* Note MC—magnesium chloride solution; FM—foam concentrate; CMB—composite magnesia binder; W—water
O. Miryuk
Cellular Structure Formation of Composite Materials 131

Fig. 2 Microstructure of cellular composite material

Strength of the cellular structure is determined by the state of interstitial parti-

tions that form the frame of the material. Crystal base of the hardened magnesia
stone, consisting mainly of fibrous magnesium hydroxychlorides (Fig. 3) provides
the composites with high strength.

Fig. 3 Microstructure of magnesia composite matrix

132 O. Miryuk

4 Conclusions

Fundamentals of a technology of composite magnesia materials with a cellular

structure have been developed.
Multicomponent composition of magnesia compositions and high reactivity of the
ingredients make it possible to implement various methods of technological impact
on raw materials and molding mass.
The first ever, influence of saline solution’s composition on foam formation and
stability was determined.
Increased foaming and stability of the foam mass is provided by mixing magnesia
compositions with a solution of magnesium chloride and using protein foam
concentrates.
Use of combined salt solutions for preparation of molding sands allows you to
control properties of foam mass and a porous composite material.
Features of porization and hardening of cellular materials using mineral fillers
of various origins were revealed. To provide reduced density of cellular mate-
rials, magnesia compositions containing polymineral waste from magnetite ores
enrichment are preferred.
A method for producing a cellular molding mixture from a composite binder has
been proposed.
Hardening of porous magnesium compositions containing the filler occurs due to
intensification of hydrate formation, which contributes to activation of the composite
binder and optimization of conditions for preparation of the molding material.
This research is funded by the Science Committee of the Ministry of Education
and Science of the Republic of Kazakhstan (Grant No. AP08856219).

References

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Mechanical Properties of Butyl Rubber
Composites with Microspheres Under
Cyclic Loading

Yuriy Yurkin, Amadeo Benavent-Climent, Pavel Kovtonyuk,

Darya Varankina, and Irina Voloskova

Abstract The development of lightweight composites to reduce noise and vibration

is a pressing challenge. One way to reduce the weight of the composite is to add hollow
microspheres to it. This article tests butyl rubber composites with hollow glass or
aluminosilicate microspheres under cyclic loading. The results showed that addition
of microspheres not only reduces the weight of the composite, but also improves its
damping properties. It was found that aluminosilicate microspheres behave like a
conventional filler, while the action of glass microspheres is more similar to that of
a plasticizer. This work could potentially contribute to understanding the properties
of microparticle-filled elastomers and contribute to the development of methods for
predicting the properties of composites.

Keywords Cyclic loading · Hysteresis · Butyl rubber · Microsphere · Polymer ·

Composite

1 Introduction

A sound and vibration damping system is widely used in the transportation, building,
aerospace and appliance industries to reduce the vibration and sounds of the mechan-
ical systems. The sound and vibration damping system is typically applied to selected
parts or areas of the structure such as door, floor, roof, etc. to prevent vibrations and
noise from being transmitted inside or outside the appliance [1, 2].
A typical damping system may include a thermoplastic or rubber layer or patch
and a constraining layer that together are effective in suppressing the extraneous
vibrations and sounds [2]. These damping systems primarily depend on the density
and mass of the thermoplastic layer for effective sound damping function. While
a denser and heavier thermoplastic layer offers better sound damping effects such

Y. Yurkin (B) · P. Kovtonyuk · D. Varankina · I. Voloskova

Department of Building Structures and Machines, Vyatka State University, 610000 Kirov, Russia
e-mail: yurkin@vyatsu.ru
A. Benavent-Climent
Department of Mechanical Engineering, Polytechnic University of Madrid, 28006 Madrid, Spain

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 135
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_13
136 Y. Yurkin et al.

damping systems often contribute to the increased weight of the vehicle or appliance.
Any attempts to reduce the overall weight of the vehicle or appliance has to occur
only at the expense of the mass of the thermoplastic layer. Hence there is a need in
the art for a light weight, sound and vibration damping system for use in the building,
automotive and appliance industries.
One of the most effective materials with vibration-absorbing properties are
polymer composite materials [3]. As a polymer base for composite materials, such
polymers as butyl rubber [4, 5], ethylene-propylene rubber [6, 7], ethylene vinyl
acetate [8, 9], bitumen [10, 11], etc. can be used. Practical application, butyl rubber
is the most promising as a polymer with high dissipative properties.
Addition of ingredients, such as a plasticizer or filler, to the polymer affects the
dynamic characteristics of the polymer [12]. Therefore, it is desirable to use filler
that has weak adhesion interactions with the polymer, which facilitates relaxation
processes and enables more fully to realize vibration damping properties of the
polymer. These requirements are met by calcium carbonate, which refers to inert
fillers and has one more important characteristic—the lowest price [13]. There is
a large number of vibration damping materials containing calcium carbonate, for
example [14–16]. Unfortunately, these inorganic fillers increase the density of the
products.
One of the most successful methods for reducing the weight of a composite is
the addition of various microspheres into its structure: glass [17–19], aluminosilicate
[20], and polymer [21, 22]. At the same time, despite the fact that there are works to
determine the mechanical properties of polymeric materials under cyclic loads [19],
similar works for butyl rubber composites filled with microspheres have not been
carried out yet. In this regard, the purpose of this article is to study the effect of the
addition of microspheres on density, tensile properties and hysteresis of composites
based on butyl rubber.

2 Methods

2.1 Materials

Butyl rubber BK-1675N (Nizhnekamskneftekhim, Russia) was chosen as the main

polymer. Curing was applied using octylphenol heat reactive resin SP-1045 (SI
Group, USA) in combination with halogen donor as Chlorobutyl rubber HBK-139
(Nizhnekamskneftekhim, Russia). Plasticizer is industrial oil I-40. Fillers were used:
calcium carbonate MiCarb (Geokom, Russia) as inert filler; Carbon black P-803
as active filler; Hollow glass microspheres HL25 (hereinafter referred to as HL)
(Hollowlite, China) and Hollow aluminosilicate microspheres (hereinafter referred
to as AS) (ASPM Group, Russia) as light weight filler.
Mechanical Properties of Butyl Rubber Composites … 137

2.2 Preparation of Composites

The butyl rubber mixture was performed in a rubber mixer with Z-shaped rotors
ZL-1.0 (Feniks, Dzerzhinsk, Russia) at temperature 80–90 °C for a mixing time of
40 min. Next, the mixture was cured in a curing press at a temperature of 180 °C
for 40 min. The composition without microspheres is designated F15 in the graphs;
formulations with AS 20, 40 and 60 g by weight are designated A20, A40 and A60,
respectively; formulations with a mass content of HL 40, 80 and 120 g are designated
H40, H80 and H120 respectively.

2.3 Measurements

The density measurements of ingredients and test specimens of butyl rubber compos-
ites were carried out using a pycnometer with a capacity of 100 cm3 (GOST 15139).
Distilled water was used as the experimental liquid. For low density materials, a
brass capsule was used for immersion (GOST R 57962).
The tensile mechanical tests were performed on a tensile tester (Shimadzu,
Japan) using a testing speed of 500 mm/min on dumbbell-shaped specimens (Fig. 1)
according to the conditions described in GOST 270.
The intermittent cyclic tensile tests at 300% elongation were performed on a
tensile tester (Shimadzu, Japan) using a testing speed of 500 mm/min on dumbbell-
shaped specimens (Fig. 1) according to the conditions described in GOST 252.
Figure 2 is a schematic diagram of an unloading-loading cycle and the hysteresis

Fig. 1 Dumbbell-shaped specimens for tests

138 Y. Yurkin et al.

Fig. 2 Schematic diagram

of an unloading-loading
cycle showing the amounts
of dissipated energy and
elastic energy [21]

loop formed, with the arrows indicating the unloading and reloading sequence [23].
The dissipated energy, or hysteretic energy loss, W D , corresponds to the area enclosed
by this hysteresis loop, while the area under the lower path (unloading part) of the
hysteresis curve represents the elastic energy stored, W E . The specific dissipated and
elastic energies were determined by dividing W D and W E by the initial volume of
the specimen. The specific damping capacity is the ratio of the energy dissipated in
a five cycle to the elastic or potential energy stored in this cycle [24, 25] and was
determined by:

WD
= . (1)
WE

Volume content of microspheres for figures was calculated by equation:

m ms /ρms
V C = n , (2)
i=1 m i /ρi

gde mms and ρ ms —mass content and density of microspheres; mi and ρ i —mass
content and density of all composite components (see Table 1).
The theoretical density of the experimental samples was determined by:
n
mi
ρ = n i=1 (3)
i=1 (m i /ρi )

Equivalent stiffness at 100% elongation (ε100 = 1) was calculated by:

Mechanical Properties of Butyl Rubber Composites … 139

Table 1 Formulation of butyl rubber composites

Ingredients Mass content (g) Density (g/cm3 )
1 Butyl rubber BK-1675N 130 0.9
2 Chlorobutyl rubber HBK-139 10 0.9
3 Industrial oil I-40 60 0.9
4 Octylphenol heat reactive resin SP-1045 15 1.4
5 Calcium carbonate MiCarb 140 2.44
6 Carbon black P-803 30 1.6
7 Hollow glass microspheres (HL) or hollow 0; 20; 40; 60 0.65
aluminosilicate microspheres (AS) 0; 40; 80; 120 0.82

σ100
E 100 = (4)
ε100

where σ100 —stress at 100% elongation.

Equivalent stiffness at 300% elongation (ε300 = 3) was calculated by:
σ300
E 300 = (5)
ε300

where σ300 —stress at 300% elongation.

3 Results

3.1 Density

The results of studying the density of mixtures with different dosages of AS and
HL are shown in Fig. 3. In this figure and further, the quantitative characteristic of

Fig. 3 Effect of microsphere 1.3

content on density of AS exp.
composite HL exp.
1.25 AS theor.
Density, g/cm3

HL theor.

1.2

1.15

1.1
0 0.1 0.2 0.3 0.4
Volume content of microspheres
140 Y. Yurkin et al.

the content of microspheres is presented in volume fractions, calculated by Eq. 2.

This is necessary in order to be able to compare the effect of fillers of different
densities. In addition, in most theoretical formulas for predicting the mechanical
properties of composites, it is the volume fraction that is used [26, 27]. As seen
from Fig. 1, the amount of added microspheres was chosen so that they occupied
a significant volume in the composite, while the total volume fraction of all fillers
reached the theoretical limiting values (0.52 for a cubic packing). The density of
composites naturally decreases with an increase in the proportion of microspheres.
Moreover, the density of composites with HL falls faster due to the lower density of
microspheres. The experimental values of the density of composites are in excellent
agreement with the theoretical values calculated from Eq. 3, which indicates that the
true density of each component in Table 1 is determined correctly.

3.2 Monotonic Tensile Tests

Considering the effect of microsphere addition on the equivalent stiffness, it can be

seen that with AS increasing, stiffness increases at 100% elongation (Fig. 4) and
practically does not change at 300% elongation (Fig. 5). The addition of HL, on the

Fig. 4 Effect of microsphere 3.5

Equivalent stiffness , MPa

content on equivalent
stiffness at 100% elongation 3
of composite 2.5
2
1.5
1
AS
0.5 HL
0
0 0.1 0.2 0.3 0.4
Volume content of microspheres

Fig. 5 Effect of microsphere 2

Equivalent stiffness , MPa

content on equivalent
stiffness at 300% elongation 1.5
of composite
1

0.5 A
S
0
0 0.1 0.2 0.3 0.4
Volume content of microspheres
Mechanical Properties of Butyl Rubber Composites … 141

Fig. 6 Effect of microsphere 6

content on tensile strength at AS

Tensile strength, MPa

break of composite HL
4

0
0 0.1 0.2 0.3 0.4
Volume content of microspheres

other hand, reduces the equivalent stiffness. Moreover, the nature of the decrease for
both 100% elongation and 300% elongation is the same: even the minimum amount
of HL sharply reduces the equivalent stiffness of composite (by 2–3 times), but with
a further increase in the share of HL, the equivalent stiffness does not change (Figs. 4
and 5). The effect of both types of microspheres on the strength of composites is
identical. An increase in the number of microspheres leads to a decrease in tensile
strength (Fig. 6). At the same time, a small amount of microspheres also leads to a
sharp decrease in the strength of the composite (by a factor of 2.5–3.5 for AS and
HL, respectively), but with a further increase in the content of microspheres, the rate
of drop in strength decreases and this dependence on the filling fraction becomes
almost linear.
Different types of microspheres have very different effects on the deformation
properties of composites. If the addition of AS leads to a decrease in elongation at
break up to 1.4 times (Fig. 7) and does not change the tensile set at break (Fig. 8),
then the addition of HL leads to an increase in elongation at break up to 2 times
(Fig. 7) and enormous increase in tensile set at break (from 20 to 420%) (Fig. 8).

Fig. 7 Effect of microsphere 2500

content on elongation at
AS
break of composite 2000
HL
Elongation, %

1500

1000

500

0
0 0.1 0.2 0.3 0.4
Volume content of microspheres
142 Y. Yurkin et al.

Fig. 8 Effect of microsphere 500

content on tensile set at AS
break of composite 400
HL

Tensile set, %
300

200

100

0
0 0.1 0.2 0.3 0.4
Volume content of microspheres

3.3 Repetitive Cyclic Tensile Tests

All butyl rubber composites are subjected to multiple cyclic loading to the same strain
300%. The cyclic tension responses of composites within microspheres is shown in
Fig. 9. For these base composite (and for all other composites), a clear Mullins
effect can be observed as that the stress in the reloading process is smaller than
the response of virgin specimens until surpassing the initial maximum applied strain
[19]. A large hysteresis loop appears between the first loading-unloading cycle, while
much smaller loops can be observed for further loading cycles. Also, after the first
loading-unloading cycle, a permanent deformation of 20% appears, which practically
does not increase with subsequent cycles. This suggests that all destruction within
the composite structure mainly occurs in the first cycle. The cyclic tension responses
of composites with AS microspheres are shown in Figs. 10, 11 and 12. The nature
of the hysteresis with an increase in AS did not change significantly compared to the
base composite.
This is especially noticeable in the loops of the fifth loading-unloading cycle
(Fig. 14). However, the first hysteresis loops undergo a clear transformation with
an increase in the AS fraction. The areas of the first loops become larger with AS
increasing. Figure 13 shows that with AS increasing, the initial modulus of elasticity
noticeably increases (the angle between the strain-stress curve originating from the

Fig. 9 Type cyclic tension 6

responses of the composites 1st 2nd
without microspheres (base 3rd 4th
composite) 4
Stress, MPa

5th

0
0 100 200 300
Strain, %
Mechanical Properties of Butyl Rubber Composites … 143

Fig. 10 Type cyclic tension 6.0

responses of the composites 1st 2nd
3rd 4th
with the mass of AS
5th

Stress, MPa
microspheres as 40 g 4.0

2.0

0.0
0.0 100.0 200.0 300.0
Strain, %

Fig. 11 Type cyclic tension

responses of the composites
with the mass of AS
microspheres as 80 g

Fig. 12 Type cyclic tension

responses of the composites
with the mass of AS
microspheres as 120 g

origin and the abscissa axis increases) and the yield area begins to be traced more
and more clearly on the curve.
The addition of HL into the composite compared to AS leads to significant changes
in the nature of the hysteresis (see Figs. 15, 16, 17, 18 and 19).
As the amount of HL increases, there is a more pronounced increase in the area
of the first loops (Figs. 15, 16 and 17). The formation of a yield point occurs even
with a small amount of HL (Fig. 18). The initial modulus of elasticity of composites
with any proportion of HL is almost identical to that of the base composite (Fig. 18).
There are also significant differences in the character of the fifth loops compared
to the composite without microspheres (Fig. 19). It is noteworthy that composites
with HL not only have a large permanent deformation after the first cycle, which,
144 Y. Yurkin et al.

Fig. 13 1st loop of cyclic 6

F15 A40
tension responses of the
5 A80 A120
composites with AS
microspheres 4

Stress, MPa
3

0
0 100 200 300
Strain, %

Fig. 14 5th loop of cyclic 4

F15
tension responses of the
composites with AS A40
3
microspheres

Stress, MPa
2

0
0 100 200 300
Strain, %

Fig. 15 Type cyclic tension 2

responses of the composites 1st 2nd
3rd 4th
with the mass of HL 1.5 5th
microspheres as 20 g
Stress, MPa

0.5

0
0 100 200 300
Strain, %

Fig. 16 Type cyclic tension

responses of the composites
with the mass of HL
microspheres as 40 g
Mechanical Properties of Butyl Rubber Composites … 145

Fig. 17 Type cyclic tension 1.5

responses of the composites
1st 2nd
with the mass of HL

Stress, MPa
1 3rd 4th
microspheres as 60 g 5th

0.5

0
0 100 200 300
Strain, %

Fig. 18 1st loop of cyclic

tension responses of the
composites with HL
microspheres

Fig. 19 5th loop of cyclic 4 F15

tension responses of the H20
composites with HL 3 H40
microspheres
Stress, MPa

0
0 100 200 300
Strain, %

depending on the proportion of HL, is 80–110% (which is much more than 20% for
the base composite), but also a further gradual increase in permanent deformation
occurs. This indicates that the composite not only receives significant damage as a
result of the first loading-unloading cycle, but also receives damage in subsequent
cycles, only much less.
Considering the effect of microsphere content on specific damping capacity of
composite, determined by loading-unloading cycle according to Eq. 1 (Fig. 20), it
can be seen that their increase leads to an increase in the ability to dissipate vibration
energy. Moreover, this ability of composites with HL is noticeably higher than that
of composites with AS.
146 Y. Yurkin et al.

Fig. 20 Effect of 0.35

microsphere content on

Specific damping capacity

specific damping capacity of 0.3
composite
0.25

0.2
AS
HL
0.15
0 0.1 0.2 0.3 0.4
Volume content of microspheres

4 Discussion

If we assess the nature of the influence of AS on the properties of butyl rubber

composites, then this nature, in principle, differs little from the nature of the influence
of any other mineral filler, for example, calcium carbonate. The effect of HL is
absolutely atypical for fillers. The nature of the introduction of HL is more similar to
the nature of the addition of the plasticizer: an increase in deformation, a decrease in
stiffness and strength. The same can be observed with other polymers [18, 19, 31, 32].
The main effect of the filler addition to the polymer is that part of the polymer
passes to the boundary layers directly interacting with the surface of the filler parti-
cles due to the high degree of adhesion with the polymer [29]. It leads to a quasi-solid
phase with a film matrix which properties differ from the properties of the polymer
matrix in general [29]. In most cases, an increase of the filler amount in the polymer,
as a result of loss of mobility and restriction of conformational transitions of macro-
molecules under the influence of a solid filler surface, results in a natural increase in
the strength and stiffness of the composite [28].
The study of the surface properties of AS [30] showed that it has a developed
surface structure and has active adsorption centers. In addition, its oil absorption
characteristic is about 30%, which, according to its lower weight, does not differ
so much from the oil absorption of calcium carbonate (18–21%). In this regard, the
nature of AS is so similar to the nature of the impact of calcium carbonate.
In contrast, HLs have a spherical shape with a hard, smooth, pore-free surface [31].
This particle shape ensures minimal oil absorption. Due to the fact that the HL surface
is deprived of the ability to absorb oil, when making a composite, this oil, falling on
the surface of microspheres, forms a thin layer, preventing the formation of physical
bonds between the filler and the polymer. This effect is sometimes deliberately used
by pretreating the filler surface with some oligomer to reduce the intensity of polymer-
filler interaction in order to ensure higher deformation properties of the composite.
In works [18, 31, 32], a weak bond of hollow glass microspheres and the polymer
is noted, the surface of the microspheres is very clean, and the interface between
the microsphere and the polymer is discrete without any signs of binding. It is the
presence of the plasticizer on the HL surface that can explain the nature of the effect
of its amount on the properties of the composite, similar to the nature of the effect
Mechanical Properties of Butyl Rubber Composites … 147

of the plasticizer. In [19], it is assumed that during the loading process, some of the
chains detach from the surface of glass microspheres, which results in a decrease
of stress in the reloading process. This can also explain the significant increase in
permanent deformation with an increase in the amount of HL compared to AS.

5 Conclusions

1. Addition of AS or HL microspheres not only reduces the weight of the

composite, but also improves its damping properties up to 1.3–1.5 times.
2. Under cyclic loading-unloading loads, an increase in the proportion of micro-
spheres leads to a significant increase in the first hysteresis loops, especially in
the case of adding HL. In this case, the composites not only increase the initial
modulus of elasticity, but also more clearly form a yield area. This property
looks promising for this type of load, such as cyclic tension-compression, when
the viscous properties of polymers are more pronounced.
3. The aluminosilicate microspheres have been found to behave like a conventional
filler and the glass microspheres have a more similar pattern of action to that of
a plasticizer. This is most likely due to the fact that a thin layer of plasticizer is
formed on the smooth surface of HL during the manufacture of the composite,
preventing the formation of physical bonds between the filler and the polymer.
As a result, during deformations, fragile polymer-HL bonds break easily, leading
to a decrease in the strength and rigidity of the composite, an increase in its
deformability and large residual deformations, which is so typical of the action
of plasticizers.

Part of this investigation was conducted under the research project MEC BIA2017 88814 R of
the Spanish Ministry of Economy and Competitivity and the European Union FEDER program.

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Reliability of Buildings and Constructions
and Safety in Construction
Experimental and Theoretical Studies
of the Concrete Static-Dynamic
Stress–Strain Curves

Natalia Fedorova, Michael Medyankin, Sergey Fedorov, and Sergey Savin

Abstract During operation, both of a residual static and dynamic load can be applied
to the structural members, and in the case of an emergency impact, such as the sudden
structural member removal, the dynamic additional loading arises in the structural
elements. Ultimate deformations and strength of concrete and reinforced concrete
under this loading mode significantly differ from their values under static or dynamic
loading mode. This paper presents the results of experimental and theoretical deter-
mination of the concrete stress–strain curve parameters for varying levels of the initial
static load and loading modes. The differential equation of the concrete deformation
model for the static-dynamic loading mode and its solution at various levels of the
initial static stress were obtained. It was established that the maximum permissible
time of impact on a concrete specimen under dynamic additional loading and the
ultimate strength of concrete depend on the level of the initial static load. Numer-
ical analysis and specimens’ tests shown that parameters of the concrete stress–strain
curves under static-dynamic loading mode substantially depend on the level of initial
relative stresses in concrete.

Keywords Concrete · Static-dynamic loading · Stress–strain curve · Special

limiting state · Deformation model · Progressive collapse · Experiment ·
Numerical analysis

1 Introduction

Design of reinforced concrete structures taking into account special impacts [1],
including emergency impacts caused by the sudden load-bearing member removal,
is associated with the use of a special limit state criteria [2].
As such criteria, the ultimate strains in compressed concrete and tensioned rein-
forcement or structural members’ ultimate deflections are currently used [3–5]. At

N. Fedorova · M. Medyankin (B) · S. Fedorov · S. Savin

Moscow State University of Civil Engineering, Yaroslavskoe shosse 26, Moscow 129337, Russia
e-mail: MedyankinMD@mgsu.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 151
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_14
152 N. Fedorova et al.

the same time, the quantitative values of these criteria in the current regulatory docu-
ments of Russia and other countries [1, 2, 6–9] are taken as approximate with a
certain margin. These criteria do not take into account loading modes, classes and
types of concrete and reinforcement, features of the structural system topology and
stress–strain state in sections of members, and etc.
When designing structural systems of buildings and structures, this inevitably
leads to an increase in the cost of their structural protection against progressive
collapse [5–11]. In this regard, it is necessary to study the ultimate characteristics
of concrete and reinforced concrete under static-dynamic loading mode caused by
structural member removal.
In the mechanics of shock loading the strain rate influence on the yield point (mate-
rial strength) necessitates the study of the physical mechanism of this phenomenon
for concrete and reinforced concrete [12–17].
Geniev developed a deformation model and a general theory of plasticity of
concrete and reinforced concrete [3, 18]. The deformation model proposed by him
is based on the hypothesis that the parameters of the stress–strain curves, ultimate
strength and deformations of concrete under static and dynamic loading conditions
depend on the type of stress state. However, the special impacts under consideration
have another loading mode, which differs from ones earlier studied. During oper-
ation, both of a residual static and dynamic load can be applied to the structural
members, and in the case of an emergency impact, such as the sudden structural
member removal, the dynamic additional loading arises in the structural elements.
Ultimate deformations and strength of concrete and reinforced concrete under this
loading mode significantly differ from their values under static or dynamic loading
mode.
The purpose of this work is the experimental and theoretical determination of the
concrete stress–strain curve parameters for varying levels of the initial static load
and loading modes.

2 Concrete Deformation Model

The proposed version of the deformation model, in addition to the hypotheses of

the theory of plasticity of concrete and reinforced concrete [18], is based on the
hypothesis of a one-parameter dependence between the ultimate deformations of
concrete on the initial stress–strain state. The physical meaning of this assumption is
that the process of microcracking in concrete during compression begins not from a
certain lower level of the stress state, but practically from the beginning of loading.
This fact was established on the latest studies on concrete structural changes during
loading [19]. Consequently, such parameters as the modulus of viscous resistance,
Poisson’s ratio, modulus of concrete dilatation change its values [20].
The objectives of this study included:
Experimental and Theoretical Studies of the Concrete … 153

• to obtain analytical dependencies for determining the main design parameters

of the concrete stress–strain curves under simple compression with a one-time
dynamic load (impact) attached at various levels of initial static load;
• to develop an experimental technique and empirically determine parameters of the
concrete stress–strain curves under simple compression with a one-time dynamic
load (impact) attached at various levels of initial static load;
• to compare and analyse the theoretical and experimental parameters’ values of
concrete stress–strain curves and evaluated different approaches its obtaining.
In accordance with the main physical dependence of the theory of plasticity of
concrete [3, 18] “stress intensity—strain intensity” T = G(Q)Q is a linear function.
The main feature of the problem under consideration is that it is solved on the basis
of the hypothesis about existence of a one-parameter dependence between ultimate
strains and the type and the initial level of the stress state under simple loading, as
well as on the hypothesis that the invariants “stress intensity T —strain intensity G”
linked by a linear function:

T = G(Q)Q. (1)

Relation (1) is relevant both of concrete initial static and dynamic additional
loading [20].
Well known [3] that the increased concrete strength is associated with the appear-
ance of viscous resistance internal forces that directly perceive the external impact
and inhibit the development of concrete deformations, which are the fracture physical
cause.
Using the hypotheses of the two-element concrete deformation model [3, 4] for
the loading mode under consideration,
√ we pass
√ from the general dependence (1)
between the invariants T = I2σ and G = I2ε to the one-dimensional (uniaxial
compression) case of testing standard concrete prisms.
Element A of this two-element concrete deformation model is described by the
dependences of the plasticity theory of concrete, the constants of which and the
magnitude of the modulus of deformation are determined by the known models
of deformation of reinforced concrete, including the nonlinear model given in the
current standards [21]:

σ A = E m ε A ; ε B = σb E b vb , (2)

where E m = is a concrete secant modulus.

The viscous element B is characterized by the modulus K of concrete viscous
resistance [3, 4].
For the considered one-dimensional stress state and two-stage static-dynamic
loading, the relationships between stresses and strains have the form (Fig. 1):

for σ A < σbst and ε A < εbst σ A = E m ε A (3)

154 N. Fedorova et al.

Fig. 1 Concrete static-dynamic stress–strain curves based on the two-element Voigt model:
mechanical two-element Voight model (a); stress–strain curves (b)

for σb < σ A ≤ Rbi

d
and ε A < εb Ri σ A = σbst + Eli (ε A − εbst ) (4)

Here E 1i = is concrete deformation modulus for the second deformation stage.

The behaviour of purely viscous element B is characterized by a constant K that
is the concrete viscous resistance modulus. Stresses at the second dynamic stage of
concrete static-dynamic deformation (section σbid ) are calculated by the formula:

∂εb
σb = K . (5)
dt
The time t, which is on the order of tenths of a second, is counted from the moment
the dynamic loading starts.
The effective work of the viscous element B does not have time to manifest itself,
but it contributes to the inhibition of deformations in element A. Thus, according to
the adopted concrete two-element deformation model, one can write:

ε A = ε B = ε; σ = σ A + σ B . (6)

It is obviously, if σ < σ R then concrete failure cannot occur for any duration of
dynamic impact.
Consequently, the relationship between the concrete dynamic ultimate strength
and the maximum permissible exposure time can be established from the given
relations (3), (4), (6) for σ > σ R :

dε Ei (σ − σb ) + E i εb
+ ε= . (7)
dt K0 K0
Experimental and Theoretical Studies of the Concrete … 155

Let us solve Eq. (1) using Bernoulli method. For this purpose, we represent strain
as a two-function product:

ε = U · V. (8)

Substituting (8) into the original Eq. (7), we obtain the following relations:

Ei (σ − σb ) + E i εb
U V + U V  +
UV = ; (9)
K0 K0
 
  Ei (σ − σb ) + E i εb
U V +U V + V = ; (9 )
K0 K0
Ei Ei
V + V = 0; or V  = − V ; (10)
K0 K0
dV Ei
= − V;
dt K0
dV Ei
= − dt; (11)
V K0
   
dV Ei
= − dt; (12)
V K0
or
Ei
ln|V | = − t; (13)
K0

from (13) we get

 
E
− Ki t
V =e 0 ; (14)

Substituting V into Eq. (3), we obtain:

Ei (σ − σb ) + E i εb
U  e K0 + U · 0 =
t
; (15)
K0

Derivation of U function take the form:

(σ − σs ) + E i εb KEi t
U = e 0 ;
K0
dU (σ − σb ) + E i εb KEi t
= e 0 ;
dt K0
156 N. Fedorova et al.

(σ − σb ) + E i εb Ei
U= ∫ e K0 t dt;
K0
 
(σ − σb ) + E i εb K 0 KEi t
U= e +C .
0 (16)
K0 Ei

Let us substitute (14) and (16) into the original Eq. (8), for concrete deformations
we get:
 Ei  K (σ − σ ) + E ε
t 0 b i b − KEi t
ε = U V = e K0 + C e 0 ;
Ei K0
 Ei  Ei (σ − σ ) + E ε
t t b i b
ε = e K0 + C e K0 ;
Ei
 E i  (σ − σ ) + E ε
t b i b
ε = 1 + Ce K0 . (17)
Ei

The general solution of the nonlinear differential Eq. (7) is carried out under the
following initial conditions: t = t b , εb = εbst .
 E 
− it
εbst E i = 1 + Ce K0 (σ − σb + E i εbst ); (18)

or after the appropriate transformations we get:

1 − KEi tb εst · E i
·e 0 = b − 1; (19)
C σb − σ
E
− K i tb
e 0
C= εbst ·E i
. (20)
σb −σ
−1

Figure 2 shows the general view of the diagram determined by dependencies (17),
(20).

3 Numerical Analysis

Using the obtained analytical dependences, we establish a relationship between the

concrete dynamic ultimate strength and the level of static loading of a concrete
sample under the static-dynamic loading mode, as well as the relationship between
the concrete dynamic ultimate strength and ultimate time of dynamic overloading
of a concrete sample. Experimental values of these dependencies were obtained in
the laboratory of NRU MSUCE [22]. Testing of prisms samples with dimensions
Experimental and Theoretical Studies of the Concrete … 157

Fig. 2 General view of

concrete static-dynamic
stress–strain curve at
different levels of initial
static compression

of 100 × 100 × 400 mm made of different concrete mixtures was carried out in
two stages. At the first stage, short-term static loading was performed under a press
continuously at a speed of 0.6 ± 0.2 MPa/s to a certain load level which equals to a
ratio of the fracture load: 0.2Rb , 0.4Rb , 0.6Rb . Due to the lack of standard equipment
that allows static testing followed by dynamic high-speed reloading of the sample on
one setup, the required level of static loading was fixed using a specially designed
device (Fig. 3), the description and operation of which are protected by the RF patent
[23]. Then the prism specimens were loaded with high-speed loading until it was
destroyed.
The test devise was equipped with an automatic control and recording system.
The maximum test load of the press was 3000 kN, the maximum recording frequency
of experimental data was 5 kHz. Experimental values of axial and transverse defor-
mations of concrete samples of prisms were recorded by the method of electric strain
gauge using strain gauges on a PLF-30 polyester substrate. The measuring base of
the sensors is 30 mm. The registration of the readings of the strain gages was carried
out using the NI PXIe-1082 equipment complex (Fig. 4). This equipment made it
possible to record readings with a sampling rate of up to 10 kHz.
Using the obtained analytical dependencies, the calculations of the ultimate
dynamic stresses and strains in the concrete of the experimental prisms were carried
out at various levels of the initial static compression (Fig. 5). Also, the influence of
the level of the initial static compression on the maximum allowable time of dynamic
reloading was investigated (Fig. 6).
The analysis of the experimentally obtained “relative stresses-strains” curves for
concrete of different strength classes and at different levels of initial static compres-
sion showed (Fig. 4) that the ultimate deformations of concrete under static-dynamic
loading mode are close to each other. The maximum discrepancy in their values
in relation to the average value does not exceed 5.6%. Comparison of the ulti-
mate dynamic stresses during the destruction of prisms shows that the coefficient
158 N. Fedorova et al.

Fig. 3 Devise for the experimental determination of the static-dynamic characteristics of concrete:
1–sample; 2–bottom plate; 3–threaded rod; 4–top plate; 5–spherical hinge; 6–bushing; 7–nut; 8–
force measuring device; 9–loading traverse; 10–longitudinal strain gauge; 11–transverse strain
gauge; 12–high speed camera

of increase in the concrete dynamic strength depends on the class of concrete and
the level of the initial relative compression of concrete. Thus, at the level of initial
compression equal to 0.4 of the concrete strength for B70 concrete the coefficient of
increase in dynamic strength was 1.4 and for B25 concrete 1.13 respectively.
Analysing the influence of the level of the initial relative compression of concrete
on the maximum permissible time of dynamic overloading, one can see (Fig. 5) that
the maximum allowable time of dynamic overloading decreases with an increase in
the level of initial compression.
Comparison of the experimental values of the studied parameters (Figs. 4, 5) with
their calculated values calculated from the analytical dependences of the proposed
version of the concrete deformation model showed their satisfactory convergence.
Experimental and Theoretical Studies of the Concrete … 159

Fig. 4 General view of test equipment: concrete prism specimen with installed strain gauges (a);
universal dynamic testing machine LabTest 6.500.5.01.1 (b); NI PXIe-1082 equipment complex
(c)

Fig. 5 Curves “relative

stresses–strains” of concrete
at levels of initial static
compression ï = 0.4 (a) and
ï = 0.6 (b): 1, 2
experimental and calculated
for B25 concrete
respectively; 3, 4 the same
for B35 concrete; 5, 6 the
same for B50 concrete; 7, 8
is the same for B70 concrete

4 Conclusions

1. The differential equation of the variant of the concrete deformation model for
the static-dynamic loading mode and its solution at various levels of the initial
static stress were obtained. It allows determining the parameters of the concrete
160 N. Fedorova et al.

Fig. 6 Influence of the level

of initial static compression
σ st
η = Rbb on ultimate time of
dynamic overloading t b : 1,2
experimental and calculated
for B35 concrete
respectively; 3, 4 the same
for B50 concrete; 5, 6 the
same for B70 concrete

static-dynamic stress–strain curves, in particular, the ultimate deformability of

concrete and the dynamic ultimate strength under dynamic loading.
2. In the work, it was established that the maximum permissible time of impact on
a concrete specimen under dynamic additional loading and the ultimate strength
of concrete depend on the level of the initial static load.
3. Numerical analysis and specimens’ tests shown that parameters of the concrete
stress–strain curves under static-dynamic loading mode substantially depend on
the level of initial relative stresses in concrete.

Acknowledgements The reported study was funded by RFBR, project number 19-38-90060.

References

1. Building Code of RF SP 296.1325800.2017 Buildings and structures. Accidental actions (with

Change No 1)
2. Building Code of RF SP 385.1325800.2018 (2019) Protection of buildings and structures
against progressive collapse. Design code. Basic statements (with Change No 1). Minstroy,
Moscow
3. Geniev GA, Kolchunov VI, Klyueva NV et al (2004) Strength and deformability of reinforced
concrete structures under accidental impacts. Publihing ASV, Moscow
4. Geniev GA, Pyatikrestovsky KP (2000) Long-term and dynamic strength of anisotropic
construction materials. CSRIBS named after V.A. Kucherenko, Moscow
5. Travush VI, Shapiro GI, Kolchunov VI, Leontiev EV, Fedorova NV (2019) Zhilishchnoe
Stroitelstvo 3:40
6. General Services Administration (GSA) (2013) Alternative path analysis and design guidelines
for progressive collapse resistance. Office of Chief Architects, Washington DC
7. Unified Facilities Criteria UFC 4-023-03 (2009) Design of buildings to resist progressive
collapse. Department of Defence, Washington DC
8. CEN Comité Européen de Normalisation (2006) EN 1991-1-7: eurocode 1—actions on
structures—part 1–7: general actions—accidental actions. CEN, Brussels
Experimental and Theoretical Studies of the Concrete … 161

9. Building Code of Ukraine DBN B. 1.2-14-2009 (2009) General principles for ensuring
the reliability and structural safety of buildings, structures, constructions and foundations.
Minregionbud, Kyiv
10. Kodysh EN, Trekin NN, Chesnokov DA (2016) Ind Civ Eng 6:8
11. Elsanadedy HM, Almusallam TH, Al-Salloum YA, Abbas H (2017) Constr Build Mat 142:552
12. Saffari H, Mashhadi J (2018) J Eng Fail Anal 84:300. https://doi.org/10.1016/j.engfailanal.
2017.11.011
13. Grigoriev AS, Shil’ko EV, Skripnya VA, Chernyavsky AG, Psah’e SG, Vestnik PNIPU (2017)
Mehanika 3:75
14. Lu D, Wang G, Du X, Wang Y (2017) Int J Impact Eng 103:124
15. Yu X, Li C, Fang Q, Ruan Z, Hong J, Xiang H (2017) Int J Impact Eng 101:66
16. Selyutina N, Petrov Y (2016) Proc Struct Integrity 2:438
17. Su H, Xu J, Ren W (2016) Ceram Int 42:3
18. Geniev GA, Kisyuk VN, Tyupin GA (1974) Theory of concrete and reinforced concrete
plasticity. Stroyizdat, Moscow
19. Bushova OB, Zinov’ev VN (2018) Classification of curves of dependence of change of speed of
ultrasound on stresses in concrete at compression. In: Proceedings of international conference
on modern methods of calculation of reinforced concrete and stone structures by limit states.
Moscow, RF, 30 Nov 2018
20. Fedorova NV, Medyankin MD, Bushova OB (2020) Ind Civ Eng 2:4
21. Building Code of RF SP 63.13330 (2018) Concrete and reinforced concrete structures. General
provisions
22. Fedorova NV, Medyankin MD, Bushova OB (2020) Build Reconstr 3:72. https://doi.org/10.
33979/2073-7416-2020-89-3-72-81
23. Fedorova NV, Medyankin MD (2019) Method of experimental determination of static-dynamic
characteristics of concrete. Patent of RF No. 2696815
Simulation of Effects the Degree
of Water-Saturation on Stress–Strain
State

Armen Ter-Martirosyan and Ahmad Othman

Abstract The ratio of the volume of water to the volume of voids is degree of
saturation (Sr ). The mechanical and physical properties of soil change due to the
changes of this degree, and may cause failure of building in the results of these
changes. The article poses and solves the problem of stress–strain state of a water-
saturated soil massif, including settlement and pore pressures of water-saturated bases
of foundation with finite width, depending on the degree of water saturation (Sr ) of
soils, taking into account the linear and nonlinear properties of the soil skeleton and
the compressibility of pore gas-containing water. The study based on results of four
models on the PLAXIS software package (Linear elastic, hardening soil, Mohr–
Coulomb, and UBC3D-PLM). Results of the simulation showed the difference in
the behavior of the soil mass under dynamic loads due to the changes of degree of
water-saturation, as well as it recommends the preferred method for studying the
saturated soil under dynamic loads.

Keywords Vibration impact · Dynamic impact · Pore pressure · Saturation ·

UBC3D-PLM · Liquefaction · PLAXIS

1 Introduction

The ratio of the volume of water to the volume of voids called degree of saturation. It’s
denoted by (Sr ). This degree generally expressed as a percentage. When this degree
equal to (0%), then the soil is fully dry and when this degree equal to (100%), then
the soil is fully saturated [1]. The stress–strain state of a water-saturated soil massif,
including settlement and bearing capacity of a water-saturated base of foundation
with finite width, depended on the degree of water saturation of soils 0.8 < Sr ≤ 1.
In the case, when a saturated soil subjected to vibration or any dynamic impacts, it
tends to compact, and decrease in volume. In the case, in the absence of drainage, the

All tests were carried out using research equipment of The Head Regional Shared Research Facilities
of the Moscow State University of Civil Engineering.

A. Ter-Martirosyan · A. Othman (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 163
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_15
164 A. Ter-Martirosyan and A. Othman

Fig. 1 Explanation the causes of soil liquefaction: a loose water-saturated sand with large pores
before the dynamic loads; b dynamic shock: a characteristic record of changes in the acceleration of
oscillations in time; c the moment of liquefaction—the bonds between the soil particles are broken.
They are suspended in the water, d compacted sand after squeezing out water and settling particles

pore water pressure increases. If the pore water pressure in the sand deposit is allowed
to build up by continuous vibration, at some time this pressure will be reached to
the overburden pressure [2, 3]. It’s means that the pressure will be equal to the: pore
water pressure (Fig. 1). Based on the effective stress principle:

σ  = σ − uw (1)

where σ  —the effective stress, σ —the total pressure, and u w —the pore water pres-
sure. If σ is equal to u w , σ is zero. In this case, In this case, the soil does not possess
any shear strength, and the particles of soils are separated and without any contact
virtually, and due to the mutual displacement of particles (down), and pore fluid (up),
the compaction occurs, and as a result the soil is transformed to a liquefied state [4].
The water-saturated (0.8 ≤ Sr ≤ 1), weakly cohesive soil are able to be liquefy
quickly under vibrations loads or any other dynamic loads. Loose water-saturated
soil temporarily turns into a state of heavy viscous fluid, which can lead to loss in the
stability of the structures at the bases on which they located. For that, it is important
to take into account the effects of the change of degree of water saturation when
designing the foundation of the structures [5, 6].
Liquefaction of saturated sands during earthquakes has been the cause of the huge
damage for many buildings, earth embankments, and retaining structures around the
world, for example the earthquake in Niigata—Japan of (1964), and the Alaskan
earthquake of 1964). Casagrande [7], based on the concept of critical void ratio, was
one of the first scientists who attempts to explain the liquefaction phenomenon in
sandy soils.
Nowadays, there are many models are used to describe the mechanical and phys-
ical properties of soils under static and dynamic effects. They are defined by a set of
equations, which determine the behavior of the soil under these effects. The nature of
the accepted hypotheses are the main difference between these models, i.e. principles
of deformation and stresses, and the conditions of the limiting state of soils under
static or dynamic loads.
This article discusses solutions for cases in which the soil is vary in the degree
of water saturation (Sr) in the range of (0.8–1.0) based on results of four models
Simulation of Effects the Degree of Water-Saturation … 165

Fig. 2 The main principle of elastoplasticity

on the PLAXIS software package (Linear elastic, hardening soil, Mohr–Coulomb,

and UBC3D-PLM), and the possibility of developing liquefaction in soils with these
degrees. The results shown that the settlement, pore pressure and bearing capacity
of the water-saturated base significantly depends on the degree of water saturation
(Sr) of soil.

2 Methods

2.1 Model Linear Elastic

The main principle of the model linear elastic is that, the strains are decomposed into
an elastic part and a plastic part (Fig. 2):

ε− = ε− e + ε− p , ε̇ = ε̇e + ε̇ p (2)

2.2 Model Mohr–Coulomb

The Mohr–Coulomb yield condition is an extension of Coulomb’s friction law to

general states of stress. In fact, this condition in this model ensures that Coulomb’s
friction law is obeyed in any plane within an element of the material. It consists of
six yield functions, formulated in terms of principal stresses [4]:
166 A. Ter-Martirosyan and A. Othman
    ⎫
1  1 
≤ 0⎪
 
f 1a = σ −σ + σ + σ sin ϕ − c cos ϕ ⎪
⎪
2 2 3 2 2 3 ⎪
⎪
    ⎪
⎪
⎪
⎪
1   1   ⎪
f 1b = σ −σ + σ + σ sin ϕ − c cos ϕ ≤ 0⎪
⎪
⎪
2 3 2 2 3 2 ⎪
⎪
    ⎪
⎪
1   1   ⎪
⎪
f 2a = σ −σ + σ + σ sin ϕ − c cos ϕ ≤ 0⎪
⎪
⎬
2 3 1 2 3 1
    (3)
1  1  ⎪
≤ 0⎪
 
f 2b = σ −σ + σ + σ sin ϕ − c cos ϕ ⎪
⎪
2 1 3 2 1 3 ⎪
⎪
    ⎪
⎪
⎪
⎪
1   1   ⎪
f 3a = σ −σ + σ + σ sin ϕ − c cos ϕ ≤ 0⎪
⎪
⎪
2 1 2 2 1 2 ⎪
⎪
    ⎪
⎪
1   1   ⎪
⎪
f 3b = σ −σ + σ + σ sin ϕ − c cos ϕ ≤ 0⎪
⎭
2 2 1 2 2 1

where ϕ—friction angle, and c—the cohesion.

2.3 Model Hardening Soil

The yield surface of a hardening plasticity model is not fixed in principal stress space
as in the elastic plastic model, but it can expand due to the plastic straining. The main
principle of the Hardening Soil model is that, it this model uses the theory of plasticity,
rather than the theory of elasticity. There are two main types of hardening: 1—Shear
hardening; 2—Compression hardening.
The Hardening Soil model is an advanced model for simulating the behavior of
different types of soil. When subjected to primary deviatoric loading, soil shows a
decreasing stiffness and simultaneously irreversible plastic strains develop.

2.4 Model UBC3D-PLM

The UBC3D-PLM model is an effective stress elasto-plastic model, which is capable

of simulating the liquefaction ehavior of soils under dynamic loading [8]. The
UBC3D-PLM model formulation is based on the original UBCSAND [9].
The UBC3D-PLM model uses the Mohr–Coulomb yield condition in a 3D prin-
cipal stress space for primary loading, and a yield surface with a simplified kinematic
hardening rule for secondary loading.
Main characteristics of the UBC3D-PLM model are given below:
• Stress dependent stiffness K Be , K Ge , me, ne, np;
• Plastic straining K GP ;
• Densification due to the number of cycles f dens ;
• Post-liquefaction stiffness degradation f E post ;
Simulation of Effects the Degree of Water-Saturation … 167

Fig. 3 Design scheme of the model

• Failure according to the criterion of Mohr–Coulomb failure ϕcv , ϕ p and c

To assess the potentiality of liquefaction of the soil, we can use the following
algorithm [9]:
• Determinate the geotechnical model of the soil deposit;
• Determine the dynamic impact according to our concerned site and the prob-
abilities of this impact, as indicated in the existing regulatory documents
[10];
• Perform the calculations using a numerical model and analysis this results.

3 Simulation of Model

As part of this study, a model of a soil, subjected to static loads (qst = 100 kPa)
and dynamic loads (qdy = 0.2qst sin(2π.ν.t), ν = 50 GPC, t = 1/ν = 0.02 s),
was modeled in PLAXIS software package. In each method, the study were for two
degrees of saturation (Sr = 0.8, Sr = 0.9999) (Fig. 3). Parameters of the soil, which
used for the four models, are presented in the Table 1, and in Fig. 4 show the dynamic
load.

4 Results

In Figs. 5 and 6, show the isolines of vertical displacements and pore pressures.
In this obtained results, it is shown that the vertical displacements in the method
UBC3D-PLM for (Sr = 0.8) do not exceed 12 cm, and do not exceed 20 cm for (Sr
= 0.9999). At the same time, the vertical displacement for the other methods do not
168 A. Ter-Martirosyan and A. Othman

Table 1 Input parameters for the calculation

Parameters Unit Linear elastic Mohr-Coulomb HS UBC3D-PLM
Drainage type – Undrained (A) Undrained (A) Undrained (A) Undrained (A)
γunsat kH/m3 19.70 19.70 19.70 19.70
γsat kH/m3 21.80 21.80 21.80 21.80
einit – 0.74 0.74 0.74 0.74
ϕ ° – 30 30 30
ψ ° – 0 0 0
C kH/m2 – 10 10 10
νun – 0.3 0.3 0.3 –
E/ E 50 r e f kH/m2 46.77 × 103 46.77 × 103 46.77 × 103 –
E oed r e f kH/m2 – – 37.34 × 103 –
E ur r e f kH/m2 – – 140 × 103 –
kx m/cyt 0.00001 0.00001 0.00001 0.00001
ky m/cyt 0.00001 0.00001 0.00001 0.00001
Rayleigh α – 0.2094 0.2094 0.2094 0.2094
Rayleigh β – 0.01061 0.01061 0.01061 0.01061
Pref kH/m2 – – 100 100
M – – – 0.5 –
Knc0 – – – 0.5 –
Rf – – – 0.9 0.9
K Be – – – – 854.6
K Ge – – – – 598.2
K GP – – – – 250
me – – – – 0.5
ne – – – – 0.5
np – – – – 0.5
ϕp – – – – 30.77
(N 1)60 – – – – 7.65
f dens – – – – 0.2
f E post – – – – 0.2

exceed 3 cm for (Sr = 0.8), and do not exceed 5 cm for (Sr = 0.9999) (Figs. 7, 8
and 9).
Results of the calculations using UBC3D-PLM model, compared with other
models, showed that the displacements increases based on the number of cycles,
which leads to the collapse as a result of developing liquefaction, and this collapse
will be faster in the degree of saturation (Sr = 0.9999) in the comparison with results
of degree of saturation (Sr = 0.8) (Fig. 7).
Simulation of Effects the Degree of Water-Saturation … 169

Fig. 4 Dynamic load: recording of dynamic stress

The results shown the effect of degree of saturation on the pore pressure, where
the pore pressure when (Sr = 0.9999)is in 10 time more the pore pressure when (Sr
= 0.8) (Fig. 10).

5 Conclusions

Based on the analytical results, we can obtain the following conclusions:

• The situation of the stress–strain state for the bases is further complicated, when
we consider the degree of water-saturation of soil, because in the absence of
drainage, in case of brief exposure to static loads or dynamic loads, it should be
taken into account that total stresses are distributed between the skeleton of the
soil and pore gas-containing water;
• The Displacement, pore pressure and the bearing capacity of the water-saturated
base significantly depends on the degree of water saturation of the soils;
• The higher the saturation degree, the faster possibility of liquefaction to appear;
• Results of the simulation showed the difference in the behavior under the impact
of dynamic loads in UBC3D-PLM model compared with the other models. Espe-
cially in the case of fully water-saturated soils. Therefore, it’s recommended to
170 A. Ter-Martirosyan and A. Othman

Method Linear Elastic Sr=0.8 Method Linear Elastic Sr=0.9999

Method Mohr-Coulomb Sr=0.8 Method Mohr-Coulomb Sr=0.9999

Method Hardening soil Sr=0.8 Method Hardening soil Sr=0.9999

Method UBC3D-PLM Sr=0.8 Method UBC3D-PLM Sr=0.9999

Fig. 5 Vertical displacements by linear elastic, Mohr–Coulomb, Hardening soil and UBC3D-PLM
methods with degree of water saturation degree (Sr = 0.8 and 0.9999) under static and dynamic
loads, pst = 100 kPa, = 25 * sin (2π * 50 * t)
Simulation of Effects the Degree of Water-Saturation … 171

Method Linear Elastic Sr=0.8 Method Linear Elastic Sr=0.9999

Method Mohr-Coulomb Sr=0.8 Method Mohr-Coulomb Sr=0.9999

Method Hardening soil Sr=0.8 Method Hardening soil Sr=0.9999

Method UBC3D-PLM Sr=0.8 Method UBC3D-PLM Sr=0.9999

Fig. 6 Excess pore pressure P-excess by linear elastic, Mohr–Coulomb, Hardening soil, and
UBC3D-PLM methods with degree of water saturation degree (Sr = 0.8 and 0.9999) under static
and dynamic loads, pst = 100 kPa, = 25 * sin (2π * 50 * t)
172 A. Ter-Martirosyan and A. Othman

Fig. 7 Dependence of the horizontal movement by linear elastic, Mohr–Coulomb, Hardening soil
and UBC3D-PLM methods with degree of water saturation degree (Sr = 0.8 and 0.9999) under
static, and dynamic loads with the dynamic time

Fig. 8 Diagram of vertical displacement by linear elastic, Mohr–Coulomb, Hardening soil and
UBC3D-PLM methods with depth under static and dynamic loads with degree of water saturation
degree (Sr = 0.8 and 0.9999)

use the UBC3D-PLM model to assessment the dynamic impact for the fully
water-saturated soils.
Simulation of Effects the Degree of Water-Saturation … 173

Fig. 9 Diagram of vertical displacement by linear elastic, Mohr–Coulomb, Hardening soil and
UBC3D-PLM methods at surface of soil (h = 0 m) under static and dynamic loads with degree of
water saturation degree (Sr = 0.8 and 0.9999)

Fig. 10 Dependence pore pressure P-excess by linear elastic, Mohr–Coulomb, Hardening soil and
UBC3D-PLM methods with degree of water saturation degree (Sr = 0.8 and 0.9999) under static,
and dynamic loads with the dynamic time

References

1. Vernay M, Morvan M, Breul P (2020) Evaluation of the degree of saturation using Skempton
coefficient B. Geomech Geoeng 15(2):79–89
2. Kererat C (2019) Effect of oil-contamination and water saturation on the bearing capacity and
shear strength parameters of silty sandy soil. Eng Geol 257, 105138
3. Dutta TT, Saride S, Jallu M (2017) Effect of saturation on dynamic properties of compacted
clay in a resonant column test. Geomech Geoeng 12(3):181–190
4. Leara A, Brinkgreve RBJ (2015) Site response analyses and liquefaction evaluation, 42
5. Minaeian V, Dewhurst DN, Rasouli V (2017) Deformational behaviour of a clay-rich shale
with variable water saturation under true triaxial stress conditions. Geomech Energy Environ
11:1–13
6. Ter-Marttirosyan ZG, Ter-Martirosian AZ (2020) Mechanics of soils, ACB, 912. Russia,
Moscow
174 A. Ter-Martirosyan and A. Othman

7. Casagrande A (1936) The determination of the pre-consolidation load and its practical
significance. In: Proceedings of ICOSMFE, vol 3. Cambridge (Mass), pp 60–64
8. Tsegaye AB (2010) Plaxis Liquefaction Model 45
9. Petalas A, Galavi V, PLAXIS liquefaction model UBC3D-PLM, 45
10. Jafarzadeh F, Ahmadinezhad A, Sadeghi H (2021) Effects of initial suction and degree of
saturation on dynamic properties of sand at large strain. Scientia Iranica 28(1):156–174
11. GOST R56353 (2015) Soils. Laboratory methods for determination of soil dynamic properties,
40
Modelling and Mechanics of Building
Structures
Elastic–Plastic Equilibrium of a Hollow
Ball Made of Inhomogeneous
Ideal-Plastic Material

Vladimir Andreev and Mikhail Maksimov

Abstract The article discusses the solution to the elastoplastic problem of the devel-
opment of the stress–strain state in an inhomogeneous thick-walled spherical shell.
It is assumed that the shell material is ideally plastic. The inhomogeneity of the
material consists in the change in the modulus of elasticity E and the yield stress σT
along the thickness of the radius, which is described by power functions with three
constants. The problem is solved in a centrally symmetric setting. Three options are
considered: (1) plastic deformations occur near the inner surface of the shell, (2)
plastic deformations occur between two surfaces of the shell, (3) an infinite array
with a spherical cavity is considered.

Keywords Hollow ball · Inhomogeneity · Plasticity · Elasticity

1 Introduction

The issues of plasticity and elastic plasticity are described in many fundamental
studies, including [1–5], etc. Publications are devoted to the statement of problems
and calculations of axisymmetric and centrally symmetric elastoplastic bodies, of
which the works [6–8], etc. can be noted. Calculations of inhomogeneous bodies
constitute a special area of mechanics. Taking into account the dependence of
mechanical characteristics on coordinates, it is rather difficult to solve such prob-
lems by analytical methods, and the development of this direction began with the
emergence and development of computer technology and numerical methods. Some
of the initiators of the development of the mechanics of inhomogeneous bodies were
G. B. Kolchin, N. A. Rostovtsev, V. A. Lomakin, W. Olszak [9–13] and other scien-
tists. The reasons for the continuous inhomogeneity of bodies are various fields or
phenomena (high or low temperature, radiation, humidity, explosive effect [14–23],
etc.).

V. Andreev (B) · M. Maksimov

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
e-mail: asv@mgsu.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 177
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_16
178 V. Andreev and M. Maksimov

The article deals with a one-dimensional problem of elastic and ideal-plasticity,

in which the modulus of elasticity and the yield point of the material can vary in a
wide range, which is the novelty of the work.

2 Statement of the Problem

Just as when considering the elastic problem for an inhomogeneous ball [13], when
solving an elastoplastic problem for this body, the equation describing its behavior,
solution methods and results are largely similar. A hollow thick-walled hollow ball
with inner and outer radii a and b, loaded from inside and outside by uniform
pressures pa and pb proportional to one parameter, is considered. The material is
considered to be ideally plastic, while the modulus of elasticity E and yield stress σT
are generally arbitrary functions of the radius. In addition, the material is considered
to be incompressible in both the plastic and elastic zones. For the first time the
formulation of such problems was given in [9, 20]. There are also solutions for the
simplest dependencies and, represented by power functions of the form Ar k . This
article provides a solution for the more general Young’s modulus and yield strength
versus radius, allowing for some practical calculations.
The problem of calculating the ball is solved in a centrally symmetric setting.
Thus, all functions depend on one coordinate the radius.

3 Derivation of Resolving Equations

In the presence of central symmetry, from the equilibrium equations in spherical

∂ ∂
coordinates, only the first remains, which, taking into account ∂θ = ∂φ = 0 and
σφ = σθ , and the absence of volume forces and forced deformations, takes the form:

dσr 2
+ (σr − σθ ) = 0. (1)
dr r
The Cauchy relations taking into account v = w = 0 are simplified, as a result
we get:

du u
εr = ; εθ = εϕ = , (2)
dr r
dεθ εθ − εr
+ = 0. (3)
dr r
The angular deformations are identically zero. Hooke’s law in spherical coordi-
nates takes the form:
Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous … 179

1 1
εr = (σr − 2νσθ ); εθ = εϕ = [(1 − ν)σθ − νσr ]. (4)
E E
Expressing σθ from (1) through σr , substituting σθ into (4), differentiating εθ with
respect to and using (3) we can come to the resolving equation
   
1 E ν r 1 1 − 2ν E  4ν 
σr + 4−r − σr − 2 · + σr = 0. (5)
r E 1−ν r 1−ν E 1−ν

In the elastic zone, the resolving Eq. (4) with allowance for ν0 = 0.5 can be
written in the form:
 
E
r σre + 4 − r σre = 0, (6)
E

where the index e denotes the solution for the elastic zone (in what follows, the index
p will be used for the plastic zone).
If we use the Huber-Mises plasticity criterion.

σθ − σr = χ σT (r ), χ = ±1. (7)

The Treska–Saint–Venant plasticity condition gives the same equality.

Then using the equilibrium Eq. (1), it is possible to obtain the resolving equation
for the plastic zone:

dσr p σT (r )
= 2χ . (8)
dr r
The integrals of Eqs. (6) and (8) essentially depend on the form of the functions
E(r )andσT (r ). Below is the calculation method for power functions E(r ) and σT (r ),
which allow approximating a very wide class of real dependences:
  a m E 
E(r ) = E 0 + 1 + (k E − 1) ; (9)
r
  a m σ 
σT (r ) = σT 0 1 + (kσ − 1) . (10)
r
Substituting (9) and (10) in (6) and (8) and integrating these equations, one can
obtain the general form of the solution in the elastic and plastic zones.
 ⎫
a3 k E − 1  a m E +3 ⎪
σr e = C2 + C1 − 3 − ;⎪
⎪
⎬
3r mE + 3 r
 3 (11)
a (k E − 1)(m E + 1)  a m E +3 ⎪
⎪
⎪
σθe = C2 + C1 + .⎭
6r 3 2(m E + 3) r
180 V. Andreev and M. Maksimov
 ⎫
r k T − 1  a m σ ⎪
σr p = D + 2χ σT 0 ln − ; ⎪
⎬
a mσ r
 (12)
r 1 (k T − 1)(m σ − 2)  a m σ ⎪
σθ p = D + 2χ σT 0 ln + + .⎪
⎭
a 2 2m σ r

Before proceeding to the definition of arbitrary C 1 , C 2 constants and D, it is neces-

sary to find the boundaries separating the zones of elastic and plastic deformations.
Calculating from (11) the difference between the principal stresses and satisfying
condition (7), we arrive at the equality

C1  a 3 1 + (k E − 1)(a /r )m E
(r ) = = 1. (13)
2χ σT 0 r 1 + (kσ − 1)(a /r )m σ

Now it is easy to find an equation for determining the radius r T , where the first
plastic deformations occur. From the condition for the maximum of the function
(r ), we arrive at the equation
 m σ  m E
a a
3 + (3 − m σ )(kσ − 1) + (3 + m E )(k E − 1)
rT rT
 m E +m σ (14)
  a
+ 3 + m E + m σ (kσ − 1)(k E − 1) = 0.
rT

Depending on the values k E , kσ , m E and m σ behavior of the function (r ) can

be significantly different.
Figure 1 shows several options for the behavior of the functional part (r ).
The dotted line shows the boundaries of the body, and note that if b → ∞,
then k2 = a b → 0. It can be seen from the four graphs that (r ) may not have
a maximum (curve 1), and in the case of an extreme point, it can lie both within
the interval (a, b) (curve 3) and outside it (curves 2 and 4). In the last two cases,
obviously, plastic deformations occur either on the inner or outer surface of the body.
In the absence of a maximum (curve 1), the highest value of (r ) at r = a should
be determined. Functions with a minimum are also possible, for example, when
m E < −(m σ + 2). In this case, plastic deformations can occur sequentially on the
surfaces of the body and close with increasing loads.

Fig. 1 Change in the nature

of the function (r ) for
different values of the
parameters of inhomogeneity
Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous … 181

To determine the pressure difference at which plastic deformations appear for the
first time, one should find a constant for a completely elastic solution, after which,
from condition (14), we obtain

1 − k23 kE − 1  
pT = ( pa − pb )T = + 1 − k2m E +3
3 mE + 3
   m 
2χ σT 0 1 + (kσ − 1) a r T σ
·   m   3 . (15)
1 + (k E − 1) a r T E a r T

From the last equality, you can also determine χ , since the sign of the expression
in the first square brackets, given k E , m E and k2 is known. So, for example, when
m E > 0, χ = sign( pa − pb ).
Depending on the place where plastic deformations occur, the further course of the
solution will be different. Two cases are considered below: r T = a and a < r T < b.
The rest of the cases will not differ significantly from those considered.
Before proceeding to the study of the development of plastic deformations, it is
necessary to write out the boundary conditions that must be satisfied by solutions
(11) and (12) for a ball. At the boundaries of the body, the stresses are:

r = a, σr = − pa ; r = b, σr = − pb . (16)

In addition, at the boundaries (r = r T,i ) separating the elastic and plastic zones
(there can be one or two such boundaries), the following conditions must be met:
⎧
⎨ σr e = σr p ;
r = r T,i σ − σr e = χ σT ; (17)
⎩ θe
ue = u p .

Here the second equality means the condition for the transition of the material
from an elastic state to a plastic one, and u– radial displacement.

4 The Appearance of Plastic Deformations on the Inner

Ball Surface (rT = a)

In this case, with an increase in the pressure difference p = ( pa − pb ), the plastic

zone extends into the depth of the ball wall. Let us denote the so far unknown radius
of the sphere separating the elastic and plastic zones through r0 (Fig. 2). Substituting
solutions (11) and (12) into (16) and into the first two equalities (17), one can obtain
four relations for determining the constants C1 , C2 , D and r0 , one of which will be
transcendental:
182 V. Andreev and M. Maksimov

Fig. 2 Layer radii

  m σ    
p r0 kσ − 1 a 1 a 3
= ln + 1− + − k23 +
2χ σT 0 a mσ r0 3 r0
      m  (18)
kE − 1 a m E +3 m E +3 r03 1 + (kσ − 1) a r0 σ
+ − k2    m  .
mE + 3 r0 a 3 1 + (k E − 1) a r0 E

Since Eq. (18) is resolved with respect to p = pa − pb , then, setting different

values r0 on the interval (a, b), it is possible to construct a dependence p(r0 ) from
which the required radius r0 is determined for any value. After that, the constants are
easily found C1 , C2 and D:
  3    m 
2χ σT 0 r0 a 1 + (kσ − 1) a r0 σ
C1 =   m ;
1 + (k E − 1) a r0 E
 3 
k2 k E − 1 m E +3
C 2 = − pb + C 1 + k ;
3 mE + 3 2
kσ − 1
D = − pa + 2χ σT 0 .
mσ

Thus, the stressed elastic–plastic state of a thick-walled cylinder and a ball in

the considered case (r T = a) can be determined without using the third boundary
condition from (17). It is necessary when determining the displacements, which are
equal u e = r εθe in the elastic zone, and εθe is determined from Hooke’s law. Having
done the appropriate calculations, we get:

C1 a 3
ue = .
4E 0 r 2

In the plastic zone, integrating the material incompressibility condition 2 du

dr
+ ur =
0, find displacement: u p = B/r . The integration constant B is determined from the
2

third boundary condition (17):

Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous … 183

C1 a 3
B= , (19)
4E 0

which allows you to write a unified formula for displacements in elastic and plastic
zones.

5 The Appearance of Plastic Deformation Inside the Ball

(a < rT < b)

With the formation of plastic deformations inside the wall, a further increase in the
pressure difference p leads to an expansion of the plastic zone in both directions, until
one of the boundaries of this zone coincides with one of the surfaces of the body, and
then until the body is completely transformed into the plastic state Denoting a smaller
radius of the plastic zone r1 , and a larger one r2 , (Fig. 3) and satisfying the boundary
conditions (16) and (17),one can find eight relations for determining the unknowns
C1 − C4 , D, B, r1 and r2 d where C3 and C4 are the constants of the solution of the
type (11) for the outer elastic zone. It should be noted that, in contrast to the previous
case, it is not possible to find stresses here without considering displacements, since
otherwise for seven unknowns (excluding B) there will be only six boundary condi-
tions. The solution of a system of eight equations is somewhat more complicated
than in the case considered above, since, along with one transcendental equation,
another nonlinear relation appears that connects r1 and r2 :
 3   m   m
r1 1 + (kσ − 1) a r1 σ 1 + (k E − 1) a r1 E
  m =   m .
r2 1 + (kσ − 1) a r2 σ 1 + (k E − 1) a r2 E

Obviously, this equation is satisfied at r1 = r2 , which corresponds to the moment

when plastic deformations appear (r = r T ). In addition, this equation must also have
solutions for a ≤ r1 ≤ r T and r T ≤ r2 ≤ b. By setting different values r1 on the
interval (a, r T ) (the value r T is determined in advance from (14)), the corresponding

Fig. 3 Layer radii

184 V. Andreev and M. Maksimov

value r2 can be numerically determined. If in this case the plastic zone reaches the
outer surface earlier, then r1 should be determined by value r2 .
Each pair of values r1 , r2 corresponds to a pressure difference p, which is
determined by the formula
 m σ  m σ    3  3 
p r2 kσ − 1 a a 1 a a
= ln + − + 1− + − k2
3
2χ σT 0 r1 mσ r1 r2 3 r1 r2
  m E +3  m E +3 
kE − 1 a a m E +3
+ 1− + − k2
mE + 3 r1 r2
   m  
r23 1 + (kσ − 1) a r2 σ
×    m  .
a 3 1 + (k E − 1) a r2 E

Having built the dependence p(r1 , r2 ), it is possible, knowingly pa and pb , to deter-

mine the boundaries of the plastic zone. After that, the rest of the constants are
determined:
  3  
2χ σT 0 r1 a 1 + (kσ − 1)(a/r1 )m σ
C1 = C3 = ;
1 + (k E − 1)(a/r1 )m E
   3 
1 kE − 1 k2 k E − 1 m E +3
C2 = − pa + C1 + ; C 4 = − pb + C 3 + k ;
3 mE + 3 3 mE + 3 2
     
1 a 3 k E − 1 a m E +3
D = C2 + C1 − −
3 r1 m E + 3 r1
  
r1 kσ − 1 a m σ
− 2χ σT 0 ln − .
a mσ r1

The constant B is found from relations (19).

6 Results

Below are some of the results of calculations  performed according to the above
method for various values of the ratio k2 = a b and parameters of inhomogeneity
m E = m σ = 2. Figure 4 shows the graphs of the dependence on k E the place of
formation of the plastic zone in the ball for several values kσ at m E = m σ = 2.
With an increase kσ at a constant value k E , the place of formation of the plastic zone
shifts from the inner surface of the ball into the depth of the wall. An increase k E ,
on the contrary, leads to a decrease in the radius r T . These two facts become clear
from Fig. 5, which schematically shows the moment of the onset of the formation of
plastic deformations in accordance with condition (7).
Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous … 185

Fig. 4 Dependence of the

place of formation of the
plastic zone on k E and kσ in
the ball

Fig. 5 A qualitative
representation of the
conditions for the appearance
of plastic deformations:
—χσT (r ), —σθ − σr

Fig. 6 Dependence the

parameters of elastic
inhomogeneity: a in a
thick-walled ball, b in an
infinite array with a spherical
cavity, ◦ − k E = k ∗E -
(formula 20)

For a plastically homogeneous and elastically inhomogeneous body, the condition

for the formation of plastic deformations on the inner surface of a cylinder or ball
can be expressed by the elementary relation:
mE
k E ≥ k ∗E = . (20)
mσ + 3
186 V. Andreev and M. Maksimov

Fig. 7 Dependence of the

size of the plastic zone on
the load

In Fig. 6 shows the dependences of the pressure difference pT = ( pa − pb )T

corresponding to the moment of appearance of plastic deformations on the parameters
of elastic inhomogeneity for a thick-walled ball and for an infinite array with a
spherical cavity. It can be noted that the influence of the parameter m E is ambiguous.
At values of k E close to zero, an increase in m E can lead to a decrease in pT , and at
higher k E < 1 values the opposite is true. In addition, in the region of small values
of k E , an increase in this parameter leads to a slight increase in pT . It is also seen
that, at k E < 1, the pressure at which plastic deformations arise can be significantly
higher than in the homogeneous case (at k E > 1, the opposite the opposite picture
is observed).
In Fig. 7 shows graphs that determine the change in the dimensions
 of the plastic
zone in the ball from the load, at kσ = 1; m E = 2; k2 = a b = 0.5 and k E = 0.1.
Using this curve for a hollow sphere as an example, a method for determining the
boundaries of the plastic zone (radii r1 and r2 ) at a known load p ∗ is shown.
In elastic–plastic problems, displacements are of considerable interest. Figure 8
shows the dependence on the pressure difference p = pa − pb of the dimen-
sionless displacement of the points of the inner contour of a thick-walled ball at
k2 = 0, 5;kσ = 1;m E = 2 for different values of k E that determine the degree of
elastic inhomogeneity of the material. With an increase in the pressure difference, all
the graphs merge, and the vertical asymptote corresponds to the complete transition
of the cylinder to the plastic state.
Figure 9 shows the dependences of the displacements of the points of the contour of
a spherical cavity in an infinite array loaded with external pressure p. The calculations
took into account both elastic and plastic inhomogeneity of the massif material.

Fig. 8 Dependence of the

displacements of the inner
contour of the ball on
pressure 1 k E = 0.5; 2
k E = 0.3; 3 k E = 0.1
Elastic–Plastic Equilibrium of a Hollow Ball Made of Inhomogeneous … 187

Fig. 9 Dependence of
displacements spherical
cavity contourin the array
from pressure: 1 -k E = 0.5,
kσ = 0.5; 2 -k E = 0.1; kσ =
1; 3 -k E = 0.4, kσ = 1; 4 -k E
= 0.6, kσ = 1; 5 -k E = 1; kσ
= 1; 6 -k E = 0.5, kσ = 2

It can be noted that for a plastically homogeneous material (kσ = 1) with an

increase in pressure, the influence of elastic inhomogeneity, as in the case of a thick-
walled cylinder, decreases. This is explained by the fact that more and more of
the massif is involved in the work, and the relative fraction of the zone of elastic
heterogeneity, which are local, decreases.
In turn, plastic inhomogeneity has a more significant effect on displacement,
which is especially noticeable at high pressures. This fact is due to the fact that
displacements are highly dependent on the size of the plastic zone, i.e. on the radius
r0 , and the latter essentially depends on the values of the parameters of the plastic
inhomogeneity kσ and m σ .

7 Conclusions

In conclusion, one should pay attention to the fact that the solution of elastoplastic
problems for elastically and plastically inhomogeneous bodies, or rather, the analysis
of the occurrence of plastic deformations in such bodies, is largely similar to the
solution of strength problems. Since the plasticity criteria of Tresk—Saint–Venant
and Huber–Mises are equivalent to two widespread theories of strength, respectively,
the theory of maximum tangential stresses and the energy theory, determining the
place of occurrence of the first plastic deformations and the corresponding loads
allows solving the strength problem at variable values of ultimate stresses.
188 V. Andreev and M. Maksimov

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Assessments of Solutions of the Uniform
Elastic Boundary Value Problem
in the Tip Area of a Boundary
Wedge-Shape Notch

Lyudmila Frishter

Abstract Assembled structures are distinguished in junction areas of elements of

materials with different mechanical properties, under impacts of forced deforma-
tions, breaking along the contact line (surface) of the elements, by stress–strain state
(SSS). A study of SSS in areas with wedge-shape boundary notches, of the junction of
structural elements under impact of forced breaking deformations is a vital practical
problem of the engineering design. The stress–strain state in boundary wedge-shape
notch areas is obtained experimentally on photoelasticity method models and distin-
guished by significant stress gradients. In experiments, the interference stripes pattern
in such areas is poorly legible. Theoretical studies of SSS in irregular boundary point
area are generally reduced to solving of the singular uniform elastic boundary value
problem. The complexity of the SSS in irregular boundary point area determines the
demand for a complex approach to the study, including experimental data extrap-
olation method development and solution assessment of the boundary value in the
irregular boundary point area. It is the purpose of the study described herein to eval-
uate the solution of the uniform elastic boundary value problem in the tip area of a
wedge-shape notch of the boundary. The area, in which the solution assessment of
the elastic value problem is given, is determined by photoelasticity method model.

Keywords Corner zone of the boundary of structure · Stress–strain state ·

Photoelasticity method · Assessments of solutions

1 Introduction

Boundary value problems for elliptical equations in areas with irregular boundary
points are subject of numerous studies [1–9]. The fundamental paper by Kondratyev
[1] presents the solution of the general elliptical problem in the irregular boundary
point area as asymptotical series and infinitely differentiable function. Members
of this series are the solutions of uniform boundary value problems for wedge or

L. Frishter (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow, Russia 129337

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 189
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_17
190 L. Frishter

cone model areas. Papers [1–7] analyze the solution of the uniform boundary value
problem in the vicinity of an irregular boundary point as ascending power series.
Papers [8, 9] consider a local curvilinear coordinate system for SSS study in the
vicinity of an irregular point in a special boundary line of an elastic body. During
the approach to an irregular boundary point in outward direction, the solution of the
elastic value problem is reduced to solving of two uniform planar problems: plane
deformation and antiplane strain or transverse shear.
Paper [4] introduces the notion of canonical singular problem characterizing the
specifics of SSS in the vicinity of an irregular boundary point, whereas the following
two theorems are proven. Any canonical singular problem has a matching transcen-
dental equation, each root of which is corresponded by a uniform solution with the
number of arbitrary valid constants therein equal to the multiplicity of the root. In the
infinitesimal vicinity of a special point, the solution of a correct boundary problem of
the elasticity theory behaves as an eigenfunction of the respective canonical singular
problem, being the asymptotical maximum by the absolute value.
Analytical solutions of elastic value problems in irregular point zone of the area
boundary line are characterized by solution singularity stipulated by the idealized
boundary shape.
Experimental and theoretical studies of the stress concentration stipulated by the
boundary shape are presented in papers by G. Neuber, N. G. Savin, R. Peterson,
B. N. Ushakov, I. P. Fomin, N. A. Makhutov, V. V. Vasilyev, V. P. Netrebko and
others.
The specifics of SSS of structures featuring “structural non-uniformity” and
forced breaking deformations is determined on polymer models of the photoelasticity
method and by unfreezing of free thermal deformations [10–13] as stress concentra-
tors. The basics of the photoelasticity method are presented in fundamental papers
[10–16]. The theoretical singular solution of the uniform boundary value problem
in the wedge-shape notch area of the boundary displays in the model as follows. In
the vicinity of the wedge-shape notch tip of the model, the striped pattern is either
illegible or poorly legible at any magnification of the vicinity fragment. At a certain
distance from the strain concentration source, there are trustworthy experimental data
continuously and monotonously changing as they approach the irregular boundary
point. For extrapolation of the trustworthy experimental data to the area where the
striped pattern gets illegible, a complex approach is required to obtain and to analyze
the deflection mode in the vicinity of the irregular boundary point of the planar area.
The tip area of the boundary notch subject to SSS assessment is determined by
the striped pattern analysis in the photoelasticity method model allowing for the
extrapolation of the experimental data [15, 17–20]. Thereby, the power nature of the
similarity of the stress and the strain is taken into account [15, 17–20].
It is the purpose hereof to evaluate the solution of the uniform boundary value
problem in the tip area of the wedge-shape notch of the boundary adjoining the
singular solution area with illegible striped pattern.
Assessments of Solutions of the Uniform Elastic Boundary … 191

2 Method of Problem Solving

2.1 Uniform Planar Elastic Boundary Value Problem

A solution of the elasticity theory problem is considered for a homogeneous or a

piecewise homogeneous body in the vicinity of an irregular boundary point of a
planar area being the point of the finite discontinuity (leap) of forced deformations.
In accordance with the theoretical analysis [4, 5, 15, 18, 20], the solution of the
elasticity theory problem η in the vicinity of the irregular point of a planar area
boundary can be presented as follows:

σi j = σioj + σilj , εi j = εioj + εil j , Ui = Uio + Uil

or

η = η o + ηl , (1)

where ηo = (σioj , εioj , Uioj ) is the solution of the uniform boundary value problem
in the area of the irregular boundary point characterizing the solution specifics;
ηl = (σilj , εil j , Uilj ) is SSS stipulated by the specified loads impact and dependent
on the geometrical parameter of the “degree of proximity” to the special point. The
presentation (1) is true also in a stereometric case for points on a special line of the
area boundary.
SSS of appearance (1) defines two self-balanced stress states in the wedge-shape
notch of the boundary. The first stressed state (SS) is obtained as a solution of the
planar uniform boundary value problem in the vicinity of the irregular point of the area
boundary transforming into a singular SS while getting closer to the irregular point
of the boundary from inside the area. A non-trivial solution of the uniform boundary
value problem characterizes the specifics of the SS in the vicinity of the irregular area
boundary point, will be called “proper”. The other part of the self-balanced planar
SS in the tip area of the wedge-shape notch of the boundary corresponds to the stress
occurring due to the impact of the specified loads to the total stress field.
The analysis of the correlation between the summands η = ηo + ηl of SSS
presentation appearance (1) allows for highlighting of the following characteristic
SSS action areas.
(a) The vicinity of the irregular point of a planar area boundary for which a singular
solution of the uniform boundary value problem is correct, which is character-
ized by σi j → σioj , σilj → 0. The specifics of the stress σioj (deformations εioj )
is of power appearance r Reλ−1 , λ ∈ [0, 0.5]. The orders of the stripes in the
area of the model’s stress concentrator (singular solution areas) are illegible at
any magnification of the irregular point vicinity.
(b) The irregular point vicinity of the area boundary where σi j ≈ σioj , σilj ≈ 0 and
the non-singular uniform elasticity problem is correct with the same “proper”
192 L. Frishter

value min Re λ, as in the singular problem. The non-singular solution area does
not contain the vicinity of the singular solution and the irregular point proper,
but adjoins it. As the deformations approach inwards to the boundary of the area
of the singular stress solution, they change continuously, their values are great
but finite. The stripe orders in the model corresponding to the non-singular
solution area are legible except for some of them.
(c) At a sufficient distance from the irregular boundary point, there is an area where
σi j = σilj , σioj = 0, and the stress is stipulated by the specified loads (total
stress field).
In the non-singular solution area of the uniform planar elasticity problem, it is
possible to apply assessments allowing for the extrapolation of the solutions over
the sections close to the irregular boundary point, under recognition of the experi-
mental data and the practical accuracy of the experimental data measurement with
the photoelasticity method.

2.2 Assessments of Non-Singular Solution of the Uniform

Problem

A small vicinity of O tip of a wedge-shape notch of the planar area boundary in a

special body border line V is considered as a self-balanced part B of the body V in the
following appearance: x 2 + y 2 < ε12 ; z 2 < ε22 ; ε1 , ε2 —small positive numbers.
The limit conditions on surface L of the area are uniform. The initial elasticity
problem in the irregular point vicinity admits the following similarity group:

x1 = t x; y1 = t y; z 1 = z;

σi j = tσi j ; εi j = tεi j ; Ui = Ui ,

where i, j = x, y, z; i1, j1 = x1 , y1 , z 1 ; t is the group parameter, t > 0.

For some small vicinity of a special boundary point, the solution of a correct
boundary problem of the elasticity theory behaves as an eigenfunction of the respec-
tive canonical singular problem, being the asymptotical maximum by the absolute
value. The power feature of the stress, the deformations has the order of r λ0 −1 , where
λ0 = min Reλ is the minimum value of the valid part of the complex root of the char-
acteristic equation of the uniform planar boundary value problem of the elasticity
theory, is determined numerically [2–4, 15].
In the tip area of the wedge-shape notch of the boundary, a vicinity c < r <
(1 + α)c, α ∈ (0, 1), θ ∈ (θ0 , θ1 ) is selected for which the non-singular planar
uniform elastic value problem with uniform limit conditions is correct. The parameter
t = 1c is sufficiently great, the parameter c is sufficiently small, in order to neglect
the influence of the total stress field σilj caused by the specified forced deformations.
Assessments of Solutions of the Uniform Elastic Boundary … 193

The stress function for the uniform elasticity value problem in a polar coordinate
system will appear as follows:

ϕ = r λ0 +1 f (θ ), (2)

where f (θ ) is the variable angle function θ , θ ∈ (θ0 , θ1 ), λ0 = min Reλ is the

minimum value of the effective part of the complex root λ of the characteristic
equation of the uniform boundary value problem in r < ε1 < c; ε1 > 0 area. We
consider the new variable r1 = r − c, r ∈ (c, (1 + α)c), r1 ∈ (0, αc). The stress
function (2) can be put down as:
 r1 λ0 +1
ϕ = (r1 + c)λ0 +1 f (θ ) = cλ0 +1 1 + f (θ ), (3)
c

where 0 < r1
c
< α < 1. For function (3), we apply the following binomial expansion:
 r1 λ0 +1 r 
1
1+ = 1 + (λ0 + 1)
c c
(λ0 + 1)λ0 (λ0 − 1) · · · (λ0 − n + 2)  r1 n
∞
+ . (4)
n=2
n! c

The stress function (3) under recognition of the extension (4) will take the
following appearance:
r 
ϕ = cλ0 +1 [1 + (λ0 + 1)
1
c
(λ0 + 1)λ0 (λ0 − 1)...(λ0 − n + 2)  r1 n 
∞
+ f (θ ), (5)
n=2
n! c

Under recognition of the stress function (5), the stress in the vicinity of the tip of
the wedge-shape notch of the boundary r1 ∈ (0, αc), r ∈ (c, (1 + α)c) will take
the following appearance:

1 ∂ϕ 1 ∂ 2ϕ cλ0 +1 λ0 + 1 (λ0 + 1)λ0
σr = + 2 2 = f (θ ) + (r − c)
r ∂r r ∂θ r c c2
∞

 (λ0 + 1)λ0 (λ0 − 1) · · · (λ0 − n + 2)
+ (r − c)n−1

n=3
(n − 1)!cn
∞

cλ0 +1   (λ0 + 1)λ0 (λ0 − 1) · · · (λ0 − n + 2)
+ 2 f (θ )[1+ n
(r − c)n ,
r n=1
n!c
194 L. Frishter
∞ 
∂ 2ϕ  (λ0 + 1)λ0 (λ0 − 1) · · · (λ0 − n + 2)
λ0 +1
σθ = 2 = c f (θ ) (r − c) n−2
,
∂r n=2
(n − 2)!cn
(6)

∂ 1 ∂ϕ λ0
τr θ = − = −cλ0 +1 f  (θ ) 2
∂r r ∂θ r

 (λ0 + 1)λ0 (λ0 − 1) · · · (λ0 − n + 2) 
∞
c  (r − c)n−1
+ n−1+ .
n=2
n!cn r r

Remainder term (5):

r  |(λ0 + 1)λ0 (λ0 − 1) · · · (λ0 + 1 − n)|  r n+1
(1 + ξ )λ0 −n
1 1
Rn = <δ (7)
c (n + 1)! c

Under recognition of λ0 ∈ (0, 1), ξ ∈ (0, 1), at n ≥ 1 (1 + ξ )λ0 −n <1, the

assessment of the remainder term (7) will take the appearance:
r  |(λ0 + 1)λ0 (λ0 − 1) · · · (λ0 + 1 − n)|  r1 n+1
1
Rn < <δ,
c (n + 1)! c

r −c
where r1
c
= c
< α  1, λ0 ∈ (0, 1); α ∈ (0, 1), or
r  |(λ0 + 1)λ0 (λ0 − 1) · · · (λ0 + 1 − n)| n+1
1
Rn < (α) <δ. (8)
c (n + 1)!

If the apex angle between the tangent lines drawn to the area boundary in the tip
of the boundary notch is decreased, then, the values are λ0 → 0, 5. We now reduce
the assessment of the remainder term Rn rc1 , e.g., for value λ0 = 0, 5:

3 2 3
At n = 1 |Rn | < (α) < = 0, 375.
8 8
1
At n = 2 |Rn | < (α)3 < 0, 0625 < 0, 1.
16
That is why the required accuracy δ = 0, 1 for the stress function is mandatory
only for the first two (three) terms of the extension series at n = 0, 1 (2), that is
  r 
ϕ(r1 , θ ) ∼
= cλ0 +1 f (θ ) 1 + (λ0 + 1)
1
, n = 0, 1,
c
or
 r
ϕ(r, c) ∼
= cλ0 +1 f (θ ) −λ0 + (λ0 + 1) . (9)
c
Assessments of Solutions of the Uniform Elastic Boundary … 195

For stress function (5), in the vicinity c < r < (1+α)c, c ∈ (0, 1), α ∈ (0, 1)
under analysis, stresses (6) will take the following appearance:

c λ0  
σr ∼
= (λ0 + 1)c1 f (θ ) + f  (θ ) , σ̄θc ∼
= 0; τ̄rcθ ∼
= 0.
r
The summands causing a significant increase of the stress, the deformations
(deformation energy increase in the vicinity r ∈ (c, (1 + α)c), αc  1, r12 1
r
) are
not considered. For example, for function f (θ ) = [c0 cos (λ0 −1)θ −c1 cos(λ0 +1) θ ],
the stress will take, at a first approximation, the following appearance:

cλ0 c1
σr ∼
= (λ0 + 1)λ0 [c2 (2 − λ0 ) cos(λ0 − 1)θ
r
+ (2 + λ0 ) cos(λ0 + 1)θ ], c1 , c2 ∈ R,

or
(λ0 + 1)λ0
σr ∼
= c[c2 (2 − λ0 ) cos(λ0 − 1)θ
r
+ (2 + λ0 ) cos(λ0 + 1)θ ], c, c2 ∈ R,
σθ = 0; τr θ ∼
∼ = 0.

For the first three terms of series (5) at n = 0, 1, 2, the stress function will take
the following appearance:
  r  (λ + 1)λ  r 2 
ϕ(r1 , θ ) ∼ λ0 +1 1 0 0 1
= c f (θ ) 1 + (λ 0 + 1) + ,
c 2! c
or
 
λ0 (1 − λ0 ) r λ0 (λ0 + 1)  r 2
ϕ(r, c) ∼
= cλ0 +1 f (θ ) − + 1 − λ20 + . (10)
2 c 2! c

For stress function of appearance (10), the stress in the analyzed vicinity of the
tip of the wedge-shape notch of the boundary will take the following appearance:
 
∼ λ0 +1 2 1 λ0 (λ0 + 1)
σr = c f (θ ) 1 − λ0 +
cr c2
 
λ0 +1  λ0 (1 − λ0 ) 2 1 λ0 (λ0 + 1)
+c f (θ ) − + 1 − λ0 + ,
2r 2 rc 2!c2

σθ ∼
= cλ0 −1 λ0 (λ0 + 1) f (θ ),
 
λ0 (1 − λ0 ) λ0 (λ0 + 1)
τr θ ∼
= −c λ0 +1 
f (θ ) + .
2 r2 2 c2
196 L. Frishter

By selection of the required number of the extension terms of stress function

(5), the stress, the deformations in the non-singular solution area of the elasticity
value problem can be obtained with accuracy corresponding to the data measurement
accuracy of the photoelasticity method. The vicinity of the tip of the boundary notch
of the area, where the assessments of non-singular solution of the uniform problem
(5), (6) are correct, is selected based on the experimental data [15, 18, 20].
In accordance with the accuracy of the photoelasticity method, it is sufficient to
take the first two (three) extension series terms (5) of the stress function at n = 0,1(2).

3 Results

The specifics of SSS of structures featuring “structural non-uniformity” and

forced breaking deformations is determined on polymer models of the photoelas-
ticity method and by unfreezing of free thermal deformations [10–15] as stress
concentrators.
In the tip area of a wedge-shape notch of the boundary adjoining the singular solu-
tion area with illegible stripes pattern, an assessment of the solution of the uniform
planar boundary elasticity value problem was obtained. The stress function for the
determined tip area of the wedge-shape notch is extended to a binomial series (5),
an assessment is given for the remainder term (5) of the series in appearance (8).
The vicinity of the tip of the boundary notch of the area, where the assessments of
non-singular solution of the uniform problem (5), (6) are correct, is selected based on
the experimental data [15, 18, 20]. It was demonstrated that it is sufficient to take the
first two (three) terms of extension series (5) in order to obtain a solution matching
the measurement accuracy of the data on models of the photoelasticity method.

4 Discussion

In the vicinity of the wedge-shape notch tip of the model, the striped pattern is either
illegible or poorly legible at any magnification of the vicinity fragment. At a certain
distance from the strain concentration source, there are trustworthy experimental data
continuously and monotonously changing as they approach the irregular boundary
point.
The vicinity of the tip of the boundary notch of the area, where the assessments of
non-singular solution of the uniform problem (5), (6) are correct, is selected based
on the experimental data. The determining of such area in a photoelasticity model
is not so much carried out based on the obtained interference stripes pattern, as by
means of a correspondence analysis of the distribution of the orders of the power
dependency bands of the distribution in accordance with the theoretical solution for
the uniform elasticity boundary value problem.
Assessments of Solutions of the Uniform Elastic Boundary … 197

5 Conclusions

For extrapolation of trustworthy experimental data into the area where the striped
pattern is illegible, assessments of SSS (5), (6) of the uniform boundary elasticity
value problem in the corresponding tip area of the wedge-shape notch boundary are
applicable. Thereby, the power nature of the similarity of the stress and the strain
should be taken into account.

References

1. Kondratyev VA (1967) Transactions of the Moscow mathematical society. M: MSU, 16:209–

292
2. Parton VZ, Perlin PI (1981) Methods of the mathematical theory of elasticity, M: Nauka, p 688
3. Timoshenko S, Gudyer J (1975) Theory of elasticity, M: Nauka, p 576
4. Cherepanov GP (1974) Mechanics of brittle failure, M: Nauka, p 640
5. Kuliev VD (2005) Singular boundary value problems, M: Nauka, p 719
6. Denisjuk IT (1995) Bulletin of the Russian academy of science. Solid Mech 5:64–70
7. Parton V, Morozov E (2017) Elastic-plastic fracture mechanics: special problems of the fracture
mechanics, M., p 192
8. Aksentyan OK (1967) Appl Math Mech 31(1):178–186
9. Chobanjan KS (1971), S. H. Bulletin of the Academy of Science of the Armenian SSR.
Mechanics. 5(XXIV):16–24
10. Frocht MM (1948) Photoelasticity, in Prigorovskiy NI (ed), Moscow—Leningrad, GITTL, vol.
2, p 432
11. Durelli A, Riley W (1970) Introduction to photoelasticity, in Prigorovskiy NI (ed), M: Mir
12. Aleksandrov AJ, Ahmetzjanov MH (1974) Polarization-optical methods for deformable solid
mechanics, M: Nauka, p 5766
13. Hesin GL et al (1975) The photoelasticity method, M: Stroyizdat, vol 3, p 311
14. Kobayashi A (1990) Experimental mechanics, M: Mir, vol 1, p 615, vol 2, p 551
15. Razumovskij IA (2007) Interference-optical methods of deformable solid mechanics, M.:
Publisher MGTU named after N. Bauman, p 240
16. Matviyenko Y (2006) Fracture mechanics models and criteria, M.: Fizmatlit, p 328
17. Pestrenin VM, Pestrenina IV, Landik LV, Bulletin TGU (2013) Math Mech 4:80–87
18. L. Yu. Frishter, Photoelasticity - based study of stress-strain state in the area of the plain
domain boundary cut-out area vertex. In: Advances in Intelligent Systems and Computing,
EMMFT 2017, Springer, vol. 692, Cham 836–844 (2017) https://doi.org/10.1007/978-3-319-
70987-1_89
19. Makhutov NA, Moskvichev VV, Morozov EV, Goldstein RV (2017) Unification of computation
and experimental methods of testing for crack resistance: development of the fracture mechanics
and new goals. Ind Lab Diagnos Mater 83(10):55–64. https://doi.org/10.26896/1028-6861-
2017-83-10-55-64
20. Frishter L (2018) Capabilities for stress-strain state generation within a stress concentration
zone using the photoelasticity method, EMMFT 2018, V1(982):692–700.https://doi.org/10.
1007/978-3-030-19756-8
Computer Programs Developing
for Solving Problems of Cylindrical
Shells Stability

Stepan Cheremnykh

Abstract Innovative methods, which are developed in a computer program for

solving stability problems in the direction of mechanics of a deformable solid
are proposed. As a result, new solutions to the stability problem according to the
theory of plates and shells under simple and complex loading at the time of bifur-
cation by V.G. Zubchaninov are obtained, which allow us to evaluate the reliability
of known solutions according to other theories. The previously known conclusion
of V.G. Zubchaninov that the classical theory of stability of shells and plates of
A.A. Ilyushina gives similar results for the critical states obtained by taking into
account simple and complex loading during bifurcation. The obtained new solutions
to the problems of stability of cylindrical shells under combined loading are verified
in the laboratory of mechanical tests of the Department of Resistance of Materials,
Theory of Elasticity and Plasticity by comparison with experimental data of various
authors. It is noted that despite the accumulated theoretical and experimental data
in the field of research on the stability and buckling of thin-walled structures, the
problem remains far from being solved and requires further study with the use of
new software systems and will be relevant in the XXI century.

Keywords Stability · Bifurcation · Shell · Trajectory · Tension · Deformation

1 Introduction

The use of the materials safety factor allows designers to admit the occurrence of
plastic deformations in structures [1]. At the same time, taking into account the
elastoplastic deformation stage, it significantly increases the reliability of engineering
calculations (even when they work within the limits of elasticity) due to a more
accurate assessment of the maximum loads and the stability margin coefficients
[2–5].
The use of shell structures in aviation, cosmonautics, construction structures and
mechanical engineering, as well as taking into consideration their operation under

S. Cheremnykh (B)
Tver State Technical University, Naberezhnaya Afanasiya Nikitina, 22, Tver, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 199
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_18
200 S. Cheremnykh

extreme conditions associated with the occurrence of plastic deformations, have

made the study of the elastoplastic deformation laws of structural materials under
complex loading and deformation among the most important and urgent problems
of the deformable solids mechanics and construction mechanics in general [6–10].
As is well known, one of the most difficult problems in the deformable solid
mechanics is the problem of constructing defining correlation beyond the elastic
limit in a complex loading process. Leaving aside purely mathematical problems,
there are two difficulties in describing the loss of stability beyond the elastic limit. The
first is to correctly determine the stress–strain state before the loss of stability. The
second is to identify ratios that determine the correlation between the increments
of the strain tensor and the increment of the stress tensor. At the same time, the
calculation of shells is a complex engineering task and requires patience from the
calculator and knowledge of the mathematical apparatus fundamentals and software
and computing systems [11, 12].
In modern software systems, there are a number of algorithms that solve stability
problems according to various theories, the most modern of which are the stability
theories of A. A. Ilyushin and the theory of stability under complex loading by V. G.
Zubchaninov. But there are no computer programs that fully take into account the
features of complex loading at the moment of shells stability loss beyond the elastic
limit [13].
Based on the regulatory framework of Article 1261 of the Civil Code of the
Russian Federation, «A computer program is a collection of data and commands
presented in an objective form, intended for the operation of a computer and other
computer devices in order to obtain a certain result, including preparatory materials
obtained during the development of a computer program, and audio-visual displays
generated by it».
In this paper, we consider the accordance of the developed computer software
package to theoretical calculations in solving the problem of stability under complex
loading for trajectories in the circle forms and circle arcs.

2 Materials and Methods

The accumulation of new theoretical and experimental data in the study of the thin-
walled elastic–plastic systems stability under complex tense state is important for
the development of effective methods of structures engineering calculation [14–20].
For real structures, the stability problem research is complicated by the complex
loading process, since the determining correlations for complex loading processes
themselves are essentially approximate.
In this case, the problem includes two parts: the theoretical construction of the
subcritical loading process and the solution of the bifurcation problem, where the
values of the stress state components and the modulus of the stress vector are
calculated at each point of the realized a complex subcritical stress trajectory.
Computer Programs Developing for Solving Problems … 201

In the mathematical representation, the equations of the stresses and deformations

dependence in the construction of the loading process image are taken in accordance
with the defining relations of the coplanarity hypothesis, which in velocities have the
form [21, 22]:

(1)

where ϑ1 —approach angle; N , dσ/d S—defining functions of the deformation

process; Ṡ—the rate of change in the arc length of the deformation path. The symbol
with a dot at the top indicates differentiation by a generalized time parameter dtd =
d dS
d S dt
.
The defining plasticity functions are described by V. G. Zubchaninov approxima-
tions [21, 22]:
 
  1 − cos ϑ1 p
N = 2G p + 2G − 2G p ; (2)
2
 
dσ 1 − cos ϑ1 q
= 2G k − [2G + 2G k ] , (3)
dS 2

where G, G k , G p —shear modulus, tangent and secant shear modulus of the material,
respectively; p, q—material approximation parameters determined from experiments
on a flat fan of two-link trajectories. These approximations were tested on flat multi-
link polyline and curved trajectories [13, 21, 22].
The Eqs. (1) in the expanded form have the form (4):

Ṡ
Ṡ11 = N ε̇11 + (dσ/d S − N cos ϑ 1 ) S11 ;
σ
Ṡ
Ṡ12 = N ε̇12 + (dσ/d S − N cos ϑ 1 ) S12 ;
σ
Ṡ
Ṡ22 = N ε̇22 + (dσ/d S − N cos ϑ 1 ) S22 , (4)
σ

where
√  
Ṡ = 2 · ε̇11
2
+ ε̇22
2
+ ε̇11 ε̇22 + ε̇12 ;

2
σ = · σ11
2
+ σ22
2
− σ11 σ22 + 3σ12 2
.
3

If in Eqs. (4) we pass from the components of the stress-deviator tensor to the
components of the stress tensor, we get [21, 22]:
202 S. Cheremnykh

σ̇11 = N (2ε̇11 + ε̇22 ) + (dσ/d S − N cos ϑ1 ) Ṡσ11 /σ ;

σ̇22 = N (2ε̇22 + ε̇11 ) + (dσ/d S − N cos ϑ1 ) Ṡσ22 /σ ;
σ̇12 = N ε̇12 + (dσ/d S − N cos ϑ1 ) Ṡσ12 /σ. (5)

To determine the angle of convergence, we have

 
σ sin ϑ1
ϑ̇ = − − χ1 , (6)
N

where χ1 —the curvature of the trajectory.

Equations (5) and (6) have the form of equations of the Cauchy problem, which we
solve by the Runge—Kutta method. We believe that the dependence
is universal for simple loading.
Stresses and deformations in the shell are calculated by the formulas:

σ11 = FP = 2 πPRh ; σ22 = qhR ; σ12 = 2 πMR 2 h ;

(7)
σ33 = 0; σ13 = σ23 = 0; σ0 = 13 (σ11 + σ22 );
l R ϕR
ε11 = ; ε22 = ; 2ε12 = γ = , (8)
l R l

where F = 2π Rh—cross-sectional area of a thin-walled tubular sample; h—wall

thickness; R—radius of the median surface of the shell (tubular sample); l—length
of the working part of the sample; ϕ—angle of twist; P—axial force; M—torque
output; q—the intensity of internal pressure.
The solution of these equations is quite lengthy, for example, the Runge–Kutta
method is most often used and implemented in various mathematical packages
(Maple, MathCAD, Maxima) to facilitate the calculation process itself.
That is why, to solve the problem of shell bifurcation, taking into account the
complex nature of deformation at the moment of loss of stability under complex
subcritical loading, for a trajectory in the form of circles and arcs of circles, it
was proposed to perform a solution in the Visual Basic for Applications (VBA)
programming language.

3 The Study Results

The program calculates the cylindrical shells stability made of materials with complex
mechanical properties, and can be used in construction structures and mechanical
engineering. The complex substantiation problem of the applied cylindrical shells
theory of stability is solved taking into account complex loading. The calculation
in the—program is performed according to various stability theories, considering
Computer Programs Developing for Solving Problems … 203

the unloading of the material. The program solves the problem of substantiating
the applied theory of stability of cylindrical shells counting complex loading, while
calculating the parameters of stability under compression, torsion and combined
loading of the material.
The calculations were performed for the experimentally implemented processes
by M. Yu. Alexandrov on steel 45 shells [22].
The two-link trajectory shown in Fig. 1 is realized when stretching to a given
process at R = 1.5% on the first link and then reaching the trajectory of radius R on
the second section [22].
During computer calculation as input a description of arrays, the values of the
curvatures of the trajectories, the number of points, the description of the coefficients
of the deformation diagram, description of the zero point (the inflection point), the
end of the description of the zero point and display the results on the breakpoint.
Next, when moving to the first section (the arc of the circle), the iterative process,
errors, and initial data for the bifurcation are set. The calculation of the initial
conditions at zero approximation, algebraic equation is solved with purely plastic
bifurcation are determined intermediate parameters of the zero-order approxima-
tion, the calculation of integrals and the solution of quadratic equations in the first
approximation, determined by the intermediate parameters of the first approximation.
For the second section (straight line—compression), the constant torsion strain
is determined, and the initial data for the bifurcation is set. The initial conditions
are calculated in the zero approximation, and an algebraic equation is solved for a
purely plastic bifurcation. The intermediate parameters of the zero approximation
are determined, the integrals are calculated and the quadratic equation is solved in
the first approximation, and, just as in the first section, the intermediate parameters
of the first approximation are determined.
Figure 2 shows diagrams based on the data obtained in the computer program
for critical voltages. The figures in Fig. 2 indicate the calculations of the shell at
the moment of loss of stability: 1, 2—according to the theory of stability of A. A.
Ilyushin; 3—taking into account the complex loading using for the approximation
functional dσ/d S(2) and (3), and for the functional N—expressions N = 2G(1−ω);
4—taking into account the complex loading at p = q = 1; 5—by p = q = 0,5. The
triangle shows the loss of stability of the sample during the experiment.

Fig. 1 Deformation
trajectory
204 S. Cheremnykh

Fig. 2 Graphs of the least

flexible shell

Fig. 3 The stress path

Figure 3 shows the loading trajectory of the shell corresponding to the performed
deformation trajectory. The solid line reflects the solution of the problem of
constructing an image of the loading process. The moment of loss of stability in
the experiment and the calculated one are shown in the figure by arrows.

4 Conclusion

The qualification of a mechanic is mainly determined by the ability to discard all

the secondary things and make the study as simple as possible. The mathematical
complexity of the statement is not an end in itself. The mathematical study should
be, if possible, extremely clear. It is for this purpose that this program was developed,
where the use of the software package used can significantly reduce the labor costs
Computer Programs Developing for Solving Problems … 205

of the researcher, eliminate unexpected errors and increase the level of automated
processing of experimental material.

References

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3. Bochkarev SA, Matveenko VP (2012) Mech Solids 47(5):560–565
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3860
5. Sofiyev AY (2018) Composite structures, 206:124–130
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Combust Explosion Shock Waves 55(4):447–455
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Vasilenkov NA, Volkova OS, Shakin A (2016) J Magn Magn Mater 398:49–53
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45 under complex loading XXII Int. Scientific Conf. «Construction the Formation of Living
Environment» (FORM-2019), 97:1–8
22. Cheremnykh SV, Skudalov PO (2020) IOP Conference Series 786:012011
Estimation of the Vibration Waves Level
at Different Distances
Mirziyod Mirsaidov, Muhammadbobir Boytemirov,
and Faxriddin Yuldashev

Abstract The article is devoted to the study of the vibrational waves propagation and
the estimation of the level of their influence on various objects. The development of
adequate mathematical models, methods for the estimation of vibration wave levels
at various distances from the earth foundation is an urgent task. In this article, a math-
ematical model was developed using a variational approach to study the propagation
of vibration waves induced by railway transport at various distances. A technique
for solving the considered problems was developed using the finite element method.
The effect of the level of vibrations propagating from the railway transport motion
on buildings located at a certain distance from the vibration source was investigated,
when the railroad bed is located at the foundation level or at a certain height from
the foundation. It was stated that if the railroad bed is located at a height of 2 m
from the level of the earth foundation of the building, then the amplitude of vibration
displacements in the building can be reduced from 1.5 to 3.5 times.

Keywords Vibration wave · Railroad bed · Half-plane · Amplitude reduction ·

Earth foundation · Building · End area · Viscous dampers

1 Introduction

An increase in the railway transport traffic, and an increase in the speed of motion,
leads to an increase in the level of ground vibrations around the railway bed. Vibra-
tions induced by railway transport lead to a number of negative phenomena, such as
a decrease in the strength of nearby located residential buildings, structures, dete-
rioration of human life, and the productivity reduction of the population living and
working in the area.
Many researchers studied the influence of vibration waves propagating from the
vehicle motion and their effect on buildings and structures.

M. Mirsaidov (B)
Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Tashkent, Uzbekistan
M. Boytemirov · F. Yuldashev
Namangan Engineering Construction Institute, Namangan, Uzbekistan

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 207
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_19
208 M. Mirsaidov et al.

For instance:
• in [1], the patterns of vibration propagation in ground near a railway bed resting
on peat soils were considered. On the basis of numerous experiments, quantitative
data were obtained for the vibrodynamic effect transferred to the foundation and
peat base by train motion. The dependencies of vibration damping over the depth
and in the transverse direction to the train motion were obtained. On the basis
of experimental data, an analytical dependence was obtained to calculate the
amplitudes of vibrations at any point of the foundation, resting on a peat base.
• in [2], the speed and acceleration of the foundation vibrations of a three-story
building, located at a distance of 10 m from the railway track, were investigated.
• the method of modeling the vibration impact of a single vehicle on the foun-
dation was considered in [3]. During the study, the type of load induced by the
vehicle motion was established and the methods of accounting for the load during
modeling and calculation were indicated. Based on the calculations and analysis,
conclusions were drawn on the nonlinear dependence of the vibration acceleration
of the foundation on the distance to the vibration source. It was shown that the
acceleration depended not on the mass of the vehicle, but on the maximum load
on the vehicle axle. It was established that the greatest vibrations were induced by
vehicles on the foundations of clayey soils, and smaller vibrations were observed
on the foundations located on sandy soils.
• vibrations induced by trains passing along the Tojala-Turku railway line (in south-
western Finland), in a residential area and adjacent areas, and their impact on the
quality of life of residents were investigated in [4]. To reduce vibration, the walls
of sheet piles and a matrix of lime-cement columns were built. The soil in this
area consisted of a thick clay layer with low shear strength, which allowed large
displacements of the railroad bed. Such a soil with a low dynamic character-
istic leads to minimal energy damping and wide propagation of vibration in the
surrounding area. The measurement results showed that the measures taken to
reduce vibration led to a 30 to 50% decrease only.
• in [5–7], a number of problems were solved on the vibration in soils, on the
influence of the railway track location on the level of propagating vibrations.
• the study in [8] simulateed a dynamic process under moving load on rails with
sleepers. The influence of the train speed on the dynamic properties of the
elements of the railway track was investigated. Stationary dynamic analysis was
conducted using the finite element method, in which the sensing functions were
used as a criterion for assessing the dynamic properties of the railway compo-
nents. The resulting displacement of the caterpillar track was compared with the
experimentally measured displacements.
Besides, in [9–15], various models, methods and results of solving some problems
related to the wave propagation from a source into the earth foundation and the wave
effect on structures, buildings, and their protection from wave effects were presented.
These are just a few publications related to the propagation of waves in the earth
foundation and their impact on structures, which consider some measures designed
to reduce the level of vibration waves on the objects under consideration.
Estimation of the Vibration Waves Level at Different Distances 209

The above review of publications shows that the modeling and study of the
process of vibrational wave propagation and their effect on various objects differ
in different studies, and each theory, model, or method used has its own advantages
and disadvantages.
Therefore, the development of adequate mathematical models, research methods
for the propagation of vibration waves in earth foundations and mitigating their effect
on various objects are an urgent task.

2 Methods

This study focuses on reducing vibration levels from railway traffic. To consider this
problem, a mathematical model is created; the system is considered, consisting of
two symmetrically located identical bearing-wall buildings on an infinite deformable
half-space (i.e., on earth foundation). Between these two buildings, a railroad track
runs 2 m above the foundation. A building is considered as a deformable body. To
formulate the problem, a finite region of a volume V 1 + V 2 with boundaries 1+ , 2+ ,
1− , 2− is cut out of half-space. Viscous dampers are set on the boundary of the finite
region 1+ , 2+ [1, 9, 10, 16, 17], excluding wave reflections from these boundaries.
The problem under consideration is posed for a plane-deformed state of the system.
To simplify the problem, symmetry conditions are used and the problems are posed
for one part (Fig. 1) occupying the volume V 1 , with the boundaries 1+ , 2+ of the
symmetric system, occupying the total volume V = V 1 + V 2 . At the same time, on
the considered part of the symmetrical foundation of volume V 1, half of the railway
line passes (Fig. 1). It is assumed that periodic load P = P0 · eiωt acts on this part
of the symmetrical foundation on the site (roadbed) Sp . The task is to determine the

Fig. 1 Design model of the symmetric system (a half) considered

210 M. Mirsaidov et al.

components of displacements at various points of the system due to the propagation

of vibration waves from a source located at the site Sp . At the boundary surfaces
between the elements of the system, (i.e., between the foundation and the building),
there are continuous displacements, stress components normal and tangential to the
boundary surface.
To describe the dynamic processes occurring in the system (Fig. 1), the principle
of virtual displacements is used, according to which the sum of the work of all active
forces, including the forces of inertia, on virtual displacements is zero, i.e.:
  
δA = − σi j δεi j dV − ρn u¨ δ u dV + σi j ν j δ udΣδ
V1 V1 Σ1+ +Σ2+
 
+ γn δ u˙ dV + pδ ud S = 0 (1)
V1 Sp

The results of experimental studies show [4, 8, 18] that during the railway transport
motion, ground vibrations of small amplitude occur according to a harmonic law.
Therefore:
• to describe the physical properties of a body, relations between the components
of stress σi j and strain εi j tensor are used in the following form [19]:

σi j = λn εkk δi j + 2μn εi j (2)

λn and μn are the Lame constants (n is the number of system elements: of the
foundation n = 1, of the building n = 2.
• the Cauchy relations are used, connecting the components of strain tensor εi j with
the components of displacement vector u [19]:
 
1 ∂u i ∂u j
εi j = + , i = 1, 2 (3)
2 ∂x j ∂ xi

• viscous dampers are used that eliminate wave reflections from the boundary of
the final region of the foundation V 1 [2, 9, 10, 20]:

x ∈ Σ1+ : σ22 = αρ1 c1 u̇ 2 , σ21 = βρ1 c2 u̇ 1

x ∈ Σ2+ : σ11 = αρ 1 c1 u̇ 1 , σ12 = βρ1 c2 u̇ 2 (4)

Here: u, εi j , σi j are the components of the displacements vector u = {u 1 , u 2 } =

{u, v}, strain and stress tensors, respectively; δ u, δεi j are the isochronous variations of
displacements and strains; {u̇ 1 , u̇ 2 }− are the velocity projections of boundary points;
γn , ρn are the attenuation coefficient and material density of the n-th element of the
system, respectively; {α, β} are the dimensionless coefficients [2, 17, 21]; p is the
Estimation of the Vibration Waves Level at Different Distances 211

periodic force generated by railway transport; ν j are the direction cosines of the outer
normal; c1 , c2 are the velocities of longitudinal and shear waves propagation in the
earth foundation; the components of the displacements vector u = {u 1 , u 2 } = {u, v}
are given in the coordinate system x = {x1 , x2 } = {x, y}; i,j = 1,2.
Now the variation problem of the steady-state forced vibrations of the system
under consideration (Fig. 1) is reduced to determining the displacement field at
various points of the system (Fig. 1) under periodic influences induced by the railway
transport motion p(t), satisfying Eqs. 1–3 taking into account Eq. (4) for any virtual
displacement δ u.

3 Results and Discussion

Now, using this model, we analyze the efficiency of railway tracks located on the
surface and at a height of 2 m from the ground level.
The objects of research are two identical reinforced concrete buildings, located
at a distance of 20 m from the railway track. The foundations of the buildings are
located at a depth of 2 m from the road level. The buildings are designed as two-story
structures with a basement (Fig. 1).
The above-considered variational problems are solved using the finite element
method. This allows us to reduce the considered problems to a system of large-order
ordinary differential equations, i.e.:

[M]{ü(t)} + [C]{u̇(t)} + [K ]{u(t)} = {P(t)} − []{u̇} (5)

Here [M], [C] and [K] are the matrices of masses, damping and rigidity of the
system, respectively; {u(t)}, {P(t)} are the displacements of nodes and vectors of
acting forces; [G] is a diagonal matrix that takes into account the boundary conditions
[3, 17, 18].
When solving the problem, the following physical and mechanical characteristics
of the foundation and the building were used: for the foundation (of gravel-sandy
soil) modulus of elasticity E = 28.5 MPa; Poisson’s ratio—ν = 0.35; density—ρ =
1850 kg/m3 . For reinforced concrete E = 0.2 • 105 MPa; ν = 0.15; ρ [M]{ü(t)} +
[C]{u̇(t)} + [K ]{u(t)} = {P(t)} − []{u̇} = 2500 kg/m3 .
If we assume that a harmonic load of angular frequency ω acts on the site Sp , i.e.

{P(t)} = {P0 }eiωt (6)

then periodic solution of the system of Eqs. (5) is sought in the following form:

{u(t)} = {u} · eiωt

{u̇(t)} = iω{u} · eiωt (7)
{ü(t)} = −ω2 {u} · eiωt
212 M. Mirsaidov et al.

The substitution of (6) and (7) into (1) leads to the solution of a system of large-
order algebraic equations with complex coefficients:

[K ]{u} + iω[C]{u} − ω2 [M]{u} = {P0 } (8)

Here {ū} is the vector of the vibrations amplitude of the points of the system;
{ P 0 }—is the vector of external load amplitude.
The system of Eqs. (8) with complex coefficients is solved by the Gauss method
[22]. As a result of solution, we obtain the components of the displacement vector
for each point of the system under consideration (Fig. 1):

{u(t)} = Re{u} cos ωt + I m{u} sin ωt (9)

The amplitudes of displacements of various points of the building (i.e., the ampli-
tudes of displacements of points a, b, c, d, k) were investigated at the load frequency
ω = 20H z. Table 1 shows the obtained amplitudes of displacements of points a, b,
c, d, k of the building (Fig. 1), located on different floors.

Table 1 Amplitudes of displacements at various points of buildings

Floors Characteristic Amplitude (A0 ) of Amplitude (A2 ) of (A2 /A0 ) ratio
points vertical the vertical
displacements of displacements of
the building points the building points
(when the rails are (when the rails are
laid at the laid at a height of
foundation level) 2 m from the
foundation level)
1 2 3 4 5
1stfloor, frequency a 0.17126 0.07057 0.41
20 Hz b 0.02154 0.03692 1.7
c 0.11169 0.05521 0.49
d 0.03397 0.02465 0.72
k 0.01044 0.00153 0.14
2nd floor, a 0.18061 0.06715 0.37
frequency b 0.04138 0.0358 0.86
20 Hz
c 0.07741 0.05465 0.7
d 0.01693 0.02036 1.2
k 0.00792 0.00259 0.32
On roofing system, a 0.19068 0.06639 0.34
frequency b 0.08796 0.03715 0.42
20 Hz
c 0.02568 0.05449 2.12
d 0.00244 0.00245 1
k 0.00773 0.00224 0.29
Estimation of the Vibration Waves Level at Different Distances 213

Fig. 2 The absolute value of

the amplitude of vertical
displacements of the
foundations located at
various distances from the
source

Fig. 3 The absolute value of

the amplitude of the vertical
displacements of the
foundations located at
different distances from the
source

As seen from the results of Table 1, when the railroad track is at a height of 2 m
from the level of the building foundation, the amplitude of displacements at various
points of the building decreases from 1.5 to 3.5 times.
The amplitude of displacements of the points of the building was investigated
when the foundation consisted of sandy-gravel soils such as loam, loess, and sandy
loam. The results obtained showed that, in this case, the amplitude of vibrations of
the points of the building was approximately reduced by five times (when the railroad
track was located at a height of 2 m from the level of the building’s earth foundation).
The results from Table 1 show that when the railroad bed is at a height of 2 m
from the level of the building foundation, there are a 1.5 to 3.5 times decrease in the
amplitude of displacements at various points of the building, compared to the case
when the railroad bed is located at the level of the foundation.
The displacements amplitudes of vibrations of various points of the structure
were investigated under vibrational influences with the frequency of vibration ω =
20 ÷ 50H z. It was determined that the amplitude of vibrations of the point of the
building had a similar pattern.
214 M. Mirsaidov et al.

Figures 2and 3 show the absolute values of vertical displacements obtained at

various points of the foundation located at different distances from the vibration
source (i.e., from the roadbed of a moving railway transport) at a vibration frequency
(ω = 20H z): ( ) blue line for the case when the railway bed is located 2 m
above the earth foundation; ( )—red line for the case when the railway bed is
located at the level of the earth foundation.
The envelope of the vibration amplitude of the ground surface damps with the
distance from the axis of the railway bed and has a non-monotonic character.
The studies conducted show that if the railroad bed is located above the earth
foundation, then it is possible to mitigate the harmful effect of the vibration amplitude
induced by the railroad transport.
As a result of the research, it was determined that when the railway track is located
at a height of 2 m from the foundation level, the amplitude of the vibration level in
nearby located structures decreases from 1.5 to 3 times, depending on the frequency
of the impact.

4 Conclusion

1. A mathematical model, a method and an algorithm were developed to assess the

level of vibration propagation induced by railway transport on various objects
located at a certain distance from the roadbed.
2. The level of vibrations propagating from a railway car on buildings located at a
certain distance from the vibration source was investigated, when the roadbed
is laid at the level of the foundation or at a certain height from the foundation.
3. If a railway bed is laid at a height of 2 m from the level of the earth foundation,
then the amplitude of vibration displacements in the building can be reduced
from 1.5 to 3.5 times.

References

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on a peat base under train motion. General technical problems and ways of their solution.
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2017/4813274
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doi.org/10.15593/2224-9826/2018.1.09
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of living. Proceedings of the 21st European young geotechnical engineers’ conference, 2011,
Rotterdam, the Netherlands
5. Ilyichev VA, Yuldashev ShS, Saidov SM (1999) Investigation of the vibration propagation
during the passage of trains depending on the location of the railway track. J Found Bases Soil
Mech 2:12–13
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6. Yuldashev SS, Matkarimov PZ (2014) Vibration propagation in soil from vehicles and vibration
protection systems. Tashkent: Fan va texnologiya markazi, p 188
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heterogeneous systems taking into account passive vibration isolation. J Found Bases Soil
Mech 2:9–11
8. Feng H (2011) 3D—models of Railway Track for dynamic Analysis. KTH, School of Archi-
tecture and the Built Environment (ABE), DiVA, id: diva2:467217 (Transport Science)., p
92
9. Mirsaidov MM, Troyanovsky IE (1980) Wave problem on the seismic stability of structures
during the propagation of a Rayleigh wave in an elastic half-space. Academy of Sciences of
the UzSSR (Ser. Tech. Sci.). 5:48–51
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account internal dissipation and wave entrainment of energy. Tashkent, FAN, p 108
11. Mirsaidov MM, Sultanov TZ, Rumi DF (2013) An assessment of dynamic behavior of the
system “structure—Foundation” with account of wave removal of energy. Mag Civ Eng 39(4).
https://doi.org/10.5862/MCE.39.10
12. Mirsaidov MM (2019) An account of the foundation in assessment of earth structure dynamics.
E3S Web of Conferences. https://doi.org/10.1051/e3sconf/20199704015
13. Mirsaidov MM, Toshmatov ES (2019) Spatial stress state and dynamic characteristics of earth
dams. Mag Civ Eng. https://doi.org/10.18720/MCE.89.1
14. Abduvaliev AA, Abdulkhayzoda AA (2020) Underground pipeline damping from the action
of Rayleigh waves. CONMECHYDRO-2020. IOP Conf. Series: materials science and
engineering. 883. https://doi.org/10.1088/1757-899X/883/1/012203.
15. Abduvaliev AA, Abdulkhayzoda AA (2020) Damping vibrations of an underground structure
using a three-mass damper. IOP Conf Ser: Earth Environ Sci 614. https://doi.org/10.1088/1755-
1315/614/1/012070
16. Persson N (2016) Predicting railway-induced ground vibrations. Dissertation. Lund University,
Sweden
17. Kumar SV (2015) Guidelines for noise and vibrations. Track design directorate research designs
and standards organisation, ministry of railways, India, p 100
18. Connolly DP, Alves CP, Kouroussis G, Galvin P, Woodward PK, Laghrouche O (2015) Soil
Dyn Earthq Eng 71:1–12. https://doi.org/10.1016/j.soildyn.2015.01.001
19. Koltunov MA, Kravchuk AS, Mayboroda VP (1983) Applied mechanics of deformable rigid
body. Higher school, Moscow, p 349
20. Kaewunruen S, Remennikov A (2008) Dynamic properties of railway track and its components:
a state-of-the-art review. New Research on Acoustics. Pp 197–200
21. Georges K, Harris P, Mouzakis K, Vogiatzis E (2016) Structural impact response for assessing
railway vibration induced on buildings. Analyse Vibratoire Expérimentale. Blois
22. Amosov AA, Dubinsky YA, Kopchenova NV (1994) Computational methods for engineers.
Moscow, Higher school, p 543
Mode Shapes of Transverse Vibrations
of Rod Protected from Vibrations
in Kinematic Excitations

Mirziyod Mirsaidov, Olimjon Dusmatov, and Muradjon Khodjabekov

Abstract This work is devoted to the mode shapes of transverse vibrations of the rod,
which is protected from vibrations under the influence of kinematic excitations. One
of the current problems is the general form of mode shapes of vibrations and analytical
expression of frequency equations, taking into account the dissipative characteristics
of vibration-protected rods. A liquid section dynamic absorber taken as a vibration
protective object in the study. The Pisarenko-Boginich model of the hysteresis type
expresses the dissipative properties of the rod material. The frequency equations and
mode shapes of vibrations of the system under consideration are generally obtained
analytically depending on the system parameters. From the frequency equation, it
is shown that which parameters, in addition to the mechanical characteristics of the
rod, depend on the resonant frequency. Effects of the liquid section dynamic absorber
have been shown to affect the mode shapes function of an elastic dissipative rod of
the hysteresis type.

Keywords Rod · Frequency · Vibrations · Hysteresis

1 Introduction

It is important to work on the reducing of harmful vibrations in mechanical systems

and identifying the factors that prevent their long-term perfect operation. In this
regard, one of the current problems is the general form of mode shapes of vibrations
and analytical expression of frequency equations based on system parameters, taking

M. Mirsaidov (B)
Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39 Kori Niyoziy Street,
Tashkent, Uzbekistan 100000
e-mail: uzedu@inbox.ru
O. Dusmatov
Tashkent State Pedagogical University, 27 Bunyodkor Street, Tashkent, Uzbekistan 100185
M. Khodjabekov
Samarkand State Architectural and Civil Engineering Institute, 70 Lolazor Street, Samarkand,
Uzbekistan 140147

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 217
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_20
218 M. Mirsaidov et al.

into account the elastic dissipative characteristics of the hysteresis type of the rod,
which is protected from vibrations in kinematic excitations.
There are a number of studies devoted to the reducing of harmful vibrations of
rods whose cross-section is constant and variable, the determination of mode shapes
of vibrations in linear vibrations under the influence of several external forces at
different points on the basis of boundary and normalization conditions.
Some of them:
• the method of internal resonances in kinematic excitations is presented in the study
[1]. In particular, the motion of the rod under the influence of harmonic excita-
tions was obtained using the Lagrange equations, and the amplitude-frequency
characteristic was analyzed. Analysis of objects in kinematic motions has been
shown to be important in the study of seismic phenomena;
• in the study [2], the frequency equation of rod with a variable cross-section under
different boundary conditions was determined and methods for its solution were
given;
• in the article [3] studied the forced vibrations of the rod under the influence of
periodic forces. Mode shape functions are proposed in several different forms and
the resonant frequency is determined;
• vibration forms of vibration-protected rods have been experimentally studied in
works [4–8]. With the change of frequencies, graphs of mode shapes of vibrations
were obtained and conclusions were made and recommendations were given;
• the work [9] deals with the determination of mode shapes of vibrations and
frequencies of vibrations in conjunction vibrations of the second rod mounted
on its free end as a dynamic absorber on a rod with one end free;
• in the study [10], the vibrations of two rods connected by a reciprocal Poyting-
Thomson model were mathematically modeled using Lagrange equations. Mode
shapes of vibrations and frequencies of vibrations were analyzed numerically;
• in the works [11–16] the theoretical basis for determining the mode shapes
of vibrations and frequencies equations of the rods was developed taking into
account the effects of various external loads and the results of the experiment
were presented;
• in the study [17], the mode shapes of combined transverse vibrations of rods and
dynamic absorbers with elastic dissipative characteristics of the hysteresis type
was obtained and analyzed as a trigonometric function, and numerical calcula-
tions, conclusions and recommendations were made for the case when the two
ends were hinged;
• the works [18–20] consider the dynamic characteristics of various high-rise struc-
tures, namely frequency, mode shape and decrements, as a one-dimensional
problem, reducing the amplitude of vibrations in them, increasing the natural
frequency of the structure by increasing the abstract part.
Although each of these works has its achievements and unexplored aspects, they
are all widely used in the development of theoretical research and in solving practical
problems.
Mode Shapes of Transverse Vibrations of Rod … 219

The results of the analysis showed that there is a need for research to determine the
mode shapes of vibration of rods with elastic dissipative characteristics of hysteresis
type, protected from vibrations, taking into account the effects of kinematic excita-
tions in general and the frequency equation based on system parameters. Therefore,
solving such problems is an urgent problem.

2 Materials and Methods

The work deals with the mode shapes of transverse vibration and the expression of the
frequency equation based on system parameters of the rod with elastic dissipative
characteristics of the hysteresis type under the influence of kinematic excitations
in conjunction with liquid section dynamic absorber mounted on it as a vibration
protection object.
The scheme of the physical model of this system is shown in Fig. 1 [21].
In this case m 1∗ and q3 are the mass and displacement of the outer body of the
dynamic absorber surrounding the liquid, respectively; m 2∗ and q4 are the mass and
displacement of solid of the dynamic absorber, respectively; m 3∗ is mass of liquid;
m 4∗ —mass of liquid attached to the body 2 with mass m 2∗ ; b F is the coefficient
of resistance of the damper; c1∗ and c2∗ are stiffness; FL (t) and FR (t) are external
forces.
The differential equations of motion of the system under consideration were
obtained using the method of bond graph and were written as follows [21]:

A∗ Q̈ + B Q̇ + C Q = F, (1)

where

Fig. 1 System with

hysteresis type elastic
dissipative characteristic rod
and liquid section dynamic
absorber
220 M. Mirsaidov et al.
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
q̈i q̇i qi u m (0)FL + u m (L)FR
Q̈ = ⎣ q̈3 ⎦; Q̇ = ⎣ q̇3 ⎦; Q = ⎣ q3 ⎦; F = ⎣ 0 ⎦;
q̈4 q̇4 q4 0
⎡ ⎤
mi 0 0
A∗ = ⎣ (m 13 + m 2∗ )u m (x1 ) (m 13 + m 2∗ ) m 2∗ + m v ⎦;
(m 2∗ − m v )u m (x1 ) m 2∗ − m v m 2∗ + m 4∗
⎡ ⎤ ⎡ ⎤
0 −u m (x1 )b F 0 ci −u m (x1 )c1∗ 0
B = ⎣0 bF 0 ⎦; C = ⎣ 0 c1∗ 0 ⎦;
0 0 bS 0 0 2c2∗

m i and qi are modal mass and displacement of accumulations of the rod; m 13 =

m 1 + m 3 ; m v is mass of liquid squeezed out by the body 2; u m (0), u m (L) and u m (x1 )
are the values of mode shapes of transverse vibrations of the rod at the points x = 0,
x = L ends of the rod and at the point where the dynamic absorber is installed
x = x1 ; b S is the viscosity coefficient of liquid; ci = c1i + jc2i ;
⎡ ⎤
L
⎢ 3E I ⎥
⎢ ρ A(1 − C0 η1 )u 2m d x − 2 η1 ⎥
⎢ ω∗m ⎥
⎢0 ⎥ 2
c1i = ⎢ ⎥ω∗m ;
⎢ L i∗ ⎥
⎢ n ∗
hi ∂ 2 ∂ 2um ∂ 2um ⎥
⎣× i∗
Ci ∗ qma ∗ um 2 dx⎦
i ∗ =1
2 (i ∗ + 3)
i ∂x ∂x2 ∂x2
0
⎡ ⎤
L
⎢ ρ AC0 η2 u 2 d x + 3E I η2 ⎥
⎢ m
ω∗m
2 ⎥
⎢ ⎥
⎢0 ⎥ 2
c2i = ⎢ ⎥ω∗m ;
⎢ L i∗ ⎥
⎢  n
hi
∗
∂ 2 ∂ 2um ∂ 2um ⎥
⎣× i∗
Ci ∗ qma i ∗ ∗ um 2 dx⎦
i ∗ =1
2 (i + 3) ∂x ∂x2 ∂x2
0

qma are amplitude values of rod vibrations; η1 , η2 = sign(ω)η22 are constant

coefficients depending on the dissipative properties of the rod material, determined
from the hysteresis curve; ω is frequency; j 2 = −1; C0 , C1 , . . . , Cn are experi-
mentally determined coefficients of the hysteresis node, depending on the damping
properties of the rod material [22]; E is Young’s module; I is moment of inertia; A
is the cross-sectional area of the rod, L is the length, ρ is the density and ω∗m are the
natural frequency.
Using the system of differential Eq. (1), the system under consideration can be
reduced to a system of algebraic equations by the differential operator S = d/dt,
and from this system of algebraic equations the variables qi , q3 , q4 are defined as
follows:
Mode Shapes of Transverse Vibrations of Rod … 221

a3 (b2 d3 − b3 d2 )
qi (S) = ;
a1 (b2 d3 − b3 d2 ) + a2 (b3 d1 − b1 d3 )
a3 (b3 d1 − b1 d3 )
q3 (S) = ; (2)
a1 (b2 d3 − b3 d2 ) + a2 (b3 d1 − b1 d3 )
a3 (b1 d2 − b2 d1 )
q4 (S) = ,
a1 (b2 d3 − b3 d2 ) + a2 (b3 d1 − b1 d3 )

where a1 = m i S 2 + ci ; a2 = −u m (x1 )(b F S + c1∗ ); a3 = u m (0)FL + u m (L)FR ;

b1 = M1 u m (x1 )S 2 ; b2 = M1 S 2 + b F S + c1∗ ; b3 = M2 S 2 ;
d1 = M3 u m (x1 )S 2 ; d2 = M3 S 2 ;

d3 = M4 S 2 + b S S + 2c2∗ ; M1 = m 13 + m 2∗ ; M2 = m 2∗ + m v ;
M3 = m 2∗ − m v ; M4 = m 2∗ + m 4∗ .

A system of differential Eq. (1) is a system of linear differential equations.

Therefore, its solutions are sought as follows:

qi = qma (t) cos(ωt + βi (t));

q3 = q3∗ (t) cos(ωt + β3∗ (t)); (3)
q4 = q4∗ (t) cos(ωt + β4∗ (t)),

where q3∗ (t) = q3∗ , q4∗ (t) = q4∗ , β3∗ (t) = β3∗ , β4∗ (t) = β4∗ are the amplitude and
initial phases of the variables q3 and q4 , respectively and they are functions of slow
variables; βi (t) is the initial phase of the slow variable function qi .
If the expression of the acceleration of the foundation is follows

W0 = εp0 cos ωt, (4)

(εp0 is the amplitude value of the foundation acceleration; ε is a small parameter).

The transverse vibrations of the rod, which are protected from vibrations under
the influence of kinematic excitations, are characterized as follows:
∞

wm (x, t) = u m (x)qma (t) cos(ωt + βi (t)). (5)
m=1

In expression (5) u m (x) is mode shapes of vibrations obtaining as a specific

solution of the differential Eq. (6) does not allow to draw general conclusions in the
study of transverse vibrations, dynamics and stability of motion of the rod protected
from vibrations and therefore limited to drawing specific conclusions.

d 4 u m (x) ρA 2
− ω u m (x) = 0. (6)
dx4 E I ∗m
222 M. Mirsaidov et al.

Overcoming this problem, that is, to be able to draw general conclusions in the
study of transverse vibrations, dynamics and stability of motion of the rod protected
from vibrations—u m (x) requires solving the problems related to boundary condi-
tions, taking mode shapes of transverse vibrations of the rod as a general solution of
differential Eq. (6).

3 Results and Discussion

Suppose both ends of the rod are fixed and liquid section dynamic absorber is installed
in the middle of it. In that case, the problem under consideration is a symmetric
problem. It will therefore suffice to look at the left half of the rod.
There is a general solution of the differential Eq. (6) [23] and it

u m (x) = A∗∗ sin km x + B∗ sin hkm x + C∗ cos km x + D∗ cos hkm x, (7)

where A∗∗ , B∗ , C∗ , D∗ are coefficients;

ρA 2
km = ω .
4

E I ∗m

In general, we express this function using the Krylov functions for convenience
in performing operations on the obtained solution (7).


4
u m (x) = Si K i∗ (km x), (8)
i=1

where S1 , S2 , S3 , S4 are coefficients;

1
K 1∗ (km x) = (cosh(km x) + cos(km x));
2
1
K 2∗ (km x) = (sinh(km x) + sin(km x));
2
1
K 3∗ (km x) = (cosh(km x) − cos(km x));
2
1
K 4∗ (km x) = (sinh(km x) − sin(km x)).
2
In determining the coefficients, S1 , S2 , S3 , S4 we use the existing boundary
conditions for the left half of the rod. These boundary conditions are as follows:
Mode Shapes of Transverse Vibrations of Rod … 223

∂wm
x = 0, wm = w0 , = 0,
∂x
(9)
L ∂wm ∂ 3 wm F0
x= , = 0, E I =− ,
2 ∂x ∂x3 2
where w0 is the displacement of the base; F0 is the amplitude value of the effect of
the liquid section dynamic absorber on the rod.
From the boundary condition x = 0, wm = w0 it is possible to write following
form:
εp0
u m (0)qma cos(ωt + βi ) = − cos ωt. (10)
ω2
From the equality of the coefficients in front of the corresponding trigonometric
functions we can write:
εp0
u m (0) = ± . (11)
qma ω2

From the boundary condition x = 0, ∂w

∂x
m
=0

∂u m (0)
= 0. (12)
∂x
L ∂wm
From the boundary condition x = 2
, ∂x =0
 
∂u m L2
= 0. (13)
∂x

E I ∂∂ xw3m = − F20 we can get following:

3
From the boundary condition x = L
2
,
 
∂ 3 u m L2 q3∗
EI qma cos(ωt + βi ) = (c1∗ cos(ωt + β3∗ ) − b F ω sin(ωt + β3∗ )).
∂x3 2
From the equality of the coefficients in front of the corresponding trigonometric
functions we can write.
  
∂ 3 u m L2 q3∗
= ± c2 + (b F ω)2 . (14)
∂x3 2E I qma 1∗

The ratio of the amplitudes to the right side of this equation can be derived from
the system of Eq. (2).

   2
q3∗ L 
 2c2∗ M1 − ω  + (M1 b S ω) ,
2 2
= ω2 u m
qma 2 h 21 + h 22
224 M. Mirsaidov et al.

where
    
h 1 = c1∗ − M1 ω2 2c2∗ − M4 ω2 − b S b F + M2 M3 ω2 ω2 ;

    
h 2 = b S c1∗ − M1 ω2 + b F 2c2∗ − M4 ω2 ω;  = M1 M4 − M2 M3 .

We put the ratio of the generated amplitudes in Eq. (14).

   
∂ 3 u m L2 L
= k 3
u
m m H, (15)
∂x3 2

where

  2  2 
ω  2
 ( 2c2∗ M1 − ω2  + (M1 b S ω)2 ) c1∗ + (b F ω)2
H =± .
2E I km3 h 21 + h 22

From the boundary conditions (11)–(15) and (13) it is possible to write the
determinant form of the following frequency equation:

1  0  0 
K 20∗ km2L K 40∗ km2L K 10∗ km2L = 0,
K 4∗ km2L K 2∗ km2L K 3∗ km2L

where
     
km L km L km L
K 10∗ = K 1∗ − H K 4∗ ;
2 2 2
     
km L km L km L
K 20∗ = K 2∗ − H K 1∗ ;
2 2 2
     
km L km L km L
K 40∗ = K 4∗ − H K 3∗ .
2 2 2

We can write the equation of frequencies from this.

        
km L km L km L km L
H 1 − cosh cos + cosh sin
2 2 2 2
    (16)
km L km L
+ cos sinh = 0.
2 2

Using the given boundary conditions and that K 1∗ (0) = 1, K 2∗ (0) = 0, K 3∗ (0) =
0, K 4∗ (0) = 0 taking into account the coefficients S1 , S2 , S3 , S4 can be determined
Mode Shapes of Transverse Vibrations of Rod … 225

as follows:
εp0
S1 = ± ;
qma ω2
S2 = 0;
       
εp0 K 10∗ km2L K 4∗ km2L − K 20∗ km2L K 3∗ km2L (17)
S3 = ∓        ;
qma ω2 K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L
       
εp0 K 20∗ km2L K 2∗ km2L − K 40∗ km2L K 4∗ km2L
S4 = ∓        .
qma ω2 K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L

We put the coefficients (17) into the general solution (8).

εp0
u m (x) = ± [K 1∗ (km x)
qma ω2
       
K 10∗ km2L K 4∗ km2L − K 20∗ km2L K 3∗ km2L
−         K 3∗ (km x) (18)
K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L
       
K 20∗ km2L K 2∗ km2L − K 40∗ km2L K 4∗ km2L
−         K 4∗ (km x)].
K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L

As a result, the transverse vibrations of the rod, which are protected from
vibrations, are expressed as follows:
∞
εp0 
wm (x, t) = ± [K 1∗ (km x)
qma ω2 m=1
       
K 10∗ km2L K 4∗ km2L − K 20∗ km2L K 3∗ km2L
−         K 3∗ (km x)
K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L (19)
       
K 20∗ km2L K 2∗ km2L − K 40∗ km2L K 4∗ km2L
−         K 4∗ (km x)]
K 10∗ km2L K 2∗ km2L − K 40∗ km2L K 3∗ km2L
×qma (t) cos(ωt + βi (t)).

The general solution obtained allows increasing the level of accuracy of the results
in the study of transverse vibrations, dynamics and stability of motion of the rod with
elastic dissipative characteristics of the hysteresis type, protected from the vibrations.
226 M. Mirsaidov et al.

4 Conclusion

1. The equation of frequencies in the transverse vibrations of the rod with elastic
dissipative characteristics of the hysteresis type, protected from vibrations, was
obtained depending on the system parameters.
2. From the frequency equation it is shown that the resonant frequency depends on
the mechanical properties of the rod, as well as on the masses of the two solids
of the dynamic absorber, the masses of the liquids squeezed out and adhered by
the internal solid, the stiffness of the springs and the damping coefficients.
3. Given the effects of the liquid section dynamic absorber on the rod with an
elastic dissipative characteristic of the hysteresis type, it was shown that the
mode shapes of transverse vibrations of the rod are directly proportional to
the amplitude value of kinematic excitations and inversely proportional to the
frequency equation.
4. The obtained results make it possible to choose the amplitude value of kinematic
excitations and the system parameters included in the equation of frequencies
in a proportional manner.

References

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Pile and Elastic–Plastic Soil Mass
Interaction

Evgeny Sobolev and Vitalii Sidorov

Abstract This paper contains a solution to the problem of interaction of a long pile
with an elastoplastic two-layer soil mass. To determine the stress–strain state, an
analytical solution and a finite element method are used. It is shown that the load
transferred to the pile is distributed between the lateral surface and the lower end.
The nature of this distribution essentially depends on the elastoplastic properties of
soils, as well as on the geometrical dimensions of the pile, in particular, on the ratio
of the diameter and length of the pile. It is noted that for a given pile length and head
load, the optimal pile diameter can be determined. The optimum diameter will be
such that the optimal distribution of the force acting on the pile head between the
lateral surface and the lower end occurs.

Keywords Pile · Elastoplastic soil · Shear stresses · Shear modulus · Lateral

resistance · Tip resistance · Finite element method · Mohr–Coulomb ·
Geotechnics · Deep foundations · Soft soils

1 Introduction

Theoretical studies and practical experience in the construction of deep foundations

show that when a long pile interacts with the surrounding soil massif, a complex
non-uniform stress–strain state (SSS) arises, which ultimately determines the bearing
capacity and settlement of the pile. Piles with a working length of more than 15 m will
be considered long. The experience of using long piles shows that the load applied to
the pile head is distributed between the lower end and the lateral surface in a ratio of
1–4. That is, the resistance of the pile consists of only a quarter of the resistance to the
lower end, while the lateral surface accounts for the rest resistance. This is not a good
situation, since long piles are used, as a rule, if there are strong and low-compressive
soils with a Young’s modulus of more than 40 MPa under the lower end. It turns out

All tests were carried out using research equipment of The Head Regional Shared Research Facilities
of the Moscow State University of Civil Engineering.

E. Sobolev (B) · V. Sidorov

Moscow State University of Civil Engineering, Yaroslavskoe shosse 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 229
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_21
230 E. Sobolev and V. Sidorov

that the resistance of these good soils is not fully utilized, since most of the load is
redistributed along the lateral surface to the weak soil layers. It is necessary, having
the length of the pile given by geological conditions, to choose the optimal diameter
(width in the case of a square section), which would make it possible to realize all
the possibilities of the bearing capacity of the soils under the lower end of the pile.
The peculiarities of the distribution of pile resistance along the lateral surface
and along the lower end, both under static and dynamic action, are considered in the
work of Sobolev and Sidorov [1]. The authors have shown that the effects of uneven
distribution of resistance and underutilization bearing capacity of soils, characterized
by the lower end of the piles and for vibratory pile penetration.
The peculiarities of the interaction of the pile and the surrounding soil using
nonlinear models are considered in the works of Ter-Martyrsyan et al. [2–4]. The
telescopic model of the interaction between the pile and the surrounding soil, used
in this work, is widely used in the works of professor Ter-Martirosyan et al. [5]. On
the basis of mathematical modelling of the reinforcement of the slab foundation with
soil–cement piles, performed by Gotman and Safiullin [6], regularities of changes in
the load on the pile depending on the thickness of the foundation slab, the geometric
dimensions of the piles, as well as the features of the technology of pile installation
were obtained. The studies carried out in [6] make it possible to extrapolate the results
of this article to piles performed by the injection method. The optimal selection of
the geometric parameters of the pile foundation affects not only the nature of the
distribution of resistance along the length of the pile, but also the final settlement of
the pile and slab-pile foundations. This aspect is discussed in detail in the works of
professor Mangushev et al. [7, 8].
The problems of uneven distribution of resistance along the pile shaft and an
increase in the bearing capacity along the lower end can be solved using widening in
the lower part, as shown in the work of Fedorov et al. [9]. However, this technique
is very laborious from a technological point of view. In the work of Chunyuk [10],
the finite element method is widely used to solve problems of the interaction of the
pile and the surrounding soil. In addition to the features of the geometric dimensions
of the pile, the properties of the soil affect the problem under study. Especially
difficult is the forecast in the presence of soft soils, which is described in the works
of Abou-Samra et al. [11], Sun et al. [12].
Let us set the task to determine the stress–strain state of a multilayer soil foundation
of a limited size and a long pile—a soil cell, the diameter of which depends on the
diameter of the zone of influence of the pile. To approximate-mate the solved problem
to the actually observed phenomena, an elastic–plastic model of soils is adopted. It is
shown that, depending on the physical and mechanical properties of soils, for a given
pile length, due to the geological structure of the construction site, it is possible to
deter-mine the optimal diameter (cross-sectional area) of the pile, at which the soil
resistance under its lower end is maximally realized.
Pile and Elastic–Plastic Soil Mass Interaction 231

2 Methods

To simplify the solution and analysis of the stress–strain state, let us assume that
the volumetric deformations of the soil in this problem will be negligible. We will
assume that shear deformation prevails under these conditions. This can be imagined
in the form of a telescopic shear mechanism of soil deformation, which is a variation
of a multi-plane shear.
For the solution, we will use an elastic–plastic model, in which stresses and strains
are related by the equation:

τ · τ∗
γ = , (1)
G 0 (τ ∗ − τ )

where: γ —deformation shear, kPa; G 0 —shear modulus, kPa, at τ → 0; τ i τ ∗ —

current and maximum values of shear stresses, respectively, kPa, and the limiting
value is determined in accordance with the Mohr–Coulomb theory of strength

τ ∗ = σ · tgφ + c, (2)

where: φ and c—soil strength parameters.

Obviously, in formula (1), when approaching the state of limiting equilibrium τ →
τ ∗ , endless growth
 of shear deformations occurs γ → ∞, while  stress relaxation
τ → 0, ratio τ γ inversely proportional to the shear modulus 1 G 0 .
It is generally accepted that the area of influence of the pile in the surrounding
soil is limited to a diameter equal to approximately six times the diameter of the pile
itself. In depth, this zone is limited by a distance of about four to five diameters from
the lower end of the pile.
Thus, the problem of determining the stress–strain state of a multilayer foundation
system and a long pile in an infinite soil space can be reduced to the problem of the
interaction of a pile with a soil massif with dimensions depending on the dimensions
of the pile (Fig. 1).
Consider a round pile. Then the problem can be solved under conditions of axial
symmetry. Let us introduce one more assumption, assuming that the shear modulus
of the pile is much higher than the shear modulus of soils. We do not take into account
the compressibility of the pile shaft (the stiffness of the pile is infinite).
The basic equation is the expression for the balance of forces acting in the
considered closed system:

N = T + R, (3)

where: N = πa 2 p1 —force causing displacement (sinking) of a pile; T = 2πalτα —

force determined by the resistance to displacement of the lateral surface of the pile;
R = πa 2 p2 —force determined by the resistance to displacement of the lower end
of the pile.
232 E. Sobolev and V. Sidorov

Fig. 1 Design diagram of

the interaction of a long pile
with a two-layer soil cylinder
(flat model)

Taking into account the accepted telescopic mechanism of interaction between

the pile and the surrounding soil, the shear stress in the soil around the pile can be
determined by the formula
a
τ (r ) = τα , (4)
2
where: a—pile radius; τα —pile lateral stress.
The accepted shear mechanism assumes that the deformation around the pile is
determined by the formula

∂V
γ =− , (5)
∂r
where: V —vertical displacement of soil on a radius r .
Determine the displacement of the long pile, which is prevented by the resistance
along the lateral surface. Transferring the parameters from formula (4) to formula
(1), taking into account formula (5), we can calculate

∂V τ (r )τ ∗
=− . (6)
∂r G 1 (τ ∗ − τ (r ))

Integration of formula (6) taking into account formula (4) and setting the boundary
condition for limiting displacements V (r = b) = 0, will result in the following
Pile and Elastic–Plastic Soil Mass Interaction 233

formula
aτα bτ ∗ − aτα
VT = ln . (7)
G1 a(τ ∗ − τα )

Let us calculate the displacement of the lower end of the pile (neglecting the
deformation of the pile shaft itself), assuming that the same multi-plane shear mech-
anism described above is implemented. In addition, let us assume that the pile acts
as a rigid stamp on the soil under the lower end. The displacement of the lower end
will be determined by the formula:

πa(1 − ν2 )ke
V R = p2 , (8)
4G 2

where: ν2 and G 2 —deformation properties of soils under the pile; ke ≤ 1—coeffi-

cient taking into account the stiffness of the lower end of the pile, determined by the
shape of the cross-section.
Equating VT = VR from formulas
 (7) and (8) taking into account dependence (3)
in the form τα = a( p1 − p2 ) 2l we get

2a G2 bτ ∗ −a · ( p1 − p2 ) 2l
p2 = ( p1 − p2 ) ln  . (9)
π ke l G 1 (1 − ν2 ) aτ ∗ −a · ( p1 − p2 ) 2l

We introduce a conventional notation for a constant value

2a G2
C= = const, (10)
π ke l G 1 (1 − ν2 )

then solution (9) will take the form:

2lbτ ∗ − a · ( p1 − p2 )
p2 = C( p1 − p2 ) ln . (11)
2laτ ∗ − a · ( p1 − p2 )

3 Results

This is a transcendental equation for the unknown p2 depending on p1 . Its solution

can be obtained using the computer-aided design (CAD) system Mathcad 15.0 M010
(developed by PTC, USA).
Let us present the results of solving expression (11) for the case when l = 50 m;
a = 0.2; 0.5; 1.0 m; b = 1.2; 3.0; 6.0 m; G 1 = 20 MPa, G 2 = 40 MPa; ν2 = 0.35;
ke = 0.5, τ ∗ = 50 kPa. Then the dependencies p2 = f ( p1 ) for different values
234 E. Sobolev and V. Sidorov

Fig. 2 Theoretical dependences p2 = f ( p1 ) (a) and sa = f ( p1 ) (b) taking into account various
geometric parameters of the pile and cell

a and b have the form shown in Fig. 2, and similar dependences V = f ( p1 ) are
obtained if we use formula (8).
Analysis of the dependencies obtained in the course of solving
 the problem p2 =
f ( p1 ), presented in Fig. 2 shows that at a constant
 ratio b a ≈ 6...7, p1 and l with
increasing pile diameter 2a the ratio of p2 p1 , which indicates the distribution  of
the load coming to the lower end of the pile, in 15 … 30% (at p1 = 4500 kN m2 ).
The obtained theoretical dependencies can be used to calculate the optimal
geometric dimensions of the piles and need further verification based on real
geotechnical problems of foundation engineering.
The problem was also studied by the numerical finite element method (FEM)
using the PLAXIS geotechnical software package in an elastic–plastic formulation
(Mohr–Coulomb soil model). The solution results are presented graphically in Fig. 3.
Analysis of the results obtained shows that the generally accepted value of the
zone of influence of a single pile on the surrounding soil mass does not exceed the
specified six diameters. As in the analytical solution, the pile shaft compressibility
was not taken into account. Soil properties were set similar to the analytical solution.
The distribution of the force acting on the pile head over the lateral surface and the
lower end completely repeats the analytical solution presented above. At the same
time, a series of numerical experiments performed on the basis of various initial data
allows one to determine the optimal value of the geometric parameters of the pile
(cross-sectional area and working length).
Finite element modelling allows you to solve the problem much faster and without
time-consuming calculations. At the same time, the reliability of the calculations
performed is confirmed by the verification of the used PLAXIS software and the appli-
cation of the soil model, which has been tested on numerous practical geotechnical
tasks.
Finite element modelling allows you to solve the problem much faster and without
time-consuming calculations. At the same time, the reliability of the calculations
Pile and Elastic–Plastic Soil Mass Interaction 235

Fig. 3 Distribution of vertical displacements (a) and stresses (b) in a two-layer massif around a
pile

performed is confirmed by the verification of the used PLAXIS software and the appli-
cation of the soil model, which has been tested on numerous practical geotechnical
tasks.

4 Discussion

The relevance of the selected research topic can be assessed by the scale of foun-
dation construction in difficult soil conditions based on the works of Zhussupbekov
et al. [13]. Difficult soil conditions usually include significant strata of weak clayey
236 E. Sobolev and V. Sidorov

soils, under which there are soils with significant bearing capacity. Under similar
conditions, the problem of interaction between the pile and the surrounding soil is
considered in the works of Troshkova et al. [14], Liu et al. [15], Wan et al. [16].
Further development of the chosen topic of this study can be noted the interaction of
a group of piles as part of a pile and slab-pile foundation. At present, there are exam-
ples of research described in the works of Wang et al. [17], Hoang et al. [18], Gautam
et al. [19], however, the problem of optimal selection of the geometric characteristics
of piles remains unresolved. The effects of group interaction of piles, considered in
the work of Safie O. and Tominaga [20], are effectively described on the basis of
mathematical modelling by the finite element method.
The design of deep pile foundations is currently at the stage of preliminary deci-
sions based on the requirements of SP 24.13330.2011. The tables of this normative
document are used to calculate the preliminary resistance values for the lateral surface
of the pile and for the lower end.
The disadvantage of this technique is that it does not take into account the actual
distribution of resistances. In addition, SP 24.13330.2011 is the heir of earlier regu-
latory documents in the field of pile foundation engineering, developed on the basis
of experimental and field studies of prefabricated piles. Long piles considered in this
work, as a rule, are made according to a technology that provides for the device
directly on the construction site. The interaction of prefabricated piles with the
surrounding soil differs significantly from the piles made directly in the soil (bored,
injection).
Taking into account these circumstances, the relevance of this work is primarily
due to a more accurate description of the mechanism for the distribution of pile
resistance along the lateral surface and along the lower end.

5 Conclusion

Based on the solution to the problem of interaction between a long pile and the
surrounding elastoplastic soil massif of limited size, the following main conclusions
can be drawn.
1. Comparison of the results of the analytical and numerical solution of the problem
showed satisfactory convergence both in terms of settlement of piles and stresses
under the lower end of the pile.
2. The load transferred to the pile is distributed between the lateral surface and the
lower end. This distribution is determined by the nonlinear properties of the soils
and the geometric parameters of the long pile (the ratio of the cross-sectional
area to the working length, in particular).
3. With a given pile length and head load, you can determine the optimal pile
diameter.
Pile and Elastic–Plastic Soil Mass Interaction 237

References

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rational immersion. Vestnik MGSU 3(114):293–300. (En Russian)
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bearing capacity of the long pile. Vestnik MGSU 5:52–61. (En Russian)
3. Ter-Martirosyan AZ, Ter-Martirosyan ZG, Manukyan AV, Chin TV, Avanesov VS (2015) Inter-
action of the pile length of finite stiffness and the surrounding soil, taking into account the
elastic-plastic properties of the soil. Nauchnoye Obozreniye 18:84–88. (En Russian)
4. Ter-martirosyan AZ, Ter-martirosyan ZG, Sobolev ES (2015) Settlement and bearing capacity
of long piles of finite stiffness with a broadened heel, taking into account the nonlinear properties
of the surrounding soil. Zhilishchnoye Stroitel’stvo 9:8–11. (En Russian)
5. Ter-Martirosyan ZG, Ter-Martirosyan AZ, Sidorov VV (2014) Initial critical pressure under
the heel of the round foundation and under the heel of the bored round pile. Nat Tech Sci
11–12(78):372–376. (En Russian)
6. Gotman NZ, Safiullin MN (2017) Calculation of the parameters of the pile field when strength-
ening the base of the foundation slab with soil-cement piles. Constr Reconstr 1(69):3–10. (En
Russian)
7. Mangushev RA (2010) Modern pile technologies. In: Mangushev RA, Ershov AV, Osokin AI
(eds) Tutorial. Publishing House ASV, Moscow, 239 s
8. Mangushev RA, Kondratyeva LN (2016) To the method of engineering calculation of the
pile-slab foundation. Int J Calculation Civ Build Struct 12(1):110–116. [En Russian]
9. Fedorov VS, Krupchikova NV, Gavrikov MD (2019) Numerical studies of the operation of
piles with terminal spherical broadening as part of a group of piles. Caspian Eng Constr Bull
3(29):100–107. [En Russian]
10. Chunyuk DYu (2002) Application of the finite element method for the calculation and design
of combined slab-pile foundations. In: Alkhimenko AI, Zertsalov MG (eds) In the collec-
tion: “Interuniversity collection of scientific papers on hydraulic engineering and special
construction”. Moscow, pp 181–187. [En Russian]
11. Abou-Samra G, Silvestri V, Desjardins SL, Labben R (2021) Drained-undrained shaft resistance
of piles in soft clays. Int J Civ Eng 19(2):115–125. https://doi.org/10.1007/s40999-020-00543-2
12. Sun H, Wang H, Wu G, Ge X (2019) The mechanical properties of naturally deposited soft soil
under true three-dimensional stress states. Geotech Test J 42(5):1370–1383. https://doi.org/10.
1007/s40999-020-00543-2
13. Zhussupbekov A, Mangushev R, Omarov A (2021) Geotechnical piling construction and testing
on problematical soil ground of Kazakhstan and Russia. Lect Notes Civ Eng 112:89–107.
https://doi.org/10.1007/978-981-15-9399-4_9
14. Troshkova N, Maltseva I, Panova A, Kochnev G (2020) Forecast of the development of sediment
pile foundations in water-saturated clay soils. IOP Conf Ser: Mater Sci Eng 972(1). Article no.
012017. https://doi.org/10.1088/1757-899X/972/1/012017
15. Liu C, Yang M, Bezuijen A (2020) Ratio of long-term settlement to immediate settlement for
piled raft on soft clay. Proc Inst Civ Eng Ground Improv 173(4):216–223. https://doi.org/10.
1680/jgrim.18.00048
16. Wan J-H, Zheng X-Z, Ouyang W-H, Liu S-W, Li X-Y (2020) Stability analysis of single pile
base on efficient finite-element method. Yantu Lixue/Rock Soil Mech 41(8):2805–2813. https://
doi.org/10.16285/j.rsm.2019.1611
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characteristics of large-diameter bored cast-in situ deep and long pile. Geotech Geol Eng
38(3):3113–3124. https://doi.org/10.1007/s10706-020-01212-w
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Regularities of Formation of Residual
Stresses in the Fatigue Crack Tip

Oleg Emel’yanov

Abstract In this paper we consider the research results of the influence of the ampli-
tude loading, tensile to tensile load ratio on the regularities of the residual compressive
stress formation ahead of the fatigue crack front. The effect of tensile to tensile load
ratio in the tension half-cycles on the fatigue crack growth is explained from the posi-
tion of interaction of residual compressive stresses with stresses from the external
load. It is shown that the use of the effective stress intensity factor range in the Paris
equation instead of its nominal range allows one to take into account the effect of
the load ratio on the material parameters to the fatigue crack growth resistance.

Keywords Fracture mechanics · Stress intensity factor · Fatigue crack growth rate

1 Introduction

Disastrous destruction of constructions for various purposes (tanks, supply pipelines,

pressure vessels, masts, towers, bridges, ships, plane etc.) caused by a large number of
cracks showed the lack of the existing classical methods of calculation. The method
application of fracture mechanics allowed obtaining more accurate results.
Particular interest causes the influence effect of the tensile to tensile load ratio
R = σmin /σmax on the fatigue crack growth rate. Numerous studies have shown that
its increase from 0 to 0.8 leads to an increase in the crack growth rate by 2–3.7
times [1–12]. It was found that the difference in the value of the stress intensity
factor range (SIF) K = Kmax –Kmin for elastic and elastoplastic deformation of the
material is not more than 4.4% with a cyclic load change [13]. It is also known that
the compressive residual stress field is formed every time at the fatigue crack tip in
the half-cycles of the load reduction with a tensile to tensile load. Their dominant
influence on the fatigue crack growth after stretch overloading is noted in [14–22].
Due to the research of the effect of the tensile to tensile load ratio on the fatigue cracks
growth rate appears to be necessary for the improvement of methods for calculating

O. Emel’yanov (B)
Department of Building and Structure Design, Nosov Magnitogorsk State Technical University,
38, Lenin Prospect, Magnitogorsk 455000, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 239
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_22
240 O. Emel’yanov

the strength and reliability of steel structures in regard to the concept of residual
compressive stresses.

2 Methods and Materials

The elastoplastic analysis of stress behavior near the crack tip was carried out by the
finite element method (FEM) for a standard compact eccentric-stretched sample, for
which formulas for the calculation of SIF are known [23]. A flat model with a fixed
crack is used as a calculation one. The final element minimum size at the crack tip
was 0.1 mm. Stepwise loading made it possible to separate the elastoplastic task into
a series of consistent elastic tasks with varying material characteristics, determined
by the elastoplastic deformation diagram [24]. This approach makes it possible to
extend the procedure for solving elastoplastic tasks to tasks in which the change in
the external load in time (unloading and subsequent cyclic loading) is taken into
account. Stress calculation is based on the work of V. V. Moskvitin [25] with a cyclic
change in the load in the n-th half-cycle of loading.
Data on the mechanical, elastic and plastic properties of steels St20, VSt3sp,
09G2S, 15G2SF (Table 1) and their deformation diagrams (Figs. 1, 2) were used

Table 1 Steel mechanical properties

Steel Plate thickness (mm) Tensile yield limit Ultimate tension σ u Ψ (%) S s (MPa)
σ y (MPa) (MPa)
VSt3sp 16 255 440 59 750
St20 8 285 511 61 997
09G2S 6 390 580 67 900
15G2SF 16 440 700 70 1200
Ψ —reduction at fracture, S s —rupture strength in the neck of the sample

Fig. 1 Deformation steel

diagrams at single-axis
stretching
Regularities of Formation of Residual Stresses … 241

Fig. 2 Generalized diagrams of steel cyclical elastic deformation (S = ST /σy )

in solving this task. Deformation diagrams were obtained by a uniform technique

using low-base strain gauges. This is presented in work [26]. All selected steels
are cyclically stable, and their mechanical properties cover the mechanical property
entire range of construction steels. The load ratio R (from 0 to 0.8) and the stress
intensity factor range (SIF) K (in accordance with the deformation diagrams in
the range 7.84–39.18 MPa m1/2 for steels St20, VSt3sp and 13.71–68.55 MPa m1/2
for steels 09G2S, 15G2SF) varied in the calculations. The monotonous plastic zone
size (the place ahead of the crack front, for which σi ≥ σy ) in the crack propagation
direction was estimated at the maximum load in the entire range of changing load
and the load ratio. We fixed the distribution and dimension of compressive residual
stress field in the process of reducing the load, and also we fixed the cyclic plastic
zone size (the place ahead of the crack front, for which, in a half-cycle of unloading,
σi ≥ ST ; ST is the cyclic yield strength).

3 Results

Inelastic deformations occur near the crack tip in the half-cycle of load increase.
This initiates residual compressive stresses σy, res ahead of the crack tip when the
load decreases. According to the obtained data, the residual compressive stresses at
the crack tip occur at the load level P/Pmax = 0.54/0.80. For fixed SIF range, the
values and size of the residual compressive stress field decrease with increasing load
ratio. The distribution σy, res in a compact sample of steel VSt3sp at minimum load
values and K = 11.33 MPa m1/2 is given as an example in Fig. 3.
Figure 4a, b illustrates the value dependences lres /rm – R and lres /rc – R for different
values of the minimum load and the SIF range (lres is the size of the compressive
residual stress field that form ahead of the crack tip in the direction of its propagation
at the minimum load, mm; rm is size of the monotonous plastic zone ahead of the
242 O. Emel’yanov

Fig. 3 Residual stress distribution in the crack tip vicinity at K = 11.33 MPa m1/2 (steel VSt3sp);
r—distance from the crack tip; t—plate thickness

Fig. 4 Relative size

dependencies of the of the
compressive residual stress
field lres /rm (a) and lres /rc
(b) from the load ratio
Regularities of Formation of Residual Stresses … 243

Fig. 5 Change of the value

|σy,res /σy | at the limits of
the cyclic plastic
deformation zone at R = 0.3

crack tip in the direction of its propagation, mm; rc is size of the cyclic plastic zone
ahead of the crack tip in the direction of its propagation, mm). It should be noted that
the dependences of lres /rm and lres /rc on R are invariant to the mechanical and plastic
properties of steels and to the values of K and Kmax . With increasing load ratio, the
size of the compressive residual stress field ahead of the crack tip lres /rm , formed in
the half-cycles of load reduction, decreases in the direction of its propagation, and
the value of lres /rc does not change and amounts t0 ≈1.8rc . It is typical for all research
steels.
The data in Fig. 5 illustrate the change of the ratio (in absolute value) of the value
residual compressive stresses σy,res formed ahead of the crack tip in the half-cycles
of the load reduction to the value of the full range of the elastoplastic stresses σy
in the half-cycle of the load increase within the cyclic plastic deformation zone at R
= 0.3 and different values Kmax . The value of |σy,res /σy | does not depend on Kmax ,
it is invariant to the mechanical and plastic properties of steels, practically does not
change at r/rc ≤ 0.3 (r is the distance from the crack tip) and gradually decreases
to the boundary of the cyclic plastic deformation zone at r/rc > 0.3. This trend is
continues throughout the research entire range of R for all selected steels. According
to the results obtained, in the cyclic plastic deformation area r/rc ≤ 0.3 the value of
|σy,res /σy | decreases with increasing load ratio (Fig. 6).
Thus, in the half-cycle of load increase, the value of |σy,res /σy | is completely
determined by the load ratio and does not depend on Kmax , K, yield strength σy
and cyclic yield stress St of steels. It is typical for cyclically stable steels within of
the cyclic plastic deformations zone at r/rc ≤ 0.3.

4 Discussion

The stress–strain behavior of the material stabilizes in the near of the crack tip
under cyclic loading, like that near of ordinary stress concentrates (holes, grooves).
However, the concentration of stresses and deformations at the fatigue crack tip is so
244 O. Emel’yanov

Fig. 6 Dependency of the

value |σy,res /σy | from the
load ratio at r/rc ≤ 0.3

high that the material is brought to destruction in the crack tip zone at small external
cyclic load range. In this case, the crack extends. The region of the alternating reverse
yielding (this is a cyclic plastic zone) is formed at the fatigue crack tip inside the
monotonically plastically deformed zone under tensile to tensile load in the half-
cycles of increasing and decreasing the load. Thus, the cyclic plastic deformation
process at the crack tip in the half-cycles of increasing and decreasing the load
determined by range K [13] justifies the advantageous use of K instead of Kmax
for creating the fatigue failure kinetic diagrams.
Since the residual compressive stress field is formed ahead of the crack tip in the
half-cycles of unloading, when the load is lowered, the crack cannot growth. This
conclusion is confirmed by fractographic research of fatigue fracture [27]. The crack
grows only during the increasing half-cycle of loading, forming grooves. When the
load is decrease, the surface of a fracture is highly deformed, created in the previous
half-cycle, forming a dark part of the groove. The ratio of the areas of the dark and
light parts of the grooves depends on the maximum and minimum load at the previous
cycle.
The phenomenological effect of the influence of the load ratio on the rate of fatigue
cracks growth can be explained as follows: in half-cycles of increasing load, the
stresses from the external load are summarized with the residual compressive stresses
formed when it is unloading ahead of the crack tip. This interaction determines the
value of the stress effective range (this is the proportion of the stresses external range
that causes accumulation of damages in the material) and the fatigue crack growth
rate (this is the increment in crack length per load cycle). Thus, the value of the
residual compressive stresses formed in the half-cycles of unloading ahead of the
crack tip, controls the fatigue crack growth rate. Therefore, in the Paris equation, the
value of the effective SIF range Keff should be used instead of the nominal SIF
range K for an integral showing of the fatigue damage accumulation process under
cyclic tensile to tensile load:
Regularities of Formation of Residual Stresses … 245

Fig. 7 Dependency of the

value U = 1–|σ y,res /Δσ y |
from the load ratio

dl
= C(K)n (1)
dN
The value of the effective SIF range Keff is determined as follows:

K e f f = U K , (2)

when U = 1–|σ y,res /Δσ y |.

As follows from the data shown in Fig. 7, the relative value of the effective stress
intensity factor range U rising with increasing R. The dependence between U and R
was approximated by the expression by regression analysis of the obtained data:

U = 1.4594R 4 − 1.3342R 3 + 0.5186R 2 + 0.174R + 0.5686 (3)

The results of processing the experimental data obtained in [1, 9, 28] using
expressions (1)–(3) confirmed such an approach expediency. This approach makes
it possible to obtain the material resistance parameters for the crack growth that do
not depend on the parameters of the external load (Fig. 8a and b). In Fig. 8b the
points are in a sufficiently narrow scatter band. These points correspond to tests with
different load ratio. This indicates that the fatigue crack growth rate does not depend
on the load ratio.

5 Conclusion

1. Residual compressive stresses in the near fatigue crack tip are formed in the
elements of the steel constructions in the half-cycles of the load reduction with
tensile to tensile load. The value and size of the residual compressive stress field
are completely determined by the stress intensity factor range and the value of
246 O. Emel’yanov

Fig. 8 Fatigue failure diagrams from the experimental data processing using a the nominal range
K in the Paris equation, b the effective range Keff
Regularities of Formation of Residual Stresses … 247

the load ratio for a rolled metal cyclically stable steels with a thickness of up to
25 mm.
2. For cyclically stable steels, the value of |σy,res /σy | at the crack tip within the
zone of cyclic plastic deformations r/rc ≤ 0,3 is completely determined in the
half-cycle of the load increase by the load ratio and does not depend on the
maximum values of the stress intensity factor, the stress intensity factor range,
the mechanical and plastic properties of the steels.
3. The effect of the load ratio on the fatigue crack growth rate is based on the
residual compressive stress interaction that are forming in the near crack tip
during unloading, with stresses from the external load in the tensile half-cycles.
The use in the Paris equation the effective stress intensity factor range instead of
the nominal stress intensity factor range makes it possible to obtain the material
resistance parameters for crack development which independent of the external
load parameters and take into account the effect of the load ratio on the crack
growth rate.

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2695.1979.tb01333.x
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02.010
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13. Emelianov OV (2015) Struct Mech Struct Calculation 3:2–6
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Sci Ukrainian SSR 21:139–143. https://doi.org/10.1007/BF01150630
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org/10.1016/j.jmps.2016.10.001
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24. Morozov YeM, Nikishkov GP (1980) Science Publishing, Moscow, 256
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Nonlinear Dynamic Analysis of Truss
with Initial Member Length Imperfection
Subjected to Impulsive Load Using
Mixed Finite Element Method

Vu Thi Bich Quyen and Dao Ngo.c Tien

Abstract This paper is concerned with the dynamic analysis of truss with initial
member length imperfection under impulsive load considering geometric nonlin-
earity. Using displacement-based finite element formulation in solving the nonlinear
problem of this truss requires incorporating the initial member length imperfection
as a dependent boundary constraint to the master stiffness equation and producing
a modified system of equations. For escaping the mathematical difficulties of treating
the initial member length imperfection this paper proposes a novel approach to formu-
late the nonlinear vibration problem based on mixed finite element formulation.
The dynamic equilibrium equation containing unknown displacements and forces is
obtained using the principle of stationary potential energy. A mixed matrix of truss
elements is established based on mixed variational formulation with length imperfec-
tion conditions considering nonlinear deformation. Combining the Newmark inte-
gration method and Newton Raphson iteration method is employed to solve the
dynamic equations with geometric nonlinearity. Based on the employed method, the
research develops the incremental-iterative algorithm and the calculation program for
determining the dynamic response of truss with initial member length imperfection
under impulsive load. The numerical results are presented to verify the efficiency of
the proposed method.

Keywords Length imperfection · Mixed formulation · Mixed finite element

method · Nonlinear analysis of truss · Impulsive load

1 Introduction

Modelling and solving nonlinear vibrations problem is becoming increasingly impor-

tant in a range of engineering applications. This is particularly true in the design of
slender structures such as truss. Nonlinear behaviour in structural dynamics arises
from a range of common material and geometric nonlinearities. The approaches and
technics for solving nonlinear dynamic problems of structures have been developed

V. T. B. Quyen (B) · D. N. Tien

Faculty of Civil Engineering, Hanoi Architectural University, Km 10, Nguyentrai, Hanoi, Vietnam

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 249
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_23
250 V. T. B. Quyen and D. N. Tien

since decades ago [1]. Rigorous analysis of geometrically nonlinear vibration of

truss structures demands creating mathematical models and solving algorithm [2–5].
In geometrically nonlinear modelling truss with initial member length imperfection
using displacement-based finite element method, the nodal equivalent force is impos-
sible to use for replacing the imperfection. In this case, the formulation of a mathemat-
ical model for nonlinear vibration analysis of truss with initial member length imper-
fection requires incorporating the initial member length imperfection as dependent
boundary constraint to the master stiffness equation and producing a modified system
of equation [6]. The treatment of initial imperfection of truss considerably increases
the difficulty in finite element formulation of nonlinear dynamical analysis such as
vibration analysis under impulsive load. For escaping the mathematical difficulties of
treating the initial member length imperfection, in [7] the author has presented novel
mixed finite truss element, including initial member length imperfection, considering
large displacement. The presented idea is developed to formulate the nonlinear vibra-
tion problem of truss with initial member length imperfection subjected to impulsive
load. The research proposes mixed formulation for constructing nonlinear dynamic
equilibrium equation of truss based on the principle of stationary potential energy. A
novel mixed finite truss element, which consists of the initial member length imper-
fection in the mixed matrix of truss element, is established based on mixed varia-
tional formulation considering large deformation of truss element. For determining
the nonlinear vibration of truss system this research establishes the incremental-
iterative algorithm based on combining the Newmark integration method and Newton
Raphson iteration method. Using the proposed algorithm, the calculation procedure
and programs for determining the dynamic response of truss with initial member
length imperfection under impulsive load are established.

2 Formulation of Dynamic Equilibrium Equation of Truss

Element Based on Mixed Finite Element Approach

Consider the truss bar (e) in the global coordinate system X0Y shown in Fig. 1.
Designate the followings.
L e —the initial length of a truss bar; L o and L—the distances between two end
nodes before and after deformation; e —initial length imperfection of truss bar;
A—the cross-sectional area of the truss element, E—elastic modulus of material;
N—the axial load of the truss element;
f e —resultant external force at the ith cross-section after deformation (i’);
u5 —force unknown at the ith cross-section after deformation, obviously u 5 =
f e = N from equilibrium condition;
u 1 , u 2 , u 3 , u 4 —nodal displacements in the global coordinate system;
m 1 , m 2 , m 3 , m 4 —nodal concentrated masses in the global coordinate system;
f 1 , f 2 , f 3 , f 4 —nodal forces in the global coordinate system;
f I , f D , p(t)—inertia force, damped axial force, external impulsive force;
Nonlinear Dynamic Analysis of Truss with Initial Member … 251

Fig. 1 Mixed truss element considering large displacement

After deformation, the length of the truss element can be calculated as follows

L= (L 0 · sin α0 + u 4 − u 2 )2 + (L 0 · cos α0 + u 3 − u 1 )2 (1)

The axial deformation of the truss element can be computed by

L = L − L e = L − L 0 + e (2)

The internal virtual work of the truss element can be computed by

  Le Le L

δ(d x)
δV = − σx δεx d V = − σx d A δεx d x = −N dx = − N δ(d x)
dx
A 0 0 0
⎛ ⎞ ⎧ ⎫ (3)
L ⎨ 4 ∂L ⎬
⎜ ⎟ ∂L
= −N δ ⎝ d x ⎠ = −N δL = −N δu i + δe
⎩ ∂u i ∂e ⎭
0 i=1

The external virtual work of truss element can be obtained by

4
δV = f 1 δu 1 + f 2 δu 2 + f 3 δu 3 + f 4 δu 4 + f e δe = f i δu i + f e δe (4)
i=1

From Eqs. (3) and (4) getting the total virtual work of truss element
252 V. T. B. Quyen and D. N. Tien

 4
  4 
∂L ∂L
δV + δV = −N δu i + δe + f i δu i + f e δe
i=1
∂u i ∂e i=1
4    
∂L ∂L
= −N + f i δu i + −N + f e δe = 0 (5)
i=1
∂u i ∂e

Based on the principle of virtual work, from Eq. (5) getting


∂L ∂L
−N + f i = 0 (i = 1, 2, 3, 4); N − fe = 0 (6)
∂u i ∂e

Adding axial deformation of truss element from Eq. (2) to Eq. (6), then expressing
axial force through deformation, having
⎧
⎪ EA ∂(L − L 0 + e )
⎪
⎨ (L − L 0 + e ) = f i (i = 1, 2, 3, 4);
Le ∂u i
(7)
⎪
⎪ EA ∂(L − L 0 + e )
⎩ (L − L 0 + e ) − fe = 0
Le ∂e

The nodal dynamic equilibrium equation of truss element can be expressed as

a fundamental differential equation describing dynamic equilibrium [6]

f i = f I,i + f D,i + pi (t) = −m i ü i − ci u̇ i + pi (t) (i = 1, 2, 3, 4) (8)

Replacing f i from Eq. (8) to Eq. (7), combining Eq. (2) and Eq. (7), getting
⎧
⎪ EA ∂L
⎪
⎨ (L − L 0 + e ) = f i = f I,i + f D,i + pi (t) = −m i ü i − ci u̇ i + pi (t)
Le ∂u i
⎪ EA
⎪
⎩ (L − L 0 + e ) − f e = 0
Le
(i = 1, 2, 3, 4)
(9)

Designate followings
Nonlinear Dynamic Analysis of Truss with Initial Member … 253
⎧
⎪ (e) EA ∂L
⎪
⎨ qi (u) = (L − L 0 + e ) (i = 1, 2, 3, 4)
Le ∂u i
;
⎪
⎪ (e) EA
⎩ q5 (u) = (L − L 0 + e ) − f e
Le
 (e)
m i = m i ; ci(e) = ci
;
m (e)
5 ≡ 0; c5(e) ≡ 0
 (e)
Pi = pi (t) (i = 1, 2, 3, 4)
P5(e) ≡ 0

Equation (9) can be written in a compact form

m (e) (e) (e) (e)

k ü k + ck u̇ k + qk (u) = Pk k = 1, 2, 3, 4, 5 (10)

where u = {u 1 , u 2 , u 3 , u 4 , u 5 }T is the vector of unknowns.

Writing the Eq. (10) in matrix form as follows

M (e) ü + C (e) u̇ + q (e) (u) = P (e) (11)

Designating
⎧  T
⎪
⎪ (e)
(u) ≡ (e)
(u), (e)
(u), (e)
(u), (e)
(u), (e)
(u) ;
⎪
⎪ q q 1 q 2 q 3 q 4 q 5
⎪
⎪
⎪
⎪  T
⎪
⎪ (e)
≡ (e)
, (e)
, (e)
, (e)
, (e)
⎪
⎪ P P 1 P 2 P 3 P4 P 5
⎪
⎨  
M (e) = diag m (e) , m (e)
, m (e)
, m (e)
, 0 ;
⎪
⎪ 
1 2 3 4

⎪
⎪
⎪
⎪ C (e) = diag c1(e) , c2(e) , c3(e) , c4(e) , 0
⎪
⎪
⎪
⎪
⎪
⎪ d 2u
⎪
⎩ ü =
du
; u̇ =
dt 2 dt
The dynamic equilibrium Eq. (11) is a nonlinear differential equation of second
order.
In finite element analysis, the approach for solving nonlinear problems is based on
dividing the total load into incremental load steps. For constructing the incremental
equation, utilizing Taylor series formula for a short to expand the function of Eq. (7)
around of variable point, keeping only linear term in δ ü, δ u̇, δu, getting incremental
equation of dynamic equilibrium of truss element

∂q (e) (u)
M (e) δ ü + C (e) δ u̇ + δu = P (e) (12)
∂u
∂q (e) (u)
Setting k (e) (u) = ∂u
, Eq. (12) can be written as follows
254 V. T. B. Quyen and D. N. Tien

M (e) δ ü + C (e) δ u̇ + k (e) (u)δu = P (e) (13)

where.
δu = {δu 1 , δu 2 , δu 3 , δu 4 , δu 5 }T —Vector of incremental unknowns (displace-
ments and incremental);
δ ü, δ u̇—Vector of incremental acceleration and vector of incremental velocity;
M (e) , C (e) —Mass and damping matrix;
P (e) —Vector of incremental dynamic load;
In Eq. (13), the mixed matrix of truss element e considering the initial length
imperfection e is given by
⎡ ⎤
k11 (u)k12 (u)...k15 (u)
⎢ k (u)k (u) ... k (u)⎥ ∂q (e) (u)
⎢ 21 22 25 ⎥
k (e) (u) = ⎢ ⎥, ki j (u) = i , (i, j = 1, 2, ..., 5) (14)
⎣ ... ⎦ ∂u j
k51 (u)k52 (u)...k55 (u)

3 Dynamic Equilibrium Equation of Truss System

Assembling all the truss element matrices to form the global system matrices, getting
the dynamic equilibrium equation and incremental equation of dynamic equilibrium
of truss system as follows

M ü + C u̇ + q(u) = P; Mδ ü + Cδ u̇ + K (u)δu = P (15)

Where
⎧
⎪
⎪ u ≡ {u 1 , u 2 , ..., u n }T ; δu ≡ {δu 1 , δu 2 , ..., δu n }T ;
⎪
⎪
⎪
⎪
⎪
⎪ δ ü ≡ {δ ü 1 , δ ü 2 , ..., δ ü n }T ; δ u̇ ≡ {δ u̇ 1 , δ u̇ 2 , ..., δ u̇ n }T
⎪
⎪
⎪
⎪
⎪
⎪ ne ne
⎪
⎪ (e) (e)
⎪
⎪ Mi, j = Mi, j ; Ci, j = Ci, j ;
⎪
⎪
⎪
⎪ e=1 e=1
⎪
⎪
⎪
⎪ ne
⎪
⎪ (e)
⎪
⎪ K i, j (u) =
⎪
⎪
ki, j (u), (i, j = 1, 2, ..., n);
⎪
⎪
⎪
⎨
e=1

⎪
q(u) ≡ {q1 (u), q2 (u), ..., qn (u)}T (16)
⎪
⎪
⎪
⎪ P ≡ {P1 , P2 , ..., Pn }T ;
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ P ≡ {P1 , P2 , ..., Pn }T ;
⎪
⎪
⎪
⎪
⎪
⎪
ne ne
⎪
⎪ qi (u) =
(e)
qi (u); Pi =
(e)
Pi ;
⎪
⎪
⎪
⎪
⎪
⎪ e=1 e=1
⎪
⎪
⎪
⎪ ne
⎪
⎪ (e)
⎪
⎩ Pi =
⎪ Pi , (i = 1, 2, .., n)
e=1
Nonlinear Dynamic Analysis of Truss with Initial Member … 255

In finite element analysis of structures, the procedure of solving nonlinear dynamic

equations is the combination of solving dynamic equations and solving nonlinear
static equations [5]. For solving dynamic equation of system under time-dependent
force such as impulsive load [7] the research employs Newmark’s method [8, 9]. This
method converts differential equations of motion of a structure to a simpler form,
which is solved algebraically and incrementally.
Using the Newmark integration method [8, 9], the dynamic equation in incre-
mental form can be written as follows
  
1 1 1 γ γ γ
δ ü = δu − u̇ − ü; δ u̇ = δu − u̇ − − 1 t ü (17)
βt 2 βt 2β βt β 2β

Adding δ ü, δ u̇ from (17) to (16), getting

!
M γC
K (u) + + δu
βt 2 βt
" #$ %
K̄ (u)
    !
1 1 γ γ
= P + M u̇ + ü + C u̇ + − 1 t ü (18)
βt 2β β 2β
" #$ %
 P̄

The incremental Eq. (18) can be written in compact form as

K̄ (u)δu =  P̄ (19)

For the solution of nonlinear static equation at current time step (19) the Newton
Raphson iteration method is adopted [10]. Equations should be centred and should
be numbered with the number on the right-hand side.

4 Numerical Investigation

4.1 Example Formulation

Investigating the truss system (shown in Fig. 2) subjected to impulsive load P(t). The
variation of the impulsive load is presented in the following functions
⎧ t
⎪
⎪ P(t) = −50. k N i f 0 ≤ t ≤ 0.025s;
⎪
⎪ 0.025
⎨ !
t
⎪ P(t) = −50. 2 − k N i f 0.025 < t ≤ 0.05s
⎪
⎪ 0.025
⎪
⎩
P(t) = 0k N i f 0.05s < t
256 V. T. B. Quyen and D. N. Tien

Fig. 2 Examined truss system

Due to the manufacturing, one truss bar has initial length imperfection (1)
e . All
of the truss bars are made of the same material and have the same cross-sectional
area. The parameters are given

E = 2.104 k N /cm 2 , A = 4cm 2 , (1) −5

e = 6 cm; m = 50.10 k N s /cm,
2

c = 10−2 k N s/cm;

For solving the dynamic equation of the truss system the research employs
the Newmark average acceleration method with γ = 1/2; β = 1/4. The investigated
time period is [t0 , t1 ] = [0, 0.75s] with time increment t = 0.0025s.

4.2 Numerical Results

The initial parameters, including internal forces and displacement of truss with initial
member length imperfection, at time t0 = 0, had been computed based on a mixed
finite element model as follows
⎧ ⎫
⎨ ⎬
u(t0 ) = −1.84, −27, −1.46, −14.05, 334.7, −327, −9.2, 26.5, 319.2 ;
⎩" #$ % " #$ %⎭
cm kN
ü(t0 ) = u̇(t0 ) = 0

The calculation results of the nonlinear vibration analysis of truss under impulsive
load are the displacement, normal force–time response, velocity, velocity of normal
force–time response and acceleration, acceleration of normal force–time response,
and phase plane of system shown in Figs. 3, 4, 5 and 6.
Nonlinear Dynamic Analysis of Truss with Initial Member … 257

Fig. 3 Displacement–time response (u 2 − t) and velocity–time response (v2 − t)

Fig. 4 Normal force–time response (N1 − t) and velocity of normal force–time response (v N1 − t)

Fig. 5 Acceleration–time response (a2 −t) and acceleration of normal force–time response (a N1 −
t)

Fig. 6 Phase plane (v2 − u 2 ) and (v N1 − N1 )

258 V. T. B. Quyen and D. N. Tien

5 Conclusions

The mixed-based formulation mathematical model for solving the nonlinear dynamic
problem of truss with initial member length imperfection has a significant advantage
over the displacement-based formulation model. Taking both unknown displace-
ments and forces gives the possibility to insert the initial member length imper-
fection into the mixed matrix of truss element and to simplify the algorithm for
solving the nonlinear dynamic problem of truss subjected to impulsive load. The
proposed method and algorithm can be effectively used to determine the dynamic
response of truss system subjected to the impulsive load.

References

1. Wilson EL, Farhoomand I, Bathe KJ (1973) Nonlinear dynamic of complex structures. Earthq
Eng Struct Dyn 1:241–252
2. Leung AYT, Yang HX, Zhu P (2014) Nonlinear vibrations of viscoelastic plane truss under
harmonic excitation. Int J Struct Stab Dyn 14(4)
3. Le Guennec Y, Savin E, Clouteau D (2013) A time-reversal process for beam trusses subjected
to impulse load. J Phys: Conf Ser 464:012001
4. Chang S-Y (2009) Numerical characteristics of constant average acceleration method in
solution of nonlinear systems. J Chin Inst Eng 4:519–529
5. Bathe KJ (2016) Finite element procedures. Prentice Hall
6. Wagg D, Neild S (2015) Nonlinear vibration with control for flexible and adaptive structures.
Springer International Publishing Switzerland
7. Quyen VTB, Tien DN, Huong NTL (2020) Mixed finite element method for geometrically
nonlinear buckling analysis of truss with member length imperfection. IOP Conf Ser: Mater
Sci Eng 960(2020):022075
8. Belytschko T, Liu WK, Moran B, Elkhodary KI (2014) Nonlinear finite elements for
continuaand structures. Wiley, Chichester, UK
9. Newmark NM (1959) A method of computation for structural dynamic. J Eng Mech Div
85:67–94
10. Crisfield MA (1981) A fast incremental/iterative solution procedure that handles snap-through.
Comput Struct 13(1–3):55-62A
Optimal Scale Modeling of Surf Zone
Waves

Izmail Kantarzhi and Alexander Gogin

Abstract The paper presents one of ways of practical application of composite

modeling technologies in hydraulic engineering—use and complementarity of phys-
ical and numerical modeling. Scale effect is studied in modeling of wave breaking
on slope using an example of real coastal protection facility which was considered
in NRU MGSU. Physical model of slope was built on the scale of 1:25 in a shallow
water basin and numerical model was created in MIKE 21 BW on scales from 1:1 to
1:50. Results showed that using scales 1:30 and smaller can lead to significant errors
in determining depth of wave breaking; at the same time wave height error does not
reach significant values and fluctuates within 1%.

Keywords Composite modeling · Numerical modeling · Physical modeling ·

Wind waves · Wave breaking · Beach slope · Hydraulic engineering

1 Introduction

Studies of onshore hydraulic structures are traditionally carried out using physical
models that reproduce movements of seawater on a reduction scale with observance of
dynamic similarity. Nowadays, numerical models are increasingly replacing physical
ones. Numerical models, realizing mathematical description of turbulent processes
in corresponding boundary conditions, can be universal and convenient. Physical
and numerical models have their strengths and weaknesses, and their use should also
take into account possibilities of theoretical analysis and outdoor measurements [1].
Composite modeling is, by definition, combined and balanced use of physical and
numerical models. Composite modeling allows modeling when solving problems that
are not solved separately by methods of physical or numerical modeling. It improves
quality of studies and reduces uncertainty. Four methods are involved in composite
modeling: physical and numerical modeling, outdoor measurements, and theoretical

I. Kantarzhi · A. Gogin (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
e-mail: alex.gogin@bk.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 259
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_24
260 I. Kantarzhi and A. Gogin

analysis. All these methods are closely related to each other in composite modeling
[2–5].
Breaking of waves in proximity of hydraulic structures is a complex phenomenon.
Accuracy of modeling such processes in a laboratory, in physical modeling, signifi-
cantly depends on an adopted modeling scale [6]. Sometimes it is possible to perform
physical modeling of one object on different scales to study a scale effect. However,
this path is costly; therefore, combined use of physical and numerical modeling is
preferable.
Runup zone after waves pass surf zone is key to coastline evolution, where erosion
is active. Since waves and storms release most of their energy in this area, predicting
wave breaking with a certain accuracy is an important issue for shore protection
projects in hydraulic engineering.
In this work we investigate an effect of wave breaking in surf zone to assess
possibilities of using small-scale physical models using an example of real coastal
protection facility which was considered in NRU MGSU; some review of this object
is presented in [7].
In accordance with hydrodynamics of coastal zone, breaking of waves near struc-
tures is the important factor of waves impact on structures. Height of waves in section
of wave breaking and position of this section can change with a change in model
scale. Therefore, in order to study the scale effect using physical modeling, it is neces-
sary to carry out series of experiments on different scales. Technique of composite
modeling is to use numerical modeling to determine a minimum scale of physical
modeling at which a level of the scale effect corresponds to the allowable one [8].

2 Methods

Let us consider the scale effect on characteristics of wave breaking section on coastal
slope using numerical modeling of propagation and breaking of waves on this coastal
slope.
A first question raised here is related to an applied hydrodynamic model of wave
propagation on slope. Next, the Boussinesq model is used.
The Boussinesq equation is an equation of a theory of long waves in which, unlike
the Saint–Venant equation, pressure distribution over depth is not hydrostatic [9]. In
[10], a hierarchy of hydrodynamic wave models of wave run-up on coastal slope is
considered.
Equations of the Boussinesq model in the form of Serre-Zheleznyak-Pelinovsky
with dispersion improved by additive terms of linear dispersion characteristics [11]
have the form:

h t + (hu)x + (hν) y = 0,
Optimal Scale Modeling of Surf Zone Waves 261
   
u t x x + νt x y + (uu x )x x + νu y x x + (uνx )x y
u t + uu x + νu y + gηx + (α2 − α1 )h 2  
+ νν y x y + gηx x x + gηx yy
     
(hu t )x x + (hνt )x y + (huu x )x x + hνu y x x + (huνx )x y + hνν y x y +
−α2 h  
+g(hηx )x x + g hη y x y
   
1 H3 H2 H
= R+ Q − hx R+Q ,
H 3 2 2
x
(1)
   
u t x y + νt yy + (uu x )x y + νu y x y + (uνx ) yy
νt + uνx + νν y + gη y + (α2 − α1 )h 2
 
+ νν y yy + gηx x y + gη yyy
     
(hu t )x y + (hνt ) yy + (huu x )x y + hνu y x y + (huνx ) yy + hνν y yy
−α2 h  
+g(hηx )x y + g hη y yy
  
1 H3 H2 H
= R+ Q − hy R+Q ,
H 3 2 2
y
(2)

where:

R = u xt + ν yt + uu x x − u 2x + νν yy − ν y2 + νu x y + uνx y − 2u x ν y , (3)

Q = h x u t + h y νt + h x uu x + h y νν y + h x x u 2 + h yy ν 2 + h y uνx + h x νu y + 2h x y uν
(4)

In the equations above u, ν(x, y, t)—horizontal velocity components;

h(x, y, t)—water depth; η(x, y, t)—free surface elevation; g—acceleration of
gravity; t—time. Dispersion relation of numerical models in terms of velocity compo-
nents contain two additive parameters α1 and α2 , which are defined from a comparison
of linear dispersion and non-linear characteristics with corresponding characteristics
according to Stokes’ wave theory [12].
The following model parameters were set: α1 = 1/17, 5; α2 = 1/10. Bottom
friction was introduced with the Chezy coefficient equal to 60. Hr ms was calculated—
the root-mean-square wave height for a period of waves:

t0 +T
2
Hr ms = η2 dt, (5)
T
t0

where T —wave period; t0 —initial wave phase.

A numerical experiment was carried out in one-dimensional formulation using
the numerical wave model MIKE 21 BW (Boussinesq Waves) based on numerical
solution of the modified Boussinesq equations in time.
262 I. Kantarzhi and A. Gogin

Fig. 1 Cross profile of breaking wave and assumed vertical profile of water particle velocities

Wave breaking in the used wave model occurs when an angle of inclination of
water surface of a wave front becomes greater than the specified one. At the same
time, a “roll” of a breaking water mass is formed on a wave crest. The onset of wave
breaking is determined by a magnitude of excess impulse, denoted by Rx x , arising
from uneven distribution of water particle velocities in body of wave. The equation
for Rx x is written as follows:
 2
δ P
Rx x =  cx − , (6)
δ
1− H H

where δ = δ(t, x)—wave crest thickness; cx —acceleration of wave crest; H —calm

water depth; P—flow rate.
A diagram of formation of wave breaking crest realized in MIKE 21 BW wave
model is shown in Fig. 1.
The most important issue in an application of some model is verification of applied
model with data from outdoor or laboratory measurements. For further analysis, we
will use results of the work carried out at NRU MGSU earlier [7].
In this work, experiments were carried out in the wave basin of the Laboratory
of Hydraulic Engineering of the Moscow State University of Civil Engineering to
study propagation of regular waves on coastal slope.
The model of coastal slope with beach (Fig. 2) was built on the scale of 1:25 in a
shallow water basin, observing geometric parameters of physical model’s similarity
to a full-scale object and recreating features of bottom relief.
Optimal Scale Modeling of Surf Zone Waves 263

Fig. 2 Physical model in the wave basin of NRU MGSU

3 Results

Wave parameters in the wave basin of NRU MGSU reproduced by wave generator
were set in accordance with conditions of a real storm and the selected scale (1:25)
of experimental studies.
In total, three series of experiments were carried out, differing in parameters of
simulated storm and position of the model. In the first series of experiments, impact
of waves on a bank protection dam of a design structure was studied, with the 1.2 m
wide beach located in front of a concrete slope. The wave parameters reproduced by
the wave generator in this series were: wave height 0.23 m, length 3.84 m, period
1.6 s. Wave parameters were measured by four wave recorders located along the
slope. Figure 3 shows the recording of wave surface on the first wave recorder from
the wave generator, and in Fig. 4—on the last one.
As waves move towards the coast, they change shape, crests become steeper and
higher.
In the second series, experiments were carried out on the scale of 1:36 with the
following wave parameters: wave height 0.18 m, length 3.4 m, period 1.4 s.
In the third series of experiments, the wave parameters reproduced by the wave
generator were: wave height 0.11 m, length 3.84 m, period 1.63 s.
The same series of experiments was carried out using the numerical wave model
described above. A typical profile of wave surface obtained is shown in Fig. 5.
The results of numerical experiments are in satisfactory agreement with the
measured wave heights. Waves become steeper as they approach breaking section,
as in the physical model. Comparison of calculations and measurements is shown in
Table 1.
264 I. Kantarzhi and A. Gogin

Fig. 3 Wave recording on the first (most seaward) wave recorder

Fig. 4 Wave recording on the last (most shoreward) wave recorder

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0.0 5.0 10.0 15.0 20.0

Fig. 5 Transformation of waves on slope for the first series of experiments based on results of the
numerical modeling

Statistical processing of all three series of measurements and calculations was

carried out, a regression dependence of the measured and calculated wave heights was
constructed. The results are shown in Fig. 6. Correlation coefficient of the measured
Optimal Scale Modeling of Surf Zone Waves 265

Table 1 Results of numerical and physical modeling of waves for the first series of experiments
Wave recorder Water depth H Distance to Measured wave Calculated RMS
No. (m) sensor (m) height hv (m) wave height Hrms
(m)
1 0.6 7.8 0.230 ± 0.005 0.231
2 0.58 4.8 0.215 ± 0.005 0.218
3 0.35 2.9 0.205 ± 0.005 0.200
4 0.17 1.4 0.186 ± 0.005 0.179

Fig. 6 Regression dependence of measured and calculated wave heights in three series of
experiments

and calculated wave heights, R = 0.985, which indicates a very good agreement
between the results of physical and numerical simulations.
After verifying the numerical model, we can proceed, in fact, to study the effect
of the modeling scale on characteristics of wave breaking on slope. A following
model was designed. Horizontal bottom with a water depth of 20.0 m is mated with
a coastal slope of 1:20 (Fig. 7). Initial wave height is 4.0 m, period is 12.0 s, which
corresponds to wavelength of 225.0 m.
Study of the scale effect was carried out on the model in one-dimensional setting.
Along the length, the model consists of 1100 one-dimensional finite elements (cells),
in which elevations of disturbed water surface are calculated. Figure 7 also schemati-
cally shows the z-axis for visualizing the bathymetric base of the model. On the lateral
boundaries of the model, condition of complete absorption of waves is set. Incoming
regular waves are generated at x = 100, form and propagate on the horizontal bottom
and break on the slope.
Wave breaking was modeled on eight scales: 1:1 (full scale), 1:5, 1:10, 1:15,
1:20, 1:30, 1:40, 1:50. Number of cells is the same at all scales. As the scale of the
model changes, the cell size, depth on the model, height and period of incoming
waves change linearly. It was important to keep the same numerical convergence
(or sensitivity) of the scaled models. The Courant number is responsible for this—a
266 I. Kantarzhi and A. Gogin

Fig. 7 Schematic representation of the flat numerical model

dimensionless quantity that describes the number of cells that wave moves in one
step in time. The Courant number is defined as:

t
Cr = c , (7)
x
√
where c = g H —wave propagation velocity; t—time step; x—cell size.
Satisfactory convergence of numerical solution of the model is achieved at Cr ≤
1. Based on this condition, the time step of the full-scale model was determined,
amounting to 0.025 s. The time step in scaled models was determined based on the
condition of equality of the Courant number in all models and depended not only on
linear scale of the model, but also on propagation velocity of generated waves. The
total simulation time was 600 s.
The general initial data and parameters of the models, as well as the parameters
of incoming waves, are presented in Table 2.
Instantaneous profiles of waves on the slope now of wave breaking beginning are
shown in Fig. 8. Beginning of breaking was determined by increase in crest mark of
wave to the maximum, followed by quick decrease.
Determination of wave height by observing of wavy water surface can be carried
out by two equivalent methods: using wave trough behind crest (zero-down-cross-
method) and using wave trough in front of crest (zero-up-cross-method) (see Fig. 9)
[13]. Choice of the method is insignificant when considering groups of waves, when
it is important to determine statistics of wave regime, but it can be essential when
considering individual waves. Height of waves in breaking section was determined
by difference between a mark of wave crest in this section and a mark of wave trough
following this crest. This approach was determined by the consideration that a trough
in front of a breaking wave had already been destroyed at that moment.
For each model, position of breaking section, water depth in breaking section,
and wave height at beginning of breaking were determined.
Based on results of the performed numerical experiments, errors caused by scale
effect were determined, separately for depth in breaking section and for wave height
Optimal Scale Modeling of Surf Zone Waves 267

Table 2 The general initial data and parameters of models to study the influence of the scale effect
on transformation of waves on coastal slope
Model scale 1:1 1:5 1:10 1:15 1:20 1:30 1:40 1:50
No. of cells 1100
Rate of slope 1:20
Cell size (m) 1 0.2 0.1 0.067 0.05 0.033 0.025 0.02
Horizontal section 20 4 2 1.33 1 0.667 0.5 0.4
depth (m)
Generated wave 4 0.8 0.4 0.267 0.2 0.133 0.1 0.08
height (m)
Wavelength (m) 225 45 22.5 15 11.25 7.5 5.63 4.5
Wave period (s) 12.0 5.4 3.8 3.1 2.7 2.2 1.9 1.7
Wave velocity 14.0 6.3 4.4 3.6 3.1 2.6 2.2 2.0
(m/s)
Courant number 0.35
Time step (s) 0.0250 0.0112 0.0079 0.0065 0.0056 0.0046 0.0040 0.0035

2 1:1
1:5
1 1:10
1:15
0 1:20
1:30
-1 1:40
1:50
-2

-3
100 200 300 400 500 600 700 800 900
4

2 1:1
1:5
1 1:10
1:15
ё

0 1:20
1:30
-1 1:40
1:50
-2

-3
700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850

Fig. 8 Instantaneous profiles of waves on the slope for all models at the moment of wave breaking
beginning given on full scale
268 I. Kantarzhi and A. Gogin

Fig. 9 Wave height determination methods for irregular waves [13]

Table 3 Results of numerical modeling to study the influence of the scale effect on transformation
of waves on coastal slope
Model scale 1:1 1:5 1:10 1:15 1:20 1:30 1:40 1:50
Breaking section, cell No 782 785 787 790 794 799 809 810
Water depth in breaking section 5.88 5.73 5.63 5.48 5.28 5.03 4.52 4.47
(m)
Wave height at beginning of 4.58 4.54 4.55 4.56 4.62 4.60 4.60 4.58
breaking (m)
Scaling error for breaking 0.0 −2.6 −4.3 −6.8 −10.2 −14.5 −23.0 −23.9
depth (%)
Scaling error for breaking wave 0.00 −0.86 −0.66 −0.37 0.85 0.59 0.48 0.10
height (%)

at beginning of breaking. For example, the errors for breaking depth were determined
by the formula:

Hn − H1
 H,n = % (8)
H1

where Hn —breaking depth at simulation scale equal to n; H1 —breaking depth at

full-scale modeling. Similarly, the scaling error for wave height in breaking section
was determined.
The results of the series of numerical experiments are shown in Table 3 and Fig. 10.

4 Conclusions

Scale errors for different modeling scales are presented in the last two lines of Table
3. The results lead to the following conclusions.
Optimal Scale Modeling of Surf Zone Waves 269

Fig. 10 Graphs of changes

in depth of breaking (left)
and wave height at beginning
of breaking (right) for all
models

In terms of depth of wave breaking, scaling error shows a direct dependence on

scale of simulation: the smaller the scale, the smaller the depth of wave breaking.
For the 1:50 model, error in breaking depth reaches 24%. A quick rise from 15 to
23% is observed between the 1:30 and 1:40 models, suggesting that using a scale
less than 1:30 can lead to significant errors in determining depth of wave breaking
on a slope.
On the contrary, change in wave height upon breaking is nonlinear. In the 1:5 scale
model, wave height decreases compared to the full-scale model, after which it begins
to grow up to the 1:20 model. In the 1:30–1:50 models, wave height upon breaking
decreases on the contrary. Wave height error does not reach significant values and
fluctuates within 1%. It can be assumed that wave height at beginning of breaking
can be reliably determined on small scale models.

Acknowledgements The reported study was particularly funded by RFBR, project number 20-38-
90169.
270 I. Kantarzhi and A. Gogin

References

1. Kantardgi IG, Zheleznyak MJ (2016) Mag Civ Eng 6

2. Hadla G, Anshakov AS, Kantarzhi IG (2020) Composite modelling in Port engineering. IOP
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p 2637
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Wellenkanal
Features of Numerical Modeling
of CFRP Steel Bars

Evgeniy Shchurov and Alexander Tusnin

Abstract Experimental studies provide objective data on the operation of struc-

tures. However, experiments take a lot of time and require special equipment. It is
not always possible to consider and simulate all the features of the structure’s oper-
ation during tests. Instead of a physical experiment a numerical experiment can be
carried out with appropriate justification. Numerical studies are based on the use of
the finite-element method. Modern computing systems allow to model structures of
any size using flat and spatial finite elements, contact interaction of different mate-
rials, and physical nonlinearity. This article describes the finite-element models used
in the design of CFRP-reinforced steel rods. The numerical results are compared
with experimental data obtained when testing CFRP-reinforced steel elements in
tension and bending. Numerical calculations were performed using the FEMAP
computer complex, considering the physical nonlinearity of the material. The exper-
imental samples were modeled by volumetric finite elements. Taking into account
the symmetry of the problem, one-eighth of the experimental element was consid-
ered in tension, while in bending, half was considered with the imposition of the
corresponding boundary conditions on the elements along the symmetry axes. The
characteristics of the materials of steel and lamella used in the calculations were
obtained when testing samples of steel and lamella. The characteristics of the glue
were obtained by a series of successive calculations when simulating experiments
to determine the strength of the adhesive bond between steel and lamella. Compar-
ison of experimental and numerical data showed that the selected parameters of
finite-element models provide reliable results.

Keywords Strengthening · Composites · FibArm

1 Introduction

Composite materials based on carbon fiber (CFRP) in construction can be used to

strengthen steel structures [1–4]. A number of studies have been carried out in this

E. Shchurov · A. Tusnin (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 271
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_25
272 E. Shchurov and A. Tusnin

direction, confirming the effectiveness of this amplification method. CFRP protects

structures well from corrosion. The use of composite materials reduces the labor
intensity of the work to strengthen structures [5–9].
At NRU MGSU, tensile and flexural tests of steel structures reinforced with carbon
fiber reinforced plastic (CFRP) were carried out. Tensile tests [10], in which a CFRP
lamella was glued on both sides of a steel plate, showed an increase in the bearing
capacity of the reinforced element by an average of 38%.
In the bending test, a single-span beam reinforced from the stretched side with
a carbon fiber lamella was investigated. The beam was loaded in the middle of the
span with a concentrated force. An increase in the bearing capacity of a reinforced
beam by an average of 10% was found experimentally [11]. The results of studies
with a similar effect of increasing the bearing capacity are given in [12–19].

2 Methods

Previously obtained experimental data were used to assess the accuracy and reliability
of finite element models reinforced with CFRP steel structures. The finite element
calculation of the experimental reinforced elements was carried out using the FEMAP
computer complex.
There were two options considered for the formation of a finite-element model.
In the first version, the glue layer was included in the design model. In the second
variant, the glue was not modeled in the design scheme, and its presence was taken
into account by the destruction of the lamella at normal stresses of 700 MPa. Such
stress levels were obtained in the lamella during the experimental determination of
the strength of the glued lamellae to steel [20].
The adhesive layer, in turn, was modeled in two ways, Fig. 1. In the first variant,
the layer consisted of one row of FE with a thickness of 0.1 mm. In the second
variant, the glue layer was modeled by two rows of FE, the total thickness of which
was 0.4 mm.
Considering the symmetry of the samples along the longitudinal and transverse
axes, a simplified model was adopted for the calculation with the imposition of
the corresponding boundary conditions along the axes of symmetry. Modeling was
performed using volumetric FE. General view of the models is shown in Fig. 2.

Fig. 1 Adhesive layer modeling schemes. a A sample without modeling the adhesive layer; b 1
layer of FE glue; c 2 layers of CE glue
Features of Numerical Modeling of CFRP Steel Bars 273

Fig. 2 General view of the FE model. a FE tensile test model; b FE bend test model

For the calculation, a volumetric element of the Solid type was adopted, which
has the form of a linear 8-node hexahedron. In all models, the interaction between
finite elements took place through joint nodes. Isotropic material type was adopted—
Isotropic. Additionally, nonlinear properties of materials were set. These properties
include stress–strain curve, Yield Criterion, Yield Stress, Hardening Rule. The Mises
criterion (0 von Mises) was adopted as the plasticity criterion. An isotropic hardening
model was adopted. For all materials, the value of maximum stresses was introduced,
upon reaching which the material fractures. Due to the fact that, during the tensile
experiments, the load was transmitted through the movements of the clamps of the test
device, it was customary in the models to transfer the load through the movement. In
the flexural test models, the transfer of forces was performed through the movement
of a separately modeled steel plate.

3 Results

Let’s consider the operation of a reinforced tensile steel element.

Figure 3 shows the experimental “displacement-load” relationships. The
following designations were adopted:
• Sample 1, 2, 3, 4—steel samples reinforced with carbon fiber lamella;
• Sample without reinforcement—steel sample without reinforcement;
• Average values—a graph displaying the average value based on the results of tests
1, 2, 3, 4 tensile samples.
Based on the results of the numerical calculation, the actual displacements
recorded for the given finite element are displayed, while the displacements issued
by the test equipment have errors. Various factors, such as backlash of moving parts
of the equipment, deformation of the clamps, etc., affect the accuracy of the fixed
movements of the test equipment.
For comparison of numerical results and experimental data, the experimental
displacements are brought in line with the results obtained in the simulation. The
274 E. Shchurov and A. Tusnin

Fig. 3 Experimental
displacement-load plots for
tensile testing. a Graphs for
the destruction of a
prototype, b graphs before
the destruction of the
amplification

conversion was carried out by multiplying the experimental movements by a correc-

tion factor. The correction factor for displacements is obtained by dividing the
displacement values in the event of plastic deformations of the test sample by the
similar values of the simulated sample. The value of the forces at which plastic defor-
mations occur was 150 kN. The displacements of the test specimen and the simulated
specimen were 1.97 and 0.27 mm, respectively. The value of the correction factor:
1.97/0.27 = 7.29.
Comparison of the numerical and averaged experimental graphs is presented in
Fig. 4, where the following designations are adopted:
• Experiment—a graph displaying averaged experimental data;
• Model “1 FE”—a graph obtained by calculating the sample, in which the adhesive
layer is modeled by one row of finite elements;
• Model “2 FE”—a graph obtained by calculating the sample, in which the adhesive
layer is modeled by two rows of finite elements;
• Model “Without glue”—a graph obtained by calculating a sample for which the
glue layer was not modeled.
The graph “Model “1 FE” practically coincides with the graph Model “Without
glue”, but the destruction of Model “Without glue” occurred earlier.
Features of Numerical Modeling of CFRP Steel Bars 275

Fig. 4 Comparison of
tensile experiment data with
numerical results

Figure 5 compares the experimental data obtained when testing a steel beam from
an unreinforced pipe with the results of a numerical calculation of the finite element
model of this pipe.
Comparison of the experimental data with the results of a numerical calculation of
an unreinforced pipe for bending made it possible to establish their good compliance.
The calculated and experimental graphs have the same maximum values, which indi-
cates the correctness of the accepted characteristics of the material and the parameters
of the models in the calculation complex. A slight discrepancy in the graphs is due
to the presence of initial stresses in a real pipe, which led to an earlier achievement
of the yield point in some zones of the pipe.
Figure 6 shows the data of simulation of bending tests of a steel pipe with
reinforcement and full gluing, where the following designations of the graphs are
accepted:
• Model with full length gluing. 1 row of FE glue—test graph of a simulated steel
pipe with full lamella gluing, in which the glue layer is modeled by one row of
finite elements;

Fig. 5 Comparison of
experimental data obtained
when testing a beam from an
unreinforced pipe with the
results of numerical
calculation
276 E. Shchurov and A. Tusnin

Fig. 6 Simulation data of

bending tests of a steel pipe
with reinforcement and full
length gluing of the lamella

• Model with full length gluing. 2 rows of FE glue—test graph of a simulated steel
pipe with full lamella gluing, in which the glue layer is modeled by two rows of
finite elements;
• Model with full sizing along the length without modeling the glue layer—test
graph of a simulated steel pipe with full gluing of the lamella, in which the glue
layer was not modeled;
• Experimental sample with full sizing—average data based on the results of testing
samples with full sizing;
• Simulated pipe without reinforcement—Graph for testing a simulated steel pipe
without bending reinforcement.
Figure 7 shows the simulation data of bending tests of a steel pipe with rein-
forcement and partial gluing, where the following designations of the graphs are
accepted:

Fig. 7 Simulated bending

test data for a steel pipe with
reinforcement and partial
sizing of the lamella
Features of Numerical Modeling of CFRP Steel Bars 277

• Model with partial gluing along the length. 1 row of FE glue—test graph of a
simulated steel pipe with partial sizing of a lamella, in which the glue layer is
modeled by one row of finite elements;
• Model with partial gluing along the length. 2 rows of FE glue—test graph of a
simulated steel pipe with partial sizing of a lamella, in which the glue layer is
modeled by two rows of finite elements;
• Model with partial gluing without modeling the adhesive layer—test graph of a
simulated steel pipe with partial gluing of the lamella, in which the adhesive layer
was not modeled;
• Experimental sample with partial sizing—average data based on the results of
testing samples with full sizing;
• Simulated pipe without reinforcement—Graph for testing a simulated steel pipe
without bending reinforcement.

4 Discussion

Let us compare the experimental and numerical data obtained in the study of stretched
reinforced samples.
In Table 1 shows the data on the forces obtained numerically and experimentally
in tension.
Based on the comparison carried out for the calculation of CFRP-reinforced tensile
elements, the models in which the glue is modeled in the design scheme are rational.
In this case, the calculation accuracy depends little on the division of the adhesive
layer by thickness.
To compare the numerical data obtained by calculating the FE models of the beam
with the experimental results, averaged graphs were generated for the experimental
samples.
Figure 8 shows a comparison of experimental data with numerical results for
reinforced beams with complete gluing of the carbon fiber lamella. Figure 8 the
following designations are adopted:

Table 1 Comparison of tensile results

Name of Maximum effort values, kN
efforts Numerical model Numerical model Numerical model Experiment
with 1 layer glue with 2 layers of “without” glue
glue
Effort in steel 149.5 149.5 150.0 –
Effort in 63.0 61.2 36.0 –
lamellas
Total effort 212.5 210.7 186.0 205.0
Number/exp. 102.8 103.7 90.7 –
Ratio, %
278 E. Shchurov and A. Tusnin

Fig. 8 Comparison of
experimental and numerical
data for a beam with
reinforcement and full sizing
along the lamella length

• Sample 1. Without reinforcement—graph of testing a steel pipe sample without

bending reinforcement;
• Sample 2, 3. Full gluing—test schedule for a steel pipe sample with full sizing of
the lamella;
• Sample Femap. Full gluing—Average test plot of a simulated steel pipe with full
sizing of the lamella.
Table 2 shows the maximum values of the forces obtained for experimental
samples and numerical models when gluing the lamella along the entire length.
The difference between the numerical results and experimental data was 2–3%,
which confirms the possibility of using the FE model for calculating such structures.
Figure 9 shows a comparison of experimental data with numerical results for
reinforced beams with partial gluing of a carbon fiber lamella. Figure 9 the following
designations for graphs were adopted:
• Sample 1. Without reinforcement—graph of testing a steel pipe sample without
bending reinforcement;
• Sample 4, 5. Partial gluing—test schedule for a steel pipe sample with partial
sizing of the lamella;
• Sample Femap. Partial gluing—Average test plot of a simulated steel pipe with
partial sizing of the lamella.

Table 2 Comparison of experimental results and models with full sizing

Experiment model name Maximum effort, kN Uncertainty compared to FE calculation
(Sample Femap. Full sizing)
Sample 2 full sizing 18.74 2.5%
Sample 3 full sizing 17.72 3.1%
Sample Femap. Full sizing 18.28 –
Features of Numerical Modeling of CFRP Steel Bars 279

Fig. 9 Comparison of
experimental and numerical
data for a beam with
reinforcement and partial
sizing along the lamella
length

Table 3 Comparison of experimental results and models with partial sizing

Experiment model name Maximum effort, kN Uncertainty compared to FE calculation
(Sample Femap. Full sizing)
Sample 4 full sizing 19.6 8.1%
Sample 5 full sizing 19.32 6.6%
Sample Femap. Full sizing 18.12 –

In Table 3 shows the maximum values of the efforts for various variants of the
model achieved in the simulation of experiments.
The difference between the numerical results and the experimental data was from
6 to 8%, which confirms the possibility of using the FE model for calculating CFRP-
reinforced beams.

5 Conclusion

A comparison of the experimental and numerical data obtained in the study of

stretched reinforced specimens showed their good compliance—the difference in
the breaking forces did not exceed 4–10%.
The use of FE modeling of experimental bendable samples, regardless of the
method of modeling the adhesive layer, ensures that the behavior of numerical models
corresponds to the work of experimental samples—for all modeling options, similar
values of the forces acting on the sample are achieved, and the difference between
them does not exceed 3–8%.
Comparison of the experimental and numerical data allows us to conclude that it
is possible to use finite-element modeling for calculating steel elements reinforced
280 E. Shchurov and A. Tusnin

with carbon fiber, glued with partial or full length gluing, steel elements operating
in tension and bending.

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40:137–142
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20. Shchurov E, Tusnin A (2017) Ind Civil Eng 6:51–55
The Concept of Bearing Capacity
Distribution in the Supports of Arches

Alexandre Danilov and Ivan Kalugin

Abstract Recently, the fundamental approach to the formation of a bearing mechan-

ical system, based on the principles of concentration and distribution of the bearing
capacity, that is, the responsibility of the elements, is rarely mentioned. An example of
the implementation of these principles is the design of the support part of a large-span
flat steel arched structure considered here. The vertical and horizontal components
of the support reaction are separated here and each one is taken by its own group
of supporting structures. The horizontal component is distributed over a group of
bearing structural systems. In this case, the load-bearing elements are included in
the work not in a sequential, but in a parallel scheme of transmission of forces. To
withstand the horizontal component of the support reaction, instead of a powerful
heavy foundation or tightening of a length equal to the span of an arch, a group of
n relatively short horizontal anchors is introduced for each support, with their own
solutions of anchoring into the ground, for example, through piles. The topology of
the anchor system can have an unlimited variety of options, which makes it possible
to use it in accordance with the specific conditions of construction and operation.
At the same time, the level of safety (level of reliability) of the structure as a whole
change fundamentally, since the failure of one or even several elements (anchors)
only leads to an increase in forces in the remaining elements.

Keywords Steel arch · Thrust · Support structure · Foundation · Tightening ·

Pile · Local failure · Safety

1 Introduction

The space freed from intermediate supports, covered with a large-span structure,
gives the building an emotional and plastic expressiveness. On the other hand, large
free internal spaces make it possible to place large-sized objects and industrial prod-
ucts inside them and use them as industrial premises, for example, for aircraft and
shipbuilding, as well as hangars and shelters for large-sized products and transport.

A. Danilov (B) · I. Kalugin

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 281
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_26
282 A. Danilov and I. Kalugin

For such purposes, the most interesting are the so-called flat vaulted roofs. At the same
time, a double-hinge scheme remains real for application, the snow load becomes
dominant, and the wind load becomes mostly negative. All this information can be
found in many sources [1–16].
On the other hand, as the flatness of the arch rises to a certain limit, the hori-
zontal component of the pressure on the foundations also increases. Depending on
the span and the intensity of the applied loads, it can approach 1000 tons. To bear it
the construction of powerful foundation blocks or high-strength internal ties between
the nodes supporting the arches on the foundations is required. For example, with
a tightening force of 900 t and a design steel resistance of 300 MPa, the tight-
ening section area will be 9000/30 = 300 cm2 or, for example, a square section of
300 × 100 mm. In this case, the relative elongation of the tightening will be 30
kN/cm2 /20,600 kN/cm2 = 0.0015, and the absolute elongation for a span of, for
example, 240 m will be 240,000 mm × 0.0015 = 350 mm. To reduce this displace-
ment of the arch on the support several times, it is necessary to proportionally increase
the tightening sectional area. It is not difficult to estimate the additional material
costs, labor intensity and installation time. Another major negative factor is the level
of reliability of such a design. Failure of one such tightening can lead to irreversible
consequences, and powerful foundation blocks seem to be the most reliable construc-
tive solution. Unfortunately, in the available publications it is not possible to find any
more acceptable proposals for solving the problem of perceiving space in arched
structures.
This article discusses an alternative approach to solving such a problem, based on
the principles of concentration and distribution of the bearing capacity, that is, respon-
sibility, of elements. Arches stand directly on foundations without retaining walls or
columns (Fig. 1). Instead of a single tightening between the support nodes of the arch
it is proposed, for each support node, to form a group of anchor devices, for each of
which (or for their subgroups) a separate connection structure with the subgrade is
provided to bear the horizontal loads. Such foundations can be, for example, piles or
groups of piles combined into blocks. For convenience of presentation, piles for each
horizontal anchor will be conventionally considered below. Then, each such anchor
will bear a corresponding number of times less load than a single tightening, and the

Fig. 1 A building fragment

with a flat arched roof
without tightening
The Concept of Bearing Capacity Distribution in the Supports … 283

length of each such anchor can be 10–20 times less, and the flexibility of the arch
support structure to horizontal displacement will correspondingly decrease.
When installed in the design position, the forces in the anchors can be adjusted.
For example, you can get approximately the same forces transmitted to the piles.
This can be achieved by appropriate selection of anchor length, cross-section and
orientation combinations. And you can also take advantage of the pre-tensioning of
the anchor rods.
In addition, it is interesting to consider the option of using polymer materials
reinforced with high-strength and/or high-modulus fibers for anchoring. Here such
properties as their several times lower specific gravity, the ratio of ultimate strength to
modulus of elasticity different from steel products (once again about prestressing), the
absence of the danger of corrosion, and even their “exotic” feature such as shortening
of carbon fiber at heating may be useful.

2 Methods

In the process of working on the topic of the article, various options for such param-
eters as the size of the span and the height arches, as well as their combinations, were
considered. Here, as an example of the implementation of the proposed solution to
the problem of bearing of thrust in arched roofs, some results of a numerical exper-
iment (calculation using the LIRA-SAPR software package) of an arch covering of
a span of 240 m and a length of 288 m (12 steps of 24 m) are given. The lifting
boom is taken as 32 m. Thus, 240/32 = 7.5. A constant load of 1.320 kN/m2 and
three versions of the design snow load (2.10 kN/m2 ) are applied. The section of the
arch is taken constant along its length in the form of a spatial four-branch truss. The
branches of a hotrolled I-beam (70SH1) are connected by lattice rods (30K1). The
axial dimensions of the cross-section of the arch are 2000 × 6000 (height) mm along
its entire length. The design vertical pressure on the foundation block is 638 tons,
horizontal—947 tons (with a pinned foundation for the bearing of thrust).
The calculation of the arch for the limit states and the check of the overall stability,
taking into account the limited flexibility in the support nodes, were performed
preliminary.
On the Fig. 2 the span of the arch is 240 m, the height is 32 m, the step of the
arches is 24 m, the horizontal elements (20 m long) at the supports of the arches are
the anchor systems.
The arches support the trusses with a span of 24 m and a height of 2 m (Fig. 3),
on which, with a step of 3 m, girders with a span of 6 m with a profiled sheet lie.
In Fig. 3, on the right, you can see the upper and lower chords of an arched truss
with a height of 6 m (the horizontal lattice is not shown in program interface), the
chords of a transverse girder with a height of 2 m (two rods to the left, the lattice is
not shown) and an anchor structure. In this version, it consists of a main stem (length
20 m) and stretched anchor elements attached to it symmetrically on both sides at
the same angle. With their free ends, they are attached to the piles. If we take the
284 A. Danilov and I. Kalugin

Fig. 2 Design scheme of the building frame with a flat arched roof

Fig. 3 Fragments of roof structures. Cross-sections of arches and the horisontal support on a
foundation with an anchor system

material of the main stem as absolutely rigid, then the forces in all anchors will be the
same and with the same materials and cross-sections all anchors will have the same
bearing capacity. Otherwise, to regulate the distribution of the bearing capacity in the
anchors, a whole set of parameters can be manipulated: material of various anchors,
cross sections, anchor lengths, inclination angles to the main stem, pretension values,
points of attachment to the trunk, and so on. This design is not unique and has the
disadvantage that in reality the main stem may have some noticeable flexibility and,
being the most loaded element, significantly affects the distribution of forces in the
anchors and the horizontal displacement of the support node on the foundation.
The Concept of Bearing Capacity Distribution in the Supports … 285

3 Results

Three cases for the parameters of this anchor system are presented, which were
modeled on three adjacent supports with a cross-section of the main stem of 800 cm2
and an anchor section of 40 cm2 , anchor pitch is 2 m, inclination to the main stem is
1:2:
1. Stem and anchors of the specified section, made of steel, without pretension.
2. The same, but in anchors pretension is modeled by symmetrically applying a
conditional temperature load to each pair of anchors from 2° to 20° with a step
of 2° at a given coefficient of thermal expansion −10–4 .
The stem is absolutely rigid, without pretension of the anchors.
For the first case, the following calculation results are shown in Figs. 4 and 5.
Support node displacement: −4.5 mm.
For the second case, the following calculation results are shown in Figs. 6 and 7.
Support node displacement: −1.5 mm.
For the third case, the following calculation results are shown in Figs. 8 and 9.
Support node displacement: −2.3 mm.
Figures 4, 5, 6, 7, 8 and 9 show that the first case as such is the least acceptable,
and the idealized third option gives the best results. The second case can easily be
approximated to the third by real mechanical pre-tensioning of the anchors.

Fig. 4 Percentage of use of bearing capacity in case 1

Fig. 5 Maximum tensile forces in anchors (t) in case 1

Fig. 6 Percentage of use of bearing capacity in case 2

Fig. 7 Maximum tensile forces in anchors (t) in case 2

286 A. Danilov and I. Kalugin

Fig. 8 Percentage of use of bearing capacity in case 3

Fig. 9 Maximum tensile forces in anchors (t) in case 3

For greater clarity, without drowning into design features, you can imagine the
anchor system in the form of n parallel anchors of the same length and calculate the
corresponding forces and elongations for each anchor.

4 Discussion

To make it clear without giving much attention to constructive details one can imagine
the anchor system in the form of n parallel anchors of the same length and calculate
the corresponding forces and elongations for each anchor.
There are also the options of branching of the arches at approaching the founda-
tions (an example of fractal geometry). This allows you to halve the pressure on each
support node.
Figures 10 and 11 show the variants of such a division of the arch supports into
several foundations. Each arch in Fig. 10 is loaded by 10 t/m, span is 240 m, height
is 32 m (for the step of arches 24 m it means 10,000 kg/m / 24 m = 417 kg/m2 ).
Support reactions in X-direction, t: −1725, −866, −445 for single, two and four
branch supports of the arch respectively. The bases of each arch should be connected
to each other in the transverse direction.

Fig. 10 Load, t
The Concept of Bearing Capacity Distribution in the Supports … 287

Fig. 11 Vertical support reactions, t

It is, of course, necessary to take into account the bearing capacity and hori-
zontal flexibility of the structures applied for fixing anchors in the ground (in partic-
ular, piles). However, this task in itself requires serious attention from the relevant
specialists in each specific case.
The application of bearing capacity distribution considered in this article based
on the use of anchor devices leads to a system of one-way springs. The issues of
stability and behavior of arched structures under temperature and dynamic influ-
ences are considered, for example, in works [7–16]. When developing constructive
solutions acceptable for practical application, it is necessary to carry out special
studies to ensure stability and analyze the influence of temperature and dynamic
effects. Consideration should also be given to the possibility of loosening or failure
of individual anchors.

5 Conclusions

The fundamental principles of concentration and distribution of bearing capacity,

that is, responsibility, of elements are applied to computer modelling of arch roof
structures.
1. The load-bearing elements are included in the work not in a sequential, but in a
parallel scheme of transmission of forces.
2. Constructive solution for the support unit of a shallow large-span arch that was
not proposed previously is considered.
3. The dimensions of the foundation for each support are reduced compared to the
case of bearing the horizontal reaction (thrust).
4. The resistance to horizontal displacement of the supports increases several times.
5. This solution provides multivariate design with the choice of the “optimal”
solution.
6. The safety level (reliability) of the structure as a whole may be regulated.
288 A. Danilov and I. Kalugin

7. There is a large free space between the supports inside the span, and not only
above but also below the zero level.
8. A number of aspects of the problem being solved require more in-depth research.

References

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tsiy bol’sheproletnykh pokrytiy [Reference book on Design of contemporary metal structures
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konstruktsiy. Chast 1: «Metallicheskie konstruktsii. Obschij kurs». Uchebnick dlja VUZov. M.:
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konstruktsiy. Chast 2: «Metallicheskie konstruktsii. Specialnyj kurs». Uchebnick dlja VUZov.
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in-plane under high temperature of fire. Adv Mater Res 243–249:3–6
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cross-section members. Eng Struct 48:155–165
Force Combinations Inducing Lateral
Torsional Buckling

Yana Makzhanova

Abstract The phenomenon of lateral torsional buckling can occur when a relatively
large transverse load is applied to a thin-walled beam. The article presents the exact
equations which give critical combinations of two transverse concentrated forces
inducing lateral torsional buckling of uniform rectangular elastic beams. The cases
of a simply supported beam, a cantilever, a beam clamped at both ends, and a two-
span continuous beam are considered. The equations, involving Bessel functions, are
obtained analytically by solving differential equations and satisfying boundary and
continuity conditions. The degenerate cases following from the equations, when one
of the two forces disappears, are identical to the well-known solutions. The derived
equations define the boundary curves of the convex feasible regions in the plane
of forces. Combinations of forces inside the feasible region correspond to lateral
torsional buckling stability. The forms of the boundary curves are analysed. The
curve is close to linear if the forces are applied to one span close to each other and
nonlinear if they are applied to different spans. The derived exact equations can be
used to estimate critical values of transverse forces or their combinations obtained by
various approximate methods. Besides, the equations can be helpful for constructing
stability conditions in structural optimization problems.

Keywords Lateral torsional buckling · Stability · Beam · Rectangular

cross-section · Critical load · Feasible region

1 Introduction

A typical problem of structural optimization is the problem of finding the structure

of minimal weight (and therefore of minimal material) which satisfies given stress,
displacement, buckling, and frequency constraints.
When we determine the minimal weight of a uniform rectangular beam under a
transverse load, taking the dimensions of its cross-section as the design variables,

Y. Makzhanova (B)
Department of Higher Mathematics, PLEKHANOV Russian University of Economics,
Stremyanny Lane 36, Moscow 117997, Russia
e-mail: makzhanova.yav@rea.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 289
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_27
290 Y. Makzhanova

and only taking into account the strength conditions, we obtain a beam with a narrow
rectangular section as a result [1]. But for beams with narrow cross-sections, lateral
torsional buckling suddenly occurs if the load reaches some critical value called
critical load. Lateral torsional buckling is a combination of lateral displacement and
twisting of the beam. This means that the condition of lateral torsional buckling
resistance must be necessarily included in the set of constraints of optimization
problems.
There exist many approaches how to find the critical load including analytical
(classical) [2–4] based on exact solution of differential equations, finite element
method [5, 6], energy method [7, 8], etc.
The critical values for one-parametric loading such as one concentrated transverse
force, or one external moment, or uniformly distributed load are well known. But
the case when several independent transverse loads are applied to a beam is a more
complicated and much less studied problem. If more than one force is applied to a
beam, then there are infinitely many combinations of allowable values of the forces
for which lateral torsional buckling does not occur. These values form a convex
feasible region in the space of forces. The boundary of the feasible region is an
infinite set of critical combinations of forces inducing lateral torsional buckling.
The lateral torsional buckling stability condition acquires special importance in
the problems of optimal loading [9], where the design variables are not only the
cross-section dimensions of the beam but also the applied forces. In such problems
the optimal solution is often defined by lateral torsional buckling stability condition.
In the article the classical approach is used to determine the boundary of the
feasible region in the space of forces for the case when two concentrated forces are
applied to uniform rectangular elastic beams with different support conditions.

2 Methods

Let us consider a rectangular uniform beam with the x-axis coinciding with the
longitudinal axis of the beam. The y-axis is in the transverse direction, and the z-axis
is in the lateral direction.
The relation between the lateral displacement and the twist angle of the beam
carrying a transverse load at the moment of lateral-torsional buckling, is given by
the following system of differential equations [2]

E I y dd xw2 + Mz θ = 0,
2

(1)
G IT dd xθ2 − Mz dd xw2 = 0,
2 2

where w is the lateral displacement of a cross-section of the beam, θ is a twist

angle of the cross-section, Mz is a bending moment acting within the section, E I y is
the flexural rigidity, G IT is the torsional rigidity, x is the distance along the beam.
System (1) is equivalent to the Prandtl equation
Force Combinations Inducing Lateral Torsional Buckling 291

d 2θ Mz2
+ θ = 0. (2)
dx2 G IT E I y

The solution of Eq. (2) depends on types of load and supports of the beam. In
most cases, it can be expressed through Bessel functions of the first kind [3]. The
critical value of a load can be found from boundary conditions as the minimal root
of an equation involving Bessel functions.
Further, we solve Eq. (2) for the cases when rectangular uniform elastic beams
with different support conditions are subjected to two independent concentrated
transverse forces. It is supposed that the forces are applied at the centers of gravity
of the cross-sections and displacements of the neutral axes of the beams are small.
The first case is discussed in detail and solutions of the others are not so detailed.

3 Results

3.1 Simply Supported Beam

Consider a simply supported beam of length l subjected to two independent concen-

trated forces as shown in Fig. 1. The cross-section is rectangular with height h and
width b.
Using the appropriate expressions for the bending moments Mz (x) acting withing
the beam, differential equation (2) turns into the system of differential equations
    2
d 2 θ1 p1 1 − ll1 + p2 1 − ll2
+ x 2 θ1 = 0 if 0 ≤ x ≤ l1 , (3)
dx2 G IT E I y
   2
d 2 θ2 − p1 ll1 + p2 1 − ll2 x + p1l1
+ θ2 = 0 if l1 ≤ x ≤ l2 , (4)
dx2 G IT E I y
 l1 2
d 2 θ3 p1 l + p2 ll2
+ x32 θ3 = 0 if 0 ≤ x3 ≤ l − l2 , x3 = l − x. (5)
d x32 G IT E I y

The boundary and continuity conditions are

if x = 0, then θ1 = 0;

Fig. 1 Simply supported

beam subjected to two forces
292 Y. Makzhanova

if x = l1 , then θ1 = θ2 , dθ
dx
1
= dθ2
dx
;
if x = l2 , then θ2 = θ3 , d x =
dθ2 dθ3
dx
;
if x = l, then θ3 = 0.
Let us introduce the notation
   2  l1 2
p1 1 − ll1 + p2 1 − ll2 p1 l + p2 ll2
=
a12 , a3 =
2
,
G IT E I y G IT E I y
 
− p1 ll1 + p2 1 − ll2 p1 l 1
α=  , β= ,
G IT E I y G IT E I y

and then solve Eqs. (3)–(5).

It can be proved that the solutions are
√ a  √ a 
1 2 1 2
θ1 (x) = A1 x · J 14 x + B1 x · J− 41 x , (6)
2 2
 
 (αx + β)2  (αx + β)2
θ2 (x) = A2 αx + β · J 4 1 + B2 αx + β · J− 4 1 , (7)
2α 2α
√ a  √ a 
3 3
θ3 (x) = A3 l − x · J 14 (l − x)2 + B3 l − x · J− 41 (l − x)2 , (8)
2 2
where A1 , A2 , A3 , B1 , B2 and B3 are integration constants, J± 14 (∗) are Bessel
functions of the first kind.
It is known that J− 41 (x) → ∞ as x approaches zero. Therefore, from the boundary
conditions θ1 (0) = 0 and θ3 (l) = 0 it follows that B1 = B3 = 0.
Satisfying the other boundary and continuity conditions, we obtain the homoge-
neous linear system in four variables A1 , A2 , B2 , A3 :

 a  
1 2  (αl1 + β)2
A1 l1 · J 14 l1 − A2 αl1 + β · J 14
2 2α

 (αl1 + β) 2
− B2 αl1 + β · J− 14 = 0,
2α
a  
3
1 2 3 (αl1 + β)2
A1 a1l1 · J− 4
2
3 l1 − A2 (αl1 + β) · J− 42 3
2 2α

3 (αl1 + β) 2
+ B2 (αl1 + β) 2 · J 34 = 0,
2α
 
 (αl2 + β)2  (αl2 + β)2
A2 αl2 + β · J 14 − B2 αl2 + β · J− 14
2α 2α
 
a3 (l − l2 )2
+ A3 l − l2 · J 14 = 0,
2
Force Combinations Inducing Lateral Torsional Buckling 293
 
3 (αl2 + β)2 3 (αl2 + β)2
− A2 (αl2 + β) · J− 2 3 + B2 (αl2 + β) · J 2 3
4 2α 4 2α

3 a3 (l − l2 )2
− A3 a3 (l − l2 ) 2 · J− 34 = 0. (9)
2

A homogeneous linear system has a nontrivial solution if its determinant is equal

to zero. Thus, setting the determinant of system (9) to be zero, and introducing
2 2
the relative (non-dimensional) forces k1 = √ p1 l , k2 = √ p2 l and lengths
G IT E I y G IT E I y
ω1 = ll1 , ω2 = ll2 , after some transformations we obtain the equation which gives
the critical combination of the non-dimensional forces k1 and k2 :

J 14 (z 1 ) −J− 34 (z 2 )J− 41 (z 3 )J− 43 (z 4 ) − J 34 (z 2 )J− 43 (z 3 )J 14 (z 4 )

+J− 43 (z 2 )J 34 (z 3 )J 14 (z 4 ) − J 34 (z 2 )J 14 (z 3 )J− 43 (z 4 )

− J− 43 (z 1 ) −J 14 (z 2 )J− 41 (z 3 )J− 43 (z 4 ) + J− 14 (z 2 )J− 43 (z 3 )J 14 (z 4 )

+J 14 (z 2 )J 34 (z 3 )J 14 (z 4 ) + J− 41 (z 2 )J 14 (z 3 )J− 43 (z 4 ) = 0, (10)

where

ω12 ω2 (k1 (1 − ω1 ) + k2 (1 − ω2 ))2

z1 = (k1 (1 − ω1 ) + k2 (1 − ω2 )), z 2 = 1 ,
2 2 (−k1 ω1 + k2 (1 − ω2 ))
(1 − ω2 )2 (k1 ω1 + k2 ω2 )2 (1 − ω2 )2
z3 = , z4 = (k1 ω1 + k2 ω2 )
2 (−k1 ω1 + k2 (1 − ω2 )) 2

1 − ω2
−k1 ω1 + k2 (1 − ω2 ) = 0 ⇒ k1 = k2 .
ω1

If the points of application of the two forces coincide, then ω1 = ω2 , k2 = 0 and

Eq. (10) becomes
 
ω12 k1 (1 − ω1 ) (1 − ω1 )2 k1 ω1
J 14 J− 43
2 2
 2 
ω1 k1 (1 − ω1 ) (1 − ω1 )2 k1 ω1
+ J− 43 J 14 = 0,
2 2

which is the same that was obtained in [3] for the case of one concentrated force
applied at the distance l1 = ω1l from the left support.
Equation (10) determines the curve which is the boundary of the region of all
feasible values of the non-dimensional forces k1 , k2 . Keeping the value of one of the
forces k1 , k2 fixed, we can find the approximate value of another one from Eq. (10).
Figure 2 shows the boundary curves of the feasible region for some values of ω1 , ω2 ;
294 Y. Makzhanova

Fig. 2 Boundary curves for

simply supported beams

that is, for various points of application of forces. These curves are typical for simply
supported beams. Lateral torsional buckling occurs if the values of k1 , k2 are outside
of the feasible region or on its boundary.

3.2 Cantilever

Consider a cantilever of length l loaded by two independent concentrated forces: p1

applied at a distance of l1 from the support and p2 applied at the cantilever tip (refer
to Fig. 3).
Using the above ideas and corresponding transformations, we obtain the equation
for the critical combinations of non-dimensional forces k1 , k2 :
  2 
(k1 ω1 + k2 )2 k2 (ω1 − 1)2 k2 (ω1 − 1)2
J 14 J− 14 J 34
2(k1 + k2 ) 2(k1 + k2 ) 2
 2 
k (ω1 − 1) 2
k2 (ω1 − 1) 2
−J 34 2 J− 14
2(k1 + k2 ) 2

Fig. 3 Cantilever loaded by

two forces
Force Combinations Inducing Lateral Torsional Buckling 295
  2 
(k1 ω1 + k2 )2 k2 (ω1 − 1)2 k2 (ω1 − 1)2
− J− 1 J− 4 3 J− 4
1
4 2(k1 + k2 ) 2(k1 + k2 ) 2
 2 
k (ω1 − 1) 2
k2 (ω1 − 1) 2
+J 14 2 J 34 = 0, (11)
2(k1 + k2 ) 2

where

l1 p1 l 2 p2 l 2
ω1 = , k1 =  , k2 =  .
l G IT E I y G IT E I y

If there is no
 force
 p1 , i.e., k1 = 0 and ω1 = 0, then Eq. (11) is converted to the
equation J− 41 k22 = 0, from which k2 = 4, 012 and therefore p2 = 4,012 l 2 G IT E I y .
This coincides with the result obtained in [3] for the case of the rectangular uniform
cantilever of length l subjected to one force at the free end.
Ifthereis no force p2 , then k2 = 0 and Eq. (11) is converted to the equation
k ω2 
J− 41 12 1 = 0, from which k1 = 4,012 ω12
and p1 = 4,012
l12
G IT E I y . This result is the
same as obtained in [3] for the case of the rectangular uniform cantilever of length
l1 subjected to one force at the free end.
Figure 4 demonstrates the boundary curves of the feasible region for relative
forces k1 , k2 applied to cantilevers.

Fig. 4 Boundary curves for

cantilever
296 Y. Makzhanova

3.3 Statically Indeterminate Beams

In the cases discussed above the bending moment had one sign throughout the entire
beam. In statically indeterminate structures the bending moment changes its sign
within one span and this must be taken into account while solving Eq. (2).

3.3.1 Beam Clamped at Both Ends

Consider a beam of length l clamped at both ends in xy plane and loaded by two
independent concentrated forces p1 and p2 as shown in Fig. 5.
For the beam, Eq. (2) can be written as the system of differential equations

d 2 θi
+ (αi x + βi )2 θi = 0, i = 1, 2, 3 (12)
dx2
where θ1 (x) is the twist angle on the segment 0 ≤ x ≤ l1 , θ2 (x) is the twist angle
on the segment l1 ≤ x ≤ l2 , and θ3 (x) is the twist angle on the segment l2 ≤ x ≤ l;
and the coefficients αi and βi are

1  
α1 =  p 1 (l − l 1 ) 2
(2l 1 + l) + p 2 (l − l 2 ) 2
(2l 2 + l)
l 3 G IT E I y
−1  
β1 =  p1l1 (l − l1 )2 + p2 l2 (l − l2 )2
2
l G IT E I y
1  2 
α2 =  p1l1 (2l1 − 3l) + p2 (l − l2 )2 (2l2 + l)
3
l G IT E I y
1  2 
β2 =  p1l1 (2l − l1 ) − p2 l2 (l − l2 )2
2
l G IT E I y
1  2 
α3 =  p 1 l 1 (2l 1 − 3l) + p 2 l 2
2 (2l 2 − 3l)
l 3 G IT E I y
1  2 
β3 =  p1l1 (2l − l1 ) + p2 l22 (2l − l2 )
2
l G IT E I y

The solution of system (12) is

Fig. 5 Beam clamped at

both ends
Force Combinations Inducing Lateral Torsional Buckling 297

1/ (αi x + βi )2
|α
θi (x) = Ai sgn((αi λi + βi )(αi x + βi )) · i x + βi | 2 J4
1
2|αi |

1 (αi x + βi ) 2
+ Bi |αi x + βi | /2 J− 41 , i = 1, 2, 3. (13)
2|αi |

Satisfying the boundary and continuity conditions θ1 (0) = 0, θ1 (l1 ) = θ2 (l1 ),

dθ1
= dθ
d x (l 1 )
θ
d x (l 1 ), 2 (l 2 )
2
= θ3 (l2 ), dθ
d x (l 2 )
2
= dθ θ
d x (l 2 ), 3 (l)
3
= 0, we obtain the
homogeneous linear system in six integration constants Ai , Bi , i = 1, 2, 3. The
system has a nontrivial solution if its determinant is equal to zero:

 = i j = 0, i, j = 1, 2, . . . , 6. (14)

The nonzero elements of the determinant in (14) can be found in the following
way:
  2
z 12 z1
11 = J 14 , 12 = J− 41 ,
2|z 2 | 2|z 2 |
 2  2
z3 z3
21 = sgn(z 1 z 3 )J 14 , 22 = J− 14 ,
2|z 2 | 2|z 2 |
 2  2
z3 z3
23 = −J 14 , 24 = −J− 41 ,
2|z 4 | 2|z 4 |
 2  2
z3 z3
31 = sgn(z 1 z 2 )J− 43 , 32 = −sgn(z 2 z 3 )J 34 ,
2|z 2 | 2|z 2 |
 2  2
z3 z3
33 = −sgn(z 3 z 4 )J− 43 , 34 = sgn(z 3 z 4 )J 34 ,
2|z 4 | 2|z 4 |
 2  2
z5 z5
43 = sgn(z 3 z 5 )J 14 , 44 = J− 14 ,
2|z 4 | 2|z 4 |
 2  2
z5 z5
45 = −J 4
1 , 46 = −J− 4 1 ,
2|z 6 | 2|z 6 |
 2  2
z5 z5
53 = sgn(z 3 z 4 )J− 34 , 54 = −sgn(z 4 z 5 )J 34 ,
2|z 4 | 2|z 4 |
 2  2
z5 z5
55 = −sgn(z 5 z 6 )J− 34 , 56 = sgn(z 5 z 6 )J 34 ,
2|z 6 | 2|z 6 |
 2  2
z7 z7
65 = sgn(z 5 z 7 )J 14 , 66 = J− 14 .
2|z 6 | 2|z 6 |

where

z 1 = β1 l = k1 ω1 (1 − ω1 )2 + k2 ω2 (1 − ω2 )2 ,
z 2 = α1 l 2 = k1 (1 − ω1 )2 (2ω1 + 1) + k2 (1 − ω2 )2 (2ω2 + 1),
298 Y. Makzhanova

Fig. 6 Boundary curves for

the clamped beam

z 3 = (α1 l1 + β1 )l = k1 · 2ω12 (1 − ω1 )2
+ k2 (1 − ω2 )2 (ω1 (2ω2 + 1) − ω2 ),
z 4 = α2 l 2 = k1 ω12 (2ω1 − 3) + k2 (1 − ω2 )2 (2ω2 + 1),
z 5 = (α2 l2 + β2 )l = (α3 l2 + β3 )l = k1 ω12 (ω2 (2ω1 − 3) + (2 − ω1 )) + k2 · 2ω22 (1 − ω2 )2 ,
z 6 = α3 l 2 = k1 ω12 (2ω1 − 3) + k2 ω22 (2ω2 − 3),
z 7 = (α3 l + β3 )l = k1 ω12 (ω1 − 1) + k2 ω22 (ω2 − 1),

lj p j l2
ωj = ,kj =  , j = 1, 2.
l G IT E I y

The boundary of the feasible region in the plane of non-dimensional forces k1 k2

for different points of application of the forces is shown in Fig. 6.

3.3.2 Two-Span Continuous Beam

Finally, consider the two-span continuous beam with the spans of length l1 and l2
loaded by two independent concentrated forces p1 and p2 as shown in Fig. 7. It is
suppos ed that the twist angle above the central support is not zero.

Fig. 7 Two-span continuous

beam
Force Combinations Inducing Lateral Torsional Buckling 299

Solving the system similar to (12) and satisfying the boundary and continuity
conditions, we obtain the equation similar to (14) with nonzero determinant elements
 
z 32 z 32
11 = J 1 (z 1 ), 12 = −J 1 , 13 = −J ,
4 4 2|z 4 | − 41 2|z 4 |

z 32
21 = J (z ), 22 = sgn(z 3 z 4 )J 3 ,
− 43 1 − 4 2|z 4 |

z 32
23 = sgn(z 3 z 4 )J 3 ,
4 2|z 4 |
 
z 52 z 52
32 = sgn(z 3 z 5 )J 1 , 33 = J ,
4 2|z 4 | − 41 2|z 4 |
 
z 52 z 52
34 = −J 1 , 35 = −J ,
4 2|z 6 | − 41 2|z 6 |
 
z 52 z 52
42 = sgn(z 3 z 4 )J , 43 = −sgn(z 4 z 5 )J 3 ,
− 43 2|z 4 | 4 2|z 4 |
 
z 52 z 52
44 = −sgn(z 5 z 6 )J , 45 = sgn(z 5 z 6 )J 3 ,
− 43 2|z 6 | 4 2|z 6 |
 
z 72 z 72
54 = sgn(z 5 z 7 )J 1 , 55 = J ,
4 2|z 6 | − 41 2|z 6 |

z 72
56 = −J 1 ,
4 2|z 8 |
 
z 72 z 72
64 = sgn(z 5 z 6 )J , 65 = −sgn(z 6 z 7 )J 3 ,
− 43 2|z 6 | 4 2|z 6 |

z 72
66 = −sgn(z 7 z 8 )J ,
− 43 2|z 8 |

where
k1 k2
z1 = (1 − ω1 )ω12 λ21 (2 − ω1 λ1 (1 + ω1 )) − ω12 ω2 (1 − ω2 )(2 − ω2 )λ1 λ22 ,
4 4
1  
z2 = k1 λ1 (1 − ω1 )(2 − ω1 λ1 (1 + ω1 )) − k2 λ22 ω2 (1 − ω2 )(2 − ω2 ) ,
2λ1
ω1  
z3 = k1 λ1 (1 − ω1 )(2 − ω1 λ1 (1 + ω1 )) − k2 ω2 λ22 (1 − ω2 )(2 − ω2 ) ,
2
1     
z4 =− k1 ω1 λ1 2 + λ1 1 − ω12 + k2 ω2 λ22 (1 − ω2 )(2 − ω2 ) ,
2λ1
1   
z5 = − k1 ω1 1 − ω12 λ21 + k2 ω2 (1 − ω2 )(2 − ω2 )λ22 ,
2
1    
z6 = k1 ω1 λ21 1 − ω12 + k2 λ2 (1 − ω2 )(2 + ω2 λ2 (2 − ω2 )) ,
2λ2
1 − ω2    
z7 = −k1 ω1 λ21 1 − ω12 + k2 ω2 λ2 (2 − λ2 (2 − ω2 )(1 − ω2 )) ,
2
1    
z8 = k1 ω1 λ21 1 − ω12 − k2 ω2 λ2 (2 − λ2 (1 − ω2 )(2 − ω2 )) ,
2λ2
300 Y. Makzhanova

Fig. 8 Boundary curves for

two-span continuous beam

pi l 2 ai li
ki =  , ωi = , λi = , i = 1, 2
G IT E I y li l

The feasible region and its boundary in the plane of non-dimensional forces k1 k2
for different points of application of the forces for the two equal spans, i.e., λ1 =
λ2 = 21 , is shown in Fig. 8.

4 Discussion

As we can see from the graphs, in all the cases the feasible region is convex. The
dependence between critical values appears to be linear for two forces applied to one
span close to each other, but it is obviously nonlinear for the two forces applied to
different spans. In the last case, the form of the boundary curve demonstrates that
the forces are actually independent of each other at the moment of buckling and the
dangerous state occurs even if one of them is relatively small.
This means that the recommended by some investigators [10] approximate linear
relation

n
pi
=1
p
i=1 i cr
Force Combinations Inducing Lateral Torsional Buckling 301

where pi are given forces, pi cr is the critical value of the single force pi (computed
under assumption that the other forces vanish), for determination of critical combi-
nations of forces gives good approximation only for the forces applied to one span
of a continuous beam close to each other.
It can be expected that the same situation will occur if more than two forces
are applied to a beam. Thus, if two forces pi and p j are applied to one span of a
continuous beam, then in the section of the space of forces by the plane parallel to
the pi p j -plane we obtain the feasible region similar to the regions shown in Figs. 2,
4 and 6; and we obtain the feasible region similar to the one shown in Fig. 8 if the
forces pi and p j are applied to different spans.

5 Conclusions

For uniform elastic beams with rectangular cross sections subjected to two transverse
concentrated forces the convex feasible regions in the plane of forces have been
constructed. The pairs of forces from the feasible region do not induce lateral torsional
buckling of the beams. The exact equations for the boundary curves of the feasible
regions were derived analytically from the solution of Prandtl equation subject to
boundary and continuity conditions for a simply supported beam, a cantilever, a
clamped beam, and a two-span continuous beam.
The exact equations of the feasible region boundaries derived in the paper can
be used for estimating accuracy of various approximate methods [11] developed
to determine critical values of single forces inducing lateral torsional buckling or
critical combinations of such forces.

References

1. Olhoff N, Taylor JE (1983) J Appl Mech 50(4b):1139–1151

2. Vlasov VZ (1959) Thin-walled elastic beams. Fizmatgiz, Moscow
3. Vol’mir AS (1975) Stability of deformable systems. Science, Moscow
4. Timoshenko SP, Gere JM (1961) Theory of elastic stability. McGraw-Hill, New York
5. Secer M, Uzun ET (2017) IOP Conf Ser: Mater Sci Eng 245:032077
6. Sahraei A, Pezeshky P, Mohareb M, Doudak G (2018) Eng Struct 174:229–241
7. Torkamani MAM, Roberts ER (2009) Thin-Walled Struct 47:463–473
8. Asgarian B, Soltani M (2011) Procedia Eng 14:1653–1664
9. Kashkovskaya YV, Lyakhovich LS (1998) News of higher educational institutions. Construc-
tion 8(476):17–22
10. Agaev NG (1990) Engineering methods for studying the stability of thin-walled structures.
Strojizdat, Moscow
11. Sahraei A, Pezeshky P, Mohareb M (2017) Lateral torsional buckling analysis and design of
steel beams with continuous spans. In: Proceedings of the 6th international conference on
engineering mechanics and materials, 31 May–3 June 2017, Vancouver, Canada (2017)
Dynamic Analysis of Reinforced
Concrete Beams on Yielding Supports

Oleg Kumpyak, Zaur Galyautdinov, Daud Galyautdinov,

and Tatiana Rakhimova

Abstract Bearing structures of buildings and constructions can be subjected to

short-term dynamic loads. These loads characterized by high intensity and short
duration of action can lead not only to significant damage of structures but also to
the death of people. The paper considers one of the methods of active safety of
structures based on the use of yielding supports in the form of crushable inserts
of an annular section. The use of yielding supports allows to reduce the intensity
of the dynamic impact. As a result, the material consumption and labor intensity
of the restoration of structures are reduced. The influence of the compliance of
the support connections on the dynamic factor of structures and consequently on
displacements, forces and stresses is considered. The influence of support compliance
on the stress–strain state of structures under short-term dynamic loading is considered
by a numerical-analytical method. On the basis of theoretical research an algorithm
for dynamic calculation of reinforced concrete beams in a conditionally elastic stage
has been developed. The calculation of structures is carried out at various ratios of
the stiffness of the structure and the yielding supports, the stages of deformation of
the yielding supports and the laws of changing of the dynamical load. Calculations
have shown that the use of yielding supports that deforming only in the elastic stage
does not lead in all cases to a decrease in the dynamic factor. In this case ranges of
values ω may arise for which there is a negative effect of yielding supports on the
dynamic factor. The greatest efficiency of the use of yielding supports is achieved
when they are deformed in the plastic stage without going into the stage of hardening.
The degree of reduction of efforts and displacements is determined by the time of
transition of the yielding support from the elastic stage to the plastic one t SY, el .

Keywords Equation of motion · Reinforced concrete beam · Dynamic factor ·

Yielding support · Natural frequency · Short-term dynamic load · Displacements ·
Efforts

O. Kumpyak (B) · Z. Galyautdinov · D. Galyautdinov · T. Rakhimova

Tomsk State University of Architecture and Building, 634003 Tomsk, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 303
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_28
304 O. Kumpyak et al.

1 Introduction

Short-term dynamic loads of an emergency nature occur as a result of various types of

explosive load sand in cases of the development of progressive destruction of build-
ings and structures, due to technological explosions at the enterprises plants, detona-
tion of explosives during hostilities and terrorist acts [1–7]. In present daysfollowing
requirements of modern regulatory documents [8], design of reinforced concrete
structures is conducted on the basis of the current dynamic load. So composite
materials can be used to strengthen structures [9]. Active methods of safety struc-
tures based on a decrease in the intensity of dynamic impact have been developing
recently. One of the methods of active protection is the use of yielding supports [10–
16]. Experimental studies of reinforced concrete beams [17] and slabs supported
along the contour [18] on yielding supports indicate the high efficiency of their use
under intense dynamic loading.

2 Methods

In papers [19, 20] the calculation of a reinforced concrete beam on yielding supports
is considered. At the same time the case of loading is considered characterized by an
instantaneous increase in the load to the maximum value followed by a drop to zero
in the course of time  (Fig. 2a). In this paper we consider the change in the dynamic
factor of beams on yielding supports kd,SY for a general load with an increase to the
maximum value over time 1 and subsequent decline over time 2 (Fig. 1b).
A single-span conditionally elastic reinforced concrete beam on yielding supports
with a span is considered l. It is loaded with evenly distributed load (Fig. 1a). The
equation of motion of a beam on non-displaceable supports has the form [21]

∂ 4 y(x, t) ∂ 2 y(x, t)
B + m = p f (t), (1)
∂x4 ∂t 2

Fig. 1 Design model of a beam on yielding supports (a) and the law of loading history (b)
Dynamic Analysis of Reinforced Concrete Beams on Yielding … 305

Fig. 2 Shows the graphs of the change in the dynamic factor of beams on yielding supports for
different
 laws of load change intime (Wel = 1); 3—yieldingsupport beam (Wel = 10); 4—provided
2 1 = 1; 5—provided 2 1 = 10; 6—provided 2 1 = 100

where B—bending stiffness of the beam considering cracks in concrete; m—beam

mass per unit length; y(x, t)—travel function; f (t)—function of the law of loading
history.
Beam displacement function on yielding supports y(x, t) is taken equal to

y(x, t) = u(t) + p F(x)T (t), (2)

where u(t)—movement function of yielding supports; F(x)—displacement distri-

bution function due to beam deformation; T (t)—dynamic function.
Substitution of expression (2) into the equation of motion (1) and the imple-
mentation of transformations according to the Bubnov-Galerkin method leads to the
equation of motion of the beam on pliable supports

T̈ (t) + ω c1
2
T (t) = ω c1
2
f (t), (3)

where ω c1 —frequency of natural vibrations of a beam on yielding supports; T (t)—

dynamic function of a beam on yielding supports.
Value ω c1 determined from the relationship
ω
ω c1 =  , (4)
π4
1+ 2
· 1
W

3
where ω—natural vibration frequency of a beam on fixed supports; W = g·lB —
parameter characterizing the ratio of the stiffness of the yielding supports of that of
the beam; g—stiffness of yielding supports.
306 O. Kumpyak et al.

3 Results

Consider structures on elastic yielding supports of constant rigidity, i.e. in expression

(4) are taken W = Wel i g = gel . In the case under consideration the solution of
Eq. (3), for the adopted law of load change f (t), is the dynamic function Tel (t),
which has the form:
 
– provided 0 ≤ t ≤ 1 f (t) = t 1

t sin(ω c1 t)
Tel (t) = − , (5)
1 ω c1 1
 
– provided 1 < t ≤ 1 + 2 f (t) = 1 − t−1
2

(1 + 2 ) sin(ω c1 (t − 1 )) sin(ω c1 t) 2 − t + 1

Tel (t) = − + , (6)
ω c1 1 2 ω c1 1 2

– provided 1 + 2 < t ( f (t) = 0)

(1 + 2 ) sin(ω c1 (t − 1 )) sin(ω c1 t) sin(ω c1 (t − 1 − 2 ))

Tel (t) = − − ,
ω c1 1 2 ω c1 1 ω c1 2
(7)

The maximum value of the dynamic function Tel (t), defined by expressions
(5)–(7), represents the dynamic factor of beams on elastic yielding supports kd,SY .
Figure 2 shows the graphs of the change in the dynamic factor of beams on yielding
supports for different laws of loading history.
From the expression (4) it can be clear that the use of elastic yielding supports cause
a decrease in the frequency of natural vibrations of the “beam—yielding support”
system which leads to the “stretching” of the dynamic coefficient function to the
right along the abscissa ω  (cm. Figure 2). The diagrammatic sketch transformation
degree kd,SY is reciprocally proportional to the parameter Wel .
For animmediately increasing load the dynamicity function has a monotonically
increasing behavior. The dynamic factor of beams on yielding supports at any value
ω , is lower than this value for a beam on fixed supports (Fig. 2a).
For a load gradually increasing in time with a subsequent drop to zero, the function
of the dynamic factor of a beam on non-displaceable supports gets the most values
in the interval ω 1 = 0 . . . 5, after which it stabilizes at the value kd ≈ 1.1 . . . 1.2
(Fig. 2b).
For the specified load the use of yielding supports that deform only in the elastic
stage, first of all, does not cause a decrease in the dynamic factor in the entire range
of the parameter area ω 1 , and at the second, the area of parameter values ω 1
Dynamic Analysis of Reinforced Concrete Beams on Yielding … 307

Fig. 3 Dynamic coefficient versus flexibility of supports when Wel = 1 (a) and Wel = 100 (b)

is expanded, within which the dynamic factor achieves its ceiling value (Fig. 2b)
[7]. So, the graph shows that at ω 1 = 1 a decrease in the dynamic coefficient
up to 65% can be noted kd,SY . At ω 1 = 5 there is an increase kd,SY up to 47%.
From the graphs it can be clear that the  ceiling value function of the dynamic factor
for structures on rigid supports (2 1 = 1) reaches at 1 < ω 1 ≤ 5, and for
structures on yielding supports (Wel = 10) at 3 < ω 1 ≤ 12. So, for structures on
elastic yielding supports we have a wider range of ω 1 , wherein kd,SY exceeds the
value kd .
The effect of the flexibility of the supports on the dynamic coefficient 
of the beams
on the yielding supports can be seen more clearly in the graphs kd,SY kd −ω 1 ,
presented in Fig. 3. On the presented graphical dependencies, the values kd,SY kd ≤
1.0 determine the area of positive influence of the support compliance,
 i.e. range of
values ω 1 , at which kd,SY ≤ kd . Accordingly, when kd,SY kd > 1.0, the presence
of elastic yielding supports results in an increase in kd,SY relative to kd , i.e. in this
case a negative effect is observed.
Dependence diagrams in  Fig. 3 show that the efficiency of yielding supports,
depending on the ratio 2 1 , becomes apparent within a rather narrow range of
the parameter ω 1 . It decreases with an increase in the stiffness of supports. The use
of elastic yielding supports is advisable for Wel = 1 when ω 1 < 10, for Wel = 10
when ω 1 < 4.5 and forWel = 100 when ω 1 < 3.2. At higher values, a negative
 is observed. kd,SY kd and the boundary value of ω 1 decrease as Wel and
effect
2 1 increase.
So, the use of yielding supports of invariable stiffness for general dynamical
loads, defined by the stage of increase and decreases can have both a positive and
negative effect on the operation of structures. This must be considered when designing
structures on yielding supports in order to avoid greater displacements and forces,
compared to structures on non-displaceable supports.
To increase the resistance of reinforced concrete structures to dynamic impacts, it
is recommended to use yielding supports of variable stiffness. One of such supports
is a crushable annular insert [22]. It is characterized by three stages of deformation:
elastic, plastic, and the stage of hardening. Consider the calculation of beams on
308 O. Kumpyak et al.

yielding supports. They are deformed in the elasto-plastic stage with no transition to
the stage of hardening.
The solution of this equation of motion (3) in the transition of yielding supports
to the plastic stage of deformation in the stage of increasing load (0 ≤ t SY,el ≤ 1 ,
where t SY,el is the end of the elastic stage of supports deformation) for the law of
load change f (t) has the form:
 
– when t SY,el ≤ t ≤ 1 f (t) = t 1

     
sin ω c1 t − t SY,el Ṫ pl t SY,el 1 − 1
T pl (t) = +
ω c1 1
     
cos ω c1 t − t SY,el T pl t SY,el 1 − t SY,el t
+ + , (8)
1 1
 
– when 1 < t ≤ 1 + 2 f (t) = 1 − t− 2
1

 
sin(ω c1 (t − 1 )) Ṫ pl (1 )2 + 1
T pl (t) = +
ω c1 2
   2 − t + 1
+ cos(ω c1 (t − 1 )) T pl (1 ) − 1 + , (9)
2

– when 1 + 2 < t ( f (t) = 0)

sin(ω c1 (t − 1 − 2 ))
T pl (t) = Ṫ pl (1 + 2 )
ω c1
+ T pl (1 + 2 ) cos(ω c1 (t − 1 − 2 )). (10)

When yielding supports move to the plastic stage of deformation in the phase of
load decrease (1 < t SY,el ≤ 1 + 2 ), the solution of the equation of motion has
the form:
 
– when t SY,el < t ≤ 1 + 2 f (t) = 1 − t− 2
1

     
sin ω c1 t − t SY,el Ṫ pl t SY,el 2 + 1
T pl (t) = +
ω c1 2

       t SY,el − 1
+ cos ω c1 t − t SY,el T pl t SY,el − 1 +
2
2 − t + 1
+ , (11)
2
Dynamic Analysis of Reinforced Concrete Beams on Yielding … 309

– when 1 + 2 < t ( f (t) = 0)

sin(ω c1 (t − 1 − 2 ))
T pl (t) = Ṫ pl (1 + 2 )
ω c1
+ T pl (1 + 2 ) cos(ω c1 (t − 1 − 2 )). (12)

When yielding supports move to the plastic stage of deformation in the stage of
zero loading (t SY,el > 1 + 2 ), the solution of the equation of motion has the
form:
– when t SY,el < t ( f (t) = 0)

  
sin ω c1 t − t SY,el       
T pl (t) = Ṫ pl t SY,el + T pl t SY,el cos ω c1 t − t SY,el .
ω c1
(13)

ω1 g pl ·l 3
Here ω c1 = 4
, W pl = B
, g pl is the stiffness of the yielding support
1+ π2 · W1
pl

in the plastic stage.

Until t < t SY,el the beam is deformed on elastic supports, the dynamic function
in this case is determined
 by
 dependences
 (5)–(7).
The values of T pl t SY,el and Ṫ pl t SY,el are determined for the equal number of
displacements and movements at the end of the elastic and the beginning of the plastic
stages:
   
yel x, t SY,el = y pl x, t SY,el , (14)

l l
   
ẏel x, t SY,el d x = ẏ pl x, t SY,el d x, (15)
0 0

   
From the condition (14) we have T pl t SY,el = Tel t SY,el , and (15) implies that
  W (W +60)  
Ṫ pl t SY,el = W pl W el +60 Ṫel t SY,el .
el ( pl )
Equations (14) and (15) represent, respectively, the equality of displacements and
quantities of motion at the end of the elastic and the beginning of the plastic stage of
deformation of the yielding supports.
The maximum displacement and force values in the beam are achieved at tmax , at
which Ṫ pl (tmax ) = 0 and the dynamic coefficient kd,SY = T pl (tmax ).
Dependences (8)–(13) show that the dynamic coefficient is affected by the tran-
sition time of yielding supports from the elastic stage to the plastic  stage t SY,el , the
ratio of the stiffness of supports in the elastic and plastic stages g pl gel as well as the
310 O. Kumpyak et al.


Fig. 4 Variation of kd,SY kd for beams when Wel = 1 (a) and Wel = 100 (b)


Fig. 5 Variation of kd,SY kd for beams at optimum values of t SY,el : a Wel = 1; b Wel = 100


values of Wel ,W pl , ω 1 and 2 1 . The effect of t SY,el on the dynamic coefficient
kd,SY is shown in Figs. 4 and 5. 
As can be seen from the given dependences, when Wel = 1 and 2 1 = 1
(Fig. 4a) for all the considered values of ω 1 at any t SY,el , there is a positive effect
of using yielding
 supports (for ω 1 = 20 when t SY,el ≤ 0.75 tmax ). With an increase
of Wel , 2 1 and ω 1 , the area of positive effect of yielding  supports decreases
in size. However, for all the considered values of Wel , 2 1 and ω 1 there is
opt
an optimal time t SY,el , at which the dynamic coefficient kd,SY decreases the most.
 opt 
Herewith, kd,SY t SY,el kd < 1 at all times.
 opt 
As can be seen from the curves of kd,SY t SY,el kd − ω 1 , based on optimal time
opt
values t SY,el , the use of elasto-plastic yielding supports can decrease kd,SY within
the considered range of the parameter ω 1 = 0 . . . 100 (Fig. 5). Structures with
ω 1 = 5 . . . 10 are least affected by yielding supports.
 Yielding supports are most
effective whenω 1 < 5. When ω 1 > 10, kd,SY kd = 0.8 . . . 0.9 for all the values
of Wel and 2 1 .
Dynamic Analysis of Reinforced Concrete Beams on Yielding … 311

4 Results

The research findings have shown that the use of yielding supports can significantly
reduce the dynamic coefficient, and, consequently, decrease displacements, forces
and stress in structures exposed to short-term dynamic impacts of emergency nature.
The most effective are yielding supports that deform in the elasto-plastic stage with
no transition to the hardening stage.
The proposed method of selecting yielding supports can determine the parameters
of flexible yielding supports for the maximum decrease in the dynamic coefficient
of a structure.

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and regulations II-11-77* dated 2013 of Ministry of Construction of Russia, p 103
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Modeling of High-Speed Interaction
of Composite Barrier and Steel Striker

Andrei Plyaskin, Nikolai Belov, Nikolai Yugov, Emil Useinov,

and Marina Savintceva

Abstract Numerous studies, carried out in the field of reinforcement of building

constructions with composite materials using carbon composites, necessitate the
creation of adequate mathematical models and methods of strength analysis of such
structures under the action of static and dynamic loads. Currently, there are spec-
ification documents and recommendations, regulating the procedure for strength-
ening load-bearing structures with external reinforcement systems under the action
of static loads. The development of mathematical models and methods for assessing
the dynamic strength of composite materials under the action of high-intensity loads
of impact and explosion of civil defence shelters and special facilities is an urgent
task. The presented work deals with the results of theoretical studies of deformation,
destruction and open penetration of single-layer and combined multilayer barriers
at high-speed impact of ball-shaped strikers. The authors have proposed a mathe-
matical model, describing within the framework of flow mechanics the processes of
deformation and destruction of composite structures of concrete slabs, reinforced
by an external reinforcement system under shock-wave loading conditions. In the
range of meeting speeds of 1000–1500 m/s, the influence of the front and rear layers
of reinforcement with carbon unidirectional fabric on the epoxide target from fine
concrete on the process of penetration with a steel indenter at the approach angle of
90˚ has been investigated.

Keywords Shock-wave loading · Carbon-fiber-reinforced plastic · Fine concrete ·

Dynamic destruction · Mathematic simulation

Currently, epoxide and carbon fiber composite materials are increasingly used in
the construction field due to their high strength and resistance to cyclic loads and

A. Plyaskin (B) · N. Belov · M. Savintceva

Tomsk State University of Architecture and Building, 634003 Tomsk, Russia
N. Yugov
Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
E. Useinov
Limited Liability Company «TEFRA», 636000 Tomsk, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 313
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_29
314 A. Plyaskin et al.

aggressive environments. The combined use of carbon composites with other struc-
tural materials such as steel, concrete and wood allow creating hybrid products and
structures with high performance [1]. The accumulated considerable experience of
application of external reinforcement systems at reinforcement of building structures
shows effective use under the action of static and cyclic loads [2–6]. The use of carbon
composites in the design of protective structures, as well as their behavior under the
action of short-term high-intensity loads, is a poorly studied area. In this regard, the
development of alternative ways to increase the stability of structures with intense
dynamic impacts in combination with other technical solutions is an interesting and
promising direction [7–16].
The mathematical model, which is proposed in this work, makes it possible to
evaluate a dynamic strength and predict the behavior of a layered barrier, made of
fine-grained concrete and reinforced on both sides with a layer of carbon plastic,
under the action of high-speed impact by a steel spherical indenter [17].

1 Mathematical Model

The specific volume of the porous medium υ is represented as the sum of the specific
volume of the matrix material υ m , the specific volume of the pores υ p and the specific
volume of υ t , formed during the crack opening: υ = υ m + υ p + υ t [18, 19]. The
porosity of the material is characterized by the relative volume of voids ξ = ξp + ξ t ,
where ξp = υ p /υ – porosity relative volumes, ξ t = υ t /υ—relative volumes of cracks.
Also, the porosity of the material may be characterized by the parameter α = υ/υ m ,
which is related by the relationship α = 1/(1 - ξ ) [19].
The simultaneous equations, describing the motion of the porous elastoplastic
environment, have the form [8, 9, 14]:
  
d d
ρd V = 0, ρud V = n · σ d S,
dt dt
V V S
 
d sj
ρ Ed V = n · σ · ud S, e = + λs,
dt 2μ
V S
 
2 2 1 c02 p0 (1 − γ0 η/2)η
s : s = σT , p = + ρ0 γ0 ε (1)
3 α (1 − s0 η)2

where t—time; V—integration scope; S—its surface; n—outer normal unit vector;
ρ—density; σ = − pg +s σ = − pg +s—stress tensor; s—its deviator; p—pressure;
g—metric tensor; u—velocity vector; E = ε + u · u/2– specific total energy; ε—
specific internal energy; e = d − (d : g)g/3– strain rate tensor deviator; d = (∇u −
∇uT )/2– strain rate tensor; s J = Ps + s · ω − ω · s– derivative of the stress deviator in
the sense of Jaumann-Knoll; μ = μ0 (1 − ξ ) 1 − (6ρ0 c02 + 12μ0 )ξ/(9ρ0 c02 + 8μ0 )
Modeling of High-Speed Interaction of Composite Barrier and Steel Striker 315

, σT = σms /α—effective shear modulus and yield strength; ω = (∇uT − ∇u)/2–

vortex tensor; ρ0 , c0 , μ0 , σms , s0 —matrix material constants; η = 1 − ρ0 υ/α.
Parameter λ is eliminated by the yield condition.
Coefficients c0 and s0 are the coefficients of linear dependence of shock wave
velocity D on mass velocity u (D = c0 + s0 u).
System (1) is enclosed by equations, linking pressure p and porosity α under
compression [19]:
 
2 α
p ≥ σT ln( ) (2)
3 α−1
 
c02 ρ0 (1 − γ0 η/2)η 2 α
ρ0 γ0 ε + − σ T ln =0 (3)
(1 − s0 η)2 3 α−1
 
α
p ≤ −as ln( ) (4)
α−1
 
c02 ρ0 (1 − γ0 η/2)η α
ρ0 γ0 ε + + as ln =0 (5)
(1 − s0 η)2 α−1

where as —model parameter.

It is believed that concrete and carbon plastics under dynamic loading before
achieving the strength condition are described by the linear elastic body model.
The criterion, proposed for concrete, is used as the strength condition [19]:
 
J3 J2 3
3J2 = [AI1 + B] 1 − (1 − C) 1 − ( )− 2 , (6)
2 3

where I1 ,J2 ,J3 —the first invariant of stress tensor, the second and the third ones of
stress deviator, respectively;

3Tc2
A = Rc − R p ; B = Rc R p ; C = ;
Rc R p

where Rc ,R p ,Tc —strength limits at uniaxial compression, tension and pure shear,
respectively.
The surface (6) for isotropic materials is obliged to meet the condition of bulge
(according to Drucker and Hill postulates), which imposes the following limitations
on the design parameters

Tc
0, 530 ≤ ≤ 0, 577
Rc R p
316 A. Plyaskin et al.

Numerical values A,B,C are determined through concrete strength limits and
carbon plastics under tensile, compression and pure shear, obtained under dynamic
loading. After the strength criterion is met, the material is considered to be damaged
by cracks.
The fragmentation process of the fractured material and the behavior of the frac-
tured material are described within the porous elastoplastic medium model, discussed
above. For a damaged material, the yield stress depends on the pressure and is
determined by the formula:

(σmax − σmin )kp

σms = σmin +
(σmax − σmin ) + kp

where σmin , σmax , k—material constants [20].

Fragmentation of fractured material, subjected to tension, occurs when the relative
volume of voids reaches a critical value ξ∗ = α∗α−1
∗
If the material, damaged by cracks,
is subjected to compression, then the fragmentation criterion is the limit value of the
intensity of plastic deformations eu∗ :
√
2
eu∗ = 3T2 − T12 ,
3
where as – model parameter; T1 i T2 - the first and the second deformation tensor
invariants [20].
The broken material is modeled by a granular medium that withstands compressive
loads, but does not withstand tensile stresses.
This model is implemented in the “RANET-3” package of compute programs,
which allows solving the problems of impact and explosion in complete three-
dimensional setting.

1.1 Calculation Results

The impact interaction of steel indenter with diameter of 5.9 mm and weight of 0.8 g
with two types of targets, made of fine concrete, was calculated within the above
model in the range of meeting speeds of 1000–1500 m/s. The first target is a concrete
tile of 20 mm thick, the second one is a concrete tile of 20 mm thick, reinforced on
the front and back side with 1 mm thick layer of carbon composite. The effect of the
layer of carbon plastics on the process of breaking through targets was investigated
by computer modeling. Parameters of the model of the investigated materials are
presented in Tables 1 and 2.
Figure 1 shows a chronogram of penetrating by a steel ball weighing 0.8 g with an
impact speed of 1000 m/s of tiles, made of fine concrete with a thickness of 20 mm.
At the time of 40 μs (Fig. 1a), the process of breakage had started in the tile.
Modeling of High-Speed Interaction of Composite Barrier and Steel Striker 317

Table 1 State equation parameters

Material ρo , g/cm3 co , km/s s0 γ0 μ0 , GPa σms , GPa
Steel 7.85 4.56 1.50 2.26 79.0 0.64
Concrete 2.2 2.33 1.51 2.0 17.0 0.0412
Coal plastic 1.46 2.9 1.22 0.48 6.67 0.88

Table 2 Model parameters

Material Rc , Rp , Tc , as , GPa α0 ξ* eu∗ σmax , σmin , k
GPa GPa GPa GPa GPa
Steel – – – 0.17 1.0006 0.3 1.0 – – –
Concrete 0.1176 0.0118 0.085 0.0125 1.01 0.013 0.45 1.1527 0.0412 0.82
Coal 0.75 0.80 0.4415 0.1051 1.004 0.05 0.40 2.4 0.88 0.82
plastic

Fig. 1 Concrete tile penetration chronogram by steel ball with impact speed of 1000 m/s: a t1 =
40 μs, b t2 = 60 μs, c t3 = 100 μs

In the time following, there was a formation of main cracks, spreading from the
crater surface to the back surface of the target at an angle of 45˚ (Fig. 1b., t = 60 μs).
Complete destruction of concrete under the deformed striker occurred at the time of
80 μs. The picture of target penetration at the moment of time 100 μs is given in
Fig. 1c. After breaking through the barrier, the deformed striker weighing 0.67 g has
a speed of 82 m/s.
Figure 2 shows the results of the calculation of the impact interaction of a steel
ball with a concrete slab, reinforced on the front and back side with carbon plastic
with a thickness of 1 mm. At a given impact rate, the steel ball penetrated the front
layer of carbon plastic and stopped in the body of the concrete tile. The process of
impact interaction lasted 58 μs. The depth of the crater in the body of the concrete
slab was 15.75 mm, and the residual mass of the steel striker after interaction with
the barrier was 0.67 g.
The increase in the initial impact velocity to 1250 m/s (Fig. 3) led to the penetration
of the front layer of carbon plastics and concrete body of the target. As a result of
impact interaction, a crater 20.87 mm deep is formed in the concrete body of the
318 A. Plyaskin et al.

Fig. 2 The pattern of impact interaction of the steel striker at the speed of 1000 m/s with barrier,
reinforced with layers of carbon composite at time t1 = 58 μs

Fig. 3 Chronogram of the impact interaction of a spherical striker with a concrete slab, reinforced
with carbon plastics, at the speed of 1250 m / s: a t1 = 50 μs, b t2 = 60 μs, c t3 = 76 μs

layered barrier. Its weight at the moment of stop (t = 76 μs) is 0.57 g. The layer of
carbon composite, glued to the back surface of the target, was preserved from the
open penetration of the concrete target.
The increase in the speed of the striker meeting the target to 1500 m/s led to the
open penetration of the layered structure. Figure 4 shows the impact interaction at
100 μs. The speed of the striker after penetrating the layered barrier is 148 m/s. As
a result of interaction with the barrier, the striker partially collapses, which leads to
a decrease in mass to 0.5 g. Simultaneously, with the penetration of the rear layer of
carbon composite, it is detached from the concrete surface.

Fig. 4 Pattern of penetration

at the time of 100 μs by the
ball at the speed of 1500 m /
s of a plate of fine concrete,
protected from the front and
back surfaces by carbon fiber
Modeling of High-Speed Interaction of Composite Barrier and Steel Striker 319

2 Conclusions

The mathematical model of the behavior of composite materials of the layered struc-
ture under the action of high-speed impact, presented in the work, makes it possible
to predict the behavior of composite structures of special buildings and structures,
subjected to high-intensity exposure of impact and explosion. A joint laboratory
experiment and mathematic simulation will make it easier to understand the test
data, give them a correct physical interpretation and, on the other hand, clarify the
parameters of the proposed model.

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Information Modeling of Wind Flows
for Object of Parametrical Architecture

Galina Kravchenko, Elena Trufanova, and Lyubov Pudanova

Abstract Relevance of the multifunctional complexes design with use of informa-

tion modeling are consider in the article. The object of the study is a 240-m-high
multifunctional complex with four underground floors in Rostov-on-Don. Calcu-
lation model is a spatial slab-rod framework which consist of reinforced concrete
columns, floor slabs, steel spatial trusses and composite binding elements of the
frame. Analytical model is a set of parametric forms which one integrated as a whole.
The analytical surfaces shaping is studied with the use of algorithmic modeling
method. The finite element method allows you to refine the calculation of the wind
load for objects of complex geometric shapes. Special algorithm for creation an
information model is proposed. This algorithm based on a numerical experiment of
the study of the analytical surfaces shaping and the rational location to the prevailing
wind flows and turbulence zones of the parametric model.

Keywords Parametric architecture · Finite element method · Information model ·

Wind flows · Wind pressure

1 Introduction

The distinctive feature of modern digital architecture are the complex and various
curved shapes. This feature offers up the new architectural paradigm, understanding
of space and particular qualities of the combination real and virtual world in one
architectural object [1–4]. The main distinction of the parametric architecture is
the attempt to bring together the spatial dynamics of natural forms and the almost
complete lack of linearity [5–9].
Parametric is currently the most important and dominant style in avant-garde
architecture. This style requiring scale in all fields: from design of framework or
interior to large-scale urban planning. The fundamental condition for the design of
unique objects of urban construction is a development of new methods of parametric

G. Kravchenko · E. Trufanova · L. Pudanova (B)

Don State Technical University, 1 Gagarin Sq., Rostov-on-Don 344000, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 321
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_30
322 G. Kravchenko et al.

modelling. Digital design creates the possibility to model a harmonious interrelation

between the object, the environment and the person [10].
Multifunctional complexes are one of the most complicated types of buildings.
Such buildings are the center of the city’s infrastructure, flows of goods and services,
as well as objects of attracting large investments to the regions. Also, multifunctional
complexes are the leading direction in the development of construction and improving
the quality of urban development because of the combine of the functions of shopping
and entertainment complexes, office centres and residential buildings. Combination
of all these ideas in one building with a clear division of various purposes zones and
improvement of the quality of their functioning at the same time is one of the most
difficult decision from the design point of view.
A special way to increase the investment attractiveness of such objects is the archi-
tectural value improvement by use of the parametric surfaces [11, 12]. Accentuating
on the central part of the complex gives the building an expressive shape.

2 Methods

In the world practice, multifunctional complexes are real estate objects with a large
area that contain offices, apartments, shopping, hotel, gaming and other spaces. Such
complexity is the one of the reasons why this type of buildings is difficult to design.
Based on these ideas the object of the study is a 240-m-high multifunctional complex
with four underground floors in Rostov-on-Don. The framework of the object of the
study is a spatial slab-rod framework which consist of reinforced concrete columns,
floor slabs, steel spatial trusses and composite binding elements of the frame (Fig. 1).
The central module is a twisted around its axis cylindrical shape. The diameter of
the levels is varying in range from 28.3 to 55.5 m on depend of the height of the level.
The peripheral units have a shape of a crescent which one is stretched vertically.
Complexity of the geometry of the parametric object causes the need to model
the wind flow in order to determine the comfort indicators and the pressure values
of wind flows [13, 14]. The module Fluent of software package ANSYS is used to
perform the numerical experiment. This module based on the finite elements method
and designed to simulate wind impacts on structures in a wind tunnel [15–17].
At the first stage of the study were developed three-dimensional information
models with use of the software package REVIT [18]. This allowed to study the
shapes and angles of surface twisting. Analytical model is a set of parametric forms
which one integrated as a whole. The central module represents a straight astroid
cylindrical shape. This form is setting with use of parametric shaping algorithm for
vertical surface. The angle of rotation relative to the center of the surface in radians
from 0 to 2 π, the distance between the extreme points of the shape on the diagonal,
which is from 28.3 to 55.5 m (Fig. 2).
The typical level of the central module is an octagon with a variable size and
rotation angle relative to the central axis with a change of 1.5°. This solution allows
you to create a total rotation of the tower in 75° around its axis along the entire height.
Information Modeling of Wind Flows for Object of Parametrical … 323

Fig. 1 3D model of
multifunctional complex

Fig. 2 The shape of the

outer surface of the central
tower in the Y-X plane

The side towers are built on the principle of the central module. But they are offset
by a sinusoid according to the function (1):

y = 0.3 · sin x 2 (1)

At the second stage of the study were created an area around the building with the
size of the surrounding space 300 × 750 × 270 m. Refinement of the finite element
grid of computational models were made with using the approximation method.
324 G. Kravchenko et al.

Fig. 3 Part of the C++ code

of the FluentApp program

Special program FluentApp used for constructing wind flow velocities in three-
dimensional space. Figure 3 demonstrate the C++ macro for calculating the wind
speed and its changes with increasing height of the calculated area.

3 Results

The main task of the study is the investigation of influence of shape to the wind flows
and comfort zones. As a result, a simple geometric model in form of parallelepipeds
was developed for central and peripheral modules. Overall dimensions of these struc-
tures corresponding to the parametric model of the object. During investigation the
values of wind flow velocity indicators for spatial models in simple (Fig. 4) and
parametric form are obtained (Fig. 5).
Analyses of these parameters shows the advantages of the analytical shape of
the outer surface. The difference between wind speeds according to the developed

Fig. 4 The values of wind flow velocity for simple geometry model in plane XY
Information Modeling of Wind Flows for Object of Parametrical … 325

Fig. 5 The values of wind flow velocity for parametric model in plane XY

models is 35, 46 and 11% which one corresponds planes X, Y, Z. Such results confirm
the advantages of a unique parametric surface [19].
The rotations of the parametric model relative to the vertical axis at angles of 0°,
22.5° and 45°are studied to determine the prevailing wind flows (Fig. 6). The analysis
of the results showed that the rational model is a rotation relative to the prevailing
wind flows by 22.5° (Fig. 6b).

Fig. 6 Search for prevailing wind flows with relative to the vertical axis at angles: a 0°, b 22.5°,
c 45°
326 G. Kravchenko et al.

4 Discussion

As the result of the study we recommend to use new parametric model for the design of
a unique multifunctional complex. The features of this model are that this parametric
model obtained on the basis of the study of analytical surface shaping relative to the
prevailing wind flows. The rational model of the object was determined. These allows
to reduce wind pressure on the main supporting structures of the framework. Also,
the values of wind speeds on the different altitudes were determined. This part is
important for performing a dynamic calculation of an object of complex geometry
(Fig. 7). It is necessary to point the impact of wind influences on the stress–strain
state of the elements of the framework. Conducting research of this kind is extremely
important in design of the unique high-rise buildings.
Basic points of algorithmic method theory for modeling analytical surfaces with
varying specified parameters are used for creation calculation model of multifunc-
tional complex building in the spatial formulation. It is important to use special
software for rational parametric modeling. The authors during investigation were
used FluentApp for modeling wind flow velocities in three-dimensional space and
specially developed C++ macro for calculating the wind speed and its changes with
increasing height of the calculated area. This allows to perform detailed analysis of
the intensity of wind flows, to determine the zones of turbulence and the distribution
of wind pressure along the height. The use of software modules for model unique
building significantly improves the quality and aesthetics of the object.

Fig. 7 Rational location of the building relative to the prevailing wind flows
Information Modeling of Wind Flows for Object of Parametrical … 327

5 Conclusions

The development of innovative technologies and methods of information modeling

significantly increases the relevance of creating objects of parametric architecture
in urban development [20–22]. Shaping of analytical surfaces of parametric objects
are investigated algorithmic modeling method with use of the finite element method.
Objects of complex geometric shapes are demand the refines calculation for wind
load to solve the task of searching for rational arrangement of the model with respect
to the prevailing wind flows.
As the result of the study it is proposed the algorithm for creating an information
model on the base of the numerical experiment. Investigation of parametric model
for the rational arrangement of the framework with respect to the prevailing wind
flows and turbulence zones is one of the most important stages in the design of unique
buildings.

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Iterative Refinement of the Boundary
Condition in the Numerical Solution
of the Thermoelasticity Problem

Filipp Sergeyev and Fyodor Kiselyov

Abstract The climatic changes observed in recent years, associated with an increase
in the average annual temperature, cause the degradation of permafrost in the northern
territories. In this regard, increasing the accuracy of numerical modelling of the
temperature regime of the soil base of structures on permafrost is of particular rele-
vance. The inclusion of insolation in the number of external factors taken into account
in the model corresponds to this trend. However, in practice, determining the absorbed
solar energy by the ground surface is not an easy task. The article proposes an algo-
rithm for a numerical solution of the inverse boundary problem for the Newton’s
boundary condition of the heat equation. Local regularization of a nonlocal boundary
condition is used. The desired value is the solar absorption coefficient. The algorithm
is applied to solve the problem of calculating the thermal insulation of a flare pit for
a gas well cluster. The influence of solar radiation is estimated for the solution of the
problem of frozen soil being heated by a working gas burner.

Keywords Thermoelasticity · Soil mechanics · Permafrost · Newton’s boundary

condition · Numerical solution · Influence of solar radiation

1 Introduction

The paper aims at forming a numerical algorithm for determining one of the coeffi-
cients in the Newton’s boundary condition in the boundary thermoelasticity problem.
The inverse boundary problem is solved based on the known additional information
about the solution. The volume of water-saturated soil at different temperature condi-
tions is considered as the model area. The change in temperature is associated with
both its seasonal transitions from positive to negative values and additional heating
at the boundary. Additional heating may be caused by a gas burner operation on the
area surface with a temperature significantly exceeding the absolute values of its
seasonal fluctuations. The temperature passing the value of water freezing–thawing
in the pores of the soil leads the problem to the Stefan formulation, which requires

F. Sergeyev (B) · F. Kiselyov

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 329
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_31
330 F. Sergeyev and F. Kiselyov

determining the movement of the boundary between the frozen and thawed phases
of the soil. A change in the soil phase state causes a change in the deformation field
and stresses in it. Since the stiffness of the thawed phase is usually much less than the
stiffness of the frozen phase, unacceptable soil deformations can occur. While in the
normal natural state, a significant part of the soil is permafrost, its foundation defor-
mations caused by a gas burner operation in practice can lead to a man-made disaster
[1, 2]. In the three-dimensional formulation of such problems, numerical methods for
their solution are practically uncontested. The mathematical model used in the work,
its thermomechanical foundations, as well as the main approaches to solving such
problems are described in [3–5]. For numerical calculations, the author’s program
for solving problems of thermoelasticity as applied to soil mechanics is used.

2 Thermoelasticity Problem Statement

We simulate a soil body with a construction (flare pit) located on it, a layered two-
phase environment occupying volume V , each layer of which is homogeneous and
isotropic. This environment is characterized by the behaviour of water-saturated soil,
one phase of which is thawed, and the other is frozen. The case of small deformations
of a continuous medium will be considered. Let the radius vector of soil points be
denoted by x, and the increment of temperature be denoted by T with respect to its
initial distribution T0 (x).
The system of thermoelasticity equations, consisting of the Duhamel-Neumann
and thermal conductivity equations, which allows one to take into account the heat
capacity jump at the phase boundary, in the so-called generalized formulation of the
Stefan problem can be written as:
⎧  
⎨ ∂
C ∂uxkl = ρ Fi + αV K ∂∂ xTi
∂ x j  i jkl ∂
(1)
⎩ ∂
λi j ∂∂xTj = ρce f f ∂∂tT , i, j = 1, 2, 3
∂ xi

   
where Ci jkl (x, T ) = K − 23 G δi j δkl + G δik δ jl + δil δ jk denotes the tensor of
soil elasticity moduli, αV (x, T ) is the coefficient of cubical thermal expansion of
soil, λi j = λ(x, T )δi j is the soil thermal conductivity tensor, ρ(x, T ) is the soil
consistency, c(x, T ) is the soil specific heat.
The eff index in the designation of the effective heat capacity is omitted. For
numerical solution of the Stefan problem in a generalized formulation the shock-
capturing method was proposed in [6, 7].
The parts of the outer boundary  of the area V , on which the boundary conditions
of the first kind (Dirichlet) and the third kind (Newton-Richmann) for temperature
are set, are denoted by  T and  q respectively.
For a boundary element  q heat exchange with the environment is defined as
follows:
Iterative Refinement of the Boundary Condition in the Numerical … 331
 
α · q = β · (T − Tair ) − λ∼
· gradT · n, (2)

where α(x, t) · q(x, t) is the solar radiation vector q flux across the boundary with
an absorption coefficient α, β(x, t) · (T − Tair ) is the boundary heat exchange with
air at a temperature Tair with the heat exchange coefficient β.
On the boundary element  T we assume the temperature T̃ (t) to be defined:

T (t) = T (3)

The standard boundary conditions for displacements and the initial conditions
complete the problem statement:

∂u k
u(x, t) = 0, x ∈  u , − αV K T δi j n j ei = S0 , x ∈  σ .
Ci jkl
∂ xl (4)
u(x, 0) = 0, T (x, 0) = T0 (x)

To form a numerical solution of system (1), spatial discretization was performed

using the finite element method. For time discretization the implicit finite difference
Euler scheme [8, 9] was applied.

3 Statement of the Inverse Problem for Determining

the Absorption Coefficient α

Accounting for solar energy absorbed by the soil surface and specified in the boundary
condition (2) is a difficult task, since the value of the parameter α depends on hard-
to-determine factors, such as vegetation and snow cover, the fraction of long-wave
radiation reflected by the atmosphere towards the Earth’s surface, which in specific
geographic areas are usually not known [10, 11]. For this reason, before solving the
thermoelasticity problem, where the temperature of the soil and its deformations are
caused by seasonal changes in temperature, solar activity and a gas burner operation,
it is necessary to solve the problem of determining the coefficient α.
The parameter α included in boundary condition (2) is determined using a two-
step process, assuming that for a given geographical point we know the values of the
ambient air temperature Tair and the solar radiation flux q 0 through the normal to
the soil surface. They are presented in Tables 1 and Table 2. When stating the inverse
problem, additional information about the solution is required. Such information

Table 1 Average monthly and annual air temperature, 0 C

1 2 3 4 5 6 7 8 9 10 11 12 Year
−21.7 −22.4 −17.8 −13.5 −5.5 2.0 7.3 7.0 3.7 −4.5 −13.0 −18.0 −8.0
332 F. Sergeyev and F. Kiselyov

Table 2 Temperature at a depth of 1.5 m, 0 C, average monthly

01 02 03 04 05 06 07 08 09 10 11 12
−9.8 −10.4 −10.7 −8.3 −3.2 −0.8 −0.5 −0.1 −0.2 −1.4 −4.1 −7.2

 
will be the time distribution of the temperature T x1∗ , x2∗ , x3∗ , t at a fixed point
with coordinates x1∗ , x2∗ , x3∗ inside the soil. It is taken based on the data from an
exploratory thermal well at discrete times t n ∈ [0, T ], n = 0, 1, 2, . . ..
The uncoupled formulation of the thermoelasticity problem makes it possible to
solve the heat equation independently of the elasticity equation. Since the sought-for
function α is included in the boundary condition just for the heat equation, we formu-
late the direct heat conduction problem. We assume the following simplifications:
the thermal conductivity tensor λi j = λδi j , i, j = 1, 2, 3, i.e. it is spherical. We
assume that the soil has a layered structure and all functions included in the thermal
conductivity equation and in the boundary and initial conditions (those that do not
depend on the gas burner operation) depend only on the vertical coordinate x3 and
on time t. We place the origin of the axis x3 on the earth surface. That means, the
problem of thermal conductivity is solved on the segment x3 ∈ [x3∗ , 0], x3∗ < 0. Then
the search for function α consists in solving the following system of equations, the
variables in which depend on one spatial coordinate x3 and time t n : T (t n ) = T n :
⎧  
⎪
⎪ ∂
λ ∂ T n+1
= ρc T −T
n+1 n
, wher e x3 ∈ [x3∗ , 0], n = 0, 1, 2, . . .
⎪
⎪ ∂ x3 ∂ x3
 t

⎨
− λ ∂ T∂ x3 = α(x3 , t n+1 ), wher e x3 = 0
n+1
β(x3 , t n+1 ) T n+1 − Tair n+1
(5)
⎪
⎪ ∂T n+1
∗
⎪ λ ∂ x3 = 0, wher e x3 = x3
⎪
⎩ 0
T (x3 ) = T0 (x3 )

In Eq. (5) to simplify the notation the product of the unknown coefficient α(t n+1 )
by the known solar energy flux q(t n+1 ) is denoted α(t n+1 ). Problem (5) with a known
function α(t n+1 ) is a direct problem. Let us now formulate the inverse boundary
problem. It is characterized by the fact that in system (5) function α(t n+1 ) is not
specified. But an additional condition is known for the temperature at depth x3∗ :

T n+1 (x3∗ ) = ϕ n+1 (6)

Then the inverse problem of searching α n+1 = α(t n+1 ) consists in solving system
(5) with the additional condition (6). Thus, we get an ill-posed problem in which both
boundary conditions are specified on the same boundary: x3 = x3∗ . The method for
its solution consists in perturbing the boundary condition (6). Instead of condition
(6), we set a nonlocal boundary condition, where the temperature value is specified
at the boundary x3 = x3∗ , and the heat flux and heat exchange with the environment
are specified at the boundary x3 = 0:
Iterative Refinement of the Boundary Condition in the Numerical … 333

  ∂ T n+1
T n+1
(x3∗ ) +δ· β · T n+1
− n+1
Tair −λ (0) = ϕ n+1 (7)
∂ x3

where δ in condition (7) is a small value called the regularization parameter. At

δ → 0 condition (7) tends to condition (6). Now we are able to formulate a correct
problem for searching the function α(t n+1 ). An approximate solution of system (5)
with additional condition (6) is the solution of the following system of equations:
⎧  
∂ ∂ T n+1
⎪ = ρc T −T , wher e x3 ∈ [x3∗ , 0], n = 0, 1, 2, . . .
n+1 n
⎪ λ
⎪ 3
⎪ ∂ x ∂ x 3  
t
 
⎨ n+1 ∗
− λ ∂ T∂ x3 (0) = ϕ n+1
n+1
T (x3 ) + δ · β T n+1 − Tair n+1
(8)
⎪
⎪ λ ∂ T∂ x3 (x3∗ ) = 0
n+1
⎪
⎪
⎩ 0
T (x3 ) = T0 (x3 )

Its solution at each time step will be carried out in two iterations. To do this, we
represent the temperature T n+1 as a sum of two functions:

T n+1 (x3 ) = V n+1 (x3 ) + W (x3 )α n+1 (0) (9)

For each of them we formulate a separate boundary problem:

⎧  
∂ ∂ V n+1
⎪ = ρc V −T , wher e x3 ∈ [x3∗ , 0], n = 0, 1, 2, . . .
n+1 n
⎪ λ
⎪
⎪ ∂
 x 3

∂ x 3

t 
⎨
− λ ∂ V∂ x3 (0) = 0
n+1
β V n+1 − Tair n+1
(10)
⎪
⎪
⎪ λ ∂ V∂ x3 (x3∗ ) = 0
n+1

⎪
⎩ 0
V (x3 ) = T0 (x3 )

and
⎧  
∂ ∂W ∗
⎪
⎨ ∂ x3 λ ∂ x3 = ρc t , wher e x3 ∈ [x3 , 0], n = 0, 1, 2, . . .
W

⎪ (βW − λ ∂ W )(0) = 1 (11)

⎩ ∂ W ∗ ∂ x3
λ ∂ x3 (x3 ) = 0

Let’s find the functions α n+1 (0) in accordance with the method of A. Tikhonov
[12, 13] from the regularizing functional minimum condition:
   2  2
Jδ α n+1 (0) = T n+1 (x3 ) − ϕ n+1 + δ α n+1 (0) (12)

We get:

ϕ n+1 − V n+1 (x3∗ )

α n+1 (0) = W (x3∗ ) (13)
δ + W 2 (x3∗ )
334 F. Sergeyev and F. Kiselyov

Thus, we obtain an approximate solution to inverse problem (5), (6) by solving

boundary problems (11) and (12) at each time step. For each time step, using V n+1 (x3 )
found in (11) and W (x3 ) found in (12) we get the function α n+1 (0) from (13) and
the temperature T n+1 (x3 ) from the formula (9). The choice of the regularization
parameter is based on the residual of the method. For this, at each time step, iterations
are carried out to solve the inverse and direct problems.

4 Numerical Experiment

To estimate the influence of solar radiation on the problem of soil being heated
by a working gas burner, a model problem was solved. The temperature field on the
foundation of the flare pit during the gas burner operation is shown in Fig. 1. Seasonal
fluctuations in air temperature are presented in Table 2, and the monthly change in
the soil temperature at a depth of 2 m is given in Table 3. Thus, it is required to
additionally set the following boundary condition on the earth’s surface x3 = 0 :

T (x1 , x2 , x3 , t)|z=o = T 0 (x1 , x2 , 0, t)

Fig. 1 Temperature, 0C
from a gas flare and heating
zones at the upper boundary
of the computational region

Table 3 The calculated radiation balance of the earth’s surface (kcal/cm2 ), monthly
01 02 03 04 05 06 07 08 09 10 11 12
−1.3 −1.0 −0.9 −0.2 1.1 5.5 8.9 4.3 1.2 −0.7 −1.2 −1.2
Iterative Refinement of the Boundary Condition in the Numerical … 335

Fig. 2 Finite element mesh

where T 0 (x1 , x2 , 0, t) is the temperature on the surface of the flare pit at the times
of horizontal flare operation in accordance with Fig. 1. At other times, the boundary
condition (2) is satisfied.
The calculation was carried out for a three-layer medium. Flare operating mode:
16 days from September 15 to 30. The computational region consists of two paral-
lelepipeds with dimensions 80 m*36 m*14.5 m and 30 m*6 m*3.45 m. Time step is
1 day; simulation period is 1 year (Fig. 2).
To simulate the thermal effect of the flare system, September was chosen as the
warmest month.
The calculation results are presented in Figs. 3 and 4. Figure 3 contains a two-
dimensional temperature field for the calculation, in which seasonal temperature
changes only considers the ambient temperature. The thermal effect of the flare
system operation does not cause thawing of underlying frozen soils. Figure 4 shows
a two-dimensional temperature field for the calculation, in which the seasonal temper-
ature changes takes into account not only heat exchange with the surrounding air,
but also solar radiation. As one can see, the picture is different here. A layer of soil
up to 3 m thick turns out to be heated to a thawed state. This can lead to undesirable
deformations of the structure foundation. Consequently, predictive analysis of the
solar radiation absorption coefficient value in the Newton’s boundary condition is
necessary when solving practically significant problems.
336 F. Sergeyev and F. Kiselyov

Fig. 3 Temperature
distribution in the region
without taking into account
the effect of solar radiation

Fig. 4 Temperature
distribution taking into
account the effect of solar
radiation on the region
surface

5 Conclusions

1. A numerical algorithm is proposed for solving the inverse boundary problem for
determining the unknown value of the solar radiation absorption coefficient in
the Newton’s boundary condition in the heat conduction problem. It uses local
regularization of a nonlocal boundary condition. The existence and uniqueness
of the solution to the inverse problem ensures self-adjointness and positive
definiteness of the auxiliary problem operator.
Iterative Refinement of the Boundary Condition in the Numerical … 337

2. The model problem of soil heating by a high temperature generated during the
actual operation of gas pipelines in the permafrost zone confirms the importance
of taking into account a factor of radiation balance in the thermal regimes of
soils.
3. This algorithm can be applied when calculating thermal loads in elements of
aerospace structures exposed to solar radiation and cosmic rays.

The work was done within the framework of Interdisciplinary Scientific and
Educational School of Moscow University “Fundamental and Applied Space
Research".

References

1. Streletskiy D.A., Anisimov O.A., Vasiliev A.A. Permafrost degradation // Snow and ice-related
risks, hazards, and disasters / Ed. by W. Haeberli, C. Whiteman. Oxford, Elsevier, Acad. Press,
2014, p. 303–344
2. Anisimov O.A., Kokorev V.A. Cities of the Russian North in the context of climate change.
In Sustaining Russia’s Arctic cities, ed. R. Orttung, 141–174. 2017. New-York: Berghahn
Publishers.
3. Sergeyev F.V., Kiselyov F.B., Gotman N.Z. Numerical model for calculation of soil deformation
during thawing // All-Russian scientific and technical conference on geotechnics “Engineering
and geotechnical surveys, design and construction of bases, foundations and underground
structures". – 2017. – pp. 6–11.
4. Kiselyov F. B., Sergeyev F.V., Stepanov R. N. The impact of coupling thermoelasticity equations
on settlement of structures on frozen soil //International Journal for Computational Civil and
Structural Engineering. – 2017. – T. 13. – №. 1. – C. 110–115.
5. F. Kiselyov, F. Sergeyev Prediction of construction bases frozen soil temperature development
under intense heating // 2019 J. Phys.: Conf. Ser. 1425 012208, DOI: https://doi.org/10.1088/
1742-6596/1425/1/012208
6. Samarsky A.A., Vabishchevich P.N. Computational heat transmission.– M.: Editorial URSS.–
2003.– 784 p.
7. Budak B.M., Solovyeva E.N., Uspensky A.B. The difference method with smoothing coeffi-
cients for solving Stefan’s problems // J. of comp. math. and math. phys. — 1965. — T. 5.
— № 5. — pp. 828—840.
8. Kudryavtsev S.A., Paramonov V.N., Saharov I.I., Shashkin A.G. Using the finite element
method in solving geotechnical problems. – Khabarovsk: Publishing house DVGUPS, 2014.
– 162 p.
9. Sheshenin S.V., Lazarev B.P. Numerical modeling of soil freezing with consideration of
moisture transfer. // Science and technology in the road industry, № 3 - 2015, pp. 27 - 30.
10. Filimonov M.Yu., Vaganova N.A. Simulation of Thermal Fields in the Permafrost With
Seasonal Cooling Devices // Proceedings of the Biennial International Pipeline Conference, IPC
Ser. “2012 9th International Pipeline Conference, IPC 2012". – 2012. – P. 133–141. Doi: https://
doi.org/10.1115/IPC2012-90287
11. Filimonov M. Y. Simulation of permafrost changes due to technogenic influences of different
ingeneering constructions used in nothern oil and gas fields / M. Y. Filimonov, N. A. Vaganova
// Journal of Physics: Conference Series. — 2016. — Vol. 754. — Iss. 11. — 112004.
338 F. Sergeyev and F. Kiselyov

12. Samarsky A.A., Vabishchevich P.N. Numerical methods for solving inverse problems of
mathematical physics.– M.: LKI publishing house.– 2009.– 480 p.
13. Filimonov M., Vaganova N. Numerical Simulation of Technogenic and Climatic Influence on
Permafrost // Advances in Environmental Research. 2017. Volume 54. Chapter 5. NY: Nova
Science Publishers. ISBN: 978–1–53610–667–1.– P. 117–142.
Pochhammer—Chree Wave Dispersion
in Hollow Cylinders

Tagibek Gadzhibekov

Abstract The closed form analytical solutions of the dispersion equation for prop-
agating modes of the Pochhammer—Chree waves in a hollow cylinder with the
traction free boundary conditions at the outer surfaces of the cylinder, are derived.
The dispersion portraits for longitudinal symmetric modes are obtained, revealing
a substantial discrepancy in dispersion of the considered waves in hollow and solid
cylinders. Along with the dispersion analysis, the analytical expressions for the elastic
harmonic displacement fields are derived.

Keywords Pochhammer—Chree wave · Hollow cylinder · Acoustic wave ·

Dispersion curves

1 Introduction

Research on Pochhammer—Cree waves in rods has been going on for over a century.
These results find applications in various fields.
The first equations describing the propagation of elastic waves in an infinite rod
were independently obtained by Pochhammer [1] and Cree [2, 3].
Dispersion curves (dependence of the phase velocity on frequency) were obtained
later on by numerical methods [4–9]; in these works, bending, torsional and
longitudinal axially symmetric modes are considered.
Using asymptotic methods, the short-wavelength limit of the phase velocity
c1,lim = c R (c R —is the velocity of the Rayleigh wave [10]) was obtained.√ And
the long-wavelength limit, which coincided with the rod velocity c2,lim = E/ρ (E
is the Young’s modulus, ρ is the density of the rod material).
In non-circular rods, Pochhammer—Cree waves were studied using the finite
element method in [11, 12].

T. Gadzhibekov (B)
Bauman Moscow State Technical University, 2-Ya Baumanskaya Ulitsa, 5, Moscow 105005,
Russia
Ishlinsky Institute for Problems in Mechanics RAS, 101 Prosp. Vernadskogo, Moscow 119526,
Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 339
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_32
340 T. Gadzhibekov

2 Basic Equations

The conclusions from [13] are taken as a basis.

For a homogeneous isotropic elastic body, the equations of motion can be
represented in the following form:

c12 ∇divu − c22 rot rotu = ü (1)

where u is the displacement field,

   
c1 = λ + 2μ ρ, c2 = μ ρ (2)

the velocities of propagation of longitudinal and transverse waves in an unlimited

isotropic medium, respectively, μ and λ are the Lame parameters, ρ is the material
density.

Applying the Helmholtz expansion theorem to the vector displacement field, the
following representation is obtained:

u = ∇ + rot (3)

here ,  are scalar and vector potentials. For the displacement components in
cylindrical coordinates, the Helmholtz representation takes the following form:

∂ 1 ∂z ∂θ
ur = + −
∂r r ∂θ ∂z
1 ∂ ∂r ∂z
uθ = + − (4)
r ∂θ ∂z ∂r
∂ 1 ∂ 1 ∂r
uz = + (r θ ) −
∂z r ∂r r ∂θ

Considering axial symmetry:

uθ = 0 (5)

Substituting expressions (3) into the equations of motion (1), we obtain:

¨
c12  = , ¨
c22  =  (6)

Potentials (6) for a harmonic wave propagating along the z-axis can be represented
as:

 = 0 (x  )e−iγ (ct−z) ,  = 0 (x  )e−iγ (ct−z) (7)

Pochhammer—Chree Wave Dispersion in Hollow Cylinders 341

here x  is the coordinate in the cross section, γ = ω/ c is the wavenumber, ω is

the circular frequency, c is the phase velocity.
Substituting (7) into (6), we obtain the Helmholtz equations for the corresponding
potentials:
   2 
c2 c
0 + − 1 γ 0 = 0, 0 + 2 − 1 γ 2 0 = 0
2
(8)
c12 c2

Passing
 to cylindrical coordinates, taking into account the axial symmetry
( ∂0 ∂θ = 0), for the scalar potential (8) we obtain the Bessel equation.
 
1 d d c2
r 0 (r ) + − 1 γ 2 0 (r ) = 0 (9)
r dr dr c12

The solution to Eq. (9) is expressed in terms of the Bessel functions of the first
and second kind:

0 (r ) = C1 J0 (q1 r ) + C2 Y0 (q1 r ) (10)

  
here C1 , C2 are unknown complex coefficients, q12 = c2 c12 − 1 γ 2 .
The axial symmetry condition for the vector potential has the form:
  
∂r ∂θ = ∂θ ∂θ = ∂z ∂θ = 0 (11)

The transition to cylindrical coordinates for the vector potential, taking into
account conditions (11), gives the Bessel equations (for each of the components):
  
1 d d c2 1
r r (r ) + − 1 γ 2
− r (r ) = 0
r dr dr c22 r2
 2  
1 d d c 1
r θ (r ) + − 1 γ − 2 θ (r ) = 0
2
(12)
r dr dr c22 r
 2 
1 d d c
r z (r ) + 2 − 1 γ 2 z (r ) = 0
r dr dr c2

The solutions to these equations have the form:

θ (r ) = C3 J1 (q2 r ) + C4 Y1 (q2 r )
r (r ) = C5 J1 (q2 r ) + C6 Y1 (q2 r ) (13)
z (r ) = C7 J0 (q2 r ) + C8 Y0 (q2 r )
  
where C3 , ..., C8 are unknown complex coefficients, q22 = c2 c22 − 1 γ 2 .
The condition of axial symmetry of the vector potential imposes one more
restriction [6]:
342 T. Gadzhibekov

r = z = 0 (14)

Taking into account (4), (5), (10), (13), (14), the vector displacement field is
represented in the form [8]:

u r = −(q1 (C1 J1 (q1r ) + C2 Y1 (q1 r )) + iγ (C3 J1 (q2 r ) + C4 Y1 (q2 r )))e−iγ (ct−z)

uθ = 0
u z = (iγ (C1 J0 (q1r ) + C2 Y0 (q1 r )) + q2 (C3 J0 (q2 r ) + C4 Y0 (q2 r )))e−iγ (ct−z)
(15)

3 Dispersion Equations

To describe the propagation of Pochhammer—Cree waves in a solid rod, it is taken

into account that the displacements at are finite, therefore, the coefficients at the
Bessel functions of the second kind (C2 and C4 ) are taken to be zero. To describe
waves in a hollow cylinder, all terms in expression (15) are retained.
The condition for the equality of forces on the lateral surface of the cylinder to
zero is as follows:

tν ≡ (λ(trε)ν + 2με · ν)|r =R = 0 (16)

here ν is the unit vector of the outward normal to the lateral surface.
Shear stresses for components trr and tr z are determined by the expressions:

trr = λIε + 2μεrr

(17)
tr z = 2μεr z

here Iε = εθθ + εzz + εrr .

The strain tensor components are determined by the formulas:
 
∂u r ∂u z ur 1 ∂u r ∂u z
εrr = ; εzz = ; εθθ = ; εr z = + (18)
∂r ∂z r 2 ∂z ∂r

Substituting the previously obtained expressions (15) into (18) and further into
(17), we obtain the boundary conditions for two surfaces of a hollow cylinder (up to
an exponential factor e−iγ (ct−z) ):
Pochhammer—Chree Wave Dispersion in Hollow Cylinders 343

⎡   ⎤
C1 r γ 2 λ + q12 (λ + 2μ) J0 (q1 r ) − 2q1 μJ1 (q1 r ) +
 2 
trr = −⎣ +C2 r γ λ + q12 (λ + 2μ) Y0 (q1 r ) − 2q1 μY1 (q1 r ) + ⎦
+2iγ μ[C3 (q2 r J0 (q2 r ) − J1 (q2 r )) + C4 (q2 r Y0 (q2 r ) − Y1 (q2 r ))] r =R
1 ,R2

=0
 
2iq1 γ μ(C1 J1 (q1 r ) + C2 Y1 (q1 r ))+
tr z = −  2  =0 (19)
+μ q2 − γ 2 (C3 J1 (q2 r ) + C4 Y1 (q2 r )) r =R
1 ,R2


Using this identity λ/μ = c12 c22 −2, conditions (19) give the dispersion equation:

det A = 0 (20)

We introduce the following functions to set the components of the dispersion

equation matrix A:

  2qμ
f 1 (q, R) = −λ γ 2 λ + q 2 (λ + 2μ) J0 (q R) + J1 (q R)
R
  2qμ
f 2 (q, R) = −λ γ 2 λ + q 2 (λ + 2μ) Y0 (q R) + Y1 (q R)
R
2iγ μ
f 3 (q, R) = − (q R J0 (q R) − J1 (q R))
R
2iγ μ (21)
f 4 (q, R) = − (q RY0 (q R) − Y1 (q R))
R
f 5 (q, R) = −2iqγ μJ1 (q R)
f 6 (q, R) = −2iqγ μY1 (q R)
 
f 4 (q, R) = − q22 − γ 2 μJ1 (q R)
 
f 8 (q, R) = − q22 − γ 2 μY1 (q R)

Taking into account functions (21), the elements of the matrix are

A11 = f 1 (q1 , R1 ); A12 = f 2 (q1 , R1 ); A13 = f 3 (q2 , R1 ); A14 = f 4 (q2 , R1 );

A21 = f 1 (q1 , R2 ); A22 = f 2 (q1 , R2 ); A23 = f 3 (q2 , R2 ); A24 = f 4 (q2 , R2 );
A31 = f 5 (q1 , R1 ); A32 = f 6 (q1 , R1 ); A33 = f 7 (q2 , R1 ); A34 = f 8 (q2 , R1 );
A41 = f 5 (q1 , R2 ); A42 = f 6 (q1 , R2 ); A43 = f 7 (q2 , R2 ); A44 = f 8 (q2 , R2 ).
(22)

The eigenvectors of this matrix dispersion equation, which are advised to be zero
eigenvalues, determine the polarization of the waves.
Solving Eqs. (20) taking into account (21), (22) at different values of the
velocity, the dispersion curves of Pochhammer—Cree waves for a hollow rod were
344 T. Gadzhibekov

Fig. 1 Dispersion curves for a hollow bar

obtained for the first time. The results were obtained for the following dimensionless
quantities:

R1 R2 = 0, 5—the ratio of the inner radius to the outer radius of the bar;
ν = 0, 2—Poisson’s ratio;
E = 1—elastic modulus;
ρ = 1—density of the rod material. 
The abscissa is the dimensionless velocity c c1 , the ordinate is the angular
frequency ω. L(0, m) is the Pochhammer—Cree longitudinal fundamental mode.
L(k, m), k = 1, ..., n higher longitudinal modes of Pochhammer—Cree. Looking
at Fig. 1 that all dispersion curves have no intersections, except for modes L(0, m)
and L(1, m), which have a common portion of the curve.

4 Conclusions

Analytical expressions are obtained for the dispersion equation describing the prop-
agation of longitudinal axisymmetric Pochhammer—Cree wave modes in an infinite
hollow cylinder.
On the basis of analytical expressions, dispersion curves for higher longitudinal
modes of Pochhammer—Cree waves in an infinite hollow cylinder are constructed.
The analysis revealed a substantial discrepancy performed in dispersion of the
considered Pochhammer—Chree waves in hollow and solid cylinders [4, 6, 14–16].
It should also be noted that the second limiting velocity (Fig. 1) in case of the
hollow cylinder has been observed apparently for the first time; in this respect see
also [17, 18].
Pochhammer—Chree Wave Dispersion in Hollow Cylinders 345

Acknowledgements The work was supported by the Russian Science Foundation (grant No.20-
49-08002).

References

1. L. Pochhammer, J. Reine Angew. Math. 81, Ueber die Fortpflanzungsgeschwindigkeiten kleiner

Schwingungen in einem unbegrenzten isotropen Kreiscylinder, 324–36 (1876)
2. C. Chree, Quart. J. Pure Appl. Math. 21, Longitudinal vibrations of a circular bar, 287– 98
(1886)
3. C. Chree, Trans Cambridge Philos. Soc. 14, The equations of an isotropic elastic solid in polar
and cylindrical coordinates, their solutions and applications, 250–309 (1889)
4. Field GS Longitudinal waves in cylinders of liquid, in hollow tubes and in solid rods. Can J
Res 11:254–263 (1934)
5. R. D. Mindlin, H. D. McNiven, Axially symmetric waves in elastic rods. Trans. ASME. J.
Appl. Mech. 27:145–151 (1960)
6. H. Kolsky (1964) Stress waves in solids. J Sound Vibr 1:88–110
7. K. F. Graff, Wave motion in elastic solids, New York: Dover, p. 692 (1991)
8. H. N. Abramson, J. Acoust. Soc. Amer. 29, Flexural waves in elastic beams of circular cross
section, 1284–1286 (1957)
9. Y. H. Pao, R. D. Mindlin, Trans. ASME. J. Appl. Mech. 27, Dispersion of flexural waves in an
elastic, circular cylinder, 513–520 (1960)
10. J. W. S. Rayleigh, Proc. London Math. Soc. 17, On waves propagating along the plane surface
of an elastic solid, 4–11 (1887)
11. L. Gavric, J.Sound Vib. 185, Computation of propagative waves in free rail using a finite
element technique, 531–543 (1995)
12. T. Hayashi, C. Tamayama, M. Murase, Ultrasonics. 44, Wave structure analysis of guided
waves in a bar with an arbitrary cross-section, 17–24 (2006)
13. M. E. Gurtin, The Linear Theory of Elasticity. In: Handbuch der Physik, Bd. VIa/2. Springer,
New York, (1972)
14. A. V. Ilyashenko, S. V. Kuznetsov, Arch. Appl. Mech. 88, Pochhammer–Chree waves:
polarization of the axially symmetric modes, 1385–1394 (2018)
15. S. V. Kuznetsov, Z. Angew. Math. Phys. 69(142), Pochhammer–Chree waves in rods:
degeneracy at the bulk wave velocities, 1–8 (2018)
16. S. V. Kuznetsov, J. Vibr. Control. 24(23), Abnormality of the longitudinal Pochhammer–Chree
waves in the vicinity of C2 phase speed, 5642–5649 (2018)
17. I. Djeran-Maigre, S. V. Kuznetsov, Comptes Rendus. Mécanique. 336(1–2), Solitary SH waves
in two-layered traction-free plates, 102–107 (2008)
18. R. V. Goldstein, Mech. Solids. 52, Long-wave asymptotics of Lamb waves, 700–707 (2017)
Prefabricated Steel Structures
with a Corrugated Web (Part 2.
Load-Bearing Capacity of a Steel Beam
with a Profiled Sheet Web)

Alexander Ibragimov, Ekaterina Zinoveva, Stanislav Rosinsky,

and Lyubov Gnedina

Abstract In the first part of the article, prefabricated metal structures with a corru-
gated wall are considered. The urgency of their development, new improvements to
existing beam structures and their application is substantiated. Proposed beams which
made of a composite I-beam, but with a split section of individual elements. Partic-
ular attention is paid to collapsible products and the possibility of their delivery to
the installation site in “unrolled” form, assembly and installation at the construction
site, dismantling (if necessary) and reuse. Fastening between themselves is carried
out on self-tapping screws or bolts, which allows assembly at the installation site.
To increase the rigidity of the profiled sheet wall without significant changes in
the geometric parameters, including the wall thickness, it will be sufficient to use
additional profiled sheet layers for strengthening. This article discusses the bearing
capacity of such beams. The software package ANSYS was used. The geometric
model of the beam has been created. To check the functionality of the program, an
initial calculation was made. The calculation of ten options of the beam has been
made. The first three options consider three types of profiled sheet used as a wall of
I-beam. Based on the calculation results, the second version of the profiled sheet was
selected for the next four calculation options for different corrugation heights and
three calculation options for different spans. The results of calculating all 10 options
are summarized in a single table for ease of analysis.

Keywords Stress–strain state · Bearing capacity · ANSYS software package ·

Displacements · Normal and shear stresses · Prefabricated steel structures ·
Corrugated wall · Profiled sheet · Corrugated sheet

1 Introduction

Today, mobile, easily transportable, pre-fabricated structures require modern

constructive and progressive solutions. Mobile structures with low resource
consumption and ease of assembly can be effectively used in the Far North, as well

A. Ibragimov · E. Zinoveva (B) · S. Rosinsky · L. Gnedina

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 347
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_33
348 A. Ibragimov et al.

as, if necessary, temporary settlements in emergencies or martial law, which proves

the importance of these buildings.
In addition, the tendencies of wide distribution of prefabricated buildings and
structures include the following:
• High competition, division of large structures into smaller ones. The creation of
more enterprises with more specific specializations makes it possible to make
short predictions.
• Lack of developed infrastructure in the regions of Siberia and the Far East;
• The need for modern temporary structures against the backdrop of natural disasters
and emergencies, when high-speed construction of structures is required right at
the scene [1–3].
One way to solve this problem is to use prefabricated steel structures with a corru-
gated web, the main advantage of which is the high bearing capacity at a relatively
low cost and universal availability of materials, as well as simplicity and speed of
assembly. An urgent task is not only the creation of such structures at the level of an
idea, but also a theoretical, calculated, experimental and economic substantiation of
their possible real-scale implementation [4–9].

2 Materials and Methods

2.1 Modeling a Steel Beam with a Profiled Sheet Web

by Finite Element Analysis in ANSYS

Based on the experience of the previous study, it was decided to use the ANSYS
software package as it is more convenient and accurate for modeling this type of
structure.
This program offers a continuously growing list of calculated means that are
capable of:
– take into account a variety of constructive nonlinearities;
– to make it possible to solve the most general case of a contact problem for surfaces;
– allow the presence of large (finite) deformations and angles of rotation;
– carry out interactive optimization and analysis of the influence of electromagnetic
fields;
– get a solution to hydrodynamics problems and much more, along with parametric
modeling, adaptive mesh rebuilding, the use of p-elements and extensive possibil-
ities for creating macros using the ANSYS parametric design language (APDL)
[10, 11].
Prefabricated Steel Structures with a Corrugated Web … 349

2.2 The Purpose and Objectives of the Calculation

Objective: To simulate a composite I-beam beam with a profiled sheet wall.

Tasks:
• Determining the finite element type for beam modeling
• Assembling the calculated model
• Derivation of dependencies of parameters and bearing capacity of a beam
• Analysis of results

2.3 Creation of a Geometric Model in the ANSYS Software

Structurally, the following stages in building a model can be distinguished: prepro-

cessor, processor, postprocessor. In the preprocessor, geometry and / or FE-mesh of
the computational area is created, boundary conditions are indicated: static, kine-
matic, etc. The processor determines the type of the problem being solved, parame-
ters and settings for the calculation. The postprocessor implements the output of the
results. ANSYS interface is presented in two types: command and graphical. In this
work, the focus is on the command interface. Some general syntax rules should be
noted:
• Each line contains one statement. It is allowed to combine operators with the $
sign, but it impairs the readability of the code.
• No description of variable types is required: all variables are accepted as valid by
default.
• There is automatic recognition of variables.
• For variable names, only Latin letters are used, the case does not differ;
• Symbols are not allowed in names:! @ # $% & ˆ * () _ - + = | \ {} [] “‘ / < > ~
• Line comments are implemented using the symbol:!
• The command line is limited to 255 characters maximum [12–14].
Thus, a parametric code was generated to simulate a steel beam with a profiled
sheet web (approximately 200 lines).
In this variant, the following parameter values are primarily set, shows in Table
1.

3 Results

To check the functionality of the program, an initial calculation was made. For the
given values of the beam parameters, the following results were obtained (Fig. 1.)
Conclusion: the program code is workable. Can continue research based on this
model.
350 A. Ibragimov et al.

Table 1 Primary calculation parameters

№ Parameter Numerical value Unit rev
1 Span of beam, L 7200 mm
2 Elastic modulus of materials (steel) 2.1 × 105 MN/m2
3 Thickness of corrugation 2 mm
4 Height of corrugation 200 mm
5 Step of cell of corrugation 100 mm
6 Step between of cell of corrugation 10 mm
7 Mast (height) of cell of corrugation 10 mm
8 Tilt angle of cell of corrugation 60 °
9 Tilt angle of corrugation relative to the Y 0 °
axis
10 Section length 1200 mm
11 Size of corner 75 mm
12 Thickness of corner 5 mm
13 Bolt diameter 5 mm
corner—corrugation—corner
14 Bolt diameter corner—plate—corner 10 mm
15 Thickness of plate 10 mm
16 Width of plate 50 mm
17 Load 10*L N/mm*mm = kN/m*m

Fig. 1 Results of calculating the beam with primary parameters (VAT)

4 Discussions

In order to track the dependence between the parameters, it is necessary to leave

some of them unchanged and fix, and consider the other as a variable.
The main parameters can be considered the load, span and height of the
corrugation.
To set the correct geometric characteristics, it is worth considering the set of rules,
GOSTs and assortments.
Prefabricated Steel Structures with a Corrugated Web … 351

On the basis of GOST 24,045–94 Steel sheet bent profiles with trapezoidal corru-
gations for construction [15], the geometrical parameters of the corrugated sheet
were adopted for the first three options. According to the purpose, profiled sheets are
divided into types:
F—for flooring coverings;
FW—for flooring and wall fences;
W—for wall protections.
In the options under consideration, a W-type profiled sheet was used for the beam
(Fig. 2).
1. Different types of profiled sheet (Tables 2, Table 3 and Table 4):

Fig. 2 W-type profiled sheet

Table 2 Different types of a W-type profiled sheet

№ Parameter Numerical value Unit rev Types of profiled sheet
1 2 3
1 Thickness of 2 mm 0.6 0.7 0.7
corrugation
2 Height of corrugation 200 mm 700 700 700
3 Step of cell of 100 mm 70 70 135
corrugation
4 Step between of cell 10 mm 35 35 35
of corrugation
5 Mast (height) of cell 10 mm 21 21 44
of corrugation
6 Tilt angle of cell of 60 ° 54.5° 54.5° 53.5°
corrugation
352 A. Ibragimov et al.

Table 3 Data for the second type of calculation

№ Parameter Unit rev Variant
4 5 6 7
1 Thickness of corrugation mm 0.7 0.7 0.7 0.7
2 Height of corrugation mm 300 500 700 900
3 Step of cell of corrugation mm 70 70 70 70
4 Step between of cell of corrugation mm 35 35 35 35
5 Mast (height) of cell of corrugation mm 21 21 21 21
6 Tilt angle of cell of corrugation ° 54.5° 54.5° 54.5° 54.5°

Table 4 Data for the third type of calculation

№ Parameter Unit rev Variant
8 9 10
1 Thickness of corrugation mm 0.7 0.7 0.7
2 Height of corrugation mm 500 500 500
3 Step of cell of corrugation mm 70 70 70
4 Step between of cell of corrugation mm 35 35 35
5 Mast (height) of cell of corrugation mm 21 21 21
6 Tilt angle of cell of corrugation ° 54.5° 54.5° 54.5°
7 Span of beam mm 3780 7560 15,120

As an example, we present the calculation results for the variant of the second
type of profiled sheet. There are presented: displacements (Fig. 3), shear stresses
(Fig. 4), normal stresses (Fig. 5).
Variable part of the parametric code Variant 2:
!VARIABLES MM, MPA, N
NN = 2 !VARIANT NUMBER
L_BEAM = 7560 !SPAN OF BEAM
EG = 2.1E5 !ELASTIC MODULUS OF MATERIALS
EYG = 2.1E5
EP = 2.1E5
TG = 0.7 !THICKNESS OF CORRUGATION
HG = 700 !HEIGHT OF CORRUGATION
SHAG_GOFRA = 70 !STEP OF CELL OF CORRUGATION
L_YACH_GORFA = 21 !MAST (HEIGHT) OF CELL OF CORRUGATION
SHIRINA_GOFRA = 35!WIDTH OF CORRUGATION
YGOL_GOFRA = 54.5 !TILT ANGLE OF CELL OF CORRUGATION
L_SEC = 1260 !SECTION LENGTH
2. The second type of profiled sheet, but different heights of the corrugation of the
beam:
Prefabricated Steel Structures with a Corrugated Web … 353

Fig. 3 Displacements (mm), Variant 2

Fig. 4 Shear stresses τ (MPa), Variant 2

3. The second type of profiled sheet, but a different span of the beam:
354 A. Ibragimov et al.

Fig. 5 Normal stresses σ (MPa), Variant 2

5 Conclusions

For convenience, will create a general summary table for all the specified beam
parameters and the results obtained (Table 5). Based on which, will draw conclusions
about the effectiveness for each calculation option.
1. Different types of profiled sheet:
The second type of profiled sheet shows itself to be the most effective, since it
has the least deflection and the lowest stresses.
2. The second type of profiled sheet, but different heights of the corrugation of the
beam:
At the second selection, in Variant 4—the deflection condition is not met. Such
the beam cannot exist. The deflection condition will be met, for a given span, with a
height of beam of more than 430 mm.
3. The second type of profiled sheet, but a different span of the beam:
At the third selection, in Variant 10—the deflection condition is not met. Such
the beam cannot exist. The deflection condition will be met, for a given span, with a
height of beam of span less than 13,700 mm.
№ Parameter Unit rev Variant
1 2 3 4 5 6 7 8 9 10
1 Span of beam, L mm 7560 7560 8160 7560 7560 7560 7560 3780 7560 15,120
2 Elastic modulus MN/m2 2.1 × 105 2.1 × 2.1 × 2.1 × 2.1 × 2.1 × 2.1 × 2.1 × 2.1 × 2.1 ×
of materials 105 105 105 105 105 105 105 105 105
(steel)
3 Thickness of mm 0,6 0,7 0,7 0,7 0,7 0,7 0,7 0,7 0,7 0,7
corrugation
4 Height of mm 700 700 700 300 500 700 900 500 500 500
corrugation
5 Step of cell of mm 70 70 135 70 70 70 70 70 70 70
corrugation
6 Step between of mm 35 35 35 35 35 35 35 35 35 35
cell of
corrugation
7 Mast (height) of mm 21 21 44 21 21 21 21 21 21 21
Prefabricated Steel Structures with a Corrugated Web …

cell of
corrugation
8 Tilt angle of cell ° 54.5° 54.5° 53.5° 54.5° 54.5° 54.5° 54.5° 54.5° 54.5° 54.5°
of corrugation
9 Tilt angle of ° 0 0 0 0 0 0 0 0 0 0
corrugation
relative to the Y
axis
10 Section length mm 1260 1260 2040 1260 1260 1260 1260 1260 1260 1260
11 Load MPa 0.0666 0.0666 0.0666 0.0666 0.0666 0.0666 0.0666 0.0666 0.0666 0.0666
(continued)
355
(continued)
356

№ Parameter Unit rev Variant

1 2 3 4 5 6 7 8 9 10
12 Displacements mm 10.106 9.9524 13.291 52.245 18.730 9.9524 6.3792 0.8757 6.3792 83.33
13 Shear stresses τ MPa 72.938 64.198 108.80 213.20 91.175 64.199 48.509 25.986 48.509 101.41
14 Normal stresses MPa 129.78 130.18 170.05 306.46 178.40 130.18 104.86 50.706 104.86 367.89
σ
Fulfillment of the deflection mm 37.8 37.8 40.8 37.8 37.8 37.8 37.8 18.9 37.8 75.6
condition L* 1/200 mm Yes Yes Yes No Yes Yes Yes Yes Yes No
A. Ibragimov et al.
Prefabricated Steel Structures with a Corrugated Web … 357

The article [1] considers the conceptual idea of a new construction, which received
a calculation justification for the creation and calculation of such structures. In subse-
quent publications, the nodes and methods of detachable connection of structural
elements into a single beam will be considered in detail, a methodology for their
calculation and design will be proposed. It is planned to create an algorithm for
calculating and designing beams with a corrugated web from profiled sheet and
chords from rolled profiles. Also, the creation of a catalog of such beams and the
selection of their parameters by key, which will allow them to carry out their typical
and experimental design.

References

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construction, real estate as a material basis for modernization and innovative development of
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(2018).
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web beams. J Mater Process Technol 3(150):242–254
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soidally corrugated steel web beams with web openings. Steel And Composite Structures
1(10):69–85
7. Xu D, Ni YS, Zhao Y (2015) Analysis of corrugated steel web beam bridges using spatial grid
modeling. Steel And Composite Structures 4(18):853–871
8. Dar MA, Subramanian N, Dar AR, Majid M, Haseeb M, Tahoor M (2019) Structural efficiency
of various strengthening schemes for cold-formed steel beams: Effect of global imperfections.
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Composite and Aluminium (ASSCCA 03) (Sydney: Australian Steel Inst), pp 807–812
10. Mansurova AR (2018) Application of the ANSYS software package in computer modeling.
Young Scientist 39(225):31–33
11. Basov KA (2006) Graphical interface of the ANSYS complex
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13. Basov KA (2006) ANSYS and LMS Virtual Lab. Geometric Modeling
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web (Part 1. Beam). IOP Conference Series: Materials Science and Engineering, 869(7):072041
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80–86
Evaluation of the Dynamic Behavior
of Multi-connected Shell Structures

Tulkin Mavlanov, Sherzod Khudainazarov, and Feruza Umarova

Abstract The dynamics of multiply connected axisymmetric shell structures

composed of thin-walled elements are made of viscoelastic materials is considered
in this article. The main result of this research is the determination of the ampli-
tude—frequency characteristics. To carry out the numerical researches and calcula-
tions, a software package has been developed that contains programs for calculating
the dynamic characteristics of multiply connected shell structures. In this case, the
problem of calculating the dynamic characteristics of multiply connected structurally
inhomogeneous axisymmetric shell structures is reduced to an effectively solvable
mathematical problem for complex eigenvalues. Using some provisions of known
methods, an algorithm for the numerical analysis of complex shell structures is
constructed, which allows avoiding the difficulties, significantly reducing the order
of resolving systems of algebraic and differential equations, and increasing the accu-
racy of the obtained solution. Based on the analysis of the results obtained, it has
been established that the implementation of the optimal value of the elastic modulus
is possible under operating conditions that are extreme in terms of strength. The
characteristics of the governing damping coefficient δ have maxima at which the
dissipative properties of the structure are most effective, i.e. the synergistic effect
of viscoelastic properties is fully manifested in the characteristics of the synergistic
effect of viscoelastic properties at the given parameters of structural heterogeneity.
It was found that the quantitative difference in the maximum amplitude values does
not exceed 12–16%. This is due to the reasons consisting in the interaction of the
axisymmetric and non-axisymmetric components of the oscillations.

Keywords Vibrations · Damping coefficient · Relaxation kernels · Stiffness

matrices · Kinematic boundary conditions · Elastic modulus

T. Mavlanov · S. Khudainazarov (B) · F. Umarova

Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39, Kori Niyoziy Str.,
Tashkent, Uzbekistan 100000

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 359
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_34
360 T. Mavlanov et al.

1 Introduction

A large place in the problem of the strength of axisymmetric structures is occupied

by the research of natural vibrations. When developing the method, the require-
ments were set to take into account the actual working conditions of the structure, to
bring the design scheme as close as possible to the full-scale design. To meet these
requirements, it was possible to create a comprehensive method of strength analysis.
Moreover, in the research of shell elements, the whole problem is reduced to solving
boundary value problems, which are solved by the direct numerical integration of
solving systems of integro—differential equations.
Various aspects of this problem were discussed in the works of A.V. Karmishin,
V. I. Myachenkov, A. N. Frolov, T. Mavlanov [1–4] and their students developed the
reliable numerical methods that allow using computers to automate solutions to prob-
lems of dynamics of structural elements with different structural forms and structure
of elements. The same methods can be extended to the analysis of complex multi-
connected shell structures, which are an arbitrary composition of shells of rotation
and rings. The peculiarity of setting and solving the problem for such constructions
is that in this case it is difficult to formulate the original boundary value problem in
the generally accepted sense, when the solving system of differential equations and
the corresponding boundary conditions are known.
In the works [5–9], free oscillations of a system of coupled shells consisting of
different segments were studied. Using the theory of the shell of first-order shear
deformation, the equations of motion of the shell and the associated boundary condi-
tions are obtained. Numerical results are presented for medium-thickness shells with
different geometric and boundary conditions.
In the papers [10–15], the spatial eigenvalues of high-rise pipes are investigated,
taking into account the variability of the slope and thickness of the structure in
the framework of the viscoelastic shell theory. It is found that when taking into
account the viscoelastic properties of the material of structures, the decrement of
their oscillations weakly depends on the value of the natural frequencies. Along with
this, not only the bending frequencies, but also the spatial natural frequencies of the
structure fall into the dangerous range of earthquake frequencies.
Nonlinear parametric oscillations of viscoelastic flat orthotropic shells of variable
thickness are considered in [16–22]. The influence of various physical, mechanical
and geometric parameters of a shallow shell of variable thickness is investigated.
In the works [23–25], the dynamic behavior of structurally inhomogeneous
mechanical systems consisting of multilayer cylinders connected to a thin viscoelastic
shell of finite length is investigated. Complex natural frequencies and amplitudes of
forced oscillations are determined using the developed methods and research algo-
rithms. It is revealed that the effect of the greatest damping ability in structurally
inhomogeneous systems manifests itself when the real parts of complex natural
frequencies approach each other due to the interaction of close natural forms with
each other.
Evaluation of the Dynamic Behavior of Multi-connected Shell … 361

Using some provisions of the known methods, it is possible to construct an algo-

rithm for numerical analysis of complex shell structures, which allows avoiding the
difficulties, significantly reducing the order of solving systems of algebraic equations
and improving the accuracy of the resulting solution.
This can be achieved by using the displacement method, the numerical integration
method, considering as such a separate link or node that is part of the structure.

2 Methods

In the general case, the behavior of the structure is described by partial differential
equations, but since the axisymmetric structure is considered, the method of sepa-
ration of variables allows us to reduce this problem to considering the behavior
of the structure for a separate harmonic expansion in the circumferential direc-
tion. The actual state will be determined by superposing the solutions for the indi-
vidual harmonics. We will consider the equilibrium equations of the structure in the
displacements of its nodes. For each node, the equation is written as
 
Gi i = Ri f0i , Nijs , n , (1)

Here Gi is the ring stiffness matrix, i is the displacement vector, and Ri is the
vector of total forces, which is determined by the forces f i 0 acting directly on the
ring, the edge forces N ijs from the shells adjacent to the ring, and the number of
harmonics n.
Suppose that for the shell of rotation there is a dependence connecting the edge
forces and displacements, in the form
      0 
N(x1 ) K11 K12 W(x1 ) N (x1 )
= + , (2)
N(x2 ) K21 K22 W(x2 ) N0 (x2 )

Then, using the condition of compatibility of the displacements of the node and
the edges of the adjacent shells and the dependence (Eq. 2), Eq. (1) can be written as


m
ijs  
Gi i = Pij Ktp , n j + Ri f0i , Nijs , n , (3)
j=1

When considering all the nodes of the structure (i = 1,2,…, m), we get the matrix
equation

ijs  
P Gi , Ktp , n  = Ri f0i , Nijs , n , (4)
362 T. Mavlanov et al.
⎡ ⎤ ⎧ ⎫ ⎧ ⎫
p11 p12 . . . p1m ⎪
⎪ 1 ⎪ ⎪ ⎪
⎪ R1 ⎪
⎪
⎢ ⎥ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎢ p21 p22 . . . p2m ⎥ ⎪
⎪  ⎪
2 ⎪ ⎪
⎪ R2 ⎪
⎪
⎢ ⎥ ⎪
⎨ ⎪
⎬ ⎪
⎨ ⎪
⎬
⎢ . . . .. . ⎥ . .
P=⎢ ⎥, = , R= , (5)
⎢ . . . .. . ⎥ ⎪ . ⎪
⎪ ⎪ ⎪
⎪ . ⎪
⎪
⎢ ⎥ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎣ . . . .. . ⎦ ⎪
⎪ . ⎪
⎪ ⎪
⎪ . ⎪
⎪
⎪
⎩ ⎪
⎭ ⎪
⎩ ⎪
⎭
pm1 pm2 . . . pmm m Rm

Thus, the research problem of the linear behavior of the design at the first stage
is reduced to solving a system of algebraic equation, where the coefficients of the
stiffness matrix of the whole structure P depend on the stiffness of the wheel G, which
are determined analytically, and from the stiffness matrices of shells of rotation K,
the coefficients of which, as will be shown below, it is possible to determine the
required degree of accuracy of the solution of boundary value problems of the theory
of shells of rotation.
At the second stage, the stress–strain state of the frames of the structure is deter-
mined from the found displacements of the nodes, kinematic boundary conditions
for the shell elements are formed, and then the complete stress–strain state of the
entire structure is determined from the solution of boundary value problems. The
existing numerical methods for solving boundary value problems make it possible to
consider the shell of rotation without excessive geometric idealization. This applies
equally to the properties of the material. Thus, using the idea of a superelement, the
method of displacements and numerical construction of solutions to boundary value
problems, it was possible to create a method that allows analyzing complex shell
structures with the accuracy of accepted theories without any simplifications in the
behavior of individual elements, while preserving all their geometric parameters and
mechanical properties of the material.
When studying the nonlinear behavior of a symmetrically loaded shell structure, it
is impossible to formulate a resolution system with the same clarity. However, using
one or another iterative process, it is possible to reduce the solution of a nonlinear
problem to a sequence of solving linear problems, where a system of linear equations
is synthesized at step t
   t−1 
Pt Gi , Ktijs yt−1 t = Ri f0i , N0t
ijs , y , (6)

Here, the shell stiffness matrices and the generalized force vector depend on some
functions y obtained in the previous iteration step. The further solution process is
completely analogous to the linear case.
From the problems discussed above, it follows that the basis of the method for
calculating multi-connected shell structures is a system of algebraic equations of the
type (Eq. 4), (Eq. 6) with respect to the displacements of the nodes of the structure.
The coefficients of the matrices P in these equations are determined by the stiff-
ness parameters of the structural elements, taking into account the acting loads, the
frequency parameter, and the number of waves in the circumferential direction.
Evaluation of the Dynamic Behavior of Multi-connected Shell … 363

Thus, in order to use this method to analyze the structure, it is necessary to be

able to calculate the stiffness parameters of the shell and ring and solve the boundary
value problem. This approach proved to be quite fruitful, as it was used in [1–4]
when constructing a method for analyzing multi-connected structures taking into
account the elastic base, physical nonlinearity in the problems of strength and stability
and under dynamic influence, and for analyzing prismatic structures with irregular
stiffness parameters and a multi-connected shape in the circumferential direction.
When drawing up the design scheme, the design was divided into Ns shell
elements, Nr nodal elements, and Ne connections (Fig. 1), a detailed classifica-
tion of which in the framework of elastic solutions is described in the works of V. I.
Myachenkov, V. P. Maltsev and T. Mavlanov [26–29].
Within each shell element (isotropic, orthotropic, or structurally orthotropic),
its geometric and mechanical characteristics, as well as the acting external loads,
change continuously along the generatrix of this element. Moreover, the Kirch-
hoff–Love hypotheses must be valid for each shell element. The behavior of shell
elements is described by the equations of shell theory in the form of V. V. Novozhilov
[30]. The cross-sections of circular frames are considered non-deformable (i.e., non-
deformable). considered according to the classical scheme of a circular ring) and

Fig. 1 Design diagram of

multi-connected structurally
inhomogeneous
axisymmetric shell structures
364 T. Mavlanov et al.

small in comparison with the distance from the axis of rotation to the line of the
centers of gravity of this section.
The relationship between generalized displacement loop shell element adjacent
to the ring, and generalized displacements of the median line of this annular element
is set by the ratio:

Wi = [
ϕi ]i , (7)

where, [(ϕi )] is the transformation matrix.

For the shell and ring elements, the relationship of the reduced forces and moments
with the components of tangential and flexural deformation is carried out using
complex symmetric viscoelasticity matrices. For a shell element
 
N̄ = C̃ ε̄, (8)

For a ring element

 
3 εk ,
Q̄ = G (9)

The physical relations for viscoelastic connections have the form:

 
 V̄ ,
Rn = K (10)

where: (Rn )—reactions in connections; V —displacements of the beginning

and end of connections; [K]—matrix of complex values of the proportionality
coefficients.
Integro-differential–algebraic equations of motion with complex coefficients for
various elements of the considered structurally inhomogeneous shell structures are
obtained from the Lagrange variational equation in combination with the Dalem-
bers principle. Moreover, two cases are considered – proper and steady-state forced
vibrations of structures.
In the simplest case, we assume that the external load changes according to the
harmonic law:

q p = q po eiω R τ , f i = f io eiω R τ , (11)

For the case of natural oscillations of the structure, the solution is sought in the
form:

u p = u po ei ω̃τ , i = io ei ω̃τ , (12)

Then the equations of motion for natural and forced oscillations of multi-
connected structurally inhomogeneous axisymmetric shell structures are assumed,
respectively [2–4]:
Evaluation of the Dynamic Behavior of Multi-connected Shell … 365

  p   
y p = f p α1 , n, y p + f p ω̃ y p , ( p = 1, 2 . . . Ns ), (13)
 2
   i js  i js  i js   i js  i js  i js
[G i ] − ω R [G ω ] i = ξi ϕi Q i + ξci ϕci Q ci ,
j s j s

(i = 1, 2, . . . , Nr ). (14)

For the case of eigenvalues of multi-connected structurally inhomogeneous shell

structures, the system of equations of the modified displacement method in the form
proposed by A.V. Alexandrov is reduced to the problem of complex eigenvalues:
 
L(n, ω̃) ¯ = 0, (15)

with the desired parameter ω entering non-linearly in (Eq. 15) and in the boundary
conditions for them. The set of parameters ω*, for which there is a nontrivial solution
of this system, is a set of complex values of the frequencies of the shell structure under
consideration. For the existence of a nontrivial solution (Eq. 15), it is necessary that
the determinant of this system is equal to zero. Therefore, the problem of determining
the frequencies of natural oscillations is reduced to finding the roots of a nonlinear
functional equation with a complex parameter
  
L n, ω̃∗  ¯ = 0, (16)

for a different number of half-waves in the longitudinal direction. The roots of

Eq. (16) are calculated by the step method in combination with the Muller method.
Matrices of complex values of stiffness, forms of natural vibrations, as well as the
stress–strain state of elements of structurally inhomogeneous multi-connected shell
structures are determined by the orthogonal run method.
Further, using the developed software package [31], the dynamics of a number of
multi-connected structurally inhomogeneous shell structures for modern engineering
products were calculated and analyzed.

3 Results and Discussion

The results of numerical modeling confirmed the presence of a synergistic effect of

viscoelastic properties in the considered complex structures, which in turn allowed
them to be optimized for dissipative characteristics and to develop the necessary
design recommendations.
The technical implementation of the tasks was to varying the numerical exper-
iment in a physically realizable limits of the geometric and physical parameters
of the components and materials of the designs, layouts dissipation heat material
and viscoelastic relations exercise aimed selection of the necessary parameters that
366 T. Mavlanov et al.

provide effective damping of resonance oscillations of a certain frequency or range

of frequencies due to the conditions you operate the product.
Consider an example for a multi-connected structurally inhomogeneous axisym-
metric shell structure consisting of 23 shell and 21 nodal elements with geometric
dimensions, shown in Fig. 1. The values of the determining damping coefficient δ
(Fig. 2) and the determining resonant amplitude of vibrations Aδ (Fig. 3) are obtained
depending on the instantaneous elastic modulus of viscoelastic conical reinforcing
shells (Eq. 4), (Eq. 8), varying within E = 109 ÷1011 N/m2 A = 0.01; α = 0.1; β =
0.05; ρ = 7.8 × 10−7 kg/sm3 . Shells under numbers 1, 2, 3, 5, 7, 9, 11, 14, 16, 18,
21, 23—elastic (E = 2 × 1011 N/m2 , ν = 0.3; ρ = 7 × 10−6 kg/sm3 ), the thick-
nesses of all shells are the same and equal to 0.03 m. As a result, the optimal value
of the variable parameter of structural inhomogeneity E o is determined, the imple-
mentation of which in the design of the structure allows you to maximally dampen
and reduce the resonant overload on the product by 1.7–2.5 times. Moreover, the
analysis of the VAT of the design shows that the implementation of E o is possible in
operating conditions that are extreme in terms of strength. The characteristics of the
determining damping coefficient δ have maxima at which the dissipative properties

Fig. 2 Changing the

vibration frequency
depending on the elastic
modulus

Fig. 3 Change in vibration

amplitude depending on
elastic modulus
Evaluation of the Dynamic Behavior of Multi-connected Shell … 367

of the structure are most effective, i.e., the synergistic effect of viscoelastic properties
is fully manifested in these parameters of structural inhomogeneity E o .
The influence of the rheological properties of materials (the coefficients of the
relaxation kernel A j , α j ,β j ), the compressibility of the material (v = var and v =
const) on the damping coefficients is studied. It is shown that taking into account the
compressibility of the material significantly increases the damping coefficients (up
to 20%).

4 Conclusions

1. Based on the mathematical theory of viscoelasticity, variation principles of

dynamics, and modern computational methods, the problem of calculating
the dynamic characteristics of multi-connected structurally inhomogeneous
axisymmetric shell structures is reduced to an effectively solvable mathematical
problem for complex eigenvalues.
2. The software package developed and implemented in the calculation practice
makes it possible to use it in research and design organizations of various indus-
tries and construction, in the training of engineering personnel by universities
with any set of computer equipment available to them.
3. The numerical studies have shown satisfactory accuracy and convergence of
the developed methods and algorithms for solving problems of dynamics of
structurally inhomogeneous shell structures.
4. The conducted studies show that taking into account the viscous properties of
the material increases the damping coefficients by up to 20%.

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Impact of Construction Seams
on the Bearing Capacity of a CVC-RCC
Combined Dam

Viktor Tolstikov and Yara Waheeb Youssef

Abstract Construction seams (joints) are adopted in CVC-RCC combined dams in

order to reduce the cracking likelihood of the concrete dam and improve the stress
distribution in the dam body. However, the arrangement of construction joints inside
a concrete dam body degrades the integrity and the ultimate bearing capacity of
the concrete dam. The purpose of this study is to investigate the impact of vertical
and horizontal construction seams on the bearing capacity of concrete gravity dams
by means of finite element method (FEM) using the software complex “CRACK”.
Bureyskaya CVC-RCC combined dam is chosen as a case study. The results of the
numerical modelling calculations show that passing from the monolithic dam model
to the dam model with weak construction seams, the strength characteristics of the
dam tend to decrease progressively. The relationship between the overloading factor
and the horizontal displacement of the dam crest is studied and analysed. In addition,
the relationship between the overloading factor and the coefficient of dam’s anti-slide
stability is analysed. Moreover, the results show that the ultimate bearing capacity of
the dam with weak construction seams is 25% less than that of the monolithic dam.

Keywords Construction seams · Ultimate bearing capacity · CVC-RCC combined

dam · Finite element model · Overloading factor · Dam failure

1 Introduction

In recent years, the new technology of the combined dam construction has become
common, as it accelerates the speed of the dam construction and saves on cost
[1]. Traditional conventional vibrated concrete (CVC) dams and roller compacted
concrete (RCC) dams have almost the same structural strength; therefore, in prac-
tice, the roller compacted concrete may be used in combination with the conventional
vibrated concrete in dam design and construction. CVC-RCC combined dams are
especially used for complex geological conditions, severe cold climate conditions
and tight construction schedules [2].

V. Tolstikov · Y. W. Youssef (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 371
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_35
372 V. Tolstikov and Y. W. Youssef

The use of roller compacted concrete has become common in the field of dam
construction in many countries of the world since the 1980s [3]. In comparison with
CVC dams, RCC dam construction method provides a lower construction cost and
a shorter construction duration, which can make up for the other delays related to
certain dam shapes or dam foundation treatment [4]. However, the difference of
deformation characteristics and material properties of RCC and CVC affects the
structural safety of the dam. Many studies have shown that high deformation and
stress gradients tend to happen at the interfaces among large-volume heterogeneous
concrete materials [1, 5].
The spaces between adjacent blocks and lifts in CVC-RCC dams are called
construction seams (joints). They are adopted in order to release the excessive temper-
ature tensile stress, improve the stress distribution and reduce the cracking likelihood
of the concrete dam [6]. However, construction seams are the weakest zones of the
dam. Their presence inside the dam body degrades the integrity (continuity) of the
dam and increases the seepage between horizontal lifts as well as through vertical
joints, which leads to a reduction in tensile and shear strengths [7]. Construction
joints not only can create special conditions in stability and stress distribution but
also can increase the potential structural hazards; hence, they can cause dam failure
under different loading conditions [8].
One of the main problems in dam design and safety monitoring is to investigate the
impact of construction joints on the behaviour of the dam [8]. On-site observations
on the behaviour of concrete gravity dams under harsh climatic conditions have
shown that opening of construction joints on the downstream face of the dam has a
considerable impact on the state of the dam [9]. Furthermore, the results of previous
experimental studies on elastic-brittle models for a dam profile (height: H = 90 m,
downstream slope: m = 0.7) show that the arrangement of vertical inter-columnar
seams in the dam body leads to a decrease in the bearing capacity of the structure,
especially in the case when inter-columnar seams have a low shear strength. However,
in the case of rough seams, the pattern of the model destruction is close to the
monolithic dam model [10]. The results of other experimental studies show that
increasing the shear characteristics (cohesion and internal friction) of the horizontal
and vertical construction seams leads to increase the ultimate bearing capacity of
the concrete gravity dam [11]. The shear characteristics of seams are increased by
filling them with cement materials (grout materials), which provide the monolithic
behaviour of the dam and help to avoid the dam cracking. Therefore, joint grouting
plays an important role in increasing the ultimate bearing capacity of the dam and
improving its stability [11, 12].
Numerical methods are currently used in different fields of science and engi-
neering because of their high efficiency and computational speed [8]. At present,
the finite element modelling is considered to be an important tool to analyse the
structural behaviour of dams [13].
The impact of vertical and horizontal construction seams on the bearing capacity
of concrete gravity dams has not been sufficiently investigated by the researchers.
Therefore, the purpose of this study is to investigate the impact of vertical and hori-
zontal construction seams on the bearing capacity of Bureyskaya dam, as a case
Impact of Construction Seams on the Bearing Capacity … 373

study, under the static loading condition. The present research is based on the finite
element modelling using the software complex “CRACK”.

2 Materials and Methods

2.1 Case Study: Bureyskaya CVC-RCC Combined Dam

Bureyskaya CVC-RCC combined dam is one of Russia’s largest hydroelectric plants

(maximum height 139 m, total crest length 789 m and crest width 19–27 m), as shown
in Fig. 1. Bureyskaya hydroelectric plant (HPP) is located on Bureya river in the
Amur region of the Far Eastern Federal District. The main works of Bureyskaya dam
construction started in 1984. Since 2015 the plant has been in permanent operation.
The main purposes of Bureyskaya HPP construction are to protect the residential
areas from catastrophic flooding of Amur and Bureya rivers, as well as to increase
the hydroelectric power generation. The site of Bureyskaya HPP is characterized
by a monsoon climate, where the average annual air temperature is 3.5 °C. The
absolute maximum air temperature is 41 °C, while the absolute minimum is 58 °C.
The average temperature of the coldest five-day period is 40 °C [14].
The combination of roller compacted concrete (RCC) and conventional vibrated
concrete (CVC) in Bureyskaya dam was due to the two following factors:
1. The low temperatures in winter, at which roller compacted concrete (RCC)
cannot be laid.
2. The desire to construct the CVC upper column of the dam in winter using a
protecting tent ahead of the construction of the RCC wedge-shaped part. This
aims to discharge the water generated from dam construction, raise the upstream
water level and gradually fill the reservoir with water.

Fig. 1 Bureyskaya
hydropower plant, Russian
Federation
374 V. Tolstikov and Y. W. Youssef

The inter-columnar seam of Bureyskaya dam formed between vibrated and roller
compacted concrete because of the dam construction technology, as follows: the
CVC upper column was constructed in winter ahead of the roller compacted concrete,
which was compacted near to the cooled CVC upper column during the warm period.
As for the horizontal construction seams of Bureyskaya dam, they formed because
of the RCC dam construction method [14].
The study of the impact of weak construction seams on the bearing capacity of
Bureyskaya dam is carried out on the right-bank cross section №16, which is shown
in Fig. 2.
The 122 m-height cross section №16 of Bureyskaya dam consists of a 14 m-wide
upper column (external zone), which was built entirely of dense water-impermeable
vibrated concrete class B15W8. The internal zone consists of roller compacted
concrete class B10. Since roller compacted concrete is not frost-resistant, it is
protected on the downstream side by a layer of frost-resistant vibrated concrete class
B15F200. The upstream slope of the dam body above the elevation of 151.00 m is
straight, while the upstream slope below this elevation is 1:0.2. The downstream dam
slope is 1:0.7 and the elevation of the starting point is 241.00 m [14].

Fig. 2 Right-bank cross section №16 of Bureyskaya dam (NWL: normal water level; MWL:
maximum water level; DSL: dead storage level) [14]
Impact of Construction Seams on the Bearing Capacity … 375

2.2 Finite Element Modelling of Bureyskaya

Dam-Foundation System

The aim of the finite element modelling of the dam-foundation system is to simulate
the real structural behaviour of the system as close as possible [12].
The research methodology is based on the numerical modelling of the dam by
means of the finite element method (FEM) using the software complex “CRACK”,
which was developed at the Department of Hydraulics and Hydrotechnical Engi-
neering of Moscow State University of Civil Engineering.
In the dam numerical modelling, it is considered the presence of the vertical
inter-columnar seam, the contact seam between the dam and the rock foundation and
9 horizontal construction seams at the following elevations: 151.00 m, 162.30 m,
174.30 m, 186.30 m, 198.30 m, 210.30 m, 219.30 m, 228.30 m and 241.00 m. The
boundary conditions are taken as follows: the boundary nodes of the bottom of the
bedrock are subjected to three-direction constraints (i.e., the nodal displacements are
equal to zero).
A two-dimensional (2D) finite element model of Bureyskaya dam-foundation
system is established using the software complex “CRACK”, as shown in Fig. 3.
The model mesh has a total of 2697 nodes and 2562 finite elements.
The physical and mechanical material characteristics of the cross section of
Bureyskaya dam-foundation system are reported in Table 1 [14].
In order to investigate the impact of construction seams on the bearing capacity
of Bureyskaya CVC-RCC combined dam, different dam parameters are analysed
according to the following two finite element models:

Fig. 3 Two- dimensional (2D) finite element model of Bureyskaya dam-foundation system (DWL:
downstream water level)
376 V. Tolstikov and Y. W. Youssef

Table 1 Physical and mechanical material characteristics of the cross section of Bureyskaya dam-
foundation system [14]
Zone name Material Density Elastic Poisson Compressive Tensile
(MN/m3 ) modulus ratio strength strength
E (MPa) (MPa) (MPa)
Dam body External CVC 0.0240 34,500 0.15 11.3 1.15
zone B15W8
Protective CVC 0.0245 34,500 0.15 11.3 1.15
layer B15F200
Transitional CVC 0.0240 30,000 0.15 7.5 0.85
zone B10
Internal RCC 0.0235 30,500 0.15 7.5 0.78
zone B10
Rock foundation Rock 0.0200 17,000 0.24 9.0 0.25

Table 2 Characteristics of the high-stiffness cement mortar used for grouting of the vertical seam
of monolithic dam [14]
Seam Seam shear Seam normal Angle of Cohesion Tensile Seam
characteristics stiffness stiffness internal (C) strength ultimate
friction closure
(ϕ)
Value 2000 MPa 6000 MPa/m 50° 1 MPa 1 MPa 2 mm
/m

1. Finite element model of the dam with weak construction seams: the quality
of the horizontal construction seams is considered to be bad (cohesion C =
0.25 MPa, angle of internal friction ϕ = 35°). Insufficient primary grouting
works are carried out along the vertical inter-columnar seam; therefore, it is
considered that the vertical seam is opened by 2 mm.
2. Finite element model of the monolithic dam (seamless dam), which ensures the
continuity of the dam body. The quality of the horizontal construction seams is
considered to be good (C = 1 MPa, ϕ = 50°). High-quality secondary grouting
works are carried out along the vertical inter-columnar seam. The characteristics
of the cement mortar used for grouting of the vertical seam are reported in Table
2.

2.3 Evaluation of the Dam Ultimate Bearing Capacity Using

the Overloading Method

The method of water overloading is often used to evaluate the ultimate bearing
capacity of dams, as well as to determine the pattern and mechanism of dam failure
[15, 16]. This method can describe the whole process from dam overloading to dam
Impact of Construction Seams on the Bearing Capacity … 377

failure. The overloading method assumes that the water bulk density increases, while
the upstream water level remains the same. The overloading factor (n) is defined as the
multiples of water bulk density and can be used as the evaluation index (criterion) for
dam stability [15]. The overloading factor is calculated by the following formulations
[13, 15, 17, 18]:

n = P/P0 (1)

P = n.γ .H 2 /2 (2)

P0 = γ .H 2 /2 (3)

where; P— applied water overload (ton/m);

P0 —normal water load (ton/m);
γ— water density (ton/m3 );
H—water height on the upstream side, which corresponds the normal water level
NWL (m).
The ultimate bearing capacity of the dam is evaluated by analysing the variation
curve of the horizontal displacement of dam crest with overloading factor. When the
curve mutation takes place, the dam loses its bearing capacity [17].
The coefficient of dam’s anti-slide stability along a horizontal seam is considered
as a criterion to judge the stability of concrete gravity dams under overloading. The
normative value of the coefficient of dam’s anti-slide stability at the contact seam
between the dam and rock foundation is equal to Ks, n = 1.32 [19].

3 Results and Discussion

The calculations are carried out under the static loading condition, i.e., considering
the self-weight of the concrete gravity dam, the hydrostatic pressure on the upstream
dam surface and the seepage uplift pressure on the base of the dam. The calculations
did not take into consideration the temperature and seismic effects and the phased
construction of the dam. Different values of the overloading factor are assumed (1.1,
1.2, 1.3, 1.4, 1.5, 1.6% ….) until the dam failure occurs. For every assumed value
of the overloading factor, the horizontal displacements of the dam crest, as well as
the values of coefficient of dam’s anti-slide stability at the contact seam (Ks ) are
calculated using the software complex “CRACK”.
378 V. Tolstikov and Y. W. Youssef

3.1 Impact of Construction Seams on the Element State

of the Dam Under Overloading

Figure 4 shows the element state of Bureyskaya dam-foundation system during the
process of overloading (from system cracking to the system overall destruction),
when considering the impact of the weak construction seams. When the dam is
overloaded from 1.2P0 to 1.3P0 , the cracks first appear on the top of the RCC wedge-
shaped part of the dam. In the process of continuous overloading up to 1.4P0 , the initial
cracks extend, meanwhile new inclined cracks appear in dam body. Moreover, biaxial
tension zones appear in the rock foundation and all the horizontal construction seams
are opened in the RCC wedge-shaped part of the dam. When the overload reaches
1.5P0 , the new and old inclined cracks connect at the middle and low elevations of the
dam body. The old biaxial tension zones extend in the rock foundation. Furthermore,
a new zone of a concrete destruction appears at the downstream dam toe. When
the overload reaches 1.6P0 , the dam cracks greatly: new inclined cracks appear in
the dam body. The old biaxial tension zones of the rock foundation extend towards
the downstream dam toe, meanwhile new biaxial tension zones appear in the low

Fig. 4 Element state of Bureyskaya dam-foundation system in the process of dam overloading,
when considering the impact of the construction seams
Impact of Construction Seams on the Bearing Capacity … 379

elevation of the dam body. The contact seam is opened by about 65.8 cm (i.e., 76%
of the length of the dam base lost its shear strength). Furthermore, decompaction of
rock foundation extends towards the downstream dam toe.
Figure 5 shows the element state of Bureyskaya monolith dam-foundation system
at the value of the overloading factor n = 2. It is observed that when the overload
reaches 2P0 , the contact seam is opened by 45.8 cm (i.e., 52% of the length of the
dam base lost its shear strength), while all the remaining horizontal construction
seams are approximately closed. A local zone of concrete destruction appears at the
downstream dam toe.

Fig. 5 Element state of

Bureyskaya monolithic
dam-foundation system at n
=2

Fig. 6 Relationship curves

between the horizontal
displacement of Bureyskaya
dam crest and the
overloading factor (n)
380 V. Tolstikov and Y. W. Youssef

3.2 Impact of Construction Seams on the Horizontal

Displacement of the Dam Crest Under Overloading

Figure 6 shows the relationship curves between the horizontal displacement of

Bureyskaya dam crest and the overloading factor (n).
As shown in Fig. 6, the horizontal displacement of the dam crest is directly
proportional with the overloading factor (n).
When the dam with weak construction seams is overloaded up to 1.3P0 , the rela-
tionship between the horizontal displacement and the overloading factor is approxi-
mately linear. From 1.3P0 to 1.6P0 , it is observed that the relationship is non-linear.
When the overloading factor reaches to n = 1.6, the dam reaches its ultimate bearing
capacity.
When the monolithic dam is overloaded up to 1.6P0 , the relationship between the
horizontal displacement and the overloading factor is approximately linear. From
1.6P0 to 2P0 , it is observed that the relationship is non-linear. When the overloading
factor reaches to n = 2, the monolithic dam reaches its ultimate bearing capacity (i.e.,
25% greater than the overloading factor, at which the dam with weak construction
seams reaches its ultimate bearing capacity).

3.3 Impact of Construction Seams on the Anti-slide Stability

of the Dam Under Overloading

Figure 7 shows the relationship curves between the coefficient of Bureyskaya dam’s
anti-slide stability at the contact seam (Ks ) and the overloading factor (n).

Fig. 7 Relationship curves

between the coefficient of
Bureyskaya dam’s anti-slide
stability at the contact seam
(Ks ) and the overloading
factor (n)
Impact of Construction Seams on the Bearing Capacity … 381

As shown in Fig. 7, the coefficient of dam’s anti-slide stability at the contact seam
(Ks ) is inversely proportional with the overloading factor (n).
Taking into consideration the impact of weak construction seams: at n = 1.6, Ks
is equal to 1.4, which is greater than the normative value (Ks, n = 1.32). Therefore,
the dam with weak construction seams loses its bearing capacity at the value of the
overloading factor n = 1.6 not because of the loss of the dam’s anti-slide stability,
but because of the concrete destruction at the downstream dam toe.
Taking into consideration the monolithic dam: at n = 1.6, Ks is equal to 1.68,
which is 20% greater than that when considering the impact of weak construction
seams. In the monolithic dam, it is observed that at n = 2, the Ks value is equal to
1.17, which is less than the normative value (Ks, n = 1.32). After that, the curve tends
to Ks = 1. Therefore, the monolithic dam reaches its ultimate bearing capacity at
the value of the overloading factor n = 2 because of the loss of the dam’s anti-slide
stability, and partially because of the destruction of concrete at the downstream dam
toe (i.e., partially because of the compressive stress failure of the downstream dam
toe).

3.4 Model Verification

The results of previous experimental studies have shown that the ultimate bearing
capacity of Bureyskaya dam with weak construction seams is 20–30% less than that
of the monolithic dam [10, 11]. It is observed that there is a good agreement between
the simulated and experimental results. Therefore, this confirms the possibility of
using the software complex “CRACK” to study the static work of concrete gravity
dams with consideration various parameters of construction seams.

4 Conclusions

The main conclusions from this study are as follows:

1. Passing from the monolithic model to the dam model with weak construction
seams, the strength characteristics of the dam tend to decrease progressively.
2. When considering the impact of weak construction seams, the dam reaches its
ultimate bearing capacity at the value of the overloading factor n = 1.6 because
of the concrete destruction at the downstream dam toe.
3. The monolithic dam reaches its ultimate bearing capacity at the value of the
overloading factor n = 2 because of the loss of anti-slide stability, and partially
because of the destruction of concrete at the downstream dam toe.
4. At the value of the overloading factor n = 1.6, the anti-slide stability of the
monolithic dam is 20% greater than that of the dam with weak construction
seams.
382 V. Tolstikov and Y. W. Youssef

5. The results of the numerical modelling calculations show that when considering
the impact of weak construction seams, the ultimate bearing capacity of the dam
is 25% less than that of the monolithic dam. These results agree well with the
results of previous experimental studies.
6. In the future, it is necessary to carry out secondary grouting works for
Bureyskaya dam in order to provide the monolithic behaviour of the dam.
7. The software complex “CRACK” can help the researchers to study the static
work of concrete gravity dams with consideration various parameters of
construction semas. This leads to reduce the need for laborious experimental
studies.

References

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304 P., Moscow (1991) (in Russian)
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(2017) (in Russian)
Assessment of Embankment Dam Slope
Stability with Consideration of Its Stress
State

Mikhail Sainov

Abstract The specific feature of embankment dam working conditions is percep-

tion of loads from hydrostatic pressure and seepage and sometimes of seismic loads.
These loads change the initial stress state of an embankment dam. Traditional engi-
neering methods of stability calculations based on analysis of conditions of limit
equilibrium of collapsing soil mass do not allow to the full extent the consideration
of stress–strain state (SSS) specific features. Therefore, the method of mathematical
modeling of loss in dam stability in the process of its SSS variation began to be used
for stability assessment. However, the method of reduction of soil strength indices,
does not meet the requirements of Russian codes of practice and is artificial. There-
fore, the author developed another methodology of slope stability analysis. It is based,
on the one hand, on engineering method of slices and, on the other hand, on consid-
eration of the dam SSS analysis results for calculation of restraining forces. The dam
discretization required is fulfilled by the finite element method. The author developed
a set of inter-related computer programs. With their aid there were performed the
embankment dam slope stability analyses with consideration of the stress state; they
allowed reaching a number of results and recommendations. It was established that
consideration of the dam SSS real character during calculations of its slopes stability
for static loads permits revealing additional safety factor. At perceiving by the dam
of the seismic forces, the most dangerous are unhardened slip surfaces, therefore,
slope stability assessment may be carried out by a simple method.

Keywords Slope stability · Embankment dam · Numerical modeling ·

Stress–strain state · Finite element method · Seismic loads

1 Introduction

Providing safety of slopes is one of the important tasks in designing embankment

dams, therefore, studies of safety is still urgent by present. Papers devoted to the
issues of refining methodology of slope stability analysis and the results of this

M. Sainov (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 383
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_36
384 M. Sainov

analysis are published every year. The examples of modern papers devoted to the
studies of embankment dam slope stability at complicated character of acting loads
are [1, 2].
There are two groups of methods for calculating the stability of slopes: analytical
methods [9–13] and methods of numerical modeling [14–23]. Each of them has its
own advantages and disadvantages, which will be discussed below.
In Russia there used a deterministic approach based on the theory of limit states.
The standard document1 (code of practice—CP) establishes the following criterion
of slope stability:
 
γc
γlc F ≤ R. (1)
γn

In this formula.
R—design value of general force effect,
F—design value of general load-bearing capacity of the system structure-
foundation,
γ c , γ n , γ lc —established by standards coefficients of work conditions, category
of structure and load combination correspondingly.
For quantitative assessment of stability this criterion is reduced to

R γn γlc
Ks = ≥ . (2)
F γc

Here K s is design value of stability factor which is determined by ratio R/F.

The Code of Practice does not require obligatory use of safety factor calculation
methodology; however, it recommends using the methods taking into account the
stress state of the structure and its foundation. It is related to specific features of
embankment dam working conditions.
Firstly, they are subject to other loads. Dams perceive not only their own weight
but also loads from water: hydrostatic pressure and seepage loads. Besides, for
high embankment dams it is required to take into account seismic forces whose
determination is a rather complicated task.
Secondly, the acting loads induce complicated distribution of stresses in the dam
body. If a dam has a heterogeneous structure the distribution of stresses may have a
very complicated character: there may be arch effect and pore pressure.
Consideration of these factors requires selection of the most suitable methodology
of embankment dam slope stability analysis. With this purpose the author analyzed
the advantages and disadvantages of the used methods of embankment dam slope
stability analysis. As a result, the author had to develop a new methodology of slope
stability assessment adapted for embankment dam design validation.

1CP 39.13330.2012. Embankment dams. Updated version of Building Code CHiP 2.06.05–84*.
Moscow. FAO «FCC», 2012. 86 p.
Assessment of Embankment Dam Slope Stability with Consideration … 385

2 Methods

A great number of methods has been developed for slope stability analysis. They
can be divided into analytical (deterministic and probabilistic) methods and methods
based on numerical modeling.
However, all the used methods are not precise. This is connected with a
complicated task on soil mass stability: it has several variables.
First of all, the shape of the slip surface is unknown and accordingly the shape of
collapsible soil mass. In analytical methods based on empirical experience the shape
of the surface is close to circular cylindrical.
Secondly, there is not a unique and theoretically justified method of determining
force R, i.e. bearing capacity of soil composing the collapsing mass. Usually for
determining the soil shear strength τlim use is made of a well-known Coulomb law:

τlim = σ tgϕ + c. (3)

Here σ —normal stress at the shear area perceived by a soil skeleton,

ϕ, c—angle of internal friction and specific cohesion of soil respectively.
The other, more complicated laws of shear strength may be used. However, even in
case of using Coulomb law, there arises one more, third complication. For calculation
of force R the value σ is necessary to be determined not for currently existing stress
state of soil mass, but for the moment of its limit state (the moment of start collapsing).
The stress state in the moment of stability loss is unknown; it is impossible to
determine it reliably and this is the main complication in analysis.
This is the reason why any method of analysis is approximate; depending on the
adopted hypothesis and assumptions, different engineering methods of analysis may
be developed. They are described in educational and scientific literature (methods of
Fellenius, Terzaghi, Cray, Bishop and others). As a rule, the engineering methods of
slope stability analysis are based on the assumption about shape of the slip surface.
In a plane problem, as a rule, a circular-cylindrical sliding surface is used. Since
the 1960s, methods have been developed for calculating the stability of slopes along
spatial slip surfaces [9]. It retains its relevance at the present time, it is considered in
publications [10–13].
Works of many authors deal with assessment of accuracy in the results of design
estimation of slope stability obtained by different methods. The main disadvantage
of certain indicated methods is absence of balance of forces acting on the collapsing
mass at observation of equilibrium by moments.
The Code of Practice (CP) does not require obligatory use of slope stability
analysis methodology; it only recommends to use the method of inclined interacting
forces. It permits conducting calculations for arbitrary shape of the slip surface. This
method is referred to analytical methods; it is based on consideration of the moment
at reaching the limit equilibrium. It is assumed that at the moment before collapsing
the restraining forces reaching their limit states are in the equilibrium with shear
386 M. Sainov

forces, as well as with forces being internal for the collapsing mass. If required the
equilibrium of forces moments may be provided.
Possibility of reaching the strict condition for equilibrium of forces in the limit
state is referred to the advantages of the inclined interacting forces method. However,
reaching this equilibrium requires additional conditions. In the methodology given
in the annexure to the Code of Practice it is recommended to select the values of
soil strength indices tg ϕ and c, providing equilibrium between shear and restraining
forces and, at the same time, the equilibrium between tilting and restraining forces.
Thus, the recommended by a standard document methodology of stability analysis
is not perfect: it does not to the full extent meet the requirement given in this particular
document. The adopted design pattern of acting forces is rather conditional. Also, it
is important that it does not permit consideration of the dam stress state.
Therefore, at designing embankment dams the absolutely different method began
to be used, which is based on numerical modeling. Numerical modeling permits
uniting stability and stress–strain state (SSS) of the structure. This method permits
direct modeling of the process of dam slopes stability loss in the process of SSS
formation. Loss of slope stability occurs at mass failure of soil shear strength, at
which formation of closed collapsing mass takes place. Therefore, it is required to
model the structure SSS in elastic–plastic formulation.
Numerical modeling of the process of loss of stability of soil slopes has been used
since the 1990s [14, 15]. This method has a number of advantages as compared to
the analytical methods. First of all, the important advantage of numerical modeling
is the fact that consideration of the stress state permits taking into account the impact
of failure zones and soil loosening existing in the structure on the structure stability.
Secondly, the structure stress analysis allows determining the shape and location of
the most probable slip surface. It may have an arbitrary but not a circular shape. The
experience in solving the problems of stability by the method of numerical modeling
shows that loss of stability is accompanied by not only one slip surface but the whole
slip zone and its shape is close to a circular shape. It corresponds to the results of
field observations over land slides of soil slopes.
For calculation of stability factor in this method there used the methodology of
reduction of shear strength (RSS) (otherwise the method of reducing the strength
indicators) [14–20]. It envisages reaching collapse (or limit state) of the structure by
gradual decrease of the soil strength indices. The stability factor is determined as
ratio between initial values of strength indices and the values corresponding to the
moment of stability loss:

tg ϕ c
Ks = = . (4)
tg ϕk ck

Here ϕk , ck —the angle of internal friction and specific cohesion respectively

corresponding to the moment of stability loss.
This approach is mentioned in the Code of Practice as the universal assessment
of stability.
Assessment of Embankment Dam Slope Stability with Consideration … 387

However, the methodology based on numerical modeling of the stability loss

process has a number of disadvantages. First of all, the method of reducing soil
properties is artificial, because sequence of the structure SSS variation occurring
in the process of calculations does not reflect the real mechanism of stability loss.
Secondly, the method of calculating the value of stability factor is not precise. It
is rather difficult to fix accurately the moment of stability loss, and contribution of
friction and cohesion forces in the total force of shear strength may be different.
The main disadvantage of this method is the fact that it does not meet the stability
criterion established in the Code of Practice. In spite of the fact that this method
permits convenient calculations of stability of complicated in design structures in
conditions of acting loads complicated by character, its use at designing embankment
dams is not justified.
Therefore, the author developed and used another method of embankment dam
slope stability analysis, which permits consideration of its stress state. To some extent
it combines the engineering method and the method of numerical modeling.

3 Results and Discussion

There are three principles in the basis of the method developed by the author.
The first principle envisages consideration of the previously assigned shape of the
slip surface and the shape of soil mass stability loss in the form of turn around the
rigid axis. Accordingly, the stability criterion is presented in the form of comparison
of shear and restraining forces.
Due to the adopted assumption the shape of the slip surface encloses the arc of a
circle. In 2D formulation the slip surface is circular (circular cylindrical), and in 3D
task it is in the form a sphere or ellipsoid of rotation.
The second principle envisages the use of all the components of the stress tensor
for calculation of shear strength.
Normal stress on the shear area is determined by formula.

σ = σx m 2x + σ y m 2y + σz m 2z + σx y m x m y + σx z m x m z + σ yz m y m z , (5)

where σx , σ y , σz , σx y , σx z , σ yz —components of the stress tensor (in soil skeleton),

m x , m y , m z —direction cosines of the slip surface inclination.
Use of all the components of the stress tensor at calculation of shear strength is
the main difference and an important advantage of the proposed methodology of
engineering methods. The structure SSS analyses are conducted for determining the
stress tensor.
The third principle at calculation of restraining forces envisages consideration of
the stress state corresponding to the existing SSS of the structure. Due to this the
equilibrium conditions of the considered collapsing mass are automatically observed.
In the proposed method the SSS analyses are conducted not for reaching collapse
of the structure but only for determining the existing stress state. This is the main
388 M. Sainov

difference of the proposed methodology from the methodology of strength properties

reduction.
Consideration of the stress state for currently existing but not limiting SSS
distinguishes the proposed methodology from the other methods (from engineering
methods and the method of numerical modeling). Of cause, it introduces some condi-
tionality in determination of the structure load bearing capacity, however, it is not
of crucial significance for stability assessment. For designing embankment dams
the case of stability loss is purely hypothetical but not planned. Certain calculations
fulfilled by the author showed that the embankment dam SSS variation at approaching
the moment of failure does not significantly affect the shear strength and it may be
really neglected.
The formula for calculation of stability factor is as follows:
N
i=1 [( σi tg ϕi + ci ) ωi ] R
Ks = K   . (6)
k=1 Vk ev,k + Tk eT,k

Here Vk , Tk —vertical and horizontal forces in element k,

ev,k , eT,k —arms (eccentricities) of vertical and horizontal forces in element k
respectively,
σi —normal stress in soil skeleton on i part of the slip surface,
ωi —area (length) of i part of the slip surface,
tg ϕi , ci —friction coefficient and soil specific cohesion respectively on i part of
the slip surface,
R—radius of the slip surface.
It is seen from the formula that for calculation of active (shear) and reactive
(restraining) forces the detail division of the structure into parts is required. Finite
elements are used for such parts. This differs the proposed methodology from engi-
neering methods of analyses, where the structure is divided into vertical portions.
Such approach permits convenient use of information about distribution of stresses
in the structure, which is obtained as a result of SSS analyses, at calculations of
stability.
For more precise calculations of stability, the finite-element model of the structure
is fulfilled in more detail than at SSS analyses. Each of the finite elements of the
initial model is divided into several parts. Their quantity is determined by a number
of integration points which are used for determination of stress distribution over the
area of the finite element. This allows convenient transfer of information about stress
distribution from one finite-element model to another.
Use of the finite-element method for discretization of the structure provides the
proposed methodology with some more additional advantages. With the aid of the
finite-element method it is possible to solve the problems of the structure seepage
regime, on determining seismic loads. Use of the general finite-element model
permits convenient transfer the results of numerical modeling of seepage regime,
seismic loads of an embankment dam for their consideration at assessing the slopes
Assessment of Embankment Dam Slope Stability with Consideration … 389

stability. This is especially important for validation of designs for embankment dams
constructed in seismic regions.
For conducting complex calculations and studies of embankment dams the author
developed a set of computer programs. Namely, program Nds_N is intended for
SSS analyses, program Otkos_N—for calculations of slope stability, Filt_N—for
calculations of seepage regime.
Program Nds_N has a number of specific features which are important for
conducting slope stability analyses. One of the specific features is possibility of
using non-linear model of soil deformation. The program permits consideration of
sequence of dam construction and reservoir impoundment. One more specific feature
is possibility of using higher-order finite elements with cube power approximation of
displacements inside the finite element. Use of non-linear function of displacements
permits obtaining non-linear and smooth distribution of stresses in the structure,
which is very important for providing accuracy in calculations of slope stability.
The specific feature of program Otkos_N is possibility of consideration of soil
internal friction angle depending on stresses, which is typical for rockfill. It is very
important for stability analyses of high rock-earthfill and rockfill dams.
A separate set of computer programs was developed by the author for calcu-
lation of seismic loads. It envisages the possibility of seismic forces analysis by
the quasi-static method (method of spectral analysis), whose use is acceptable by
building standards. Calculation by the quasi-static method envisages determination
of structure dynamic characteristics: modes and periods of self-oscillations.
Calculation is conducted in the following sequence. First of all, dynamic
properties of the dam body materials are determined with consideration of their
compaction under the action of static forces and water saturation. Consideration of
soil compaction is of great importance for the structure dynamic characteristics: due
to it the values of soil dynamic modulus sharply increases. The results of SSS dynamic
modeling at static loads are used for determination of soil compaction impact.
Then matrices of masses and stiffness of finite-element model are plotted. Search
of self-frequencies and modes of oscillations is fulfilled by the iteration method of
subspace of proper vectors of stiffness and masses matrices. As a rule, it is sufficient
to determine 20–50 principal modes of self-oscillations. Then calculation of seismic
forces is fulfilled separately for each mode of oscillations and then the total seismic
force is determined.
General sequence of embankment dam slope stability analyses with consideration
of acting seismic forces is as follows:
(1) Dam SSS analysis at action of seismic forces with consideration of construction
sequence and application of loads.
(2) Determination of dam dynamic characteristics with consideration of currently
existing SSS.
(3) Calculation of seismic loads acting on the dam by the method of spectral
analysis.
(4) Dam SSS analysis with consideration of adding seismic loads to static forces.
390 M. Sainov

(5) Development of a detail finite-element model for slope stability calculations

with referencing to the model of information about the dam SSS.
(6) Analysis of embankment dam slope stability.
Slope stability analysis at action of static forces is simpler and includes fulfillment
of steps 1, 5, 6 from the indicated algorithm.
The author used the developed methodology for conducting a number of studies on
the dam slope stability with consideration of the stress state [22, 23]. The following
conclusions and recommendations are the result of these studies:
(1) Use of all stress tensor components at determination of normal stresses on the
slip surface results in increase of shear strength. It permits revealing additional
stability factor of embankment dam slopes. Its value is estimated to be up to
10%.
The example may be the results of slope stability analysis of 330 m high rock-
earthfill dam at acting static forces. At using Terzaghi method the downstream
slope stability factor amounted to 1.67, and that of the upstream slope to 1.64.
At analysis with consideration of SSS the stability factors amounted to 1.68
and 1.78 respectively. These results were confirmed by the results of numer-
ical modeling of slope stability loss calculated by the method of soil strength
properties reduction.
(2) The results of the dam slope stability assessment are affected by presence of
arch effect in distribution of stresses in the dam body. In the near-slope zones
the stresses increase and in the internal zones they are somewhat decreased.
Due to this fact at using the linear model of deformation the considerable
tangent stresses arise in the near-slope zones, which distorts the assessment of
slope stability factor. To prevent distortion, it is recommended to use a non-
linear model taking into account the effect of soil linear deformation modulus
increase with growth of compressive stresses.
(3) Consideration of the stress state at slope stability analysis with acting seismic
forces shows that the most probable are subsurface slip surfaces [23]. There-
fore, the simplest design diagram on stability of the layer on the slope surface
is quite suitable to be used for the dam slope stability assessment.
Consideration of SSS at acting seismic forces results in reduction of slope
stability factor design value as compared to using engineering methods. This
is explained by unfavorable soil stress state at perception of horizontal seismic
forces.
For example, structural stability assessment was carried out at 330 m high
dam at acting horizontal seismic forces, induced by an earthquake of 9 points
intensity (foundation acceleration was 0.32 g). At using Terzaghi method the
downstream slope stability factor amounted to 1.42, that of the upstream slope
to 1.26. At analysis with consideration of SSS the stability factor comprised
1.21.
(4) At conducting slope stability analysis by finite-element method it is necessary
to use finite elements with non-linear function of displacements, because this
is required to provide proper accuracy in distribution of stresses.
Assessment of Embankment Dam Slope Stability with Consideration … 391

These are new results of theoretical and practical importance.

4 Conclusions

(1) The standard document in Russia requires consideration of the stress state at
slope stability analysis of embankment dams. It is rather difficult to meet this
requirement at using traditional engineering methods of analysis.
Therefore, the author developed a special methodology of analysis adapted to
solving the problems of slope stability of high embankment dams. It permits
considering the impact on slope stability of such factors as the dam stress
state, seismic and seepage loads. The proposed method for the stress state
allows conducting stability analysis for spatial slip surfaces.
Based on the finite-element method the author developed a set of inter-related
computer programs for solving the problems related to SSS, seepage regime,
seismic loads and dam slope stability.
(2) The results of slope stability analysis by engineering methods and approaches
with consideration of SSS differ from one another. Consideration of an embank-
ment dam stress state permits revealing additional stability factor of its slopes
at acting static forces, however, this factor is not large. At stability analysis
with action of seismic forces the consideration of SSS results in decrease of
slope stability factor estimate.
(3) At using dam slope stability analysis with consideration of SSS it is necessary
to observe a number of rules to provide accuracy in stability assessment of
the dam near-slope zones. Thorough discretization of these zones into finite
elements is required as well as use of non-linear models of soils.

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Interaction of Long Piles
with a Multilayer Soil Mass, Taking
into Account the Elastic and Rheological
Properties

Zaven Ter-Martirosyan and Aleksander Akuletskii

Abstract When weak clay soils lie at the base, the settlement of the building can
continue for a long period of time. In the design of foundations on such soils, the
forecast of settlement over time is of great importance. The strength and stability of
structures depends both on the rate of development of the settlement over time and on
the final settlement of the structure. Therefore, the approach to describe the process
of foundation settlement must be considered as rheological. This article discusses
the statement and solution of the problem of the interaction of a long pile with
the surrounding multilayer and underlying soils. The problem was considered in a
linear setting, taking into account the rheological properties of the surrounding soil
mass. The solution is presented by analytical method. It has been established that the
rheological properties of a multilayer soil mass have a significant effect on the nature
of the redistribution of forces on the pile between the lateral surface and the lower
end. The dependence of the pressure change under the lower end of the pile on time
is obtained. An expression is found for determining the reduced shear modulus for a
multilayer soil mass. The solutions obtained can be used to preliminary determining
of the movement of a long pile in time. The analytical solutions in the article are
supported by the graphical part. The graphs of the dependence of the force on the
pile heel on time are given for a variable load on the pile head and for a variable pile
length. The selection of the optimal ratio of the pile length and its diameter allows
the most effective usage of the bearing capacity of the pile.

Keywords Multilayer and underlying soils · Reduced modulus · Rheological

properties · Settlement rate

1 Introduction

Most of the construction sites are characterized by difficult engineering and geolog-
ical conditions, represented by the presence of several layers at the base, including

Z. Ter-Martirosyan · A. Akuletskii (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
e-mail: akula.92@inbox.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 393
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_37
394 Z. Ter-Martirosyan and A. Akuletskii

weak clay water-saturated soils. Under these conditions, as a rule, the following is
used: soil consolidation [1–5], soil reinforcement [6, 7], significant deepening of the
underground part of buildings, etc. But the pile foundation is considered as the main
type of foundation on such sites [8–14]. When weak clay soils lie at the base, the
settlement of the building can continue for a long period of time. There are cases
when the settlements of buildings and structures didn’t subside for several decades.
The most famous example is the Leaning Tower of Pisa, the slope of which devel-
oped over several centuries. In the design of foundations on such soils, the forecast
of settlement over time is of great importance. The strength and stability of structures
will depend both on the rate of development of the settlement over time and on the
final settlement of the structure. Therefore, the approach to the description of the
process of foundation settlement should be considered as rheological [15–23]. It is
known that when a long pile interacts with the surrounding multilayer and underlying
soils, a complex inhomogeneous stress–strain state (SSS) arises. In this article, we
consider the problem of the interaction of a long pile with a multilayer soil mass in a
linear setting, which has rheological properties, as well as the problem of determining
the reduced shear modulus for a soil mass.
Investigations of the stress–strain state around a long pile by the numerical method
show that the effect of the length of the pile on the surrounding soil mass extends to
a distance of no more than 6–7 pile diameters, and under its lower end of the same
order in depth [24]. When the distance between the piles is less than six diameters,
the displacement of the pile and the soil in the interpile space occurs simultaneously,
and the pile foundation and soil are displaced as a united mass [25]. These studies
allow us to consider the problem of the interaction of a long pile with a soil mass as
the problem of the interaction of a pile with a soil mass of limited dimensions in the
form of a cylinder with a diameter 2b and height L > l, where is l—the length of the
pile (Fig. 1).
When analyzing the stress–strain state of soils around the pile and under its end, it
was found that shear deformations prevail in the interaction of the pile with the soil,
volumetric deformations can be ignored [26]. We will consider the solution to the
problem for a round pile. We also assume that the stiffness of the pile significantly
exceeds the stiffness of the soil E pile  E soil .
Let’s write down the equilibrium equation for the considered case (Fig. 1):

N =T+R (1)

where

N = π · a 2 · p1 (2)

T = 2π · a · l · τ (3)

R = π · a 2 · p2 (4)
Interaction of Long Piles with a Multilayer Soil Mass, Taking … 395

Fig. 1 Design scheme for

the interaction of a pile with
a multilayer soil cylinder

Substituting Eqs. (2), (3), (4) into Eq. (1), we obtain expression for τ :
a
τ = ( p1 − p2 ) · (5)
2l
Since E pile  E soil the pile settlement of each layer under consideration is equal,
i.e.

S1 = S2 = Si = S (6)

where Si —pile settlement of the ith layer; S—total pile settlement.

The shear deformation of the elementary soil layer around the pile can be
determined by the following dependence:

τi (r )
γi (r ) = − , (7)
Gi

where G i —shear modulus of the ith layer; i = 1, 2, ..., n- layer number.

Based on Eq. (6), we can write an expression for the shear stresses of the i—th
layer:
τ
τi = · Gi (8)
G

where G—reduced shear modulus of a multilayer soil mass.

396 Z. Ter-Martirosyan and A. Akuletskii

Based on the equation for the distribution of shear stresses along the length of the
pile, we obtain:

τ · l = τ1 · l1 + τ2 · l2 + τi · li (9)

Considering (9) and (8) together, we obtain an expression for determining the
reduced shear modulus for a multilayer soil mass:

l1 · G 1 + l2 · G 2 + li · G i
G= (10)
l

where l—pile length; i = 1, 2, ..., n—layer number.

Let’s write the dependence for the rate of change in the shear deformation around
the pile, taking into account the rheological properties of the surrounding soil mass:

τ̇α τα
γ̇ = − + (11)
G η(t)

.
where τ —shear stress rate; τα = T /2 · π · a · l; η(t)—weighted average viscosity
α
index.
Since the forces transferred to the pile are constant (p1 = const) the rate of change
of pressure on the pile head does not change ( ṗ1 = 0). Based on this, we determine
the rate of change of shear stresses:
a
τ̇α = − ṗ2 · (12)
2·l
The rate of settlement of the pile from the action of shear stresses on the lateral
surface, taking into account the elastic-viscous characteristics of the surrounding soil
mass:
   
a · τα b a · τ̇α b
V̇T = · ln + · ln (13)
η(t) a G a

where G—we determine by the Eq. (10).

Let’s determine the rate of settlement of the pile due to the deformation of the
soil under the lower end of the pile, assuming that the pile acts as a flat round stamp.
The equation has the next form:

π · a · (1 − ν0 ) · K
V̇R = ṗ2 · (14)
4 · G0
Interaction of Long Piles with a Multilayer Soil Mass, Taking … 397

where ṗ2 —rate of pressure change under the heel of the pile; ν0 i G 0 —deformation
parameters of the soil under the lower end of the pile; K ≤ 1—coefficient taking
into account the depth of application of the load on the pile heel.
Based on the fact that the stiffness of the pile is much greater than the stiffness
of the surrounding soil E pile  E soli , the rate of settlement from forces on the lateral
surface is equal to the rate of settlement from the action of forces at the level of the
lower end of the pile. Equating (13) and (14), and also taking into account (5) and
(12), we obtain:
   
a2 b a2 b π · a · (1 − ν0 ) · K
( p1 − p2 ) · · ln − ṗ2 · · ln = ṗ2 ·
2 · l · η(t) a 2·l ·G a 4 · G0
(15)

Performing certain transformations of expression (15), we obtain a linear

differential equation of the 1st order:

1 p1
ṗ2 + p2 · = (16)
η(t) · A η(t) · A

where
π · (1 − ν0 ) · K · l 1
A= b + (17)
2 · G 0 · a · ln a G

The general solution of the differential Eq. (16) is found by the formula [27]:

 

− dt p1 dt
p2 (t) = e η(t)·A · ·e η(t)·A dt + C (18)
η(t) · A

Let’s consider the solution to Eq. (18) where η(t) = η = const. In this case:
  
p1
p2 (t) = e− η·A · = e− η·A · p1 · e η·A + C = p1 + C · e− η·A
t t t t t
· e η·A dt + C
η(t) · A
(19)

The constant of integration C is determined from the initial condition t = 0. Then:

C = p2 (0) − p1 (20)

Considering (19) and (20) together, we finally obtain an expression for deter-
mining the pressure under the heel of the pile at a certain point in time t:

p2 (t) = p1 + ( p2 (0) − p1 ) · e− η·A

t
(21)
398 Z. Ter-Martirosyan and A. Akuletskii

Pile settlement at a specific point in time t can be obtained by the equation:

π · a · (1 − ν0 ) · K
VR (t) = p2 (t) · (22)
4 · G0

where p2 (t) is found by the formula (21).

Consider the solution to Eqs. (21) and (22) with the initial condition p2 (0) = 0 at
variable values of the pile radius η1 = 1 × 1012 P; η2 = 5 × 1012 P; η3 = 1 × 1013
P; η4 = 5 × 1013 P, and l = 30 m; b = 6.5 · a; E 1 = 30 MPa; E 2 = 10 MPa; E 3 =
25 MPa; E 0 = 50 MPa; a = 0.5 m; K = 0.7; ν 1 = ν 2 = ν 3 = ν 0 = 0.35.
Consider the solution to Eqs. (21) and (22) with the initial condition p2 (0) = 0
at variable values of the pile radius a1 = 0.3 m; a2 = 0.5 m; a3 = 0.8 m; a4 = 1.0 m,
and l = 30 m; b = 6.5 · a; E 1 = 30 MPa; E 2 = 10 MPa; E 3 = 25 MPa; E 0 = 50 MPa;
η = 5 × 1012 P; K = 0.7; ν 1 = ν 2 = ν 3 = ν 0 = 0.35.
Consider the solution to Eqs. (21) and (22) with the initial condition p2 (0) = 0
at variable values of the pile radius l1 = 24 m; l 2 = 30 m; l 3 = 36 m; l 4 = 45 m, and
a = 0.5 m; b = 6.5 · a; E 1 = 30 MPa; E 2 = 10 MPa; E 3 = 25 MPa; E 0 = 50 MPa;
η = 5 × 1012 P; K = 0.7; ν 1 = ν 2 = ν 3 = ν 0 = 0.35.

2 Results of the Research

Analysis of the dependences obtained (Fig. 2) shows that the stress under the pile
bottom and the pile settlement change at different rates and tend to a constant value
over the time (at t → ∞, p2 (t) → p2∞ = const, VR (t) → VR∞ = const).
Therefore, proceeding from Eq. (5), the shear stresses on the lateral surface of the
pile decrease with time. According to the dependences obtained (Fig. 3), with an
increase in the pile diameter, the rate of pressure changes under the bottom and

Fig. 2 Dependency graphs of p2 (t) (left) and V R (t) (right) at different viscosity parameters of the
surrounding soil
Interaction of Long Piles with a Multilayer Soil Mass, Taking … 399

Fig. 3 Dependency graphs of p2 (t) (left) and V R (t) (right) at different values of the pile radius

Fig. 4 Dependency graphs of p2 (t) (left) and V R (t) (right) at different values of the pile length

settlement of the pile increases. At the same time, with variable diameters of the pile,
the settlement tends to different values. The dependences shown in Fig. 4 showed
that with an increase in the length of the pile, the rate of pressure changes under the
base and settlement decreases. In addition, at different pile lengths, the settlement of
the pile tends to a constant value.

3 Conclusion and Discussion

1. When the pile interacts with the surrounding multilayer soil mass, which has
elastic-viscous characteristics, a complex stress–strain state occurs, in which
the stress p2 under the pile base changes over time.
2. According to the obtained dependences, with an increase in the pile diameter,
the rate of pressure changes under the foot and settlement of the pile increases.
400 Z. Ter-Martirosyan and A. Akuletskii

At the same time, with variable diameters of the pile, the settlement tends to
different values.
3. The obtained dependences of the pressure under the pile base and the settlement
of the pile on the length of the pile showed that when the length of the pile
increases, the rate of change in pressure under the base and settlement decreases.
In addition, at different pile lengths, the settlement of the pile tends to a constant
value.
4. The rheological properties of a multilayer soil mass have a significant impact on
the nature of the redistribution of forces on the pile between the lateral surface
and the lower end.
5. The dependence for determining the reduced shear modulus of a multilayer soil
mass is obtained.
6. The solutions obtained can be used to preliminary determining the movement
of a long pile in time. Selection of the optimal ratio of pile length and diameter
allows you to use the bearing capacity of the pile as efficiently as possible.

Acknowledgments All tests were carried out using research equipment of The Head Regional
Shared Research Facilities of the Moscow State University of Civil Engineering.

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Propagation of a Spherical Wave
in Elastoplastic Medium with Complex
Equations of State

Sherzod Khudainazarov and Burkhon Donayev

Abstract The problem of the propagation of a spherical wave in a soil is solved

in an analytically inverse way for soils with more complex equations of state. The
results are obtained for the propagation of a spherical shock wave in soil with a more
complex equation of state for the shape change in the medium. The study shows that
taking into account the nonlinear elastic shock waves of the annular stress leads to
an increase in comparison with the elastic medium. Note that in the case of using a
complicated equation of state of the soil, a spherical shock wave propagates in the
soil, behind the front of which, in the disturbance region, the medium is unloaded.

Keywords Loading · Unloading · Shock wave · Caverns · Soil · Stress-strain

state · Transcendental

1 Introduction

A shock wave is an area of compression of a medium, which in the form of a

spherical layer propagates at a supersonic speed in all directions from the source of
its formation. Depending on the medium in which the shock wave propagates (in air,
water or soil), it is respectively called an air shock wave, a shock wave in water, a
seismic explosion wave in the ground.
Distinguish between a shock wave of natural and anthropogenic origin. Natural
waves include shock waves caused by volcanic eruptions, earthquakes, hurricanes,
tornadoes, falling meteorites, etc. The anthropogenic shock waves are those that
occur as a result of explosions of nuclear devices, chemical explosions, explosions
at nuclear power facilities, explosions at oil refining and petrochemical industries,
explosions of gas-air mixtures or mixtures of flammable liquids and gases with
air. Most of the destruction and damage to buildings and structures, equipment of

S. Khudainazarov (B)
Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Kori Niyoziy St, 39,
Tashkent, Uzbekistan 100000
B. Donayev
Karshi Engineering-Economics Institute, Mustakillik street, Karshi, Uzbekistan 225

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 403
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_38
404 S. Khudainazarov and B. Donayev

industrial facilities, as well as damage to people, as a rule, is caused by the action of

a shock wave.
In [1, 2], a methodology for three-dimensional reconstruction of the position and
shape of the blast shock wave as a function of time was developed. A series of blasting
tests is performed, during which the blasting process is displayed by several high-
speed digital cameras scattered over a large area. High speed images are processed
using the Schlieren method for shock wave imaging. Shock propagation is measured
and corresponds to the Dewey equation. Analysis of the position and propagation
of the shock wave makes it possible to reveal asymmetries at the shock front due
to the asymmetric explosion process. The methods developed here are shown to
be useful tools that can be implemented to complement traditional point tools in
current explosive research testing and provide improved explosion performance over
traditional field testing methods.
In [3–6], the propagation of a strong spherical shock wave in a dusty gas with
or without self-gravity effects in the case of isothermal and adiabatic flows was
investigated. It is assumed that the dusty gas is a mixture of fine solid particles and
ideal gas. Analyzes show that after the effects of the gravitational field are turned on,
the impact force unexpectedly increases and noticeable differences in the distribution
of flux variables are found. Increasing the time also increases the power of the blow.
It is investigated that taking into account the isothermal flow increases the impact
force and eliminates the singularity in the density distribution.
In [7–11], nonlinear relations based on the laws of Winkler and Coulomb are
proposed to describe the process of shear interaction between structures and soil.
Their advantages and disadvantages are shown, as well as the suitability of the exper-
imental results. And also one-dimensional non-stationary wave problems for soil and
structure are numerically solved using the method of characteristics and the method
of finite differences. Analysis of the obtained numerical solutions shows a signifi-
cant dependence of longitudinal stresses on wave processes in the soil, the dynamic
stress state of the soil and the mechanical properties of the soil and the material of
the structure. The results obtained are the basis for the development of a new stan-
dard calculation of the strength of underground engineering structures under seismic
impact.
In works [12–15], studies were carried out to analyze the behavior of a high-
rise structure with various kinematic effects, taking into account the real geometry,
dissipative and nonlinear properties of the structure material. A generalized approach
to the dynamic calculation of high-rise structures has been developed, frequency
characteristics have been constructed at various points of the structure. It was found
that the nonlinear properties of the material of structures appear when the impact
of a spherical wave can cause significant deformations in the structure. This applies
not only to the magnitude of the impact force, but also to its frequency content. If
nonlinear elastic deformation of the material manifests itself in the structure, then
this leads to a decrease in the displacement amplitudes of the points and an increase
in the oscillation period compared to a linear elastic structure with similar kinematic
effects.
Propagation of a Spherical Wave in Elastoplastic Medium … 405

In [16], the problem of the propagation of a spherical shock wave in an elastoplastic

medium is solved analytically and numerically by the method of characteristics. The
results show that taking into account the diagrams of nonlinear elastic impact leads to
an increase in the circular stress wave as compared to the elastic medium. It turned out
that the concentration of stresses on a spherical cavity is higher than on a cylindrical
one.
In [17–19], a variational statement, methods and algorithms for solving various
dynamic problems for a viscoelastic system are given, taking into account the condi-
tions of non-reflection on the boundary of a finite base. The dynamic behavior of
an inhomogeneous viscoelastic system under a short-term intense load on the foun-
dation is investigated. A weak dependence of energy dissipation in a system with
hereditary viscoelastic properties of a material on the frequency of natural vibrations
is revealed, as well as a dependence of the wave removal of energy on the main
natural frequencies of the system vibrations.
In [20, 21], the propagation of a spherical wave in linear-elastic and viscoelastic
media was investigated. Here, a new model and a new approach to the analytical
solution of the problem have been developed to model viscoelastic damping. Wave
propagation is achieved by cascading separate geometric and viscoelastic damping
mechanisms. Comparison of the analytical model with the results of dynamic finite
element modeling shows that the method of cascading individual transfer functions
is a suitable approach for wave propagation in viscoelastic media.
In [22, 23], the propagation of axisymmetric viscoelastic waves in extended multi-
layer cylindrical structures was investigated. The classical methods of mathematical
physics are used to solve boundary value problems in a cylindrical coordinate system,
the spatial Fourier transform, the method of complex amplitudes for the variable
components of the displacement vector and the stress tensor. The low-frequency
resonances of the cylindrical shell are investigated and the physical regularities of
the formation of the pitch of the sound of its fundamental tone are determined.
Dynamic analysis and the results of engineering analysis on the nature of the oper-
ation of structures during strong earthquakes indicate that the rigidity of structures
does not always remain constant. Therefore, the parameters of the actual response of
structures must be determined only with the help of nonlinear analysis, which allows
developing more reasonable design and construction methods, increasing the effi-
ciency of structures while maintaining the required level of reliability. However,
the problem under consideration is three-dimensional and non-stationary; there-
fore, methods for calculating underground structures for dynamic effects due to
the complexity of the physical and mechanical properties of the soil, the nature of
seismic and seismic explosive effects, the shape and geometry of structures have not
yet been developed enough. In this direction, a certain success has been achieved with
an obstacle of various shapes within the framework of the linear theory of elasticity,
but a limited number of works have been devoted to elastoplastic deformations.
In this regard, in this work, a number of specific problems have been solved in
relation to the study of the interaction of waves with a spherical cavity in soils, taking
into account their complex elastoplastic deformations.
406 S. Khudainazarov and B. Donayev

The main goal of this article is to study the one-dimensional and two-dimensional
nonstationary problem of the dynamic theory of plasticity as applied to the calcu-
lation of the parameters of the medium in cases of propagation and deformation of
waves from different surfaces, based on the deformation theory with more complex
equations of state.
The goal of this article is to study the one-dimensional and two-dimensional
nonstationary problem of the dynamic theory of plasticity as applied to the calculation
of the parameters of the medium in cases of propagation and deformation of waves
from various surfaces, based on the deformation theory with more complex equations
of state.

2 Methods

In this work, the problem of the propagation of a spherical wave in the soil is solved
in an analytically inverse way for soils with more complex equations of state. The
complexity of the equations of state is based on the assumption that the shape change
function (stress intensity) of the deformation theory [24, 25], which is one of the
equations of state of the medium, depends on the first invariants of the strain tensor, i.e.
This function in the process of loading the soil (Fig. 1), according to the experimental
data [26], is represented as:

σ (ε) + 25  b 
σi = (ε, εi ) = σiH (εi ) − σi (εi ) − σiH (εi ) , (1)
15
 
σiH,b = a j 1 − exp(−b j εi ) + c j εi , j = 1, 2 , (2)

In formulas (2), the coefficients a j , b j , c j are assumed to be known constant

values. For the function σ (ε) included in (1), we have the expression

Fig. 1 Change of the first

and second invariants of the
stress tensors σ , σi and
deformation ε, ε1
Propagation of a Spherical Wave in Elastoplastic Medium … 407

Fig. 2
Graph of the change in the shock wave in the ground
r = R(t)

σ (ε) = (α1 + α2 |ε|)ε , σi (εi ) = (β1 + β2 εi )εi (3)

In the process of solving the problem, it is assumed that when using

∂ 2u ∂σrr (σrr − σφφ )

ρ0 = +2 (4)
∂t 2 ∂r r

and (1), a spherical shock wave propagates in the soil (Fig. 2), at the front of which the
medium is instantly loaded in a nonlinear manner, followed by a linear irreversible
unloading with Young’s moduli E1 and E2 .This assumption is confirmed by the
solution and numerical calculations of the problem. For the analytical construction
of the solution to the problem, the surface of the shock wave r = R(t), as in (1), is
given as a polynomial of the second degree with respect to time, and the load profile
σ0 (t) is determined from the solution of the problem. Then from the condition
•
σrr∗ = −ρ0 R (t)u ∗t , u ∗t = −R(t)εrr
∗
, u(r, t) = 0 at r = R(t) (5)

taking into account the first Eq. (3), (4) and (1), all the parameters of the medium,
including the mass velocity u ∗ and deformation ε∗ = εrr ∗
(since at the mixing wave
∗
front u ≡ 0) at the shock wave front, are obtained by the known quantities depending
on time t or coordinates r . In this case, to determine the deformation ε∗ (t) at the
shock front, instead of the first equation,
•
ρ0 R 2 (t) − (α1 + 49 β1 )
ε∗ (t) = εrr (t) = − ,
(α2 − 8
β )
27 2
(6)
•
u ∗t = − R (t)ε∗ (t),

we obtain a transcendental equation of the form, which is solved numerically

using the standard procedure.
408 S. Khudainazarov and B. Donayev
   
4 (α1 ε − α2 ε2 + 25) 1 − exp(−b2 εi )
α1 − α2 ε + a2 1 + +
9 15 ε1
 
α1 ε − α2 ε2 + 25 α1 ε − α2 ε2 + 25 (7)
+ c2 − (c1 − c2 ) − a1 ·
15 15εi
  2
· 1 − exp(−b2 εi ) − ρ0 R 2 (t) = 0, wher e εi = − εrr , εrr = ε ≺ 0,
3
Let Eq. (7) have a solution

ε = ε∗ (t) (8)

If the equations of the shock wave surface r = r0 + R1 t − R2 t 2 /2 are represented

in the form t = t (r ), then the wave velocity is written relative to the coordinate and
from (7) we find

ε = ε∗r (9)

Thus, by analogy with (6), at the front of a spherical shock wave, we have the
following conditions:

∂u ∂u
= ε∗ (t), = −R(t) · ε∗ (t) (10)
∂r ∂t
In this case, in the region of perturbation, the equation
 2  
∂ 2u 2 ∂ u 2 ∂u u Q(r )
= a0 + − + (11)
∂t 2 ∂r 2 r ∂r r ρ0 a02

with decision

ψ  (r − a0 t) + ϕ  (r + a0 t) ψ(r − a0 t) + ϕ(r + a0 t)
u(r, t) = −
r r2
r r
r 1
− Q(r )dr + + Q(r )r 3 dr (12)
3(λ + 2G 0 ) 3(λ + 2G 0 )r 2
r0 r0

where ψ i ϕ—unknown functions.

Here is the function Q(r ), expressed by the formula

∂   2 
Q(r ) = σrr (r ) − ρ0 a02 ε∗ (r ) + σrr∗ (r ) − σφφ (r ) − 2Gε∗ (r ) , (13)
∂r r
Propagation of a Spherical Wave in Elastoplastic Medium … 409

ρ0 a02 = λ + 2G 0 λ0 = E 1 − 29 E 2 G 0 = 13 E 2 ; a0 = (E 1 + 49 β1 )/ρ0 directly

depends on the solution of the transcendental Eq. (7), represented in the form (1.9).
To find the unknown functions ψ and ϕ included in (1.12), substituting (1.12) into
(1.10), we obtain a system of two equations in the form:

ψ  (R − a0 t) + ϕ  (R + a0 t) 1 R(t)
1
− Q(ξ )dξ − ·
R(t) (λ0 − 2G 0 ) r0 (λ0 − 2G 0 ) (14)
   
· σrr (R(t)) − ρ0 a02 ε∗ (R(t)) − σrr (r0 ) − ρ0 a02 ε∗ (r0 ) = ε∗ (t);
  R 2 (t) R(t)
2R(t)ψ  (R − a0 t) − ψ  (R − a0 t) − ϕ  (R + a0 t) − ·· Q(ξ )dξ −
(λ0 + 2G 0 ) r0
R 2 (t)  ∗   
σrr (R(t)) − ρ0 a02 ε ∗ (R(t)) − σrr (r0 ) − ρ0 a02 ε ∗ (r0 ) =
(λ0 + 2G 0 )
= R 2 (t)(1 + R(t)/a0 ε ∗ (t))
(15)

Now, substituting (15) into (14), after some transformations, we have:


 1 1 R(F(z)) R(F 1 (z)) R(F 1 (z))
ψ (z) = Q(ξ )dξ + · ·
2 (λ0 + 2G 0 ) r0 (λ0 + 2G 0 ) [R(F(z) − a0 ]
1  ∗ 
· Q[R(F 1 (z))] + σrr (R(F 1 (z))) − ρ0 a02 ε ∗ (R(F 1 (z)))
(λ0 + 2G 0 )
1  ∗  R(F 1 (z)) · R(F 1 (z))
− σrr (r0 ) − ρ0 a02 ε ∗ (r0 ) +
(λ0 + 2G 0 ) (λ0 + 2G 0 )[R(F1 (z)) − a0 ]
 ∗  2R(F 1 (z)) − a0 • 
· σrr (R(F1 (z))) − ρ0 a02 ε ∗ (R(F 1 (z))) + 1 + R (F 1 (z))/a0
R(F 1 (z)) − a0
••
∗ R(F 1 (z)) · R(F 1 (z)) ∗
· ε (F 1 (z)) + ε (F 1 (z))
a0 [R(F 1 (z)) − a0 ]
 • 
R(F 1 (z)) •
+ 1 + R (F 1 (z))/a0 ε ∗ (F 1 (z)) (16)
R(F 1 (z)) − a0

where F 1 (z) is the root of equation R − a0 t = z with respect to t. Denoting the

right-hand side of (16) through functions ϕ(z) and appropriately substituting (16)
into (15), we obtain an expression for the desired function ϕ(r + a0 t).. Then, taking
with considering (12), the solution of the spherical problem in this case is represented
as:
⎧ r −a t ξ1 r +a0 t R(F2 (ξ ))−a0 F2 (ξ )
1⎨
0

u(r, t) = dξ1 ϕ(ξ )dξ − dξ1 ϕ(ξ )dξ

r⎩
R−a0 t z 10 R+a0 t z 10
410 S. Khudainazarov and B. Donayev

r +a0 t R(F2 (ξ )) r +a0 t

R(F2 (ξ )) 
+ dξ1 Q(ξ )dξ + R(F2 (ξ )) σrr∗ (R(F2 (ξ )))
λ0 + 2G 0
R+a0 t r0 R+a0 t
r +a0 t
 R(F2 (ξ )) 
− ρ0 a02 ε∗ (R(F2 (ξ ))) dξ − [σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ
λ0 + 2G 0
R+a0 t
r +a0 t
⎧ r −a0 t ξ2 ξ1
1⎨
+ R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ } − 2 dξ2 · dξ1 · ϕ(ξ )d(ξ )
r ⎩
R+a0 t R−a0 t z 10 z 10
r +a0 t ξ2 R(F2 (ξ ))−a0 F2 (ξ )

− dξ2 dξ1 ϕ(ξ )dξ

R+a0 t z 20 z 10
r +a0 t ξ2 R(F2 (ξ1 ))
R(F2 (ξ ))
+ dξ2 dξ1 Q(ξ )dξ
λ0 + 2G 0
R+a0 t z 20 r0
r +a0 t ξ1
R(F2 (ξ ))  ∗ 
+ dξ1 · σrr (R(F2 (ξ ))) − ρ0 a02 ε∗ (R(F2 (ξ ))) dξ
λ0 + 2G 0
R+a0 t z 20
r +a0 t ξ1
R(F2 (ξ ))
− dξ2 dξ
λ0 + 2G 0
R+a0 t z 20
r +a0 t ξ1
 
· [σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ + dξ1 R(F2 (ξ )) · ξ ∗ (F(ξ ))dξ
R+a0 t z 20

1 1 1 + −
+ − C35 − C24 a0 t
2 R(t) r
R(t)−a0 t ξ2 ξ1 R(t)−a0 t ξ2
R(F2 (ξ1 ))−a0 F2 (ξ1 )
+ dξ2 dξ1 ϕ(ξ )dξ − dξ2 dξ1 ϕ(ξ )dξ
z 10 z 10 z 10 z 20 z 20
R(t)+a0 t ξ2
R(F2 (ξ1 ))
+ dξ2 dξ1 ·
λ0 + 2G 0
z 20 z 20
R(F2 (ξ1 )) R(t)+a0 t ξ1
R(F2 (ξ ))
· Q(ξ )dξ + dξ1
λ0 + 2G 0
r0 z 20 z 20
 ∗ 2 ∗

· σrr (R(F2 (ξ ))) − ρ0 a0 ε (R(F2 (ξ ))) dξ
Propagation of a Spherical Wave in Elastoplastic Medium … 411

R(t)+a0 t ξ1
R(F2 (ξ ))  ∗ 
− dξ1 · σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ
λ0 + 2G 0
z 20 z 20
R(t)+a0 t ξ1

+ dξ1 R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ }

z 20 z 10

 R(t) R(t)
1 R 2 (t) 1
+ Q(ξ )dξ − ξ 3 Q(ξ )dξ
r 3(λ0 + 2G 0 ) 3(λ0 + 2G 0 )R(t)
r0 r0
R(t)
1 
+ · ξ 2 [σrr (ξ )− ρ0 a02 ε∗ (r0 ) dξ
λ0 + 2G 0 R(t)
r0

R 2 (t)  r03
− [σrr (r0 )− ρ0 a02 ε∗ (r0 ) + ·
3(λ0 + 2G 0 ) 3(λ0 + 2G 0 ) · R(t)
r
  r 1
· σrr∗ (r0 )− ρ0 a02 ε∗ (r0 ) − Q(ξ )dξ +
3(λ0 + 2G 0 ) 3(λ0 + 2G 0 )r 2
r0
r r
1  
· ξ 3 Q(ξ )dξ − · ξ 2 σrr∗ (ξ )− ρ0 a02 ε∗ (ξ ) dξ
(λ0 + 2G 0 )r 2
r0 r0

r  ∗  r03
+ σrr (r0 )− ρ0 a02 ε∗ (r0 ) −
3(λ + 2G 0 ) 3(λ0 + 2G 0 )r 2
 ∗ 0 2 ∗

· σrr (r0 )− ρ0 a0 ε (r0 ) , (17)

where F2 (ξ ) is the root of the equation with respect to R(t) + a0 t = z. From

(17), taking with considering (10), to determine the mass velocity u t and the radial
component of the soil deformation εrr , we obtain the expressions:
⎧
⎪ r −a0 t R(F2 (r +a0 t))−a0 F2 (r +a0 t)
a ⎨
u t (r, t) = u ∗t (r, t (r )) +
0
− ϕ(ξ )dξ − ϕ(ξ )dξ
r ⎪ ⎩
r −a0 t (r ) R(F2 (r +a0 t (r )))−a0 F2 (r +a0 t (r ))
R(F2 (r +a0 t))
R(F2 (r + a0 t)) R(F2 (r + a0 t (r ))
+ Q(ξ )dξ −
(λ0 + 2G 0 ) (λ0 + 2G 0 )
r0
R(F2 (r +a0 t (r )))
R(F2 (r + a0 t))  ∗
Q(ξ )dξ + σrr (R(F2 (r + a0 t)))
(λ0 + 2G 0 )
r0

− ρ0 a02 ε ∗ (R(F2 (r + a0 t)))
R(F2 (r + a0 t (r ))  ∗ 
− σrr (R(F2 (r + a0 t)))− ρ0 a02 ε ∗ (R(F2 (r + a0 t))))
(λ0 + 2G 0 )
412 S. Khudainazarov and B. Donayev

(R(F2 (r + a0 t)) − R(F2 (r + a0 t (r )))  ∗ 

− · σrr (r0 )− ρ0 a02 ε ∗ (r0 )
(λ0 + 2G 0 )
+ R(F2 (r + a0 t)) · ε ∗ (F2 (r + a0 t))

− R(F2 (r + a0 t (r ))) · ε ∗ (F2 (r + a0 t (r )))
⎧
⎪ r +a0 t ξ r +a0 t R(F2 (ξ )−a0 F2 (ξ )
a0 ⎨
− 2 − dξ · ϕ(ξ )dξ − dξ1 ϕ(ξ )dξ
r ⎪⎩
r −a0 t z 10 r +a0 t (r ) z 10
r +a0 t R(F2 (ξ1 ))
R(F2 (ξ )
+ dξ · Q(ξ )dξ
(λ0 + 2G 0 )
r +a0 t (r ) r0
r +a0 t
R(F2 (ξ ))  ∗ 
+ σrr (R(F2 (ξ )))− ρ0 a02 ε ∗ (R(F2 (ξ ))) dξ
(λ0 + 2G 0 )
r +a0 t (r )
r +a0 t
R(F2 (ξ )  ∗ 
− σ (r0 )− ρ0 a02 ε ∗ (r0 ) dξ
(λ0 + 2G 0 ) rr
r +a0 t (r )
⎫
r +a0 t ⎪
⎬
+ R(F2 (ξ ))ε ∗ (F2 (ξ ))dξ , (18)
⎪
⎭
r +a0 t (r )
⎧ r −a t R(F2 (r +a0 t)−a0 F2 (r +a0 t)
∂u 1 ⎨ 0
∗
εrr (r, t) = = εrr (R(t), t) + ϕ(ξ )dξ − ϕ(ξ )dξ
∂r 2⎩
z 10 z 10
R(F2 (r +a0 t))
R(F2 (r + a0 t))
+ Q(ξ )dξ
(λ0 + 2G 0 )
r0
R(F2 (r + a0 t))  ∗
+ σrr (R(F2 (r + a0 t)))
(λ0 + 2G 0 )
 R(F2 (r + a0 t))
− ρ0 a02 ε∗ (R(F2 (r + a0 t))) −
(λ0 + 2G 0 )
 ∗  
· σrr (r0 )− ρ0 a0 ε (r0 ) + R(F2 (r + a0 t)) · ε∗ (F2 (r + a0 t))
2 ∗
⎧ R(t)−a t R(F2 (R(t)+a0 t)−a0 F2 (R(t)+a0 t)
1 ⎨
0

− ϕ(ξ )d(ξ ) − ϕ(ξ )d(ξ )

R(t) ⎩
z 10 z 10
R(F2 (R(t)+a0 t))
R(F2 (R(t) + a0 t R(F2 (R(t) + a0 t))
+ Q(ξ )dξ +
(λ0 + 2G 0 ) (λ0 + 2G 0 )
r0
 ∗ 
· σrr (R(F2 (t) + a0 t))− ρ0 a02 ε∗ (R(F2 (R(t) + a0 t)))
R(F2 (R(t) + a0 t  ∗ 
− σrr (r0 )− ρ0 a02 ε∗ (r0 )
(λ0 + 2G 0 )

+ R(F2 (R(t) + a0 t)) · ε∗ (F2 (R(t) + a0 t))
Propagation of a Spherical Wave in Elastoplastic Medium … 413
⎧ r −a t ξ1
2⎨
0

− 2 dξ1 ϕ(ξ )d(ξ )

r ⎩
z 10 z 10
r +a0 t R(F2 (ξ ))−a0 F2 (ξ1 ))

− dξ1 ϕ(ξ )dξ

z 20 z 10
r +a0 t R(F2 (ξ1 ))
R(F2 (ξ )
+ dξ1 Q(ξ )d(ξ )
(λ + 2G 0 )
z 20 r0
r +a0 t
R(F2 (ξ )  ∗ 
+ σrr (R(F2 (ξ ))) − ρ0 a02 ε∗ (R(F2 (ξ )))
(λ + 2G 0 )
z 20
r +a0 t
R(F2 (ξ ))  ∗ 
dξ − σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ
(λ + 2G 0 )
z 20
r +a0 t

+ R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ

z 20
⎧ R(t)−a t ξ1
2 ⎨ 0

+ 2 dξ1 ϕ(ξ )dξ

R (t) ⎩
z 10 z 10
R(t)+a0 t R(F2 (ξ1 ))−a0 F2 (ξ1 ))

− dξ1 · ϕ(ξ )dξ

z 10 z 10
R(t)+a0 t
R(F2 (ξ ))  ∗
+ σ (R(F2 (ξ )))
(λ0 + 2G 0 ) rr
z 20
R(t)+a0 t
 R(F2 (ξ ))  ∗ 
− ρ0 a02 ε∗ (R(F2 (ξ ))) dξ − σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ
(λ0 + 2G 0 )
z 20
⎫
R(t)+a0 t
⎬ 2 − +
+ R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ + − C24 a0 t + C35
⎭ r3
z 20
r +a0 t ξ2 ξ2 r +a0 t ξ2 R(F2 (ξ1 ))−a0 F2 (ξ1 ))

+ dξ2 dξ1 ϕ(ξ )dξ − dξ2 dξ1 ϕ(ξ )dξ

z 20 z 20 z 10 z 20 z 20 z 10
414 S. Khudainazarov and B. Donayev

r +a0 t ξ2 R(F2 (ξ1 ))

R(F2 (ξ ))
+ dξ2 dξ1 Q(ξ )d(ξ )
(λ0 + 2G 0 )
z 20 z 20 r0
r +a0 t ξ1
R(F2 (ξ1 ))
+ dξ1
(λ0 + 2G 0 )
z 20 z 20
r +a0 t ξ1
⎫
  ⎬
· σrr∗ (r0 )− ρ0 a02 ε∗ (r0 ) dξ + dξ1 R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ
⎭
z 20 z 20
R(t)−a0 t ξ2 ξ1
2 − +
− 3 −C24 a0 t + C35 + dξ2 dξ1 ϕ(ξ )dξ
R (t)
z 10 z 20 z 10
R(t)+a0 t ξ2 R(F2 (ξ1 ))−a0 F2 (ξ1 ))

− dξ2 dξ1 ϕ(ξ )dξ

z 20 z 20 z 10
R(t)+a0 t ξ2 R(F2 (ξ1 ))
R(F2 (ξ ))
+ dξ2 · dξ1 Q(ξ )d(ξ )
(λ0 + 2G 0 )
z 20 z 20 r0
R(t)+a0 t ξ1
R(F2 (ξ ))  ∗ 
+ dξ1 σrr (r0 )− ρ0 a02 ε∗ (r0 ) dξ
(λ0 + 2G 0 )
z 20 z 20
ξ1
⎫
R(t)+a0 t
⎬
+ dξ1 R(F2 (ξ )) · ε∗ (F2 (ξ ))dξ
⎭
z 20 z 20
 r
1 1
− Q(ξ )d(ξ )
(λ0 + 2G 0 ) 3
R(t)
r R(t)
2 2
+ 3 ξ Q(ξ )dξ −
3
· ξ 3 Q(ξ )dξ
3r 3R 3 (t)
r0 r0
r
2  
− ξ 2 σrr∗ (ξ )− ρ0 a02 ε∗ (ξ ) dξ
r3
r0
⎫
2
R(t)
  ⎬
+ ξ 3 σrr∗ (ξ )− ρ0 a02 ε∗ (ξ ) dξ
R 3 (t) ⎭
r0
1  ∗   
− σrr (r )− ρ0 a02 ε∗ (r ) − σrr∗ (R(t))− ρ0 a02 ε∗ (R(t))
(λ0 + 2G 0 )
Propagation of a Spherical Wave in Elastoplastic Medium … 415

2r 3 1 1  ∗ 
+ − 3 σrr (r0 )− ρ0 a02 ε∗ (r0 ) , (19)
3(λ0 + 2G 0 ) r 3 R (t)

where

z 10 = r0 − a0 t0 , z 20 = r0 + a0 t0 ,
•
u t (r, t (r )) = − R (t (r )) · ε∗ (r ), εrr (R(t), t) = εrr
∗
(t) = ε∗ (t),
(20)
•
+ − r 2 R (0)ε∗ (0)
C35 = 0 , C24 =− 0 .
a0

Further, according to

σrr = λε + 2Gεrr ,
σφφ = σ00 = λε + 2Gεφφ ;
 
∂u u σ 2 σi 1 σi (21)
ε= +2 ; λ= − ; G=
∂r r ε 9 εi 3 εi

and
   
σrr = σrr∗ (r ) + λ0 ε − ε∗ (r ) + 2G εrr − εrr ∗
(r ) ;
∗ (22)
σφφ = σφφ (r ) + λ0 ()ε − ε∗ (r )) + 2G 0 (εφφ − εφφ
∗
(r )) ,

taking with considering (20), the volumetric deformation ε (r, t) and stress compo-
nents σrr (r, t). σφφ (r, t) of the soil in the disturbance region are determined
(Fig. 3).

Fig. 3 Graph of the change in the shock wave in the soil r = R (t)
416 S. Khudainazarov and B. Donayev

Substituting the expression for σrr (r, t), obtained in the course of solving the
problem, into the boundary condition

(23)

we obtain a formula for determining the load profile σ0 (t).

3 Results and Discussion

Specific calculations on a computer were carried out for the following initial data:

kN · s2
E 1 = 1.4 × 103 MPa, E 2 = 0.0 × 103 MPa,ρ = 2.0 ,
m4

In this case, the shock front is given in the form

R(t) = r0 + R1 t − R2 t 2 /2, R(t)  0 (24)

The calculation results are presented in Figs. 4, 5 and 6 for and

in Fig. 7 for . Moreover, in Fig. 4, curves 1, 2, 3 refer to sections
, curves 4—to the front of a spherical shock wave r = R(t) ,
and curves 5—to surface r = f (t) where the radial stress σrr vanishes.
Figure 5 shows that surface r = f (t), where σrr = 0, turns out to be elongated
towards the front of the shock wave, and the load σ0 (t) is concave to the axis ot. The
speed of the shock wave R, depending on the coordinate r , linearly decays (Fig. 6).
The calculation results show that the load profiles σ0 (t) = −σrr (r0 , t) obtained
using the inverse method for cases (3) (Fig. 7 dashed lines) and (3) (Fig. 7 solid
lines) are significantly different and have a decaying character depending on the
time t. At In this case, the σ0 (t) curve for (3) is located higher in absolute value than
the σ0 (t) curve for (1). In the latter case, the law of decay of the curve σ0 (t) turns
out to be steeper with the smallest time interval of action on the boundary of the
•
spherical cavity than for (3). In contrast to case (3) at (1) the mass velocity u of the
soil gradually increases depending on the time T (Fig. 7). The radial stress σrr at the
Propagation of a Spherical Wave in Elastoplastic Medium … 417

Fig. 4 Curves of changes in

•
stress σrr , speed u,
volumetric deformation ε
and displacement u(r, t)
depending on time t at
(curves 1, 2, 3), where curves
4 and 5 correspond to the
front of the wave r = R(t)
and surface r = f (t), where
σrr = 0

Fig. 5 Change in the load

profile σ0 (t) on the cavity
with radius r0 = 0, 1 m and
surface shape r = f (t),
where σrr ≡ 0, depending on
time t

Fig. 6 Change in the speed

of the front of a spherical
•
wave R depending on the
distance r
418 S. Khudainazarov and B. Donayev

Fig. 7 Change in stress

σrr , mass velocity
•
u , volumetric deformation ε
and displacement u(r, t)
on a spherical cavity of radius
depending on time t
for cases σi = σi (ε, εi )
(solid) and σi = σi (εi )
(dotted lines)

cavity boundary decreases with time faster than at the front of the spherical wave
r = R(t) , (Fig. 4 curves 1–4).
Thus, we note that in the case of using the complicated equation of state of the
soil (1.1), a spherical shock wave r = R(t) propagates in the soil, behind the front of
which, in the disturbance region, the medium is unloaded. Similarly, you can conduct
research for the case when σ = σ (ε, εi ).

4 Conclusions

1. The problem of the propagation of a spherical wave in the soil is solved analyt-
ically in the opposite way on the basis of the deformation theory, taking into
account the generalized equations of state of the medium.
2. The results are obtained by the inverse method of propagation of spherical
shock waves in soil with complex equations of state. Concrete calculations on a
computer show that the curves to the front of the spherical shock wave r = R(t),
and the curves to the surface r = f (t), where the radial stress σrr vanishes.
3. It was found that in the case of using the complicated equation of state (1.1),
a spherical shock wave r = R(t) Propagates in the soil, behind the front of
which, in the perturbed region, the medium is unloaded.
Propagation of a Spherical Wave in Elastoplastic Medium … 419

4. The calculation results show that the load profiles σ0 (t) obtained using the
inverse method for cases (1.1) (Fig. 1.7 dashed lines) and (1.2) (Fig. 1.7 solid
lines) are significantly different and have a decaying character depending on
from the time of t.
5. In the process of solving the problem, it is assumed that when using (1.4)
and (1.1), a spherical shock wave propagates in the soil, at the front of which
the medium is instantly loaded in a nonlinear manner, followed by a linear
irreversible unloading with Young’s moduli E 1 and E 2 .

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Identification Vibration Characteristics
of Structures by Operational Modal
Analysis (OMA) Technique

Trung Duc Tran, Anh Tuan Le, and Dinh Huong Vu

Abstract The paper presents how to identify natural frequencies and mode shapes of
structures by Operational Modal Analysis (OMA) technique, in which the Frequency
Domain Decomposition (FDD) method is used. This method is an experimental
method only base on the data of measuring the dynamic response of the structures
under the excitation due to ambient forces and operational loads to determine the
vibration characteristics. Measure vibration (acceleration) and determine spectral
density matrix, using the singular values decomposition method of spectral density
matrix to determine the natural frequencies and mode shapes of structures. The calcu-
lation results show that the natural frequencies, the mode shapes form determined by
the OMA technique is consistent with the calculation results according to the theory
and show the reliability of the method.

Keywords Natural frequency · Mode shape · Identify · EMA · OMA · FDD

1 Introduction

The use of experimental tests to obtain information about the dynamic response of
buildings is an important content in the inspection of the structure and monitoring
of the building’s health. The activity of the building structure is expressed as a
combination of modes, each of which is characterized by a set of parameters (natural
frequency, damping ratio, mode shape) and depends characteristics of geometry,
materials and boundary conditions [3, 4, 7].
Experimental Modal analysis (EMA) determines these parameters from measure-
ments of applied force and structural response [7]. EMA have been applied in various
fields, such as automotive engineering, aerospace engineering, industrial machinery
and construction engineering. The determination of dynamic parameters by EMA
technology becomes more difficult in the case of building structures because of their
large size and low frequency range.

T. D. Tran (B) · A. T. Le · D. H. Vu
Le Quy Don Technical University, 236, Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
e-mail: trungductran@lqdtu.edu.vn

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 421
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_39
422 T. D. Tran et al.

Applying measurable and controllable stimuli is often a difficult work that requires
expensive and complex equipment. For this reason, researchers have recently focused
more on the advantages provided by Operational Model Analysis (OMA) techniques
[3, 4]. The OMA allows the testing of estimating structural dynamics parameters only
from vibration response measurements. The idea behind OMA is to take advantage of
the natural excitation that is available from surrounding forces (wind, vehicle, shock,
etc.) to replace artificial stimulation. Since the OMA only requires the measurements
of the structure’s dynamic response under operating conditions, when subjected to
ambient stimulation, it is also called different names, such as identifying surrounded
vibration pattern or analyze only the output model (Output-only). OMA techniques
include methods such as frequency domain decomposition method (FDD) [4, 5],
stochastic subspace identification (SSI) method [8].
The paper presents the theoretical basis to determine the dynamic parameter of
the structure according to the theory, vibration measurement test of structure and
determine the natural frequencies, the mode shapes of the steel beam structure by
OMA technique uses frequency domain decomposition (FDD) method.

2 Methods

2.1 Analytical Method to Determine the Dynamic Parameters

of Cantilever Beams

Consider a beam structure have any distribution mass m(x), with distributed load q
(x,t) [1] (Fig. 1).
Differential equation for free vibration without considering the effect of resistance
is written in the form.
 
d2 d2 X
E J (x) 2 = ω2 m(x)X (1)
dx2 dx

In which, E is the elastic modulus of the beam material, J (x) is the moment of
inertia of the beam cross-section, X is the bending form of beam structure (mode
shape) only depends on x, ω is natural frequency, m(x) is the mass per unit length,
x is the distance from the fixed end.

Fig. 1 Analytical diagram q (x,t)

x
m(x), EJ(x)
y
Identification Vibration Characteristics of Structures … 423

If beams have constant stiffness and mass evenly distributed, we have:

d4 X m
4
− ω2 X =0 (2)
dx EJ
With the above equation and the boundary conditions corresponding to the
cantilever beam, we can write the formula to calculate the specific vibration frequency
as follows:

EJ
ωi = αi2
(3)
ml 4

In which, E is the elastic modulus of the beam material, J is the moment

of inertia of the beam cross-section, m is the mass per unit length, l is the
length of the cantilever beam. αi is the coefficient, get the values αi =
1, 875; 4, 694; 7, 885 ; ...; π(2i + 1)/2.
Corresponding to the natural frequency ωi , we have the ith mode shape.

2.2 Frequency Domain Decomposition (FDD) Method

Frequency domain decomposition is proposed by Brincker et al. [5]. This method

decomposes the spectral density matrix at each frequency into singularity values and
singularity vectors by the singular value decomposition (SVD). Frequency domain
decomposition is an extension of the basic frequency domain technique or commonly
known as the Pick Peaking technique, in which natural frequencies is identified by
finding peaks in the spectral density matrix.
The relationship between unknown input x(t) and measured response output y(t)
can be expressed as follows:

[G yy (ω)] = [H (ω)]∗ [G x x (ω)][H (ω)]T (4)

where:
[G x x (ω)] is the Power Spectral Density (PSD) matrix of the input;
[G yy (ω)] is the PSD matrix of the responses;
[H (ω)]∗ is the complex conjugate matrix of Frequency Response Function (FRF);
[H (ω)]T is the transpose matrix of FRF.
The FRF can be written in partial fraction


N
[Rk ] [Rk ]∗
[H (ω)] = + (5)
1
jω − λk jω − λ∗k

λk = −σk + jωdk (6)

424 T. D. Tran et al.

where:n is the number of modes, λk is the pole of the kth mode shape, σk is minus
the real part of the pole and ωdk is the damped natural frequencies of the k th mode
shape.
[Rk ] is the residue expressed as follows.

[Rk ] = φk .γkT (7)

where: φk is the mode shape vector, γk the modal participation vector.

Suppose the input is white noise, its power spectral density is constant or.
[G x x (ω)] = C, (C is constant). Formula (4) is rewritten as follows:

 N 
N    
[Rk ] [Rk ]∗ [Rk ] [Rk ]∗ T
[G yy (ω)] = + .C. + (8)
1 1
jω − λk jω − λ∗k jω − λk jω − λ∗k

Multiplying the two partial fraction factors and making use of the Heaviside
partial fraction theorem, after some mathematical manipulations, the output PSD
can be reduced to a pole/residue form as follows:


N
[Ak ] [A∗k ] [Bk ] [Bk∗ ]
[G yy (ω)] = + ∗ + + (9)
1
jω − λk jω − λk − jω − λk − jω − λ∗k

where: [Ak ] is the kth residue matrix of the output PSD.

At a certain frequency ω only a limited number of modes will contribute signifi-
cantly, typically one or two modes. Thus, in the case of a lightly damped structure,
the response spectral density can always be written:

 dk φk φkT d ∗ φ ∗ φ ∗T
[G yy (ω)] = + k k k∗ (10)
k∈Sub(ω)
jω − λk jω − λk

where: k ∈ Sub(ω) is the set of modes be denoted at a specific frequency, φk is the

mode shape vector and λk is the pole of the kth mode shape.
The Frequency domain decomposition technique is based on the singular value
decomposition of the Hermitian response spectral density matrix.

[G yy (ω)] = [U][S][U] H (11)

where: [S] is a diagonal matrix holding the scalar singular values, [U] is a unitary
matrix holding the singular vectors and [U ] H is a Hermitian matrix.
From vibration measurement data of the structure (acceleration), we calculate
the spectral density matrix [G yy (ω)] and decompose the singular value according to
formula (11) to determine the natural frequencies of the structure.
Identification Vibration Characteristics of Structures … 425

Table 1 The physical

No. Parameter Value Unit
parameters of the test
structure 1 Length 710 mm
2 Density weight 7850 kg/m3
3 Modulus of elasticity 2.03 × 105 Mpa
4 Width 60 mm
5 Height 8 mm

3 Test on Real Structures

3.1 Test Objectives

The test to obtain dynamic responses (acceleration) of steel beam structures at

nodes over time. The result of vibration measurement is used to identify the natural
frequencies, mode shapes of the structure.

3.2 Test Model

Test structure is a steel beam. The physical parameters of the structure are shown in
Table 1.

3.3 Test Equipment

The equipment used in the test is listed in Table 2.

Table 2 The physical parameters of the test structure

No. Equipment Code Company Measuring range Quantity
name
1 Vibration NI National Instrument Multi -channel 01
measurement cDAQ-9137
equipment
2 Accelerometer PCB 352C68 PCB Group ± 50 g (100 mV/g) 01
3 Accelerometer PCB 353B33 PCB Group ± 50 g (100 mV/g) 01
426 T. D. Tran et al.

3.4 Test Layout

The test layout for determining the natural frequencies of the steel beam is arranged as
shown in Fig. 2. In which, using two accelerometer sensors to measure the vibration
of the beam, the position of the sensors is shown in Fig. 3, the NI cDAQ-9137
Connected with accelerometer sensors and display. Accelerometer measurements
are collected and displayed through the NI Signal Express software pre-installed.
Proceed with the installation and install parameters for measuring equipment,
Create vibration for the structure by any stimulus is large enough for the structure
to work in the elastic stage. The measured data are recorded as the value of the
acceleration overtime at the location where the acceleration is mounted.

Fig. 2 Experiment setup of

the real structure

Fig. 3 The position of the 710

sensors 50 290 340 30 BOLTS
30 30
60

PCB352C68 PCB353B33

- Note: Unit of measure is millimeter

Identification Vibration Characteristics of Structures … 427

4 Results

4.1 Vibration Results of the Structure

After measuring the vibration of the structure, acceleration at the nodes on the steel
girder structure is obtained over time. The data of one measurement is shown in
Figs. 4 and 5.

Fig. 4 Results of acceleration at the middle of the beam

Fig. 5 Results of acceleration at the free position of the beam

428 T. D. Tran et al.

Fig. 6 Power spectral density (PSD)

Table 3 Comparison of natural frequencies between methods

No. Mode FDD (Hz) EMA (Hz) Error (%) Theory (Hz) Error (%)
1 1 12.75 12.8 0.4 12.9 1.2
2 2 81.0 79.8 1.5 80.9 0.1
3 3 227.3 228.6 0.6 226.6 0.3
4 4 439.5 446.1 1.5 444 1.01
5 5 733.5 735.6 0.3 734 0.07

4.2 The Identification Results of Natural Frequencies

With the acceleration data obtained from the experiment, calculate and estimate
the power spectral density according to Welch’s estimation method and resolve the
singularity values by SVD algorithm according to formula (8). We determine the
natural frequencies of the structure corresponding to the positions of the maximum
power spectral density function. Results of identifying the five natural frequencies
are shown in Fig. 6.
Comparing the natural frequencies obtained by the FDD method and the results of
the calculation of the natural frequencies by the experimental modal analysis (EMA)
method [2] and according to theory [1] are shown in the Table 3.

4.3 Identify Mode Shapes

Most OMA methods provide their results in the form of complex eigenvalues and
complex eigenvectors. Since the estimates of specific vibrational-form are in the form
Identification Vibration Characteristics of Structures … 429

of complex vectors, a distinction is needed between the real modes, characterized by

the real oscillator vector real values and the complex modes. From SVD singularity
resolution, we can determine the complex eigenvectors corresponding to the corre-
sponding frequencies, at the specific vibration frequency values there are specific
vibrations of the structure, real part of the vector particularly is the amplitude of the
structure vibration at the locations put the accelerometer head.
To accurately determine the specific vibration pattern of the structure, it is neces-
sary to use many vibration probes located at different positions. Because only two
accelerometers are used, it is necessary to carry out many measurements, the fixed
sensor is used as a reference and move the other sensor at different positions. Through
the measurements, determine the amplitude of the vibration at the positions and
standardize and determine mode shapes of the structure.
Take three measurements, the position of the accelerometer in the measurements
is shown as the Fig. 7.
From the measured data, calculated according to FDD, we get the value of
amplitude of variation corresponding to the types of vibration in Table 4.
Carry out the combination of amplitudes of the same vibration form separately
and draw on the proportions, we get the mode shapes as follows (Fig. 8).
From the results of identifying the natural frequency and mode shapes form by
OMA technique, it shows that the natural frequency is very close to the results
calculated by the forced excitation method and analytical method, the mode shapes
as calculated according to theory. Thus, shows the consistency between theory and
experiment and confirms the reliability of the method.

710
BOLTS 30 340 320 20

PCB353B33 PCB352C68

(a) Measurement on real structures (b) Diagram of 1st measurement 01

710 710
BOLTS 30 480 180 20 BOLTS 30 170 490 20

PCB353B33 PCB352C68 PCB353B33 PCB352C68

(c) Diagram of 2nd measurement (d) Diagram of 3rd measurement

Fig. 7 Experimental diagram to determine the specific vibration pattern

430 T. D. Tran et al.

Table 4 The amplitude value

Measured Mode Natural Range of vibration Amplitude normalization
times frequency Sensor Sensor Sensor position Sensor
(Hz) position 01 position 02 01 position 02
1 1 12.75 − 0.9327 − 0.3606 1 0.387
2 81 − 0.7887 0.6132 1 − 0.777
3 227.3 − 0.9935 0.1141 1 − 0.115
2 1 12.75 − 0.8204 − 0.5659 1 0.69
2 81 − 0.9918 0.1253 1 − 0.126
3 227.3 − 0.8143 0.5802 1 − 0.713
3 1 12.75 − 0.9908 − 0.1352 1 0.136
2 81 − 0.8361 0.5466 1 − 0.654
3 227.3 − 0.7323 − 0.681 1 0.93

Fig. 8 Mode shapes of the

beam a 1st mode shape,
b 2nd mode shape, c 3rd (a)
mode shape

(b)

(c)

5 Conclusion

The paper presents the content of the Operational model analysis (OMA) method,
conducting tests on real structures, and identifies the natural frequencies and mode
shapes of the steel beam structure.
The results of identifying are consistent with the natural frequency obtained by
the forced excitation method and theoretically calculated, with small errors and
mode shapes consistent with the calculation theory. This shows the reliability of
the experimental and the identification method.
Operational model analysis technique can be developed for the identification of
the damping ratio of structures, and for application in monitoring, diagnosing the
health of structures and applications in the optimization of shock absorbers. Passive
fluctuations, reduce construction damage when it is affected by earthquakes.
Identification Vibration Characteristics of Structures … 431

References

1. Ba PD, Trung NT (2010) Dynamics of structures. Construction Publishing House, Ha Noi, Viet
Nam
2. Tuan TD, Tuan LA, Huong VD (2017) Identify natural frequencies of structures by the forcing
vibration method. J Constr Sci Technol 1:27–31
3. Rainieri C, Fabbrocino G (2014) Operational modal analysis of civil engineering structures.
Springer, New York
4. Brincker R, Ventura C (2015) Introduction to operational modal analysis, 1st edn. Wiley, New
York
5. Brincker R, Zhang L, Andersen P (2001) Modal identification of output-only systems using
frequency domain decomposition. Smart Mater Struct 10:441–445
6. Zhang L, Brincker R, Andersen P (2005) An overview of operational modal analysis: Major
developments and issues. In: Proceedings of the 1st international operational modal analysis
conference (IOMAC), April 26–27, Copen-hagen
7. Ewins DJ (2000) Modal testing, theory practice and application, 2nd edn. Research Studies Press
Ltd, Hertfordshire, England
8. Van Overschee P, De Moor B (1996) Subspace identification for linear systems, theory, imple-
mentation, application. Kluwer Academic Publishers, P.O. Box 17, 3300 Dordrecht, The
Netherlands.
Pile Rows for Protection from Surface
Waves

Aleksandr Dudchenko, Daniel Dias, and Sergey Kuznetsov

Abstract Numerical studies of surface Rayleigh wave interaction with piles using
Finite Element Method are presented in this article to show the attenuation effect
of such wave barrier along with the possibility to implement pile rows as a method
of vibration protection of buildings and underground structures from surface waves
of Rayleigh type. Spatial FE models are used to analyze the influence of pile field
parameters such as pile length, pile diameter number of rows and pile spacing on
vibration reduction effect of the field, with respect to the wavelength that depends on
frequency characteristics of vibration loading and soil conditions. Apart from that,
it is shown how additional pile rows can decrease internal forces in the piles inside
the protected zone which can be important for deep foundations.

Keywords Numerical simulation · Ground vibration · Vibration isolation · Pile

wave barrier · Rayleigh wave scattering

1 Introduction

The study of piles as a vibration barrier started from the work of Richart and Woods
[1], where the performance of this type of protection is investigated experimentally.
In addition to that, the authors suggested initial design guidelines for pile barriers.
Later, Woods [2] confirmed screening effect of cylindrical hole barriers on Rayleigh
waves using holography. One of the first theoretical studies is performed by Javier
Aviles and Sanchez-Sesma [3, 4] who theoretically analysed interaction of pile rows
with body waves [3] as well as Rayleigh waves [4] by using planar and spatial

A. Dudchenko (B)
LTD. PIK-Project, Barrikadnaya, D. 19, Str. 1, et/pom/kom 6/II/6 6/II/6, Moscow 123242, Russia
D. Dias
Antea Group, ANTONY PARC 1, 2/6 Place du General de Gaulle, 92160 Antony, France
S. Kuznetsov
Institute for Problems in Mechanics Russian Academy of Sciences, 101 Prosp. Vernadskogo,
Moscow 119526, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 433
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_40
434 A. Dudchenko et al.

models. The authors suggested values of pile length, spacing and width of the barrier
for effective vibration isolation.
In [5], Kattis et al. adopt Boundary Element Method (BEM) in the frequency
domain to analyze vibration isolation by pile rows. Further development of this
approach in [6] allowed to model a pile row as an infilled effective trench using
the homogenization method which is applied in the mechanics of fibre-reinforced
composite materials. In that work, interaction of a pile row with Rayleigh waves is
considered accounting for one of the most important factors, which is the volume
fraction of piles. It is worth noting, that this simulation method slightly overestimates
the reduction effect of a pile row comparatively to the modelling of independent piles.
Additionally, the authors show that trench barriers have a better reduction effect than
pile rows and the type of a pile cross-section has virtually no effect on the vibration
reduction. Afterwards, this solution technique is extended for spatial simulation of
row’s interaction with Rayleigh waves in frequency domain in [6]. In addition to that,
BEM is utilized in the work of Tsai [7] in order to study active vibration protection
for different types of piles, pile lengths and spacing.
Another interesting approach based on the periodicity theory and Finite Element
Method (FEM) is implemented by Jiankun Huang [8] for the analysis of horizontal
vibration attenuation by pile rows. Then, this method is further developed for plane
waves in [9] and pile barriers with initial stress in [10]. In these works, the authors
propose the concept of dispersion curves and analyse the attenuation zones for pile
fields. The waves with the frequencies within the attenuation zone cannot propagate
through the periodic pile barriers. It is shown that the reduction ratio of pile rows
relates to relative Young’s modulus, the relative density of piles (ratio of pile material
density to that of the soil) and pile fraction [8]. Meanwhile, initial stress affects [10]
the width as well as the lower and upper bounds of the attenuation zone, having
practically no effect on the final reduction effect.
Vibration attenuation properties of pile rows in porous media are analysed in
the works [11] and [12] of Yuan-Qiang Cai et al. for the cases of surface Rayleigh
and body waves respectively using Fourier–Bessel series. In that research, such key
factors as pile spacing, relative pile Young’s modulus and density are underlined.
Moreover, it is shown that vibration isolation from Rayleigh waves in porous media
is less effective than that in non-porous elastic media which is not in the agreement
with the study carried out by Lu [13] presenting better effectiveness of pile barrier
for the case of two phase media.
Multiple body wave scattering by several pile rows is analysed in [14] with the
method proposed by the authors. It is shown that an increase in the number of rows
improves vibration reduction properties of a pile barrier. At the same time, such
method is found to have better screening effect for the case of low-frequency body
waves.
Most of the previous researches deal with the parameters of pile field inde-
pendently, regardless their complex effect on vibration attenuation properties. The
parameters of pile fields ensuring maximum reduction effect are difficult to imple-
ment in practice accounting for current technology and construction codes. The
Pile Rows for Protection from Surface Waves 435

present work is targeted to the full-scale spatial analysis of Rayleigh wave inter-
action with piles using FE method implemented in Abaqus software. The energy
distribution in the protected zone along with the estimation of the extension to which
the parameters of a pile field affect the resulting vibration reduction are analysed.

2 Problem Formulation

Interaction of surface Rayleigh waves with pile fields are considered within the
framework of linear elastic constitutive law. This approach can be appropriate for
artificial vibrations except blasts as well as for earthquakes and blasts in the areas
that are remote from their epicentres. In that case, small strain constitutive equation
for homogeneous isotropic media can be written in the following form:

1
c2p ∇div u(x, t) − cs2 rot rot u(x, t) + b(x, t) = R
u(x, t), (1)
ρ

where u(x, t) and b(x, t) are the functions of displacements and body forces
respectively c p , cs are the compressive and shear wave speeds accordingly:
 
λ + 2μ μ
cP = , cS = , (2)
ρ ρ

where λ, μ are Lame’s parameters.

The following initial conditions are considered:

u(x, t)|t=0 = 0, ∂t u(x, t)|t=0 = 0 (3)

In that case, the initial stress distribution in the half-space is neglected as it has
virtually no effect on the displacements, velocities and accelerations of the points in
the observation zone.
For isotropic media on the free surface of the half-space (Fig. 1), the boundary
condition of zero stress is used:

tξ = σ · ξ = 0, x ∈ ξ , (4)

where I is the unit diagonal matrix, ξ is the unit outward normal to the free surface
ξ and tξ is a surface stress vector. In the case of elastic media Eq. (4) takes the
following form:

tξ ≡ (λtr (ε)I + 2με) · ξ = 0, x ∈ ξ , (5)

436 A. Dudchenko et al.

Fig. 1 Boundary conditions and unit normals to the free surface of the half-space and contact
interface between the pile and soil respectively

where ε—small strain tensor.

In the case of a pile barrier installed in the medium (Fig. 1), the condition of
perfect mechanical contact is applied to the contact surfaces between the piles and
the soil:

t pile x·η∈ η = tsoil |x·η∈ η

u pile x·η∈ η = usoil |x·η∈ η
(6)


where t pile x·η∈ η , tsoil |x·η∈ η are the stresses on the contact surface from a pile and
soil sides respectively; η is the unit normal to the interface between the pile and soil

η ; u pile x·η∈ η , usoil |x·η∈ η are the displacement vectors on the contact surface on
the pile and soil sides respectively; the indexes pile and soil correspond to the pile
and soil accordingly.

3 Simulation Methods and FE Models

3.1 Finite Element Model Configurations

Mathematical formulation including constitutive equations as well as boundary and

initial conditions for the considered problem is defined by Eqs. (1–6). The analysis is
Pile Rows for Protection from Surface Waves 437

Fig. 2 The scheme of the simplified spatial model of pile field

performed in time domain for surface Rayleigh waves, generated by fully harmonic
surface line loading with vibration amplitude A and frequency ω:

f (x, t) = A eiωt δ(x)ν, (6)

Time integration is performed using explicit finite-difference procedure which is

based on the second order explicit central difference integration scheme involving
Lax-Wendroff method [15, 16]. Time increment size is selected automatically by
the program satisfying Courant–Friedrichs–Lewy (CFL) condition [15, 16]. Spatial
discretization is performed using finite element method (FEM) in Abaqus 2016 soft-
ware [16]. The standard finite element library of Abaqus/Explicit software package
is used in the calculations [16]. The region is meshed with finite elements of the
C3D8R type which are eight-node hexahedral elements with a linear shape function
with control of deformations. Maximum length ratio is in the range [0.95,1].
In the subsequent analysis two types of 3D models shown in Figs. 2 and 3 are
used. The first model represents a piece of a pile field with several rows and three
planes of symmetry that are used to decrease the model size (Fig. 2). The first plane of
symmetry passes through the wave source perpendicular to the direction of Rayleigh
wave propagation and parallel to the pile row. It is assumed that there are several
piles in a row located along the same straight line at the same distance from each
other(the length of the row can be compared with the dimensions of the wave front
or larger than that, so the effect of the row length can be neglected).This allows to
438 A. Dudchenko et al.

Fig. 3 The scheme of spatial model for real full scale pile field

introduce two additional planes of symmetry passing through the pile axis and middle
of the interval between the piles parallel to the direction of propagation of surface
waves thus, substantially reducing the number of elements. On the free surface at the
top of the symmetry plane fully harmonic line loading defined by Eq. (6) is applied.
Meanwhile, the remaining part of the top surface is free. On the bottom and right
planes of the model non-reflecting boundaries for P waves are used. This model is
used to analyse the influence of pile diameter, spacing and length on the reduction
ratio of a pile field as well as to determine the optimal values of these parameters for
the following analysis involving the model of a more realistic pile field (Fig. 3).
At the second stage, a full scale spatial model of the regular pile field is adopted
to simulate a real finite size pile field which may surround a construction or be the
foundation of a structure (Fig. 3). For this model the main parameters are set based
on the results obtained from the analysis using the first model (Fig. 2). Basically, the
full scale 3d model is used to confirm the main results and trends identified in the first
calculation stage. Similarly, to the first model the second one is a three-dimensional
with the condition of symmetry applied on the left surface. Top surface of the model
is free while non-reflecting boundaries are applied on the other surfaces (Fig. 3).
The dimensions of the models are chosen in a way that the waves reflected from
the boundaries of the models should not return to the observation zone with the length
equalling to 2l, where l is the wavelength of Rayleigh wave. The mesh size is less
than 0.1l. Additionally, pile rows are created at a distance from the symmetry plane
(left plane) so that the interaction of the waves and piles would occur remote enough
from the source with account of symmetry condition.
The models presented in Figs. 2 and 3 allow to analyse the influence of pile field
planar configuration (quadratic or triangular cells (Fig. 4) on vibration reduction
effect along with the interaction with surface waves. Vertical and horizontal sizes
of the first model (Figs. 2) equals to 9l and 18l respectively (l is the wavelength of
Rayleigh wave), while the width of the model varied according to the pile distance.
Pile Rows for Protection from Surface Waves 439

Fig. 4 The types of pile field configuration: square cells (on the left) and triangular cells (on the
right)

The sizes of the second model (Fig. 3) along the X,Y and Z axis equals to 9l, 6l and
5l. The size of the protected zone  is l, while the observation zone size equals to
2l.
Primary calculations show that there is hardly no difference in vibration reduction
effect for plane and spherical waves. Therefore, taking into account the requirements
to the model size, Rayleigh waves with a plane front are considered in the following
text. In addition to that, two main assumptions are also made: (1) the size of the
protected zone does not change; (2) the same soil conditions are used for all the
calculations.
It is worth noting that a pile field can act as a barrier if the wavelength is comparable
or less than the pile length and the dimensions of the pile field in plane. For low
frequency range corresponding to earthquakes f = 2÷10 Hz the wavelength of
Rayleigh wave varies from 100 to 10 m in soft soils, while in rigid soils it can
exceed 200 m. At the same time, pile depth which is more than 50 m is difficult
to implement in practice. Therefore, the lowest frequency f = 2 Hz is chosen as
it generates Rayleigh waves with large enough wavelength corresponding to real
vibration sources both natural and anthropogenic nature. While, construction of a
pile field providing reasonable vibration reduction effect is not possible even in soft
soils for lower frequencies as it will require larger pile lengths. At the same time,
higher frequencies correspond to shorter wavelengths and require smaller protective
pile barriers. The results are shown in relation to the maximum Rayleigh wavelength
l equalling to 50 m and corresponding to minimum vibration frequency f = 2 Hz.
Young’s modulus and density for soft soils are chosen according to the seismic
shear wave velocities that are given in Eurocode 8 [16]. The attenuation effect of the
field is analysed using the value of the kinetic energy reduction ratio:

K pile
kr ed,E = , (7)
K hom
440 A. Dudchenko et al.

Table 1 Dynamic
Material Density (kg/m3 ) Poisson’s ratio Young’s
parameters of materials
modulus (MPa)
Soil 1800 0.25 55
Concrete 2450 0.23 30,000

where K pile is the kinetic energy field of the surface layer in the protected zone and
K hom is the kinetic energy for the same layer in the model without pile field. The
observation layer is placed behind the pile rows at a surface level.
According to the results obtained by Kattis et al. in [6], it is possible to replace a
pile row with an effective trench, thus basic qualitative results obtained in the works
[1, 18–21] etc. regarding the influence of the depth, width and mechanical material
parameters can be extrapolated to pile rows. Which means, the higher the difference
in the mechanical parameters of the piles and the soil the better vibration reduction
effect can be observed. However, the range of materials for a pile field is quite narrow.
Therefore, further analysis is limited by piles made of reinforcement concrete, which
are more widely used. Mechanical parameters of concrete and a possible soft soil
are shown in Table 1 in agreement with [16].
This work concerns interaction of Rayleigh waves with piles and pile fields outside
of the source vicinity. This is primarily due to the fact that the behaviour of waves
in the source zone has difficult to predict complex nature which is strongly affected
by geological conditions along with the source itself. Additionally, it is possible to
distinguish the major waves that carry the energy of vibration source. As a result,
the distance between the pile row and the source gives virtually no effect on the final
reduction effect in the protected zone.
Hereinafter, if the variable dimension is not explicitly specified, it is presented
in the dimensionless form. The main dimensionless complex is given in the section
below by default, geometrical variables are given in relation to the Rayleigh wave
wavelength.

3.2 Dimensional Analysis

Kinetic energy and displacement fields of the area beyond the pile field can be
described by the following group of dimensionless parameters:
 
E pile ρ pile D H S
K pile = f ; ; ; ; ; ν pile ; νsoil , (8)
E soil ρsoil l l l
 
E pile ρ pile D H S
u pile =g ; ; ; ; ; ν pile ; νsoil (9)
E soil ρsoil l l l
Pile Rows for Protection from Surface Waves 441

where the index soil indicates the soil material of the half-space, while the index pile
corresponds to the parameters of the pile field; l is the wavelength of the Rayleigh
wave in a half-space (this wavelength can be solved from the Bergmann-Viktorov’s
equation); E pile , E soil correspond respectively to Young’s modulus of the piles and
soil respectively; ν pile , νsoil are corresponding Poisson’s ratios; ρ pile , ρsoil are the
corresponding densities; D, H, S are the diameter, length and spacing of the pile field
accordingly. A pile field interacts with seismic waves as a uniform composite barrier,
thus it is convenient to introduce the value of pile fraction—α = π4SD2 showing the
2

density of the pile field. Afterwards, all the geometric values are normalized in
relation to the wavelength of Rayleigh’s wave.

4 The Computed Results

As a starting point, the influence of pile diameter (d̃ = Dl ), pile fraction (α = π4SD2 )
2

and distance between piles (s̃ = Sl ) is considered. In order to estimate the influence
of these parameters, the reduction ratios are calculated at the surface level in the
protected zone . Figure 5 represents the reduction ratios for the surface layer.

Fig. 5 Reduction ratio for the surface layer Ẽ = 550,ρ̃ = 1.3 and h̃ = 1
442 A. Dudchenko et al.

Fig. 6 The influence of row number on the vibration reduction for low diameter piles d̃ = 0.01
(left plot) and high diameter piles d̃ = 0.06 (right plot) Ẽ = 550,ρ̃ = 1.3,h̃ = 1 and α = 0.162

E pile ρ pile
Contour plots in Fig. 5 are plotted at Ẽ = E soil
= 550,ρ̃ = ρsoil
= 1.3, ν pile =
0.2,νsoil = 0.25 and h̃ = Hl = 1.
The obtained results reveal that for a single row pile barrier, both diameter and
pile fraction play and important role as the maximum vibration decrease is observed
at the values of pile fraction and diameter equalling to α = 0.16 and d̃ = 0.1
respectively. However, as it will be shown in the following text, pile diameter is less
important if a pile barrier is composed of more than 2 rows (Fig. 6). In addition to
that, it can be seen from Fig. 5 that the reduction ratio for the surface layer declines
with an increase in the diameter at the constant alpha significantly up to the value
of normalized diameter equalling to d̃ = 0.06. Then it maintains the same level
slightly fluctuating around it. At the same time, pile fraction significantly affects the
reduction effect which is growing with an increase of alpha. If d̃ is located in the
range d̃ ∈ [0, 0.03] such one row pile barrier is not effective even if pile fraction is
high.
Plots in Fig. 6 show the influence of pile row number on the reduction effect at
different pile configurations and two pile diameters—small and large which corre-
spond to d̃ = 0.01 and d̃ = 0.06 respectively. Curves in the right and left plots in
E pile ρ pile
Fig. 6 are plotted at Ẽ = Esoil = 550,ρ̃ = ρsoil = 1.3, ν pile = 0.2,νsoil = 0.25,
h̃ = Hl = 1 and α = 0.162.
Figure 6 shows that the pile configurations (triangular and quadratic cells) have
virtually no effect on the reduction ratio. Therefore, the curve in the left plot in Fg.
6 is plotted only for the quadratic configuration. Apart from that, an increase in the
number of rows leads to a better vibration reduction effect of the pile field and even
the barriers designed of low diameter piles but having several rows can give the same
reduction effect as a single row pile barrier with high diameter piles. However, high
diameter piles give better reduction effect at the same number of rows (Fig. 6, right
plot). Therefore, it is important to estimate the optimal configuration of pile field
in terms of material volume, designed vibration reduction level and technology for
each practical case.
Pile Rows for Protection from Surface Waves 443

Fig. 7 The change of screening effect with increase in the pile length for low diameter piles
d̃ = 0.01 (left plot) and high diameter piles d̃ = 0.06 (right plot) Ẽ = 550,ρ̃ = 1.3,h̃ = 1 and
α = 0.162

It is clear that the pile length should be comparable with the wavelength, otherwise
there will be virtually no diffraction and scattering of Rayleigh waves by the piles.
Hence, the field itself cannot be used as a vibration barrier. Therefore, it is important
to determine the relation between pile length and the attenuation effect. The plots
in Fig. 7 show the change in the reduction ratio with the increase in the pile length.
E pile ρ pile
The curves in this figure are plotted at Ẽ = Esoil = 550,ρ̃ = ρsoil = 1.3, ν pile =
0.2,νsoil = 0.25.
According to the graphs in Fig. 7, reduction effect increases with the pile length
significantly resulting in the reduction ratio of kr ed,E = 0.4 reaching an asymptotic
limit for low diameter piles although, continue decreasing for high diameter piles.
It means that further increase in the pile length will not change the reduction effect
noticeably in the case of low diameter piles while it can slightly increase the perfo-
mance of high diameter piles. Additionally, for pile length which is less than the
wavelength h̃ < l better reduction is observed at the surface layer while for longer
piles underground layer shows better vibration reduction in the case of high diameter
piles.

5 Conclusion

Pile field can be an effective measure to protect structures from surface Rayleigh
waves as it decreases the transmission of wave energy, that is carried out by the surface
waves into the protected region, thus, declining the amplitude of displacements,
velocities and accelerations of the points in this zone. Simplified and full scale spatial
models are used in the calculations and the results obtained using the both models are
in a good agreement. Thus, it is possible to extrapolate the results from the simplified
model to the full scale pile field that may surround a structure.
This way of protection shows good effectiveness when the maximum possible
wavelength is comparable with the planar dimensions of a protected area along with
444 A. Dudchenko et al.

the geometrical parameters of the pile field. This is the case for seismic waves in soft
soils, such as clays with low plasticity index, loose and medium sands etc. as well as
high frequency artificial vibration sources generating vibrations in stiffer soils, like
clays with high plasticity index, dense sands etc. At the same time, for both cases of
application, acoustical density of the pile barrier must be different to that of the soil.
In that case, the pile field satisfying this condition can provide up to 50% decrease
in the vibration energy transmitted to the protected zone. It is possible to improve
vibration reduction effect of a pile field increasing pile diameter, length and fraction.
However, further rise of these values may lead to inappropriate cost of the structure
along with the additional complexity in the construction technology.
The main parameters that affect vibration reduction are the pile fraction, length,
diameter as well as the number of pile rows. It is shown that pile length should be
more than half of the wavelength to ensure at least 20% reduction in the kinetic
energy, meanwhile the influence of the pile fraction and diameter is strongly affected
by the number of rows. It means that for a single row pile barrier, the diameter of
piles plays an important role up to the value of diameter equalling to 0.06 l. Then
it has virtually no effect on the reduction ratio of the surface layer, while for the
underground layer it affects the vibration decrease up to the diameter of 0.08 l (here
l is the design wavelength).
In the case of multi row pile barriers, the effect of pile diameter still exists, but
becomes less important because the reduction ratio of low diameter piles installed
in several rows can be the same as that of high diameter piles but designed as one
row barrier. Therefore, there are no strict limitations on pile diameters. However, the
volume of the material for the pile field will be equal for a single and multi-rows
pile fields if the same vibration reduction is provided. Therefore, it is possible to use
lower diameters for the piles which is a better solution from technological point of
view.
An additional important result from the use of such barrier is a decrease in bending
moments in the inner piles, that can be used as a deep foundation. It is shown that
the possible reduction effect in bending moments of the inner piles can reach 80%.
A pile field is a less effective measure than seismic barriers in terms of vibra-
tion reduction. Although, they can protect constructions from body waves which,
however, is beyond the scope of this research.
As a perspective of this work, the calculations involving current models of elasto-
plastic media, that are relevant for soils, will be performed to estimate the effect of
pile—soil interaction more precisely.

Acknowledgements The authors (AVD and SVK) thank the Russian Science Foundation Grant
20-49-08002 for financial support.

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21. Qiu B (2014) Numerical study on vibration isolation by wave barrier and protection of existing
tunnel under explosions (Doctoral dissertation, INSA de Lyon)
Longitudinal-Transverse Bending
of Physically Nonlinear Rods
by Quasi-Static Loads and Mass Forces

Yury Nemirovskii and Sergey Tikhonov

Abstract The problems of longitudinal-transverse bending of physically nonlinear

concrete multilayered rods of arbitrary cross-sectional shape by quasi-static loads
and mass forces are considered. The deformation law of each layer of the rod is
adopted as an approximation by a polynomial of the second order. It is assumed that
the rod can withstand the applied loads if deformation in all layers of the rod does
not exceed the corresponding limit values at tension or compression. The considered
examples illustrate the influence of gravity, the angle of inclination of gravity to the
axis of the rod, and the parameters of the cross-section shape on the ability of rods
to withstand the applied loads. The ranges of parameters at which there is no loss of
bearing capacity of the rod are determined by numerical calculations.

Keywords Longitudinal-transverse bend · Rod · Limit deformation · Concrete ·

Concrete deformation diagram · Tension · Compression · BUBNOV-Galerkin
method · Systems of nonlinear algebraic equations · Systems of nonlinear
differential equations

1 Introduction

At present, technologies for the production of reinforced concrete products, for

example, surface (layer-by-layer) vibro-formation of concretes by slipformers, allow
to create the multilayered construction products (columns, cross-bars, beams, hollow
flooring, industrial wall panels) with various cross-section shapes [1]. Rods are some
of the most common structures in construction industry. There are a lot of Russian
and foreign scientific studies devoted to the calculation of beams under the quasi-
static loads at longitudinal-transverse bending, but the calculations are limited to

Y. Nemirovskii
Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch, Physics of Fast
Processes Laboratory, Russian Academy of Sciences, 630090 Institutskaya str., 4/1, Novosibirsk,
Russia
S. Tikhonov (B)
Faculty of Information and Computer Systems, I. Ulianov Chuvash State University, 428015
Moskovskiy pr., 14, Cheboksary, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 447
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_41
448 Y. Nemirovskii and S. Tikhonov

the cases of a rectangular cross-section and the cases of multilayered rods are not
considered [2–5].
The cases of complex and longitudinal-transverse bending of multilayered rods
of an arbitrary cross-section are considered in the mathematical package Maple
by the Bubnov-Galerkin method in [6, 7]. The beam functions are taken as basic
functions of the Bubnov-Galerkin method. In the case of longitudinal-transverse
bending, the appropriate beam functions are difficult to select in order to satisfy the
necessary conditions of fastening. The polynomials of degree N, which coefficients
are determined by the fastening conditions, are taken in this work as basic functions
instead of beam functions.

2 Methodology

Consider the multilayered concrete rods of constant cross-section. The law defor-
mation for each layer of the rod is represented as an approximation by a polynomial
[8–12] of the second order (1), where the coefficients A1i , A2i are coefficients which
can be found from real diagrams of concrete tension and compression.
− +
σi = A1i ε + A2i ε2 , −ε∗i ≤ ε ≤ ε∗i (1)

− +
where ε∗i , ε∗i are modules of ultimate deformations of the materials in corresponding
layers of the rod relatively under compression and tension.
The examples of calculating the coefficients A1i , A2i for the concrete grades B30,
B50, B70 on reference values [13], using approximating dependences [14], are given
in Table 1.
The maximum deformation of the limit value in the ith layer under tension or
compression is taken as a criterion for the conditional limit state.
We take the Cartesian coordinate system with the x axis directed along the rod
axis with a certain reference to the cross section and with the origin at one of the
ends of the rod.
The resulting stress fields σ j will generate the internal force factors

Table 1 Results of calculation of the approximation coefficients

Concrete grade ε∗− (%) ε∗+ (%) A1 (MPa) A2 (MPa)
B30 0.1046 0.0035 32,500 15,533,088
B50 0.1447 0.0042 38,000 13,127,273
B70 0.1805 0.0046 41,000 11,358,108
Longitudinal-Transverse Bending of Physically Nonlinear Rods … 449

n 
 n 

N= σi d S, M y = σi zd S, (2)
i=1 S i=1 S
i i

which must satisfy the equilibrium equations

d 2 My dm y d N
2
= qz − , = −qx . (3)
dx dx dx
If we accept the Kirchhoff-Lyav hypotheses as valid, then we have

ε(x, y, z) = ε0 (x) − zκ y (x), (4)

du 0 d 2 w0
ε0 (x) = , kappa y = , (5)
dx dx2

where u 0 , w0 are components of the displacement vector of center line points of the
rod.
Assuming that there is a case of the rod pinched at both ends, we have

u 0 (0) = w0 (0) = u 0 (l) = w0 (l) = 0,

   
dw0  dw0  du 0  du 0  (6)
= = = = 0.
d x x=0 d x x=l d x x=0 d x x=l

From Eqs. (1)–(5), we obtain a system of differential equations with respect to

displacements with boundary conditions (6).
This system of differential equations is solved by the Bubnov-Galerkin method
[15].
Assume that


N 
N
u 0 (x) = Byk (x), w0 (x) = Crk (x), (7)
k=1 k=1

where Bk , Ck are constants.

If we take for ϕi (x) the expressions


N
ϕi (x) = Di,k x k−1 , (8)
k=1

where Di,k are determined by the boundary conditions.

Assume yk (x), rk (x) in the expression (8) to be equal
450 Y. Nemirovskii and S. Tikhonov

yk (x) = rk (x) = ϕk (x). (9)

If we substitute the displacements (7) into the obtained differential equations

and denote the left sides of corresponding equations as L 1 (x), L 2 (x), taking as the
basic functions f k (x) in the Bubnov-Galerkin method the same functions as in the
expansion (8), then we obtain a system of 2 N algebraic equations in relation to 2 N
unknowns Bk , Ck , k = 1 . . . N

l
L i (x) f k (x)d x = 0, i = 1 . . . 3, k = 1 . . . N (10)
0

The values Bk , Ck can be determined from the obtained system of equations by

any numerical method.

3 Results

Assume that the functions bounding the ith layer in cross-section have the form bi (z)
and h i is a height of the corresponding layer.
Let the rod be affected by the distributed loads q1 , q2 , q3 , q4 and the gravity force
which projection on the corresponding axes is equal to qsx , qsz .
Then for the stresses and moments we have.
b1 (0) bn (0) 3 h i 
bi (z)

qz = 2 q2 dy+2 q1 dy+2 dz qsz dy,
0 0 i=1 h 0
i−1

b1 (0) bn (0) 3 h i 

bi (z)

qx = 2 q3 dy+2 q4 dy+2 dz qsx dy,
0 0 i=1 h 0
i−1

b
n (h n ) b1 (0) b
n (h n ) 3 h i 
bi (z)

m y = −2 xq1 dy − 2 xq2 dy+2h n q3 dy − 2 dx qsx dy,
0 0 0 i=1 h 0
i−1

The acting loads q1 , q2 , q3 , q4 , qsx , qsz are assumed to be equal

q1 = t11 + t12 x, q2 = t21 + t22 x, q3 = t31 + t32 x, q4 = t41 + t42 x, (11)

3 h i
 3 h i

qsx = gx ρ bi (z)dz, qsz = gz ρ bi (z)dz,
i=1 h i=1 h
i−1 i−1
Longitudinal-Transverse Bending of Physically Nonlinear Rods … 451

where t11 , t12 , t21 , t22 , t31 , t32 , t41 , t42 are constants, gx , gz are projections of the
gravity acceleration vector.
As the values of load parameters, we take

N N
t11 = −32200 k , t12 = 5060 k 2 , t21 = 0, t22 = 0,
m m
N N
t31 = 184000 k , t32 = 18400 k 2 , t41 = 0, t42 = 0 (12)
m m
Consider a rod of T-beam cross-section as an example (Fig. 1). The concrete
grades of corresponding layer are indicated opposite each cross-section.
At the absence of gravity and the value k = 1 for loads (12), we obtain a solution
corresponding to Figs. 2 and 3.

Fig. 1 A cross-section of
the rod if y1 = 0.3 m

Fig. 2 Distribution of deformation of the materials at the upper contact boundaries of the layers
along the rod length
452 Y. Nemirovskii and S. Tikhonov

Fig. 3 Distribution of deformation of the materials at the upper contact boundaries of the layers
along the rod length

Figures 2 and 3 show that the rod is able to withstand the applied loads and even
has some additional bearing capacity.
Assume that the rod is affected by the gravity acceleration which projection on
the corresponding axes is determined by the ratios

gx = g cos α, gz = g sin α, (13)

where g = −9.81 sec m

2 , the angle α determines the direction of gravity action, the

density ρ of all concrete grades is considered equal to 1800 mkg3 .

Then, if the loads are similar and gravity acts vertically downwards (α = 0), the
solution takes the form given in Figs. 4 and 5.
For this case, we reduce the loads by three times, taking k = 13 in the ratio (12),
then we get a solution that has the form given in Figs. 6 and 7.
Figures 6 and 7 show that deformation at the lower contact boundaries of the rod
layers is significantly lower than the corresponding limit deformations at compression
under the specified loads. Therefore, we will analyze the distribution of deformation
of the materials at the upper contact boundaries of the layers in further calculations
(Fig. 6). According to Fig. 6, the deformation is closest to the corresponding limit
value in the first rod layer, but does not exceed it.
+
Further, we will consider the number γ = εmi /ε∗i for each layer, equal to the ratio
of maximum deformation εmi along the rod length at the upper contact boundaries
+
of the layers to the limit deformation ε∗i of the corresponding layer material.
To estimate the influence of the inclination angle α of the gravity vector to the Oz
+
axis, we calculate the values γ = εmi /ε∗i for each value α from 0 (gravity is directed
π
vertically downward) to 2 (gravity acts along the rod axis). In external loads (12),
we assume that the coefficient k = 1.
Longitudinal-Transverse Bending of Physically Nonlinear Rods … 453

Fig. 4 Distribution of deformation of the materials at the upper contact boundaries of the layers
along the rod length

Fig. 5 Distribution of deformation of the materials at the upper contact boundaries of the layers
along the rod length

According to Fig. 8, deformation exceeds the corresponding limit values only in

the first layer of the rod at 0 ≤ α ≤ 1, but when 1 < α ≤ π2 , deformation does not
exceed the corresponding limit values in all layers of the rod.
To estimate the effect of cross-sectional parameters on the bearing capacity we
consider the case of a cross-section, given in Fig. 1, from the value with variable values
of the cross-section parameter y1 . Moreover, the height of the corresponding layer
h 1 is considered when the product is y1 h 1 = const, i.e., the material consumption
and the mass of corresponding layer are equal. In loads (12), we take the coefficient
k = 1/3 and assume that gravity acts vertically downwards.
454 Y. Nemirovskii and S. Tikhonov

Fig. 6 Distribution of deformation of the materials at the upper contact boundaries of the layers
along the rod length

Fig.7 Distribution of deformation of the materials at the lower contact boundaries of the layers
along the rod length

Figures 8 and 9 show that if the values are y1 < 0.3 m, then the indicated structure
is able to withstand the applied loads. Furthermore, at values y1 > 0.3 m deformation
begins to exceed the limit values in the first layer of structure.
Consider a rod of I-beam cross-section (Fig. 10) of the same area as in the previous
cases.
In this case, we change the values of the cross-section parameter y1 so that the area
of the second layer in cross-section remains the same, that is y1 (h 2 − h 1 ) = const.
All external loads are assumed to be similar to the previous considered case.
Longitudinal-Transverse Bending of Physically Nonlinear Rods … 455

Fig. 8 Dependence of the ratio γ on the inclination angle α

Fig. 9 Dependence of the ratio γ on the parameter y1 in the case of T-beam cross-section

Fig. 10 A rod of I-beam

cross-section
456 Y. Nemirovskii and S. Tikhonov

Fig. 11 Dependence of the ratio γ on the parameter y1 in the case of I-beam cross-section

The solution obtained in Fig. 11 shows that deformation stops exceeding the limit
values in all layers of the rod in the case of y1 ≤ 0.5 m.

4 Conclusions

Thus, the considered examples let us conclude that gravity essentially affects the
bearing capacity, and its underestimation can lead to significant errors in the calcula-
tions. The bearing capacity is also influenced by the parameters of a cross-sectional
shape, which changed values can significantly increase the ability of a structure to
withstand the applied loads.

Acknowledgements This work is carried out with the partial financial support of RFBR grant
(project №19-01-00038).

References

1. Mut AR (2009) Concrete and reinforced concrete. Equip Mater Technol 20–23
2. Pashanin AA (2011) Beton i Zhelezobeton 6:15–18, http://rifsm.ru/en/editions/journals/
3. Almazov VO, Zabegaev AV, Popov NN, Rastorguev SV (1994) News of higher educational
institutions. Construction 11:10–15
4. Gemlering AV (1974) Calculation of rod systems. Stroitelstvo, Moskow
5. Mishenko AV, Nemirovskii YV (2013) Beton i Zhelezobeton 4:5–12, http://rifsm.ru/en/edi
tions/journals/
6. Nemirovskii YuV, Tikhonov SV (2020) PNRPU Mech Bull 1:60–73
7. Nemirovskii YuV, Tikhonov SV (2020) Model Struct Mech 12:11–49
Longitudinal-Transverse Bending of Physically Nonlinear Rods … 457

8. Mishenko AV (2014) Struct Mech Eng Construct Build 4:42–51. http://journals.rudn.ru/struct

ural-mechanics/article/view/10902
9. Mishenko AV (2014) Bulletin of SUSU. Ser Constr Eng Archit 14:12–16
10. Mishenko AV (2005) Direct and nverse problems of deformation of layered rods taking into
account the physical nonlinearity. In: XIX All-Russian Conference Numerical Methods for
Solving Problems of the Theory of Elasticity and Plasticity, EPPS, 28–31 August 2005, Biysk,
Russia
11. Nemirovskii YV, Tikhonov SV (2020) Vestnik of the Yakovlev Chuvash State Pedagogical
University. Ser Mech Limit State 3:247–253
12. Nemirovskii YuV, Tikhonov SV (2020) Mech Solids 55(6):767–775
13. SP 63.13330.2012 (2012) Concrete and reinforced concrete structures. Basic provisions.
Updated edition of SNiP 52-01-2003
14. Nemirovskii YU (2018) Vestnik of the Yakovlev Chuvash State Pedagogical University. Ser
Mech Limit State 3:26–37
15. Goloskokov DP (2017) Space, time and fundamental interactions 1:7–85, http://www.stfi.ru/
en/main.html
The Stress–Strain State of Reinforced
Concrete Arches with a View of Concrete
Viscoelasticity

Serdar Yazyev, Vladimir Andreev, and Leysan Akhtyamova

Abstract The resolving equations for determining the stress–strain state of a rein-
forced concrete element undergoing the action of a bending moment and a longitu-
dinal force, taking into account the creep of concrete on the basis of a viscoelastic
model, are obtained. These equations allow for a known value of internal forces to
determine the stress–strain state in arbitrary sections of statically definable arches.
Internal forces in the arches are calculated analytically, and a step-by-step calculation
is used to determine the stresses. Also, the development of the finite element method
for the case of viscoelasticity of concrete for a reinforced concrete element has been
carried out. Comparison of the results obtained by means of numerical-analytical
calculation and FEM is performed. The calculation by the finite difference method
was carried out with the subsequent comparison of the results with the FEM.

Keywords Stress–strain · Reinforced concrete · Concrete viscoelasticity

1 Introduction

Since the arches are small curvature bars, they can be calculated using the formulas
for eccentrically compressed reinforced concrete bars. We consider a reinforced
concrete element subjected to a bending moment and axial force.
The cross-section, as well as the design scheme are shown in Fig. 1. Tensile
stresses are assumed to be positive.
According to the hypothesis of flat sections, the total deformation of concrete
is the sum of the axial deformation ε0 and the deformation due to the change in
curvature:

εb = ε0 − yχ , (1)

S. Yazyev · L. Akhtyamova
Gagarin Sq, Don State Technical University, 344010 Rostov-on-Don, Russia
V. Andreev (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 459
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_42
460 S. Yazyev et al.

Fig. 1 To the calculation of a reinforced concrete element

where χ —is bar curvature change.

From the condition of the reinforcement and concrete work compatibility, we
write down the expressions for reinforcement deformations:
 
ε S = ε0 + yS χ ε S = ε0 − yS χ . (2)

Distances yS and yS are substituted into the formula (2) by the absolute value.
According to the viscoelastic body model, the total concrete deformation is the
sum of elastic deformation εbel and creep strain εb∗ [1]:

σb
εb = + εb∗ . (3)
Eb

From (3), the stresses in concrete are written in the form:

 
σb = E b (εb − εb∗ ) = E b ε0 − yχ − εb∗ . (4)

Reinforcement stresses are determined as follows:

 
 

σ S = E S ε S = E S (ε0 + yS χ ) , σ S = E S ε S = E S ε0 − yS χ . (5)

Let us write the equation of the sum of moments about the axis z:

  
−M + σ S A S yS − σ S A S yS − σb y d A = 0. (6)
A

Having compiled the sum of the projections of all forces on the longitudinal axis
of the bar, we obtain:
The Stress–Strain State of Reinforced Concrete Arches … 461

 
N = σS A S + σS A S + σb d A. (7)
A

Substituting (4) and (5) to (6), for the symmetrical reinforcement case.
 
(A S = A S , yS = yS ) we get:
⎛ ⎞

1 ⎝
χ= M − E b εb∗ y d A⎠, (8)
E Ir ed
A

where EIr ed = E S I S + E bIb —is the reduced bending stiffness of a cross section;
   2 3
I S = E S A S yS2 + A S yS ; Ib = bh 12
.
Admeasurement ε0 is found from the Eqs. (4), (5), (7):
⎛ ⎞

1 ⎝
ε0 = N + E b εb∗ d A⎠, (9)
E Ar ed
A

  
where E Ar ed = E S A S + A S + E b Ab —reduced stiffness of a cross-section under
axial tension (compression). The Eqs. (4), (5), (8), (9) can be used to calculate
the creep of statically definable arches. At the first stage, a static calculation is
performed—the internal force factors M and N are determined. In statically definable
systems with constant external loads, they do not depend on time. The cross-section
in height is divided into m parts y, and the time interval is n steps t. For the given
cross-sections at each point, stresses in concrete are calculated without taking creep
into account.
If the creep law is given in differential form, then the calculated stresses can
∂ε∗
be used to determine the growth rates of creep deformations ∂tb , as well as creep
deformation at time t + t using linear approximation [2–5]:

∂εb∗
εb∗ (t + t) = εb∗ (t) + t. (10)
∂t
Time intervals τi may not be equal to each other. If the section of the arch is
rectangular, then the integrals entering into (8) and (9) are also calculated numerically
using the trapezoidal method:
h
 2 ∗ ∗ m−1
εb0 y0 + εbm ym
εb∗ y d A = b εb∗ (y)ydy = by + ∗
εbi yi . (11)
2 i=1
A − h2
462 S. Yazyev et al.

2 Methods

Calculation of arches by the finite element method. Derivation of resolving equations.

It is assumed that the behavior of a continuous curved beam is sufficiently
accurately characterized by the behavior of a broken bar composed of small recti-
linear elements. From physical considerations it follows that with a decrease in the
elements’ size, the solution should converge and, as experience shows, convergence
is observed [6].
At the same time, special attention should be paid to the method of specifying
the nodal loads: the distributed load is more correctly represented in the form of
statically equivalent concentrated nodal forces [7].
The calculations will use the bar finite element shown in Fig. 2. Each node of
this element has 3 degrees of freedom: 2 linear displacements u and v, as well as
a rotation angle ϕ. The vector of nodal displacements will be written as: {U } =
 T
u i u j vi ϕi v j ϕ j .
The deflection of a finite element will be approximated as follows:
  T
v(x) = α0 + α1 x + α2 x 2 + α3 x 3 = 1 x x 2 x 3 α0 α1 α2 α3
  . (12)
= 1 x x 2 x 3 {α} .

The vector {α} can be found from the following condition:

 
dv  dv 
v(0) = vi ; ϕ(0) = −  = ϕi ; v(l) = v j ; ϕ(l) = −  = ϕ j .
d x x=0 d x x=l

In matrix form, these conditions take the form:

Fig. 2 Bar finite element

The Stress–Strain State of Reinforced Concrete Arches … 463
⎧ ⎫ ⎡ ⎤
⎪
⎪ vi ⎪ 1 0 0 0
⎨ ⎪ ⎬ ⎢
ϕi 0 −1 0 0 ⎥
=⎢
⎣1
⎥ · {α} = [C] · {α}. (13)
⎪
⎪ vj ⎪
⎪ l l 2
l3 ⎦
⎩ ⎭
ϕj 0 −1 −2l −3l 2

Let us express the vector {α} from (13) by nodal movements:

 T
{α} = [C]−1 · vi ϕi v j ϕ j
⎡ ⎤
00 1 0 0 0
⎢0 0 0 −1 0 0 ⎥
=⎢ ⎣0 0
⎥
1 ⎦{U } = [F]{U }. (14)
− l32 2l 3
l2 l
00 2
l3
− l12 − l23 − l12

Then the deflection function will be written as:

 
v(x) = 1 x x 2 x 3 [F]{U }. (15)

And the second derivative of the deflection takes the form:

d 2v  
2
= χ = 0 0 2 6x [F]{U }. (16)
dx
For axial displacements u, we take linear dependence on x:
 x x
u = 1− ui + u j . (17)
l l
Then the axial deformation ε0 will be defined as follows:

du  
ε0 = = − 1l 1
l
0 0 0 0 {U }. (18)
dx
The expressions for the stiffness matrix and the load vector will be obtained based
on the Lagrange variational principle. The total energy E is the sum of the deformation
potential energy and the external forces work:

E = P + A. (19)

The potential energy of deformation is the sum of the concrete and reinforcement
potential energy:

P = Pb + PS + PS . (20)

The potential energy of concrete is determined by the following expression:

464 S. Yazyev et al.


1
Pb = σb εbel d V, (21)
2
Vb

where εbel denotes elastic deformation of concrete, which is the difference between
total and creep deformation:

εbel = εb − εb∗ . (22)

We will assume that the creep strain is independent of x within the element.
Substituting (1) in (22) and then (4) and (22) in (21), we get:
 2  2 2
1 d 2v 1 d v
Pb = E b ∫ ε0 − y 2 − εb∗ d V = E b Ab ∫ ε02 d x + Ib ∫ 2
dx
2 Vb dx 2 (l) (l) d x

d 2v
+ ∫ (εb∗ )2 d V − 2 ∫ ε0 d x ∫ εb∗ d A + 2 ∫ 2
d x ∫ εb∗ yd A, (23)
Vb (l) A (l) d x A

3
where Ib = bh
12
is the moment of concrete inertia; Ab = bh is the concrete section
area.
The potential deformation energy of the reinforcement located at the bottom face
can be found as follows:
   2 2
1 1 d 2v 2 d v
PS = σS εS d V = E S AS (ε02 + 2ε0 yS + y S ) d x. (24)
2 2 dx2 dx2
VS (l)

For top edge reinforcement similarly:

   2 2
 1   1  d 2v
 
2 d v
PS = σS εS d V = E S A S (ε02 − 2ε0 yS 2 + (yS ) ) d x. (25)
2 2 dx dx2
 (l)
VS

 
In the case of symmetrical reinforcement (A S = A S , yS = yS ) for the potential
energy of all reinforcement deformation, we obtain:
⎛ ⎞
   2
 1 ⎜ d v
2
⎟
PS + PS = E S ⎝ A S, gen ε02 d x + I S d x ⎠, (26)
2 dx2
(l) (l)

   2
where I S = A S yS2 + A S yS is reinforcement inertia moment.
The Stress–Strain State of Reinforced Concrete Arches … 465

Substituting (16) and (18) in (24) and (26), we obtain the following expression
for the potential energy of a reinforced concrete element:

1   1
P= {U }[K ]{U } − {U } Fb∗ + E b (εb∗ )2 d V, (27)
2 2
V

# $
[K c ]  
where [K ] = —stiffness matrix, which has a block structure; Fb∗ —
[K i ]
contribution of concrete creep deformations to the vector of nodal loads;
# $
E Ar ed 1 −1
[K c ] = ;
l −1 1
⎡ 12 ⎤
l3
− l62 − 12l3
− l62
⎢ − 62 4 6 2 ⎥
[K i ] = E Ir ed ⎢ l l
⎣ − 123 62 123
l2 l ⎥
6 ⎦;
l l l l2
− l62 2l 6
l2
4
l
⎛ ⎧ ⎫ ⎧ ⎫⎞
⎪
⎪ −1 ⎪ ⎪ ⎪
⎪ 0 ⎪⎪
⎜ ⎪
⎪ ⎪ ⎪ ⎪⎟
⎜ ⎪
⎪
⎪ 1 ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪ 0 ⎪⎪
⎪
⎪⎟
 ∗ ⎜ ⎨ ⎬  ⎨ ⎬⎟
⎜ ∗ 0 ∗ 0 ⎟
Fb = E b ⎜ εb d A + εb yd A ⎟. (28)
⎜ ⎪
⎪ 0 ⎪ ⎪ ⎪
⎪ −1 ⎪
⎪⎟
⎜A ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪⎟
⎝ ⎪
⎪ 0 ⎪
⎪
A ⎪
⎪ 0 ⎪
⎪⎠
⎪
⎩ ⎪
⎭ ⎪
⎩ ⎪
⎭
0 1
 
The work of external forces is the product of the external nodal loads Fq vector
 
on the vector of nodal displacements: A = {U }T Fq .
From the condition of the total energy functional minimum, we obtain:


∂E ∂P ∂A ∂ 1  
= + = {U }[K ]{U } − {U }T Fq
∂{U } ∂{U } ∂{U } ∂{U } 2

1 ∗ 2 ∂   
+ E b ∫(εb ) d V − {U }T Fb∗ = 0.
2 V ∂{U }

Finally, the problem is reduced to a system of linear algebraic equations of the

form:
   
[K ]{U } = Fq + Fb∗ . (29)

The calculation was performed for a three-pivot circular arch loaded with a
uniformly distributed load q. The design scheme is shown in Fig. 3.
466 S. Yazyev et al.

Fig. 3 Design scheme of the

arch

The equation for the arch axis outlined along the circular arc is:
%  2
L f L2
y= R2 − −x − R+ f; R = + ; (30)
2 2 8f
L − 2x y+R− f
sinϕ = ; cos ϕ = .
2R R
Internal forces in the section K of the arch are calculated by the formulas:
 
k = Mkδ − H yk ; Nk = − Q δk sin ϕk + H cos ϕk , (31)

where Mkδ , Q δk denote the moment and shear force in the section K in a beam with a
similar span and load. In case of uniformly distributed load: Mkδ = q2x (L − x); Q δk =
2
q
2 (L
− 2x); H = q8Lf .
The problem was solved with the following initial data: q = 50 kmN , arch span
L = 16 m, elevation f = 3.2 m, cross-sectional dimensions: b = 20 cm, h = 40 cm,
τ0 = 28 days, E b (τ0 ) = 3×104 MP, E S = 2 · 105 MP, reinforcement ratio
A 
μ = S,Abgen = 0.02, yS = yS = 15 cm. Concrete aging was taken into account, i.e.,
the increase in its elasticity modulus over time. The time
& dependence of the concrete
'
elasticity modulus was taken as: E b (t) = E b (τ0 ) · b1 + (1 − b1 )e−b2 (t−τ0 ) , b1 =
1.282, b2 = 0.019.
The graph of the change in the concrete elasticity modulus is shown in Fig. 4.
The equation of the viscoelastic model of concrete hereditary aging was used in
the calculation and has the form:

t
σb (t) ∂C(t, τ )
εb (t) = − σb (τ ) dτ. (32)
E b (t) ∂τ
τ0

The Eq. (32) can also be represented in the form (3), introducing the following
notation:
The Stress–Strain State of Reinforced Concrete Arches … 467

Fig. 4 The graph of the

change in the concrete
elasticity modulus

T, days

t
∂C(t, τ )
εb∗ =− σb (τ ) dτ. (33)
∂τ
τ0

It is recommended to take the creep measure in [28] in the form:

eαt − eατ  
C(t, τ ) = C αt
+ B e−γ τ − e−γ t . (34)
e −1

To determine the creep deformations described by the expression (33), it is

possible to use the formula (11); however, if the creep measure is a sum of
exponentials, it is more convenient to represent the creep law in differential form.
Substituting (34) into (33), we get:

t ∂ eαt − eατ  
εb∗ = − ∫ σb (τ ) (C αt + B e−γ τ − e−γ t ) dτ
τ0 ∂τ e −1
Cα t t
= ∫ σb (τ )eατ dτ + Bγ ∫ σb (τ )e−γ τ dτ. (35)
eαt − 1 τ0 τ0

We represent the concrete creep deformation as the sum of two components:

εb∗ = εb1
∗ ∗
+ εb2 ;

t t
∗ Cα ατ ∗
εb1 = αt σb (τ )e dτ ; εb2 = Bγ σb (τ )e−γ τ dτ. (36)
e −1
τ0 τ0

∗ ∗
The component εb1 characterizes hereditary creep properties, and εb2 char-
acterizes the influence of the growing environment on its deformative properties
[28].
468 S. Yazyev et al.

Let us find the time derivative of each component:

∗    
∂εb1 ∂ Cα t
ατ Cα ∂ t ατ
= ∫ σb (τ )e dτ + αt ∫ σb (τ )e dτ
∂t ∂t eαt − 1 τ0 e − 1 ∂t τ0
Cα 2 eαt t ατ Cα αt αeαt  ∗

=− αt
∫ σb (τ )e dτ + αt
σb (t)e = αt
Cσb (t) − εb1 ;
(e − 1) τ02 e −1 e −1
(37)
∗  
∂εb2 ∂ t
= Bγ ∫ σb (τ )e−γ τ dτ = Bγ σb (t)e−γ t .. (38)
∂t ∂t τ0

Using the expressions (37) and (38) together with (10), it is possible to determine
∗ ∗
the components of creep deformation εb1 and εb2 at every moment of time.
The values of the rheological constants in the calculation were taken equal to:
α = 0.032, γ = 0.062, C = 3.77 · 10−5 MPa−1 , B = 5.68 · 10−5 MPa−1 .

3 Results and Discussion

For the calculations using the FEM, a software package was developed in the Matlab
complex. To check the correctness of the program operation, a test problem was
solved for a statically definable arch.
Figure 5 shows a graph of the change in stress in the reinforcement depending on
x and t.

The upper mesh surface corresponds to the stress σ S in the reinforcement at the
upper edge. Bottom shaded surface corresponds to the stresses σ S in the reinforcement
at the bottom edge.

Fig. 5 Reinforcement stress

change
MPa

x, cm
The Stress–Strain State of Reinforced Concrete Arches … 469

Fig. 6 Change in stresses in

concrete at y = h / 2 and y =
–h / 2

MPa

x, cm

Figure 6 shows a graph of the change in stresses in concrete depending on x and

t.
The upper surface corresponds to the stresses at y = h/2, the lower corresponds
to the stresses at y = −h/2.
Figure 5 shows that, due to concrete creep, the stresses in the reinforcement in
absolute value increase. In concrete, stresses, on the contrary, decrease, as evidenced
by Fig. 6. The most significant redistribution occurs at the points where the bending
moments are maximum (x ≈ 2.1 m and x ≈ 13.9 m). Figure 7 shows the distribution
of stresses in concrete in the section x = 2.1 m at the beginning of the creep process
(dashed line) and at the end of the creep process (solid line). Figure 8 shows the
change in stresses σ S in the reinforcement of the lower edge at x = 2.1 m.

Fig. 7 Stress distribution in

concrete in the section x =
2.1 m at the beginning and at
the end of the creep process
MPa

y, cm
470 S. Yazyev et al.

Fig. 8 Change in stresses in

the reinforcement of the
lower edge with time in the
section x = 2.1 m

MPa

t, days

Due to the creep of concrete, the compressive stresses in the reinforcement of

the lower face in the section with the maximum bending moment increased from
91.7 MPa to 173.1 MPa, i.e., 1.9 times.
In concrete, due to creep, the change in stresses is not so significant: for a more
compressed face at x = 2.1 m, the stresses decreased from 16.2 MPa to 13.6 MPa,
i.e., by only 20%. The distribution of stresses in concrete along the section height,
both at the beginning and at the end of the creep process, is linear.
Figure 9 shows the distribution of creep deformations εb∗ depending on x and y at
t = 100 days. The greatest inelastic deformations are observed in the sections with
maximum bending moments.
Table 1 shows a comparison of stresses in concrete and reinforcement at the bottom
edge at x = 2.1 m at different points in time, obtained numerically-analytically, as
well as numerically using the FEM.
The table shows that the results practically coincide, which indicates the reliability
of the technique developed.

x, cm
y, cm

Fig. 9 Distribution of creep deformations depending on x and y at t = 100 days

The Stress–Strain State of Reinforced Concrete Arches … 471

Table 1 Comparison of the results of numerical-analytical calculation with FEM

t, days 30 40 50 60 70 80 100
σb , M P Numerical-analytical − − − − − − −
calculation 15.84 14.70 14.16 13.87 13.72 13.64 13.56
FEM − − − − − − −
15.81 14.67 14.17 13.87 13.71 13.62 13.54
σS , M P Numerical-analytical − − − − − − −
calculation 102.8 137.7 154.6 162.7 167.8 170.4 173.2
FEM − − − − − − −
102.6 137.4 154.3 163.1 168.9 170.3 172.8

Summarizing the above-said, it can be noted that resolving equations have been
obtained to determine the stress–strain state of a reinforced concrete element expe-
riencing a bending moment and longitudinal force, taking into account the creep of
concrete on the basis of a viscoelastic model.
These equations allow for a known value of internal forces to determine the stress–
strain state in arbitrary sections of statically definable arches. Internal forces in the
arches are calculated analytically, and a step-by-step calculation is used to determine
the stresses.

References

1. Tamrazyan AG, Yesayan SG (2012) Mechanics of concrete creep: monograph. MGSU, Moscow
2. Chepurnenko AS, Andreev VI, Yazyev BM (2013) Energy method in the calculation of the
stability of compressed bars taking into account creep MGSU Herald 1:101–108
3. Kozelskaya MY, Chepurnenko AS, Litvinov SV (2013) Application of the Galerkin method
for calculating the stability of compressed bars taking creep into account [Electronic resource]
Engineering Bulletin of the Don, v. 2 (2013). Information on http://ivdon.ru/magazine/archive/
4. Andreev VI, Yazyev BM, Chepurnenko AS (2014) On the bending of a thin plate at nonlinear
creep. Adv Mater Res 900:707–710
5. Andreev VI, Chepurnenko AS, Yazyev BM (2014) Energy method in the calculation stability
of compressed polymer bars considering creep. Adv Mater Res 1004–1005:257–260
6. Zenkevitch O (1975) The finite element method in technology. Mir, Moscow
7. Zenkevitch O, Chang I (1974) The finite element method in the theory of structures and in
mechanics of continuous media. Nedra, Moscow
Differential Equations with Fractional
Derivatives for Studying an Oscillator
with Viscoelastic Damping

Alexander Andreev, Temirkhan Aleroev, Mohammad Khasambiev,

and Hedi Aleroeva

Abstract The paper considers differential equations with fractional derivatives used
for describing the functioning of an oscillator with viscoelastic damping as well as
diffusion processes. On the basis of a mathematical model with fractional derivatives,
the present research deals with the qualitative parameter of the model to confirm
the consistency of the given process with experimental data. An analysis of recent
publications on this topic has been carried out and a method of obtaining a qualitative
assessment of the obtained mathematical model is given. The presented approach was
tested in a numerical experiment using the developed software.

Keywords Differential equations · Mathematical model · Fractional derivatives

1 Introduction

At the moment, there are several definitions of a fractional order derivative. This fact
complicates the choice of numerical methods for solving problems accompanying
equations with fractional derivatives, since they directly depend on the form of the
selected derivatives, therefore, it creates necessity to compare the results that were
obtained by using different definitions and numerical methods.
Mathematical models with different base [1–10] are widely used in different fields,
as well as models based on differential equations with fractional derivatives in the
study of physical processes occurring in an inhomogeneous fractal medium and some

A. Andreev
Ulyanovsk State University, Lev Tolstoy Street, 42, Ulyanovsk 432017, Russia
T. Aleroev
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
M. Khasambiev (B)
Grozny State Oil Technical University Named After Academician M. D. Millionshchikov, Kh. A.
Isaev Avenue, 100, Grozny 364051, Russia
H. Aleroeva
Moscow Technical University of Communications and Informatics, Aviamotornaya str., 8a,
Moscow 111024, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 473
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_43
474 A. Andreev et al.

biological and chemical processes [11–20]. When solving the problems listed above
in differential equations with fractional derivatives, fractional derivatives are used
[11], defined in Caputo sense:
β t
Dt f (t) = (m−b)
1
(t − τ )m−β−1 f (m) (τ )dτ , m − 1 < β < m, m ∈ N.
0

or in the Riemann–Liouville sense:

x
Dxα g(t) = (m−b)
1 dm
dxm
(x − ξ )m−α−1 g(ξ )dξ , m − 1 < α < m, m ∈ N.
0
The relationship between fractional derivatives in Caputo and Riemann–Liouville
sense is determined by the following relation:

γ γ

n−1
f (k) (0)t k−γ
C
Dt f (t) = Rl
Dt f (t) − .
k=0
(k + 1 − γ )

It should be noted that the fractional differentiation operator in Caputo sense takes
a number of advantages over the fractional differentiation operator in Riemann–
Liouville sense. The main advantage of the fractional differentiation operator in
Caputo sense is that when it is used in the formulation of a problem for a fractional
differential equation, it’s possible to set the usual initial and boundary conditions. In
addition, the fractional derivative in Caputo sense, taken from the constant, is equal
to zero, while the fractional order derivative of the constant in Riemann–Liouville
sense is not equal to zero.
In fractional calculus, the Mittag–Leffler function plays a special role [21]:
∞
 zk
E α (z) = ,
k=0
(1 + kα)

which is a generalization of the series for the exponent at n ! = (n+1) on (α n+1)
and remains invariant if it is acted upon by the fractional differentiation operator:
D α E α (z α ) = E α (z α ).
In the works of most authors, mathematical models are built for the study of
processes based on fractional order differential equations, in which only derivatives
in Riemann–Liouville [12–15] sense or only in Caputo [16–20] sense are used. At
the same time, publications should be noted in which derivatives are combined in
Riemann–Liouville sense and in Caputo sense [11].
The use of equations with fractional derivatives in the modeling of these processes
and phenomena leads to the problem of determining the indicator of the adequacy of
the built mathematical model, which can be further used to obtain a forecast of the
course of this process or phenomenon.
Differential Equations with Fractional Derivatives for Studying … 475

2 Research Methods

The indicator of the adequacy of the mathematical model of an oscillator with

viscoelastic damping can be implemented in two ways:
(1) by studying the influence of a perturbation of the fractional derivative order on
the solution of Eq. (1);
(2) by introducing the indicator presented in the works [1–10] based on the analysis
of variance of experimental data obtained and based on the solution of Eq. (1).
Let’s demonstrate the first option.
The study was carried out on samples of polymer concrete mixture prepared on
the basis of polyester resin. The main strain and strength characteristics of polymer
concrete are shown in Table 1.
Methods for parametric identification of the fractional derivative order in the
so-called Bagley-Torvik model.
Let’s consider the Cauchy problem

Table 1 Strain and strength characteristics of polymer concrete based on diane and
dichloroanhydride-1,1-dichloro-2,2 di (n-carboxyphenyl) ethylene
Type of asphalt Temperature, Elasticity Strain modulus Plasticity Viscosity
concrete °C modulus (MPa) (MPa) (MPa, s)
1 0 5000 2300 0.20 2 × 107
2 0 3000 1000 0.28 1 × 106
3 0 2500 650 0.35 5 × 104
4 0 3000 1000 0.28 1 × 106
5 0 2500 650 0.35 5 × 104
6 0 2000 420 0.40 5 × 103
7 0 5000 2300 0.20 2 × 107
8 0 4000 1600 0.23 8 × 106
9 0 3000 1000 0.28 1 × 106
10 0 2000 420 0.40 5 × 103
1 20 2500 650 0.35 5 × 104
2 20 1750 330 0.43 1 × 103
3 20 1250 170 0.50 1 × 102
4 20 1750 330 0.43 1 × 103
5 20 1250 170 0.50 1 × 102
6 20 1000 130 0.52 5 × 101
7 20 2500 650 0.35 5 × 104
8 20 2000 420 0.40 5 × 103
9 20 1750 330 0.43 1 × 103
10 20 1000 130 0.52 5 × 101
476 A. Andreev et al.

d 2u
+ cD α u + λu = 0, (1)
dx2

u(x)|x=0 = 0, u  (x)x=0 = 1, (1a)

x u(τ )dτ
where D α u(x) = 1 d2
(2−α) d x 2 (x−τ )α−1
(1 < α < 2) is the fractional order
0
derivative in Riemann–Liouville sense.
Equation (1) was used by Bagley and Torvik [22, 23] to solve the problem of
modeling the damping properties of various viscoelastic materials (polymers, glasses,
etc.). The physical systems modeled by Eq. (1) are more exposed to fluctuations in
the fractional derivative order, which in turn generates a quite significant problem of
parametric identification of the value of this fractional derivative order.
The solution to the problem (1), (2) in this paper is found by using the Laplace
integral [24] and a technique is proposed for parametric identification of the fractional
derivative order on the basis of experimental data.
Let’s successively integrate Eq. (1) twice from 0 to x and transform the expression:
⎡ t ⎤
x x x  x  t
c d⎣ u(τ )dτ ⎦
u  (t)dt − dt + +λ u(τ )dτ dt = 0
(2 − α) dt (t − τ )α−1
0 0 0 0 0 0
(2)

Since the function u(x) has a summable fractional x derivative of the

t  t u(τ )dτ x
order α, taking into account (2−α) c d u(τ )dτ  = c d  –
dt (t−τ )α−1  (2−α) dt (t−τ )α−1 
0 0
 0 0
t u(τ )dτ 0 x
c d  = c
(x − t) u(t)dt,
1−α
(2−α) dt (t−τ )α−1  (2−α)
0 0 0
We obtain:

x x  t
c
u(x) − x + (x − t) 1−α
u(t)dt + λ u(τ )dτ dt = 0. (3)
(2 − α)
0 0 0

To solve the Eq. (3), we use the Laplace transform. Let’s suppose that U (s) is an
image of a function u(x), i.e.

∞
U (s) = e−st u(t)dt.
0

We assume that the solution to the Eq. (4) is in the class of functions for which
the Laplace integral converges. Function
Differential Equations with Fractional Derivatives for Studying … 477

x
(x − t)1−α u(t)dt
0

is a convolution of functions u(x) and x 1−α . Indeed, by definition, the convolution

of two functions f 1 (x) and f 2 (x) is the function

x
F(x) = f 1 (t) f 2 (x − t)dt,
0

for which there is an image formula

x
f 1 (t) f 2 (x − t)dt. =. F1 (s)F2 (s),
0

where F1 (s) and F2 (s) are the images of functions f 1 (x) and f 2 (x), respectively.
Taking into account that, the image of the power function x μ at μ > −1 equals
(μ + 1)s −1−μ , we have

x
u(t)dt U (s) · (2 − α)
α−1
= . (4)
(x − t) s 2−α
0

As
⎡ ⎤
x t x t
⎣ u(τ )dτ ⎦dt = dt u(τ )dτ = U (s) · s −2 , (5)
0 0 0 0

then, by applying the operational Laplace calculus to (3), we obtain the equation for
the image of the solution.

c λ 1
1+ + · U (s) = 2 .
s 2−α s 2 s

where we get the formula from for the image

1
U (s) = . (6)
s2 + cs α + λ
478 A. Andreev et al.

The formula (6) allows expressing the solution to problem (1)–(1a) in terms of
the Laplace integral

σ+i∞
1
u(x) = est U (s)ds. (7)
2πi
σ −i∞

3 Discussion

It’s possible to plot the graphs of the solutions numerically by using the formula (7).
In this case, numerical calculations were carried out in the Mathcad 14 environment.
Figure 1 shows the graphs of the solution u(x) of Eq. (1) for various values of the

Fig. 1 Graphs of solutions

to the boundary value
problem for the equation of
the oscillator model with
viscoelastic damping at
1<α<2
Differential Equations with Fractional Derivatives for Studying … 479

parameter (the fractional derivative order) α. The values of the remaining parameters
(modulus of elasticity and modulus of rigidity of the system) are taken with the
following values: = 1, 2, λ = 89. These values of the parameters were obtained in
the course of experiments on polymer concrete samples [24, 25]. With the help of
numerical verification, it is possible to verify the correctness of the limiting behavior
of the solution, which, when the parameter α values are sufficiently close to the value
equal to two, turns into harmonic oscillations.
To make sure that the formulation of the parametric identification problem is
correct, it is necessary to investigate the stability of the solution with regard to
inaccuracies of the parameter α. For this, in the vicinity of the point, let’s consider
the relative increment of this parameter by δ, i.e. α  = α(1 + δ). The norm deviation
function in L 1 is defined as follows,

 
ρ(α, δ) =  u(x, α) − u(x, α  )d x (8)

where u(x, α) is the solution to the fractional differential Eq. (1) with the order of
the fractional derivative α.
It should be noted that the value of the expression ε(α, δ) = ∂ρ(α,δ) ∂δ
determines
the sensitivity of the solution to Eq. (1) to a possible error concerning the value of the
order of the fractional derivative α. The values of the quantity ε(α, δ) corresponding
to different values of the order of fractional differentiation α and levels δ = 0, 05;
0, 1; 0, 15 were determined numerically and presented graphically in Fig. 2.
The obtained values ε(α, δ) allow concluding that the sensitivity increases with
the growth of α. The maximum value does not exceed 0.2, which allows to conclude
that the solutions to problem (1) are stable with regard to a small parameter error α
and the correctness of the identification problem for this parameter.
In the paper [26], the solution (1), for the case 0 < α < 1, was calculated by
using a sequence of recurrent kernels and written out in the form of a power series.

Fig. 2 To the question of the

sensitivity of solutions to
problem (1), (2) to the
parameter error (the
fractional differentiation
order) α
480 A. Andreev et al.

Fig. 3 Definition of a
function by formula (9) in
Mathcad and its comparison
with the solution obtained by
using the Laplace transform

We, for the case 1 < α < 2, in the same way, found that
∞ 
 n
Cnm cm λn−m x 2n+1−mα
u(x) = x − (−1)n+1 . (9)
n=1 m=0
(2n − mα + 2)

By comparing the graphs of solutions obtained numerically by formulas (7) and

(8), it can be concluded that they are identical (Fig. 3). To plot the graphs of the
solution (10), we will use the Mathcad 14 software package:


50 
n
combine(n, m) · cm · λn−m · t2n+1−m·α
h(t, α) := t − (−1)n+1 ·
n=1 m=0
(2n − m · α + 2)

In [27], the possibility of calculating the solution at any point was given, which
enabled to develop a simple and effective method for parametric identification of the
parameter α from experimental data, on the assumption that the remaining parameters
of the equation are known (with varying degrees of accuracy). Let’s suppose several
experimental points are known u(xi ) = Ui , i = 1, . . . , N . The unknown parameter
α can be selected by minimizing the deviation of the theoretical curves from the
experimental ones. Theoretical points can be calculated by the formula (9) u(xi , α).
Let us determine the deviation function by the method of least squares.


N
F(α) = (Ui − u(xi , α))2 (10)
i=1

This function represents the sum of deviations of theoretical points from exper-
imental ones. The value α, that minimizes this function can be approximately
considered the desired one. The identification accuracy depends on the number of
experimental points, as well as the accuracy of other system parameters.
Differential Equations with Fractional Derivatives for Studying … 481

Table 2 Experimental points for polymer concrete samples

xi (c) 0,27 0,4 0,68 1,1 1,3 1,6
Ui 0,06 – 0,038 – 0,0098 0,018 – 0,0097 – 0,01

Fig. 4 Comparison of
experimental data and
theoretical curve

For the purpose of approbation of the technique, let us take the experimental data
obtained in [24, 25]. The values for polymer concrete samples based on polyester resin
(dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) are presented in Table 2.
Figure 4 puts on display experimental points and theoretical curve.
Comparison of the experimental data with the model allows to conclude that the
modeling is adequate and the parametric identification technique is highly accurate.
Knowledge of the model parameter allows, for example, to predict the stain and
strength characteristics of a material (polymer concrete, asphalt concrete, etc.) under
loading.

4 Conclusion

Deformation and strength criteria are decisive in the calculation of non-rigid road
surfaces. Actually, significant statistical data on changes in these indicators have
been stored. Despite this, there is no reliable method to determine the change in the
elastic modulus of polymer concrete during its exploitation. In this paper, on the
basis of a large number of empirical data, a model is built to describe the change in
the strain-strength characteristics of asphalt concrete under loads using the methods
of fractional calculation. Various ways of solving the problem of parametric identi-
fication of this model are also considered. The development of the quality indicator
of the built mathematical model, when compared with experimental data obtained
empirically, allows to conclude about the adequacy of modeling and a sufficiently
high accuracy of the used parametric identification technique.
482 A. Andreev et al.

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sel’skom khozyaystve (Methodology for assessing human resources in agriculture), Ekonomika
sel’skogo khozyaystva Rossii № 7. S. 47–53
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Engineering and Smart Systems
in Construction
Macrokinetic Model of Biochemical
Oxidation

Vladislava Nikolaevna Volkova and Viktor Leontievich Golovin

Abstract In the Russian Federation, the current standards oblige to carry out not
only complete biological treatment of wastewater, but also their additional treatment.
The main task of post-treatment technological processes is the removal of biogenic
elements and some specific pollutants from water that has undergone biochemical
treatment. Thus, the improvement of post-treatment remains an urgent environmental
task, and the requirements for it can be defined as follows: ensuring the most complete
removal of biogenic and other specific pollutants, including those contained in post-
treatment wastewater in the form of colloids; ensuring the safety of discharges into
water bodies by reducing the formation of carcinogens during disinfection; ensuring
a high cleaning effect with increased instability of waste management conditions;
ensuring technological simplicity and reliability of post-treatment devices; ensuring
a reduction in the consumption of clean water for restoration work. This paper inves-
tigates a method for post-treatment of wastewater by a slow filmless filter with a
vertical filtering surface. The properties of the slow filter cassette with respect to
ammonium and phosphorus ions have been studied. The analysis of the integral
kinetic curves is carried out. The efficiency of additional purification of wastewater
from biogenic elements by a filmless slow filter for ammonium and phosphates has
been investigated.

Keywords Deep cleaning · Biochemical destruction · Slow filter · Biogenic

elements · Substrate inhibition · Stationary processes

1 Introduction

Progress is visible all over the world is noticeable in the field of environmental protec-
tion, the most important result of which is the reduction of the negative impact on its
condition, in particular, when wastewater is discharged into water bodies. The results
of a survey of treatment facilities show that after cleaning, the permissible limits of

V. N. Volkova (B) · V. L. Golovin

Federal State Autonomous Educational Institution of Higher Professional Education “Far Eastern
Federal University” (FEFU), Polytechnic Institute, Vladivostok, Russian Federation
e-mail: vladavibi@bk.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 487
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_44
488 V. N. Volkova and V. L. Golovin

effluent quality indicators, standardized for fishery reservoirs, can be significantly

exceeded (5–10 times) [1]. This is due not only to the excess of the permissible load on
water bodies, when, with unstable hydrological parameters, the intensity of dilution
of treated effluents changes significantly and, therefore, it is not possible to achieve
the required degree of their dilution. Often, a more important factor determining the
excess of the permissible load is the inconsistency of the cleaning technologies used
in practice with modern environmental requirements for the prevention of environ-
mental pollution. During traditional disinfection of insufficiently treated wastewater
with chlorine or ozone, chlorine or ozone organic substances, which are carcino-
gens, enter the water body. Such toxins cause almost more harm to the biocenosis
of natural water bodies and watercourses than the discharge of untreated wastewater
at all, and the content of residual chlorine, which is incorporated into the structure
of organic compounds and loses its active oxidizing properties, does not guarantee
absolute epidemiological safety.

2 Methods

In the Russian Federation, the standardization of the load on water bodies is used,
with the provision of control of the dispersion or dilution of pollution, which is
determined, in particular, by the value of the biological oxygen demand (BODtot) of
the treated wastewater. At the same time, this indicator can be assessed as depending
on a number of treatment conditions, for example, such as temperature, the content of
toxic substances in the treated water, the ripening and aging time of activated sludge,
changes in the species composition of microorganisms, as well as the effectiveness
of the applied post-treatment methods—tertiary treatment. The main task of the
technological processes used in post-treatment is to remove biogenic elements and
some specific pollutants from the water that has undergone biochemical treatment.
In particular, the role of destruction of microorganisms in the formation of colloidal
systems when changing the conditions of their vital activity from aerobic in aeration
tanks to anaerobic in secondary sedimentation tanks is not entirely clear. However,
like other organic matter, from which only coarse particles are removed in secondary
sedimentation tanks without destructive effects. Thus, the main burden of removing
the residual amount of nutrients, after biological treatment, which are mainly in the
form of a colloidal solution, falls on tertiary treatment.
The technologies used in the world for such purification are very diverse and
usually very complex. In countries such as the USA, Australia, and Spain, membrane
methods of wastewater treatment are actively used [2, 3]. Such technologies are
designed to ensure a more complete removal of pollutants, including biogenic
substances. This allows the treated wastewater to be reused, in particular, by
discharging it into the reservoirs of the drinking water supply systems. According to
the results of a survey of wastewater treatment systems in the United States [4], it can
be noted that filtering devices are used quite often in tertiary treatment. However, the
greatest effect of removing nutrients is achieved on devices with a floating filter bed
Macrokinetic Model of Biochemical Oxidation 489

(Placerville) and in adsorbers (“Tahoe Truck” complex). The Taho Trucki complex
previously used a slow filter made in the form of a soil tank with a sandy load
with a horizontal filtering surface. Despite the high effect of removing impurities
of different dispersion during film filtration, this structure had to be abandoned due
to technological difficulties during the regeneration of the filter layer. Considering
that the main reason for the difficulties of regeneration in such filters is the need to
remove silty biofilm from a horizontal surface and, therefore, it was necessary to
empty them, dry and replace the top layer of the load, an attempt was made to use
removable filter elements with a vertically located filtering surface (radial blocks and
flat cassettes) [4]. This design of slow filters allowed to eliminate important techno-
logical problems of regeneration of the filter material and to reduce the amount of
clean water used.
Despite the high effect of removing impurities of different dispersion during
film filtration, this structure had to be abandoned due to technological difficulties
in regenerating the filter layer. Considering that the main reason for the difficulties
of regeneration in such filters is the need to remove silty biofilm from a horizontal
surface and, therefore, it was necessary to empty them, dry and replace the top layer
of the load, an attempt was made to use replaceable filter elements with a vertically
arranged filtering surface (radial blocks and flat cassettes) [4]. This design of slow
filters eliminated important technological problems of filter media regeneration and
reduced the amount of clean water used.
In the Russian Federation, in the course of tertiary wastewater treatment, the desire
to simplify the process of deep wastewater treatment has determined the widespread
use of filtering devices, in particular, rapid filters with granular loading and frame-
filling filters [1]. In a filmless slow filter with a filter material thickness of 0.2–0.3 m,
biofilm does not form on the frontal surface, and the biomass is relatively evenly
distributed throughout the pore volume. Due to this effect, the period of biomass
accumulation and, consequently, the duration of the filtration cycle can be increased
to one year or more, depending on the content of the residual amount of suspended
and colloidal particles after the secondary sedimentation tanks. Thus, in comparison
with slow filters with a horizontally located filtering surface, the regeneration period
increases by 10–14 times. In this case, when the maximum saturation of the pore
space with pollutants is reached, the cassettes are replaced with reserve ones, and
the filtering capacity of the material is restored outside the device by drying the
accumulated biomass and blowing the cassette with a fan without using purified
water. Filmless slow filters with a high degree of additional treatment of effluents
and more complete removal of organic matter ensure the ecological safety of water
bodies during discharges even after disinfection by chlorination or ozonation. The
circuit of a filmless slow filter is shown in Fig. 1.
In order to reveal the effectiveness of a filmless slow filter with a vertical filtering
surface, it is necessary to conduct an experiment.
The aim of the experiment was to study the filtering properties of a slow filmless
filter cassette with a vertical filtering surface in relation to ammonium and phosphorus
ions. To achieve this goal, it is necessary to the following tasks:
490 V. N. Volkova and V. L. Golovin

1 - case;
2 - branch pipes of the purified liquid supply; 2,3 - cassettes; 4.5 - filter material;
5 - waterproof partitions;
7 - supply pipes; 8 - supply manifold; 9 - outlet turbine;
10 - branch pipes; 11 - drain collector of purified water

Fig. 1 Schematic of a filmless slow filter with a vertical filter surface

• examine the composition of wastewater for the presence of ammonium and

phosphates, before and after filtration;
• to build a kinetic model based on research data, to analyze the integral kinetic
curves of destruction.

3 Results

Technological parameters of biochemical destruction of ammonium and phosphorus

ions in waste water after secondary sedimentation tanks were worked out on a slow
filter cassette with a vertical filtering surface of 0.05 m3 volume. The unit includes a
system for supplying and discharging waste water; the cassette consists of two paired
cassettes with a filtering load. The first cassette contains a polypropylene fiber filter
with a mesh size of 3–5 mm. The second cassette consists of expanded polystyrene
granules with a grain size of 2–3 mm. The concentration of ammonium and phos-
phorus ions is determined according to the methods recommended in the specialized
literature: ammonium—with Nessler’s reagent, phosphates—with a mixed reagent.
Experiments on biological purification of the aquatic environment from ammonium
Macrokinetic Model of Biochemical Oxidation 491

and phosphates were carried out with free-floating microflora. Stationary processes
of biochemical destruction were simulated without additional supply of pollutants
during the experiment. The primary results of the experiment were points in concen-
tration–time coordinates. All points were averaged over three or four values. The
experimental data are given in Table 1.
The shape of the curve allows various statistical descriptions. The dependence
presented below is quite complex for analytical presentation. For data analysis, the
rate of oxidation is taken, which is defined as the amount of harmful substances
removed per unit of time, for any pair of values of concentration and time:

ρi
= −μ0 Vi , (1)
ti

where μ0 —is the initial concentration of biomass;;

ρi = ρi + 1–ρi —concentration increment;
ti = ti + 1 – ti —time increment;
Vi —specific oxidation rate.
For Vi values, it should be understood as the average values of the specific velocity
in the interval ρί interact with the average values of the concentration in the same
interval. The statistical significance of the curves of the “specific rate-concentration”
type shows a functional dependence with a maximum [5]. The kinetic model is
derived from the parametrization of experimental data rather than from theories
explaining the results of the observed parameters.
The parameters received:
1. Interaction of the elements ammonium and phosphorus with microorganisms,
without which biochemical the process is not possible.
2. The influence of the environment on the ability of microorganisms to decom-
pose, which is determined by the concentration of ammonium and phosphates
in wastewater.
The first parameter is determined by the power law depending on concentration.
The second parameter is the distribution oxidation rate with increasing substrate

Table 1 Results of experimental data

№ Filtration Before processing, after After processing on a Vi is the specific
rate, l / h Tue. sedimentation tanks, biofilter, mg / l oxidation rate
mg / l
Ammonium Phosphates Ammonium Phosphates Ammonium Phosphates
1 0.4 2.53 2.51 2.45 1.46 2.33 12.86
2 0.8 2.53 2.51 2.38 1.27 2.15 7.03
3 1.2 2.53 2.51 2.5 0.92 2.45 5.42
4 1.6 2.72 1.32 2.52 1.33 2.18 0.82
5 2 2.72 1.32 2.21 1.28 1.33 0.69
492 V. N. Volkova and V. L. Golovin

concentration. The coefficient taking into account this dependence is equal to unity
at zero concentration of substances tending to zero with increasing pollutant. With
this coefficient, we obtain the dependence [6]:

Vρ = α · ρb · e−cρ , (2)

where a, b, c—empirical coefficients;

Vρ—dependence of the specific oxidation rate on the concentration of the
pollutant, l/h;
ρ—concentration of a pollutant, g / m3 ;
e—constant = 2718.
Finding empirical coefficients is straightforward. Taking the logarithm of formula
(2) and replacing the variables, we obtain the linear regression equation, the definition
of which is possible using data analysis, the regression function in EXCEL.
This dependence (2) is necessary, but insufficient for calculating the stationary
process of biochemical destruction [7].
When passing from expression (1) at t → 0, we obtain an equation describing
the process of biochemical purification in differential form:

∂ρ
= −μ0 V, (3)
∂t

where ρ—is the increase in the concentration of the pollutant;

t—is the time increment;
V—specific rate of oxidation.
Expression (3) describes the kinetics of the biochemical oxidation process, where
the dependence of the pollution concentration on time is the specific rate. The kinetic
model of biochemical oxidation is a system of two functions showing the interaction
the concentration of the removed substance in the time of the cleaning process,
obtained on the basis of interaction of the same parameters in differential form:

ρ = fp (t); V = fv (ρ). (4)

where f—is the antiderivative function.

The results of statistical processing are shown in Fig. 2 and in Table 2. The form
of the dependence for V adequately describes the experimental model. The obtained
data of biochemical purification from ammonium and phosphates are statistically
significant according to the F criterion and the coefficient of determination. Thus,
the proposed dependence (2) is a universal kinetic model of biochemical destruction
[6].
The minimum coefficient of determination was R2 = 0056.
The resulting kinetic model is universal for calculating specific, non-stationary
processes in biofilters.
Macrokinetic Model of Biochemical Oxidation 493

Influence of the concentration of ammonium and phosphates in

water on the rate of biosorption oxidation

V - specific rate of oxidation of nitrogen and

14
12
10
phosphorus, l / h

8
6
phosphates
4
ammonium
2
0
0 0.4 0.8 1.2 1.6 2
ρ --concentration of ammonium and phosphates in water, mg / l

Fig. 2 Influence of the concentration of ammonium and phosphates in water on the ate of
biosorption oxidation

Table 2 Results of statistical processing of experimental data for relationships of the form “specific
rate—concentration”
V—function, Regression equation Confidence Determination The ratio of
ρ- argument coefficients interval of coefficient, R2 the calculated
a b c deviations of value of the
the regression F-criterion to
equation from the table
the experiment
V—specific 2.514 0.136 0.001 0.130 0.184 0.678
rate of
nitrogen
oxidation, l / h
ρ- ammonium 2.514 0.136 0.001 0.130 0.184 0.678
concentration
in water, mg / l
V—specific 1.342 0.235 0.011 0.225 0.056 0.178
rate of
phosphorus
oxidation, l / h
ρ- 1.342 0.235 0.011 0.225 0.056 0.178
concentration
of phosphates
in water, mg / l
494 V. N. Volkova and V. L. Golovin

According to the data obtained, the efficiency of destruction was obtained.

It was revealed that the efficiency of additional purification of wastewater from
biogenic elements by a filmless slow filter in relation to ammonium is 63.3%, in
relation to phosphates—18.8%.

4 Conclusions

1. The parameters of the dependence of the oxidation rate on concentration of

harmful substances, the contact of ammonium and phosphates with microor-
ganisms was determined, without which the biochemical process is impos-
sible. Obviously, the depressing effect of the environment on the ability of
microorganisms to decompose.
2. A kinetic model of the biochemical oxidation process has been built, showing
a system of two functions that determine the dependence of the concentration
of the removed substance in time.
3. The efficiency of additional purification from ammonium by 18.8% and phos-
phates by 63.3% was obtained, which made it possible to reduce the concentra-
tion of ammonium and phosphates below the established norms of discharges
of harmful substances.

Acknowledgements The reported study was funded by RFBR according to the research project
№ 20-38-90004.

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treatment and reuse. Inżynieria Ekologiczna 24(P):107–119
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Proceedings of the fao expert consultation on water desalination for agricultural applications,
Rome P, 19–28
4. Golovin VL (2011) Tertiary waste water treatment. Problems of land reclamation and water
management in the Far East of Russia Sat scientific works of DalNIIGiM. Vladivostok, Dalnauka,
17(P), 155–173
5. Krichkovska LV, Vaskovets LA, Gurenko IV (ed) (2014) in—Kharkiv design solutions for the
development of biological devices for biological purification of gas-like wikids. NTU “KhPI”,.
p 208
6. Bakharava GY et al. (2015) Development of macrokinetic models for the process of biological
purification of gas accumulated sums. Sci Rise 2(7). S:12–15
7. Bakhareva AY et al (2016) Development of a universal model of the kinetics of the stationary
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Buildings Enclosures Coupling by Its
Energy Efficiency, Seismic Resistance
and Microclimate

Erkin Boronbaev, Berikbay Unaspekov, Aigul Abdyldaeva,

Kamoliddin Holmatov, and Nurbubu Zhyrgalbaeva

Abstract Features of a constructive solution of coupling an external brick wall and

reinforced concrete attic flooring of existing low-rise residential buildings forced
with a reinforced concrete monolithic frame are studied. A photograph presented
shows mold on the intersection zone of the inner surfaces of the external wall
and the attic flooring. The calculation of the building frame carried out using the
program LiraCAD 2013 on a seismic load intensity of 9 points on the MSK-64 scale.
Two-dimensional temperature distributions in the cross section of the enclosures, in
particular in the area of thermal bridges, presented as isotherms using the ArchiCAD
20 software package. The multidisciplinary task of ensuring the required seismic
resistance, energy efficiency and microclimate of the building has been solved. An
expedient constructive solution of the coupling external brick wall and reinforced
concrete attic flooring for the reconstruction of existing and design of new buildings
is proposed. The dimensions of the cross section of monolithic reinforced concrete
columns and crossbars of the building frame and thickness of an additional layer of
thermal insulation of the thermal bridge zones are determined. A new mounting unit
is developed to attach the pitched roof’s Mauerlat to the anti-seismic belt. Practical
recommendations are given that aim to reduce the negative temperature and thermal
effects of the thermal bridges. The proposed constructive solutions made it possible
to exclude the main causes of violations of sanitary conditions in the premises caused
by mold growth on the surfaces of hygroscopic enclosures materials.

Keywords Building energy efficiency seismic resistance microclimate wall

flooring thermal bridges

E. Boronbaev (B) · A. Abdyldaeva · K. Holmatov · N. Zhyrgalbaeva

Department Heat-Gas Supply and Ventilation, Kyrgyz State University of Construction
Transportation and Architecture Named After N. Isanov, 34 b, Maldybaev str, 720020 Bishkek,
Kyrgyzstan
e-mail: boronbaev@elcat.kg
B. Unaspekov
Department Engineering Systems and Networks, Kazakh National Technical University Named
After K.I. Satpayev, 22, Satbaev str, 050013 Almaty, Kazakhstan

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 495
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_45
496 E. Boronbaev et al.

The main goal of this research is to consider and solve the interdisciplinary practical
problem of the expedient coupling of building’s external enclosures. The problem of
ensuring the efficiency and safety of building operations while creating favourable
conditions in the internal space is considered.
The proposed path is aimed at reducing the energy consumption of buildings,
which leads to a decrease in greenhouse gas emissions into the atmosphere—and so
to a decrease in the rate of warming of the global climate of our planet. In particular,
the task of increasing the energy efficiency of buildings is being solved in order to
provide the required internal microclimate with lower energy consumption.
The energy efficiency and microclimate of a building primarily depends on the
characteristics of the building enclosures. In areas with high seismicity, buildings
have special design solutions such as a reinforced concrete frame. In many cases, such
a frame significantly reduces the heat-shielding capacity of the building envelope. In
this article, the authors for the first time considered the multidisciplinary problem of
ensuring the required level of energy efficiency, seismic resistance and microclimate
of low-rise residential buildings. This task is relevant in many regions, where, on the
one hand, the climate features require large energy consumption for the formation
and maintenance of comfortable conditions of the building microclimate, and on the
other hand, there is a high seismic hazard. This problem requires a mutually agreed
consideration in the planning, design, reconstruction and operation of buildings in
many parts of the world; in particular, in the vast regions of the Russian Federation
and the mountainous countries of Central Asia.
The article [1] presented scientific and technical foundations of mutually coordi-
nated provision of energy efficiency and seismic resistance of low-rise civil build-
ings. Studies of the features of construction of seismic resistant and energetically
passive houses are rare [2]. On the other hand, many authors note the importance of
implementing constructive solutions to achieve the required thermal protection [3,
4], optimization [5] and increase [6–8] of the energy efficiency of buildings. It is
especially important to solve the problems of ensuring energy efficiency of external
walls [4, 5, 7, 9].
We used software packages for numerical methods of research, calculation and
visual presentation of results. The program LiraCAD 2013 on a seismic load intensity
of nine points on the MSK-64 scale was used to calculate the seismic resistance of a
two-story residential building with a monolithic reinforced concrete frame structure
and brick-infill exterior walls.
To determine the two-dimensional temperature distribution at the cross section
of the junction of the exterior wall with the building covering, the ArchiCAD 20
software package was used. In this case, it was found, that at the specified junction
area, such interior surfaces of the corners are observed on which the temperature is
lower than the one required by the microclimate standards.
As a result of the adopted method of visual inspection of existing buildings, it
was established that, due to the indicated lower temperature, mold appears in this
intersecting area of the interior surfaces of the exterior wall and attic floor.
In regions with high seismicity hazard, where scaffolding is less accessible, low-
rise civil buildings are constructed from a monolithic reinforced concrete frame
Buildings Enclosures Coupling by Its Energy Efficiency … 497

Fig. 1 Exterior view of a

two-story residential
building at address: 56,
Tunguch str, Bishkek,
Kyrgyzstan

Fig. 2 Photo of mold in the

junction area of the exterior
wall and attic floor surface

structure and brick-infill exterior walls. A representative of such buildings is a two-

story residential building, a fragment of the appearance of which is shown in Figs. 1
and 2 is a photo of mold in the junction area of the exterior wall and attic floor surface.
Such frame buildings are widespread in rural areas and small towns of Central Asia
where, for example, more than half of the population of Kyrgyzstan and Tajikistan
lives. Consequently, a large number of residential and public buildings in high seismic
hazard regions have problems associated with the presence of iron-concrete elements
in the exterior wall, which are thermal bridges [6].
The authors of this article have established that the features of the structural unit
of the junction between the exterior wall and the attic floor have been not widely
studied from the standpoint of ensuring the standard indicators of energy efficiency
and the microclimate of the building. The inner corner area of this junction is exposed
to low outside air temperatures, both from the walls and from the floor. In addition,
this angle is formed by reinforced concrete elements with high thermal conductivity.
Accordingly, two-sided intense cooling of the corner zone is observed.
Field surveys of residential buildings showed mold growth at the intersection zone
of the inner surfaces of the outer walls and the attic floor. It established that mold
appears (see Fig. 2) even in cases when the temperature and relative humidity of
498 E. Boronbaev et al.

Fig. 3 Coupling of the exterior wall and the attic floor of an existing residential building in Khujand,
Tajikistan; a—cross section; b—isotherms in the cross section

the air are near the standard values. The main reason for mold growth is the low
temperature and high humidity of building paint and plaster.
We studied a two-story occupied house without thermal insulation (as in Fig. 1)
in Khujand, Tajikistan, with a total area of 140.2 m2 . The documentation for the
development and design of this building’s reconstruction showed that the thermal
insulation of the building enclosure will have a gap at the junction of the exterior
wall and the attic floor (Fig. 3a).
The specified gap is associated with the widespread fastening of the Mauerlat
wooden structure of the pitched roof to the outgrowth part of the brick wall, the
height of which is usually 300–600 mm higher than the upper level of the ceiling
(Fig. 3a). The desire to eliminate the specified gap by completely covering this wall
branch with a layer of thermal insulation does not give the desired result, since the
position of the Mauerlat does not allow this.
A two-dimensional temperature distribution in the zone of the considered coupling
of two enclosures in the form of isotherms (Fig. 3b) was obtained using the ArchiCAD
20 software package. As can be seen, before the thermal protection reconstruction, the
difference between the room air temperature and the temperature on the inner surface
of the main surface of the exterior wall and the attic floor is greater than their standard
values, equal to 4 °C and 3 °C, respectively (according to SP 50.13330.2012. Thermal
protection of buildings. Updated edition of SNiP 23–02-2003). This difference is
5.4 °C for the exterior wall and 4 °C for the attic floor.
At the intersection of the inner surfaces of the two enclosures, the temperature is
6.6 °C, and the specified temperature drop is 13.4 °C. Under these conditions, there
is a high probability of moisture condensation and mold growth on these surfaces.
It is known that mold growth begins when the temperature on the inner surface of
the enclosure is 12.6 °C at an air temperature of 20 °C, its relative humidity is 50%
and the humidity of the surface layer of hygroscopic material is about 80% (see SP
Buildings Enclosures Coupling by Its Energy Efficiency … 499

Fig. 4 Constructive solution of the coupling exterior wall and the attic floor. a—structure for the
Mauerlat attachment; b—constructive scheme: 1—attic floor insulation; 2 and 3—additional and
main layer of thermal insulation; 4—plaster

KR 23–101-2013. Thermal performance design of buildings. The authors Boronbaev

E and Abdyldaeva A are members of the development team of this national code).
With the aforementioned data, a structural unit was developed and proposed
(Fig. 4) for attaching the pitched roof Mauerlat to a monolithic anti-seismic belt.
On each pair of steel threaded rods embedded in a monolithic anti-seismic belt,
an equal-flange steel angle is put through the holes, the level of which is adjusted
using nuts. It serves as a support for the Mauerlat, which is attached from above with
a similar steel corner and nuts. The step between these attachment points (Fig. 4b),
the diameter of the pins and the dimensions of the corners are taken depending on
the magnitude of seismic, wind and snow loads.
The temperature distributions are studied at the designated coupling of the
enclosures where the proposed attachment unit is not used (Fig. 5a) and used (Fig. 5b).
The thickness of the insulation layers in Fig. 5, made from mineral wool slabs
(with a density of 75 kg/m3 and a thermal conductivity coefficient of 0.04 W/m2
°C), was determined in accordance with SP 50.13330.2012 at degree days of the
calculated heating period 1984 °C·day for the city Khujand. For the exterior wall,
the thickness of the thermal insulation layer is 40 mm, and for the attic floor, 70 mm.
It was found that even with the provision of standard thermal protection of both
building enclosures, the temperature requirements of SP 50.13330.2012 are not met
at the thermal bridge zone. The observed temperature on the corner is 11.4 °C, and
the aforementioned required temperature difference is 8.6 °C (instead normative of
4 °C for the walls and 3 °C for the ceiling).
The calculation for the seismic resistance of a monolithic reinforced concrete
frame of the two-story building under consideration was carried out (according to
the design scheme developed in [1]) using the LiraCAD 2013 program with a seismic
load of 9 points on the MSK-64 scale. In this case, it was found that the accepted
dimensions of the cross-section of the middle and extreme girders and columns (see
500 E. Boronbaev et al.

Fig. 5 Temperature distribution at the cross-section of the coupling zone of the exterior wall and
the attic floor. a—with the main layer of thermal insulation; b—with the main and additional layers
of thermal insulation

Fig. 1) can be reduced to 340 × 340 mm. Accordingly, with a brick wall of 1.5
bricks, the thickness of the additional layer of thermal insulation of the thermal
bridge zone is determined to be 40 mm. With a thickness at the additional and main
thermal insulating layers of the exterior wall of 40 mm, with a reduced cross-section
of the crossbar and the using of the proposed Mauerlat attachment unit (Fig. 4b), the
temperature values on the inner surfaces (Fig. 5b) of the junction area correspond to
the requirements of 50.13330. 2012. In this case, the temperature difference on the
corner of the wall, equal to 3.6 °C, is less than the required 4 °C.
The values of temperatures achieved on the inner surfaces of the thermal bridges
zones and on the main surface of the exterior walls and the attic, above 16.4 °C,
provide, firstly, significantly improved conditions of the microclimate of the room
by increasing its radiation temperature (according to GOST 30,494—2011. Resi-
dential and public buildings. Parameters of indoor microclimate), secondly, prevents
violations of sanitary-hygienic conditions in the premises which occur when mold
grows on room enclosures.
The main indicators of practical recommendations on seismic resistance and
energy efficiency of low-rise civil buildings in six characteristic settlements of the
Russian Federation, Kyrgyzstan and Tajikistan, are presented in the Table 1.
For the building under consideration in Khujand, the following has been achieved:
(1) improvement of microclimate parameters and sanitary-hygienic conditions; (2) an
increase in the economic indicators of construction while reducing the consumption
of concrete by 5.3 m3 ; (3) energy savings of 117.7 kWh per year. Certainly, the
indicated heat energy savings are significantly higher for areas with colder climates.
The results of the authors’ research presented above do not contradict the conclu-
sions in works [10, 11] that the heterogeneity of the building envelope leads to a
discrepancy between the actual level of thermal protection of modern buildings and
Buildings Enclosures Coupling by Its Energy Efficiency … 501

Table 1 Recommended indicators for typical localities of three countries

Cities By seismic resistance According to energy efficiency, thermal insulation
layer thickness, mm
Point Column mm Main layer Additional layer
x mm calculated accepted calculated accepted
Russian Federation
Krasnodar 8 340 × 340 62 70 23 40
Magadan 9 134 140 23 40
Kyrgyzstan
Osh 9 340 × 340 64 70 23 40
Kyzyl-Jar 8 120 120 23 40
Tajikistan
Dushanbe 9 340 × 340 33 40 23 40
Khujand 8 39 40 23 40

regulatory requirements. This circumstance is also observed when solving the prob-
lems of seismic resistance of buildings by reinforcing brick walls with composite
materials [12] and using a reinforced concrete frame structure [1, 13]. The authors
of works [4, 6, 14, 15] similarly established that it is important to study the processes
of heat exchange in the zone of thermal bridge areas of external enclosures, which
leads to an increase in heat loss of a building.
It should be noted that the results of our research were obtained when considering
the influence of elements of seismic resistant monolithic reinforced concrete frame on
the energy efficiency of a building, the levels of microclimate and sanitary-hygienic
condition in the premises. The mentioned concrete elements of the building enclosure
are also considered as thermal bridges.
The analysis of the published works of many authors showed that in modern
conditions the results of those studies that are also devoted to solving practical prob-
lems to reduce the negative effect of thermal bridges in enclosing structures are of
big interest [6, 14, 16, 17].
For the first time, in [16] scientifically and practically substantiated and published
both the definition of thermal bridges and their classification, which were presented
as architectural, structural, and operational thermal bridges. When optimizing of
year-round building’s thermal regimes to ensure its microclimate [18], it is required
to study the negative impact of thermal bridges on the economic performance of
a building in the cold season [16]. To solve such a practically important problem,
numerical methods of studying [17] and thermo-graphic visualization of the results
[19] are also used. Moreover, the results of the study are based, as in [20, 21], on
a graphical representation of the temperature distribution in the cross section of the
thermal bridge zone as a set of isotherms.
In the cold period, all three types [16] of thermal bridges cause not only an
increase in heat losses, but also deterioration in the microclimate conditions in
502 E. Boronbaev et al.

the corresponding rooms. Such deterioration manifests itself through relatively low
temperatures on the inner surfaces of the outer enclosure in the thermal bridge zone.
The authors of works [22–24] also note the significant influence of such local
temperatures on the formation of microclimate parameters. The author of the works
[25, 26] also notes the need to constantly maintain the regulatory parameters of the
microclimate and prevent an increase in the concentration of impurities in the indoor
air above the permissible level. Articles [27, 28] also discuss the harmful effects of
mold on human health.
Conclusions
1. The solution of a multidisciplinary task to ensure the required energy effi-
ciency, seismic resistance and microclimate of a low-rise frame building made
it possible to pro-pose a construction solution for coupling an external brick
wall and a reinforced concrete attic floor for reconstruction of existing and the
design of new buildings.
2. Expedient solutions are proposed: (a) the dimensions of the cross-section of
monolithic rein-forced concrete columns and crossbars of the building frame;
(b) the thickness of the additional layer of thermal insulation of the thermal
bridge zone; (c) a new construction solution for fixing the pitched roof Mauerlat
to the exterior wall seismic belt. Practical recommendations for reducing the
negative impact of thermal bridges on the microclimate and energy efficiency
of buildings are presented.
3. The recommended constructive solutions made it possible to exclude the causes
of violation of sanitary-hygienic conditions in the premises caused by the growth
of mold on the inner surfaces of hygroscopic materials of the building enclosures.

References

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2. Kilar V, Azinovic B, Koren D (2014) World acad of sc engine and technol. Int J Arch Envir
Eng 8(4):365–371
3. Ctaxov AE, Kadokova C (2018) Bectn gpad in. 3(68):219–222
4. .K. Boponbaev (2010)Bect BGTU im B.G Xyxova. 4:127–130
5. Ivanqenko BT, Bacov EB (2019) Ppom i gpad ctp 7:23–27
6. Norbert L (2001) Heating, cooling, lighting: design methods for architects. Wiley, New York
7. MycopinaTA, Gamanova OC, Petpiqenko MP (2017) Bectn MGCU, 11(110):1269–1277
8. Loginova EB, Evdokimenko MO (2017) Bectn Xakacck Gy 20:26–31
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10. Lobov OI, Ananev AI, Pymapov AG (2016) Ppom i gpad ctp 11:67–71
11. Tycnina BM, Fazov DX (2017) Ppom i gpad ctp 4:19–24
12. Gpanovcki AB, Damyev BK, Ocipov PB, Cimakov OA (2017) Ppom i gpad ctp
4:44–49
13. Bedov AI, Gacin AM, Gabitov AI (2018) Izv vyzov Texn tekct ppom 6(378):188–195
14. Danny Harvey LD (2006) A handbook on low-energy buildings and district-energy systems:
fundamentals, techniques and examples. London, James and James
15. IvanqenkoBT, Bacov EB (2016) Bectn BGTU im B.G Xyxova 7:12–17
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16. Boponbaev K (2013) Bectn. KGUCTA, Bixkek 4:130–136

17. Karabulut K, Buyruk E, Fertelli A (2016) Therm Sci 20(1):185–195
18. Boponbaev K (2004) Izv vyc yq 3av Ctp 10:60–64
19. Ucadcki DG, Kovylin AB, Lepilov BI, Kapapyzova H (2017) Bectn Bolgogp.
Goc apx-ctp yniv.. Cep. Ctp. i apx. 50:63–72
20. Hallik J, Kalamees T (2020) A new method to estimate point thermal transmittance based on
combined two-dimensional heat flow calculation. In the e3s-conference proceedings of 12th
nordic symposium on building physics (NSB 2020), 6–9 September 2020, Tallinn, Estonia.
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21. Abdykalykov A, Boronbaev E, Begaluev U, Holmatov K, Zhyrgalbaeva N (22–24, Apr 2021)
Building wall corner structures, its microclimate and seismic resistance. In the e3s-conference
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27. Tepl kova HA, Omelqenko EB (2016) Molod yqen 18–1:23–25
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Kyrgyzstan. In the e3s-conference proceedings of 12th nordic symposium on building physics
(NSB 2020), 6–9 September 2020, vol 172, Tallinn, Estonia. E3S Web of Conferences, p 19010
Pre-ammonization in the Preparation
of Chromaticity Water for Drinking
Purposes

Zhanna Govorova, Uliana Rudich, and Oleg Govorov

Abstract The natural water sources of the North-West region of Russia are charac-
terized by low turbidity, medium and high chromaticity, the presence of organic and
bacterial pollution, and an increased anthropogenic load. As part of the technological
scheme for the purification of such water, disinfection with chlorine and its deriva-
tives is used. When water is chlorinated, volatile organochlorine compounds are
formed, such as trihalomethanes (chloroform, bromoform, bromodichloromethane,
and others). Some of them are carcinogenic. The results of the correlation analysis,
which showed that the formation of chloroform is influenced by the chromaticity
(R2 = 0.24) and permanganate oxidizability (R2 = 0.43) of river water, as well
as the dose of chlorine (R2 = 0.65). The article substantiates the use of prelimi-
nary ammonization using ammonium sulfate. The description of the experimental
stand simulating the operation of an industrial waterworks and its operating modes
is given. The experiments were carried out on real water with permanganate oxidiz-
ability of 10–11.8 mgO2 / l, chromaticity—59–76 deg., turbidity—4.5–4.7 EMF, pH
from 7.4 to 7.6 and alkalinity of 4.6 mmol / dm3 . Ammonium sulfate was dosed
into water 1–2 min before chlorine. It was found that at doses of ammonium sulfate
0.3–0.6 mg / l and the ratio of ammonium to chlorine (1:4, 1:6, 1 8), chloroform
is formed in minimal quantities, but its concentration does not exceed the hygienic
standard. At the same time, the concentration of residual chlorine in the purified
water was—0.94–1.18 mg/l.

Keywords Preammonization · Trihalomethanes · Chloroform · Chromaticity water

Z. Govorova (B) · U. Rudich

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow 129337, Russia
e-mail: GovorovaZhM@mgsu.ru
O. Govorov
JSC Scientific-Technical Center «FONSVIT», 4, Lesnyye polyany, 143982 Balashikha, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 505
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_46
506 Z. Govorova et al.

1 Introduction

The natural waters of the North-West region (Russia) are characterized by low miner-
alization, a small content of suspended solids, and the colored by of humic substances
of bog origin. Sources of water supply in the region differ in terms of technolog-
ical features depending on the value of chromaticity and alkalinity into several main
groups.
The first group—sources of water supply with low turbid water, medium chro-
maticity and low alkalinity. The second group—sources of water supply with very
low alkalinity and high chromaticity and, the third group—sources of water supply
with highly chromaticity and a relatively high alkaline reserve (Table 1). Often, in
water sources, a significant amount of phytoplankton develops in certain periods of
the year, an increased bacterial pollution and the presence of organic pollution of not
only natural, but also technogenic origin [1–3].
When using the water of the listed groups of water supply sources for drinking
purposes, it is required to improve its quality. For this purpose, a reagent two-stage
technology is used, which provides for the treatment of water in sedimentation tanks
or clarifiers with a layer of suspended sediment and filters. An important stage
of purification, on the effectiveness and safety of which the health and sanitary
and epidemiological well-being of consumers depends, is the disinfection of water.
Various chlorine-containing reagents are used to oxidize organic compounds and
disinfect water [1, 2].
When disinfecting colored water containing natural and industrial pollution,
volatile organochlorine compounds (mainly trihalomethanes) are formed [4–
11]. Trihalomethanes (THM) are toxic. These include chloroform, bromoform,
bromodichloromethane, dibromochloromethane and others. Chloroform is most
commonly found in drinking water. The maximum permissible concentration of
chloroform is no more than 0.06 mg / l [13].
The study of the quality of drinking water and its comparison with the concentra-
tion of chloroform showed that the latter correlates with the main indicators. Figure 1
shows as an example the dynamics of variability in terms of water quality indicators
of the river. Vologda and the content of chloroform in purified drinking water at
the pumping station of the second lift for the period 2012–2017. The determining
factors of influence on the formation of chloroform content in water at the output of
the water treatment plant are permanganate oxidizability (R2 = 0.43), the color of

Table 1 Sources of water supply for the North–West region

Indicators Groups
First Second Third
Chromaticity (degrees platinum-cobalt scale) 35–60 80–200 80–250
Alkalinity 0.1–0.14 0.08–0.15 (0.4–0.5) 0.8–1.5
(mmol /dm3 )
Suspended substances (mg / l) 0.15–0.36 5.6 (20–40) 6–15 (35–70)
Pre-ammonization in the Preparation of Chromaticity Water … 507

Fig. 1 Dynamics of variability of indicators of river water quality and chloroform content in
drinking water

river water (R2 = 0.24), as well as the chlorine dose (R2 = 0.65) [14]. The curve of
the relationship between chromaticity of river water and the chloroform content in
drinking water is shown in Fig. 2.
Trihalomethanes have high mutagenic and carcinogenic activity, and therefore
require a decrease in their concentration. Methods for reducing the concentration of
THM in drinking water include: prevention of contamination of the water supply
source, removal of THM in the process of water treatment and prevention of the
formation of THM [6, 8, 12, 15].
Preference is given to methods aimed at preventing or minimizing the formation
of trihalomethanes. These methods include:
• preliminary cleaning from organic suspended and dissolved impurities,
• changing the chlorination mode,
• replacement of chlorine with other oxidizing agents such as ozone, potassium
permanganate, etc.

Fig. 2 The curve of the relationship between the chromaticity of river water and the chloroform
content in drinking water
508 Z. Govorova et al.

• application of treated water ammonization [15–20].

The last method is easy to implement and affordable. In the presence of ammonia
or ammonium salts in the water, the chlorine introduced into it forms chloramines,
the oxidation potential of which is much lower than that of free chlorine. However,
the products of hydrolysis and decomposition of chloramines are strong oxidants [2].
Depending on the pH of the aqueous medium and the concentration of chlorine and
ammonia (ammonium ions), monochloramines (NH2 Cl), dichloramines (NHCl2 ) or
trichloramines (NCl3 ) are formed.
The reaction for the formation of monochloramine is as follows:

NH+ −
4 + ClO → NH2 Cl + H2 O (1)

An increase in the ClO– :NH 4 + ratio and a decrease in pH values lead to the
formation of dichloramine and trichloramine:

NH2 Cl + ClO− → NHCl2 + OH− (2)

NHCl2 + ClO− → NCl3 + OH− (3)

Trichloramines can only form in an acidic environment (pH < 4,4).

In water, chloramines hydrolyze to form ammonia and hypochlorous acid:

NH2 Cl + H2 O → NH3 + HClO (4)

NHCl2 + 2H2 O → NH3 + 2HClO (5)

Data on the bactericidal effect of chlorination with ammonization show that

although at first chloramine chlorine is somewhat inferior in bactericidal activity to
pure chlorine, but after 1–2 h its effect on coli bacteria levels off, and then ammonia
fixes the bactericidal effect of chlorine. Thus, chloramines preserve active chlorine
and are its carriers.
Various reagents are used to ammonize water: an aqueous solution of ammonia
NH3 . H2O (GOST 9–92), ammonium chloride NH4 Cl (GOST 2210–73) and ammo-
nium sulfate (NH4 )2 SO4 (GOST 10,873–73). Ammonium sulfate has advantages
over other reagents. It is safe and highly soluble in water [2]. In the case of prelimi-
nary ammonization, the reagent is introduced before the chlorine is introduced into
the river water conduits.
Evaluation of the effectiveness of the process of ammonization of natural chro-
maticity water aimed at reducing the concentration of trihalomethanes and ensuring
the hygienic standards of drinking water is an urgent scientific and practical task.
Pre-ammonization in the Preparation of Chromaticity Water … 509

Fig. 3 Experimental stand: a—reagent farm unit, b—clarifier and filter

2 Materials and Methods

The object of the study is chromaticity hydrocarbonate-calcium waters of low and

medium mineralization with chromaticity up to 200 degrees, low turbidity (0.48–12.5
EMF), high content of organic pollutants (chemical oxygen consumption—60–83
mgO2 / l) and the level of microbiological indicators. The water source is subject to
anthropogenic pressures. Chlorine absorption of water, depending on the season of
the year, varies widely from 0.2 to 8.6 mg / l.
An experimental stand (Fig. 3) was mounted at the operating waterworks to study
the process of preliminary ammonization, which simulated the operation of industrial
facilities.
It included a block of reagent facilities with supply tanks and reagent metering
pumps; mechanical mixer; suspended sediment clarifier model and a filter, process
pipelines, instrumentation and control valves.
The reagents used were coagulant—aluminum sulfate; flocculant—acrylamide
polymer; reagent for ammonization—ammonium sulfate; disinfection reagent—
chlorine water. The ammonium sulfate solution was dosed into the river water
pipeline 1–2 min before the chlorine water was introduced. The coagulant was fed
in front of the mixer, and the flocculant in its last compartment.
The preammonization process was monitored by residual chlorine and ammo-
nium, while the efficiency was assessed by bacteriological parameters and chloro-
form.
The efficiency of the clarifier and filter was evaluated in terms. The efficiency of
the clarifier and filter was assessed according to the main indicators of water quality
using standard methods.

3 Results

At the initial stage, it was necessary to determine the optimal ratios of chlorine and
ammonium sulfate for the disinfection and purification of river water. Pretreatment
510 Z. Govorova et al.

of water with 0.5% ammonium sulfate solution with doses of 0.28–0.93 mg / l,

which corresponded to the ratio of ammonium to chlorine 1:3–1:10 showed that
chloroform was formed in quantities of 54–75 µg / l at ratios 1:3 → 1:5, and was
absent in samples at ratios 1:6–1:10. For comparison, in a sample of chlorinated water
without ammonization, the concentration of chloroform was 125 µg / L, which is
1.7–2.3 times higher than in the case of preliminary ammonization.
The studies of the ammonization process were carried out during the winter low-
water period, when the water temperature did not exceed 0.8° C, the permanganate
oxidizability was 10–11.8 mgO2 / dm3 , the chromaticity value was 59–76 deg., The
turbidity was 4.5–4.7 EMF, the pH of water varied from 7.4 to 7.6, and the alkalinity
reached 4.6 mmol / dm3 . The total microbial count did not exceed 105–245 CFU
/ ml, and the total coliform bacteria were at the level of 141.5–550 CFU / 100 ml.
The mode of reagent treatment and doses (Table 2) were taken as close as possible
to those adopted at water treatment facilities.
The dose of ammonium sulfate and its injection point, the ratio of ammonium
to chlorine (1:4–1:8) were taken on the basis of preliminary laboratory tests. The
ascending flow rate of the clarifier with a layer of suspended sediment was in the
experiments from 0.5 to 0.8 mm / s, the rate of water filtration through the layer of
inert loading was 4.5–5.8 m / h. In this case, the operation of the clarifier did not stop
during the entire test period, and the filter was washed after each filtration cycle.
The research results are shown in Figs. 4 and 5.
It was experimentally established that only with one chlorination of water with
chlorine doses of 2.0 and 2.5 mg / dm3 with low chlorine absorption (1.25–1.5 mg
/ dm3 ) and its processing using a two-stage technology, the chloroform content in

Table 2 Doses of reagents

Reagent Dose (mg / l) Solution concentration
(%)
Ammonium sulfate 0.25–0.625 0.5
Chlorine water 1.6–3.74 0.12…0.071
Aluminum sulfate 45–53 0.85…0.96
Acrylamide polymer 0.2–0.4 0.095…0.097

Fig. 4 Chloroform content in purified water after clarifier and filter (DCl = 2.5 mg / l)
Pre-ammonization in the Preparation of Chromaticity Water … 511

Fig. 5 Chloroform content in purified water after clarifier and filter (DCl = 2.0 mg / l)

clarified and filtered water was 54–106 µg / l and 64–67 µg / L, respectively. The
concentration of residual chlorine in the filtrate varied from 0.5 to 0.67 mg / l. At the
same time, ammonization of water with a similar chlorine absorption rate minimized
the formation of chloroform in purified water. After the first stage, the concentration
of chloroform in the purified water was 15–57 µg / l, the second stage—5–55 µg
/ l. The concentration of residual chlorine in the purified water after the first stage
ranged from 0.8 to 2.2 mg/l, and the second stage—did not exceed 0.94–1.18 mg / l.
The concentration of ammonium in the treated water according to the scheme:
ammonization → chlorination → clarification → filtration did not exceed 0.34 mg
/ l with its initial content in river water of 0.16–0.29 mg / l. Under the accepted
treatment modes and operating parameters of the main structures, the quality of the
treated water in all respects corresponded to the standards for drinking water [13].

4 Conclusions

Analysis of literature data and our own research has shown that the chlorination
of low-turbid chromaticity waters containing organic and bacterial contaminants
produces toxic trihalomethanes. The preferred method to prevent their formation is
preammonization using ammonium sulfate.
The obtained research results indicate that when water is chlorinated without
ammonization, chloroform is formed in concentrations that exceed the MPC by
1.8 times, the addition of ammonium sulfate 2 min before chlorine, depending on
its dose, made it possible to reduce the chloroform concentration by 50–60% and
ensure it standard in purified water, stabilize the concentration of residual chlorine
in drinking water, reduce chlorine consumption by 40–50%, which is confirmed by
the experience of introducing this method at many water treatment facilities in the
country.
512 Z. Govorova et al.

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Verification of Heat Supply System
Telemetry Data

Elena Kitaytseva

Abstract Purpose. The aim of the study is to develop a system for verifying the
telemetry data of the heating network based on the consistency of physical laws.
Methods. The article discusses existing approaches to data verification. A system of
checks is proposed, which includes 11 conditions that check the fulfillment of the
laws of conservation of mass, mechanical and thermal energy. The checks involve
measured and calculated parameters. Each checked parameter participates in several
checks, which increases the probability of error localization. Results. The proposed
system of checks was used for retrospective analysis of heat supply system telemetry
data. The analysis of the telemetry data showed a large difference in the number
of violations-from their complete absence for individual objects and conditions to
almost 100% of “defects” for other objects when checking other conditions. The
article presents the frequency of occurrence of violations and quantitative indicators
of deviation from acceptable values. Explanations are given for the increase in the
number of violations associated with the calculated parameters. Examples of the
frequency distribution of the measurement error of the volume flow rate at the input
of the consumer and the temperature difference between the heating (at the output)
and the heated (at the input) coolant in the heat exchanger are presented. Conclusions.
The obtained results of using the proposed verification system have confirmed its
viability. The proposed system of checks can be used as an integral part of the
analytical subsystem for monitoring district heating systems. The data verification
system should be constantly expanded.

Keywords Telemetry data verification · Heat supply systems · “Smart” networks

1 Introduction

The Smart City system includes a large number of subsystems, including systems
that control and manage the operation of engineering systems. One of these systems

E. Kitaytseva (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow, Russia 129337
e-mail: KitaytsevaEH@bk.ru

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 513
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_47
514 E. Kitaytseva

is the heat supply system. The principle of centralized heat supply, adopted in the
heat supply of large cities of the Russian Federation, makes heat supply systems life-
support systems. Improving the efficiency of production, transmission and consump-
tion of heat energy and heat carriers is one of the tasks that must be solved by the
“Smart Heat Networks” systems [1–4].
The world is moving from the concept of Automatic Meter Reading (AMR) to
the new concept of Smart Metering (smart, intelligent measurement). It is no longer
enough to read data, build graphs, and automatically prepare reports [5–7]. We need
automatic analysis, conclusions, and system response. The analytical system [8]
should solve these problems.
The requirements for improving the reliability of the obtained measurement results
are caused by several reasons:
• the inadmissibility of controlling the heat supply system based on unreliable data;
• the inadmissibility of using unreliable data as source information for real-time
modeling of hydraulic and thermal processes;
• the inadmissibility of using unreliable data for comparison with the simulation
results;
• the inadmissibility of using unreliable data for the instrument accounting of heat
energy consumption.
The problem of data verification arises regardless of the object of monitoring-a
building [9] or engineering systems [10–12]. The object type plays an important role
in the development of data verification algorithms. Storing the monitoring results in
the archives allows you to conduct a retrospective analysis and use the results of the
retrospective analysis in the express analysis [13]. Express analysis is carried out in
real time.
When verifying the data, there are 2 fundamental questions: what to compare the
measurement data with and what is the permissible deviation. Let us consider the
existing approaches to the verification of telemetry data in heat supply systems.
In the first approach [14], the measurement results are compared with the calcu-
lated parameters recalculated for the current operating conditions. The reference
points are the contractual indicators—design heat load, the calculated flow rate, the
graph of the change in the temperature of the coolant depending on the outside air
temperature.
The design heat load is the main parameter, its value is determined as a result of
calculating the heat losses of the building and the standard consumption of hot water.
Two buildings built on the same project in the same city may have equal design heat
load, but the actual amount of heat required to ensure comfortable living conditions
in them may differ if their indicators for solar radiation and wind load differ [15].
The actual heat load for the needs of hot water supply is determined by the actual
consumption of hot water, which in most cases is significantly less than the standard.
Therefore, the provision of the calculated heat load may in some cases not guarantee
a comfortable living environment.
Another approach [16] when verifying telemetry data in heat supply systems is to
compare the measurement parameters obtained from the same type of objects. It is
Verification of Heat Supply System Telemetry Data 515

proposed to compare the relative temperature difference at the input, the difference
in mass flow rates of the coolant, and the relative difference in heat consumption.
Buildings are considered to be of the same type if they are connected to the same
source and built according to the same project.
This combination of objects in one group does not take into account a number of
important factors that can lead to a scatter of data: the time of construction of the
object, the state of its internal heat-consuming systems, the distance of the object from
the source, the state of thermal insulation of pipelines, and the most important—the
lack of regulation of the heat supply system. You can agree with the idea of comparing
the parameters of the same type of objects, but the list of criteria by which they
can be considered the same type needs to be expanded. With the correct choice of
objects of the same type, the analysis of parameters can be carried out in the express
analysis mode. A retrospective comparative analysis of the parameters is possible
if the conclusion about the similarity of objects at the time of data acquisition is
reliable.
Determination of the permissible accuracy of deviations is especially important
for express analysis, the essence of which is checking whether the measurement
results are within the permissible range. Most often, the permissible deviation is
associated with the measurement error [15]. In [16], it is proposed to use the following
numerical deviation values: for the relative temperature difference at the input—1%
of the average temperature, for the difference in mass flow rates of the coolant—
10% of the average flow rate, the relative difference in heat consumption-5% of the
average heat consumption.

2 Methods

Regardless of the approach to data analysis, the overall data verification system
should include the following groups of checks:
• Checking data for consistency with physical laws;
• Checking data for outliers [17];
• Checking the data for deviations from the constructed regression lines [18, 19].
This article covers only the first group of checks.
Checking the data for consistency with physical laws (laws of conservation
of mass, mechanical and thermal energy) includes checking the values directly
measured (pressure, temperature, volume flow) and calculated from the results of
measurements.
The mass flow rate M, t/day, was determined by the formula:

M = ρ(t)V, (1)
516 E. Kitaytseva

where ρ(t)—is the density of the coolant at temperature t was approximated by the
dependence:

ρ(t) = 1000 − 3.872 · 10−2 (t − 4) − 5.638 · 10−3 (t − 4)2

+ 2.008 · 10−5 (t − 4)3 − 4.29 · 10−8 (t − 4)4 (2)

V, m3 /day, volume flow rate.

Heat energy Q, Gcal / day, was determined by the formula:

Q = c(M1 t1 − M2 t2 ), (3)

where M1 , t/day, t1 , °C, respectively, the mass flow rate and temperature at the outlet
of the object in the supply pipe; M2 , t/day, t2 , °C, respectively, the mass flow rate
and temperature at the entrance to the object in the return pipeline.
The following criteria were developed:
• For any object on which measurements were taken:
• Pressure P1 , temperature t1 , volume V1 and mass M1 flow rate in the supply
pipe must not be less than the pressure P2 , temperature t2 , volume V2 and mass
M2 flow rate in the return pipe, respectively:

P = P1 − P2 > 0; (4)

t = t1 − t2 > 0; (5)

V = V1 − V2 ≥ 0; (6)

M = M1 − M2 ≥ 0. (7)

• For heat exchangers installed at consumers premises:

• The amount of heat transferred by the heating heat carrier Q 1 must not be less
than the amount of heat received by the heated heat carrier Q 2 :

Q = Q 1 − Q 2 = c(M11 t11 − M12 t12 )

− c(M21 t21 − M22 t22 ) ≥ 0; (8)

where c is the heat capacity of the coolant, Gcal / (t °C), M11 , M12 , M21 ,
M22 , t / day, are the mass flow rates, respectively, at the inlet and outlet of
the heat exchanger in the heating circuit and at the outlet and inlet to the heat
exchanger in the heated circuit; t11 , t12 , t21 , t22 , °C,—the temperature of the
Verification of Heat Supply System Telemetry Data 517

coolant, respectively, at the inlet and outlet of the heat exchanger in the heating
circuit and at the outlet and inlet to the heat exchanger in the second circuit.
• In the case of countercurrent flow in the heat exchanger, the temperature of the
heating coolant at the outlet of the heat exchanger must not be lower than the
temperature of the heated coolant:

t1−2 = t12 − t22 ≥ 0; (9)

• For the “source- consumer” connection (in the absence of booster and mixing
pumps in the network):

– The total pressure in the supply pipeline at the outlet of the source P1s must
be higher than the total pressure in the supply pipeline at the input of any
consumer P1c :
 
P1 = P1s − max P1c > 0. (10)

– The total pressure in the return line at the source inlet P2s must be less than the
total pressure in the return line at the outlet of any consumer P2c :
 
P2 = min P2c − P2s > 0. (11)

– The available pressure of the source must be greater than the available pressure
of any consumer P c :
 
P s > max P c . (12)

– The temperature of the coolant in the supply pipeline at the outlet of the source
t1s must be higher than the temperature of the coolant in the supply pipeline at
the inlet of any consumer t1c :
 
t1 = t1s − max t1c > 0. (13)

– The temperature of the coolant in the return pipe at the source inlet t2s must be
the same as the highest temperature of all the temperatures in the return pipe
at the outlet of the consumers t2c
 
t2 = max t2c − t2s > 0. (14)

In the proposed system of conditions, each parameter measurement is involved in

several checks. For example, the pressure in the supply line at the outlet of the source
is used to check the conditions (1), (10), and (12). The temperature of the coolant at
the input to the consumer is used to check the conditions (5), (8) and (9). The volume
518 E. Kitaytseva

flow rate of the coolant at the input to the consumer is used to check the conditions
(6) and (8). This approach increases the probability of finding “abnormal” values.

3 Results

The proposed system of checks (1–14) was used to verify the results of measurements
of the parameters of the heat network. The sample covered the period from 1.01.2018
to 25.03.2019. The summer period of operation of the heat network was not specified
explicitly, so it was calculated from the average daily outdoor temperature. As a result,
the data of measurements carried out from 18.06.2018 to 29.09.2018 were removed
from the initial sample. The frequency of data collection is a day. The information
was collected at the heat supply source and 6 central heating points. The total design
heat load of six central heating points was more than 80% of the attached load. There
were no booster and mixing pumps in the network.
Measurements of the following parameters were presented for all objects:
• overpressure at inlet P1 and outlet P2 ;
• coolant temperature at the inlet t1 and outlet t2 ;
• volume flow rate in the supply pipeline V1 and in the return pipeline V2 .
The results of the verification of telemetry data using the proposed verification
system are presented in Tables 1 and 2. The source is not presented in the tables,
since no violations of the conditions were noted for it (1–14). There are no rows
in the table that match 4 conditions (5, 10, 12, 14). There were no violations of
these conditions for all consumers. Table 1 shows the number of detected condition
violations, expressed as a percentage of the sample size. Table 2 shows the maximum
absolute violations of the conditions: for the conditions (4, 9, 11, 13)—minimum
values, for conditions (6–8), the ratio of the parameter differences to their average
value is presented.
The most common condition violation (Table 1 row 2 consumer 5) of the condi-
tions checked for directly measured parameters is the violation of condition (2).
Figure 1 shows the frequency distribution.
The violation of condition (9) (Table 1, row 5) for consumers 3 and 4 practically
coincides with the total number of measurements. The data presented in Table 2 (row
5) for consumer 4 shows that the temperature at the outlet of the heat exchanger on
the heating side is significantly lower than the temperature on the heated side. The
frequency distribution t1−2 is shown in Fig. 2.

4 Discussion

The analysis of the presented results showed the following.

Table 1 Number of violations, %, of the total number of measurements
№ Condition Consumer
(formula) 1 2 3 4 5 6
Heat exchanger circuit (1-heating circuit, 2-heated circuit)
1 2 1 2 1 2 1 2 1 2 1 2
1 P > 0(4) – – – – – – – – 0.9 – – –
Verification of Heat Supply System Telemetry Data

2 V > 0(6) 22 2.3 – – 43 – 4.1 8.1 75 34 43 –

3 M > 0(7) 51 2.3 51 – 87 – 22 31 100 58 99 0.9
4 Q > 0(8) 85 99 94.5 19.5 0.87 0.29
5 t1−2 > 0(9) 38 42 93.6 98 – 0.58
6 P2 > 0(11) 5.2 – – – 1.7 – – – 99.4 – – –
7 t1 > 0(13) 0.9 – 0.3 – – – 0.3 – 0.3 – 0.3 -
519
 520

Table 2 Maximum values of deviations

Condition Dimension Consumer
1 2 3 4 5 6
Heat exchanger circuit (1-heating circuit, 2-heated circuit)
1 2 1 2 1 2 1 2 1 2 1 2
1 P > 0 at – – – – – – – – −12.6 – – –
2
2 200 VV11 −V
+V2 > 0 % −7.8 −3.4 – – −7.1 – −1.9 −1.3 −2.0 −2.8 −7.8 –
1 −M2
3 200 MM1 +M2 > 0 % −9.8 −3.7 −1.4 – −9.8 – −3.8 −1.5 −2.9 −1.8 −9.3 −0.3
1 −Q 2
4 200 QQ 1 +Q 2 > 0 % −12.4 −6.5 −1.6 −20.4 −3.3 −1.3
5 t1−2 > 0 °C −0.2 −0.3 −1.9 −7.1 – −0.01
6 P2 > 0 at −9.7 – – – −0.2 – – – −21 – – –
7 t1 > 0 °C −7.2 – −0.7 – – – −4.3 – −3.8 – −6.3 –
E. Kitaytseva
Verification of Heat Supply System Telemetry Data 521

0.25
ni/N
0.20

0.15

0.10

0.05

0.00
-2.0 -1.6 -1.4 -1.1 -0.8 -0.5 -0.2 0.1 0.3 0.6 0.6

Fig. 1 Frequency distribution of the relative volume flow for the consumer 5

0.300
ni/N
0.250

0.200

0.150

0.100

0.050

0.000
-6.8 -6.3 -5.9 -5.5 -5.1 -4.7 -4.3 -3.9 -3.5 -3.0
t1-2, C

Fig. 2 Frequency distribution of the temperature difference t1−2 for the consumer 4

Some violations are isolated in nature and are associated with a failure in the oper-
ation of the equipment that was detected and eliminated. For example− checking
condition (4) (Table 1, row 1) revealed violations only in one consumer within
3 days (0.9%). Checking the condition (13) (Table 1, row 7) revealed a failure when
measuring the temperature in the return pipeline at the source inlet. At the same
time, for consumer 5, checking condition (11) (Table 1, row 6) indicates systematic
failures in measuring the pressure in the return pipeline at the outlet of the consumer.
The number of violations of condition (6) is always less than the number of
violations of condition (7) (rows 2 and 3 of Table 1 and 2). The maximum absolute
error max|V | (7.8%) (Table 2, row 2) does not exceed the data presented in [16]. The
increase in the error in determining the difference in mass flow rates in comparison
522 E. Kitaytseva

with the difference in volume flow rates does not exceed 2% (compare rows 2 and 3
of Table 2). A correlation of the errors is explained as follows. The mass flow rate
M is related to the volume flow rate V by the dependence (1). The mass flow rates
difference is equal to:

M = M1 − M2 = ρ(t1 )V − V2 [ρ(t2 ) − ρ(t1 )]. (15)

At V > 0, the difference in mass flow rates M can become negative, depending
on the ratio of the density of the coolant in the supply ρ(t1 ) and return ρ(t2 ) pipelines.
The temperature of the coolant in the supply and return pipelines varies throughout
the day, so the use of a single temperature value increases the error in calculating the
mass flow rates M1 and M2 and their difference M.
The number of violations of the conditions for direct measurements is less (Table
1, rows 1, 2, 5–7) than for conditions in which the results of calculations are compared
(Table 1, rows 6, 7), which is explained by “error transfer” [20].
Violations of conditions (6–9) are massive (Table 1, rows 2–5). This is due to
the fact that the tested values must be equal to each other in the limit. Due to the
measurement error (conditions 6 and 9) and calculations (conditions 7 and 8), the
error in calculating the difference in parameters increases.
The frequency distributions, presented in Fig. 1 and 2, show that it is necessary
to pay attention not only to the frequency of violations of the conditions, but also to
the calculation error.
Volume flow and temperature are directly measured parameters. The maximum
relative difference in volume expenditures was −2.0% (Table 2, row 2). The most
common values of the relative volume flow belong to the range -0.8% to 0.1% (Fig. 1).
All values belong to the area defined by the accuracy of the flow meter measurement.
The number of violations of condition (9) (Table 1 row 5) for consumers 3 and
4 is almost the same as the total number of measurements. The data presented in
Table 2 (rows 5 and 4) show that the temperature at the outlet of the heat exchanger
on the heating side is significantly lower than the temperature on the heating side.
The frequency distribution t1−2 (Fig. 2) confirms the conclusion that this deviation
cannot be explained by the measurement error.

5 Conclusions

1. In “smart” systems, control and regulation is based on the values of the measured
parameters obtained in real time. Therefore, comprehensive verification of
telemetry data is a mandatory component of such systems.
2. The proposed system of checks is based on the consistency of the measurement
results with physical laws. The tests use both the direct results of measurements
and the results of calculations. The system provides for checking the measure-
ment data for each object separately, as well as joint checks for all objects of
the heat network.
Verification of Heat Supply System Telemetry Data 523

3. The conducted analysis of the telemetry data showed a large variation in the
number of violations-from their complete absence for individual objects and
conditions to almost 100% “marriage” for other objects when checking other
conditions.
4. Due to the “transfer” of the error when checking conditions that include calcu-
lated parameters, the number of violations is greater and the error is higher than
when checking conditions with directly measured parameters.
5. The reliability of the data can be increased as a result of their verification.. It is
necessary to create and expand the system of checks. The examples above are
only part of them.
6. The proposed system of checks can be used as an integral part of the analytical
part of monitoring systems for district heating systems.

References

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Groundwater Treatment Plants
as a Sustainable Source
of Iron-Containing Nanopowders

Lev Maksimov, Rowan Baker, Ruslan Safargaliev, Svetlana Maksimova,

and Viktor Mironov

Abstract The presence of global water scarcity, as induced by anthropogenic

contaminants, has led to the rise of wastewater treatment facilities. However, in
middle and low—income countries and regions in particular, the technogenic waste
“sludge” produced after water purification has served little particle application, and
has even posed an environmental hazard. In this article, the authors examine ground-
water treatment plants’ (GWTP) as an alternative source of raw materials, specif-
ically for iron-containing powders, which itself has a wide range of applications.
The Velizhanskaya GWTP in Tyumen, Western Siberia, Russian Federation, served
to provide the waste samples for this study. The main features of such technogenic
waste, such as the chemical composition, the shape, and the size of its particles,
are described and analyzed. In order to consider wastes and related products, the
technical and economic properties of nanoscale state imparting through ultrasonic
radiation impact have been investigated. The key potential areas of consideration for
technogenic wastes and related applications have been identified. The environmental
risks of studied technogenic wastes possible inappropriate utilization and possible
industrial safety rules violations have been analyzed.

Keywords Wastewater · Groundwater · Water treatment plant · Wastes ·

Utilization · Ecology · Nanopowders · Iron · Iron oxide · Fe

L. Maksimov (B) · R. Safargaliev · S. Maksimova · V. Mironov

Industrial University of Tyumen, Volodarskogo Street, 38, Tyumen, Russia 625000
L. Maksimov
FERRME GROUP Ltd., Yalutorovskaya Street, 21, Tyumen, Russia 625000
R. Baker
University of California, 619 Charles E. Young Drive East, Los Angeles 90095-1496, USA
R. Safargaliev
Tyumen State University, Volodarskogo Street, 6, Tyumen, Russia 625003

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 525
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_48
526 L. Maksimov et al.

1 Introduction

1.1 Global and Environmental Implications

Modern challenges of the water supply, such as increased pollution and anthropogenic
induced water scarcity, have become an area of increasing concern for many experts
in the field of waste treatment. In the twenty-first century, the world population’s
access to clean water through the use of waste treatment technologies will only
continue to increase. However, as the mass application of complex water treatment
technologies increases the number and geographic scope of water treatment plants,
the amount of technogenic wastes will only continue to increase.
This is a global concern, but has the most impact on middle and low—income
countries. Most middle and low—income countries have a lack of waste manage-
ment facilities, burdened by waste utilization technologies inaccessibility [1]. Conse-
quently, waste management in these countries presents a different set of problems in
the long-term as compared to high—income countries with a strong base for waste
management.
Groundwater treatment plants (GWTPs) belong to one of the leaders among water
supply system infrastructure facilities for technogenic wastes generation. During
the water treatment process GWTPs generate the sludge-containing wash water—
about 10% on the average compared to the total amount of pure water received.
Groundwater treatment plant’s sludge is a fine powder, consisting of more than 70%
of iron oxides and other associated compounds, such as compounds of calcium,
arsenic, magnesium and etc. [2]. Considering the specific focus of the research,
further for its naming simplicity word “sludge” is used in this paper.
Deposition of dewatered sludge at landfills poses a serious environmental hazard
due to the threat of atmospheric and soil contamination with iron-containing dust
[3]. Recycling this waste has the potential to increase its chemical homogeneity. One
of the main reasons of received data further expanding applicability is similarity
of the hydrogeological regime throughout all Western Siberia and a lot of other
regions all over the world. It is necessary to search for effective and environmentally
friendly methods of disposal of sludge from deironing stations, in particular, as a raw
material for production [4, 5]. Involvement of sludge from GWTPs in production for
the creation of new types of products corresponds to the principles of a circular
economy [6].

1.2 Chemical Composition and Application

One of the distinguishing features of the sludge is the particle size composition, that
consists of nano- and micro-size particles. Considering the full range of accompa-
nying properties for similar size-type particles, this sort of technogenic waste can
be classified as the PM 2.5 particles. Submitted fact shows us that studied sludge
Groundwater Treatment Plants as a Sustainable Source … 527

class is more hazardous to biocenoses than it’s established in the current hazardous
wastes’ classification in many countries [7, 8].
Nanoscale zero-valent iron is used in sorption materials designed to extract
contaminants from aqueous solution. The advantage of magnetized nanocomposites
is that it can be easily insulated and removed with external magnets [9].
Sorption material based on sludge of groundwater de-ironing plants showed high
oil-absorbing capacity. Thermal activation of the material leads to an increase in the
efficiency of oil extraction from aqueous solutions [10]. An increase in the activa-
tion temperature of the iron-containing sludge leads to an increase in the magnetic
susceptibility and specific magnetization of the samples [11].
The main physical methods for the synthesis of powders consisting of nanopar-
ticles include: arc discharge, laser evaporation, condensation in an inert gas atmo-
sphere, laser and jet pyrolysis, plasma methods [12]. All physical methods for the
synthesis of nanoparticles are high-tech and complex processes. This is a disadvan-
tage of physical methods for the synthesis of nanopowders. For this reason, the cost
of most nanoparticle materials is very high.
The technological process for obtaining nanopowders based on chemical methods
is simpler. However, chemical methods for synthesizing nanopowders have several
disadvantages. The first disadvantage is that chemical reactions generate reaction
by-products and the nanopowders are chemically impure. The second disadvantage
is that the particle sizes vary widely from several tens of nanometers to microm-
eters. Additionally, nanoparticles synthesized by chemical methods are prone to
aggregation, for example, when obtaining nanopowders Fe3 O4 [13].
Mechanical methods for obtaining nanopowders are based on the structural
decomposition of coarse-grained materials as a result of deformation while main-
taining the original chemical composition [14].
The preparation of iron oxide nanopowders Fe2 O3 in [15] was carried out in a
planetary mill. Hematite was used as a raw material. The authors of the study point
to the dependence of the particle size on the time and speed of the ball mill. The
minimum particle size that could be achieved with this grinding is 17.1 nm. In this
case, 10 h of operation of the installation were spent, which is very labor and energy
intensive.
In the study [16], iron oxide nanopowders Fe3 O4 with sizes ranging from 30 to
80 nm were obtained in a planetary mill. The starting material was iron metal powder.
It can be concluded that iron oxidation occurs during grinding.
Technological simplicity is characteristic for the production of nanopowders by
methods based on the use of ultrasonic vibrations that create cavitation in liquids [17].
This advantage makes the method applicable for the production of nanopowders from
the sludge of de-ironing plant.
Previously, we have already carried out a number of studies on the potential
hazard of such wastes’ disposal in dry state at MSW landfills and on the possibility
of its’ useful recycling [18, 19]. According to the existing research results in sphere
of powder obtaining we hypothesize that iron-containing nanopowders with a wide
range of end uses can be obtained.
528 L. Maksimov et al.

2 Methods

2.1 Groundwater Treatment Plants’ Wastes Sampling

Conditions

2.1.1 Location of Sampling Points

The studied samples were taken from the Velizhanskaya Groundwater Treatment
Plant that supplies Tyumen with purified water. This station is located in the Southern
part of Western Siberia in Russian Federation. Existing hydrological regime can be
described as highly polluted by iron compounds with low levels of other pollutants.
This explains the high level of iron oxide in chemical composition of the studied
sludges. Presented composition is widespread. Subsequently, the obtained data is
applicable for many regions across the world despite their distance from each other.

2.1.2 Procedural Conditions

The samples were picked from iron removal filters side surfaces in dry solid condition.
For further studies similar samples could be taken in state of GWTP’s filters’ wash
water for further dehydration and dry solid samples obtaining. Studied samples have
a form of ultrafine powders.

2.2 High Temperature Redox Reaction Conditions

2.2.1 Temperature Conditions

The reaction temperature was selected based on the following requirements:

• The temperature is sufficient to activate the reaction of obtaining carbon monoxide
(CO) from carbon dioxide (CO2 ) and carbon-containing solid reagent in situ;
• The temperature is sufficient to activate the reaction of obtaining iron oxides (II)
and (II, III)—(FeO and Fe3 O4 ) from iron oxide III (Fe2 O3 ) and carbon monoxide
(CO);
• The temperature does not lead to solid mixtures melting;
• Temperature provides the highest energy efficiency of the process;
• Ease of implementation of engineering measures to ensure compliance of real
laboratory and experimental-industrial conditions with design.
Relying on the listed requirements the range of temperatures from 450 to 600
degrees Celsius with 50 degrees steps will be taken.
Groundwater Treatment Plants as a Sustainable Source … 529

2.2.2 Chemicals and Reagents

For the full course of the reduction reaction ensuring and further implementation
in industry one of the most abundant and easy to use reagents was taken. Carbon
monoxide can be generated in situ using coal (or similar carbon-containing solid-state
reagents) and carbon dioxide. Its properties afford us to use quite low temperatures—
starting from 400 °C. Compared with hydrogen, this gas environment has less risks
of explosion.

2.2.3 Reactor Chamber Processing Scheme

According to previously chosen reagents, the reactor chamber must provide imper-
meability for the external gas environment and the capacity for gas convection.
Convection will lead to reagents recuperation simultaneously with redox reaction.
This process can be described by system of two equations:

Fe2 O3 + CO => Fex Ox + CO2 (1)

CO2 + C => 2CO (2)

where Fex Ox —one of the iron oxides or metallic iron depending on the reaction’s
temperature.
This process is possible because of two factors:
(1) Significant density difference of gas reagents (1.98 g/cm3 for carbon dioxide
versus 1.25 g/cm3 for carbon monoxide);
(2) Very high level of powder-state sludge mass porosity (5.25 g/cm3 for iron oxide
(III) versus 2.15 g/cm3 for dry GWTPs’ sludge)
Using this data we can conclude that the most efficient reaction processing requires
treating sludge placing on top of solid compounds mixture. The source of solid carbon
must be below the treating sludge.

2.3 Ultrasonic Treatment Conditions

2.3.1 Ultrasonic Milling Device Properties

Ultrasonic dispersion was carried out by ultrasonic disperser MEF93.T. Disperser’s

intensity is 250 W/cm2 , switching power is 600 W, operating frequency is 22 kHz.
All samples were treated for 1 h sharp.
530 L. Maksimov et al.

2.3.2 Liquid Medium Selection

For appropriate research results of ultrasonic influence on GWTP’s sludge, three

liquid mediums were used:
• Distilled water;
• Aqueous solution of synthanol (0.1%);
• Aqueous ethanol solution (1%).
Distilled water was chosen for its application simplicity and as a standard basis for
solutions. Moreover, water is the initial environment of studied sludge. Synthanol as
a case of surfactant in the form of aqueous solution was chosen for its cracks closure
and reaglomiration prevention ability. Ethanol as a case of alcohol in the form of
aqueous solution was chosen for its ability to increase the fluidity of solution. It
could conduct surface tension overcoming for liquid filling in pores with nanoscale
slots. Liquid filled slots proportion increment is directly proportional to the ultrasonic
dispersion effectiveness.

2.3.3 Particle Size Measurement Methods

The granulometric composition of raw and processed sludge were measured using
two method of Scanning electron microscope (SEM);
As an SEM device we used MIRA3 TESCAN. Energy range of the electron beam
incident on the sample: from 200 eV to 30 keV. Its possible top resolutions are 1.2 nm
at 30 keV and 3.5 nm at 1 keV.

3 Results

3.1 Ultrasonic Treatment Results

3.1.1 Liquid Medium Influence on Results

Considering our focus for iron-containing powders obtained with highest dispersion
capacity and tendency of spherical shape, 3 liquid medium particles were taken. For
proper processed samples comparing, the initial state (Fig. 1) and ultrasonic treated
(Fig. 2) sludge are provided.
The reached effect, achieved under conditions of 1 h of ultrasonic treatment for
samples obtained with the temperature of 550 °C is shown as the most signifi-
cant examples for water, syntanol (0.1%) and ethanol (1%) at SEM-microphotoraps
(Figs. 3, 4 and 5).
Microphotographs indicate the significant difference between water and other
liquids. Particles with quasi spherical shape can be found in the first picture, but in
Groundwater Treatment Plants as a Sustainable Source … 531

Fig. 1 Initial sludge without ultrasonic treatment

Fig. 2 Initial sludge after ultrasonic treatment

Fig. 3 Sludge treated in distilled water

Fig. 4 Sludge treated in aqueous solution of synthanol (0,1%)

532 L. Maksimov et al.

Fig. 5 Sludge treated in aqueous solution of ethanol (1%)

the next two multiple wrecked structures are shown. As such, all further data will be
provided only for distilled water medium. Other mediums were considered as less
effective for the set goals achieving.

3.1.2 Temperature’s Influence on Results

Adding to the already shown picture of sludge, processed at 550 °C, micropho-
tographs of other samples are given below in order of processing temperature
declining.

4 Discussion

As the results demonstrate, we can conclude that application of synthanol and ethanol
solutions together with ultrasonic treatment leads to instable structure formation. This
could possibly be attributed to the presence of calcium compounds. The content of
calcium compounds can reach 30% of the total dry matter mass of sludge. These
compounds can be partly dissolved and recrystallized on the sludge surface.
The other issue is highly crystallized structures formation. Microphotographs indi-
cate the dependence of temperature. The higher the temperature—the more fragile
the sludge structure will be. Samples with processing temperatures of 500 and 450
degrees Celsius have many spherical and quasi spherical particles with an agglom-
erate structure and consisting of smaller crystals, about 70–100 nm in size. The
efficiency of ultrasonic action on the sludge after heat treatment is higher due to the
embrittlement of the structure caused by a change in the crystal lattice during the
transition from Fe2 O3 to Fe3 O4 and FeCO3 (Figs. 3, 6, 7, and 8).
After treating the sludge at a temperature of 450 °C in a carbon monoxide environ-
ment, the particles have a pronounced spherical shape, extremely low open porosity,
and the lowest level of adhesion with neighboring particles relative to all other heat-
treated samples (Fig. 8). These properties are key for most of the developed products
based on micro- and nanopowders.
Groundwater Treatment Plants as a Sustainable Source … 533

Fig. 6 Sludge processed at temperature 600 °C

Fig. 7 Sludge processed at temperature 500 °C

Fig. 8 Sludge processed at temperature 450 °C

An increase in the treatment temperature of the sludge to 550 °C led to the devel-
opment of the process of adhesion of particles into developed polymorphic systems,
in which developed open porosity is also not observed, but the ultrasonic effect of the
applied power is no longer sufficient for their deagglomeration (Fig. 3). This indi-
cates the formation of groups of particles that have undergone fusion, and, therefore,
their subsequent grinding can lead to the formation of particles with a higher degree
534 L. Maksimov et al.

of fragmentation and a lower degree of sphericity. In addition to many cleavages

on the surface of the particles, open porosity is also clearly visible, which was not
observed in previous samples. The significant heterogeneity of shapes, sizes, and
the degree of surface curvature caused by the development of open porosity and the
appearance of chips force us to talk about a potential decrease in the quality of the
reaction products obtained at such temperatures under given conditions.

5 Conclusion

The best result in all these parameters is achieved when processing in a carbon
monoxide medium at a temperature of 450 °C and post-processing with ultrasonic
radiation for 1 h with a power of 250 W/cm2 in distilled water. This approach also
makes it possible to abandon the use of reagents such as surfactants and alcohol
(ethanol), as well as to avoid an irrational increase in the processing temperature
of raw materials. The cumulative effect of this makes it possible, with the lowest
energy and material costs, to achieve the required properties of products based on
metal-containing highly dispersed powders that meet the requirements of key areas
of consumption.
The resulting set of scientific data provides the foundation for further research and
development to improve the properties of the final product, including when changing
the parameters of raw materials that are not identical at different water treatment
plants, depending on hydrogeological conditions and water treatment conditions.

References

1. Nanayakkara N, Arambepola IG, Aluthwatte M, Rajasinghe C, Herath G (2020) Groundwater

Sustain Develop 11:100414
2. Maksimov LI, Mirinov VV (2020) Vestnik Tomskogo gosudarstvennogo arkhitekturno-
stroitel’nogo universiteta. J Construct Architect 22(2):162 (In Russ.)
3. Popov VK, Pasechnik EY, Karmanova A (2016) MATEC Web Conf. 85:01013
4. Meng L, Chan Y, Wang H, Dai Y, Wang X, Zou J (2016) Environ Sci Pollut Res 23:5122
5. Zou JL, Xu GR, Li GB (2009) J Hazard Mater 165:995
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37:649
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(2011) Procedia Environ Sci 4:198
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Develop. 7:470
10. Novovselova LY, Sirotkina EE (2009) Rus J Phys Chem A 83:2127
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12. Rao CNR (2007) Nanocrystals: synthesis, properties and applications. Springer, Berlin
13. Uschakov AV (2017) J Supercond Nov Magn 30:311
14. Yadav TP (2012) Nanosci Nanotechnol 2:22
Groundwater Treatment Plants as a Sustainable Source … 535

15. Arbain R (2011) Miner Eng 24:1

16. Ding C (2007) China Part 5:357
17. Pakharukov YV, Shabiev FK, Mavrinskii VV, Safargaliev RF, Voronin VV (2019) Jetp Lett
109:609
18. Maksimov LI, Kuskov KV, Maksimova SV, Kulemina AA (2019) Proc of D. I. Mendeleev’s
conference, 332. (In Russ.)
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resources … in Siberia and the Arctic in the XXI conference, 180. (In Russ.)
Global Environmental Challenges
Particles Transport with Deposit Release
in Porous Media

Liudmila Kuzmina and Yuri Osipov

Abstract Models of transport and filtration of fine particles in porous media are used
in the design of foundations, tunnels and hydraulic structures. During the particles
transport, some of the particles get stuck in narrow pores and form a sediment. The
suspension or colloid flow washes out the retained particles and increases the concen-
tration of suspended particles. Consider a model of the suspension/colloid transport in
one-dimensional homogeneous porous media with the simultaneous action of forces
aimed at sedimentation and release of particles. The model consists of an equation
for mass exchange between particles and an equation describing the rate of sediment
formation, taking into account the retention and release of particles. Exact solutions
are obtained at entrance of the porous medium, on the front separating the injected
fluid and the clean water, and ahead of the front. A solution to the problem is found
in the form of a traveling wave.

Keywords Particles transport · Particles retention and release · Porous medium ·

Exact solution

1 Introduction

Transport and filtration of small particles of suspensions and colloids in porous media
are common in nature and technologies [1–5]. In the construction industry, to create
a solid foundation, a liquid grout is pumped into the soil. The grout filters in the
pores of the rock and, when solidified, strengthens the foundation [6, 7].
The transport of particles through a porous medium is accompanied by the forma-
tion of a deposit. Various mechanisms of particle capture are associated with elec-
trical, hydrodynamic and gravitational forces and complex pore geometry [8, 9].
When approaching a small pore, the particle is stopped in its throat (size-exclusion
mechanism of particle retention) [10]. When transported through large pores, some

L. Kuzmina
National Research University Higher School of Economics, Moscow, Russia
Y. Osipov (B)
Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow, Russia 129337

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 539
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_49
540 L. Kuzmina and Y. Osipov

particles are deposited on the pore walls. As a rule, models of filtration and particle
transport assume that the retained particles are stationary Impact of other particles
and the fluid pressure cannot release precipitated particles and lift them into the flow
[11–14].
A model of suspension/colloid filtration in a porous medium is considered [15]. At
the initial moment, the porous medium with constant deposit concentration is filled
with suspension/colloid. A clear fluid without suspended particles is pumped into
the porous medium entrance. The formation of a deposit during particle transport
is accompanied by the rise of part of the retained particles under the influence of a
fluid flow. The gradual rise of the retained particles can be explained by the layered
structure of the deposit. Water captures the deposited particles of the upper layer,
when they rise, the next layer of deposit opens, which also breaks off from the porous
medium framework. The deposit is gradually washed out of the pores and transported
to the porous medium outlet.
The equations for the exchange of particles masses and for the change in the
deposit concentration form the filtration model [16]. The change in deposit in the
kinetic equation depends on the difference between the terms responsible for the
growth and reduction of the retained particles concentration. The growth of deposit
is proportional to the fluid speed and to the volumetric concentration of particles in
the suspension or colloid; the coefficient of proportionality is the filtration function
[17, 18]. The washout of the deposit is proportional to the velocity of the carrier
fluid, the coefficient of proportionality is called the particle release function. Both
functions depend on the volumetric concentration of retained particles.
Due to the washing out of the retained particles, the injected clean water turns into a
suspension of low concentration. The front of the injected fluid moves from the porous
medium inlet to the outlet, separating suspensions of high and low concentration.
Over time, more and more suspended and retained particles are transported to the
outlet and the concentration of suspended particles decreases behind the front.
The exact solution to the problem is obtained ahead of the injected water front,
as well as on the front and at the porous medium inlet. A travelling wave solution is
found.

2 Mathematical Model

In the domain Ω = {x ≥ 0, t ≥ 0}, consider the system of one-dimensional

filtration model
∂C ∂C ∂S
+ + = 0, (1)
∂t ∂x ∂t
∂S
= Λ(S)C − λ(S). (2)
∂t
Particles Transport with Deposit Release in Porous Media 541

Here C(x, t) and S(x, t) are the unknown suspended and retained volumetric
concentrations. Λ(S) called the blocking filtration function is continuous, differ-
entiable, positive and decreasing at 0 ≤ S < Sm , Λ(S) = 0 at S ≥ Sm . The
particle release function λ(S) is continuous, differentiable, and increases with S ≥ 0,
λ(0) = 0. The fluid velocity v = 1.
The boundary and initial conditions

C|x=0 = 0, (3)

C|t=0 = C0 , S|t=0 = S0 , S0 < Sm . (4)

guarantee a unique solution to the system (1), (2).

The injected fluid moves with constant velocity v = 1, its boundary forms a
concentration front t = x dividing the domain  into subdomain Ω1 = {x ≥
0, 0 ≤ t ≤ x} ahead of the front and subdomain Ω2 = {x ≥ 0, t ≥ x} behind
the front. At all points of the porous medium, the sediment concentration S(x, t)
changes continuously, while the solution C(x, t) is discontinuous on the front.
In the domain Ω1 , the solution takes the form C = C(t), S = S(t). Integration
of the Eq. (1) over time yields:

C + S = K, K = const. (5)

Using the condition (4) and the Eq. (5), we find the constant K = C0 + S0 and
obtain the relation

C = C0 + S0 − S. (6)

Substitute the relation (6) into the Eq. (2)

∂S
= Λ(S)(C0 + S0 − S) − λ(S). (7)
∂t
Solution of the Eq. (7) with the condition (4)

S1 (t)
ds
= t. (8)
Λ(s)(C0 + S0 − s) − λ(s)
S0

To study the dependence of the solution (8) on the value of the initial sediment
consider the denominator of the integrand in formula (8)

z(s) = (s)(C0 + S0 − s) − λ(s). (9)

542 L. Kuzmina and Y. Osipov

The denominator (9) decreases on the segment [0, Sm ], at the ends of the segment,
the function takes values of different signs z(0) > 0, z(Sm ) < 0. Therefore, the
denominator has a single root S∗ ∈ (0, Sm ). From formula (8) it follows that at
0 < S0 < S∗ the solution S1 (t) increases and tends to S∗ at t → ∞; at S0 > S∗
the solution decreases and also tends to S∗ at t → ∞; when S0 = S∗ the solution is
constant: S = S∗.
At the front t = x, the solution S1 = S∗ is given by the formula

S1 (x)
ds
= x. (10)
Λ(s)(C0 + S0 − s) − λ(s)
S0

Let us set the condition on the front of the injected fluid

S|t=x = S1 (x) (11)

To consider the problem Eq. (1)–(4) in the domain Ω2 , we replace condition (4) by
condition (11). In the case λ(S) ≡ 0 (filtration without deposit release), the solution
in the domain Ω2 does not depend on time: C = 0, S = S1 (x) [19]. If λ(S) > 0 at
S > 0, then the solution depends on the time t and the coordinate x.
According to the condition (3), at the inlet x = 0 the Eq. (2) has the form

∂S
= −λ(S). (12)
∂t
The solution to (12) with the condition (4)

S0
ds
= t. (13)
λ(s)
S2 (0,t)

Since λ(0) = 0, the solution S2 (0, t) decreases and tends to zero at t → ∞.

3 Traveling Wave Solution

To study solutions to complex nonlinear systems, it is useful to consider traveling

waves [20]. We will seek a solution to the system (1), (2) in the form

C = C(w), S = S(w), w = x − ut, (14)

where the constant u is the unknown wave velocity.

The Eq. (1) takes the form
Particles Transport with Deposit Release in Porous Media 543

−uC + C − u S = 0, (15)

Integration of the Eq. (15) yields

−uC + C − u S = K , K = const. (16)

Let us set the conditions at infinity

w → +∞ : C → 0, S → 0; w → −∞ : C → C0 + S0 − S∗, S → S ∗ .
(17)

Substitution of the conditions (17) into the Eq. (16) gives

uS C0 + S0 − S∗
C= , u= . (18)
1−u C0 + S0

From formulas (18)

C0 + S0 − S∗
C= S. (19)
S∗

From the Eq. (2) and the formula (19) it follows that the function S(w) satisfies
the equation

C0 + S0 − S∗ C0 + S0 − S∗
− S = Λ(S)S − λ(S). (20)
C0 + S0 S∗

The Eq. (20) with the conditions (17) at infinity has an infinite number of solutions.
Let the right-hand side of the Eq. (20) be positive at 0 < S < S∗. Then the
additional condition

w = 0 : S = s0 , 0 < s0 < S∗ (21)

determines a unique solution to the Eq. (20)

S
S ∗ (C0 + S0 − S∗) ds
− = w, (22)
C0 + S0 (C0 + S0 − S∗)Λ(S)S − S ∗ λ(S)
s0

satisfying the conditions (17) (see Fig. 1).

A traveling wave S(w) is a continuous function decreasing from maximum limit
value S∗ = 1 to zero.
544 L. Kuzmina and Y. Osipov

Travelling wave S
S

1.0

0.8

0.6

0.4

0.2

w
4 2 2 4

Fig. 1 The traveling wave S(w) at s0 = 0.5, S∗ = 1

4 Numerical Solution

Numerical calculation is performed for Sm = 2, Λ(S) = 2 − S, λ(S) = S, S0 =

0.5, C0 = 1.5. In this case the denominator (9) takes the form

z(s) = 4 − 5s + s 2 ,

Its smaller positive root S∗ = 1. Figures 2 and 3 show the concentrations of

suspended and retained particles in the domains Ω1 and Ω2 .

a Suspended concentration C b Retained concentration S

C S
1.5
1.0

1.4
0.9

1.3 0.8

1.2 0.7

1.1 0.6

0.5 1.0 1.5 0.5 1.0 1.5

Fig. 2 Particles concentrations in the domain Ω1 a suspended C b retained S

Particles Transport with Deposit Release in Porous Media 545

Retained concentration S at the inlet x 0

S

0.5

0.4

0.3

0.2

0.1

t
1 2 3 4 5

Fig. 3 The concentration S(0, t) in the domain Ω2

In the domain Ω1 , the suspended particles concentration C(t) decreases and tends
to zero at t → ∞; it reaches its maximum at the initial moment: C(0) = C0 = 1.5.
The retained particles concentration of deposited particles S(t) increases starting
from the initial deposit S0 = 0.5 and tends to S∗ = 1 at t → ∞.
In the domain the retained particles concentration S(0, t) decreases and tends to
zero at t → ∞.

5 Discussion

During slow filtration of a suspension/colloid in a porous medium, the particles

are deposited on the framework of the porous medium and do not detach from
it. If the direction of the fluid flow suddenly changes, then with a sharp change
in pressure, some of the retained particles are detached from the framework and
become suspended. It is assumed that during filtration of the resulting suspension,
the remaining precipitated particles can be detached from the framework.
The particle transport model with simultaneous capture and release of particles
provides a balance between lifting and settling mechanisms. If the suspended parti-
cles concentration is low, and the retained particles concentration is high, then the
retained particles are detached from the framework of the porous medium. When the
suspended particles concentration is high, and the retained particles concentration is
low, the mechanism of particle retention prevails and the deposit increases.
The model with simultaneous capture and release of particles differs from the
standard particles transport model with fixed deposit. In contrast to the model with
546 L. Kuzmina and Y. Osipov

a stationary deposit [21], at a fixed point in a porous medium, with increasing time,
both concentrations decrease and tend to zero.
The considered model allows the possibility of complete washout of the deposit.
A more complex model assumes that the deposit consists of fixed and moving parts.
The movable part can be washed out by the fluid flow, while the stationary part
is firmly fixed on the frame of the porous medium. This model will be considered
separately.

6 Conclusions

For a model of particles transport in a porous medium with the particle retention and
release of the deposit.
• the front of the injected fluid separates suspensions of different concentrations,
• the exact solution at the porous medium inlet is constructed,
• the exact solution is obtained on the concentration front,
• the exact solution ahead of the front is found,
• the exact solution in the form of a traveling wave is obtained.

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Physically-Based Streamflow Predictions
in Ungauged Basin with Semi-Arid
Climate

Dmitry Kozlov and Anghesom Ghebrehiwot

Abstract Rainfall-runoff models that utilize reanalysis datasets as driving vari-

ables have been widely applied for generating hydrological responses in data-sparse
regions. Apparently, there are various requirements that affect the choice of a partic-
ular method of hydrologic investigation. In the present study, Soil and Water Assess-
ment Tool (SWAT) and precipitation-runoff (MIKE 11-NAM) models were selected
to simulate streamflows from a small watershed with semi-arid climate, using Climate
Forecast System Reanalysis (CFSR) as driving inputs. As such, models that provide
reliable streamflow predictions from regions with similar climate settings, whose
errors and uncertainties are within acceptable ranges, can be identified. The main
characteristics of performance criteria indicate that the SWAT model relatively
outperform the MIKE 11-NAM model. However, while most of the statistical evalu-
ations prove the acceptable performance of the SWAT model, broad range of predic-
tion uncertainties during calibration and validation were also reflected. Among the
possible sources of errors, errors due to forcing data are most likely to be accounted
for the unsatisfactory portions of both models. Therefore, to minimize model uncer-
tainty and thereupon improve its performance, in-situ data collection need to be
incontestably boosted up. The study also highlights the need for further investiga-
tion on the possible mechanisms of proper application of CFSR that avoid erroneous
streamflow predictions from similar regions.

Keywords Eritrea · MIKE 11-NAM · Rainfall-runoff · Reanalysis datasets ·

Sensitivity analysis · Simulation · SWAT model

1 Introduction

Physically-based mathematical models are used to analyse and predict hydrolog-

ical and biogeochemical processes within river catchments, including the flow of
water, sediment, chemicals, nutrients, and microbial organisms, within watersheds,
as well as quantify the impact of human activities on these processes. Such models

D. Kozlov · A. Ghebrehiwot (B)

Moscow State University of Civil Engineering, Yaroslavskoe shosse, 26, Moscow, Russia 129337

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 549
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0_50
550 D. Kozlov and A. Ghebrehiwot

have been useful tools in underpinning our understanding about the dynamic inter-
actions between climate and land surface hydrology [1–3] and providing the missing
information as a basis for decision-making. As such, a broad spectrum of critical
environmental and water resources problems have been addressed with the support
of physically-based mathematical models. Nevertheless, a bulk of evidence demon-
strates that there are limitations in existing models due to lack of full representation of
the complex hydrologic systems and spatio-temporal variability of the hydrological
and meteorological components. As a result, recent studies [3–5] emphasized the
need for the development of watershed models that make use of the widest possible
information available and underpin the current practices of sustainable manage-
ment of river basins, as well as new techniques that integrate economic, social and
environmental perspectives.
Rainfall-runoff models are one of the extensively applied predictive tools for
generating hydrological responses. Whichever rainfall-runoff model we select for
whatsoever purpose it may be, it remains to be only an approximate representation
of the real processes. Despite the efforts put in to overcome the fundamental problem
of extensive difference in the spatial and time scales of hydrological models through
the application of downscaling of model outputs, selection of an appropriate hydro-
logic model is yet one of the critical issues. Conspicuously, the effectiveness of a
model largely depends on availability and reliability of historical ground information;
the efficacy of a model is lower and more uncertain in ungauged regions and vice-
versa. At global scale, river basins in many parts of the world are not only ungauged
but also experience a significant reduction in the ground hydrometric networks [3,
6–9]. Regional studies in East Africa1 [10] have shown that the political and socio-
economic situation in this part of the African continent has not been conducive
to conventional hydrological data collection. Moreover, these problems are enor-
mously exacerbated by the consequences of anthropogenic and climatic changes.
To overcome the gap in shortage of data from conventional hydrometric networks,
numerous researchers in the field [11–14] have investigated the application of remote
sensing-based information.
The use of global climate reanalysis datasets for modelling streamflow has shown
that the effectiveness of the model depends on the source and resolution of the
input datasets and climate of the region of interest. For example, the CFSR of the
National Centres for Environmental Prediction (USA) and ERA-Interim were used
to model daily and monthly streamflows using the SWAT model of a river basin,
located in the Sudan-Sahel region [15]. They found that the ERA-Interim datasets
generated better results compared to the former. Similarly, the use of SWAT and
MIKE 11-NAM in the conditions of South Africa [16] and Eritrea [11] showed low
statistical representativeness between precipitation data from the CFSR and field
rainfall measurements, as well as overall water imbalance. An assessment of the
applicability of CSFR for modelling hydrological processes within the boundaries
of five river basins with different hydrological and climatic conditions in Ethiopia

1WMO and GWP, Integrated Drought Management Programme Handbook of Drought Indicators
and Indices, no. 1173. 2016.
Physically-Based Streamflow Predictions in Ungauged Basin … 551

and the United States was carried out [13]. They found that the use of input variables
from the CFSR provides modelling of streamflow as good as the outputs from the
use of model inputs from ground-based weather stations. Thus, despite the fact that
conventional in-situ hydrometric data remain the most accurate and reliable sources
of input information, the use of reanalysis datasets as alternative source for modelling
runoff in ungauged or poorly gauged river basins has been proposed. One of the most
sophisticated and widely used models that make use of reanalysis datasets is SWAT
model. It is “a conceptual, continuous-time model developed to assist water resource
managers in assessing water supplies and non-point source pollution on watersheds
and large river basins” [17] and operates at a daily time step. The SWAT model has
got worldwide recognition. For example, SWAT and global climate models were
used to study the formation of streamflow in Russia [18], the United States [19], the
hydrological situation of Africa [20], including the impact of climate change on the
availability of fresh water on the African continent [21]. But, certain shortcomings
of SWAT model were noted [22], especially in terms of comparing the simulation
results with long-term in-situ data on daily runoff and/or discharge of pollutants.
The SWAT model is complex with semi-distributed parameters, so its use requires
a large amount of input data, which makes it difficult to parameterize and calibrate.
For the SWAT model, special computational algorithms were created, which are based
on the method of multidimensional mathematical optimization, including the SWAT-
CUP software module [23, 24]. SWAT-CUP is designed for the purpose of auto-
calibration and uncertainty analysis for the SWAT model and combines five different
optimization algorithms: sequential analysis of all possible sources of uncertainty
SUFI-2, the genetic algorithm of swarm intelligence (PSO), as well as methods of
general probabilistic uncertainty estimation (GLUE), parametric solution (ParaSol)
and Monte Carlo with Markov chains (MCMC), which allows us to use various
objective functions and criteria. The advantage of SWAT-CUP is that it combines
several calibration and uncertainty analysis procedures into a single interface, making
the model calibration procedure more understandable and faster.2 Despite the fact
that the SUFI-2 algorithm was quite effective for large-scale models, the equifinality
problem is still one of the most acute for the calibration of the parameters of the
hydrological model [25].
River basins in Eritrea are characterised by the spatial and temporal variability of
climate and geophysical characteristics, landuse and climate changes. The majority
of river basins do not have regular observation network, or characterized by a lack
of high-quality field data. Under such circumstances, the development of models
and schemes for water management planning remains to be a complex task [26]. A
recent survey shows that the management, collection and processing of hydrometric
networks at national level are declining. In the contrary, there are a lot of ongoing
nation-wide water resources related development projects [26, 27], including the
construction of reservoirs, diversion structures, expansion of agriculture and settle-
ments. Thus, given the lack of high-quality field observational data, on the one hand,

2SWAT-CUP 2012: SWAT Calibration and Uncertainty Programs - A User Manual,” Sci. Technol.,
2014.
552 D. Kozlov and A. Ghebrehiwot

and the ongoing intensive water management activities, on the other, the applica-
bility of satellite-based climate data becomes a timely, important and urgent task in
the region. As such, we recently evaluated the applicability of the conceptual model
“precipitation-runoff” (MIKE 11-NAM) for streamflow simulations from the Mereb-
Gash river basin using CFSR [11]. Nevertheless, the results failed to satisfy the model
acceptance criterion and accordingly three tasks were suggested as future courses of
action: the transformation of CFSR information into a more realistic one; the evalua-
tion of reanalysis data from other sources at different time scales and resolutions; and
the study of the effectiveness of other software systems. Therefore, the objectives of
the study were as follows: (i) to use the SWAT to establish a hydrological model of
the Debarwa subbasin in the upper reaches of the Mereb-Gash River basin with a
monthly estimated time interval; and (ii) to evaluate the effectiveness of the SWAT
in comparison with MIKE 11-NAM for the purposes of modelling streamflow in the
conditions of the specified subbasin.

2 Materials and Methods

The major work of the study required physically-based SWAT and MIKE 11-NAM
models establishment in order to evaluate the various water balance components at
subbasin level and monthly time intervals. The latter model’s setup, details of the
procedures and working principles can be referred to the authors’ recent publication
[11]. Thus, the discussion of this section mainly focused on the SWAT model. These
models were predominantly established using freely available data. Other than the
topographic, soil and landuse data, SWAT requires climatic data at daily or sub-
daily time steps. Major input data for SWAT include digital elevation model (DEM),
landuse, soil properties, and daily weather data. These data were complemented
by additional sources, provided by the Ministry of Land, Water and Environment,
Department of Water Resources, Eritrea. Finally, the findings of both models were
evaluated and intercompared using different statistical evaluation techniques, which
is presented in the ensuing section. Generally, the modelling procedures include
model setup, calibration, uncertainty analysis, sensitivity analysis, validation, and
analyses such as climate change, best management practices, risk analysis, etc.
Debarwa is a small watershed in the upper reaches of Mereb-Gash basin. The
location, landuse and other hydrologic features of the subbasin are depicted in Fig. 1.
The total area of the catchment is approximately 200 km2 , with an altitudinal range
from 1905 to 2550 m above msl. It is a mountainous (50% of area has a slope greater
than 10%) covered by sparse shrubs and agriculture. The soil type in the area is
dominated by Eutric Nitosols of clay soils, followed by Humic Cambisols of clay-
loam. Both soil categories fall under the third hydrologic group (C); that is, the soils
have a slow infiltration and water transmission rates when thoroughly wetted, as well
as a layer that impedes downward movement of water or have moderately fine to fine
texture. The Debarwa watershed lies in moist highlands zone where temperature
varies from 0 °C to 32 °C and an average annual rainfall of 547 mm. Climate in
Physically-Based Streamflow Predictions in Ungauged Basin … 553

Fig. 1 Location and landuse maps of the study area

the catchment can be characterized as moderate with December-January being the

coldest and March–April the hottest. Maximum precipitation occurs in the summer
season, specifically in the months of July and August with a monthly mean rainfall
of 185 mm and 175 mm, respectively. The watershed has one global weather station
and one flow measuring station at the outlet.
In the SWAT program, a watershed is divided into multiple subbasins, which are
then further subdivided into hydrologic response units (HRUs). HRUs are defined as
“lands with similar spatial datasets, namely topography, landuse, and soil types and all
the components of the soil water balance could be determined on an HRU basis, with
the assumption that similar HRUs would have similar hydrologic characteristics”
[24]. Simulation of watershed hydrology is done in the land phase, which controls
the amount of water, sediment, nutrient, and pesticide loadings to the main channel in
each subbasin, and in the routing phase, which is the movement of water, sediments,
etc., through the streams of the subbasins to the outlets. Besides, the soil processes
include lateral flow from the soil, return flow from shallow aquifers, and tile drainage,
which transfers water to the river; shallow aquifer recharge, and capillary rise from
shallow aquifer into the root zone, and finally deep aquifer recharge, which removes
water from the system. The climate-driven hydrological cycle provides moisture and
energy inputs. In this regard, global CFSR data from the National Centre for Atmo-
spheric Research (USA) was utilized. The QSWAT 2012 interface was used to set up
and parameterize the model. On the basis of DEM and the stream network, a threshold
drainage area of 3 km2 was chosen to discretize the watershed into 13 subbasins,
which were further subdivided into 61 HRUs based on soil, landuse, and slope. Each
HRU is normally thought to be a uniform unit where water balance calculations are
made. A schematic representation of the model setup is shown in Fig. 2. The climate
554 D. Kozlov and A. Ghebrehiwot

Fig. 2 Schematic representation of the Debarwa subbasin

variables based on the CFSR as well as data on daily and monthly water consump-
tion in the catchment area were entered into the model as input information. For the
estimated time period of the simulation, the interval from 1994 to 2010 was consid-
ered. Approximately two–thirds of the data was used for calibration and the rest for
validation. The initial and final runs were performed using SUFI-2. The calculations
didn’t consider point and distributed sources of pollution, bottom sediments, nitrogen
and phosphorus loads, reservoir regulations, and the spatial variability of some other
parameters.
The initial selection of parameters depends on the behaviour of the initial model
result before any calibration. The SWAT-CUP program has the provision of ten inde-
pendently evaluated objective functions and an additional multi-objective function—
a combination of two or more objective functions. As has been clearly articulated
in various literatures [23, 25], the outputs corresponding to each objective function
are normally unique, leading to the conditionality of objective functions. As such,
multi-objective function has been suggested to overcome the problem of condition-
ality. On the other hand, model uncertainty could be minimized if and only if we
clearly identify the sources of uncertainty. Possible sources of uncertainty in hydro-
logic modelling [11, 23, 24, 29] can be categorized as follows: (i) model input data;
Physically-Based Streamflow Predictions in Ungauged Basin … 555

(ii) model assumptions and simplifications; (iii) the science underlying the model;
(iv) stochastic uncertainty also known as variability; and (v) code uncertainty, such
as numerical approximations and undetected software errors. It would be unrealistic
to expect a perfect model performance at the end because of the aforementioned
sources of errors as well as many activities that occur in the watershed.
Successful application of hydrological models largely depends on the calibra-
tion and sensitivity analysis of the parameters [23, 28]. Calibration and validation
procedures are effectively used only with field observations. The information about
the measured daily or monthly streamflow data is important for these procedures.
SUFI-2 in the SWAT-CUP module [30] was employed for calibration and validation
procedures. The SUFI-2 algorithm covers a wide range of parameter uncertainties
at the beginning of calculations, as a result of which the observational data initially
falls into the 95% uncertainty forecast (95PPU-confidence probability). 95PPU is
the interval between 2.5% and 97.5% of the total distribution of the output simu-
lated variable (water flow) obtained using an efficient Latin hypercube sampling
algorithm, excluding 5% of the worst simulations [30]. Then, with each iterative
step, the uncertainty interval narrows, and simultaneously two indices are checked
that determine the degree of agreement and uncertainty of the model: the P-factor
(the percentage of measurement results that fall into the 95PPU), ranging from 0
to 1, and the R-factor (the ratio of the average width of the 95PPU interval to the
standard deviation of the corresponding measured value). In an ideal situation, when
the simulation results are exactly (100%) consistent with the observational data, the
P-factor is 1. A P-factor value of 0.70 or higher is considered sufficient for the results
of streamflow modelling. The P-factor and R-factor of 1 are iterations that exactly
match the measurement results. The desired value of the R-factor, determined by
Eq. (1), is considered as acceptable if its value is less than 1.50 [30].
n j  
1
nj ti =1 xsti ,97.5% − xsti ,2.5%
R − f actor j = (1)
σoj

where xsti ,97.5% and xsti ,2.5% are the upper and lower boundary of the 95PPU at time-step
t and simulation i, nj —the number of data points, and σ oj —the standard deviation
of the jth observed variable.
As has been discussed, the SUFI-2 optimization algorithm allows the use of
various objective functions, out of which the Nash and Sutcliffe efficiency (NS)
was used (NS = 1.0 being optimal value and 0.75 < NS ≤ 1 being acceptable). In
addition, the coefficient of determination (0.70 < R2 < 1.0), modified coefficient of
determination (bR2 ), per cent bias (PBIAS < ± 25), and ratio of the root mean squared
error to the standard deviation of measured data (RSR ≤ 0.6), whose corresponding
equations are represented by Eqs. (2–6), were also additional criteria for statistical
model evaluations.
556 D. Kozlov and A. Ghebrehiwot

   2
Q m,i − Q m Q s,i − Q s
i
R = 
2
2   2 (2)
i Q m,i − Q m i Q s,i − Q s
 
|b|R 2 i f |b| ≤ 1
bR =2
(3)
|b|−1 R 2 i f |b| > 1

(Q m − Q s )i2
N S = 1 −  i 2 (4)
i Q m,i − Q m
n
(Q m − Q s )i
P B I AS = 100 × i n (5)
i Q m,i

n n  2
RS R = (Q m − Q s )i2 / Q m,i − Q m (6)
i=1 i=1

where Q—a variable (e.g., discharge); m and s—stand for observed and simu-
lated variables; b—slope of the regression line between the observed and simulated
variables; and i—the ith observed or simulated data.

3 Results and Discussion

Rainfall and corresponding simulated daily streamflow from SWAT model prior to
calibration in SWAT-CUP program, as well as observed streamflow at the outlet
of the watershed was analysed and evaluated. As such, absolute overlapping in the
seasonality of rainfall and corresponding simulated and observed streamflows were
noticed; a large amount of rainfall produced high flows and vice versa. However, a
considerable quantitative mismatch between the simulated and observed streamflows
(R2 = 0.10) was realized at this stage. This disparity was in fact a signal that our
calibration may not yield a perfect fit by all means possible.
During parameterization process, SWAT-CUP provides two different methods of
sensitivity analysis: one-at-a-time and global. In this study, the latter method was
applied, where all selected parameters change at a time and uses multi-regression
computation. The SUFI-2 program permits up to 1000 iterations for one complete
iterative run. The global sensitivity uses the P-value and t-stats for analysing the sensi-
tivity of selected parameters to prioritize them; large t-stat and lower P-value indicate
higher parameter sensitivity and vice versa. The study area is a watershed character-
ized as ungauged or poorly gauged with a limited in-situ hydrometric data. Besides,
SWAT contains a large number of variable parameters involved in the calibration
process. In such conditions, calibration of all parameters causes great difficulties.
Therefore, first we need to select the most significant parameters, which are thought
to represent the hydrological processes, for the calibration procedure. To this end,
the sensitivity analysis of randomly selected 15 parameters was carried out within
Physically-Based Streamflow Predictions in Ungauged Basin … 557

the SUFI-2 procedure (Table 1), out of which those that have the greatest influence
on the formation of streamflow in the study area are identified. After a series of
tests in SWAT-CUP, it was found that the top most sensitive parameters include CN,
SHALLST, and RCHRG_DP.
Considering the dynamics and radical uncertainty of daily flows, calibration was
limited to monthly flows. Accordingly, the performance of the best parameter sets
chosen during the sensitivity analysis was evaluated by two statistical evaluations:

Table 1 Parameter sensitivity analysis and calibrated monthly streamflow values

Parameter Description Sensitivity Simulated values
t-stat P-value Fitted Min Max
CN2* Soil conservation service −32.53 0.00 −0.29 −0.30 0.10
(SCS) curve number for
moisture condition II
SHALLST Initial depth of water in −7.69 0.00 3308 1000 5000
the shallow aquifer (mm
of H2 O)
RCHRG_DP Deep aquifer percolation −5.62 0.00 0.06 0.00 0.80
fraction
ALPHA_BF Baseflow alpha factor −3.44 0.00 0.30 0.00 0.50
(1/days)
EPCO Plant uptake −1.83 0.07 0.90 0.30 1.00
compensation factor
CH_N2 Manning’s “n” value for −1.30 0.20 0.24 −0.01 0.30
the main channel
SURLAG Surface runoff lag −0.21 0.84 21.46 6.00 24.00
coefficient
REVAPMN Threshold depth of water 0.69 0.49 355.6 0.00 400
in the shallow aquifer for
“revap” or percolation to
the deep aquifer to occur
(mm of H2 O)
OV_N.hru Manning’s “n” value for 0.82 0.41 0.54 0.10 1.00
overland flow
GW_DELAY Groundwater delay time 1.49 0.14 468.85 150 500
(days)
CH_K2 Effective hydraulic 1.55 0.12 190.30 100 400
conductivity in main
channel alluvium
(mm/hour)
FFCB Initial soil water storage 2.22 0.03 0.38 0.20 1.00
expressed as a fraction of
field capacity water
content
(continued)
558 D. Kozlov and A. Ghebrehiwot

Table 1 (continued)
Parameter Description Sensitivity Simulated values
t-stat P-value Fitted Min Max
ESCO Soil evaporation 2.40 0.02 0.25 0.20 1.00
compensation factor
GW_REVAP Groundwater “revap” 4.73 0.00 0.09 0.02 0.20
coefficient
GWQMN Threshold depth of water 6.58 0.00 4982.50 1500 5000
in the shallow aquifer
required for return flow to
occur (mm of H2 O)
* The change is relative whereas the change in all other parameters is replacement with other value

(i) model prediction uncertainty and (ii) model performance evaluation. Uncertainty
analysis refers to the propagation of all model input uncertainties to model outputs,
which stem from the lack of knowledge of physical model inputs to model parameters
and model structure. Identification of all acceptable model solutions in the face of all
input uncertainties can provide us with model uncertainty in SWAT-CUP as 95PPU.
Once the model is parameterized and the ranges are assigned, the model is normally
run some 300–1000 times [23]. After all simulations are completed, the provision
of post-processing option in SWAT-CUP calculates the objective function and the
95PPU for all observed variables in the objective function. The prediction uncertainty,
which is represented by the shaded regions for the calibration (Fig. 3) and validation
(Fig. 4) processes, is expressed by the 95PPU in SUFI-2. As a result, P-factor values
were estimated to be 0.34 and 0.43 for calibration and validation, respectively (Table
2). In other words, only 34% and 43% of the observed streamflows are bounded by
the 95PPU during calibration (1997–2001 and 2007–2010) and validation periods
(2002–2006), respectively. On the other hand, the R-factor values are also equal
to 2.56 and 3.48 for calibration and validation periods, respectively (Table 2). The

Fig. 3 Comparison of observed and simulated monthly streamflows during calibration period
(1997–2001 and 2007–2010)
Physically-Based Streamflow Predictions in Ungauged Basin … 559

Fig. 4 Comparison of observed and simulated monthly streamflows during validation period (2002–
2006)

Table 2 Summary of statistics for calibration and validation procedures

Process Uncertainty prediction Objective function
P-factor R-factor R2 bR2 NS PBIAS RSR
Calibration 0.34 2.56 0.80 0.79 0.73 −42.0 0.52
Validation 0.43 3.48 0.32 0.18 0.12 −9.8 0.94

calibrated and validated values of P-factor and R-factor are clearly outside of the
recommended ranges [30], i.e., P-factor > 0.70 and R-factor < 1.50.
Five model performance indicators were employed, out of which NS was used
as the major objective function as has been described above. The other four perfor-
mance indices include R2 , bR2 , PBIAS, and RSR. Results as tabulated in Table 2
clearly show that all the performance indicators for the calibration period (R2 , bR2 ,
and NS > 0.70, and RSR < 0.60) are in fairly acceptable ranges. In other words, the
statistical indices indicate that there is a good agreement between the observed and
simulated streamflows. On the contrary, the corresponding model performance indi-
cators for validation (R2 and bR2 < 0.40, NS < 0.50, and RSR > 0.70) are evaluated
as unsatisfactory. PBIAS measures the average tendency of the simulated data to be
larger or smaller than their observed counterparts. Positive values represent model
underestimation bias and negative values indicate model overestimation bias [31].
So, PBIAS values-based model performance during calibration could be evaluated as
unsatisfactory (PBIAS > ± 25), whereas that of validation is evaluated as acceptable
(PBIAS < ± 10). PBIAS-values show model overestimation by 42% and 9.8% during
calibration and validation, respectively.
To understand the issue of conditionality, an investigation on the effect of objective
function choice on the model performance was explored by running SUFI-2 post-
processing alone. This procedure does not require the running of the SWAT model
again. Accordingly, three objective functions were tested, namely NS, PBIAS and R2
against other indicators. The graphical visualization (Fig. 5) and model performance
indicators (Table 3) clearly illustrate how the choice of objective function affects the
560 D. Kozlov and A. Ghebrehiwot

Fig. 5 Effect of objective function selection on calibration solutions

Table 3 Effect of choice of objective function on calibration solutions

Type of function Uncertainty prediction Objective function
P-factor R-factor R2 NS bR2 PBIAS RSR
NS 0.21 2.03 0.71 -1.41 0.44 -225.4 1.55
PBIAS 0.05 0.48 0.76 -3.14 0.33 -185.39 -0.58
R2 0.05 0.35 0.76 -3.31 0.32 -193.37 -0.61

calibration solution. While each objective function produced unique solutions, which
was also reported by many researchers [23, 25], overestimation of simulated flows,
especially peak flow and baseflow, could be clearly detected in all of the outputs in
this particular case.
In the preceding section, we realized that overall performance of the SWAT model,
verified by the use of statistical evaluations, was unsatisfactory. Unsatisfactory perfor-
mance of the SWAT model was specifically magnified during the analysis of model
prediction uncertainty in calibration and validation processes (Table 2). At this stage,
it was necessary to think of possible sources of errors and uncertainties. Accordingly,
we arrived at the conclusion that errors due to input climate data (e.g., precipitation)
had considerable influence on the unacceptable model outputs. Because, considerable
overestimation of the CFSR-based precipitation as compared to field observations
had been reported in the authors’ recent works [10, 11]. This situation directed us to
compare the outputs from physically process-based distributed SWAT with a semi-
distributed MIKE 11-NAM so as to come up with a model with relatively better
performance. While the former is discussed in the preceding sections, the latter’s
analyses are briefly discussed in the ensuing paragraph.
Physically-Based Streamflow Predictions in Ungauged Basin … 561

MIKE 11-NAM model has less number (9) of basic parameters than that of SWAT
model. The list of these parameters, their descriptions, lower and upper limits and
fitted values during calibration are presented in Table 4. The fitted values are the
optimal values that were obtained through iterative process and manual and automatic
calibrations. Having seen these values, we were able to realize that some of them are
far beyond our realistic expectations (e.g., runoff coefficient, baseflow, etc.). Because,
Debarwa catchment is characterised by mountainous, low infiltration rate as a result
of poor soil conditions and vegetation cover. Besides, it remains dry for much of
the year due to its ephemeral nature. During rainy days, the watershed experiences
flash floods [26] with short durations of flows (time to peak, time base, time lag) and
lower or almost zero baseflows. Thus, a runoff coefficient of 0.10 and high values of
baseflows, in some cases, are deemed to be quite irrelevant. At this point in time, it
is very difficult to verify the other fitted values owing to the absence of field data.
The intercomparison between simulated monthly streamflows of SWAT and
MIKE 11-NAM models, as well as observed flows for calibration (Fig. 6 and Fig. 7)
and validation (Fig. 8), respectively, were analysed. Moreover, these outputs were
evaluated using various objective functions whose values for calibration and valida-
tion are summarized in Tables 5 and 6, respectively. All of the performance indicators
discernibly show that MIKE 11-NAM is far less satisfactory; the statistical indictors

Table 4 MIKE 11-NAM model basic parameters for calibration and validation procedures
Parameter Unit Description Fitted Lower bound Upper bound
U max mm Upper limit of the amount of water in 20 10 20
the surface storage, representing
interception, depression, and surface
storages
L max mm Maximum water content in the lower 300 100 300
zone storage, representing the soil
moisture below the surface from which
plants draw water for transpiration
CQOF – Overland flow runoff coefficient that 0.10 0.10 1.00
determines the distribution of excess
rainfall into overland flow and
infiltration
CK IF hour Time constant for interflow from the 967.46 500 1000
surface storage
CK 12 hour Time constant for overland flow and 44.13 10 50
interflow routing, routed through two
linear reservoirs in series
TOF – Threshold values for overland flow 0 0 0.99
TIF – Threshold values for interflow 0 0 0.99
TG – Threshold values for groundwater 0.98 0 0.99
recharge
CK BF hour Time constant for routing baseflow 4000 1000 4000
562 D. Kozlov and A. Ghebrehiwot

Fig. 6 Comparison of observed and simulated monthly streamflows during calibration

Fig. 7 Correlation between observed and simulated monthly streamflows during calibration: SWAT
(left) and MIKE 11-NAM (right)

Fig. 8 Comparison of observed and simulated monthly streamflows during validation

Table 5 Performance evaluation of selected models during calibration period

Type of model Objective function
R2 bR2 NS RSR PBIAS
SWAT 0.80 0.77 0.73 0.52 -42
MIKE 11-NAM 0.20 0.19 -1.62 1.61 188
Physically-Based Streamflow Predictions in Ungauged Basin … 563

Table 6 Performance evaluation of selected models during validation

Type of model Objective function
R2 bR2 NS RSR PBIAS
SWAT 0.32 0.22 0.12 0.93 9.80
MIKE 11-NAM 0.17 0.05 -0.01 1.00 -64.10

are less than the allowable ranges and the visual graphical comparisons of observed
and simulated do not fairly coincide. In addition, Fig. 7 shows a better correlation
between observed and simulated streamflows in SWAT (R2 = 0.80) than MIKE 11-
NAM (R2 = 0.20). Therefore, based on the statistical evaluations and visual graphical
comparisons, it is fair to say that the SWAT model, without forgetting the issue of
uncertainty as has been described above, strikingly outperformed MIKE 11-NAM
during calibration and validation procedures.
Physically-based models play an important role in obtaining hydrological and
biogeochemical information in catchments that are not sufficiently studied from
a hydrological point of view in arid and semi-arid regions. While some models
are complex others are fairly simple. The former types of models normally require
significant amounts of reference information and have a large number of parameters,
whereas the latter require less reference information and have fewer parameters. The
effectiveness and suitability of physically-based models for hydrological predictions
in ungauged and/or poorly gauged river basins depends on numerous factors such as
data availability and computational facility, knowledge and experience of the user,
the type of the problem, and economics. It is understandable that a given approach
will seldom satisfy all of these requirements, and consequently one approach will
seldom be uniformly better than the other under all circumstances. Each model,
regardless of its complexity, has its own strengths and weaknesses. A choice among
approaches depends on their systematic evaluations, which, in turn, entails construc-
tion of an objective function, use of goodness-of-fit criterion, sensitivity analysis,
error analysis, and comparison and ranking.
In view of the above facts, physically-based models with semi-distributed and
lumped parameters, namely SWAT and MIKE 11-NAM, which are widely used for
hydrological response predictions in arid and semi-arid regions, were studied. As
noted earlier, to overcome the limitation of reference information, the technology
of using satellite climate reanalysis datasets (e.g., CFSR) has drawn the attention
of researchers in the field. However, these applications are mainly constrained by
lack of in-situ data for calibration and validation procedures and significant amounts
of model uncertainty. Thus, cautious application of reanalysis datasets has been
suggested. SWAT model, which uses reanalysis datasets as well as other databases,
which are available in the public domain as driving inputs without any modifications,
was employed. To ascertain the model efficiency and identify models with acceptable
uncertainty, it was necessary to intercompare with other models, out of which MIKE
11-NAM was selected. In this respect, based on the performance evaluations of
both models, promising results have been achieved. However, the current approach
564 D. Kozlov and A. Ghebrehiwot

requires additional endeavours and verifications that ensure the required level of
certainty is attained. In this regard, some possible insights have been proposed.
Sensitivity analysis shows the portion of parameters in the model output uncer-
tainties. More sensitive parameters have a higher share of model uncertainties than
less sensitive ones in the model output if that parameter is left uncalibrated. There-
fore, sensitivity analysis is the first step that should be taken into consideration
in model calibration. However, not all sensitive parameters may be calibrated in
ungauged catchments. In this study, there were no measured parameters and hence,
it is recommended that further efforts should be made to use all available data sources
of the catchment under study. This helps to exclude less sensitive parameters from
calibration and avoid unnecessary and arbitrary adjustments of parameters. Gener-
ally, the SWAT model uncertainty, represented by P-factor and R-factor, were found
to be outside of the acceptable limits for calibration and validation periods. Thus,
other approaches that intend to make CFSR and other reanalysis datasets suitable for
hydrologic and environmental investigations in the region need to be investigated.

4 Conclusion

The choice of a particular method of hydrologic investigation depends on various

requirements such as data availability, approach, economic, and the like. It is conceiv-
able that a given approach will seldom satisfy all of these requirements. This fact
was substantiated by our recent work [25] on simulation of streamflows from Mereb-
Gash river basin in Eritrea using MIKE 11-NAM model and reanalysis datasets as
model inputs. As such, a comparable research was necessary in order to identify the
most effective and suitable models for the conditions of the region under considera-
tion. SWAT and MIKE 11-NAM physically-based were applied to simulate stream-
flows from a Debarwa subbasin within the Mereb-Gash river basin. The reason for
focusing at subbasin level was to reduce the accumulated errors that we had expe-
rienced when the larger river basin was considered. Findings indicated that SWAT
relatively outperformed MIKE 11-NAM in terms of overall model efficiency. Never-
theless, while most of the objective functions proved the acceptable efficiency of the
former model, it also reflected a lot of uncertainties during calibration and valida-
tion procedures. Yet the uncertainties due to the use of SWAT model were greatly
decimated as compared to that of MIKE 11-NAM. Among the different sources of
model errors, we believe, errors due to forcing data are highly likely to be accounted
for lower performances. However, this does not mean that the plausible scenario that
the model’s performance could be influenced by other sources of errors is totally out
of consideration.
Even though reanalysis datasets have apparently great advantage over in-situ
observations in terms of their simplicity, the findings from this study underscored
the need for critical re-examination of the former. In this respect, we would like to
Physically-Based Streamflow Predictions in Ungauged Basin … 565

suggest the following approaches. Firstly, to minimize model uncertainty and there-
upon improve its performance, ground data collection systems need to be strength-
ened as much as possible. Secondly, further investigation on the applicability of
CFSR datasets to simulate streamflows shall be carried out in the near future; for
example, downscaling or upscaling of the forcing datasets, depending on the overall
situation of projects, would be a possible option in this direction. This could be done
with the help of local hydrometric information, for example, long-term annual rain-
fall. Otherwise, using the CFSR datasets without any modifications are likely to end
up in erroneous predictions in semi-arid regions. Finally, an intercomparison of the
currently addressed models and other models, irrespective of their complexity, are
suggested as future course of work.

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Author Index

A E
Abdyldaeva, Aigul. See 495 Emel’yanov, Oleg. See 239
Akhtyamova, Leysan. See 459
Akuletskii, Aleksander. See 393
Aleksandrova, Olga. See 29 F
Aleroeva, Hedi. See 473 Fedorova, Natalia. See 151
Aleroev, Temirkhan. See 473 Fedorov, Sergey. See 151
Andreev, Alexander. See 473 Frishter, Lyudmila. See 189
Andreev, Vladimir. See 177. See 459
Anisimov, Sergey. See 53. See 111
G
Gadzhibekov, Tagibek. See 339
B Galyautdinov, Daud. See 303
Baker, Rowan. See 525 Galyautdinov, Zaur. See 303
Belov, Nikolai. See 313 Ghebrehiwot, Anghesom. See 549
Benavent-Climent, Amadeo. See 135 Gnedina, Lyubov. See 347
Bichaev, Maxim. See 73 Gogin, Alexander. See 259
Bobyleva, Tatiana. See 3. See 13 Golovin, Viktor Leontievich. See 487
Boronbaev, Erkin. See 495 Govorova, Zhanna. See 505
Boytemirov, Muhammadbobir. See 207 Govorov, Oleg. See 505
Bulgakov, Boris. See 29
Bunkov, Victor. See 99
H
Holmatov, Kamoliddin. See 495
C
Cheremnykh, Stepan. See 199
I
Ibragimov, Alexander. See 347
D
Danilov, Alexandre. See 281
Dias, Daniel. See 433 K
Dobshits, Lev. See 53. See 111 Kalugin, Ivan. See 281
Donayev, Burkhon. See 403 Kantarzhi, Izmail. See 259
Dorzhieva, Elizaveta. See 45 Khasambiev, Mohammad. See 473
Dudchenko, Aleksandr. See 433 Khodjabekov, Muradjon. See 217
Dusmatov, Olimjon. See 217 Khudainazarov, Sherzod. See 359. See 403
© The Editor(s) (if applicable) and The Author(s), under exclusive license 567
to Springer Nature Switzerland AG 2022
P. Akimov and N. Vatin (eds.), Proceedings of FORM 2021, Lecture Notes in Civil
Engineering 170, https://doi.org/10.1007/978-3-030-79983-0
568 Author Index

Kiselyov, Fyodor. See 329 S

Kitaytseva, Elena. See 513 Safargaliev, Ruslan. See 525
Kovtonyuk, Pavel. See 135 Safina, Galina. See 21
Kozlov, Dmitry. See 549 Sainov, Mikhail. See 383
Kravchenko, Galina. See 321 Savin, Sergey. See 151
Kumpyak, Oleg. See 303 Savintceva, Marina. See 313
Kuzmina, Liudmila. See 539 Semenov, Vyacheslav. See 45
Kuznetsov, Sergey. See 433 Sergeyev, Filipp. See 329
Shamaev, Alexey. See 3. See 13
Shchurov, Evgeniy. See 271
L Sidorov, Vitalii. See 229
Lam Van, Tang. See 85 Silman, Yulia. See 99
Le, Anh Tuan. See 421 Smirnov, Aleksandr. See 111
Leshkanov, Andrei. See 53 Sobolev, Evgeny. See 229
Sodomon, Marc. See 45
Stepina, Irina. See 45
M
Maksimov, Lev. See 525
Maksimov, Mikhail. See 177 T
Maksimova, Svetlana. See 525 Ter-Martirosyan, Armen. See 163
Makzhanova, Yana. See 289 Ter-Martirosyan, Zaven. See 393
Mavlanov, Tulkin. See 359 Tien, Dao Ngo.c. See 249
Medvedev, Vyacheslav. See 65 Tikhonov, Sergey. See 447
Medyankin, Michael. See 151 Titova, Irina. See 45
Minh, Nguyen Dac Binh. See 85 Tkach, Evgeniya. See 73
Mironov, Viktor. See 525 Tolstikov, Viktor. See 371
Mirsaidov, Mirziyod. See 207. See 217 Tran, Trung Duc. See 421
Miryuk, Olga. See 123 Trufanova, Elena. See 321
Tusnin, Alexander. See 271

N
Nemirovskii, Yury. See 447 U
Ngan Van, Pham. See 85 Umarova, Feruza. See 359
Unaspekov, Berikbay. See 495
Useinov, Emil. See 313
O Ustinov, Artem. See 99
Osipov, Yuri. See 539
Othman, Ahmad. See 163
V
Varankina, Darya. See 135
P Volkova, Vladislava Nikolaevna. See 487
Plevkov, Vasilii. See 99 Voloskova, Irina. See 135
Plyaskin, Andrei. See 99. See 313 Vu, Dinh Huong. See 421
Pudanova, Lyubov. See 321
Y
Q Yazyev, Serdar. See 459
Quang, Nguyen Duc Vinh. See 29 Youssef, Yara Waheeb. See 371
Quyen, Vu Thi Bich. See 249 Yugov, Nikolai. See 313
Yuldashev, Faxriddin. See 207
Yurkin, Yuriy. See 135
R
Rakhimova, Tatiana. See 303
Rosinsky, Stanislav. See 347 Z
Rudich, Uliana. See 505 Zhyrgalbaeva, Nurbubu. See 495
Zinoveva, Ekaterina. See 347