applied

sciences
Article
Experimental Investigation of Concrete Transverse
Deformations at Relatively High Loading Rates for
Interpretation of High Strength Concrete Behavior
Iakov Iskhakov , Ilya Frolov * and Yuri Ribakov

Department of Civil Engineering, Ariel University, Ariel 40700, Israel; yizhak@ariel.ac.il (I.I.);
ribakov@ariel.ac.il (Y.R.)
* Correspondence: ilyafrolov0711@gmail.com

Abstract: Loading rates affect the behavior of concrete specimens from the beginning of the loading
process until failure. At rather high loading rates, longitudinal deformations in concrete specimens
under a compressive load are practically elastic up until the ultimate limit state. It has been previously
demonstrated that transverse deformations effectively indicate high-strength concrete behavior in
the entire static loading process range. A theoretical model for cylindrical concrete specimen failure
under compressive load, based on a structural phenomenon, has also been proposed. The aim of
the present research is experimental verification of using transverse deformations in addition to
longitudinal ones for investigating high-strength concrete behavior at the non-elastic stage. This
research is based on testing normal-strength concrete cylindrical specimens under compression
at relatively high loading rates. The theoretical model of the cracking and failure scheme of the





 cylindrical specimens are experimentally confirmed. The obtained results demonstrate that it is
Citation: Iskhakov, I.; Frolov, I.;
possible to use transverse deformations for the interpretation of initiation and development of
Ribakov, Y. Experimental inelastic deformations in high-strength concrete up to class C90 based on the data for normal-strength
Investigation of Concrete Transverse concrete specimens of class C30 subjected to relatively high loading rates.
Deformations at Relatively High
Loading Rates for Interpretation of Keywords: normal-strength concrete; high-strength concrete; transverse deformations; stress–strain
High Strength Concrete Behavior. relationship; compressed concrete specimen failure scheme
Appl. Sci. 2021, 11, 8460. https://
doi.org/10.3390/app11188460

Academic Editor: Junwon Seo 1. Introduction

The transverse deformation analysis of structures is usually limited to obtaining Pois-
Received: 17 August 2021
son’s ratio, which is only applicable to the range of elastic concrete behavior. At the same
Accepted: 9 September 2021
Published: 12 September 2021
time, for compression, this range can reach half of the concrete strength. Consequently, the
elastic compressed concrete stage is an essential part of concrete behavior investigations [1].
Publisher’s Note: MDPI stays neutral
However, according to modern design codes, the elastic stage is limited to about 40% of
with regard to jurisdictional claims in
its strength [2,3]. Further concrete behavior is characterized by elastic–plastic deforma-
published maps and institutional affil-
tions, which can be graphically represented by a convex square parabola relative to the
iations. deformation’s axis. If instead of a square parabola, a sinusoidal approximation is used, the
difference in potential energy is about 7% [4].
The relationship between stresses (σc ) and strains (ε c ) in compressed concrete is
characterized by the following three parts [1]:
Copyright: © 2021 by the authors. – A linear part from 0 to 0.5 f ck , where f ck is concrete strength;
Licensee MDPI, Basel, Switzerland. – A convex parabolic part bound by f ck ;
This article is an open access article – A descending branch.
distributed under the terms and Deformations range from 0 to 0.5‰, from 0.5 to 2‰, and from 2 to 3.5‰ in the first,
conditions of the Creative Commons second, and third parts, respectively.
Attribution (CC BY) license (https://
This relationship was proposed to describe the behavior of normal-strength concrete
creativecommons.org/licenses/by/
(NSC) up to class C60. High-strength concrete (HSC) represents classes C70–C90 [2] and
4.0/).

Appl. Sci. 2021, 11, 8460. https://doi.org/10.3390/app11188460 https://www.mdpi.com/journal/applsci

Appl. Sci. 2021, 11, 8460 2 of 16

exhibits mainly elastic behavior with weakly expressed elastic–plastic deformations [5].
For the more accurate identification of HSC non-elastic deformations, the use of transverse
deformation graphs of the compressed specimens was proposed [6].
At the same time, an additional important factor affecting compressed concrete be-
havior is the specimen’s loading rate [7–9]. The loading rate used in the present research
was selected according to standard requirements of modern codes. Following [10], it is
recommended to load a concrete specimen at a rate within (0.2 ... 1.0) MPa/s; according
to [11], the recommended rate is (0.25 ± 0.05) MPa/s. However, at such loading rates,
longitudinal deformations, even for NSC, mainly exhibit elastic behavior [7–9].
Twenty-two cylindrical NSC specimens were tested under uniaxial loading [7]. The
specimen diameters were between 10 and 80 cm, and their heights were from 20 to 320 cm
(which corresponds to a real column length). The specimens were loaded at a rate of about
0.1 MPa/s. The obtained σc vs. ε c relationships, when σc was the compressed concrete
stresses and ε c was the longitudinal deformations, were characterized by an almost linear
performance up until specimen failure. Similar results were obtained for uniaxial loaded
15 × 30 cm cylindrical specimens made of steel fibered reinforced concrete with a strength
of 27.6 MPa [8]. In this case, the loading rate was 1.3 mm/min (which corresponds to about
0.1 MPa/s). Comparable results were obtained for twenty-six cylindrical NSC specimens
loaded at a rate of (0.25 ± 0.05) MPa/s [9].
It was shown that in certain cases, particularly in steel-fibered high-strength concrete
(SFHSC), transverse deformations efficiently indicate elastic–plastic concrete behavior,
which is less pronounced when longitudinal deformations are used [6]. A corresponding
theoretical model for the type of transverse cracks and their development as well as
a consequent scheme for specimen failure was proposed. Following this concept, the
transverse crack development process can be divided into three stages:
– Linear specimen behavior;
– Nonlinear specimen behavior;
– Ultimate limit state (ULS).
The first stage continues until the initiation of cracks (ε trans ≤ ε ct ul ), where ε trans is
the transverse deformation value, and ε ct ul is the ultimate elastic transverse deformation.
At this stage, the cylindrical specimen’s cross section is nondamaged and has an original
circular shape. The second stage begins from the initiation of cracks, corresponding
to ε ct ul , and continues until a twofold ultimate elastic transverse deformations value
(ε ct ul < ε trans ≤ 2ε ct ul ) is reached. In this case, it is assumed that the specimen’s cross
section has an inner circular part (unbroken core) and several outer wedge-shaped segments
separated by transverse cracks. The last stage corresponds to transverse deformations that
exceed 2ε ct ul . Similar concrete and mortar cylindrical specimen failure patterns were also
obtained under cyclic uniaxial compression [12].
The present research is focused on the experimental verification of the proposed
theoretical model based on the dependence of transverse deformations on longitudinal
ones. With this aim, cylindrical NSC specimens were tested, and transverse deformations
were analyzed. The specimens were tested at a loading rate of 0.2 MPa/s. Longitudinal
and transverse deformations were measured and recorded simultaneously.
The obtained experimental results were further evaluated from the point of view of
confirming the principles of the structural phenomenon [13] (a brief description of this
phenomenon is given in the following sub-section).

The Essence of the Structural Phenomenon

The structural phenomenon was proposed as a result of an extensive experimental
and theoretical database analysis related to concrete structures [13]. The essence of the
phenomenon is that structural parameters in elastic state increase or decrease twice at
failure. This behavior usually occurs for symmetrical structures under symmetrical loading.
The phenomenon has been proven at the material level and is also valid for reinforced
concrete structures subjected to static and dynamic loads.
Appl. Sci. 2021, 11, 8460 3 of 16

The structural phenomenon is based on two theories: the quasi-isotropic state of a

structure at the ultimate limit state (ULS) [14] and the mini-max principle [15]. Theoretical
and experimental confirmation of the phenomenon is obtained for different structure types,
such as beams, frames, spatial structures, and structural joints. The structural phenomenon
allows additional equation(s) for calculating structural parameters to be obtained. This, in
turn, can lead to the development of efficient methods for structural analysis that allow
using less empirical coefficients in design procedures.

2. Research Aims
A method for analyzing the weakly expressed inelastic behavior of HSC using com-
pressed NSC subjected to relatively high loading rates is investigated in this study. Since
longitudinal inelastic deformations in compressed HSC specimens are practically not mani-
fested, transverse deformations can be used for concrete inelastic behavior analysis [6]. The
value of these deformations is approximately five times less than that of the longitudinal
ones (Poisson’s ratio is approximately equal to 0.2). However, it also does not provide a
good solution for the HSC problem since the initiation of longitudinal cracks caused by
transverse deformations practically coincides with specimen failure.
For this reason, another solution is considered in this paper. As mentioned above,
NSC behaves elastically almost until failure at rather high loading rates, as does HSC.
However, maximal longitudinal deformations in NSC are almost two times higher than
in HSC. At the same time, concrete tensile strength practically does not grow when the
concrete class increases to over C70. Therefore, transverse deformation development in
NSC is much more significant than in HSC, which leads, in turn, to the longitudinal crack
development that determines inelastic concrete behavior in the transverse direction of a
specimen. Thus, NSC subjected to relatively high loading rates can be considered as a
model of HSC behavior at normal loading rates based on transverse deformation analysis.
The main purpose of this research is to investigate transverse deformation devel-
opment in a concrete specimen under axial compression throughout the entire concrete
behavior range until failure. The influence of transverse deformations on the nature of crack
development and the specimen failure pattern is also considered based on the structural
phenomenon [13]. For this purpose, transverse deformation dependence on longitudinal
deformations is investigated throughout the entire loading process, from the beginning of
loading until failure.
The objectives of this study are:
– Detailed analysis of the elastic–plastic behavior of cylindrical NSC specimens and
revealing the advantages of using transverse deformations as one of the main features
of compressed concrete behavior;
– Analysis of the nature of specimen failure based on transverse crack development;
– Verification of a theoretical model of the specimen’s upper surface failure pattern [6]
based on the structural phenomenon;
– Confirming that the unbroken cylindrical core area is twice as small as the initial one;
– Checking the possibility of interpreting HSC behavior using relatively high NSC
specimen loading rates.

3. Experimental Program
3.1. Testing Stages
In the framework of the present study, the first three 15 × 15 × 15 cm cubic specimens
were tested to obtain the concrete strength. Measuring equipment suitability was prelim-
inarily verified by applying step loading and unloading to three 15 × 30 cm cylindrical
specimens. For this reason, ten load steps were selected (each load step was 40 kN). After
that, two test series were conducted to analyze elastic–plastic behavior of 15 × 30 cm
cylindrical specimens under a constant loading rate of 0.2 MPa/s. The first and the second
test series included four and three specimens, marked as Sp1.1, Sp1.2, Sp1.3, Sp1.4 and
Sp2.1, Sp2.2, Sp2.3, respectively.
Appl. Sci. 2021, 11, 8460 4 of 16

3.2. Material Properties

Concrete proportioning was based on the absolute volumes method [16]. The follow-
ing materials were used to prepare the concrete mixture:
– A locally produced composite 42.5N CEM II/B-LL Portland cement [17] with portions
of limestone between 21 and 35%;
– A locally available quarry sand fraction of 0 to 4.75 mm;
– A locally available quarry crushed dolomite limestone coarse aggregate fraction of
19/14 according to SI 3-1998 [18].
Basic properties of the concrete components are presented in Table 1. The final
composition is given in Table 2. The concrete mixture was prepared using a 0.045 m3
rotating pan mixer when the calculated mixture volume was 0.043 m3 . Dried aggregates
were used for all of the mixtures.

Table 1. Concrete component properties.

Components Units Values

Cement density, ρcem kg/m3 2900
Standard cement consistency, Kscc % 27.5
Sand fineness modulus, FM - 1.85
Sand particle density, ρs kg/m3 2644.5
Crushed stone bulk density (oven-dry), ρcsb kg/m3 1270.9
Crushed stone particle density, ρcs kg/m3 2586.7

Table 2. Concrete mixture proportioning.

Components Units Values

Cement, C kg/m3 448.4
Water, W kg/m3 224.2
Water-cement ratio, W/C - 0.5
Sand, S kg/m3 602.2
Crushed stone, CS kg/m3 1017.8
Fresh concrete density in compacted state, ρc kg/m3 2293

3.3. Concrete Mixture and Specimen Preparation

The concrete mixture was prepared according to ASTM C192/C192M [19]. Cubic and
cylindrical specimens were cast. Internal and external vibrations were used to compact
the fresh concrete in cylindrical and cubical molds, respectively. All of the specimens
were removed from the molds 24 h after casting and were cured for 27 days at constant a
23 ± 2 ◦ C temperature and at a relative humidity of 95 ± 5%.

3.4. Motivation for Selecting the Cylindrical Specimens

The cylindrical specimens are obviously more suitable to achieve the research goals
because there is no local stress concentration problem as is the case, for example, in the
corners of the prismatic ones. Additionally, cylindrical specimens are characterized by
uniform deformation development along the entire perimeter of the specimen section.
Thus, the edge effect problem is avoided in the tests.
The ratio between the diameter and the height of the concrete cylindrical specimen
was 1:2 (15:30 cm). This ratio allows the modeling of real full-scale compressed structural
element behavior. Additionally, the chosen ratio guarantees the presence of a relatively
large unrestrained area in the middle of the specimen’s height [20]. In this case, the element
buckling problem is practically prevented. At the same time, due to some inaccuracy in the
transmission of external compressive forces and concrete heterogeneity, additional stresses
associated with small (technological) eccentricities may appear. This problem was solved
by selecting a proper test setup.
 large unrestrained area in the middle of the specimen’s height [20]. In this case, the ele-
ment buckling problem is practically prevented. At the same time, due to some inaccuracy
in the transmission of external compressive forces and concrete heterogeneity, additional
stresses associated with small (technological) eccentricities may appear. This problem was
Appl. Sci. 2021, 11, 8460 5 of 16
solved by selecting a proper test setup.

3.5. Test Setup

TheSetup
3.5. Test measurement system was selected according to the following main behavioral
features of a specimen that
The measurement needed
system was to be obtained:
selected according to the following main behavioral
− Loading
features force; that needed to be obtained:
of a specimen
−
– Loading force;deformations;
Longitudinal
−
– Transverse deformations.
Longitudinal deformations;
– To conductdeformations.
Transverse the axial compression test, the class 1 testing machine (with a maximum
errorToofconduct
1% FS) and the a maximum
axial compressionload oftest,
3000 kN,
the which
class was controlled
1 testing by a PC-unit,
machine (with a maximum was
used.ofThe
error 1%applied
FS) and load was measured
a maximum by ankN,
load of 3000 external
which pressure sensor,bywhich
was controlled had been
a PC-unit, was
installed
used. Theinapplied
the oil line
loadofwas
the measured
testing machine.
by an external pressure sensor, which had been
Fourinlinear
installed the oildisplacement transducers
line of the testing machine.(LDT) with a 50-mm stroke and a high-measure
resolution (< 1 displacement
Four linear μm) were used to measure
transducers the with
(LDT) longitudinal
a 50-mm deformations. The sensors
stroke and a high-measure
were located in the specimen’s central zone in order to exclude the
resolution (< 1 µm) were used to measure the longitudinal deformations. The sensors influence of the were
edge
effect associated
located with thecentral
in the specimen’s frictionzone
forces
in occurring at the contact
order to exclude area between
the influence the effect
of the edge speci-
men and the
associated withtesting machine
the friction plates.
forces The base
occurring of contact
at the the longitudinal deformation
area between measure-
the specimen and
ment
the was 15
testing cm. The
machine upperThe
plates. viewbaseofof
the tested
the specimen
longitudinal and the measurement
deformation measurement devices
was
was
15 given
cm. The in Figure
upper view1. Figure
of the 2tested
shows the location
specimen and of
thethe devices on the
measurement specimen’s
devices surface
was given in
development.
Figure 1. Figure 2 shows the location of the devices on the specimen’s surface development.

Appl. Sci. 2021, 11, 8460 Figure 1.

Figure 1. Measurement
Measurement device
device position
position scheme:
scheme:1.1.Cylindrical
Cylindricalconcrete
concretespecimen;
specimen;2.2.LDT
6 devices;
LDT 3.
ofdevices;
16
Strain-gauge (SG) sensors; A, B, C, and D–sectors.
3. Strain-gauge (SG) sensors; A, B, C, and D–sectors.

Figure 2.
Figure 2. Specimen
Specimensurface development:
surface 1. LDT
development: devices;
1. LDT 2. SG 2.
devices; sensors; A, B, C A,
SG sensors; andB,D–sectors (ac-
C and D–sectors
cording to Figure
(according 1). 1).
to Figure

As mentioned above, there was a technological eccentricity problem caused by inac-

curacy in the applied force transmission. This problem was solved by using four sensors
located on the tested specimen’s opposite sides (see Figure 1). Such sensor distribution
allowed more accurate data to be obtained that revealed the presence of eccentricity in
Appl. Sci. 2021, 11, 8460 6 of 16
Figure 2. Specimen surface development: 1. LDT devices; 2. SG sensors; A, B, C and D–sectors (ac-
cording to Figure 1).

As
Asmentioned
mentionedabove,above,there
therewas
wasaatechnological
technologicaleccentricity
eccentricityproblem
problemcaused
causedby byinac-
inac-
curacy in the applied force transmission. This problem was solved by using
curacy in the applied force transmission. This problem was solved by using four sensors four sensors
located
locatedononthethetested
testedspecimen’s
specimen’s opposite
opposite sides (see
sides Figure
(see 1). Such
Figure sensor
1). Such distribution
sensor al-
distribution
lowed more accurate data to be obtained that revealed the presence of eccentricity
allowed more accurate data to be obtained that revealed the presence of eccentricity in in stress
distribution. The LDT
stress distribution. Thesensors were attached
LDT sensors by originally
were attached developed
by originally mounts,mounts,
developed allowing al-
independent deformation
lowing independent transfer to
deformation each sensor.
transfer to each sensor.
Two
Twostrain-gauges
strain-gauges(SG)(SG)with
withaa6060mmmmbasis,
basis,aameasure
measureresolution
resolutionup uptoto0.1
0.1micro-
micro-
strains,
strains, a resistance of 120 Ohm, and a gauge factor of 2.07 were used to measurethe
a resistance of 120 Ohm, and a gauge factor of 2.07 were used to measure the
transverse
transversedeformations.
deformations.The TheSGSGlength
lengthwaswas chosen
chosenbased
basedon on
thethe
requirement
requirement of exceeding
of exceed-
the
ingcoarse aggregate
the coarse maximum
aggregate maximumdimensions by at least
dimensions by atthree
least times
three [21].
timesThe general
[21]. view
The general
ofview
the test setup is presented in Figure 3.
of the test setup is presented in Figure 3.

Figure3.3.Measurement
Figure Measurementdevice
devicesystem:
system:1.1.LDT
LDTdevices;
devices;2.2.SG
SGsensors;
sensors;3.3.load
loadsensor;
sensor;4.4.data
datalogger;
logger;
5. Wheatstone bridge.
5. Wheatstone bridge.

AAthin
thin layer
layer of
of sifted
siftedfine
finegrain
grainsand
sandwas
wasused
usedto level the the
to level upper surface
upper of theofcylin-
surface the
drical specimens. It also allowed for a reduction in the friction effect between the
cylindrical specimens. It also allowed for a reduction in the friction effect between the testing
machine
testing plate and
machine the
plate specimens.
and the specimens.

4. Experimental Results and Discussion

4.1. Evaluation of Concrete Parameters in the Elastic State
The experimentally obtained dependences between the longitudinal deformations and
the stresses in uniaxially loaded cylindrical concrete specimens, which were tested in the
first test series, are presented in Figure 4. As evident from this figure, the dependences are
close to linear and allow for the disclosure of the concrete’s ultimate elastic deformations
and its modulus of elasticity.
The dependences of the transverse deformations on stresses are shown in Figure 5.
These dependences enable the concrete’s Poisson’s ratio as well as the pattern of inelastic
concrete deformation development (which cannot be done using only longitudinal defor-
mations) to be obtained. It should be mentioned that this possibility becomes available
when the transverse deformations exceed the Poisson’s ratio validity range, i.e., after the
initiation of longitudinal cracks, when ε trans > ε ct ul .
 4.1. Evaluation of Concrete Parameters in the Elastic State
The experimentally obtained dependences between the longitudinal deformations
and the stresses in uniaxially loaded cylindrical concrete specimens, which were tested in
the first test series, are presented in Figure 4. As evident from this figure, the dependences
Appl. Sci. 2021, 11, 8460 7 of 16
are close to linear and allow for the disclosure of the concrete’s ultimate elastic defor-
mations and its modulus of elasticity.

Figure 4. Experimental dependences 𝜎 vs. 𝜀 .

The dependences of the transverse deformations on stresses are shown in Figure 5.

These dependences enable the concrete’s Poisson's ratio as well as the pattern of inelastic
concrete deformation development (which cannot be done using only longitudinal defor-
mations) to be obtained. It should be mentioned that this possibility becomes available
when the transverse deformations exceed the Poisson’s ratio validity range, i.e., after the
initiation
Figure
Figure 4. 4. of longitudinal
Experimental
Experimental cracks, when
dependences
dependences vs.𝜀𝜀ε c. .
𝜎 σcvs. 𝜀 .

The dependences of the transverse deformations on stresses are shown in Figure 5.

These dependences enable the concrete’s Poisson's ratio as well as the pattern of inelastic
concrete deformation development (which cannot be done using only longitudinal defor-
mations) to be obtained. It should be mentioned that this possibility becomes available
when the transverse deformations exceed the Poisson’s ratio validity range, i.e., after the
initiation of longitudinal cracks, when 𝜀 𝜀 .

Figure dependences𝜎σc vs.

Experimentaldependences
Figure5.5.Experimental vs. 𝜀ε trans ..

FollowingFigure
Following Figure5,5,the
thetested
testedspecimens
specimensexhibited
exhibitedelastic
elasticbehavior
behaviorininthe
thetransverse
transverse
direction up until about 70% of the concrete strength (concrete specimen
direction up until about 70% of the concrete strength (concrete specimen strength ranges strength ranges
from 20.9 to 28.2 MPa). For example, the concrete strength of Sp1.1 is equal
from 20.9 to 28.2 MPa). For example, the concrete strength of Sp1.1 is equal to 24 Mpa, and to 24 Mpa, and
the strength value corresponding to the elastic stage reaches 17 MPa, which
the strength value corresponding to the elastic stage reaches 17 MPa, which is about 70% is about 70%
ofofthe
thespecimen
specimenstrength.
strength.Further,
Further,transverse
transversedeformations
deformationsgrowgrowinina anonlinear
nonlinearmanner,
manner,
which
which
Figure indicates
5.indicates thethe
Experimental development
development
dependences of inelastic
𝜎of inelastic
vs. 𝜀 deformations
.deformations that that areevident
are not not evident
in thein the
lon-
gitudinal direction (see Figure 4). The experimental results are summarized in Table 3. 3.
longitudinal direction (see Figure 4). The experimental results are summarized in Table
Following Figure 5, the tested specimens exhibited elastic behavior in the transverse
Table 3. Experimental values of maximum elastic and ultimate transverse concrete strains, modulus
direction up until about 70% of the concrete strength (concrete specimen strength ranges
of elasticity, Ecm , and Poisson’s ratio, ν.
from 20.9 to 28.2 MPa). For example, the concrete strength of Sp1.1 is equal to 24 Mpa, and
the strength
Specimen value corresponding
εc el , ‰ to the elastic
εct ul ,stage
‰ reachesE17 MPa, which is about
ν 70%
cm , MPa
of the specimen strength. Further, transverse deformations grow in a nonlinear manner,
Sp1.1 0.678 0.100 27700 0.15
which indicates
Sp1.2
the development
0.453
of inelastic0.100
deformations that are not evident in0.22
27600
the lon-
gitudinalSp1.3
direction (see Figure
0.556 4). The experimental
0.086 results are summarized
26740 in Table
0.16 3.
Sp1.4 0.605 0.092 32000 0.15
Average 0.573 0.095 28510 0.17

The obtained experimental data correspond to those available in Eurocode 2 [2]. A

comparison of the known Poisson ratios data [6] is presented in Table 4. As evident from
this table, the Poisson’s ratios for all of the tested specimens are close to the known results.
 Average 0.573 0.095 28510 0.17

The obtained experimental data correspond to those available in Eurocode 2 [2]. A

comparison of the known Poisson ratios data [6] is presented in Table 4. As evident from
Appl. Sci. 2021, 11, 8460 this table, the Poisson’s ratios for all of the tested specimens are close to the known results.
8 of 16

Table 4. Poisson’s ratios.

Table 4. Poisson’s ratios. References

Bondarenko and
SI 466 [3] BR [23]
References Eurocode 2 [2] Iskhakov [24]
Suvorkin [22]
SI ...
4660.25
[3] Bondarenko and BR [23] Eurocode
0.15 0.20 [22]
Suvorkin 0.20 0.20 2 [2] Iskhakov
0.15 [24]
... 0.20
0.15 ... 0.25 0.20 0.20 0.20 0.15 ... 0.20
It should be also noted that according to the previously proposed theoretical model
[1], theIt maximum elastic
should be also longitudinal
noted deformations
that according are within
to the previously 0.5‰,
proposed and the ultimate
theoretical model [1],
elastic deformations are equal to 1‰, which is also confirmed
the maximum elastic longitudinal deformations are within 0.5‰, and the by the obtained
ultimateexperi-
elastic
mental data.
deformations are equal to 1‰, which is also confirmed by the obtained experimental data.
The
Thedata
dataobtained
obtainedininthe
thesecond
secondtest
testseries
series show
show the
the similar tendencies in
similar tendencies in the
thebehavior
behav-
ior of the specimens and confirms the previously presented results (Figures
of the specimens and confirms the previously presented results (Figures 6 and 7). 6 and 7).

Appl. Sci. 2021, 11, 8460 9 of 16

Figure
Figure6.6.Experimental dependences𝜎σc vs.
Experimentaldependences vs. 𝜀ε c.

Experimental
Figure7.7.Experimental
Figure dependences𝜎σcvs.
dependences vs. 𝜀ε trans .

4.2. Evaluation of Cylindrical Specimen Behavior in the Inelastic State

4.2. Evaluation of Cylindrical Specimen Behavior in the Inelastic State
Following modern design codes [2,3], Poisson’s ratio determines concrete behavior
Following modern design codes [2,3], Poisson’s ratio determines concrete behavior
in the transverse direction. At the same time, based on the experimental results obtained
in the transverse direction. At the same time, based on the experimental results obtained
in the framework of the present study, it is very important to consider the transverse
in the framework of the present study, it is very important to consider the transverse de-
deformations over the range of the Poisson’s ratios. With this aim, the dependence of the
formations over the range of the Poisson’s ratios. With this aim, the dependence of the
transverse deformations on longitudinal ones was analyzed. The relationship between
transverse deformations on longitudinal ones was analyzed. The relationship between
these deformations for the first test series is shown in Figure 8. As evident from this figure,
these deformations for the first test series is shown in Figure 8. As evident from this figure,
the dependence is nonlinear in the interval when ε trans > ε ct ul (see Table 3).
the dependence is nonlinear in the interval when 𝜀 𝜀 (see Table 3).
 in the transverse direction. At the same time, based on the experimental results obtained
in the framework of the present study, it is very important to consider the transverse de-
formations over the range of the Poisson’s ratios. With this aim, the dependence of the
transverse deformations on longitudinal ones was analyzed. The relationship between
Appl. Sci. 2021, 11, 8460 9 of 16
these deformations for the first test series is shown in Figure 8. As evident from this figure,
the dependence is nonlinear in the interval when 𝜀 𝜀 (see Table 3).

Dependencesofoftransverse
Figure8.8.Dependences
Figure transversestrains
strainson
onlongitudinal
longitudinalones.
ones.

Table55presents
Table presentsthe
thevalues
valuesof ofthe
thePoisson's
Poisson’sratios
ratiosas
aswell
wellas
asthe
therelationship
relationshipbetween
between
the maximum transverse,
the maximum transverse, 𝜀 ε trans1 , and longitudinal deformations, , at ULS.
, and longitudinal deformations, 𝜀 , at ULS. Following
ε c1 Following
/ε c1
this table,
this table, the
theaverage
averageratio between
ratio between these two parameters
these and theand
two parameters Poisson’s ratio, ε trans1
the Poisson's ratio,
ν ,
is about
⁄ 1.8. It should be noted that, in accordance with the structural phenomenon [13],
, is about 1.8. It should be noted that, in accordance with the structural phenom-
this value should be close to 2 [6]. The difference between the obtained experimental and
enon [13], this
theoretical valueisshould
results about be10%,close to 2 [6].
which The difference
is explained between
by concrete the obtained and
heterogeneity experi- the
mental and theoretical results is about 10%, which is
probable technological eccentricity that was described above. explained by concrete heterogeneity
and the probable technological eccentricity that was described above.
Table 5. Relationship between transverse and longitudinal strains at ULS and Poisson’s ratio values.

Specimen ν εtrans1 εtrans1 /εc1

εc1 ν
Sp1.1 0.15 0.28 1.86
Sp1.2 0.22 0.35 1.57
Sp1.3 0.16 0.26 1.63
Sp1.4 0.15 0.32 2.13
Average 0.17 0.30 1.80

The dependence between ε trans /ε c and the longitudinal deformations, ε c , is shown in

Figure 9. This dependence is an additional tool that demonstrates the correspondence of
concrete behavior with the previously developed theoretical model [6]. Following Figure 9,
the value of ε trans /ε c increases very slowly, increasing approximately up to ε c = 0.9 ‰,
and then a sharp increase in this ratio is evident. This confirms that concrete behavior in
this area is inelastic and corresponds to the theoretical model. According to this model, the
ultimate elastic concrete behavior is limited by a longitudinal strain value of 1.0‰, which
correlates with the experimental value of 0.9‰. At the longitudinal crack development
stage caused by transverse deformations, the ratio ε trans /ε c tends to be 0.4‰, which is
approximately twice as high as it is in the elastic stage. This statement corresponds to the
structural phenomenon [13].
 0.9 ‰, and then a sharp increase in this ratio is evident. This confirms that concrete be-
havior in this area is inelastic and corresponds to the theoretical model. According to this
model, the ultimate elastic concrete behavior is limited by a longitudinal strain value of
1.0‰, which correlates with the experimental value of 0.9‰. At the longitudinal crack
development stage caused by transverse deformations, the ratio 𝜀 ⁄𝜀 tends to be
Appl. Sci. 2021, 11, 8460 10 of 16
0.4‰, which is approximately twice as high as it is in the elastic stage. This statement
corresponds to the structural phenomenon [13].

Figure 9. Dependences
Figure between
9. Dependences average
between average ⁄𝜀/ε cand
𝜀 ε trans and𝜀ε c. .

4.3.
4.3. Analysis
Analysis of Specimen
of Specimen Failure
Failure Progress
Progress Caused
Caused byby Transverse
Transverse Deformations
Deformations
ToTo analyze
analyze thethe entire
entire process
process of longitudinal
of longitudinal crackcrack development,
development, frameframe by frame
by frame im-
images were used (Figure 10). According to [6], this process is divided into three
ages were used (Figure 10). According to [6], this process is divided into three stages. In stages.
In the first stage, the concrete is exposed to tension until the initiation of longitudinal
the first stage, the concrete is exposed to tension until the initiation of longitudinal cracks,
cracks, when ε trans ≤ ε ct ul (Figure 10a), where, according to the results presented in
when 𝜀 ≤𝜀 (Figure 10a), where, according to the results presented in Table 3, the
Table 3, the average value of ε ct ul for three the specimens is equal to 0.095‰. In the
average value of 𝜀 for three the specimens is equal to 0.095‰. In the second stage,
second stage, when ε ct ul < ε trans ≤ 2ε ct ul (that is, transverse deformations develop until
when 𝜀 <𝜀 ≤ 2𝜀 (that is, transverse deformations develop until they reach a
they reach a value of about 0.2 ‰ (Figure 5)), there is the process of longitudinal crack
value of about 0.2 ‰ (Figure 5)), there is the process of longitudinal crack initiation and
initiation and development (as shown in Figure 10b). At the selected loading rate, this
development (as shown in Figure 10b). At the selected loading rate, this stage continues
stage continues for about 6 seconds. The crack propagation process ends with specimen
for about 6 seconds. The crack propagation process ends with specimen failure, when
failure, when ε trans > 2ε ct ul (Figure 10c). The obtained longitudinal crack development
𝜀 2𝜀 (Figure 10c). The obtained longitudinal crack development pattern corre-
pattern corresponds to the deformation process and failure in the columns. For example, a
sponds to the deformation process and failure in the columns. For example, a similar be-
similar behavior in the reinforced concrete columns of residential building was observed
havior in the reinforced concrete columns of residential building was observed during the
during the Mexico earthquake in 2017 [6].
MexicoCrack
earthquake in 2017in[6].
propagation the specimens occurs parallel to the compressive force direction
that proves deformation development in the transverse direction, which is caused only by
pure tension due to the cylindrical form of the specimens. As a result, the longitudinal
crack development process in specimens leads to their separation into several vertical
wedge-shaped segments and an unbroken circular inner core, which was theoretically
predicted earlier [6].
The following important investigation step is to analyze the specimen failure type after
unloading (Figure 11). Following the structural phenomenon [13], it can be theoretically
predicted that the cross-section area of the destroyed specimens is approximately half
that of the initial one. Experimental studies confirmed this prediction for all of the tested
specimens. The area of the inner core A and of the outer segments B were formed on the
upper surface of all of specimens at failure (see Figure 11).
Appl. Sci. 2021, 11, 8460 11 of 16
Appl. Sci. 2021, 11, 8460 11 of 16

(a)

2.0 s 3.0 s 4.5 s

(b)

(c)
Figure 10.
Figure 10. Typical
Typical crack
crackpropagation
propagationininthe
theinvestigated
investigatedspecimens: (a)(a)
specimens: End of of
End stage 1–crack
stage development
1–crack initiation;
development (b)
initiation;
Stage 2–crack development process; (c) Stage 3–specimen failure.
(b) Stage 2–crack development process; (c) Stage 3–specimen failure.

Crack propagation in the specimens occurs parallel to the compressive force direction
that proves deformation development in the transverse direction, which is caused only by
 predicted earlier [6].
The following important investigation step is to analyze the specimen failure type
after unloading (Figure 11). Following the structural phenomenon [13], it can be theoreti-
cally predicted that the cross-section area of the destroyed specimens is approximately
Appl. Sci. 2021, 11, 8460 half that of the initial one. Experimental studies confirmed this prediction for all12ofofthe
16
tested specimens. The area of the inner core A and of the outer segments B were formed
on the upper surface of all of specimens at failure (see Figure 11).

Sp1.1 Sp1.2 Sp1.3 Sp1.4

Figure11.
Figure 11.Failure
Failurepattern
patternof
ofthe
theupper
uppersurface
surfaceof
ofthe
thespecimens.
specimens.

The ratios
The ratios between
between the
the entire
entirecross-section
cross-sectionarea
areaof
ofthe
thespecimens
specimens(A(A++B)B)and
andthat
thatof
of
the inner core A are presented in Table 6. The obtained data show that the average
the inner core A are presented in Table 6. The obtained data show that the average ratio is ratio
is close
close to which
to 2, 2, which confirms
confirms thethe structural
structural phenomenon
phenomenon [13].
[13].

Table6.6.Measured
Table Measuredvalues of(A𝐴+ B𝐵) and
valuesof andAAand
andtheir
theirratio.
ratio.

2𝟐 𝐀A+B 𝐁
Specimen
Specimen 𝐀 𝐁 ,cm
(A+B), 𝐜𝐦 A, cm2𝟐
𝐀, 𝐜𝐦 A
𝐀
Sp1.1
Sp1.1 176.5
176.5 91.6
91.6 1.9
1.9
Sp1.2 176.4 75.1 2.3
Sp1.2 176.4 75.1 2.3
Sp1.3 176.4 74.9 2.4
Sp1.3
Sp1.4 176.4
177.1 74.9
99.4 2.4
1.8
Sp1.4
Average 177.1
176.6 99.4
85.3 1.8
2.1
Average 176.6 85.3 2.1
Appl. Sci. 2021, 11, 8460 Thus, the typical failure pattern of a compressed concrete specimen
13 of 16 is presented in
Thus,
Figure 12. the typical failure pattern of a compressed concrete specimen is presented in
Figure 12.

Figure 12.
Figure 12.Typical failure
Typical pattern
failure of compressed
pattern concrete specimen.
of compressed concrete specimen.

4.4. Correspondence
4.4. Correspondence of theofObtained Experimental
the Obtained Results to the
Experimental Available
Results toData
the Available Data
Eight cylindrical NSC specimens with a height-to-diameter ratio equal to 2 were
Eight cylindrical NSC specimens with a height-to-diameter ratio equal to 2 were tested
tested under uniaxial loading [7]. Diameters of the specimens differed from 10 to 80 cm.
under uniaxial
The presented dataloading [7].development
report crack Diameters in of
thethe specimens
vertical direction anddiffered from 10 to 80 cm. The
the formation
presented
of an unbrokendata report
circular crack
inner core development
that corresponds in thefailure
to the vertical direction
pattern and
described the formation of an
in this
paper (see Figure
unbroken 2: 20-40-02
circular inner andcore40-80-02 [7]). Additionally,
that corresponds all of
to the the specimens
failure patterndemon-
described in this paper
strateFigure
(see elastic behavior
2: 20-40-02in the and
longitudinal
40-80-02 direction practically until failure,
[7]). Additionally, regardless
all of the of
specimens demonstrate
the size difference (see Figure 3 [7]).
elastic behavior in the longitudinal direction practically until
Thirty-three 15 × 30 cm fibered-concrete cylindrical specimens with a minimum 28-
failure, regardless of the size
difference (see Figure 3 [7]).
day concrete compressive strength of 27.6 MPa were uniaxially loaded. The typical failure
patternThirty-three 15 × 30 cm
with vertical longitudinal fibered-concrete
cracks is presented in Figure cylindrical specimens
8 [8]. The obtained stress–with a minimum
longitudinal strain relationships for compressed concrete specimens
28-day concrete compressive strength of 27.6 MPa were uniaxially loaded. are linear, almost un- The typical
til peak load (Figure 15 [8]). Thus, it should be mentioned that the results obtained for
failure pattern with vertical longitudinal cracks is presented in Figure 8 [8]. The obtained
concrete mixes with different amounts and types of fiber correspond to the results ob-
stress–longitudinal
tained in this paper for NSC strain relationships for compressed concrete specimens are linear, al-
specimens.
Six cylindrical NSC specimens with different end conditions were subjected to uni-
axial compression [9]. In this case, transverse deformations were additionally measured.
The obtained data show a good convergence with the present research results. The stress–
longitudinal strain relationships are linear until failure when the stress–transverse strain
relationships have well pronounced non-elastic behavior until a peak load (see Figure 2
Appl. Sci. 2021, 11, 8460 13 of 16

most until peak load [8]. Thus, it should be mentioned that the results obtained for concrete
mixes with different amounts and types of fiber correspond to the results obtained in this
paper for NSC specimens.
Six cylindrical NSC specimens with different end conditions were subjected to uni-
axial compression [9]. In this case, transverse deformations were additionally measured.
The obtained data show a good convergence with the present research results. The
stress–longitudinal strain relationships are linear until failure when the stress–transverse
strain relationships have well pronounced non-elastic behavior until a peak load (see
Figures 2 and 3 [9]). The earlier described failure pattern is obtained for specimens with
reduced friction conditions (see Figure 5 [9]).
Thus, the results presented for 47 specimens [7–9] correspond to the data obtained by
testing 7 specimens within the framework of the present research. This, in turn, significantly
increases the reliability of the obtained results.

5. Comparison of the Results Obtained for NSC with Available Data for HSC
Based on the data obtained in the framework of the present study (see Section 4),
the behavior of NSC, which corresponds to class C30 at relatively high loading rates,
can be compared with that of HSC, which is described in [6]. The available data on
HSC (class C90) includes the test results for three cylindrical specimens. Longitudinal
deformations were measured using linear variable displacement transducers (LVDT) and
transverse deformations measured by strain gauges. The dependences of these two types
Appl. Sci. 2021, 11, 8460 of deformations on stresses under uniaxial load are shown in Figure 13 and 14 Figure
of 16 14,
respectively. Corresponding data for the comparison of the elastic and nonelastic behavior
of HSC and NSC are provided in Tables 7 and 8.

Figure
Figure13.13.
Experimental 𝜎 –𝜀
Experimental σc –εdependences forfor
c dependences HSC (following
HSC [6]).[6]).
(following

Figure
Figure 14.14.
Experimental 𝜎 –𝜀
Experimental σc –ε transdependences
dependencesforfor
HSCHSC (following
(following [6]).[6]).

Table 7. Experimental values of maximum elastic strains, ultimate transverse strains, and Poisson’s
ratios for HSC (following [6]).

Specimen No. 𝜺𝒄 𝒆𝒍 , ‰ 𝜺𝒄𝒕 𝒖𝒍 , ‰ 𝝂

Appl. Sci. 2021, 11, 8460 14 of 16

Table 7. Experimental values of maximum elastic strains, ultimate transverse strains, and Poisson’s
ratios for HSC (following [6]).

Specimen No. εc el , ‰ εct ul , ‰ ν

1 0.780 0.143 0.18
2 0.829 0.161 0.19
3 1.004 0.162 0.16
Average 0.871 0.155 0.18

Table 8. Relationships between experimental Poisson’s ratio values, transverse, and longitudinal
strains at ULS (following [6]).

Specimen No. ν εtrans1 εtrans1 /εc1

εc1 ν
1 0.18 0.34 1.89
2 0.19 0.29 1.53
3 0.16 0.29 1.81
Average 0.18 0.31 1.74

The limit transverse deformation of the HSC specimens corresponding to concrete

elastic behavior is 0.15 ‰. In this case, the external load value is approximately half of
the concrete strength, which corresponds to the structural phenomenon [13]. The average
Poisson’s ratio value is equal to 0.18. Further load increases are accompanied by the linear
development of longitudinal deformations (Figure 13), while the transverse deformations
begin to exhibit elastic–plastic behavior (Figure 14). Therefore, it is logical to analyze the
ratio between the longitudinal and transverse deformations for further interpretation of
the HSC behavior. Following Table 8, at ULS, this ratio is equal to 0.31, which is close to
the results obtained in the frame of the present study (see Table 5). It should be mentioned
that the relative transverse deformation growth, ε trans1ν /ε c1 , in both cases (for HSC and NSC)
is equal to about 1.7.
A comparison of the results obtained for the NSC and HSC (Tables 5 and 8) shows suf-
ficient convergence of the standard deviations and the coefficients of variation for Poisson’s
coefficient and ε trans1 /ε c1 . The standard deviations are 0.024 and 0.031 and the coefficients
of variation are 13.7% and 10.1% for Poisson’s coefficient and ε trans1 /ε c1 , respectively. Thus,
the obtained data confirm that it is possible to use experimental results for NSC specimens
obtained at rather a high loading rate for the interpretation of HSC behavior. This, in turn,
allows for the disclosure of the presence of the non-elastic behavior stage in compressed
HSC, which is important for developing proper and effective design approaches.

6. Conclusions
The importance of this research is attributable to the fact that the inelastic behavior of
compressed high-strength concrete specimens is practically not pronounced in longitudinal
deformations and therefore is not even analyzed. Consequently, high-strength concrete is
considered to be a pure brittle material, which significantly limits its applications, especially
under dynamic loads. For this reason, the experimental investigation of normal strength
concrete at relatively high loading rates was conducted in order to prove the effectiveness
of using transverse deformations as an indicator for the interpretation of high-strength
concrete behavior.
The obtained experimental results and their comparison with high-strength concrete
specimen behavior showed good convergence of such parameters such as Poisson’s ratio
and the ratio of transverse to longitudinal deformations in the inelastic stage of concrete
behavior. The proposed theoretical failure pattern for compressed cylindrical concrete
specimens, which includes an unbroken circular core surrounded by separate vertical
wedge-shaped elements, was also experimentally confirmed. Additionally, the ratio be-
tween the initial specimen’s cross-section area and the destroyed one was obtained as
approximately equal to 2, which corresponds to the structural phenomenon.
Appl. Sci. 2021, 11, 8460 15 of 16

Thus, the approach of using transverse deformations for analyzing concrete perfor-
mance proves that from the viewpoint of transverse deformations, high-strength concrete
behaves inelastically and not as a brittle material. In our opinion, further investigation
of this phenomenon can form a basis for the more accurate and effective design of high-
strength concrete structures.

Author Contributions: Conceptualization, I.I.; methodology, I.I.; I.F. and Y.R.; validation, I.I.; I.F. and
Y.R.; investigation, I.F., I.I., and Y.R.; resources, I.I. and Y.R.; data curation, I.F., I.I., and Y.R.; formal
analysis, I.F.; writing—original draft preparation, I.F., I.I., and Y.R.; writing—review and editing, I.I.,
I.F., and Y.R. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments: This research was conducted with the assistance of the Center for Integration in
Science, Ministry of Aliyah and Integration, the State of Israel.
Conflicts of Interest: The authors declare no conflict of interest.

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