Studia Geotechnica et Mechanica, 2024; 46(3): 244–258

Original Study Open Access

Bilel Boualleg*, Nadjet Bouacha

Analysis of the behavior of structures under the

effect of progressive rupture of a cavity
https://doi.org/10.2478/sgem-2024-0019
received November 22, 2023; accepted July 15, 2024.
and the safety of their users. Various studies have been
conducted to investigate the issues caused by the collapse
Abstract: The ground movements related to the presence of of underground cavities and the impact of these collapses
old underground cavities are often damaging to structures on surface structures.
and infrastructures. Considering these ground movements Table 1 summarizes the main research studies
in calculations will prevent considerable human loss and analyzing cavity collapse using experimental and
material damage. Many areas, both in Algeria and in numerical methods, classified chronologically.
abroad, are prone to instability caused by ground rupture Our work focuses on analyzing the interaction
and the phenomenon of sinkhole progression. The between the soil and the structure, first examining the
objectives of this work are first to numerically simulate progressive collapse of an underground cavity beneath a
the process of cavity collapse and second to analyze the structure and then evaluating its impact on the stability
impact of cavity properties on structure stability. A finite of the structure. Finally, we conduct a parametric study
element model was established to analyze the influence to understand how cavity volume, depth, and spacing
of several cavity parameters (dimensions, volume, and influence the stability of the structure.
spacing). Validation of the model relied on comparing To predict ground movements caused by cavity
numerical results with experimental data from scientific degradation, geotechnical engineers have various
research, as well as those from analytical approaches. methods at their disposal. These methods include
Adequate correlation was achieved. The study allowed empirical approaches using detailed field data, analytical
deriving mathematical equations relating to several methods based on mechanical equations, and numerical
parameters, including cavity dimensions and position methods. These approaches are documented in the
in the soil, soil characteristics, and footing width. These scientific literature notably by Deck et al. (2006) and
results will be considered to reduce the risk of surface Dolzhenko (2002).
structure instability. The main objectives of this study are to minimize the
consequences of cavity collapse, maintain the stability of
Keywords: Soil-structure interaction; cavity; structure; the structure, and reduce deformations observed at the
modeling; rupture; footing; displacement. structural elements level. Our methodological approach
involved initially validating a numerical model, followed
by calculating the collapse values using an analytical
1 Introduction method. Test results were presented using a scaled-down
model. In practice, empirical methods are often guided
The frequent presence of underground cavities in certain by analytical approaches or finite element calculations.
developable areas poses a potential collapse risk that can These methods are then adjusted based on experimental
be detrimental to the proper functioning of infrastructures curves, as highlighted by Aftes (1982).
The empirical approaches described by Peck (1969)
*Corresponding author: Bilel Boualleg, Department of Civil for vertical displacements and by Lake et al. (1992) for
Engineering, University of Souk-Ahras, BP 1553, 41000 Souk-Ahras, horizontal displacements are used to predict ground
Algeria Laboratory of Management, Maintenance and Rehabilitation of surface movements after the excavation of a circular
Urban Equipment and Infrastructure (InfraRES), E-mail: b.bouallag@ tunnel (see Figure 1). Equations 1 and 2 express vertical
univ-soukahras.dz; http://orcid.org/0000-0002-3594-1914
and horizontal displacements, respectively:
Nadjet Bouacha, Department of Civil Engineering, University of
Souk-Ahras, BP 1553, 41000 Souk-Ahras, Algeria Laboratory of −x2�
Management, Maintenance and Rehabilitation of Urban Equipment Vertical displacement: S(x) = Smax × e 2i2 (1)
and Infrastructure (InfraRES); http://orcid.org/0000-0001-2345-6789

Open Access. © 2024 Bilel Boualleg, Nadjet Bouacha, published by Sciendo. This work is licensed under the Creative Commons
Attribution alone 4.0 License.
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 245

Table 1: Summary of conducted research studie.

Author Year Objective Type

Nakai et al. 1997 Investigate the effect of 3D and expansion on ground movements during Experimental
tunnel excavation

Dyne 1998 Analyze the different parameters: the opening of the cavity, the width of the Experimental 2D scale
cavity, and the height of the covering model

Burd et al. 2000 Study soil-structure interaction during tunneling under masonry structures Numerical
and analysis MEF-OXFEM

Laefer 2001 Study the damage to structures on shallow foundations subject to soil Experimental (a small-
movements induced by excavation scale model of 1/10th).

Mahamma 2002 Study the soil-structure interaction phenomena during the collapse of a mine Experimental
gallery. The collapse of the mine gallery was modeled by successive sinking of
a cylinder along the axis of propagation of the rupture

Shanin et al. 2004 The study of the effect of ground movements and their mechanical behavior Experimental trap model
during tunnel excavation.

Boumalla 2005 Vary a number of parameters such as the opening of the cavity, the height of Experimental
the cover, the rate of initiation of a melt, or the subsidence of the ground

Sung et al. 2006 Analyze the settlements and ground pressure at the surface due to the tunnel Experimental
in the cases without and with the foundation structure in the vicinity.

Castro et al. 2007 Study the “block caving” mining method, not the movements that occur on the Experimental
surface of the land large-scale 3D model
Trueman et al. 2008

Lee & Bassett 2007 Simulate the deformation of the tunnel by changing its diameter, to Experimental
investigate the behavior of existing foundations located near the tunnel
Kikumoto et al. 2009

Caudron 2007 Characterize the influence of soil-structure interaction during the formation of Experimental
a sinkhole and numerical

Deck and Anirudth 2010 To investigate the phenomenon of soil-structure interaction due to mine Numerical 2D model
subsidence, taking into account the influence of length, rigidity of the CESAR LCPC
structure, mechanical properties of the soil, and intensity of subsidence.

Boramy Hor 2012 Simulate ground movements and their consequences on the surface. Experimental/numerical
3D physical model

Al Heib et al. 2013 Understanding sinkhole consequences on masonry structures using a large Experimental
small-scale physical modeling. The paper presents the main results of the
small-scale physical model designed to study the consequences of subsidence
on structures. Present the transfer of movements from the soil to the structure.
The objective is to understand and then to predict the real behavior and the
damage of structures on subsidence areas .

Nghiem et al. 2014 Physical model for damage prediction in structures due to underground Experimental
excavations: a small-scale physical model (1/40 scale factor on the
dimensions) under normal gravity. It has been designed for developing and
validating experimentally new methods of prediction of damages to masonry
structures induced by subsidence (generally resulting from underground
excavations of tunnels and mines)

Keawsawasvong 2021 Limit analysis solutions for spherical cavities in sandy soils under overloading. Numerical
An investigation on the stability of spherical cavities in sandy soils under
overloading at the ground surface is carried out in this study. By using finite
element limit analysis, a spherical cavity is numerically simulated under an
axisymmetric condition, and the lower and upper bound solutions of the
stability of spherical cavities can be obtained
246 Bilel Boualleg, Nadjet Bouacha

Table 1: Summary of conducted research studie.

Continued

Author Year Objective Type

Yongyao et al. 2023 A numerical simulation study on the evolutionary characteristics of the Numerical
damage process of karst soil cavity under positive pressure effect

Keba and Isobe 2024 Bearing capacity of a shallow foundation above the soil with a cavity based Numerical
on a rigid plastic finite element method. Based on the rigid plastic finite
element method (RPFEM), this study investigates the performance of the
footing on the soil with a cavity. The RPFEM is used in plane strain conditions
and necessitates only a few materials to predict the bearing capacity: the unit
weight of the soil, the cohesion, the shear resistance angle, and the dilation
angle

Table 2: Empirical formulas for determining i (Dolzhenko, 2002).

Authors Proposed expression Soil type Calculated i value

Atkinson & Potts. (1977) i = 0.25(1.5C + D) Dense sands with surcharge 3.65 m
Oteo & Sagaseta. (1982) i = 0.525H + 0.42R Granular soils 5.67 m
Dyer et al. (1986) i = 0.29H Loose to medium dense sand 2.60 m
Al Abram (1998) i = 0.15H + 0.5D Analogical soil 3.60 m

They manage to propose both equations related to

vertical and horizontal displacements such that i is derived
from the empirical formulas (Dolzhenko, 2002), where (i =
0.15H + 0.5D). The formulas then become as follows:

0.667Vcavity −x2�
Vertical displacement: S(x) = ×e 2i2 (3)
2.5i

(x�1m)
Horizontal displacement: V(x) = S(x) ×
(H�1m)γ
(4)

with an optimal value of γ equal to 0.87 and Vcavity=a×b

Figure 1: Schematic diagram of the empirical approach by Peck (1969).
(a and b the height and width of the cavity).

(2)
2 Case Study
x
Horizontal displacement: V(x) = S(x) × H

The parameter i, representing the width of the

The work of Caudron et al. (2007), which particularly
collapse trough, can be approximated by various empirical
focused on soil-structure interaction phenomena during
relationships taking into account the diameter D, the cover
the formation of a cavity near a surface frame, was used
C, and the nature of the soil (Table 2).
as a reference as it perfectly aligns with the concept of
Caudron et al. (2006) demonstrated that with some
this research. The authors chose a real case (the Malakoff
modifications, the two expressions (eqs 1 and 2) can
limestone quarry in the Paris region) involving a cavity of
be used to predict surface displacements as long as
10 meters wide and 2 meters high with a depth of 8 meters
the movement curves remain continuous. This is the
formed by 9 laminated soil layers (see Fig. 2) (Caudron,
case when the upper part of the lining is made of the
2007; Caudron et al., 2006; Caudron et al., 2004).
granular material and the cavity is rectangular in shape.
The characteristics of the actual materials are detailed
Mathematical treatment and integration of the value of
in Table 3.
the increment i depending on the soil type are presented
in Table 2.
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 247

Table 3: Geo-mechanical characteristics of different materials (Caudron, 2007).

Layer Materials E (MPa) ʋ Rtraction (MPa) Cohesion (MPa) φ (°)

8 Marls 70 0.25-0.30 0.30 0.80 28

2 and 5 Stones 100 0.25-0.30 0.30 0.80 29

6 Clay sand 130 0.25-0.30 0.20 1.2 30

3 and 9 Limestone 20 0.25-0.30 0.80 2.00 31

7 Stones 200 0.25-0.30 01 1.00 35

4 Marls 50 0.25-0.30 0.1 0.20 26

1 Stones 50 0.25-0.30 0.20 0.40 27

the thermal effect has no influence on the series of tests,

which makes the problem somewhat simpler. Then, three
scale factors must be determined in order to establish
the entire relationship between the full-scale case study
and the small-scale model. These scale factors concerned
gravity, density, and length. The tests are carried out under
normal gravity, so the corresponding scale factor is 1.
The analogous soil has a unit weight of 65 kN/m³,
so the scale factor on density is 3. The final scale factor
concerns
length. It was set at 1/40 in order to have a test bench
Figure 2: Real model of the cavity. with practical dimensions. From this point, all other scale
factors can then be deduced from the laws of similarity
The main source of problems and complexity in and these three values. Therefore, it is not possible to
designing a physical model lies in the importance of adhere to all similarity rules, particularly those concerning
adhering to similarity rules with respect to the original stress states. The results of small-scale tests will then
phenomenon. Obviously, for a full-scale model, this is not be qualitative rather than quantitative. The value 1/40 is
limiting. The main limitation in this case comes from the chosen, which remains within the limits of the commonly
cost and feasibility of tests. accepted range of values for the Schneebeli material used
This is why it is common to resort to reduced (Ovesen, 1979) (see Table 5).
models, which present a number of advantages: speed, The experimental reduced model is simpler,
reproducibility, and the possibility of working until failure. represented at a scale of 1/40, with a soil mass of 1000 mm
However, to ensure that the phenomenon obtained in the width for a covering height of 200 mm above the cavity
reduced model exhibits behavior similar to that observed (see Fig. 3).
in full scale, it is necessary to ensure compliance with The Schneebeli analog material, used to represent the
a number of rules. These are the laws of similarity, as soil, consists of a mixture of assemblies of steel rods with
presented by Dehousse and Arnould (1971) and by Bazant diameters of 3, 4, and 5 mm in precise proportions (see
(2004). Since then, Garnier has specified their application Fig. 4) (Schneebeli, 1956 ; Schneebeli, 1957). The lining
to the field of geotechnics (Garnier, 2001a; Garnier, 2001b). consists of a cohesive bench with a thickness equivalent to
Each letter accompanied by an asterisk (*) represents 50 mm located at the top of the cavity. This is topped with
the scale factor associated with the change in scale for the a layer of pulverulent material equivalent to 150 mm in
respective quantity. Table 4 provides the meanings of each thickness. The cavity, with a height equivalent to 50 mm, is
quantity. gradually created up to a maximum width corresponding
Caudron et al. (2006) extensively presented the small- to 250 mm in five steps (Caudron et al., 2004; Caudron et
scale physical model: design and limitations of the model. al., 2006; Caudron et al., 2007).
The first step in defining the small-scale physical model The structure used for the study of soil-structure
was the laws of similarity. The assumption is made that interaction is of the steel beam-post type. The load
248 Bilel Boualleg, Nadjet Bouacha

Figure 3: Experimental scale model.

Table 4: List of similarity laws.

Number Similarity law Meaning of scale factors

1 x*/L*=1 Equality of coordinates relative to length scale

2 U*/L*=1 Equality of displacements relative to length scale
3 U0*/L*=1 Equality of displacements at origin relative to length scale
4 g*/γ*=1 Equality of acceleration scale to gravity scale
5 E*L*2/F*=1 Conservation of the ratio of elasticity modulus scale by length squared to force scale
6 γ*t*2/L*=1 Identity of acceleration and length scales as time cannot be altered
7 P*L*2/F*=1 Conservation of the ratio of pressure scale times length squared to force scale
8 (σ0*L*2)/F*=1 Conservation of the ratio of stress scales times length squared to force scale
9 (ρ*γ*L3*)/F*=1 Conservation of the ratio between scales of quantities determining inertia force relative to
force scale

Table 5: List of scale factors.

the structural elements respects the scaling: the behavior
obtained is purely elastic. The rules of similarity govern
Symbol Scale factor concerned Dimension Value
the relationship between a full-scale object and its scale
L* Length of reference L 1⁄40 model. In the context of a similarity that must account for
x* Coordinates L 1⁄40 the mechanical behavior of the object, a number of scale
factors must be considered. Fig. 5 shows a schematic view
E* Modulus of elasticity ML t -1 -2
3⁄40
of the full-size building as well as its scale model.
ρ* Density ML -3
3 Table 6 shows the characteristics obtained from the
g* Acceleration of gravity Lt -2
1 predimensioning of the structure with the assumed data
and the equivalent characteristics for the scale model
F* External punctual force MLt-2 3⁄64000
with the similarity laws. However, the use of a coherent
p* Superficial force ML-1 t-2 3⁄40 material is a prerequisite for representing the resistant
U* Displacement L 1⁄40 bench needed to create the cavity.
σ* Constraint ML t -1 -2
3⁄40 All conducted characterization tests by researchers on
the Schneebeli material show that its behavior is identical
γ* Inertia acceleration Lt -2
1
to that of dense sand (Dolzhenko, 2002; Kastner, 1982).
Therefore, the material was modified by Caudron (2006)
considered, uniformly distributed over the beam elements, to represent a cohesive material.
corresponds to 10 kPa, comparable to permanent loads and Cohesion was achieved using an aqueous adhesive.
service loads. The stiffness (ES and Eiz with E the Young’s Laboratory tests were conducted to determine the
modulus, S the section, and Iz the moment of inertia) of mechanical characteristics of the modified material, and
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 249

Table 6: Structure characteristics in real size and scale model.

Characteristics Values

Real model Scale model

Module (MPa) 33000 2475

Section (m )
2
0.04 25×10-6
Inertia (m4) 1.33×10-4 52×10-12
Loading (kPa) 10 0.75

Table 7: Geo-mechanical characteristics of scale model soils.

Characteristics Unit Pulverulent Coherent

soil soil
Figure 4: Overview of Schneebeli rolls.
Young’s modulus (E) MPa 50–100 50–100

Friction angle (φ) ˚ 26 28–30

Cohesion (c) KPa ≈0 200

Poisson’s ratio (ν) / 0.3 0.3

Density (ρ) kg⁄m 3

2200 2200

fractions), gradually diluting it until the desired behavior

was obtained. Two concentrations of adhesive, those
at C/4 and C/8, seem to allow obtaining a material that
exhibits mechanical characteristics close to those desired.
The modified material exhibits characteristics close to
those desired, so they are more finely characterized in
Table 7.

3 Numerical Modeling
The finite element method is employed to model the soil-
structure interaction during the formation of a cavity.
Once the model is established, a comparison between
the numerical results and the experimental data from
(Caudron, 2007), as well as those from the analytical
method, will be performed. The modeling involves
several successive and distinct steps, such as data input,
definition of boundary conditions, meshing, calculation
phases, simulation startup, and results analysis.
To model the structure, it is important to present
all the data related to the different materials: building
Figure 5: Real and scale model of the structure (Caudron et al, 2007).
geometry, material properties (powdery and cohesive
soil, air, column-beam structure, footings), load, cavity
the improvement concerns the friction angle, cohesion, dimension, and depth. The schematic representation of
and volumetric behavior from a pulverulent soil to a the model is shown in Figure 6. It consists of a soil block
cohesive one. The biaxial tests, therefore, began with with a height of 25 m and a width of 40 m. Comprising two
different concentrations of adhesive at C/2, C/4, and C/8 superimposed soil layers, the upper layer is of powdery
(C: glue concentration; C/2, C/4, and C/8: concentration type with a height of 6 m overlying a cohesive soil layer
250 Bilel Boualleg, Nadjet Bouacha

Structure

Pulverulent
Structure soil

Cavity
Pulverulent soil

Coherent soil
Cavity

Coherent soil
Blocking

Blocking

-a-

Load
-a-

Load
Air

Air
Footing

Footing

-b- -c-

Figure 6: Model geometry:

-b-(a) global geometry, (b) structure, and (c) cavity diagramming. -c-

with a height of 19 m. A two-level structure is embedded positioning the structure at a sufficiently distant distance
in the upper layer at a depth of 1.2 m. The lower layer from the edges to allow for good stress distribution in the
encompasses a cavity with a height of 2 m and a length of soil.
10 m in a rectangular shape. The soil mass has been discretized entirely by 15-node
Based on the values of the properties of the various triangular finite elements. The same type of elements
materials used in the experimental study, the same has been adopted for meshing both the soil body and the
values are incorporated into the material database of the structure to ensure correct assembly. The mesh consists
numerical model. Tables 8 and 9, respectively, present the entirely of 561 elements and 4631 nodes. Local mesh
properties of the soils and structural elements. Among refinement has been performed in areas where strong
the problems encountered during the modeling is the gradients are likely to appear, i.e., around the cavity, to
representation of the rupture process and the simulation obtain a good estimation of stress and displacement fields
of the void present during the progression of the cavity. (see Fig.7).
We chose the same properties for air to simulate the The reasoning process adopted for the calculation of
void existing between the structural elements. Air is such a model led us to establish 7 phases:
represented as a material with low physical properties. Phase 0: Initiation of stresses (K0 procedure) to
For the calculations, we assume that the interfaces determine initial effective stresses.
between the different soil layers are perfectly adherent, Phase 1: Excavation at the depth of the footings.
implying continuity of vertical stresses and vertical Phase 2: Installation of the structure.
displacements. Boundary conditions are ensured both by This phase involves activating the structure (footings
embedding the soil at the base and on the sides and by and framework) and backfilling to ensure the stability of
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 251

Table 8: Soil properties.

Parameters Name Unit Pulverulent soil Coherent soil Air

Material model Model - Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb

Material type Type - Drained Drained Drained

Soil unit weight above phreatic level γunsat kN⁄m3 17 20 5

Soil unit weight below phreatic level γsat kN⁄m 3

19 22 5

Permeability in horizontal direction kx m⁄day 1 0 1

Permeability in vertical direction ky m⁄day 1 0 1

Young’s modulus E kN⁄m2 100000 100000 5

Poisson’s ratio ν - 0.3 0.3 0.1

Cohesion c kN⁄m2 2 200 1

Friction angle φ ° 26 26 5

Dilatancy angle ψ ° 7 9 1

Strength reduction factor interne Rinter - 1 1 1

Table 9: Properties of structural elements.

Parameters Name Unit Value

Type of behavior Material type - Elastoplastic

Normal stiffness EA kN⁄m 132000

Flexural rigidity EI KNm2⁄m 4389

Equivalent thickness d m 0.632

Weight w KN⁄m/m 10

Poisson’s ratio ν - 0.35

Figure 8: Calculation phases.

We triggered the rupture by initiating a thin layer of

air void, which we then gradually extended until complete
collapse.
After several attempts and corrections, the sequence
of calculation phases has been established without
interruption, and we have reached the phase of result
exploitation (launch of calculations) as shown in Figure
11:
According to the experimental results (Fig. 12a), we
Figure 7: Model meshing. observed that the rupture of the stiff bench occurs shortly
after degradation. The latter fall into the cavity, and the soil
the structure. At this stage, the soil is loaded only under reduced to powder on the surface follows their movement.
its own weight. The foundations of the structure are then evaluated,
Phase 3: Loading. revealing some smaller voids in the fracture areas in the
Phases 4, 5, 6, 7 represented in Figure 10 express the stiff bench and significant detachment of the right footing
process of rupture numerically simulated according to the of the structure. Figure 12b presents the visualization
collapse mode of the cavity. of the last phase of the numerical model (total cavity
252 Bilel Boualleg, Nadjet Bouacha

-a-

Figure 11: Calculation launch.

rupture). In this phase, the structure is tilted to the right,

and consequently, the maximum displacement at footing
1 is recorded, as it is located at the cavity axis.
To demonstrate the validity of our model, we
compared the results of vertical and horizontal collapses
of different footings with the vertical and horizontal
displacements at the surface of the cavity. The validation
of the numerical model relies on comparing the results
with those derived from experimental data and compared
-b- -c- analytical results. Figures 13 and 14 display the results
of the structure collapse comparison using analytical,
Figure 9: Phases 1, 2, and 3: (a) Phase 1 (excavation), (b) Phase 2 numerical, and experimental methods. A good correlation
(soil + structure), and (c) Phase 3 (loading).
is obtained between numerical and experimental results.
A slight deviation of 5 mm from the analytical results
is observed. The maximum collapse occurs at footing 1,
which is the most affected due to its positioning with the
axis passing through the center of the cavity, unlike the
other footings, which are slightly inclined.
The maximum displacement is observed on the
right side of the structure, corresponding to the cavity
axis (Position x = 20 m) (see Figure 13). Horizontal
-a- -b- displacements are depicted in the graph (Figure 14),
showing three curves with two types of positive and
negative displacements. A significant influence of the
structure is observed on the left side.
There is a good agreement between the curves of
numerical, experimental, and analytical collapses, where
the maximum value of numerical collapse is almost equal
to that of experimental collapse (with a difference of a few
millimeters). The maximum slope of collapse is located at
the axis of the cavity. Gradients or breaks in the curve of
-c- -d- the numerical method are attributed to the simulation of
cavity rupture (see Figure 10c). In the numerical method,
the cavity is subdivided into small squares, and the higher
Figure 10: Cavity rupture
Figure 10:process.
Cavity rupture process the number of squares, the more the gradient of the curve
(a) Phase 4 (initial cavity rupture)
(b) Phase 5 (2nd cavity rupture) attenuates.
(c) Phase 6 (3rd cavity rupture) Figure 15 shows that the vertical displacements of the
(d) Phase 7 (total cavity rupture) footings are identical for the experimental and numerical
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 253

a: Experimental b: Numerical

Figure 12: Final phase of rupture.

0,01
position [m]
0
Displacement [m]

0 10 20 30 40
-0,01 Anal
Figure 15: Differences in displacements of each footing among the
-0,02 Num
three methods.
-0,03 Exp

-0,04
4 Analysis of Influencing
Figure 13: Vertical displacements (numerical, experimental, and Parameters
analytical) of the footing.
To study the gradual degradation of cavities and their
impact on structure stability, a comparative method was
employed to develop influential parameters such as
volume, spacing, and depth of cavities. This approach
will evaluate the effect of these parameters using variable
Anal
0,015 ratios, including a ratio of b/a ranging from 0.5 to 3, a ratio
Num of H/B varying from 0.5 to 3, and a ratio of L/B also ranging
Displacement [m]

0,01
Exp from 0.5 to 3. The dimensions B, H, a, b, and L are shown
0,005
in Figure 12 (Djamel Saadi et al., 2020).
0 position [m] Figure 17 illustrates the behavior of the soil and the
-0,005 0 10 20 30 40 soil-structure interaction concerning the increase in
-0,01
cavity volume and structure instability. After analyzing
the behavior of the footings, it is observed that those
aligned with the cavity axis undergo more deformations
Figure 14: Horizontal displacements (numerical, experimental, and
analytical) of the footings. than those away from this axis. Stress levels peak as the
cavity volume increases. Vertical stresses are evident on
the sides of the cavity, with a compression value of -450
methods, although there is a slight difference for the kN/m².
analytical method (8.3 mm as the highest estimate for The impact on the structure is negligible as the stress
footing 4). It can also be observed that there is a slight approaches zero, but the footings are exposed to stresses
difference in the horizontal displacements of the footings ranging from -150 kN/m² to -300 kN/m² depending on their
using the three numerical, experimental, and analytical position relative to the cavity.
methods (5.67 mm as the highest estimate for footing 2). The relative exploitation of results is presented in Fig. 18.
254 Bilel Boualleg, Nadjet Bouacha

From observing the curves of vertical and horizontal

displacements (Fig. 18), it can be seen that each curve is
symmetrical with respect to the cavity axis.
For the H/B ratio: The displacement value increases
as the cavity depth decreases.
For the L/B ratio: The displacement value increases as
the distance between the cavities decreases.
For the b/a ratio: We also notice that the displacement
Figure 16: Models used in this study: (B) width of the footing, (H) value increases with the increase in dimensions of cavity.
depth of the cavity, (L) cavity spacing, (a) cavity height, and (b)
This led us to search for an equation expressing the
cavity width.
relationship between the cavity volume and the vertical
and horizontal displacements that occur in the soil.

b/a = 0.5 b/a = 1

b/a = 1.5 b/a = 2

b/a = 2.5 b/a = 3

Figure 17: Stress in the yy plane.

 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 255

0,005 b/a = 0.5 b/a = 1 0,003 b/a = 0.5 b/a = 1

b/a = 1.5 b/a = 2 Positon [m] b/a = 1.5 b/a = 2
Vertical displacement [m] 0 b/a = 2.5 b/a = 3 b/a = 2.5 b/a = 3
0,002

Horizontal displacement [m]

0 10 20 30 40
-0,005
0,001
Position (m)
-0,01
0
-0,015 0 10 20 30 40
-0,001
-0,02
Ratio b/a Ratio b/a
-0,002

0,1 H/B = 0.5 H/B = 1 0,025

H/B = 0.5 H/B = 1
0,08 H/B = 1.5 H/B = 2 H/B = 1.5 H/B = 2
0,02

Horizontal displacement
H/B = 2.5 H/B = 3
Vertical displacement [m]

H/B = 2.5 H/B = 3

0,06 0,015
0,04 0,01

[m]
0,02 Position [m] 0,005 Position [m]
0
0
0 10 20 30 40
-0,02 0 10 20 30 40
-0,005
-0,04 Ratio H/B
Ratio H/B -0,01

0,005 L/B = 0.5 L/B = 1 L/B = 0.5 L/B = 1

0,004
L/B = 1.5 L/B = 2 L/B = 1.5 L/B = 2
0
Horizontal displacement

L/B = 2.5 L/B = 3

Vertical displacement [m]

0,003 L/B = 2.5 L/B = 3

0 10 20 30 40
-0,005 0,002
Position [m]
[m]

-0,01 0,001
-0,015 0
0 10 20 30 40
-0,02 -0,001 Position [m]
Ratio L/B
-0,025 -0,002 Ratio L/B

Figure 18: Vertical and horizontal displacements according to the three ratios.

The different profiles were analyzed based on the are exposed due to the rupture of cavities beneath the
parameters a, b, H, B, and L, with trend lines plotted. foundations.
The correlation coefficients R², ranging from 0.90 to 0.98, According to Figure 20, we notice that the most
demonstrate a good fit of the data. The equations obtained influential ratio on the stability of the structure is the
from the curves in Figure 19 allow for accurate prediction depth ratio of the cavity. This prompts us to focus our
of the horizontal or vertical collapse value based on the analysis on the significance of the void depth and its
characteristics of the void beneath the structure (volume, impact on structural instability. We also observe that the
depth, and spacing), as well as the initial displacement significance of volume, depth, and dimensions is more
(without void), as illustrated in Equations 5 and 6. significant when the ratio lies between 0 and 2.5. Beyond
a ratio of 2.5, all three parameters will have an equivalent
H−0.766 L −0.17
Uy = Uy0 (0.347 + 0.9383 + 4.0803
b
a B
+ 4.0803
B
) (5) impact.

H−1.016 L −0.34
) (6)
b
Ux = Ux0 (0.3525 + 0.9514 + 8.3075 + 1.8262
5 Conclusion
a B B

Equations 5 and 6 obtained are specific to the type of soil

used in this study and provide an initial prediction. They In this study, we verified the validity of the numerical
can be used to anticipate the risks to which structures model by relying on both the results from an experimental
256 Bilel Boualleg, Nadjet Bouacha

Ux/Ux0
3

Uy/Uy0
2 2

1 1 y = 0.347x + 0.9383
y = 0.3525x + 0.9514
R² = 0.9675
R² = 0.9566
0 0
0 0,5 1 1,5 2 2,5 3 0 0,5 1 1,5 2 2,5 3
Ratio b/a Ratio b/a

10 20

Ux/Ux0
y = 4.0803x-0.766
Uy/Uy0

8 15
R² = 0.9021
6 y = 8.3075x-1.016
10
R² = 0.9807
4
5
2
0
0
0 0,5 1 1,5 2 2,5 3 0 0,5 1 1,5 2 2,5 3
Ratio H/B Ratio H/B

2,5 3
Ux/Ux0

Uy/Uy0

2
1,5 2
y = 1.8262x-0.34
1 R² = 0.9861 y = 2.1087x-0.17
1
0,5 R² = 0.9783
0 0
0 0,5 1 1,5 2 2,5 3 0 0,5 1 1,5 2 2,5 3
Ratio L/B Ratio L/B

Figure 19: The variation in displacements under the three ratios.

b/a
0,1 H/B model conducted by one of the researchers and analytical
vertical displacement [m ]

0,08
L/B laws. We furthered our investigation through a parametric
Without cavity
study based on varying the ratio between the volume,
0,06
depth, and distance between cavities.
0,04 – The obtained results are reliable, providing the
0,02 developed model with a solid foundation for future
0
case simulations.
0 0,5 1 1,5 2 2,5 3 – The stability of the footing above the cavity is
Ratio influenced by several parameters related to it, notably
its volume, depth, and the distance between two
0,03 cavities.
b/a
horizontal displacement

H/B – A correlation has been established between the footing

0,02 L/B displacement and the cavity properties (Equations
Without cavity
5 and 6). Although specific to the studied soil type,
[m]

0,01 this relationship can be considered to assess the risk

posed to structures due to the fracture of underlying
0 cavities.
0 0,5 1 1,5 2 2,5 3
Ratio In this regard, this research was primarily proposed to
analyze the cavity failure process leading to structural
Figure 20: Displacement assembly according to the three ratios.
 Analysis of the behavior of structures under the effect of progressive rupture of a cavity 257

instability due to the effect of differential displacement. References

This results in high stresses on load-bearing elements
and the consequent appearance of cracks and fractures. [1] Aftes. The convergence confinement method.
Recommendations, Tunnels, and underground structures, July
Through the use of finite element methods, other
1982, pp. 98-105.
parameters such as the position of the structure,
[2] Al Abram, I. Study on a two-dimensional reduced model of
anchoring of the footings above the void, and shape the displacement field induced by the digging of a tunnel at
can be addressed to better understand the behavior of shallow depth. Interaction with existing works. PhD thesis,
structures facing abrupt void collapses. These findings INSA Lyon 1998.
necessitate, in a subsequent study, proposing technical [3] Al Heib, M., Nghiem, H. L., and Emeriault, F.” Understanding
sinkhole consequences on masonry structures using large
solutions to reduce potential disasters and analyzing their
small-scale physical modeling.” 7th international conference
effectiveness in mitigating settlement and maintaining on Case Histories in Geotechnical Engineering The westin
maximum structural stability until human intervention. Chicago North Shore Wheeling Illiois 2013.
The next phase of this study will involve predicting the [4] Atkinson, J. and Potts, D. “Subsidence above shallow tunnels
bearing capacity of the soil based on the three parameters, in soft ground.” Journal of the Geotechnical Engineering
Division,1977,103, 307– 325.
dimensions, depth, and spacing, and establishing their
[5] Bazant, Z. Introduction to scale effects on structural strength.
relationship with footing design principles. This risk
Paris: Hermes, 2004.
management approach aims to prevent any significant [6] Boramy Hor. Assessment and mitigation of the consequences
damage and strengthen the margin of safety. of ground movements on buildings: experimental and
numerical approaches. PhD thesis, INSA Lyon 2012.
[7] Boumalla, E. Definition and feasibility study of a reduced
physical model for the study of the impact of subsidence and
Symbols sinkholes on surface buildings Dissertation of the internship
of the engineer of the Mines of Douai. Rapport de Master
Recherche, Ecole des Mines de Douai, France 2005.
H: height of the soil cover put at the end
[8] Burd, H J., Houlsby, G T., Augarde, C E. and Liu G. Modelling
D: tunnel diameter tunnelling induced settlement of masonry buildings. Proc Inst
x: distance to the center of the bowl Civil Eng: Geotech Eng, 2000, 143(1), 17–29.
S(x): vertical settlement at the abscissa x [9] Caudron, M., Emeriault, F. and al Heib, M. Collapse of
Smax: maximum surface settlement underground cavities and interaction with surface structures:
i: distance to inflection point experimental approach on a two-dimensional analogue model.
National Geotechnical and Engineering Geology Days, Lille,
Z0: distance between surface and tunnel axis
France 2004.
E: Young’s modulus [10] Caudron M., Emeriault F., Kastner R. & Al Heib M. 2006.
ν: Poisson coefficient Sinkhole and soil-structure interactions: Development of an
γ: volume weight of the soil experimental model. ICPMG, Hong-Kong, 04-06 Aug. 2006, pp
λ: rate of un-confinement 1261-1267.
[11] Caudron, M., Emeriault, F., al Heib, M.and R. Kastner. Numerical
V(x): horizontal displacement at abscissa x
modelling of the soil structure interaction during sinkholes.
O: opening of the cavity National Geotechnical and Engineering Geology Days Lyon,
W: width of the cavity France 2006.
Ux: horizontal displacement [12] Caudron, M. Ground movements and deformations of a
Uy: vertical displacement building following a sinkhole: experimental and numerical
Ux0: horizontal displacement without cavity approach. INSA, Lyon, France 2007.
[13] Caudron, M. Experimental and numerical study of soil-structure
Uy0: vertical displacement without cavity
interaction during the occurrence of a sinkhole. Thèse, INSA,
a: height of the cavity Lyon, France 2007.
b: width of the cavity [14] Castro, R., Trueman, R. and Halim A. A study of isolated draw
B: width of the soles zones in block caving mines using a large 3D physical model.
L: length of the soles International journal of rock mechanics & mining sciences,
44, Julius Kruttshmitt Mineral Research Center, University of
n: porosity
Queensland, Australia 2007.
C: cohesion [15] Deck O., Al Heib, M., Homand, F. & Wojtkowiak, F. Synthesis
φ: friction angle of methods for predicting the consequences of mining
ψ: angle of expansion subsidence on buildings. Application to the case of the
C: initial glue concentration Lorraine iron basin. Mineral industry techniques, 2006, 29,
RTraction: tensile strength 83-10.
258 Bilel Boualleg, Nadjet Bouacha

[16] Deck, O., Anirudth, H. Numerical study of the soil-structure [33] Nakai, T., Xu, L. and Yamazaki, H. 3D and 2D model tests and
interaction within mining subsidence areas. Computers and numerical analyses of settlements and earth pressures due to
Geotechnics, in press, 2010. tunnel excavation. Soils and Foundations, (1997), 37, 31-42.
[17] Dehousse and Arnould. Scale models of structures in Civil [34] Nghiem, H. L., Heib, M. A., & Emeriault, F.. Physical model
Engineering. Paris: Dunod, 1971, 183. for damage prediction in structures due to underground
[18] Djamel Saadi, Khelifa Abbeche, Rafik Boufarh. Model excavations. International conference on geotechnical
experiments to assess effect of cavities on bearing capacity of engineering (Geoshanghai 2014), May 2014, Shanghai, China.
two interfering superficial foundations resting on granular soil, pp.155-164. ffineris-01863823.
2020. [35] Oteo, C. and Sagaseta, C. “Prediction of settlements due
[19] Dolzhenko, N. Experimental and numerical study of models, to underground openings.” International Symposium on
Two-dimensional reduction of tunneling. Development of Numerical Models in Geomechanics, Ghent, Belgium,1982,
a specific law of behavior. PhD thesis, National Institute of 653–659.
Applied Sciences, Lyon 2002. [36] Peck, R. B. Deep excavation and tunneling in soft ground.
[20] Dyer, M., Hutchinson, M. and Evans, N. Geotechnicals aspects Proceeding of the 7th International Conference of Soil
of underground construction in soft ground, chapter Sudden Mechanics, Mexico: State-of-the-Art, 1969, Vol.3, pp.225-290.
valley sewer : a case history, 671–676. Rotterdam 1996 : A.A. [37] Schneebeli, G. “A mechanics for cohesionless soils” Minutes
Balkema. of the Academy of Sciences, Paris: 1956, Tome 243, pp. 2647-
[21] Dyne, L. The prediction and occurrence of chimney subsidence 2673.
in southwestern Pennsylvania. Master of Science in mining and [38] Schneebeli, G. A mechanical analogy for the study of the
minerals engineering, Virginia Polytechnic Institute and State stability of two-dimensional earth structures. Proceedings
University 1998. of the 4th International Conference on Soil Mechanics &
[22] Keba Lukueta E ., Isobe K. Bearing Capacity of a Shallow Foundation Engineering (I.C.S.M.F.E.). London 1957.
Foundation above the Soil with a Cavity Based on Rigid Plastic [39] Shahin, H., Nakai, T., Hinokio, M., Kurimoto, T. and Sada,
Finite Element Method. Applied Sciences. 2024; 14(5):1975. T. Influence of surface loads and construction sequence on
https://doi.org/10.3390/app14051975. ground response due to tunneling. Soils and Foundations,
[23] Garnier, J. Physical models in geotechnics. Evolution of 2004, 44, 71-84.
experimental techniques and fields of application. French [40] Sung, E., Shahin, H., Nakai, T., Hinokio, M. & Makoto, Y.
Journal of Geotechnics, 2001a, 97, 3–29. 2006. Ground behavior due to tunnel excavation with existing
[24] Garnier, J. Physical models in geotechnics. Method validation foundation. Soils & Foundations, 2006, 2:189-207.
and application examples. Revue Française de Géotechnique, [41] Yongyao Wei and al. A numerical simulation study on the
2001b, 98, 5–28. evolutionary characteristics of the damage process of karst soil
[25] Kastner, R. Deep Excavations in Urban Sites. PhD thesis, INSA, cavity under positive pressure effect. Geohazard Mechanics.
Lyon 1982. Volume 1, Issue 4, 2023, Pages 288-296, ISSN 2949-7418,
[26] Keawsawasvong, S. Limit analysis solutions for spherical https://doi.org/10.1016/j.ghm.2023.10.002.
cavities in sandy soils under overloading. Innov. Infrastruct.
Solut. 6, 33 (2021). https://doi.org/10.1007/s41062-020-
00398-5
[27] Kikumoto, M., Shanin, H. M., Nakai, T., Nagata, M. and Toda,
K. Influences of tunnel excavation on the neighboring group-
pile foundation. Proc. of 54th Geotechnical Engineering
Symposium, 2009, 54, 355-362.
[28] Kratzsch, H. Mining subsidence Engineering. Spring-Verlag,
Berlin/Heideilberg, New York 1983.
[29] Laefer, D. F. Predicting and assessment of ground movement
and building damage induced by adjacent excavation, PhD
thesis, The University of Illinois, Urbana, Ill. (2001).
[30] Lake, L., Rankin, W,and Hawley, J. Prediction and effects of
ground movements caused by tunneling in the soft ground
beneath urban areas. Construction Industry Research and
Information Association, London 1992, UK: CIRIA Project Report
30.
[31] Lee, Y., Bassett, R. Influence zones for 2D pile-soil-tunnelling
interaction based on model test and numerical analysis.
Tunneling and underground space technology 22 (2007) 325-
342.
[32] Mahamma, F. Behaviour of structures built on the surface
during subsidence of old careers. DEA thesis, INSA de Lyon
2002.