
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72220
10.1038/s41598-024-72220-6
Article
2D discrete element analysis of the footing above excavated circle in soil
Sarfarazi Vahab 1
Tabaroei Abdollah a.tabaroei@eshragh.ac.ir

2
Dias Daniel 3
Abedi Mohammadmahdi 4
Vakili Amir Hossein 5
Zhao Yang 6
1 https://ror.org/01hgb6e08 grid.459564.f 0000 0004 0482 9174 Department of Mining Engineering, Hamedan University of Technology, Hamedan, Iran
2 Department of Civil Engineering, Eshragh Institute of Higher Education, Bojnourd, Iran
3 grid.5676.2 0000000417654326 3SR Lab, Univ. Grenoble Alpes, CNRS, Grenoble INP, 38000 Grenoble, France
4 https://ror.org/037wpkx04 grid.10328.38 0000 0001 2159 175X Department of Civil Engineering, University of Minho, ISISE, ARISE, Campus de Azurém, 4800-058 Guimarães, Portugal
5 https://ror.org/04wy7gp54 grid.440448.8 0000 0004 0384 3505 Department of Environmental Engineering, Faculty of Engineering, Karabuk University, Karabuk, Türkiye
6 grid.9227.e 0000000119573309 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071 China
13 9 2024
13 9 2024
2024
14 213994 2 2024
4 9 2024
© The Author(s) 2024
2024
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The bearing capacity and settlements of surface foundations located on a soil slope are the important issues that have to be considered by geotechnical engineers for the design. The presence of an underground void beneath the footing can affect the foundation stability and can lead to serious structure damages. In this study, the results of two-dimensional (2D) discrete element (DE) and finite element (FE) analyses of a surface footing on a soil slope above a void are presented. To validate the numerical model results, the DE results obtained have been compared with experimental test presented in the previous study. After validation of the DE numerical model, parametric studies were carried out to evaluate the effect of important factors on the surface footing performance. The studied parameters include the horizontal spacing of the void axis relative to the slope edge (SH), the vertical spacing of the void crown relative to the footing base (SV), the horizontal spacing of the footing edge relative to the slope edge (De) and the void diameter (Dv). The effects of these parameters on the pressure-settlement curves and the contact force distributions in the soil slope are presented and discussed. The results showed that the footing bearing pressure increases with an increase of SH, SV and De but decreases when Dv increases. The behavior of a surface footing on a soil slope above a void significantly depends on the SV value.

Keywords

Soil slope
Void
Discrete element method
PFC 2D
Footing pressure
Subject terms

Engineering
Mathematics and computing
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The bearing capacity and settlements of surface foundations located on soil slopes are important issues that have to be considered by geotechnical engineers. When large manmade structure was built on the soil slope, the soil was compacted based on the weight of structure. The effective radius may be reach to an underground structures where situated below the huge foundation. Therefore, the foundation can cause local failure and or completely disaster of underground structure. This phenomenon will be bold by decreasing the distance between foundation and underground excavation, too. Wei et al.1 conducted a comprehensive analysis of the disaster progression in the slope-tunnel system. They emphasized the similarity between the stress state of the slope and the displacement-time curve of the system during the entire failure process. The overall disaster process consists of three stages: initial creep, accelerated development, and damage, as illustrated in Fig. 1.Fig. 1 Catastrophic process surface foundations located on soil slopes above the tunnel1.

The situation of the underground tunnel related to surface foundation and its distance from the soil slope affect the stability of these structures. In other word, large displacement concurrent with huge disaster will be occurred under critical stationing configuration of pile-tunnel and slope. The bearing capacity of surface foundations can change with the presence of different factors such as the foundation type, the soil type beneath the foundation, the groundwater level presence, De and β. In some cases such as gas and water networks or old conduits and tunneling, voids can be located under the building foundations. The presence of an underground void under a footing can affect the foundation stability and can lead to serious structure damages. Results of the previous studies showed that the interaction between underground voids and shallow foundations is of great importance on the behavior of shallow foundations.

The use of DEM and FEM in solving of geotechnical problems was reported in previous studies2–5. In the literature some experimental, numerical and analytical studies were performed to study the performance of soil slopes6–15. In addition, several studies were performed on the stability of footing located above voids16–26. Baus and Wang16 evaluated the void position influence and its geometry on the bearing capacity of a footing located above a void. Badie and Wang17 and Wang and Badie18 conducted experimental and numerical analysis to study the surface footings behavior resting on a void. Jao and Wang21 by conducting numerical analysis, focused on the effects of the concrete lining stiffness, dimension and position of the void on the bearing capacity of shallow foundations. Based on numerical analyses, they proposed a mathematical expression to determine the reduced footing bearing capacity due to the void. Rodriguez-Roa22 measured the variation of the ground surface settlement profile due to the digging of a void in a granular soil. Kiyosumi et al.23 by conducting some loading tests on shallow foundations founded on sedimentary rocks considered rectangular and square voids. They found that the failure mechanisms depend upon the void location. It is then important to know if the void is located at the footing center with an eccentricity from the footing center. Lavasan et al.25 applied 2D FE numerical analyses to evaluate the effect of twin voids on the bearing capacity of a shallow foundation. They showed that the foundation failure mode depends on the size and position of the voids and the footing. According to the results, they suggested some critical values for the bearing capacity variation of a surface footing.

Nowadays, the use of the soil reinforcement techniques is widely used in geotechnical engineering. The performance of the footings on geosynthetic-reinforced soil with voids was reported in previous studies27–33. Das and Khing27 carried out a model test to study the effect of a void on the bearing capacity of unreinforced and geogrid-reinforced foundations and found that the presence of a void reduced the foundation bearing capacity, but by adding one geogrid layer the capacity can substantially increases. Tafreshi and Khalaj29 by conducting experimental tests evaluated the effect of a geogrid reinforcement on pipes deformation and the soil surface settlements under repeated loadings. They concluded that geogrid layers can significantly reduce the vertical pipe diameter impact and the soil surface settlements. Tafreshi et al.32 carried out different experimental tests to investigate the effects of some important factors on the bearing capacity and settlements of unreinforced and geogrid-reinforced footing above a void. They reported that for a certain bearing pressure, the settlement of the footing decreases as RD of the soil beneath the footing and the number of geogrid layers or embedment depth of the void increases. Some analytical studies were presented in the literature to design geosynthetic-reinforced soil techniques overlying voids and sinkholes28,34,35.

From previous studies, it was clearly showed that no comprehensive studies were conducted to understand the void effect on the performance of a surface footing on a soil slope. The determination of the bearing capacity and settlements of a surface footing on a soil slope above a void is a challenging issue for geotechnical engineers and many questions remain about it. In this paper, a number of 2D DE and FE analysis are presented to evaluate the effective parameters on the performance of a surface footing on a soil slope above a void. These parameters include SH, SV, De and Dv. Effects of these parameters on the pressure-settlement curves and interpretation of the results obtained from DE and FE analysis are presented in the later sections.

Problem definition

The geometry and definition of the problem parameters including the soil slope, the surface footing and void is presented in Fig. 2. As it can be seen, a rigid surface footing with a B = 100 mm is placed on the soil slope. Single circular void with variable diameters are considered to be located below the footing base. The effect of the parameters SH, SV, De and Dv are evaluated on the performance of the footing. It is important to specify an appropriate boundary for the geometric model so that boundary conditions have the minimum effect on the numerical analysis results. According to the values suggested by Wang and Badie18 and Jao and Wang 21, the dimensions of the numerical model should be extended between three and 5Dv from the void axis. According to the results obtained from parametric study, a value of 3Dv is considered in this study. The values of variable parameters consider in DE analysis are illustrated in section "Results and discussion".Fig. 2 Problem definition and notation used in DE analysis (not to scale).

DE modeling

The software PFC2D V5.2 (discrete elements) was used to complete the numerical analysis presented in this paper36. In this code, at each step of time, the Newton's second law is applied for each particle in order to correct the velocity and location of the particle. The result of these calculations at each step is a series of contact forces for all the particles (Fig. 3). These forces are shear and normal forces.Fig. 3 Contact force model36.

Based on the new particles positions, the bonding forces are created by the relative displacements of each particles pair. The radii of the disc-shaped particles were created using a different value. At each step, using a specific coefficient, the particle radius can be increased to achieve the desired soil porosity 37–42. In order to prepare the boundaries around the model, two types of walls were used37:Infinite walls: In this case, boundaries of the sample are extending indefinitely in all directions.

Finite walls: In this case, boundaries are convex surfaces such as cylinders, spheres or other special geometric shapes.

By using the walls motion that are boundaries, the particles are affected by forces. Special connection conditions occur at the wall edges. The movement speed of the walls (transfer or rotation) as well as the forces and moments on each wall can be controlled. The micro-parameters of the disks are related to the contact bonds, contact Young modules, and ratio of normal stiffness to shear stiffness.

Material production process

The soil mass is considered as a compact set of particles of non-uniform size considering spherical and circular shapes in which the particles are bonded together. The forces between the grains and the compaction of the particle set are of an arbitrary macroscopic scale and isotropic. The material production process has the following five stages37: initial compaction of the particles, application of the isotropic stress, reduction of the number of floating particles, contact between the particle and removing the model walls.

Particle generation in PFC 2D

The minimum and maximum radius of the particle assembly are Rmin = 0.25 cm and Rmax = 0.415 cm, respectively. They were inserted at random positions at a given area. The particle radii were chosen to have a relatively uniform distribution in the sample, Fakhimi et al.43.

Determination of the proper micro-parameters (calibration)

The constitutive model at a ball-ball contact consists of three parts: a stiffness model, a slip model, and a bonding model. The stiffness model provides an elastic relation between the force and relative displacement in a contact. The slip model is a relation between the shear and normal contact forces. The contact-bond can only transmit a force in a contact. A biaxial test by confining a rectangular sample within four walls was simulated (Fig. 4). The left and right walls simulate the stress confinement and the top and bottom walls simulate loading platens. The sample is loaded in a strain-controlled mode by specifying the velocities of the top and bottom walls. The stresses and strains applied on the sample were determined in a macro-mode by summing the forces acting upon, and the relative distance between the appropriate walls. The material response is evaluated by tracking the various stress and strain quantities. The axial deviatoric stress versus the axial strain for the biaxial test on bonded granular material is presented, and the Mohr’s circle is drawn to reach the failure envelope. In Table 1, the soil physical parameters in Lee and Manjunath44 experimental tests were considered. A biaxial test by confining a rectangular sample within four walls is simulated (Fig. 4).Fig. 4 Biaxial test by confining a rectangular sample.

Table 1 Soil parameters used in Lee and Manjunath44 experimental tests.

Parameter	Gs (-)	Cu (-)	Cc (-)	γdmin (kN/m3)	γdmax (kN/m3)	φ (degree)	
Value	2.60	4.60	1.10	16	18.20	38	

Numerical model parameters

Using DEM, the soil slope, surface footing and void are considered. Figure 5, shows the DE numerical model that is considered for the comparison with the Lee and Manjunath44 results. The dimensions of the model are 1800 mm long and 1000 mm high and B = 100 mm. The average number of particles used for the soil slope simulation were 19,987. A vertical pressure of 1 kPa with an increasing range was directly applied to the bonded particles till the numerical samples failed. The pending process of loading in at equilibrium state of each loading step, the applied vertical pressure, and consequent footing settlement were recorded. In all DE analyses, the pore-water pressure effect is not considered. Tables 2 and 3, present the soil input parameters and the footing particle used in the DE analysis, respectively.Fig. 5 DE numerical model of the problem validated by the Lee and Manjunath44 results.

Table 2 Parameters of the soil used in the DE analysis (obtained from calibration).

Parameter	γ (kN/m3)	ω (%)	kn (N/m)	ks (N/m)	n-bond (N/m)	s-bond (N/m)	
Value	26.50	0	5 × 105	2.5 × 105	0.04	0.08	

Table 3 Parameters of the footing used in the DE analysis.

Parameter	γ (kN/m3)	Rmin/Rmanx (-)	kn (N/m)	ks (N/m)	n-bond (N/m)	s-bond (N/m)	μ (-)	
Value	30	0.25/0.415	1 × 106	1 × 106	50	50	0.20	

FE modeling

The FE simulations of the void positions on the footing response that located on a soil slope were carried out using the program PLAXIS 2D V8.6. The software is worked based on FEM and developed for the analysis of stress-deformation and stability of different geotechnical problems45. For simulation the soil behavior in PLAXIS, different constitutive models has been presented. One of these models is an elasto-plastic hyperbolic model that called the HS model can simulate the non-linear behaviour of the sand used in this study. The plate element used to model the footing and the void lining, respectively. An interface element used to model SSI effects. A load is located on the footing surface and applied to that until the failure occurred.

Validation of the numerical model

In this study, the results obtained from the DE numerical model were verified by experimental results reported in the literature. Lee and Manjunath44 carried out laboratory model tests and numerical analysis to study the performance of a surface footing on an unreinforced and geosynthetic-reinforced soil slope. In that study, the dimensions of the testing tank were 1800 mm (long) × 900 mm (wide) × 1200 mm (high). The rigid strip footing has a length equal to the width of the tank made from steel box. They used the Mumbra sand in their experimental tests. The values of De and β were 100 mm and 26.56°, respectively. In this study, only the unreinforced test carried out by Lee and Manjunath44 is numerically simulated. The results obtained from the numerical modeling and those from laboratory model test carried out by Lee, and Manjunath44 are presented in Fig. 6. In this figure, the footing pressure is plotted against the footing settlement. As shown in Fig. 6, the results obtained from the DE analysis are lower than those from the laboratory model test conducted by Lee and Manjunath44 (except for the values 6 mm ≤ s ≤ 8 mm from which two pressure-settlement curves are consistent with each other). For the footing settlement values lower than 6 mm (s ≤ 6 mm), the increase in the slope of the footing pressure-settlement curve (footing stiffness) is more evident. Beyond this value, the footing stiffness remains constant. By comparing the results obtained from the numerical analysis and the laboratory model test, it can be seen that there is a good agreement between them. This reasonable agreement indicates the appropriate accuracy of the numerical modeling.Fig. 6 Comparison between the results obtained from the current study and the results of Lee and Manjunath44 study.

Results and discussion

In this section of the paper, the DE analysis results in terms of pressure-settlement curves and contact force distributions in the soil slope are presented. Four series of DE and FE analysis were performed to evaluate the surface footing behavior on the soil slope above a void. Under these four series of DE analysis, the effects of different values of SH, SV, De and Dv parameters on the footing performance are investigated. In all of the analysis β = 25°. The value of variable parameters which are considered in the DE analysis, are presented in the following sections. The results obtained from the numerical modeling of this study can help the engineers to have a better understanding of the surface footing behavior on the soil slope above a void.

Effect of the SH values

To investigate the SH effect on the surface footing performance on a soil slope above a void, six DE analyses were performed (analysis series one). In the analysis of this series, the value of SV, De and Dv were kept constant and equal to 400 mm, 100 mm and 100 mm, respectively, but the SH value was considered variable from 150 to 400 mm. The position of the void under different values of SH is shown in Fig. 7.Fig. 7 Position of the voids under different SH values; (a) SH = 150 mm, (b) SH = 250 mm and (c) SH = 400 mm.

The variations effect of the SH value on the footing pressure-settlement curves are presented in Fig. 8a. As it can be seen in Fig. 8a, for values of s ≤ 1 mm, the footing response to the SH value variation is similar and all of the pressure-settlement curves are consistent with each other. By increasing in the SH value, the bearing pressure of the footing increases, where the best performance of the footing can be seen, as the value of SH = 400 mm. As the value of SH increases, the applying pressures from the footing on the void walls decreases. For the values SH ≥ 350 mm, the changes in increasing the bearing pressure of the footing become slower.Fig. 8 (a) Footing pressure-settlement curves for different SH values, (b) effect of axial pressure on the axial displacement for different void configuration (analysis series one).

Figure 8b shows the effect of axial pressure on the boundary displacement for each cavity configuration. When SH has lowest value (SH = 150 mm), lowest axial displacement was occurred in the boundary. As it can be seen, axial displacement was increased in a large rate by increasing the axial pressure. Its increasing rate was decreased up to 1.2 mm while axial pressure was 22.5 kPa. In constant axial pressure, the axial displacement was increased by increasing the SH. When SH has largest value (SH = 400 mm), largest axial displacement was occurred in the boundary. As it can be seen form Fig. 8b, axial displacement was increased in a large rate by increasing the axial pressure. Its increasing rate was decreased to 1.2 mm while axial pressure was 27.5 kPa. It’s clear that for two different void configurations, displacement of 1.2 mm were occurred in two different configurations of void. In fact situation of void affect the boundary displacement. The more be SH, the more be axial pressure to gain similar displacement.

Figure 9, shows the compression and tension contact forces chains distributions in soil slopes at the ultimate bearing pressure for the SH = 150 mm, 250 mm and 400 mm values, respectively. In this figure and in all numerical results presented here, the black and red colors represent the compression and tension forces, respectively. It can be seen that, in the soil elements beneath the footing, many compression forces are created and these forces can extend to the void crown top. Generally, the maximum tension forces are concentrated in the void roof and bottom. However, the maximum compression forces are concentrated in the void right and left. At a certain SV value and with increasing the SH value, the tension force amounts distributed on the void roof and bottom and the compression forces are distributed on the void right and left reduces. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing in the SH value.Fig. 9 Contact force chains distributions in soil slopes for different SH values obtained from DE models; (a) SH = 150 mm, (b) SH = 250 mm and (c) SH = 400 mm.

Figure 10a–i shows the horizontal displacement distribution, vertical displacement distribution and vertical stresses distribution created in soil slopes for different SH values obtained From FE analyses, respectively. It can be seen from Fig. 10a–c that the horizontal displacement of slope surface was decreased by increasing the SH value. Also, from Fig. 10d–f it’s clear that the vertical displacement of ground surface was decreased by increasing the SH value and soil slope. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing the SH value (Fig. 10 g–i).Fig. 10 (a, b and c) Horizontal displacement distribution; (d, e and f) Vertical displacement distribution and (g, h and i) Vertical stresses distribution in soil slopes for different SH values obtained from FE models.

Effect of the SV values

In the series two analysis, three DE analyses were performed to study the SV effect on the surface footing performance on the soil slope above a void. In the analysis of this series, only the SV value was varied while the other parameters including SH, De and Dv were kept constant and equal to 150 mm, 100 mm and 100 mm, respectively. The SV value was considered variable from 100 to 400 mm. In Fig. 11, the void position under different values of SV is illustrated.Fig. 11 Position of the void under different SV values; (a) SV = 100 mm, (b) SV = 200 mm and (c) SV = 300 mm.

The footing pressure-settlement curves variations for different SV values presents in Fig. 12. From Fig. 12, it is clear that the surface footing behavior on the soil slope above a void significantly depends on the SV value. The footing bearing pressure increases with increasing the SV value. For s = 1.5 mm, when the SV value increases from 100 to 400 mm, the footing bearing pressure increased of 42%. When the SV value increases, the void has a little effect on the footing bearing pressure. In this case, the failure mechanism at the soil beneath the footing is a shear failure.Fig. 12 Footing pressure-settlement curves for different SV values (analysis series two).

The compression and tension contact forces chains distributions in the soil slopes at the ultimate bearing pressure for the values of the SV = 100 mm, 200 mm and 300 mm are presented in Fig. 13. When SV = 100 mm and by applying the footing load, significant compression forces occurred in the void walls. They can lead to an instability of the whole system and consequently a lower footing bearing pressure (footing pressure-settlement curve for SV = 100 mm in Fig. 13). Such an effect reduces with increasing the SV value. At certain SH value and with increasing the SV value, the tension force amounts distributed on the void roof and bottom and the compression forces distributed on the void right and left decreases. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing in the SV value.Fig. 13 Contact force chains distributions in soil slopes for different SV values; (a) SV = 100 mm, (b) SV = 200 mm and (c) SV = 300 mm.

Figure 14a–i shows the horizontal displacement distribution, vertical displacement distribution and vertical stresses distribution created in soil slopes for different SV values obtained From FE analyses, respectively. It can be seen from Fig. 14a–c that the horizontal displacement of slope surface was decreased by increasing the SV value. Also, from Fig. 14d–f it’s clear that the vertical displacement of ground surface was decreased by increasing the SV value. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing the SV value (Fig. 14 g–i).Fig. 14 (a, b and c) Horizontal displacement distribution; (d, e and f) Vertical displacement distribution and (g, h and i) Vertical stresses distribution in soil slopes for different SV values obtained from FE models.

Effect of the De values

One of the important parameters in the design of a surface footing on a soil slope is De value. To investigate the effect of De on the surface footing behavior on soil slope above a void, three DE analyses were performed (analysis series three). In the numerical analysis of this series, the value of the SH, SV and Dv parameters were kept constant equal to 150 mm, 400 mm and 100 mm, respectively. The value of De was varied from 100 to 400 mm. The position of a surface footing under different values of De, is shown in Fig. 15.Fig. 15 Position of the void under different De values; (a) De = 200 mm, (b) De = 300 mm and (c) De = 400 mm.

The footing pressure-settlement curves variations at different value of De, are shown in Fig. 16. The results show that for the value s ≤ 1 mm, the footing response to variation in the De value is similar. As a result, the footing bearing pressure increases with increasing the De value. In s = 4 mm, the footing bearing pressure increases up to 15.13% when the value of De increases from 100 to 400 mm.Fig. 16 Footing pressure-settlement curves for different De values (analysis series three).

The compression and tension contact forces chains distributions in the soil slopes at the ultimate bearing pressure for the values of the De = 200 mm, 300 mm and 400 mm are illustrated in Fig. 17. In the soil elements beneath the footing, many compression forces are created, but as the depth increases, the compression forces intensity reduces. From Fig. 17 with increasing the value of De, the intensity of compression and tension forces applied to the void walls decreases (see Fig. 17c in comparison to Fig. 17a). Finally, the vertical stresses distribution created in soil slopes was decreased by increasing in the De value.Fig. 17 Contact force chains distributions in soil slopes for different De values; (a) De = 200 mm, (b) De = 300 mm and (c) De = 400 mm.

Figure 18a–i shows the horizontal displacement distribution, vertical displacement distribution and vertical stresses distribution created in soil slopes for different De values obtained from FE analyses, respectively. It can be seen from Fig. 18a–c that the horizontal displacement of slope surface was decreased by increasing the De value. Also, from Fig. 18d–f it’s clear that the vertical displacement of ground surface was decreased by increasing the De value. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing the De value (Fig. 18 g–i).Fig. 18 (a, b and c) Horizontal displacement distribution; (d, e and f) Vertical displacement distribution and (g, h and i) Vertical stresses distribution in soil slopes for different De values obtained from FE models.

Effect of the Dv values

Dv value has important effect on the soil slope response. To investigate the Dv effect on the behavior of a surface footing on a soil slope above a void, three DE analyses were performed (analysis series four). In the numerical analysis of this series, the value of SH, SV and De parameters were kept constant and equal to 150 mm, 400 mm and 100 mm, respectively, but the value of Dv varied from 50 to 125 mm. The position of the surface footing under different values of Dv, is shown in Fig. 19.Fig. 19 Position of the void under different Dv values; (a) Dv = 50 mm, (b) Dv = 75 mm and (c) Dv = 125 mm.

Figure 20 displays the effect of the different value of Dv on the footing pressure-settlement curves. This figure clearly shows how the footing bearing pressure varies with the settlement under different value of Dv. At the same void position and slope geometry, the footing bearing pressure decreases by an increase of the Dv value. Increasing Dv can leads to an increase of the void walls displacements and finally a void instability.Fig. 20 Footing pressure-settlement curves for different Dv values (analysis series four).

The compression and tension contact forces chains distributions in the soil slopes at the ultimate bearing pressure obtained from the DE analysis for the Dv = 50 mm, 75 mm and 125 mm values are shown in Fig. 21. In the soil elements beneath the footing, many compression forces are created, but as the depth increases, the compression forces intensity reduces.Fig. 21 Contact force chains distributions in soil slopes for different Dv values; (a) Dv = 50 mm, (b) Dv = 75 mm and (c) Dv = 125 mm.

Figure 21 also shows that when Dv = 125 mm and by applying a load to the footing, a significant compression forces occurred to the void walls. It can lead to an instability of the whole system and consequent lower the bearing pressure in the footing. In addition, with decreasing the Dv value, the compression and tension forces intensity applied to the void walls reduces (Fig. 21a in comparison to Fig. 21c). Finally, the vertical stresses distribution created in soil slopes was decreased by increasing in the Dv value.

Figure 22a–i shows the horizontal displacement distribution, vertical displacement distribution and vertical stresses distribution created in soil slopes for different Dv values obtained from FE analyses, respectively. It can be seen from Fig. 22a–c that the horizontal displacement of slope surface was decreased by increasing the Dv value. Also, from Fig. 22d–f it’s clear that the vertical displacement of ground surface was decreased by increasing the Dv value. Finally, the vertical stresses distribution created in soil slopes was decreased by increasing the Dv value (Fig. 22g–i).Fig. 22 (a, b and c) Horizontal displacement distribution; (d, e and f) Vertical displacement distribution and (g, h and i) Vertical stresses distribution in soil slopes for different Dv values obtained from FE models.

Discussion

Both of the 2D DE and FE model used in this research can predict the surface footing behavior on a soil slope above a void with good accuracy. When SH has lowest value (SH = 150 mm), lowest axial displacement was occurred in the boundary. As it can be seen, axial displacement was increased in a large rate by increasing the axial pressure. Its increasing rate was decreased to 1.2 mm while axial pressure was 22.5 MPa. In constant axial pressure, the axial displacement was increased by increasing the SH. When SH has largest value (SH = 400 mm), largest axial displacement was occurred in the boundary.

Axial displacement was increased in a large rate by increasing the axial pressure. Its increasing rate was decreased to 1.2 mm while axial pressure was 27.5 MPa. Its clear that for two different void configurations, displacement of 1.2 mm were occurred in two different configurations of void. In fact situation of void affect the boundary displacement. The more be SH, the more be axial pressure to gain similar displacement. For s = 4 mm, the footing bearing pressure increases up to 15.13% by an increase of the De value from 100 to 400 mm. With increasing the De value, the intensity of the compression and tension forces applied to the void walls decrease. With decreasing the Dv value, the compression and tension forces intensity applied to the void walls reduces.

Both of the DE results and FE results show that when a void was situated exactly below the footing, large tensile stresses were distributed at the roof and floor of a void. Also, biggest compressive pressures were concentrated at the left and right sides of a void. Both of the tensile stress and compressive stress were decreased by increasing the distance between a void and surface footing. When a void was situated exactly below the footing, large vertical displacement was occurred at the roof of it. Also, biggest horizontal displacement was induced at the left and right sides of a void. Both of the vertical displacement and horizontal displacement were decreased by increasing the distance between a void and surface footing.

When tunnel was situated exactly below a void, large footing settlement was occurred at the ground surface. Footing settlement were decreased by increasing the distance between a void and surface footing.

Conclusions

To investigate the surface footing behavior on a soil slope above a void, four series of 2D DE and FE analysis were performed. The effects of important parameters such as SH, SV, De and Dv on the pressure-settlement curves and the contact force distributions in the soil slope were evaluated. The following conclusions can be drawn from this study:The footing bearing pressure increases by an increase of the SH value. The best performance of the footing can be seen, for a value of SH = 400 mm.

The footing bearing pressure increases with an increase of the SV value. Additionally, the surface footing behavior on a soil slope above a void significantly depends on the value of SV.

By applying a load to the footing with a void located at SV = 100 mm, significant compression forces occurred to the void walls. They can lead to an instability of the whole system and consequently lower the footing bearing pressure.

The footing bearing pressure increases with an increase of the De value.

At the same void position and slope geometry, the footing bearing pressure decreases with an increase of the Dv value. Increasing Dv can leads to a displacement increase in the void walls and finally can lead to a void instability.

In this paper the influence of cavity diameter, its situation and foundation position on the settlement and horizontal displacement of tunnel and soil slope were examined. The authors will study the influence of interaction between foundation and underground space on the soil slope failure in new future.

Abbreviations

DE Discrete element

DEM Discrete element method

FE Finite element

FEM Finite element method

HS Hardening soil

RD Relative density

SSI Soil-structure interaction

2D Two-dimensional

List of symbols

B Footing width

Cc Coefficient of curvature

Cu Uniformity coefficient

De Horizontal spacing of the footing edge relative to the slope edge

Dv Void diameter

Gs Specific gravity

kn Normal stiffness

ks Shear stiffness

n-bond Bond normal strength

Rmax The maximum radius of the particle assembly

Rmin The minimum radius of the particle assembly

s Footing settlement

SH Horizontal spacing of the void axis relative to the slope edge

SV Vertical spacing of the void crown relative to the footing base

s-bond Bond shear strength

γ Unit weight

γdmax Maximum dry unit weight

γdmin Minimum dry unit weight

ω Water content

φ Friction angle

β Slope angle

μ Friction coefficient

Author contributions

(1) Vahab Sarfarazi: Definition of the problem, analysis (2) Abdollah Tabaroei: Validation, Interpretation of the results, Formal analysis, Investigation, writing paper (original draft) (3) Daniel Dias: Writing (review and editing) (4) Mohammadmahdi Abedi: Writing (review and editing) (5) Amir Hossein Vakili: Writing (review and editing) (6) Yang Zhao: Writing (review and editing).

Data availability

All data generated or analysed during this study are included in the paper.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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