1 Introduction

The underground voids may be formed due to different reasons such as chemical activities in the areas dealing with water soluble material (e.g. limestone and dolomite) or subjected to mining, digging the aqueducts, urban installation and sewer networks creation, tunneling, subway excavation and similar activities leading to excavation of underground layers. Due to the population growth and in response to existing needs, the demand of tunnels for urban transportation has rapidly increased. These tunnels usually excavated close to the soil surface and their effects will develop to the ground level and can significantly affect the performance of shallow foundations located above these voids. Results of prior researches in this field indicated that the interaction between shallow foundation and underground voids has significant effects on the performance of shallow foundations.

Almost all of existing studies on the interaction between shallow foundation and tunnel focused on a single void. In practice, there may exist twin voids in the soil deposit which affects the bearing capacity of shallow foundations located above them. Atkinson and Cairncross (1973), Atkinson et al. (1975) and Atkinson and Potts (1977) were the first who conducted some research works on the interaction between a single void and the bearing capacity of shallow foundation. They considered the stability of a circular shallow tunnel in homogeneous and cohesionless soil. Baus (1978) numerically evaluated the stability of strip shallow footing placed above a single underground void in stiff silty clay. Wang and Baus (1980) investigated the settlement behavior of footing constructed on the single void. Baus and Wang (1983) studied the effect of void location and its geometry on the bearing capacity of the footing located above a single void. Badie and Wang (1984) and Wang and Badie (1985) considered the stability of surface strip footings on a single void in cohesive soils. They performed both experimental and numerical research work and then compared the results. Jao and Wang (1998) conducted a numerical study on the stability of a surface footing on lined underground tunnel in different soils. They focused on the effect of location and dimension of void and stiffness of concrete lining on the bearing capacity of shallow foundation. They presented a mathematical expression to calculate the reduced bearing capacity of footing due to the void installation. In addition, they introduced a critical depth in which the void no longer affects the bearing capacity of the surface footing.

Rodriguez-Roa (2002) measured the variation of settlement profile of ground surface due to digging a void in layered granular soil in Santiago city (Chile) and performed some numerical simulations for a comprehensive comparison. In recent years, the influence of dynamic loading and soil reinforcing technique (Asakereh et al. 2013) and undrained conditions (Sloan and Assadi 1991; Osman et al. 2006; Lee et al. 2014) on the stability of the voids has been studied. Kiyosumi et al. (2007) evaluated the effect of multiple square shaped voids on the yielding pressure of strip footing by keeping the shape of footing and void fixed while four different configurations for the location of the voids are assumed. To the best knowledge of the authors, the effect of the size of the footing and void on bearing capacity and failure mechanism and their correlations have not been evaluated yet. Therefore, this research focuses on investigating the bearing capacity and failure mode of the system of footing and twin voids for different widths of footing while the voids diameter, embedment depth, the eccentric distance, and the spacing between voids change.

2 Numerical Analysis Procedure

The commercially available finite element program PLAXIS was used to model the footing-twin voids system. The behavior of soils was numerically simulated considering Mohr–Coulomb failure criterion in conjunction with a non-associated flow rule (φ ≠ ψ). Since this paper mostly focuses on bearing capacity analysis, an ideal elastic–perfect plastic constitutive model is sufficient. In these types of analyses, the loading path has no significant influence on the failure state (Ghazavi and Lavasan 2008; Lee and Salgado 2005; Lee et al. 2005; Ukritchon et al. 2003; Schanz 1998). However, usage of more sophisticated constitutive model might be essential if deformation analysis is expected to be carried out for underground opening problems (Zhao et al. 2015; Vakili et al. 2014). In all analyses performed in present study, it is assumed that a fully rough and rigid strip footings are constructed on the ground surface with no embedment depth. Two different types of soil are considered. Since the friction angles for both material are considerably smaller than 30°, the dilatancy angle is assumed to be equal to zero an all analyses (Bolton 1986). In addition, the voids are assumed to have no lining.

According to the studies performed by Wang and Badie (1985) and Jao and Wang (1998), the boundaries from the void axis should be extended between 3 and 5 times of the void diameter. Therefore, by some trial analyses, the vertical and horizontal boundaries of model are chosen sufficiently far to have no effect on the results. According to the results of trial analyses, the soil media from void borders has been continued five times the void diameter. Since the width of the footing is always smaller than the void diameter (B/D < 1), this boundary provides sufficient lateral space not to restrict the formation of general failure mechanism beneath the footing. The vertical borders have been fixed by roller and the bottom of the model has been fixed in both horizontal and vertical directions.

To determine the initial state of the soil deposit, the model was geostatically analysed considering no void-footing in the domain. Accordingly, the stresses developed in different levels were obtained from these gravity analyses by taking lateral earth pressure K 0 = 1 − sin φ (Jaky 1944) into account. Subsequently, the void was created in the model and the stresses were updated in the model. Thereafter, the displacement in the nodes were set to zero and loading the footing was attempted. This procedure is exactly identical to what happened in the experimental tests and real cases. To avoid the difficulties arisen from numerical simulation of the foundation and the interface between footing and soil, the rigid strip shaped footing is simulated by applying constant prescribed settlement distribution to the soil elements at the contact surface through an incremental load calculation process. In all analyses, the numerical calculation is stopped when the total settlement of footing reaches to 0.3B, where B is the width of the footing. However, in some cases the system of the footing and twin voids gets unstable and the calculation is automatically halted by the code. In post processing, the ultimate bearing capacity is determined as the average vertical stress generated in the gauss points next to the soil-footing contact zone. To avoid the probability of singularity at the footing’s edges, a local mesh refinement is conducted around the footing.

To investigate the mesh dependency of the failure load, a number of trial analyses were conducted through the verification study. To achieve that, the soil-footing-twin voids domain was numerically simulated considering different number of elements uniformly distributed in the model. The dependency of failure loads on the number of the elements obtained from numerical simulation of two experimental studies as shown in Fig. 1. Since these studies are later taken into account as benchmarks to verify the numerical model, their details are explained in Sect. 3.

Fig. 1
figure 1

Dependency of failure pressure to mesh size for the verification analyses

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As seen, the failure load is mesh dependent while the mesh is coarsely generated (the number of the elements are <10,000). Further than this range, although the influence of element number on the failure load is vanished, the time consumed for each analysis is considerably increased. In addition, results of existing numerical researches demonstrate that the mesh size plays an important role in evaluating the size and location of the shear band. In the case of generating coarse mesh, the numerical shear bands become unconventionally thick and it is not easy to be recognized (Ebrahimian and Noorzad 2013; Huang et al. 2013; Heaney et al. 2013; Cerato 2005; Alsalieh 2004). This can affect the capability of the numerical results in accurate estimation of failure mechanism which is partly aimed in this research. Results of the trial analyses conducted in this research indicated that the model is discretized having more than 10,000 elements, the bearing capacity is no longer mesh dependent; however, the geometry of the shear band is still dependent on mesh shape. Generating finer mesh leaded to convergence of the shear band and failure mechanism as well. Accordingly, the further trial analyses indicated that the mesh independency of failure load and convergence of shear band would be fulfilled when the numerical domain comprises 45,000 elements.

Figure 2 shows a schematic sketch of appropriate mesh pattern considered in present numerical analyses.

Fig. 2
figure 2

Schematic mesh shape for numerical analyses in present study

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3 Verification of Numerical Model

In order to verify the capability of numerical method, two most relevant case studies have been numerically simulated and obtaining results are compared. Badie and Wang (1984) conducted some experimental and numerical attempts to consider the bearing capacity of a strip footing with a width of 51 mm located above a void with a diameter of 122 mm. It is assumed that the crest of the void was located 102 mm below the footing bottom and the center of the void was exactly under the center of the foundation. The tests were conducted in a rigid tank with the size of 1524 by 366 mm. A Kaolin clay type soil was used. The soil modulus of elasticity, unit weight, cohesion, and friction angle were reported to be equal to 19.87 MPa, 16.28 kN/m3, 158.7 kPa, and 8°, respectively. In this section, the tests performed by Badie and Wang (1984) are numerically simulated and the results are compared, as illustrated in Fig. 3. As seen, there is a generally reasonable agreement between results, demonstrating the capability and accuracy of numerical modeling.

Fig. 3
figure 3

Comparison between present studies and Badie and Wang’s studies

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Jao and Wang (1998) investigated numerically the influence of the underground concrete lined void on the bearing capacity of a surface footing located above the void. In this case, a strip footing with a width of 914 mm was considered. The void was placed at a depth of 914 mm from the foundation base. Three different types of soil were used by Jao and Wang. In addition, they performed a number of experiments on a Kaolin soil with the elasticity modulus, cohesion, and dry unit weight equal to 19.84 MPa, 158.8 kPa, and 14.1 kN/m3, respectively. The void had a concrete lining with a thickness of “t” varying between 24.5 and 152.4 mm. The ratio of D/B was taken between 1.0 and 4.0 and the ratio of W/D was between 1.0 and 3.0, where B, D, and W are footing width, distance between the footing base and the void crest, and diameter of the void, respectively. In order to verify the capability of numerical method used in present study to model the footing-lined void system, a void with a diameter of 2742 mm and concrete lining with a thickness of 24.4 mm was numerically modeled in present study. The distance between void crown and the footing bottom was 914 mm. The mechanical parameters of soil were considered to be identical with those reported by Jao and Wang (1998). The influence of concrete lining element was numerically simulated by the use of fully elastic shell element considering elastic modulus and thickness equal to 17 GPa and 25 mm, respectively. The interaction between outer face of the lining (concrete material) and soil was regarded by generating an interface having 60 % of the soil strength. The numerical results obtained from this verification analysis and those reported by Jao and Wang (1998) are compared in Fig. 4. As shown, this comparison demonstrates the accuracy of results obtained from numerical solution.

Fig. 4
figure 4

Comparison between present studies and those from Jao and Wang’s studies

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Comparison of final settlements in Figs. 3 and 4 can be related to the mode of failure in unlined void and the void with rigid (concrete) lining. As seen, the instability occurs at lower settlements in system with unlined voids (Fig. 3) whereas the failure in the soil beneath footing is more likely for voids with rigid lining (Fig. 4) that obviously corresponds to larger settlements.

To better clarify the system response for the case of footing on a single unlined void, the distribution of principal, mean and deviatoric stresses in the system obtained from verification analysis (based on Jao and Wang 1998) are shown in Fig. 5. As seen, before loading, arching occurs around the void (Fig. 5a); however, the arching effect disappears by applying load to the footing and the principal stresses rotate towards the failure mechanism. The distribution of mean and deviatoric stresses in Fig. 5c, d indicate that the local shear failure and instability in the void walls coincide. This is consistent with what reported by Jao and Wang 1998. According to Figs. 3, 4 and 5, the accuracy of assumptions as well as initial and boundary conditions made in numerical modeling of a shallow footing on a soil media with void (such as boundary condition, initial stress, voiding stages and etc.) is satisfactory and therefore, it can be used further to investigate the effect of twin voids on the bearing capacity of surface footings constructed above them.

Fig. 5
figure 5

Principal, mean and deviatoric stress in system of footing and single void; a direction of principal stress before loading; b direction of principal stress after loading; c distribution of mean stress after loading; d distribution of deviatoric stress after loading

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4 Parametric Studies

In the presence of twin underground voids beneath a single shallow foundation, the following three modes of soil failure may occur. In post processing, if one of them is encountered either by controlling the stability of the model or the shape of the shear band, the corresponding average vertical stress generated at gauss points beneath the footing is defined as ultimate bearing pressure.

  1. (a)

    The unlined twin voids become unstable before forming the shear failure mechanism beneath foundation. The results of some trial analyses indicated that if the strains in the voids exceed 5 % of void diameter, instability occurs in the voids. In this situation, the stress created below the footing is nominated as the ultimate bearing capacity of the footing.

  2. (b)

    The failure mechanism approaches to the ground surface and thus general shear failure occurs.

  3. (c)

    A punch failure mechanism forms in the soil between shallow foundation and void crown. Obviously, such mechanism belongs to shallow voids and occurs before abovementioned mechanisms. In this condition, the punching stress occurring below the footing is nominated as the ultimate bearing capacity.

A thorough investigation of numerical results at different steps of analysis indicated that the instability in the system coincides with a dramatic variation of the slope of load-deformation curve, whilst the slope of load-settlement curve gradually varies in case of general shear failure. Thus, the ultimate bearing load is determined as the stress corresponds to the point with highest curvature in load-settlement curve that can capture all of above-mentioned failure modes.

The schematic sketch of the three mainly possible failure modes for a single strip footing constructed on soil deposit with single and twin voids are shown in Figs. 6 and 7, respectively. In this section, a numerical parametric study performed to investigate the effect of underground twin voids size and location on the bearing capacity of the surface footing with a rough base for two soils.

Fig. 6
figure 6

Schematic sketch of possible failure mechanisms in footing-single void domain. a Instability in void walls. b General shear failure forming

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Fig. 7
figure 7

Schematic sketch of possible failure mechanisms in footing-twin void domain. a Instability in void walls. b General shear failure forming. c Punch failure under footing

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The mechanical properties of the soils considered in parametric study are presented in Table 1, where the parameters E, c, φ, ψ, γ d and ν denote the Elasticity modulus, cohesion, friction angle, dilatancy angle, dry unit weight and Poisson’s ratio of the soil, respectively.

Table 1 Mechanical properties of the soils in numerical analyses
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Figure 8 illustrates the geometry of a single strip footing resting on the soil with twin identical circular voids and the geometrical parameters such as B, D, Z, and L which are footing width, void diameter, depth of void crown from the bottom of the footing, and spacing between internal sides of two voids, respectively. Since the system of footing and twin voids can be asymmetric, the parameters X1 and X2 stand for the horizontal distances of voids 1 and 2 to the center line of footing. To investigate the variation of bearing pressure, a dimensionless parameter q v/q nv namely bearing pressure ratio was defined, where q v represents the current bearing pressure of surface footing and q nv denotes ultimate bearing capacity for an identical footing resting on the soil having no void. Thus, the bearing pressure ratio (q v/q nv) for no void condition tends to 1 while the current bearing pressure at ultimate load is equal to ultimate bearing capacity.

Fig. 8
figure 8

Geometry of a shallow strip footing on two underground voids

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4.1 Effect of Footing Width

Figure 9 shows the bearing pressure ratio-settlement curves for a strip footing located above underground twin voids for various footing widths in soils 1 and 2.

Fig. 9
figure 9

Load-settlement curve for footings with various widths (Z/D = 1, L/D = 1, D = 6.0 m). a Soil 1. b Soil 2

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As shown in Fig. 9, the mode of failure changes with loading footing is increased by an increase in the ratio of footing width to void diameter. Whereas the depth of void is considered to be constant, the failure mechanism approaches to the excavated zoned more rapidly for wider footings. This event leads to a significant decrease in the bearing capacity of footings with greater widths in comparison with a footing on the soil without void. Therefore, it can be concluded that the bearing pressure ratio decreases by either an increase in the width of foundation resting on identical voids at the same depths or by an increase in the diameter of void for footings with identical width. To better explain the mode of the failure in such cases, the variation of the shear bands are shown in Figs. 10, 11 and 12.

Fig. 10
figure 10

Shear strain distribution beneath footing for various void diameters (B = 1.5 m, soil 1). a B/D = 0.75. b B/D = 0.25

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Fig. 11
figure 11

Shear strain distribution under the footing located above twin void (half of media, soil 1)

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Fig. 12
figure 12

Shear strain distribution under the footing with presence of twin voids in one side of footing at collapse load (soil 1)

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As seen in Fig. 10a, footing on void with small diameter (B/D = 0.2) does not interact with the void and therefore the failure mechanism develops toward the ground surface. Obviously, the failure mechanism for systems with no void or small void is general shear failure for soil 1. Furthermore, the transient zone in general shear failure has a circular shape which is consistent with that expected for cohesive soils. By considering shear strain distribution in Fig. 10 for different ratio of footing width to void diameter, it is observed that the mechanism of failure varies from general shear failure to punch by an increase in the width of strip footing on the soil with twin voids. Figure 11 illustrates the shear strain distribution for footing on twin voids. The strain has concentrated beneath single footing and approaches toward the voids sides. There is a shear band which has been formed between the footing and void. The thickness of the shear band decreased by increasing the distance between two voids until that the stressed zone under the footing and around the voids do not interfere. If this occurs, there is no effect on the bearing capacity of footing due to the void existence.

Figure 11 also shows that with the presence of voids beneath the foundation, the failure develops towards the voids while the shear wedges cross the voids. Therefore, the local shear failure beneath the footing causes an instability in the wall of the voids.

Figure 12 illustrates the shear strain distribution underneath the strip footing on soil having twin voids with eccentricity.

According to Fig. 12, since the closer void to the footing is located exactly beneath the footing, a punch failure is originating from footing to the adjacent void wall. By an increase in applying load to the footing, a significant instability has occurred in the void which is close to the footing. Thereafter, the shear band is passing through the voids and the second void starts to collapse. Therefore, the crest of second void becomes unstable and the shear band extends to the ground surface. Apparently, a block of the soil including footing-twin voids and the soil between them punches and therefore, the voids would not stand stable at this situation.

4.2 Effect of Void Diameter

Figure 13 illustrates the pressure-settlement variations of a strip footing with a width of 1.5 m which is located above the underground two identical voids with various diameters in soil types 1 and 2. It is to be noted that both identical voids have been assumed to be located at the same depth regarding the ground surface.

Fig. 13
figure 13

Variation of load-settlement curve for various void diameters (Z/L = 1, B/L = 0.25, L = 6 m). a Soil 1. b Soil 2

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Figure 13 indicates that by an increase in the diameter of void, the bearing pressure ratio decreases while the distance of void crown and footing base is kept constant. In addition, with decreasing the void diameter, the load-settlement characteristics of footings approaches to the no void condition. In these analyses, the void diameter increases to a limit value. This is because, voids are unlined and an increase of the void diameter leads to instability of voids.

4.3 Effect of Embedment Depth of Voids from Footing Base

Figure 14 shows the effect of distance between footing and the crown of the void on the bearing capacity of shallow footing.

Fig. 14
figure 14

Variation of load-carrying capacity of footing for various void crown depth (L/D = 1, B/D = 0.25, D = 6 m). a Soil 1. b Soil 2

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According to Fig. 14, the bearing capacity of footing increases by an increase in the depth of void crown from the footing bottom where the load-settlement curve of footing approaches to that corresponds to no void condition. It is also interesting to note that if a void crown depth increases to a specified depth, the effect of voids on the bearing capacity tends to be negligible. This is because the general shear failure occurs in the soil beneath the footing which is identical to the reference cases where there is no void. This specified depth is called “critical depth”.

4.4 Effect of Spacing Between Adjacent Voids

The effect of variation in the edge to edge spacing between two identical voids on bearing capacity of surface footing is investigated and obtaining results has been presented in Fig. 15.

Fig. 15
figure 15

Variation of load-settlement curve for various spacing between two voids (Z/D = 1, B/D = 0.25, D = 6 m). a Soil 1. b Soil 2

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According to Fig. 15, it could be found that the bearing pressure for footing above twin voids will be reduced by a decrease in the internal distance between two voids. It occurs due to the interference between failure mechanism formed underneath the footing and the voids leading to instability of the whole system and consequent lower bearing capacity. On the other hand, when two voids are placed at close spacing, the adjacent lateral sides of voids become unstable very rapidly by increase in footing pressure. Therefore, the ultimate bearing capacity of footing is decreased when two voids constructed at close spacings. Also, increasing the distance between two voids affects the bearing pressure ratio up to specified limitation which is nominated “critical distance” and over this distance the voids won’t affect on bearing capacity and analysis will converge to the no void condition.

5 Critical Values of Geometrical Parameters

For all of abovementioned geometrical parameters of a strip footing on the soil with twin voids, there are some critical values in which considering greater values for that particular parameter (i.e. B/D, D/L, Z/D, L/D) has no significant effect on the dimensionless ratio of (q v/q nv). In this section, the variation pattern of these parameters versus the ratio of (q v/q nv) has been investigated. With respect to the small differences of the geometrical parameters corresponding to soils 1 and 2, the values for both soils have been inserted to a single chart as one series of data and a particular curve has been fitted for each parameter. In addition, the numerical formulas for fitted curves are presented.

Variation of bearing pressure ratio (q v/q nv ) for various B/D is presented in Fig. 16. It should be noted that other geometrical parameters of system (i.e. D/L, Z/D, and L/D) are kept constant. As shown in Fig. 16, the bearing pressure ratio of the strip footing on soil having twin shallow voids (Z/D = 1) decreases by an increase in the width of footing. This event occurred due to expanded size and depth of failure mechanism developed in the soil underneath footings with greater width. This led to a considerable instability in the voids and therefore the system of voids collapsed before developing the ultimate bearing capacity of footing. With respect to Fig. 16, it seems that increasing the width of footing further than about B/D = 0.5 has no significant effect on the ultimate bearing capacity. Therefore, the critical value for footing width on twin voids located at shallow depth is about B cr  = 0.75D.

Fig. 16
figure 16

Variation of bearing pressure ratio for various B/D (Z/D = 1, L/D = 1)

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Variation of bearing pressure ratio (q v/q nv) for various embedment depths of voids has been illustrated in Fig. 17. As discussed before, the ultimate bearing capacity of footing tends to increase by an increase in the depth of voids. It may be due to getting voids away from the zone in which the failure mechanism is formed. Therefore, the footing acts as a single strip foundation on the soil having no void. By an extrapolation in the results presented in Fig. 17, it can be expected that the influence of voids on the bearing capacity of shallow foundation is disappeared at about (Z/D) cr  = 4.

Fig. 17
figure 17

Variation of bearing pressure ratio at various Z/D (L/D = 1, B/D = 0.25)

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Variations of (q v/q nv) versus L/D and D/L ratios for soils 1 and 2 are shown in Fig. 18. It is to be noted that the parameter L is assumed to vary as long as the parameter D remains constant for L/D curve. However, for the curve corresponding to D/L, the parameter D is the variable and L is kept unchanged. Regarding Fig. 18, the ultimate bearing capacity of shallow footing is decreased by increasing the diameter of voids while the spacing between void’s crest and the footing remains unchanged. According to the results of analyses performed in this section, the system of soil with twin voids does not stand stable for D/L > 1. It may be due to considerable decrease in the confining stress in the block of soil situated between adjacent voids and therefore, this block becomes unstable before loading procedure of footing. Thus, the critical value for the ratio of (D/L) cr is approximately equal to 1. As seen in Fig. 18, an increase in the spacing between voids having constant spacing causes a significant increase in the bearing capacity. The natural pattern for variation of L/D curve shows that there is a critical value for this ratio in which the effect of void on the bearing capacity of the footing is vanished and the bearing pressure ratio (q v/q nv) approaches that of no void condition. It seems that if two voids constructed at edge to edge distances greater than (L cr ), underground voids will no longer affect the bearing capacity and the ratio of q v/q nv remains close to unity. According to the equation illustrated in Fig. 18 for the curve fitted to (L/D) data, the critical value for edge to edge spacing between voids L cr is about 6D.

Fig. 18
figure 18

Variation of q v /q nv versus L/D and D/L for B = 1.5 m (Z/L = 1, B/L = 0.25 for D/L curve and Z/D = 1, B/D = 0.25 for L/D curve)

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It should be noted that the equations and critical values mentioned in this section is reliable for a single strip footing with 1.5 m width resting on assumed materials and twin void geometry.

5.1 Effect of Eccentricity of Voids with Respect to Axis of Footing

In all analyses, two underground voids were situated symmetrically with respect to the axis of surface footing. In this part, twin underground voids are located at one side of the surface footing axis. Moreover, the distance between the voids, their diameter, and the depth of void embedment are kept constant.

Figure 19 shows the variations of the contact pressure of strip surface footing located above twin underground voids versus settlement for different values of void eccentricity for soils 1 and 2.

Fig. 19
figure 19

Load-settlement curves of footing for various void eccentricity (Z/D = 1, B/D = 0.25, D = 6 m). a Soil 1. b Soil 2

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In Fig. 19, X1 and X2 are horizontal distances between centers of voids 1 and 2 to the center of the footing, respectively. Both voids are placed at one side of the surface footing vertical axis (Fig. 8). According to Fig. 19, by an increase in the eccentricity of voids, the footing bearing capacity increases. This can be explained by to forming a general shear failure in expanded zone for the situation in which two voids placed far from each other. However, by a decrease in spacing between voids, the failure mechanism varies to a combination of punch failure in the void which is closer to the footing and unstable sides for both voids simultaneously. Obviously, such postulation strongly depends on the size of void and foundation as well as depth of void. In addition, with increasing the horizontal distance of void axis from the footing axis, the interference of footing failure wedges becomes far from void vicinity. Moreover, the distance between stressed zone due to the footing loading and stressed zone caused by void excavation becomes greater and thus the bearing capacity of surface strip footing increases.

6 Conclusions

In present study, the effect of underground twin voids on the bearing capacity of a shallow strip foundation located at the ground surface has been numerically evaluated. Two different types of the cohesive-frictional soils have been taken into account. To assess the accuracy of numerical modeling, some comprehensive verifying analyses were conducted on available experimental and numerical studies. Thereafter, the parametric studies were performed to investigate the effect of contributing parameters (such as size and location of footing and voids) on load-settlement response of strip footing located on the soil having twin voids. It is to be noted that the results presented in this study correspond to a strip footing with the width of 1.5 m and twin void geometry and the materials considered in this study. The parametric study indicated the variation of the bearing capacity corresponding to different geometries of footing and voids. The investigation of failure mechanism indicated that the mode of failure is dependent on the size and location of the voids as well as the footing. Based on the results, the following remarks may be cited:

  1. 1.

    Basically, forming twin voids beneath a surface footing leads to decrease the bearing capacity of footing depending on the geometry of system.

  2. 2.

    In the case of shallow voids, the bearing capacity of the footing located above twin underground voids decreases by an increase in the footing width.

  3. 3.

    Increasing the void diameter causes a remarkable decrease in the bearing capacity of the footing located above them while the relative vertical distance between voids and footing is kept constant.

  4. 4.

    An increase in the depth of voids beyond Z cr , which is equal to 4D for the materials examined in this study, the bearing capacity of the footing located above the voids remains unchanged and the load-settlement response of the footing follows that in case of no void.

  5. 5.

    Considering the horizontal distance between two voids greater than the “critical distance”, the bearing capacity of the footing no longer changes. For the soil considered in present study, the critical distance (L cr ) is equal to about 6D.

  6. 6.

    The system of soil-twin voids does not stay stable when the edge to edge spacing between voids is taken less than D.

  7. 7.

    The bearing capacity of surface footing increases by an increase in the eccentricity of twin voids with respect to the axis of surface footing, and the ultimate bearing capacity for twin voids at far spacing approaches the value similar to the case of no void.

  8. 8.

    Locating the void exactly beneath footing causes a punch failure in the soil. If the other void situated in close spacing, the voids walls become unstable and a block punch happens.