1 Introduction

The bearing capacity of soils has extensively been investigated for a long time [1,2,3,4,5,6,7,8]. The existence of a single footing in a semi-infinite half-space is unrealistic due to the complex nature and the wide variety of engineering structures for which footings might be constructed in groups of multiple configurations in large buildings or as a series of parallel sleepers (footings) in the railway. According to Terzaghi’s theory concerning the failure mechanism of footings, the lateral distance of passive zones extends from three to five times the footing width based on the internal friction angle of the foundation soil. Therefore, if an adjacent footing is constructed within this distance, interaction between the two footings takes place affecting the bearing capacity of soil. Another fundamental difference is that an asymmetrical failure occurs underneath the two neighboring footings due to the imbalanced forces. Whereas in the case of a group of multiple footings or a series of footings, a symmetrical failure in the soils must occur. As a result, substantial influence occurs on the bearing capacity behavior. Thus, it is imperative that better understanding is developed for the effect of interaction of multiple footings in a series.

A number of studies were conducted to compute the ultimate bearing capacity of a group of footings and it is understood from these studies that the ultimate bearing capacity of multiple footings in a group is always greater than that of a single footing. This is due to the interaction between multiple footings. This interaction caused an increase in the confining pressure around the failure zones leading to increase shear resistance and bearing capacity.

Some of these investigations were performed on two adjacent strip footings on unreinforced sand beds [9,10,11,12]. Investigations to assess the bearing capacity of multiple footings in a series that are in close proximity are limited [13,14,15,16].

In recent decades, reinforced soils are frequently used to support footings in order to increase the footing bearing capacity and reduce potential settlement. Several experimental, analytical and numerical investigations have been performed to determine the response of footings rested on reinforced soils [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The effect of interaction between two neighboring foundations above reinforced soil on the ultimate bearing capacity has received increasing attention [26, 31,32,33,34,35]. Ghazavi and Lavasan [33] employed the definition of interference factor (If) which was defined as the ratio of ultimate bearing capacity of interfering footings on reinforced sand to that of a single footing resting on unreinforced sand to describe the beneficial effect of interaction of two closely spaced footings on the reinforced sand. The influence of geometry and orientation of geogrid layers on footings performance was also studied. In their study, the reinforcement under two shallow footings caused bearing capacity to increase by 150 and 200% after the addition of one and two reinforcement layers respectively. Lavasan and Ghazavi [26] evaluated the effect of interference on the behavior of closely spaced two square and circular footings on reinforced sand beds. It was found that tilting of footings was significantly influenced by the number of reinforced layers. Based on laboratory experiments on reinforced sand overlying soft clay, Roy and Deb [35] suggested that the interference factor is a function of the dimensions of the rectangular footings. It should be noted that the above studies were conducted on two closely spaced footings. Due to the scare of data for all variable parameters, the influence of various influential parameters e.g. type of soil, degree of soil packing and number of reinforcement layers on the behavior of multiple footings in a series that are in close proximity needs to be assessed.

In this paper, the performance of both single and multiple strip footings in a series resting on reinforced and unreinforced sand is studied. A series of Finite Element (FE) analyses was performed to evaluate the influence of (1) clear spacing between footings, (2) angle of internal friction, (3) angle of dilation and (4) number of reinforcement layers on the ultimate bearing capacity. In order to verify the FE model utilized in this study, the results of preliminary investigations are compared with those attained based on theoretical and experimental investigations which are available in the technical literature. The results are presented in a dimensionless form as illustrated thereafter. Additionally, the effect of dilation angle on the ultimate bearing capacity and the changing of stress distribution in vertical and horizontal direction due to reinforcement existing is also presented in this study. Furthermore, the achieved data for footing-reinforcement systems are used to assess an equivalent approach which relies on assuming an apparent cohesion value for the reinforced soil zone. The proposed approach simplifies the FE solution since the requirement to model the interaction between reinforcement and surrounding soils is no longer needed.

2 Problem definition and numerical model

Figure 1 shows a schematic representation of the problem geometry for a multiple rigid strip footings with width B on cohesionless soil with a specific unit weight (γ). It shows that the footings are equally loaded (P) and constructed at the same clear space (S). The finite element simulations compute the ultimate load (Pu) of all the footings when they are loaded simultaneously to failure. The ultimate bearing load is determined as a function of the angle of internal friction (ϕ) and the spacing between footing on both reinforced and unreinforced sand. Furthermore, this study assumes an infinite number of footings collapse at the same ultimate load. In order to normalize the results, the performance of a footing in a group of footings is compared to that of a single footing for both reinforced and unreinforced sand scenarios.

Fig. 1
figure 1

Multiple strip footings: case identification and parameters

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The numerical investigation in this study is based on two-dimensional finite element modelling using Plaxis-2D V8.2 [36]. Because this study sought the failure load of footing irrespective of its settlement, Mohr–Coulomb failure criterion with friction angle ϕ is assumed to be applicable for soil modeling. The soil is modelled as 15-noded triangular plane strain elements. Although, Plaxis enables the automatic mesh generation, the mesh size was refined in the area beneath the footing and around the reinforcement layers to enhance the accuracy of the simulations. The selection of the element size is a key factor in such study. Therefore, different finite element meshes were examined to reduce the mesh effects on the finite element results particularly due to changing the number of reinforcement layers and the model size. Due to the geometry of the group of footings in a series as presented in Fig. 1, the stresses and strains must be symmetrical around the vertical axes that pass through the center of footings (B/2) and the middle distance between footings (S/2). Consequently, the rectangular domain bounded by those two axes of symmetry is modelled to perform the analysis of a group of footings in a series.

The depth of soil in the numerical model is kept constant at 10B as the collapse load remained unchanged irrespective of the increasing soil depth beyond this value. The vertical boundaries of model are restrained horizontally while the bottom boundary is constrained in both horizontal and vertical directions. On the other hand, full model is developed to model the single footing resting on unreinforced and reinforced sand with the same layer spacing shown in Fig. 1.

In this paper, the ultimate bearing capacity is sought irrespective of footing settlement therefore, the effect of soil modulus (E) in predicting the interaction behavior has not been dealt with in this study [16], Different values for the angle of internal friction of sand of ϕ = 25°, 30°, 35°, 40° are studied to determine its effect on the system performance. The value of Poisson’s ratio of 0.3 is kept constant through the analyses. Load control method is used accompanied with iterative analysis up to failure using internal friction angle as a failure criteria [37]. The dilation angle (ψ) is also taken into consideration. Therefore, in all analyses, Rowe’s stress-dilatancy relationship was used in order to determine the angle of dilation as follow [38, 39]:

$$\uppsi = \, 1.25(\upphi -\upphi_{\text{cri}} )$$
(1)

where ϕ and ϕcri are the angle of friction and the critical friction angle respectively. The critical friction angle for quartz sand ranges from about 28° to 36° depending on several parameters such as uniformity and shape of particles and mineralogy. Generally, the friction angle at the critical state is assumed to be constant and independent of density. For granular materials, the recommended value of critical friction angle is about 30° [40].

For reinforced sand, the performance of footings is investigated with one and two reinforcement layers (N = 1, 2). A stiffness (k) of 100 kN/m′ is assumed to be maintained in the current study for all reinforcement material. The interaction between soil and reinforcement layers is modelled by means of interface elements at the both sides of each layer. The effect of interface reduction factor (reduction of friction compared to the soil friction angle) is presented hereinafter. For the case of a single footing on reinforced sand, the width of reinforcement is taken as 7B for all analyses.

In the finite element analyses, the footing is modelled by rigid plate element. The plate is homogenous, isotropic and the normal and flexural stiffness of footing are kept constant in all analyses and the pressure is applied on the top of the footing.

The number of elements and mesh size are maintained fixed through the analyses of model which have the same geometrical dimensions and soil properties. The reinforcement layers are modelled. However, they can be activated or deactivated to simulate the existing of reinforcement layers and unreinforced bed cases respectively. Of note, prior to the application of footing load, the initial conditions including the initial geometry configurations and initial effective stresses were established. The initial effective stresses are generated by means of k0 procedure.

The ultimate bearing capacity for a strip footing is generally computed by Terzaghi’s equation which combines the effect of cohesion (c), surcharge (q) and soil unit weight (γ). The equation is typically written as follow:

$${\text{q}}_{\text{u}} = {\text{cN}}_{\text{c}} + {\text{qN}}_{\text{q}} + 0.5\upgamma{\text{BN}}_{\upgamma}$$
(2)

The present study, for a single footing, deals with the determination of ultimate bearing capacity in the form of Nγ. Therefore, in order to isolate the contribution of the soil unit weight factor Nγ, the effects of other terms Nc and Nq are eliminated by assuming the soil is cohesionless and the footings are to be constructed at ground surface i.e. without surcharge (q).

3 Verification of numerical model

In order to verify the developed numerical model in this study, data from previous numerical and experimental studies that are available in the technical literatures were used. The verification phase included comparison of results under different conditions. The proposed model was first validated using data for the bearing capacity of a single footing on unreinforced sand. The present results were compared with those obtained using the lower bound finite element limit analysis solution reported by Kumar and Bhattacharya [16]; Kumar and Khatri [6]; Ukritchon et al. [4]; and Hijaj et al. [5]. Furthermore, it was also compared with the method of characteristic solution obtained by Kumar [7]. Concerning to the case of multiple footings on unreinforced sand, results of the numerical model were verified using the results of Kumar and Bhattacharya [16]. On the other hand, for a single footing on reinforced sand, the validity of the current study was conducted using the experimental results of Das et al. [20]. Moreover, the numerical model is examined and compared by the results reported by Ghosh’s [41] which computed the response of two closely footings on reinforced sand. Hereafter results of the verification are presented and discussed prior to conduction of the comprehensive parametric study.

3.1 Single footing on unreinforced sand

Table 1 illustrates the values of Nγ for a single footing on unreinforced sand for different values of friction angle (5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°) where;

$${\text{N}}_{\gamma } = \, \frac{{q_{u} }}{0.5\gamma B}.$$
(3)
Table 1 Nγ values for single footing on unreinforced sand
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As presented in Table 1, except for ϕ < 15°, the values of Nγ predicted from the current analysis matched quite well with reported results.

3.2 Multiple strip footings on unreinforced sand

Kumar and Bhattacharya [16] determined ultimate bearing capacity of multiple strip footings resting on unreinforced sand. All footings were identically spaced with clear distance (S) and equally loaded to failure load. Their study was based on the finite element with lower bound limit analysis and results were presented in terms of an efficiency factor (ξ); where ξ is the ratio of ultimate load carried by a single strip footing in a group to the ultimate load of isolated strip footing at the same conditions. The current numerical procedure is functioned to solve the same problem of Kumar and Bhattacharya [16]. Figure 2 shows the comparison between the magnitude of (ξ) conducted by their experimental study and present numerical analysis. The efficiency factor is determined with respect to changes of spacing between footings. The figure illustrates that satisfactory match between results of the numerical model in this study and those by Kumar and Bhattacharya [16] was attained.

Fig. 2
figure 2

Efficiency factor against footing spacing ratio (S/B) for different angle of internal friction

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3.3 Single footing on reinforced sand

Das et al. [20] performed an experimental study to investigate the ultimate bearing capacity of a strip footing with width (B) of 76.2 mm on geogrid-reinforced sand beds. The tests were conducted on sand beds that were prepared with a unit weight of 17.14 kN/m2 and had an angle of internal friction of 41°. Geogrid layers were positioned at equal spacing of h = B/3. In addition, the top most layer was placed at u = B/3 beneath the footing. The tensile strength of geogrid reinforcement layers at 2% strain was 182 kN/m′. Eight tests were performed with changing number of reinforcement layers (N = 1–8). The current model is created with the same conditions of Das et al. [20] experiments. Furthermore, the numerical model is employed to study the impact of interface element factor (R) on the numerical results.

A detailed comparison between the results of the developed numerical model without interface elements, Rigid interface (R = 1, R = 0.9 and R = 0.8) and experimental data is presented in Fig. 3. The results are presented in terms of bearing capacity ratio (BCR), which is defined as the ratio of the bearing capacity of footing on geogrid-reinforced sand to that attained on unreinforced soil bed. It is clear that substantial degree of improvement in the value of BCR is observable with increasing number of reinforcement layers up to 3 layers. For reinforced sand beds with 4 layers or more, no significant increase in the bearing capacity was noticed. The results in Fig. 3 shows that the numerical results obtained without consideration of an interface element between reinforcement and surrounding soil are in good agreement with the experimental results. This could be attributed to that the full bond counteracts the missing two mechanisms which are passive resistance membrane resistance. In comparison with test results, the BCR that is determined based on interface element was found to be lower in particular at lower values of R.

Fig. 3
figure 3

Effect of number of reinforcement layers and interface elements on BCR in comparison with results of Das et al. [20]

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3.4 Two strip footing on reinforced sand

Ghosh and Kumar [41] conducted a series of model test on two adjacent strip footing on reinforced sand. The spacing between footings was examined in order to investigate its influence on the value of bearing capacity. A comparison between results of the Finite Element analysis in this study and those obtained experimentally for a system with a single reinforcement layer that was inserted at a depth of 7.5 cm in a soil with an angle of internal friction of 38.9° and has a stiffness of 64.5 kN/m′ were presented in Fig. 4. It shows the improvement ratio (ξ)r in bearing capacity against the footing spacing in terms of (sc/B) where, (ξ)r is the ratio between the bearing capacity of single footing in case of two adjacent footing over reinforced sand to that of single footing and sc is center to center footing spacing. The numerical results indicated that similar behavior is obtained with 8% difference between the results of the numerical and those experimentally attained. Besides, the results indicates that the maximum enhancement in bearing capacity is obtained at S/B = 2. The numerical model was therefore considered quite capable of simulating the behaviour of footings on unreinforced and reinforced sand beds under different conditions.

Fig. 4
figure 4

finite element result for two adjacent strip footings on reinforced sand compared with results reported by Ghosh and Kumar [41]

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4 Results and discussion

The finite element model is employed to discover the impact of angle of dilation, internal friction angle and spacing between footings on the performance of strip footing supported by unreinforced and reinforced sand. Besides, the changing in stress and settlement distribution through the different cases is also presented.

4.1 Effect of angle of dilation (ψ) on value of Nγ for a single footing on reinforced and unreinforced sand

This section presents the results of the study on the effect of dilatancy angle on the ultimate bearing capacity of a single footing on unreinforced and reinforced sand beds. It is well known that during shearing, the positive dilation angle refers to soil expansion and a negative one means that the soil in which the net movement of particles causes contraction [42]. Determination of soil dilatancy generally extracted from existing stress-shear strain relationships. The peak strength of soil is usually associated with the maximum rate of dilation. A great attention has been given for the relationship between the friction angle (ϕ) and the dilation angle (ψ) [38, 39, 43]. Disparate understanding concerning the determination of soil dilatancy was recorded due to several affecting factors. Most relationships showed significant effects of stress state, soil density, particles’ shape and fine content on soil dilatancy. In addition, the interaction between soil reinforcement and adjacent soil changes the soil dilatancy behavior in which increased volume of soil occurs at failure plane leading to an increase in the dilation angle [44]. Therefore, in this section, a range of angle of dilation is examined to assess its impact on the footing response. Values for the bearing capacity factor Nγ are presented in Fig. 5 for different values of ϕ due to changing in the dilatancy angle. Although in many studies, the dilation angle of zero has been assumed, the negative dilation angle, as shown in Fig. 5d, is acceptable for rather loose sand due to its contractive behaviour under shearing. Figure 5 shows a significant increase in Nγ with the increase of dilatancy angle for the case of reinforced sand. This could be attributable to increasing in dilatancy due to increasing confining effect of reinforcement. Obviously, the impact of changing of the dilatancy angle in the case of reinforced sand is greater than that observed for unreinforced sand beds. Careful inspections of data presented in Fig. 5 illustrates that higher values of Nγ was experienced with increasing number of reinforcement layers. In addition, the relationship between Nγ and angle of dilation follows a three stages. At the first and third stages, a slight increase in Nγ was observed as the dilatancy increased. The third stage seems to initiate at a dilation angle of about 20°, 15°, 10°, and 5° for ϕ = 40°, 35°, 30° and 25° respectively. Whereas the second stage is apparently a transition zone that was characterized by a significant increase in Nγ with increasing angle of dilation but it was dependent on the angle of friction of soil and number of reinforcement layers. The sharp increase in Nγ during the transition zone could be attributed to expansion of soil volume upon shearing which led to a reduction in the slack effect [45]. Hence, there would be minimum values for the dilation angle to overcome the slack effect in different reinforced grounds as a function of the packing condition of soil and number of reinforcement layers.

Fig. 5
figure 5

Influence of angle of internal friction and dilation angle on the bearing capacity factor, Nγ, for a single footing resting on reinforced and unreinforced sand

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4.2 Efficiency factor (ζ) for multiple strip footing on reinforced sand

Figure 6 shows the effect of interference between footings on the ultimate bearing capacity which is assessed using an efficiency factor (ζ). The efficiency factor (ζ) is a dimensionless factor and determined as the ratio of ultimate bearing capacity of one footing in a group of strip footings over reinforced sand beds to that observed for a single footing under the same conditions. Of note, the efficiency factor was expressed as a function of the spacing ratio which is often taken as the ratio of clear spacing to the footing width. Figure 6 illustrates the magnitude of (ζ) for different values of friction angle with changing spacing ratio (S/B). It can be noticed that, for all cases, the value of (ζ) is greater than 1 and increases with the decrease in the value of (S/B). Very limited interaction between adjacent footings was observed for a clear spacing that was twice or more the footing width. The results suggest that the angle of friction plays a major role in the interaction between footings and hence the efficiency factor. The efficiency factor always increases with increasing the angle of friction. In case of sand bed with an angle of friction of 40o, the efficiency factor varied between 204 and 1 for the case of N = 1 and between 232 and 1 for sand beds with two layers of reinforcement. On the other hand, for other values of angle of frictions (ϕ), the values of the efficiency factor were found to range between 1 and 6.8 for case of N = 1 and varies between 1 and 18 for case of N = 2. It can be noticed that the increasing number of reinforcement layers did not help in loose sands whereas it worked well in medium to dense sand with ϕ > 30°. The same results are illustrated in other form in Fig. 7 whereas the efficiency factor is related to the angel of internal friction and the same trend can be observed. It is clear that the efficiency factor increases with the reduction in the spacing between multiple footings, number of reinforcement layers and friction angle.

Fig. 6
figure 6

Efficiency factor for reinforced sand with variation of angle of internal friction and footing spacing (S/B)

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Fig. 7
figure 7

Efficiency factor for unreinforced and reinforced versus angle of internal friction

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Figure 8 illustrates an example of shear stress distribution of unreinforced and reinforced sand beds. It can be noted that the shear stress txy along the vertical planes at boundary condition (axis of symmetry) becomes to zero.

Fig. 8
figure 8

shear stress distribution for group of strip footing

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The distribution of normal stress (σy) underneath closely spaced strip footings for both reinforced and unreinforced soil beds are presented in Figs. 9 and 10. It can be noticed that the reinforcement layers play an important role in redistribution of the stress. At the same level of applied loading, Fig. 9 shows a comparison between unreinforced and reinforced sand (N = 1, 2) in terms of σy for the case of ϕ = 30° and S/B = 0.3. All of three cases are loaded by the ultimate bearing pressure that was determined on unreinforced sand bed. As shown in, the maximum σy for reinforced soil reduced by 39.7% and 42.6% for the cases of sand beds with one and two layers of reinforcement respectively in comparison with those on unreinforced sand beds. This could be attributed to the effect of reinforcement on lateral spreading of induced stress than that occurs on unreinforced soil i.e., the volume of soil which withstands the footing load is bigger due to the apparent cohesion caused by reinforcement. In other words, for unreinforced sand, the applied footing pressure distributes over a relatively small area which is dependent on the angle of friction and depth from the bottom of the footing. On the other hand, in the case of reinforced sand, the load transfer mechanism is greatly affected by the existence of reinforcing layers. The generation of shear stresses on both sides of reinforcement layers results in the redistribution of stresses over larger zone. In addition, insertion of the reinforcement layers increase the confining stress around loaded area in comparison with that in unreinforced sand at the same load level and depth.

Fig. 9
figure 9

Normal stress distribution for reinforced and unreinforced sand at the same load level for case of ϕ = 30°, S/B = 0.3

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Fig. 10
figure 10

Normal stress distribution for reinforced and unreinforced sand at ultimate bearing capacity for case of ϕ = 30°, S/B = 0.3

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Figure 11 shows the distribution of horizontal soil movement Ux for the same cases at the same conditions in order to highlight the confining effect induced by the reinforcement. It shows clearly that the horizontal movement under strip footing highly influenced by soil reinforcement. Whereas, the horizontal displacement, in comparison with that on unreinforced sand bed, reduced by 57.6% and 61.8% on reinforced sand beds with one and two layers of reinforcements respectively. It can be concluded that the presence of reinforcement layers increases the interaction between closely spaced footings and causes remarkable confinement which in turn, enhanced substantially the soil resistance to applied bearing pressure.

Fig. 11
figure 11

Horizontal displacement (Ux) distribution for reinforced and unreinforced sand at the same load level for case of ϕ = 30°, S/B = 0.3

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On the other hand, Fig. 10 shows the distribution of σyu inside the soil mass at ultimate bearing capacity of each case. It can be observed that the ratios between the maximum normal stress on reinforced sand bed and that for unreinforced one are found to be 1.57 and 2.74 for one and two layers of reinforcement respectively. In addition, due to the reinforcement materials, the interlocking between soil particles increases leading to a deeper stress distribution in the case of reinforced beds than that observed on unreinforced sand bed.

4.3 Equivalent cohesion for reinforced sand

In this section, an equivalent approach is presented to estimate the ultimate bearing capacity of strip footing on reinforced sand so as to avoid simulation of complex interactions between soil and reinforcement layers. The improvement in ultimate bearing capacity by reinforcement is achieved by assuming an apparent cohesion. Whereas, the reinforcement depth (d) is replaced by an equivalent layer with homogenous properties. The strength characteristics are defined by both of friction angle (ϕ) and cohesion (c). The cohesion parameter has been termed as apparent cohesion by some researchers to account for the additional confinement caused by inclusion of reinforcement layers [46]. Therefore, the shear strength, due to reinforcement, for granular material can be expressed by:

$$\uptau_{\text{r}} = {\text{c}}_{\text{r}} +\upsigma^{\prime } \tan\upphi^{\prime }$$
(4)

where r stands for reinforced composition, τr = shear strength; cr = apparent cohesion, σ′ = effective normal stress. Several studies were performed to investigate the characteristics of shear strength of reinforced soil by conducting shear box and triaxial tests [47, 48].

In this numerical analysis, an apparent cohesion was added alongside with the internal friction angle of sand to simulate the benefits of reinforcement in attempt to simplify modelling and computational cost of interactions between reinforcement layers and adjacent soils.

Relies on the obtained results of Das et al. [20] which were discussed in Sect. 4.2, the applicability of equivalent cohesion approach is evaluated. Their experimental study is simulated by performing the present numerical model without reinforcement to predict the equivalent cohesion which represents the rise in ultimate bearing capacity caused by reinforcement. Table 2 shows the values of equivalent cohesion (cre) through changing of the number of reinforcement layers for reinforced sand (ϕ = 41°, u = h=25.4 mm). Figure 12 shows a good agreement between results predicted by and equivalent approach. It therefore suggested that the equivalence approach seems promising and might shorten the computational time significantly. Further research is underway to assess it fully using experiment data.

Table 2 shows the values of equivalent cohesion (cre) through changing of the number of reinforcement layers
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Fig. 12
figure 12

Equivalent approach compared by Das et al. [20]

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5 Conclusions

A comprehensive numerical study was conducted to assess the effects of interference and interactions between closely spaced strip footings on unreinforced and reinforced sand beds. The developed numerical model was verified using data from multiple studies on a single footing on unreinforced and reinforced sand beds. An efficiency factor was introduced and utilized to assess the degree of improvement on bearing capacity. A number of conclusions could be drawn from the numerical study;

  1. 1.

    The verification exercise illustrated that the developed numerical model is capable of modelling the behavior of footings on unreinforced and reinforced sand beds.

  2. 2.

    The numerical results for a single footing on unreinforced and reinforced sand beds illustrated clearly that the bearing capacity factor Nγ is dependent on the angle of friction, angle of dilation and number of reinforcement layers.

  3. 3.

    The efficiency factor for multiple strip footings on reinforced sand beds improves markedly with decreasing the spacing between adjacent footings. The interaction between adjacent footings seems minor when the value of S/B is greater than 2.

  4. 4.

    Normal stress underneath multiple strip footings is reduced by around 40% with the inclusion of layers of reinforcement in comparison with the unreinforced sand bed. In addition, stresses are distributed over larger area in case of reinforced sand beds.

  5. 5.

    An alternative approach is assessed that relies on the use of apparent cohesion. The use of apparent cohesion seems promising but further experimentations are required so as to provide concrete conclusions.