Numerical analysis of bearing capacity of multiple strip footing on unreinforced and reinforced sand beds
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In this paper, the ultimate bearing capacity of multiple strip footings on reinforced and unreinforced sand beds is investigated using finite element method. The study utilizes efficiency factor to assess the change in the ultimate bearing capacity due to footing interference. The impacts of angle of internal friction, clear footing spacing and number of reinforcement layers on the efficiency factor are presented and evaluated. In addition, the effect of dilatancy angle and stress distribution in soil-reinforcement system is examined. The developed numerical model was verified against available theoretical and experimental data from literature prior to its use in this study. It is noted that for both reinforced and unreinforced sand, the ultimate bearing capacity of strip footings in a group is always greater than that of a single footing when the footing spacing is less than twice the footing width. Furthermore, the efficiency factor was found to increase as the footing spacing decreased and as the number of reinforcement layers increased. The degree of enhancement in ultimate capacity is, therefore, dependent on the angle of internal friction and number of reinforcement layers. The paper also proposes and assesses the use of alternative approach to modelling the effect of reinforcement using an apparent cohesion to reflect the addition of reinforcement layer. The use of apparent cohesion seems reasonable but further investigations would be required.
KeywordsGeosynthetics Soil reinforcement Bearing capacity Finite element analysis Soil dilatancy Apparent cohesion
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  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  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  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
The numerical investigation in this study is based on two-dimensional finite element modelling using Plaxis-2D V8.2 . 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.
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 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 ; Kumar and Khatri ; Ukritchon et al. ; and Hijaj et al. . Furthermore, it was also compared with the method of characteristic solution obtained by Kumar . Concerning to the case of multiple footings on unreinforced sand, results of the numerical model were verified using the results of Kumar and Bhattacharya . 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. . Moreover, the numerical model is examined and compared by the results reported by Ghosh’s  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
Nγ values for single footing on unreinforced sand
Angle of friction, ϕ (°)
Kumar and Bhattacharya 
Kumar and Khatri 
Ukritchon et al. 
Hijaj et al. 
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
3.3 Single footing on reinforced sand
Das et al.  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.  experiments. Furthermore, the numerical model is employed to study the impact of interface element factor (R) on the numerical results.
3.4 Two strip footing on reinforced sand
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
4.2 Efficiency factor (ζ) for multiple strip footing on reinforced sand
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 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.
shows the values of equivalent cohesion (cre) through changing of the number of reinforcement layers
Reinforcement layers (N)
Reinforcement depth (d) (mm)
Equivalent cohesion (Ce) (kN/m2)
The verification exercise illustrated that the developed numerical model is capable of modelling the behavior of footings on unreinforced and reinforced sand beds.
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.
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.
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.
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.
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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