Introduction

Most of foundations especially with industrial applications are subjected to horizontal seismic and wind forces that causes eccentric loading. Regarding the cases where loading is applied with eccentricity, a few studies have been developed experimentally to identify the critical values of reinforcement layers for reinforcing of the soil under the strip and rectangular foundations. Sawwaf [1] conducted a series model tests on eccentrically loaded strip footing resting on geogrid reinforced sand and proposed that the effect of reinforced soil on the bearing capacity ratio is greater at lower values of eccentricity and in greater relative densities. They found out that maximum improvement occurred at a depth ratio of u/B = 0.33 and h/B = 0.5 where u is the depth of first layer, h is vertical spacing between layers and B is the width of footing. In the same vein, Patra et al. [2] proposed an empirical relationship from model loading tests on an eccentrically loaded strip foundation in geogrid reinforced sand bed. Also, Sadoglo et al. [3] reported that reinforcement increased ultimate loads in comparison with the unreinforced cases and this contribution can become much larger with increasing of load eccentricity. Many researchers [4,5,6,7,8,9,10,11,12] reported that for strip footings the optimum values of u/B for the first geogrid layer varies from 0.35 to 0.45, depending on the value of load eccentricity. In addition, several researchers such as Michalowski and You [13], Mahiyar and Patel [14], Patra et al. [2] and Saran et al. [15, 16] investigated the behavior of the square and rectangular foundations under eccentrical loading. Sawaf and Nazir [17] studied the behavior of eccentrically loaded small scale ring footings resting on sand. They reported that the behavior of an eccentrically loaded ring footing was significantly improved with an increase in depth and relative density of the replaced compacted sand layer. Furthermore, Boushehrian and Hataf [18] focused on the behavior of circular footing on sand and reported that the circular foundation was more suitable and economical for axi-symmetric structures. Taiebat and Carter [19, 20] suggested the shape of the failure in (V, H, M) space using the results of a finite element study for circular foundations on undrained clay subjected to eccentric loads where V and H are vertical and horizontal loads and M is the moment applied to the foundation. They claimed that the effective width method commonly used in the analysis of foundations provided good approximations to the collapse loads.

Few studies have been performed for analyzing of the circular footing under eccentrical loading. The experimental studies explained above focused on eccentrically loaded strip footing on reinforced soil and no previous study has so far investigated the behaviour of eccentrically loaded circular and square foundation resting on reinforced sand. Thus, prediction of the eccentrical bearing capacity circular and square footings with varying number of reinforcement layers is still a challenging subject in foundation engineering. In this sense, in this paper the behavior of a circular and square foundation on unreinforced and reinforced sand under central and eccentrical loading are studied and the effect of load eccentricities on rotation of footing and ultimate bearing capacity is presented. Hence, new equations are proposed for reinforced and unreinforced condition to determine the ultimate bearing capacity under eccentrically loadings. In this work, many sets of foundation loading data extracted from nine published works, are processed and analyzed. Since the aim of the research is to verify the prediction of proposed method on the behaviour of eccentric loading of reinforced sand, the raw data have been picked from the works that have reported the bearing capacity data at different sand and reinforcement condition. Also the details of the processed datasets are summarized in each analysis. Besides, the proposed equations have been compared in unreinforced conditions with the Effective Area Method proposed by Meyerhof [21] for the evaluation of the previously published laboratory model test results and numerical analyses in the literature review. Finally, the large-scale (field) and laboratory tests results of Saran et al. [15, 16] have been used for validating the eccentrical ultimate bearing capacity, which indicate the accuracy of the proposed method, too.

Eccentrical Loadings Background

Meyerhof [21] studied behavior of rectangular footing with eccentrical loading and suggested the semi-empirical concept of effective width that can be used to estimate the ultimate bearing capacity of a shallow foundation subjected to one-way load eccentricity. He assumed that the contact pressure of footing decreases linearly from toe to heel, when subjected to a loading with eccentricity. According to this, the bearing capacity of a strip foundation (with width B) can be determined assuming that the load acts centrally along the effective contact width (B′ = B−2e) as shown in Fig. 1a where e is load eccentricity. Prakash and Saran [22] provided a comprehensive mathematical formulation to estimate the ultimate bearing capacity and settlement for strip foundations in a c-ϕ soil subjected to eccentric loading. It was concluded that, for high silos, refinery towers, wind turbines and chimneys, the circular foundation was more economical than any other form of footing, because the direction of wind and earthquake overturning moments were not fixed; and moreover, the load eccentricity is always one-way. In case of circular foundations under eccentric loading, Highter and Anders [23] suggested non-dimensional curves for calculation the effective area Aʹ and the effective width Bʹ. Effective area was defined as the equivalent area of footing, which can be loaded centrally when a vertical load was applied on a location other than the centroid of footing or when a foundation was subjected to a centric load and momentum. Similarly, in this case, Taiebat and Carter [20] suggested that the area of the fictitious foundation, A′, can be calculated using Eq. 1. The width (b e ) and length (l e ) of effective equivalent rectangular are shown in Fig. 1b that can also be determined by Eqs. 2 and 3, respectively.

Fig. 1
figure 1

Effective area method for a square section in one-ways eccentricity and b circular section

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$$A'={l_e}.{b_e}=\frac{{\pi {R^2}}}{4} - \frac{1}{2}\left[ {e{{\left( {{R^2} - {e^2}} \right)}^{0.5}}+{R^2}{{\sin }^{ - 1}}\left( {\frac{e}{R}} \right)} \right]$$
(1)
$${l_e}={\left[ {A'{{\left( {\frac{{R+e}}{{R - e}}} \right)}^{0.5}}} \right]^{0.5}}$$
(2)
$${b_e}={l_e}{\left( {\frac{{R - e}}{{R+e}}} \right)^{0.5}}$$
(3)

where R is the footing radius, e is the load eccentricity, l e and b e are length and width of effective area, respectively. It is well shown by many researchers that for strip and rectangular foundations the effective area method of Meyerhof [21] is an appropriate method for calculating the ultimate bearing capacity of eccentrically loaded foundations. However, many methods have been developed by researchers for the investigation of circular footings under eccentrically loading situations. For instance, Purkayastha and Char [24] carried out stability analyses on eccentrically loaded continuous foundations supported by sand using the slices method proposed by Janbu [25]. Slices method which is the most common limit equilibrium technique, discretized the soil mass into horizontal or vertical slices. They assumed that failure surface under strip footing subjected to eccentrically loading develops to the side of the eccentricity away from the footing. Based on their analysis, they found that the bearing capacity of strip foundation under eccentrical loadings was a coefficient of centric bearing capacity of foundation and proposed a reduction factor (R k ) as Eq. 4:

$${R_k}=1 - \frac{{{q_{u(eccentric)}}}}{{{q_{u(centeric)}}}}$$
(4)

where R k is the reduction factor, q u(eccentric) is the ultimate bearing capacity of eccentrically loaded and q u(centric) is the ultimate bearing capacity of centrally loaded footing. Based on a statistical analysis, it was also shown that internal friction angle (ϕ) have not any influence on R k [2]. The magnitude of R k can be expressed for shallow foundation with embedment depth of footing equal zero (D f = 0) as Eq. 5:

$$R_{k} = 1.862\left(\frac{e}{B}\right)^{{0.73}}$$
(5)

Also, based on the reduction curves of Meyerhof, the following equation is proposed by Bowles [26] for obtaining the reduction factor for footing under eccentric loading resting on non-cohesive soil and for 0 < e/B < 0.3 as Eq. 6:

$${R_k}=\sqrt {\frac{e}{B}}$$
(6)

where e is the loading eccentricity and B is the width of footing. It reported that for circular footing B is the diameter of footing. It is worthy to note that the reduction factor proposed by Purkayastha and Char [24] provides larger values than Bowles [26]. Also, it is should be mentioned that the difference between two equations increases with the increment of load eccentricity.

Test Materials and Setup

Sand

The used soil in this investigation includes uniformly graded sand with effective size D 10 = 0.18 mm, coefficient of uniformity C u  = 10.55 and coefficient of curvature C c  = 4.21 and is classified as SP (poorly graded). The grain size distribution curve is determined by dry sieve analysis method on three sand specimens. The average particles size distribution curve is illustrated in Fig. 2. On representative sand specimens, laboratory tests were conducted to determine specific gravity, maximum and minimum dry densities, and direct shear tests under normal stress ranging between 40 and 160 kPa in relative density of 60%. The sand properties are given in Table 1.

Fig. 2
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Particle size distribution curve of sandy soil

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Table 1 Sand properties used in tests
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Geogrid

In order to provide reinforcement material for the model test, geogrid with tensile strength 7.68 kN/m was used. This reinforcement is made of high density polyethylene (HDPE). The main reason for selecting this type of geogrid was the same peak tensile strength in every direction. The properties of reinforcement layers are given in Table 2.

Table 2 The engineering properties of the geogrid used in the tests
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Laboratory Model Tests

The size of the used tank for conducting the model tests was decided based on the sizes of footings and the influenced zoning. As the Fig. 3 the test tank has inside dimensions of 600 mm × 600 mm in plan and 600 mm in height. The tank is made from Plexiglas, which is supported directly on two steel beams. To record the footing settlement, a LVDT (Linear Variable Differential Transducer) with 25 mm travel range and 0.001 mm accuracy is utilized. In tests with eccentricity load, two LVDTs are used to measure tilt of the footing. The first LVDT was placed on the footing center and the second LVDT was located at the loading point. The tests were conducted in a controlled strain condition. During the tests, a data logger and a computer was used to record the load and settlements. The used model footings in the present tests were circular with 120 mm in diameter and square with 120 mm in width, which are made of rigid steel plates. Bottom of the models were became rough by a layer of geonet that was fixed with epoxy glue to ensure that the rough condition was same as in the all tests. Diameter of the circular footing was selected <0.2 tank width to ensure that failure planes under footing would be within the tank limit. When the load was applied within a limited area which is known as Kern rhombic [23], a maximum pressure occurs at the corners or edges of the footing and minimum pressure becomes zero. Kern boundary is R/4 for circular foundation. In a loading inside the Kern boundary, whole footing area is under pressure. Outside the Kern boundary, the footing edge is subjected to uplift pressure. The loading on the Kern boundary caused that the edge pressure becomes zero. In this study, five different load eccentricities are selected: e/D = 0.0417 and 0.0833 for inside the Kern boundary, 0.125 for on the Kern boundary, and 0.1667 and 0.2083 for outside the Kern boundary. The load eccentricities can be located on the model footing by small holes. Also, five different load eccentricities are considered, in one-way and two-ways namely, for the square footing: e = 0.5, 1, 1.5, 2 and 2.5 cm equal to e/B = 0.0417, 0.0833, 0.125, 0.1667 and 0.2083.

Fig. 3
figure 3

Geometric parameters studied in the laboratory model tests

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Experimental Program

In order to provide experimental control and repeatability of the tests, the sandy soil was placed in the tank using raining technique. The height of raining to achieve the desired density was determined by performing a series of trials with different heights of raining. Then, sand was rained from 40 cm height to maintain a stable relative density of 60% in all the tests. By placing a metal cup in different heights in the tank, the relative density of sand is monitored, too. After preparing the tank soil up to 50 cm height, the surface of sand is leveled by special ruler. Later on, the footing is placed on the sand and the rate of displacement sets at 1 mm/min. In the reinforced cases a square shaped geogrid layer, 4.5 times the diameter of footing (L/D = L/B = 4.5) with depth ratios of u/D = u/B = 0.42 and h/D = h/B = 0.42 based on researches [27,28,29,30,31,32] was placed after leveling the surface of the sand and pouring was continued to the selected surface of footing.

Results of Experimental Models

The results include of load-settlement curves from 68 experimental tests which are carried out on centrically and eccentrically loaded circular and square footings in both reinforced and unreinforced conditions are presented in this section. For both loading the number of geogrid layers is increased from 1 to 3. Several tests are repeated twice to verify the repeatability and consistency of the test result and the average of each of two tests are considered. It should be noted that the difference between load-settlement curves are less than 4% by the same load-displacement patterns. The load-settlement curves of circular and square footings under various load eccentricity are illustrated in Fig. 4. As expected, when the load eccentricity increases, the ultimate load decreases. In addition, by increasing the load eccentricities, the settlement at maximum load (i.e. vertical displacement values at failure) decreases for both circular and square footings. The ultimate loads of foundation for centrically and eccentrically loading are obtained from the curves of load-settlement according to suggestions of Boushehrian and Hataf [18] and Sawwaf and Nazir [17]. In all the tests, two LVDTs are used to record the settlement of footing at two points. Footing tilt is calculated to be the difference ratio of two LVDTs to the distance between them. The ultimate loads at centrically and eccentrically loadings for unreinforced and one, two and three layers of reinforcement are displayed in Tables 3, 4, 5 and 6. The efficacy of reinforcement in increasing the ultimate load is expressed as a non-dimensional quantity called the Improvement Index, which is defined as the ratio of ultimate load of reinforced soil to ultimate load of unreinforced soil. Improvement Index increases with increase of number of reinforcement layers for all the tests, but the improvement is more significant for circular footing. Also, by increasing the load eccentricity, reinforcement layers have more influence on circular footing than square footing.

Fig. 4
figure 4

Load-settlement curves for various load eccentricities, a circular and b square footing

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Table 3 Ultimate load for unreinforced condition
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Table 4 Ultimate load for one layer of reinforcement
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Table 5 Ultimate load for two layers of reinforcement
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Table 6 Ultimate load for three layers of reinforcement
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When the load is transmitted at the base of footing the moment of the soil particles in horizontal and vertical direction occurs. Due to eccentric loading the footing tilt and pressure below the footing does not remain uniform. The edge near to the load settles more than the other edge. Thus two edges of the footing settle by different amount that causes rotation of footing. The tilt will occur in a direction in which the settlement is higher. The amount of tilt and the pressure at the base depends upon the value of eccentricity width ratio. Footing tilt curves (in degree) at ultimate load versus load eccentricities for both footings are presented in Fig. 5. When eccentricity to width or diameter ratio (e/B or e/D) is greater than the Kern boundary, the edge of the footing away from load will lose it’s contact with the soil which will result in the reduction of effective width of footing and hence reduction of ultimate load of foundation. For circular footing, when load eccentricity is outside the Kern of footing the rate of increment affects larger than inside the Kern in comparison with square footing for both one-way and two-ways load eccentricities. Using of the reinforcement layers the tilt of footing decreases in comparison with the unreinforced tests for both footings, but the reinforcement layers has more influence on circular footing than square footing. The tilt of footing increases with increment of load eccentricity and the footing rotation increases linearly with increment of displacement as shown in Fig. 6, that investigated the effects of different settlements on footing rotation in reinforced sand with three layers of geogrid at the same load eccentricity. The main reason for the lower tilt of the circular footing in comparison with square footing in reinforced condition may be attributed to reinforcement mechanism which limits the spreading and lateral deformations of sand. The circular footing causes a bigger mobilized tension in reinforcement layers that enables the reinforcement to resist the imposed horizontal shear stresses built up in the mass beneath the loaded area by transferring the footing load to deeper layers of soil. Therefore, the failure wedge becomes larger and the frictional resistance on failure planes becomes greater.

Fig. 5
figure 5

Tilt of footing versus load eccentricity

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Fig. 6
figure 6

Settlement and footing rotation for different footing on reinforced with three layers for a circular, b square one-way and c square two-way

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Proposed Method for Prediction of Eccentric Ultimate Bearing Capacity

Unreinforced Sand

It is appear that the presented ultimate bearing capacities in Table 3 can be expressed in a form similar to those proposed by Purkayastha and Char [24] as reduction factors. With regard to data sets, a curved line is used to graphically display trend in average data to help predicting of reduction factors in this condition. Based on the increase of measurements at a specific rate a power trend line is considered. New reduction factor is proposed based on the trend line from test results for shallow (embedment depth of zero) circular and square foundations under eccentric loading as given in Eq. 7 for unreinforced condition, where e is the load eccentricity and D or B is the circular footing diameter or width of square footing.

$${q_{u(e)}}={q_{u(e=0)}}(1 - {R_k})={q_{u(e=0)}} - {q_{u(e=0)}} \times \alpha {\left(\frac{e}{{{B^{}}o{r^{}}D}}\right)^k}$$
(7)

The reduction factors (R k ) versus load eccentricity for both footings are presented in Fig. 7. Based on this figure it is obvious that the rates of increase in R k have the same behavior for the both of footings (circular and square footings) under different load eccentricities. For R k , average values of each footing are taken for the plot of the results graph. The average values of α and k are 2.313 and 0.945, respectively. The experimental reduction factor increases by increasing the load eccentricity ratio. To verify the proposed formulation, the reported experimental or numerical results in several investigations [33,34,35,36,37] are compared with the calculated bearing capacities of Meyerhof’s method and the proposed reduction factor. The details of the processed datasets and comparison between the measured and estimated bearing capacities for the circular and square footings on unreinforced sand are summarized in Table 7. The ultimate bearing capacity equation of Meyerhof [21] for non-cohesive soils on shallow footing can be expressed by q u = 0.5BʹγN γ λ γs , where q u is the ultimate bearing capacity per unit area of the footing effective surface, Bʹ is the effective width of the footing, γ is the effective soil unit weight and N γ and λ γs are respectively the bearing capacity factor and shape factor which are defined in Eqs. 8 and 9, where for circular and square footing B = L = D.

Fig. 7
figure 7

Reduction factor versus load eccentricity

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Table 7 Measured versus estimated bearing capacities on unreinforced sand
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$${N_\gamma }=[{2.72^{\pi \tan \phi }}\left(\frac{{1+\sin \phi }}{{1 - \sin \phi }}\right) - 1]\tan (1.4\phi )$$
(8)
$$\lambda _{{\gamma s}} = 1 + 0.1\left(\frac{B}{L}\right)\tan ^{2} \left(45 + \frac{\phi }{2}\right)$$
(9)

According to the presented error in Table 7, it is evident that the predicted values by using the effective area of Meyerhof’s method and the proposed formulation are in good agreement with experimental or numerical results for both footings in terms of eccentric ultimate bearing capacity. For some models, the difference between the ultimate bearing capacity (q u ) and that in the proposed method is lesser than effective area of Meyerhof. It can be seen from Table 7 that the effective width concept generally overestimated the performance of unreinforced sand. However, the predicted ultimate bearing capacity values with the proposed method have a maximum error less than Meyerhof’s method. It is worth to note that, based on this investigation, one can find that increasing of the footing diameter or footing width (D or B) and internal friction angle (ϕ) have not any influence on the reduction factor which has been also reported by Purkayastha and Char [24] and Patra et al. [2].

Reinforced Sand

Based on the laboratory results, a solution for estimating the eccentric ultimate bearing capacity of geogrid reinforced sand for the both of footings in one-way and two-way load eccentricities are developed. As Eq. 4, R k is defined as Eq. 10, where q u(eccentric,reinforced) is the eccentric ultimate bearing capacity of reinforced soil foundation and q u(ccentric,unreinforced) is the centric ultimate bearing capacity of unreinforced soil foundation.

$${R_k}=1 - \frac{{{q_{u(eccentric,re\inf orced)}}}}{{{q_{u(centeric,unre\inf orced)}}}}$$
(10)

It is worth to note, the advantage of the proposed method in reinforced condition is that eccentric ultimate bearing capacity in reinforced sand predicts from centric ultimate bearing capacity in unreinforced sand which is well known by many researchers and can be predicted simply with theoretical methods. As reduction factor in reinforced sand may contain negative values and cannot create an exponential trend line, therefore a constant value (π) is added to the R k . A non-linear regression model is utilized to illustrate the increasing amount of reduction factor values in an object as it loaded eccentrically. The linear relationship lead to the nearly linear increase of reduction factor with the increase of number of reinforcement layers, and no convergence can be obtained, while the proposed power best fit relations overcomes this shortcoming and converges as the number of reinforcement layers increases. This exponential trend line is based on the Eq. 11:

$${R_k}=\alpha '{\left(\frac{e}{{{B^{}}o{r^{}}D}}\right)^{k'}}$$
(11)

where the α′ and k′ are constants. The experimental values of R k versus different load eccentricities for every number of reinforcement layers are given in Fig. 8. As demonstrated in this figure, in contrast to unreinforced sand, circular and square footings result different values of R k in the same test condition. It is worth to note that the R-squared values are larger than 0.95, meaning that the lines fit the data perfectly. The circular footings have a lower value of reduction in comparison with square footings. Besides, for square footing, two-way eccentricities demonstrated higher reduction to one-way eccentricities. While, by increasing the load eccentricity the difference between related reduction factors becomes more with an almost linear increment. The new bearing capacity formula (Eq. 12) that incorporated the contribution of reinforcements to the increase in eccentrically bearing capacity is developed as follow for both type of footing:

Fig. 8
figure 8

Reduction factor versus load eccentricity in reinforced condition

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$${q_{u(eccentric,re\inf orced)}}={q_{u(centric,unre\inf orced)}} \times (1+\pi - {R_k})$$
(12)

The values of α′ and k′ versus the number of reinforcement layers for circular, square with one-way and two-way eccentricity are shown in Fig. 9. An order of two polynomial trend lines is used to predict influence of N on the values of R k for circular and square footings with one-way and two-way load eccentricities, respectively. Results show the R-squared values for them are one, which indicate acceptable fit of the lines to the data. The parameters of reduction factor can be easily evaluated with these equations to determine the ultimate bearing capacity on reinforced sand. A comparison is made between large-scale work of Saran et al. [15] and the proposed method for square footing width B = 1000 mm and load eccentricities of 0.1 B placed at a depth zero below the ground level in reinforced condition which is shown in Table 8. The unit weight of sand was 15 kN/m3 and angle of internal friction was 35°. The used geogrid in model tests had tensile strength 20 kN/m at 5% strain in both the machines and the cross-machine directions and sand-geogrid frictional angle was 22°. Also the proposed solutions are verified by the results of laboratory model tests conducted by Badakhshan and Noorzad [2] for circular footing on geogrid-reinforced sand under eccentrically loadings with reinforcement tensile strength 7.68 kN/m at 2.5% strain. It is clear that the predicted values by using the proposed solution are in good agreement with those obtained in model test results of Saran et al. [15] and Badakhshan and Noorzad [39]. In order to ensure the capability of the proposed equations, the obtained test results by Saran et al. [16] on square footing subjected to one-way eccentric load resting on reinforced sand, are also compared with the calculated bearing capacities. The related test soil is classified as poorly graded sand (SP) and physical properties of test sand such as maximum void ratio (e max), minimum void ratio (e min), relative density are determined as 0.913, 0.509 and 65%, respectively. The unit weight of sand is about of 16.0 kN/m3. The angle of internal friction is obtained from triaxial tests about 39°. Tensar SS20 geogrid was used to reinforce the sand throughout the model test program. The tensile strength of geogrid both in the longitudinal and transverse directions is equal to 20 kN/m. The square footing measures 200 mm x 200 mm in plan. Depth of the top-most layer (u) and vertical spacing between adjacent layers of reinforcement (h) were kept as 0.25B, where, B is the width of the footing. Size of each reinforcement layers (L r ) was taken as B, 2B and 3B. These tests were conducted on the footing for e/B = 0.1 and 0.2. It can be seen from Table 8 that the proposed method are in good agreement with the test results of Saran et al. [16] in tests with L r bigger than two. It is worth to note, the advantage of the proposed method in reinforced condition is that eccentric ultimate bearing capacity in reinforced sand predicts from centric ultimate bearing capacity in unreinforced sand which is well known by many researchers and can be predicted simply. Also, as in experimental tests that conducted by the authors, based on previous researches [27,28,29,30,31,32] the optimum length of reinforcement is selected L r = 4.5B, and the proposed equation is compatible with this condition and in others un-optimum lengths such as some of the test results of Saran et al. [16], the proposed equation has an unacceptable error. On the other hands, the proposed method provided a good prediction of eccentric ultimate bearing capacity for models with a maximum error less than 14% in optimum condition of reinforced soil.

Fig. 9
figure 9

Values of α′ and k′ versus the number of reinforcement layers, a square footing with two-way load eccentricity, b square footing with one-way load eccentricity and c circular footing

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Table 8 Measured versus estimated bearing capacities on reinforced sand
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Finite Element Procedure

Numerical analysis based on finite element method (FEM) in a three-dimensional space is provided for two purposes as verifying the laboratory tests conducted by the authors and studying the behavior of reinforcement layers. The constructed numerical model is validated using experimental data obtained for circular footing on three layers of reinforcement in load eccentricity e/D  = 0.125. Essential and geometric boundary conditions which were prescribed displacement equal to zero at the boundary of the simulation domain are utilized in modeling. Thus, supports were used to restrain foundation soil bed boundaries against relative rigid body motions. For the modeling, boundary conditions were chosen such that nodes at the base of the model are restricted in all directions (full fixity of displacement in both directions x and y), while vertical boundaries were restricted horizontally (fixity of displacement in directions x) and were free to move in the vertical direction. The properties of sand and reinforcement layers were chosen such that those used in the laboratory test. A first step towards accurate modeling of gesynthetic was choosing a suitable material model. In previous works the only material property of reinforcement was elastic normal (axial) stiffness but in the current study the reinforcement layers are modeled as a linear elastic material. The soil was discretized using 10-noded tetrahedron solid elements while the reinforcement was modeled with 4-noded rectangular membrane elements which having negligible bending stiffness. Membrane elements were used to represent thin surfaces in space that offer strength in the plane of the element but have no bending stiffness. The thickness of reinforcement layer was assumed 4 mm in all models as actual condition. Secant modulus of elasticity for the reinforcement per unit length which chosen from the manufacturer’s manual is about 219 kN/m. Also analysis was performed under displacement control. In modeling the yielding of frictional material (sand), the elastic-perfectly plastic Drucker–Prager constitutive model with a non-associated flow rule was used. The parameters adopted for the analysis are given in Table 9. The parameters (d) and (β) are the cohesion and internal frictional angle used in the Drucker–Prager constitutive model. More details on these parameters can be found in Helwany [40]. The model was divided into four main domains; the top soil (above the reinforcement), the bottom soil (below the reinforcement), foundation and the reinforcement layers. It is worth to say that a small cohesion value (1 kN/m2) is used in this study to improve the stability of the analyses and avoid any singularity that may arise. The interaction between reinforcement and sand was modeled at both sides by using interface elements. For define interface between the sand (solid elements) and the reinforcement layers (membrane elements) a parametric study is conducted and indicates that using of the surface to surface contact model with the traditional node to surface contact formulation is suitable. It should be noted that the behavior of the soil-reinforcement interface is simulated using the Coulomb friction model with two material parameters such as friction coefficient (μ) and tolerance parameter (E slip). Figure 10 compares the obtained results from the laboratory model test and numerical analysis. As seen, this comparison is satisfactory, thus the results of a larger footing may be determined. Also, the numerical analysis is taken up to understand the reinforcement behavior mechanism further in light of stress and displacement distributions underneath the footings. Vertical displacements contours of embedded reinforcement layers in sand are shown in Fig. 11 from numerical analysis. This figure clearly indicates that the utilized reinforcement layers lengths are appropriate for this analysis. As be expected, the vertical displacement of the upper layer (located closer to the footing) is larger than that of the lower one. Also most of the geogrid deformations occur mainly in the area immediately below the footing with very small deformation away from the loaded area. The comparison between experimental test and numerical model demonstrates that the proposed approach for modeling unconfined geogrid in 3D is suitable for solving geogrid-reinforced soil systems and this agreement can confirm the accuracy of proposed equations. For comparison of the proposed method with a large scale circular footing in reinforced sand with load eccentricity, in numerical analysis of large-scale the diameter of footing is selected 2.5 m. Figure 12 shows 3D finite element mesh used in this analysis. The sand and reinforcement layers are chosen as same as the above. The results of numerical large scale models are compared with results of current study to find out the ability of the proposed method to calculate eccentric bearing capacity of real scale footing. For this purpose the different load eccentricity ratio (e/D = 0.05, 0.1 and 0.15) in N = 3 condition (u/D = h/D = 0.42) were considered. The ultimate bearing capacity of footing in desired load eccentricities is determined about 6303, 5877 and 4833 kPa for numerical models and calculated about 6798, 6012 and 5262 kPa for proposed method, respectively. Comparison of estimated and numerical values of bearing capacity for the large scale analyses on reinforced sand present that there are very good agreement between measured values with numerical models and proposed methods.

Table 9 Material properties used in numerical models
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Fig. 10
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Validation of laboratory tests for circular footing

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Fig. 11
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vertical displacement contours for reinforcement layers

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Fig. 12
figure 12

Finite element mesh for large scale models

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Design Example

It is required to predict the ultimate bearing capacity of a square footing resting on reinforced sand to carry safely, a central load with eccentricity of 0.1. The unit weight of sand is 15 kN/m3 and angle of internal friction is 35°. The sand below the footing is reinforced with geogrid of 20 kN/m strength. Taking footing of width B = 500 mm, placed at a depth zero below the ground level and sand is reinforced in three layers with L r = 4.5 and embedment depth ration u/B = h/B = 0.42. In this condition for prediction the ultimate bearing capacity of reinforced sand, at the first time the centric-unreinforced ultimate bearing capacity of footing must be determined by an appropriate theoretical method for example Meyerhof equation. Thus, the value of q u(ccentric,unreinforced) calculates about 189 kN/m2. By using the value of reduction factor related to the e/B = 0.1 and N = 3 from Fig. 8, can obtain the ultimate bearing capacity on reinforced sand under one-way eccentrical loading based the Eq. 12. In this example the value of R k calculate about 2.226 and consequently the q u(ecccentric,reinforced) determine about 361 kN/m2.

Conclusions

The current paper presents obtained data from laboratory tests carried out on shallow circular footing with 120 mm in diameter and square footing of 120 mm in width under central and eccentral loadings resting on unreinforced and reinforced sandy soil. An equation is proposed for eccentrically loaded circular and square footings resting on reinforced sand. The results are in fair agreement with those of Meyerhof’s method in unreinforced condition. Also, finite element method (FEM) in a three-dimensional space is used for verifying the laboratory tests and studying the behavior of reinforcement layers. Based on the obtained results, the following main conclusions were drawn:

  1. 1.

    By increasing the load eccentricity, ultimate bearing capacity decreases and this reduction is more when the load eccentricity is larger than footing Kern for both circular and square footings. The footing tilt increases linearly with load eccentricity and this rate for load eccentricities inside the Kern is smaller than outside the Kern. Also, the tilt of footing at ultimate bearing capacity, from center to Kern increase about 18%, but while this increase is about 53% to outside the Kern.

  2. 2.

    Improvement Index increases with increase of number of reinforcement layers for all the tests, but the improvement is more significant for circular footing and by increasing the load eccentricity, reinforcement layers have more influence on circular footing than square footing. The circular footing causes a bigger mobilized tension in reinforcement layers that enables the reinforcement to resist the imposed horizontal shear stresses built up in the mass beneath the loaded area by transferring the footing load to deeper layers of soil in comparison with square footing.

  3. 3.

    Based on this method, the eccentrically ultimate bearing capacity for both of circular and square footings can be calculated which is only dependent on the load eccentricity ratio (e/D or e/B) and unreinforced central ultimate bearing capacity.

  4. 4.

    The process of calculating the bearing capacity has been simplified by presenting suitable charts in non-dimensional form that can be directly used for the same purpose. It is shown that the new reduction factor that is capable of predicting the one-way and two-way eccentrical ultimate bearing capacity of circular and square footings on geogrid reinforced soil is depended to the two parameters of α′ and k′. The measured values by using the proposed solution are compared with the large scale tests and large scale numerical model and show a good agreement with a maximum error less than 14%.