1 Introduction

The ultimate bearing capacity of strip footing on granular layer overlying clay is estimated using conventional bearing capacity models (refer to Shoaei et al. 2012 for a review). Extensive studies by analytical approaches, experimental investigations, and numerical techniques have been used in the past. Among them, two semi-empirical approaches based on the equilibrium concept are widely used.

The first one is the projected area or load spread method, which assumes that the granular layer spreads load uniformly to a larger area at the top of the underlying clay layer. This approach, first proposed by Terzaghi and Peck (1948) and expanded by Yamaguchi (1963), ignores any shear resistance provided by the granular layer. The bearing capacity of strip footing is determined from the ultimate bearing capacity of the underlying clay and the base area of the block, as expressed in Eq. (1).

$${q}_{u}=\left(1+\frac{2H}{B}\mathrm{tan}\alpha \right)\left({c}_{u} {N}_{c}+{p}_{0} \right){\le q}_{g}$$
(1)

These earlier methods assumed a constant angle of load spread (α) irrespective of the strength of the soil layers and geometric conditions. However, the selection of α is crucial. Subsequently, a number of recommendations are made regarding α, as summarised in Table 1.

Table 1 Recommended values of the angle of load spread (α)
Full size table

Meyerhof (1974) suggested that the value of α varied within the depth of the granular layer and proposed to use an average value of 2ϕ/3. Lawton (1995) numerically integrated the curves developed by Burmister (1958) and Fox (1948) for the two-layer elastic soil system and produced a design chart of α versus the modulus ratio (Egranular/Eclay). The value of α was dependent on the shear strength of the clay, stiffness ratio between the layers, and the depth-to-width ratio (H/B) (Kraft and Helfrich 1983; Brocklehurst 1993; Kenny and Andrawes 1997; Ballard et al. 2011; Lees 2020). Lyon and Fannin (2006) observed significant variations in α with granular layer thickness with α between 20 and 40°, and higher α values appearing at greater thickness. Eshkevari et al. (2019) used the limit equilibrium method to develop a model and found that α could be negative or zero. In practice, α ranges from 11.3 (5 V:1H) to 18.4° (3 V:1H) (Tang et al. 2017). This shows that there is no consensus in the selection of α despite it having a significant influence on the estimation of bearing capacity. Therefore, the uncertainty about α is the main shortcoming of this approach.

The second equilibrium approach assumes a granular punching shear failure mechanism with a vertical side block (α = 0). The shearing resistance along the side surface of the block having an approximately truncated pyramidal shape is pushed into the clay as proposed by Meyerhof (1974) and Hanna and Meyerhof (1980) are shown in Fig. 1a. The granular layer is assumed to behave in the state of passive failure along the vertical plane underneath the footing perimeters. The results of the analysis are presented in the form of charts showing the relationship between the coefficient of punching shear (Ks) and undrained shear strength of clay (cu) for a range of δ/ϕ. These charts are not presented in a dimensionless form and are applicable only to the value of the sand unit weight and layer thickness that were adopted during the test. Furthermore, the detailed limit equilibrium calculation used by Hanna and Meyerhof (1980) is not clear to reproduce their analysis (Burd and Frydman 1997). The main drawback with this semi-empirical method is the difficulty in estimating the forces along the vertical shear planes, and empirical Ks charts provided are limited to ϕ = 40 to 50° range. Interpolation or extrapolation is required for cases beyond the given value, which may be incorrect.

Fig. 1
figure 1

adapted from Meyerhof 1974)

Failure of soil below footing on a dense granular layer above soft clay (

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This shows the uncertainty of these approaches as they have simplified failure mechanisms and empirical coefficients (α for the load spread method and Ks for the punching shear method).

These load spread and the punching shear models have some common characteristics that the bearing capacity of the clay is unaffected by the applied vertical stress to the clay surface outside the loaded area. Madhav and Sharma (1991) showed that the bearing capacity of the granular layer may be significantly enhanced up to 30% by considering gradual decay of the vertical stresses horizontally across the clay surface. They found that the vertical stresses decay could extend up to 5B from the edge of the footing. Perkins and Edens (2002), who studied the distress models for geosynthetics-reinforced pavements using FEM, made similar observations for the unreinforced case.

Michalowski and Shi (1995) used the kinematic approach of limit analysis for upper bound estimates of the bearing capacity of strip footings for sand-clay layers. They carried out extensive parametric studies, and the results were presented in the form of design charts. Burd and Frydman (1997) studied the Michalowski and Shi (1995) solutions over a limited range of variables and found that the assumed sand was a fully associated material which might lead to overestimation of the bearing capacity. In reality, the dilation of granular soils is substantially lower than the angle of friction. Therefore, the adoption of a dilation angle equal to the friction angle will provide an upper bound solution for a nonassociated material. For nonassociated material, the friction angle needs to be modified to obtain an acceptable kinematic solution (Davis 1968; Drescher and Detournay 1993).

In the 60 centrifuge model tests conducted by Okamura et al. (1997, 1998), they found cu increased with depth which needs to be transformed into an equivalent value. The bearing capacity of footings on soft clay of strength increasing with depth using dimensionless ratio ρB/cu0 (where ρ = dcu/dz) is the linear increase of cu with cu0 was characterised by Tani and Craig (1995). Furthermore, the limit of shear stresses generated at the interface between the layers of different stiffness is governed by cu at the surface of the clay rather than depth-dependent. The effect of ρ on the bearing capacity of strip footing resting on a granular layer over clay with increasing strength of clay was further investigated by Shiau et al. (2003). They found that the bearing capacity increased with H/B, eventually, the failure mechanism is entirely within the deeper granular layer (did not penetrate the underlying clay bed). In addition, the bearing capacity of strip footing on a granular layer over undrained clay is unaffected by clay unit weight.

Milligan et al. (1989a and b) developed a new method for the design of unreinforced and reinforced unpaved roads using the load spread model. The applied vertical stresses to a granular layer also develop horizontal stresses within this layer. These stresses, held in equilibrium by shear stresses at the surface of the clay, have an adverse impact on the bearing capacity. They observed that the geogrid reinforcement at small surface deflection can enhance the development of stresses at the interface of granular and clay layers, which had a positive impact on bearing capacity. By considering only the shear stresses developed at the base of the granular layer and not the stabilising function of geogrid, the Milligan et al. (1989a and b) method may under-estimate the full benefit of geogrid performance, particularly over soft clay.

Abu-Farsakh et al. (2016) used a field plate load test (PLT) to evaluate the performance of geosynthetics-reinforced material in unpaved roads. They found that geosynthetics-reinforcement leads to an increase in bearing capacity and reduction of surface deformation with double geogrid layers yielded the best performance. Gabr and Hart (2000) conducted laboratory PLT to evaluate the effect of geogrid-reinforcement on the elastic modulus of sandy soil and found the improvement of bearing capacity at relatively low displacement.

A field PLT is one of the most reliable methods of ultimate bearing capacity for footing design. However, the results are based on one particular size, and other tests are required for other sizes. Due to the requirement of specific equipment, difficulties in the test setup, and time has taken is to complete the field, PLTs are comparatively expensive and seldom used if reasonable information could be obtained from other methods. Laboratory tests, on the other hand, are quicker, cheaper, and easier than field PLT and they also can be performed under controlled conditions (Hugo et al. 1991; Goud et al. 2018). However, it is difficult to replicate the actual behaviour of the two-layered soil system with a small-scale laboratory test. Large-scale laboratory tests can simulate the two-layered soil system behaviour without the detrimental effects of the boundary conditions, scale effects, and coarse grains materials frequently used in foundation construction in lieu of sand which is more reflective of field conditions. These shortcomings can be overcome by using numerical methods.

Numerical techniques are alternative methods for calculating the bearing capacity of footings on layered soil, as demonstrated by past studies (Griffiths 1982; Brocklehurst 1993; Manoharan and Dasgupta 1995; Burd and Frydman 1997; Merifield et al. 1999; Shiau et al. 2003; Zhu 2004; Helwany 2007; Ballard et al. 2011; Mosadegh and Nikraz 2015; Khatri et al. 2017; Naeini et al. 2018; Pham and Ohtsuka 2021). These techniques can deal with complicated loadings, geometries of complex shape, boundary conditions, idealise soil behaviour, anisotropy, layered soils, and complex stress–strain relationships. They have added the advantage of combining all the parameters into a single problem, less time consuming to set up than experimental techniques, and removes the characterisation of model uncertainty in the two semi-empirical methods (e.g., α for the load spread method and Ks for the punching shear method). However, depending on the type of simulation, 3D (three-dimensional) models can take a huge amount of computation time. The model confined boundary conditions (geometry, roller, fixed, etc.) may apply some extra restrictions, which can unintentionally affect the results (Ghadimi 2015).

Past studies (e.g., Perkins and Edens 2002; Abu-Farsakh et al. 2016) have used FEM and laboratory plate load tests to demonstrate the effect of aggregate confinement provided by geogrid reinforcement through interlocking. Reinforcement stiffens the aggregate layer, thereby increasing the bulk stress, which in turn increases the stiffness of the granular layer. McDowell et al. (2006) used a pull-out test and found the confinement influence zone of 100 mm on either side of the geogrid. Bussert and Cavanaugh (2010) found that the stabilising effect of geogrid extends over 300 mm from the geogrid plane. Therefore, geogrid and aggregate composite can be considered a mechanically stabilised layer.

This study proposes a new numerical technique to address the limitations of some of these previous approaches to estimate the bearing capacity of strip footings on non-stabilised and geogrid-stabilised granular layers over clay. The parametric study has identified various parameters and dimensionless groups that influence the bearing capacity used to formulate this new approach.

2 Problem Definition

The 2D plane strain bearing capacity problem considered in this study is illustrated in Fig. 2. A strip footing of width B rests on a granular layer that has a thickness H, a unit weight γ, an internal friction angle ϕ, and a surcharge q. The granular layer is underlain by a clay layer of the infinite depth of undrained shear strength cu. For this study, the short-term stability of the footing is considered, and it is reasonable to assume the granular layer is fully drained, and the clay bed is undrained.

Fig. 2
figure 2

Schematic diagram and terminologies of the problem

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Past studies (Brown and Meyerhof 1969; Meyerhof and Hanna 1978; Michalowski and Shi 1995) found that the limiting bearing capacity for a strong granular layer over a weak clay soil system becomes constant when the critical depth-to-width ratio (H/B) is around 2.5 to 2.7. In this study, the FEM solutions have been computed for problems where H/B ranges from 0.5 to 2 and cu vary from 15 to 120 kPa with constant footing width B. These model geometry and the soil strength parameters provide coverage for most practical problems.

The bearing capacity of a shallow strip footing on a clay layer can be expressed as

$${q}_{c}= {c}_{u}{N}_{c }{S}_{c}+ {p}_{0}$$
(2)

where \({p}_{0}= \gamma D + \gamma H\).

Nc equals the well-known Prandtl’s solution of (2 + π) or 5.14. However, for a two-layer soil system with a dense granular layer overlying soft clay, Eq. (2) can be substituted into Eq. (1) to form Eq. (3).

$${q}_{u}=\left(1+2\frac{H}{B}\mathrm{tan}\alpha \right) {q}_{c}{\le q}_{g}$$
(3)

where \({q}_{g}\) is the ultimate bearing capacity entirely due to a thick granular layer without clay. In the case of no embedment, \({q}_{g}\) can be determined from Eq. (4).

$${q}_{g}=0.5 B\gamma {N}_{\gamma }{S}_{\gamma }$$
(4)

By dividing all the items in Eq. (3) by qc and the equation can be written in the form

$$\frac{{q}_{u}}{{q}_{c}}=\left(1+2\frac{H}{B}\mathrm{tan}\alpha \right) \le \frac{{q}_{g}}{{q}_{c}}$$
(5)
$$\frac{{q}_{u}}{{q}_{c}}=\left(1+m \frac{H}{B}\right) \le \frac{{q}_{g}}{{q}_{c}}$$
(6)

Equation (5) suggests that the value of bearing capacity ratio qu/qc depends on α and H/B. It is shown later in the study that H/B has a significant influence on the bearing capacity of the granular layer over clay. It is worth noting that when qu/\({q}_{g}\) equals unity, the bearing capacity is fully dependent on the granular layer alone and a critical H/B is reached at which the clay has no effect on the bearing capacity as illustrated in Fig. 1b. Equation (5) can be presented in the form of linear Eq. (6). It shows that qu/qc is linearly related to H/B with m being the gradient (slope) and intercept on the qu/qc axis at 1. This fundamental idea is the basis for this new approach to estimate the bearing capacity of strip footings on a granular layer over clay.

By assuming the load from the footing is spread over a total width B at the base of the granular layer and the bearing capacity qc at the clay surface is cuNc + γH, the punching shear model proposed by Meyerhof (1974) and Hanna and Meyerhof (1980) can be compared with the load spread method as shown by Burd and Frydman (1997). Then, the punching shear model becomes

$${B}^{^{\prime}}=B+ \frac{\gamma {H}^{2}{K}_{s} \mathrm{tan}\phi }{{c}_{u}{N}_{c}}$$
(7)

This matches the value of load spread angle expressed as

$$\mathrm{tan}\alpha =\left(\frac{\gamma H}{{c}_{u}}\right)\left(\frac{{K}_{s}\mathrm{tan}\phi }{2{N}_{c}}\right)$$
(8)

Equation (8) implies that α depends on ϕ and cu/γH.

Considering the incompressibility of clay and the geometry of the problem, the net work done by the weight of the clay must be zero and the bearing capacity solution has been shown to be independent of the unit weight of clay (Michalowski and Shi 1995).

The relationships described in Eqs. (5), (6) and (8) imply that qu/qc for strip footing resting on a two-layered soil system is related to two dimensionless parameter groups H/B and cu/\({p}_{0}\), internal friction angle ϕ and geogrid stiffness k when the geogrid is used, is given by

$$\frac{{q}_{u}}{{q}_{c}} =f\left(\frac{H}{B},\frac{{c}_{u}}{{p}_{0}},\phi ,k\right)$$
(9)

This means that the presence of ground water table in the granular layer is accounted for by subtracting the pore water pressure from the total pressure to obtain the effective vertical stress po at the base of granular layer. In addition, the effective vertical stress po is inclusive of overburden or surcharge stress where applicable.

3 Analysis of Data from Past Studies for Linearity

The proposed Eq. (6) showing the linear relationship between qu/qc and H/B with m being the slope (gradient) and intercept on the qu/qc axis at 1 is investigated using the data from published literature. Extensive past studies of load spread method, experimental results, limit analysis methods, finite element limit analysis, and finite element method for the bearing capacity of strip footing on granular layer overlying clay have been collated and processed.

The qu/qc and H/B values determined from Hanna and Meyerhof (1980) for cu equal to 15 and 30 kPa with different ϕ values are plotted in Fig. 3a and 3b (denoted HM). It shows a nonlinear relationship, but linear approximation also fits the data strongly as explained by higher R2 values. This can be attributed to the increasing influence of shear strength of the granular layer at higher values of H/B resulting in an overestimation of bearing capacity at higher H/B values and underestimation of bearing capacity at lower H/B and cu values (Lees 2017).

Fig. 3
figure 3

Bearing capacity ratio (qu/qc) versus H/B (data source: Hannah and Meyerhof 1980)

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The design charts presented by Michalowski and Shi (1995) using the kinematic approach of limit analysis (designated LA) is used to calculate the bearing capacity of strip footings over a two-layer soil system. These design charts are used to interpolate the qu/qc and H/B values. The results show a reasonable linear relationship (strong R2 values) between qu/qc and H/B as depicted in Fig. 4.

Fig. 4
figure 4

Bearing capacity ratio (qu/qc) versus H/B (data source: Michalowski and Shi 1995)

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Kenny and Andrawes (1997) performed 10 laboratory plate load tests to simulate the performance of strip footing on a sand layer overlying soft clay. The experimental test results are presented in Fig. 5a. The laboratory test (denoted LT) results show some variability as expected. Overall, there is an approximately linear relationship between qu/qc and H/B with an intercept on the qu/qc axis at 1 and the statistical fit is also reasonable.

Fig. 5
figure 5

Bearing capacity ratio (qu/qc) versus H/B (data source: a Kenny and Andrawes (1997), b Okamura et al. (1997, 1998), and c Huang and Qin (2009))

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Okamura et al. (1997, 1998) carried out 60 centrifuge model tests on the strip and circular footings on unreinforced Toyoura sand with ϕ = 45° overlying Kawasaki clay with different cu by varying H/B. The laboratory tests are limited to clay strength ranging from 21.9 to 22.4 kPa for strip footing. The results are plotted in Fig. 5b. The best-fit line is closely linear between qu/qc and H/B.

Huang and Qin (2009) used the limit analysis (LA) to obtain the upper bound multi-rigid-block solutions for the bearing capacity of two-layered soils. They produced design charts of the bearing capacity ratio (qu/γB) versus the shear strength ratio (cu/γB) to compare with Michalowski and Shi (1995) and Shiau et al. (2003). These design charts have been interpolated to obtain qu/qc and H/B values and shown in Fig. 5c. The plot also shows an approximately linear relationship.

The dimensionless design charts to calculate the ultimate bearing capacity of a strip footing investigated using finite element method limit analysis (FELA) for upper and lower bound solutions by Shiau et al. (2003) are used to test the validity of the linear relationship between qu/qc and H/B. The FELA results also show an approximately linear relationship between qu/qc and H/B as shown in Fig. 6.

Fig. 6
figure 6

Bearing capacity ratio qu/qc versus H/B (data source: Shiau et al. 2003)

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Zheng et al. (2019) proposed discontinuity layout optimisation (DLO) method with a wide range of geometric and soil strength parameters to develop design charts and to calculate the bearing capacity of vertically loaded strip footings on sand overlying clay. These design charts have been interpolated to obtain qu/qc and H/B values and are plotted in Fig. 7. It also closely follows a linear relationship.

Fig. 7
figure 7

Bearing capacity ratio (qu/qc) versus H/B (data source: Zheng et al. 2019)

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Pham and Ohtsuka (2021) used the rigid-plastic finite element method (RPFEM) to investigate the ultimate bearing capacity of strip footing on sand-clay layered soils. They developed a series of design charts showing the relationship between q/γB and cu/γB. These design charts have been used to interpolate the qu/qc and H/B values. RPFEM results show approximately a linear relationship between qu/qc and H/B as illustrated in Fig. 8. The results are generally similar to Shiau et al. (2003) in terms of qu/qc and H/B trend. It is worth noting that the gradient of the linear relationship m can be a negative value as illustrated in Fig. 8a.

Fig. 8
figure 8

Bearing capacity ratio (qu/qc) versus H/B (data source: Pham and Ohtsuka 2021)

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By assuming the load spread angles of 2 V:1H (26.6°) and 1 V:1H (45°), the bearing capacity of strip footing on a granular layer over soft clay is calculated. The results are then used to generate qu/qc and H/B values. The results evidently follow a linear relationship between qu/qc and H/B, as illustrated in Fig. 9.

Fig. 9
figure 9

Bearing capacity ratio (qu/qc) versus H/B (data source: load spread methods)

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These analyses of data from various past studies clearly demonstrate that there is a clear indication of the linear relationship between qu/qc and H/B with an intercept on the qu/qc axis at 1.

4 Finite Element Model (FEM)

Strip footing is a rectangular shape of infinite length (L) and finite width (B) (thus, B/L ≈ 0). Normally, footings with L/B > 10 are regarded as strip footings. It can be analysed assuming 2D plane strain conditions. The 2D plane strain model is developed using ABAQUS software to analyse the non-stabilised and geogrid-stabilised granular layer over the clay, as illustrated in Fig. 10. The dimensions of the model geometry are selected in such a way that the boundary effects are minimised. The geogrid reinforcement is placed at the interface of granular and clay layers. Conventional kinematic boundary conditions are adopted where the vertical sides are fixed in a horizontal direction with vertical displacement, and the bottom of the model is fixed in both vertical and horizontal directions. The footing is assumed to be in perfect contact with the granular layer (that is, they share the nodal points in between). The static load is applied using the displacement method up to 20 mm to the footing of width B.

Fig. 10
figure 10

2D plane strain model of a strip footing with geometry and boundary conditions

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The elements chosen for the granular and clay layers are an eight-node biquadratic plane strain quadrilateral shape function and a reduced integration (CPE8R) element, while a two-node linear 2D displacement truss element type (T2D2) was used to discretise the geogrid-reinforcement with a thickness of 0.003 m based on SIMULIA (2013). The top granular layer was assumed to be stronger than the bottom clay layer. The mesh topology and convergence studies have been performed in the previous case studies (Chua et al. 2021). Mohr–Coulomb plasticity models are selected to represent elastoplastic material with the hardening behaviour of granular and clay layers, whereby the shear strength developed in soil is a function of internal friction, cohesion, and applied stress.

The modulus of the granular layer is dependent on the underlying clay. Soft clay does not provide the firm support needed to obtain good compaction of the granular layer thereby resulting in lower modulus. The modulus ratio of granular to clay soil is usually taken to be 1 to 5 as recommended by Heukelom and Klomp (1962) and a ratio of 2 to 5 is selected in this study.

The influence of the associated flow rule is significant for a higher angle of friction (ϕ > 30°) up 15% as demonstrated by past studies (Manoharan and Dasgupta 1995; Potts et al. 2001; Loukidis and Salgado 2009; Mosadegh and Nikraz 2015). In this study, the associated flow rule is replicated by selecting the dilatancy angle (ψ) equal to ϕ less than 30° (Shiau et al. 2003; Ameratunga et al. 2016).

Considering the induced strain in the geogrid is very small and is considered within the elastic range, a linear elastic model is used to characterise the behaviour of geogrid material. Similar models have been used successfully by other researchers (Dondi 1994; Perkins 2001; Perkins and Edens 2002; Ling and Liu 2003; Abu-Farsakh et al. 2014). The geogrid selection is based on commonly used in engineering practice and commercially available in the market. An arbitrary axial stiffness of 600 kN/m is used for geogrid in this study. The stabilising effect of geogrid that extends over 300 mm from the geogrid plane is replicated in the model by assigning stress concentrations of 20 kPa at the geogrid elevation and decreases at a constant rate to a nominal of 0.1 kPa at 300 mm above the geogrid plane (Lees 2017). Other presumptive input parameters of material constitutive models considered in this parametric study are based on published data from Ameratunga et al. (2016), BRE (2004), and Erickson and Drescher (2001), as summarised in Table 2.

Table 2 Material properties for FEM analysis of bearing capacity of strip footings
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A Coulomb friction model is used to simulate the shear resistance interaction between geogrid reinforcement and the granular layer. The geogrid-soil interaction simulation consists of two components. In the normal direction to the interface, hard contact is assumed with no separation is allowed in the model. Whereas in the tangential direction, full interlocking is assumed between the geogrid and soil interface by using tie constraint condition in the ABAQUS interaction feature. The friction model requires the input of two material properties comprising the coefficient of friction and elastic slip. For the case of small surface deformation considered in this study, small elastic slippage of 0.001 m is likely to occur (Perkins and Cuelho 1999; Perkins 2001). The coefficients of friction for geogrid-granular and geogrid-clay layers are 0.84 and 0.62, respectively, are based on BOSTD (2007).

5 Results and Discussions

5.1 Effect of Depth-to-Width Ratio (H/B) in Layered Soil

The results from the developed FEM model are plotted in Figs. 11 and 12. It is evident from these plots that when the internal angle of friction (ϕ) increases for non-stabilised and geogrid-stabilised granular layers, shear resistance increases and hence the bearing capacity. By comparing Figs. 11 a, b, and c, we can see that the strength of the granular layer has a greater influence on the bearing capacity for larger values of H/B for the non-stabilised granular layer. This trend is similar for the geogrid-stabilised granular layer as depicted in Figs. 12 a, b, and c. It reveals that the internal angle of friction (ϕ), for both non-stabilised and geogrid-stabilised granular layers, has a considerable influence on the bearing capacity (qu). These results are consistent with Meyerhof and Hanna (1978) in that the bearing capacity increases with H/B up to approximately H/B ≈ 5 for the strip footing. It is interesting to note that for soft underlying clay, the possibility of punching failure increases with an increase in the strength of the granular layer (Shiau et al. 2003).

Fig. 11
figure 11

Bearing capacity ratio (qu/qc) versus H/B for a non-stabilised granular layer

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Fig. 12
figure 12

Bearing capacity ratio (qu/qc) versus H/B for a geogrid-stabilised granular layer

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The effects of H/B on qu/qc for various values of cu and ϕ for both non-stabilised and geogrid-stabilised granular layers for allowable settlement of 20 mm are illustrated graphically in dimensionless forms in Figs. 13 and 14. The bearing capacity is obtained by averaging the vertical stress component in the first row of integrating elements below the displaced nodes from FEM analyses with different H/B values ranging from 0.5 to 2.0 for soil parameters summarised in Table 1. It is clear from Figs. 13 and 14 that the qu/qc increases with H/B for a constant cu. This is attributed to the granular layer, which plays a major role in bearing capacity when the angle of load spread is increased within the punching shear mechanism for the non-stabilised granular layer. The load spread capacity is further enhanced with a geogrid-stabilised granular layer. This is evident that the bearing capacity for the geogrid-stabilised granular layer is superior to the non-stabilised granular layer. This is attributed to lateral movements at the base of the granular layer and restrained on the top of clay by the geogrid through interlocking between the geogrid and the surrounding soil, thereby improving load spread, as illustrated in Fig. 2. This finding demonstrates that the bearing capacity is controlled by both the thickness of the granular layer and the undrained shear strength of the clay layer.

Fig. 13
figure 13

Bearing capacity ratio (qu/qc) versus undrained shear strength ratio (cu/γB) for non-stabilised granular layer

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Fig. 14
figure 14

Capacity improvement factor (CIF) versus H/B for a geogrid-stabilised granular layer

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5.2 Effect of Clay Strength in Bearing Capacity

The effect of the strength of underlying clay on the qu/qc of a granular layer with H/B = 0.5 and 2.0 for a range of ϕ is illustrated in Fig. 13. This shows that the bearing capacity ratio qu/qc decreases with the increase in undrained shear strength ratio (cu/γB) in a nonlinear trend. Although the bearing capacity increases with the increase in shear strength ratio (cu/γB), the bearing capacity improvement diminishes significantly when it reaches qu/qc = 1 for stiff clay. For a lower H/B, the effect on qu/qc is significantly lower than higher H/B values. This is to be expected as the thickness of the non-stabilised and geogrid-stabilised granular layer has a significant influence on the qu/qc. This suggests that the significant improvement of the bearing capacity occurs when the clay is soft and the granular layer is very dense. Overall, the greatest influence of qu/qc was encountered at cu/γB value less than 5.

5.3 Effect of Geogrid-Stabilised Granular Layer on Bearing Capacity

The improvement of bearing capacity due to the geogrid-stabilised granular layer can be expressed in terms of capacity improvement factor (CIF) as in Eq. (10).

$$CIF={q}_{R}/{q}_{UR}$$
(10)

Figure 14 shows that a single layer of geogrid reinforcement at the interface of the compacted granular layer and the clay yields improvement on CIF values ranging from 1.1 to 1.6., which is a substantial improvement over the non-stabilised case. This improved CIF is attributed to an increase in granular stiffness resulting from aggregate interlock between the geogrid and the granular layer. In addition, the granular-geogrid-clay soil system forms a composite material that inhibits the development of the soil failure wedge. These findings are consistent with the laboratory PLT tests reported by Gabr and Hart (2000) using the biaxial geogrid. It is worth noting that the beneficial effect of geogrid on the CIF is modest for higher cu values. This infers that the geogrid-stabilised granular layer provides smaller improvement on CIF for firm to stiff clays. The effect of ϕ of the granular layer shows a similar trend with bigger improvement on CIF for low cu and marginally higher CIF for a dense compacted granular layer.

In all cases of non-stabilised and geogrid-stabilised granular layers, the relationship between CIF and H/B are closely linear with an intercept on the CIF axis at 1, and the geogrid-stabilised granular layer outperforms the non-stabilised granular cases. Additionally, the impact of reinforcement on bearing capacity for firmer clay is smaller.

5.4 Comparison of FEM Results with Published Results

The parametric results in the FEM study are plotted in dimensionless form with qu/qc and H/B and are compared with published studies: the analytical kinematic predictions of Michalowski and Shi (1995), laboratory centrifuge model tests by Okamura et al. (1997, 1998), the advanced upper- and lower-bound techniques by Shiau et al. (2003), the FEM upper-bound multi-block solutions by Huang and Qin (2009), the discontinuity layout optimisation method by Zheng et al. (2019) and the rigid-plastic FEM by Pham and Ohtsuka (2021). It is important to note that most of the past studies have limited coverage in terms of ϕ of the granular layer and cu of the clay layer.

For the non-stabilised granular layer for ϕ = 35° and cu = 15 kPa, Fig. 15 shows that the results are comparable with Michalowski and Shi (1995), Shiau et al. (2003), and Pham and Ohtsuka (2021). When ϕ is equal to 40°, the results are in good agreement with Shiau et al. (2003) and Pham and Ohtsuka (2021) for cu of 15 and 30 kPa, respectively. For dense non-stabilised granular layer (ϕ = 45°) over soft clay (smaller cu values), the FEM results in this study are in good agreement with Shiau et al. (2003), but lower than Okamura et al. (1997, 1998) and Michalowski and Shi (1995). Michalowski and Shi (1995) assumed the sand layer as a fully associated material, which might have led to the overestimation of the bearing capacity.

Fig. 15
figure 15

Comparison of qu/qc versus H/B for non-stabilised granular layer

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For the geogrid-stabilised granular layer with ϕ = 45° and cu = 30 kPa, Fig. 16 shows that the results are in good agreement with Milligan et al. (1989a, b). However, the Milligan et al. approach, similar to load spread methods, takes no account of qu/qc, H/B, and cu. As a result, the Milligan et al. (1989a, 1989b) outputs are similar only for cu = 30 kPa but no other cu values.

Fig. 16
figure 16

Comparison qu/qc versus H/B for geogrid-stabilised granular layer

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It can be concluded that the proposed Eq. (6) showing the linear relationship of qu/qc and H/B with m being the slope (gradient) and intercept on the qu/qc axis at 1 is synonymous with past studies.

When m values derived from the linear relationship of qu/qc and H/B for the non-stabilised granular layer are superimposed in the design curves, it shows that the proposed lower bound power trend lines for various ϕ values fitted very well as illustrated in Fig. 17. This outcome verified the proposed approach to estimate bearing capacity for strip footing.

Fig. 17
figure 17

Comparison of m values with published data for non-stabilised granular layer

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These design curves are compared with the load spread method for angles of load spread of 26.6 (2 V:1H) and 45° (1 V:1H) with the corresponding m values of 1.2 and 2.8 as shown in Fig. 18. Due to the increased influence of the self-weight of the granular layer at lower values of cu in Eq. (1), the m values show an increase at lower cu/\({p}_{0}\) values. This comparison clearly demonstrates that the adoption of a single angle of load spread in all cases could result in over- or under-estimation of bearing capacity.

Fig. 18
figure 18

Comparison of m versus cu/\({p}_{0}\) for non-stabilised and geogrid-stabilised granular layers

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In the case of the geogrid-stabilised granular layer, the design curves show a similar trend when compared with Lees and Matthias (2019). Figure 18 shows that the power trend line by Lees and Matthias (2019) is shown to have higher m values for all cases of cu/\({p}_{0}\) than the design curves because of the denser geogrid stabilised granular layer with internal friction angle ϕ = 50° as compared with ϕ = 35 to 45° used in this study.

5.5 Proposed Design Curves for Strip Footings on Granular Layer over Clay

The m values derived from the linear relationship of qu/qc and H/B have been used to derive the design charts. In order to accommodate the variability of data, a lower bound fitted with a power trend line is used to express the relationship of m and cu/\({p}_{0}\) as shown in Figs. 19a and b. The power-rule expression with various values of ϕ for non-stabilised and geogrid-stabilised granular layers ranging from 35 to 45° are shown in Figs. 19a and b. This allows direct determination of m for a given ϕ and cu/\({p}_{0}\) graphically or by using the appropriate power-rule equation. The m value is then applied to Eq. (6) to determine the bearing capacity of strip footing.

Fig. 19
figure 19

Design curves for strip footing for non-stabilised and geogrid-stabilised granular layer over clay

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The design curves show a similar trend with a significant shift towards a higher m value for all cases of cu/\({p}_{0}\) for geogrid-stabilised granular layer. In addition, there is an exponential increase in m values for cu/\({p}_{0}\) less than approximately 2.5. This shows that the m value is very sensitive when the shear strength of clay is very low. It could be attributed to the unstable characteristics of very soft clay.

5.6 A Design Example

The use of the proposed m and cu/\({p}_{0}\) design charts for a strip footing on a non-stabilised and geogrid-stabilised granular layer is illustrated by a typical design example. The objective of this is to show how the design curves are applied in the design of strip footing and to compare against established design approaches (e.g., Hanna and Meyerhof 1980; Okamura et al. 1998; Eshkevari 2018; Lees and Matthias 2019; Lees 2020 and load spread methods) and to determine the minimum thickness of the granular layer. The input soil parameters and the strip footing loading conditions used in this design example are summarised in Table 3.

Table 3 Soil parameters used for a design example
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The load spread method is dependent on the assumed α to estimate the vertical compressive stress below a uniformly loaded area of the granular layer. The assumed α has a significant influence on the estimation of vertical compressive stress, which in turn affects the bearing capacity. By using α equal to 26.6 (2 V:1H) and 45° (1 V:1H), the predicted bearing capacities are lower than other methods, including the method proposed in this study, as shown in Fig. 20. This observation explains the reason why smaller α values between 11.3 (5 V:1H) and 18.4° (3 V:1H) are recommended in practice (Tang et al. 2017).

Fig. 20
figure 20

Thickness of granular fill (NS: non-stabilised, GS: geogrid stabilised)

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The thickness of the non-stabilised granular layer in this study shows a good agreement when compared with the T-value method by Lees (2020) and results are also comparable with Hanna and Meyerhof (1980) for clay conditions for cu = 25–35 kPa. The bearing capacities in this study are higher when compared with Okamura et al. (1998), resulting in a thinner non-stabilised granular layer. This may be attributed to the correction of the self-weight of the block in Okamura et al.’s formulation. The bearing capacities in this study are lower when compared with Eshkevari (2018) for lower values of cu and higher for larger values of cu. The results are the same for cu, around 27 kPa. The bearing capacity of the geogrid-stabilised granular layer in this study is lower than the T-value method by Lees and Matthias (2019), yielding a thicker granular layer. They converge at around cu equal to 40 kPa. This can be attributed to the use of ϕ = 50° in Lees and Matthias (2019).

Overall, the geogrid-stabilised granular layers are thinner than non-stabilised granular layers in all cases except thicknesses design by load spread method. This can be attributed to the effect of aggregate confinement provided by geogrid reinforcement through interlocking that stiffens the aggregate layer thereby increasing the bulk stress, which in turn increases the granular layer stiffness resulting in an overall increase in bearing capacity of the geogrid-stabilised granular layer. The confinement effect constrained the lateral movement of aggregate within 300 mm above the geogrid plane in the granular layer and down within the clay layer.

6 Research Assumptions

The proposed method to determine the bearing capacity of strip footing on a non-stabilised and geogrid-stabilised granular layer over clay by FEM has been developed using some assumptions. First, the clay is assumed to be saturated fine-grained soil with low permeability. It behaves in an undrained manner under vertical loading, which is practically incompressible and frictionless. Second, the geogrid is laid on a horizontal plane and placed at the interface of the granular layer and clay. Third, the granular layer is assumed to be compacted well and is of the right quality to provide enough strength and stiffness to support the strip footing before the clay fails. In addition, proper compaction of the granular layers is important to ensure sufficient granular layer strength and stiffness. The minimum required thickness of the granular layer is 150 mm. Furthermore, the deformation of the granular layer is governed by the associative flow rule. Fourth, the generic geogrid stabilised granular layer is established with a minimum 200 MPa tensile modulus for the geogrid. In all cases, the geogrids should meet the minimum AASHTO (2018) guidelines to comply with geosynthetics survivability requirements. Fifth, uniform vertical loading without eccentric and inclined loading conditions will apply. Finally, environmental effects such as degradation and loss of surface shape due to moisture changes in expansive soils are excluded. Hence, it is recommended to be cautious if the results are used other than these conditions.

7 Conclusions

The bearing capacity of a strip footing placed on a non-stabilised and geogrid-stabilised granular layer over clay has been simulated using a 2D plane strain finite element model in ABAQUS. The modelling results are found to compare reasonably well with the available published data in the literature. The FEM results are first validated with published results of past studies: upper and lower bounds limit analysis, finite element limit analysis, finite element method, and laboratory physical modelling. For the non-stabilised granular layer, the results are in good agreement when compared with the T-value method by Lees (2020) and comparable with the classical semi-empirical method by Hanna and Meyerhof (1980) for clay conditions greater than cu = 25 kPa. In the case of the geogrid-stabilised granular layer, the results compare well with the T-value method by Lees and Matthias (2019). The geogrid-stabilised granular layer is thinner than the non-stabilised granular layer in all cases except for the load spread method.

The parametric results from the model are then used to generate dimensionless design charts: slope (m) versus strength ratio (\({c}_{u}/{p}_{0}\)), for a wide range of undrained shear strengths for the clay commonly encountered in practice. A power trend line equation is derived to express the relationship of m and \({c}_{u}/{p}_{0}\) for a range of internal angles of friction (ϕ) of the granular layer. These simple equations are convenient and simple to use even in the spreadsheet to estimate the bearing capacity. The proposed new approach can be a convenient alternative to complex established methods. It can be applied easily to a strip footing under embedded and surface loading conditions including the effect of groundwater where the strengths of granular layer and clay are characterised by both internal friction and cohesion.

The use of geogrid reinforcement in the granular layer yields CIF values ranging from 1.1 to 1.6 for the strip footing bearing capacity depending on the soil parameters and the granular layer thickness. These results show that the geogrid layer is mobilised even at low strains with 20 mm surface deformation. This demonstrates that the geogrid-stabilised granular layer is a viable solution in ground improvement techniques to create a composite material with improved performance characteristics. The results of this study can be used for preliminary concept design to estimate granular layer thickness as the approach is derived from relatively small literature-based data, and it would be beneficial to verify them with field tests.