Introduction

One of the most commonly used foundations in constructing railway tracks, retaining walls, transmission towers, and bridge piers is the strip foundation. These buried structures are typically supported by transferring their load to soils with enough bearing capacity and acceptable settling properties. Since the strip footing stability plays a significant role in practice, several researchers have considered the stability of strip footings on slopes by determining solutions from many different techniques including semi-empirical methods (e.g., [1,2,3]), limit equilibrium techniques (e.g., [4,5,6,7]), slip-line solutions (e.g., [8, 9]), limit analysis (e.g., [10,11,12,13]), finite element methods (e.g., [14,15,16,17]), finite element limit analysis [18], and discontinuity layout optimization (DLO) approaches [19,20,21]. However, there are a few works to study the influence of seismic events on the bearing capacity of footings on slopes, which should be a concern in earthquake areas due to the destructive effects of the footing during seismic situations. One conventional and widely used approach for determining the stability of embedded structures is the pseudo-static approach, where the seismic forces are simply considered as horizontal and/or vertical seismic coefficients (kh and kv) and functions of gravity acceleration.

The seismic responses of a footing on soil have been one of important issues in geotechnical engineering [22,23,24]. Several research have been conducted to calculate the bearing capacity of strip footings on slopes with pseudo-static seismic forces considerations. The theoretical approaches, including limit equilibrium methods (e.g., [25,26,27]), lower bound solutions (e.g., [28]) and upper bound solutions (e.g., [29,30,31,32,33]), and the stress characteristic method (e.g., [34]) which delivered an effective solution to evaluate the problems. In comparison to the analytical techniques discussed previously, a prior assumption regarding the failure mechanisms is not required, therefore, it can provide excellent predicted performance with a wide range of parameters considered. Shiau et al. [35] and Raj et al. [36] investigated the seismic bearing capacity of sloped footings by employing finite element limit analysis (FELA) which provided the upper and lower bounds solutions.

By utilizing the lower bound FELA, Kumar and Chakraborty [37] also computed the bearing capacity factor Ng for a rough strip footing in cohesionless slopes under seismic scenarios. Subsequently, Chakraborty and Kumar [38] and Chakraborty and Mahesh [39] have studied the seismic bearing capacity of strip footings on a sloping ground surface and embankments utilizing the same methodologies. Using the Discontinuity layout optimization (DLO) technique, Zhou et al. [21] studied the ultimate seismic bearing capacity and failure mechanisms for strip footings placed close to the cohesive-frictional soil slopes. In addition, Cinicioglu and Erkli [40] investigated the seismic bearing capacity of strip footings lying on or adjacent to a slope by employing the finite element program PLAXIS in undrained conditions. Recently, the FELA technique was employed by Luo et al. [41], Beygi et al. [42], and Zhang et al. [23] to solve the seismic bearing capacity of strip footings on cohesive and cohesive-frictional soils, in spite that their solutions are limited to the cases of footings resting on the surface of slopes. Due to its popularity, the FELA technique has also been used to many other geotechnical problems [43,44,45,46,47,48,49,50,51,52,53].

In this paper, rigorous solutions of seismic bearing capacity of strip footing in cohesive soil slope are investigated by employing the finite element limit analysis (FELA) and a pseudo-static technique to evaluate the seismic loadings. The upper bound (UB) and lower bound (LB) outcomes attained by the FELA are compared with previously published results. The influences of the seismic acceleration coefficient, soil characteristics, and geometrical parameters on the seismic bearing capacity and the associated failure mechanisms of this problem are investigated. A comprehensive set of design tables and charts are also provided for the uses in design practices. The associated sensitivities are further assessed using multivariate adaptive regression splines (MARS) model, which is capable of accurately capturing the nonlinear relationships between a set of input variables and output variables in multi-dimensions. The MARS-based design equations used for forecasting the solutions of the seismic bearing capacity of strip footings embedded in cohesive slope are finally proposed using the artificial data set generated from FELA. The MARS-based design equation of the current study may be used to perform more precise and reliable evaluations of the seismic bearing capacity of this problem, while considering the coupled influences of the seismic acceleration coefficient, soil characteristics, and geometrical configurations.

Problem Statement and Modelling Technique

The problem definition of a strip footing on a slope is shown in Fig. 1. Under plane strain conditions, the slope has an inclination (β) and a height (H). The footing is assumed to be rigid material with width (B), depth (D), and the distance from top slope to edge of the footing (L). Defining the soil to be a rigid-perfectly plastic Tresca material with a unit weight (γ) and undrained shear strength (cu), the horizontal seismic acceleration (kh) is applied to the footing and the slope under seismic forces. The vertical seismic acceleration is ignored in this study.

Fig. 1
figure 1

Schematic diagram of the model

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A typical model of the footing on slope problem is shown in Fig. 2. The boundary condition is determined using the standard fixity tool in OptumG2 [54], in which all boundary conditions are created as follows: both left-hand and right-hand boundaries are kept stationary in the x direction and the bottom of the boundary is fixed in both x and y directions. The movements at other boundaries are set to be freely moved in both x and y directions. The footing is model as a rigid elastic material and the footing-soil interface is considered as a perfectly rough condition. The size of the domain of this problem is chosen to be sufficiently large so that the plastic zone is contained within the domain and does not intersect with the right and bottom boundaries. An automatically adaptive mesh refinement technique in OptumG2 is used to improve the accuracy of upper and lower bound solutions [55]. Using this technique, the number of elements is automatically increased in the zone with high shear power dissipation that requires sensitivity analyses. The five iterations of the adaptive meshing are used in this study, where the number of elements is set to be automatically increased from 5000 to 10,000 elements [56,57,58,59,60,61,62,63,64,65,66,67].

Fig. 2
figure 2

Numerical model, boundary condition, and failure mechanism

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The seismic bearing capacity of strip footing on a slope can be represented by dimensionless parameters [68] as follows:

$$N_{c} = \frac{{q_{u} }}{{c_{u} }} = f\left( {\beta ,\frac{H}{B},\frac{L}{B},\frac{D}{B},\frac{{c_{u} }}{\gamma B},k_{h} } \right),$$
(1)

where Nc is the undrained seismic bearing capacity factor; qu is the ultimate bearing capacity, H/B is the normalized slope height; L/B is the normalized distance from top slope to edge of the footing; D/B is the normalized depth of footing; cu/γB is the strength ratio. kh is the horizontal seismic acceleration. The seismic bearing capacity factor can be normalized as Nc=qu/cu in Eq. (1). Numerical results are averaged values from UB and LB solutions of OptumG2. The study covers a range of five dimensionless parameters, which comprise the value of cu/γB varies from 1.5 to 5.0, while H/B varies from 1 to 4, L/B values have the ranges of 0–4, D/B varies from 1 to 2, four β values of 15°, 30°, 45°, and 60° and the horizontal seismic acceleration coefficient kh is taken into account at three distinct values of 0.1, 0.2, and 0.3, respectively. These ranges are decided based on the related published literature in [40,41,42].

Comparison of Results

The comparisons are for the investigations into the variation of Nc with kh considering the changing angle slope β with a set of values of remaining dimensionless parameters of cu/γB, H/B, L/B. Figures 3, 4, 5 and 6 show such comparisons. A comparison of Nc variation is presented in Fig. 3 using the case of (cu/γB = 5, D/B = 0, H/B = 4, and L/B = 0) for the four various slope angles. Also shown in Fig. 4 is for the case of L/B = 1. All other parameters are the same as in Fig. 3. In general, the bearing capacity factor Nc decreases linearly with increasing kh. The numerical comparisons have shown that the present results are slightly larger than those lower bound results in Lou et al. [41], despite the fact that they are in good agreement. Seeing the results of kh = 0.3 in Fig. 4, there is a tendency that all curves merge into one point, i.e., one Nc value. One possible reason may be due to the fact that the slopes become unstable as kh increases, resulting in one small value of Nc. This comparison exercise has provided good confidence in producing all later parametric results in the paper.

Fig. 3
figure 3

Comparison of variation in Nc with kh (cu/γB = 5, D/B = 0, H/B = 4, and L/B = 0)

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Fig. 4
figure 4

Comparison of variation in Nc with kh (cu/γB = 5, D/B = 0, H/B = 4, and L/B = 1)

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Fig. 5
figure 5

Variation of Nc with L/B (cu/γB = 2.5, H/B = 4, and D/B = 1)

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Fig. 6
figure 6

Variation of Nc with cu/γB (L/B = 1, H/B = 4, and D/B = 0)

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Results and Discussion

The relationship between Nc and L/B for the various values of kh and β is presented in Fig. 5 for the case of (cu/γB = 2.5, H/B = 4 and D/B = 1). Numerical results have shown that, for all values of kh, the bearing capacity factor Nc increases nonlinearly with the increasing L/B. The larger the pseudo-static acceleration factor kh, the less the value of Nc. The trend is the same for all slope angles β. Nevertheless, the rate of increase (gradient of line) is different from one to the other where the larger the slope angle, the greater the increase of Nc as L/B increases.

The variation study of Nc with cu/γB is presented in Fig. 6 for the case of (L/B = 1, H/B = 4 and D/B = 0). Three values of kh and four values of slope angles β are included in the study. For β = 15° and 30°, the value of Nc is almost constant as the value of cu/γB increases. However, the relationship between Nc and cu/γB becomes nonlinear as β increases (see for example, β = 45° and 60°). In particular, for β = 60°, the increase in Nc stops approximately at a value of cu/γB = 3.5, after which a slight decrease of Nc is attained. It can, therefore, be concluded that the effect of cu/γB on Nc is conspicuous at a higher value of β. Numerical results have also shown that an increase of kh or β leads to a decrease in Nc.

The next study is for the relationship between Nc and β. This is shown in Fig. 7 for three kh values and four different values of L/B of the case (cu/γB = 2.5, H/B = 4 and D/B = 1). In general, Nc decreases as β increases for all values of L/B. The rate of decrease (gradient) becomes smaller as L/B increases, and the relationship between Nc and β becomes nonlinear. The results have also shown that an increase of kh results in a decrease of Nc and an increase of L/B leads to an increase in Nc. Using the same data in Fig. 7, the figure presented in Fig. 8 shows the effect of kh on Nc. As expected, the increase of kh is to decrease the bearing capacity factor Nc. The relationship is a linear reduction, and the gradients of the lines are almost the same for all values of β and L/B.

Fig. 7
figure 7

Variation of Nc with β (cu/γB = 2.5, H/B = 4, and D/B = 1)

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Fig. 8
figure 8

Variation of Nc with kh (cu/γB = 2.5, H/B = 4, and D/B = 1)

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The final study is for the variations of Nc with the normalized footing depth ratio D/B. This is presented in Fig. 9 using the case of (cu/γB = 2.5, L/B = 1 and H/B = 4). As D/B increases, so as the Nc. The relationship is a nonlinear one. This trend is similar to a standard bearing capacity problem of a shallow foundation. In addition, the smaller the slope angle β, the larger the bearing capacity factor Nc. The larger the kh, the smaller the bearing capacity factor Nc.

Fig. 9
figure 9

Variation of Nc with D/B (cu/γB = 2.5, L/B = 1, and H/B = 4)

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All numerical results of the bearing capacity factor Nc corresponded to the investigated dimensionless input parameters are presented in Tables 1, 2 and 3. These data will be used for MARS study in a later section. In regard to the associated failure mechanisms, selected studies are presented in Figs. 10, 11 and 12 for the effects of cu/γB, D/B, and kh, respectively.

Table 1 Seismic bearing capacity Nc (H/B = 1)
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Table 2 Seismic bearing capacity Nc (H/B = 2)
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Table 3 Seismic bearing capacity Nc (H/B = 4)
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Fig. 10
figure 10

Comparison of failure mechanisms (β = 30°, L/B = 2, H/B = 4, D/B = 1, and kh = 0.1)

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Fig. 11
figure 11

Comparison of failure mechanisms for (β = 15°, cu/γB = 2.5, L/B = 0, H/B = 4, and kh = 0.1)

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Fig. 12
figure 12

Comparison of failure mechanisms for (β = 30°, cu/γB = 2.5, L/B = 0, H/B = 4, and D/B = 1)

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The upper bound shear dissipation contour plots are normally used to represent failure mechanisms of geo-stability problems. The actual values of the colored contour are not important for a perfectly plasticity constitutive model, and, therefore, the contour bars for these plots are not normally shown in a technical document. Figure 10 shows the effects of cu/γB on the associated failure mechanisms. The selected case is for (β = 30°, L/B = 2, H/B = 4, D/B = 1 and kh = 0.1). Note that, as the value of cu/γB increases, the overall area of the slip zone reduces, and the failure type transforms from a toe-failure mode to a face-failure mode. The reduction in the area of failure zone indicates an increase in seismic bearing capacity as the shear strength ratio cu/γB of the soil slope increases.

The effect of D/B on the associated failure mechanisms is presented in Fig. 11. The selected case for this study is for (β = 15°, cu/γB = 2.5, L/B = 0, H/B = 4 and kh = 0.1). The seismic bearing capacity increases as D/B increases, and there appears that the local failure mechanism is similar to a single sided Prandtl type of failures. For studying the effect of kh on the associated failure mechanisms, the case of (β = 30°, cu/γB = 2.5, L/B = 0, H/B = 4 and D/B = 1) is chosen. This is shown in Fig. 12, where a slight decrease in the area of failure zone is depicted as the value of kh increases. This makes sense, as the larger seismic forces would enable a search of the shortest path of the slip line to the slope surface, and, therefore, it leads to a decrease in the seismic bearing capacity, as well as a reduction in the area of the failure zone.

Sensitivity Study Using MARS Model

Multivariate adaptive regression splines (MARS) model is a nonlinear and non-parametric regression approach that can be used to capture nonlinear relationships between the input variables and the output results using a series of piecewise linear segments (splines) with differing gradients. The MARS technique does not require any specific assumptions to build functional correlations between the input variables and the output results. The different splines are connected using a knot representing by the end of one spline and the beginning of another. The fitted basic functions (BFs) have the flexibility to a studied model where the bends, thresholds, and other derivations from linear functions are allowed. The basic function can be generally written as in the following equation:

$${\text{BF}} = \max \;(0,x - t) = \left\{ {\begin{array}{*{20}l} {x - t} \hfill & {{\text{if}}\;x > t} \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} }, \right.$$
(2)

where x is an input variable and t is a threshold value.

MARS model produces BFs by searching in a stepwise manner, of which the knot locations will be automatically determined using the adaptive regression algorithm. MARS model is presented by a two-step procedure. The first (forward) step gives BFs and finds their potential knots to optimize the model performance and fitting accuracy. The second (backward) step uses pruning algorithm based on the generalized cross validation (GCV) value to delete the unimportant terms, leading to a final generation of an optimal model. The value of GVC can be determined by Eq. (3), where N indicates the number of basic functions, k indicates the penalty factor, RMSEi indicates the root mean square error for the training dataset, and R indicates the number of data points.

$${\text{GCV}} = \frac{{{\text{RMSE}}}}{{\left[ {1 - \left( {N - kN} \right)/R} \right]^{2} }}$$
(3)

To measure the important of each parameter on the output results, the value of relative important index (RII) would be determined by Eq. (4). This equation calculates the different in GCV values between the models before and after delete the unimportant terms [69, 70]

$$RII(i) = \frac{\Delta g(i)}{{\max \left\{ {\Delta g(i),\Delta g(2),\Delta g(3), \ldots ,\Delta g(n)} \right\}}},$$
(4)

where Δg is the increase in GCV when ith parameter is deleted.

To demonstrate the complex relationship between input variables and output results, MARS proposed the correlation equation by merging all linear basic functions (BFs), as shown in Eq. (5), where a0 is the constant, N is the number of BFs, gn is the nth BF, an is the coefficient of gn.

$$f(x) = a_{o} + \sum\limits_{n = 1}^{N} {a_{n} g_{n} } (X)$$
(5)

Compare to another machine learning approach (i.e., artificial neural networks (ANN), least-square support vector regression, extreme learning machine, Gaussian process regression [71,72,73,74,75,76,77]) MARS are considered as an effective approach [74, 78]. Moreover, MARS model has been successfully applied to a number of geotechnical applications (see, e.g., [79,80,81,82,83,84,85,86,87,88,89,90,91,92,93]). Further details of MARS model can be found in Zhang [94].

This aforementioned MARS model was utilized to perform sensitivity analyses of each input variables (i.e., L/B, β, cu/γB, D/B, H/B, and kh) and an empirical prediction of Nc value introduced by considering the coupling effects of input variables. All FELA numerical results presented in Tables 1, 2 and 3 are used as the artificial training data for MARS model. In that, the sets of dimensionless variables (i.e., L/B, β, cu/γB, D/B, H/B, and kh) and the corresponding Nc values are assigned as input data and target value in MARS model. Totally, 1296 data sets are used for MARS model.

In engineering practice, a careful design requires sensitivity analysis of each input variable [95, 96]. This is best presented through the relative importance index (RII) for the design output, i.e., the Nc values. As mentioned above, the value of RII shows the degree of influence, i.e., a RII of 100% indicates that the corresponding input variable has the most significant influence on the output Nc. Figure 13 shows the RII of each dimensionless parameter from MARS analysis. Numerical results have shown that the normalized embedded depth D/B has the greatest effect on seismic bearing capacity of footings placed on slope with a RII of 100%. This is followed by kh, H/B, β, L/B, and cu/γB with RII of 39.38%, 38.83%, 35.16%, 29.15%, and 23.24%, respectively. These results indicates that, for a general shallow foundation, the width and depth of the foundation is most important to determine bearing capacity. Although the present study considers the influence of other parameters, they cannot replace the most importance of width and depth of the shallow foundation. Moreover, this RII study has improved our understanding on the level of importance of each design parameter for the problem considered. The confidence level in practical design can, therefore, be enhanced greatly with these RII values.

Fig. 13
figure 13

Relative importance index of each input variable for the design Nc value

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Table 4 presents an empirical prediction equation provided by MARS model where the 30 BFs are listed. They can be written as in the following equation:

$$\begin{aligned} N_{c} & = { 6}.{984 } + { 1}.{3}0{1 } \times {\text{ BF1 }}{-}{ 2}.{357 } \times {\text{ BF2 }}{-}{ 4}.{13}0 \, \times {\text{ BF3 }}{-} \, 0.0{36 } \times {\text{ BF4 }}{-} \, 0.00{4 } \times {\text{ BF5 }} + \, 0.0{14} \\ & \quad \times {\text{ BF6 }} + \, 0.00{6 } \times {\text{ BF7 }}{-} \, 0.00{7 } \times {\text{ BF8 }}{-} \, 0.{179 } \times {\text{ BF9 }}{-}{ 2}.{3}00 \, \times {\text{ BF1}}0 \, + \, 0.0{16 } \times {\text{ BF11 }} + \, 0.{24}0 \\ & \quad \times {\text{ BF12 }}{-} \, 0.{383 } \times {\text{ BF13 }}{-} \, 0.{158 } \times {\text{ BF14 }}{-} \, 0.{375 } \times {\text{ BF15 }}{-} \, 0.{388 } \times {\text{ BF16 }} + \, 0.{278 } \times {\text{ BF17 }}{-} \, 0.{246 } \times {\text{ BF18 }}{-} \, 0.{7}0{1 } \times {\text{ BF19 }}{-} \, 0.{1}0{5 } \times {\text{ BF2}}0 \, {-} \, 0.{1}0{8 } \times {\text{ BF21 }}{-} \, 0.{17}0 \, \times {\text{ BF22 }} + \, 0.{347 } \times {\text{ BF23 }} + \, 0.0{37 } \times {\text{ BF24 }}{-} \, 0.0{81 } \times {\text{ BF25 }} - \, 0.0{73 } \times {\text{ BF26 }} + \, 0.0{3}0 \, \times {\text{ BF28 }}{-} \, 0.{332 } \times {\text{ BF29 }} + \, 0.{321 } \times {\text{ BF3}}0. \\ \end{aligned}$$
(6)
Table 4 Basis functions and mathematical equations in MARS model
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To validate the accuracy of the proposed MARS-based design equation, a comparison of Nc value between the Eq. (6) solutions and the actual FELA solutions (average of UB and LB solutions) are presented in Fig. 14. The comparison results have shown that both solutions are in a good agreement, where the coefficient of determination of R2 is 92.21%. The comparison shows that the proposed MARS-based can be well applied with reasonable accuracy in practices.

Fig. 14
figure 14

Comparison of the Nc value between the proposed MARS equation and FELA

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Conclusions

This study has examined the seismic bearing capacity performance of a strip footing resting on undrained cohesive slopes using the robust finite element limit analysis with upper and lower bound theorems. The following conclusions are drawn based on the study.

  1. 1.

    The extended parametric studies for the individual dimensionless parameters, i.e., (L/B, β, cu/γB, D/B, H/B, and kh) were performed. The bearing capacity factor Nc increases with a rise of (L/B, D/B, and cu/γB), while it decreases as the values of (β and kh) decrease. Comprehensive results were reported in both graphical and tabular forms for design practices.

  2. 2.

    Based on MARS model, the results of sensitivity analyses showed that the normalized depth of footing D/B has the most influential effect on the seismic bearing capacity factor Nc with important index (RII) of 100% while cu/γB is the least importance parameter with RII of 23.24%. Other investigated parameters are followed by kh, H/B, β, and L/B with RII of 39.38%, 38.83%, 35.16%, and 29.15%, respectively

  3. 3.

    An empirical equation with good accuracy (R2 = 92.21%) based on MARS model was proposed to determine the seismic bearing capacity factor Nc.

This study has paved the road for future geo-stability research to include sensitivity analysis of multi-variable problems with the useful relative importance index (RII) and design equation. This has many practical implications in the seismic design of soil structures in geotechnical engineering. However, it still has some limitations and should be investigated further in the future. For example, the proposed equation for the seismic bearing capacity factor Nc is appropriate for the ranges of dimensionless input parameters specified in the paper. The accuracy of the equation can be guaranteed if the input values are out of these ranges. Besides, the present solutions cannot be used for multi-layered soils. Further research work can be expanded to study the layered effects.