Research Article  Open Access
Seismic Bearing Capacity of Strip Footings on Cohesive Soil Slopes by Using Adaptive Finite Element Limit Analysis
Abstract
By employing adaptive finite element limit analysis (AFELA), the seismic bearing capacity of strip footing on cohesive soil slopes are investigated. To consider the earthquake effects, the pseudostatic method is used. The upper and lower bounds for the seismic bearing capacity factor (N_{ce}) are calculated, and the relative errors between them are found within 3% or better by adopting the adaptive mesh strategy. Based on the obtained results, design tables and charts are provided to facilitate engineers use, and the effects of footing position, undrained shear strength, slope angle, slope height, and pseudostatic acceleration coefficient are studied in detail. The collapse mechanisms are also discussed, including overall slope failure and foundation failure.
1. Introduction
The stability problem of strip footing on slopes often occurs in engineering practice, which has been investigated by many researchers. To determine the bearing capacity of strip footing on slopes, different methods are used, ranging from semiempirical methods [1, 2], limit equilibrium techniques [3–6], slipline solutions [7, 8], limit analysis [9–11], finite element methods [12, 13], finite element limit analysis [14], and discontinuity layout optimization (DLO) approaches [15–17]. However, the effect of seismicity is not considered for all the studies mentioned previously, which should be treated carefully in earthquake areas due to the devastating influence of the footing under seismic conditions.
Considering the pseudostatic seismic forces, a variety of studies have been performed to calculate the bearing capacity of strip footing on slopes. Theoretical solutions provide an efficient way to analyze the problem, such as limit equilibrium approaches [18–21], lower bound [22] and upper bound [23–27] solutions, and stress characteristic method [28]. Compared to the analytical methods mentioned above, numerical modeling does not require assumptions of failure mechanisms to be made, which may provide good accuracy results and consider a wider range of parameters [29]. By using finite element limit analysis (FELA), Shiau et al. [30] and Raj et al. [31] examined the seismic bearing capacity of footings placed on slopes, and the upper and lower bounds have been presented. Kumar and Chakraborty [32] used the lower bound FELA to compute the bearing capacity factor N_{γ} for a rough strip footing on cohesionless slopes under earthquake conditions. Later, using the same methods, the seismic bearing capacity of strip footings on a sloping ground surface and embankments has been investigated by Chakraborty and Kumar [33] and Chakraborty and Mahesh [34]; respectively. More recently, Zhou et al. [35] determined the ultimate seismic bearing capacity and the collapse mechanisms for strip footings located adjacent to cohesivefrictional soil slopes by using the DLO method. Under undrained conditions, Cinicioglu and Erkli [36] applied finite element software PLAXIS to investigate the seismic bearing capacity of strip footings resting on or near slopes.
In this work, an AFELA program developed by the authors is employed to examine the seismic bearing capacity of strip footings on cohesive soil slopes. The pseudostatic approach is applied to consider the earthquake loadings, which has been incorporated into the current AFELA program. Based on the AFELA program, the upper bound (UB) and lower bound (LB) results obtained by the present solution are first compared to those conducted by the previous studies. Then, the effects of seismic acceleration coefficient, soil properties, and geometrical parameters on the seismic bearing capacity factor are explored. To facilitate engineers use, design tables and charts are provided and the failure modes are also discussed.
2. Statement of the Problem
Under plane strain conditions, Figure 1 illustrates a general layout of the problem analyzed. A weightless, rough strip footing of width B is constructed near a slope with a slope angle β and a slope height H. The normalized footing distance to the crest is given as λ (=footing distance/footing width). The soil with unit weight γ is assumed as a homogeneous media with undrained shear strength c_{u}, following the Tresca yield criterion with an associated flow rule. Based on the pseudostatic approach, a same horizontal seismic acceleration coefficient k_{h} is applied to the footing and the slope [17, 36]. The vertical seismic acceleration coefficient is ignored owing to its insignificant influence on the seismic bearing capacity [36, 37].
The seismic bearing capacity of strip footings on cohesive slopes can be represented by the dimensionless seismic bearing capacity factor N_{ce}, which is defined aswhere q_{u} is the ultimate seismic bearing capacity.
In this paper, the magnitude of c_{u}/(γB) ranges from 0.625 to 7.5 and H/B ranges from 1 to 4 [12, 36]. The value of λ varies from 0 to 4 corresponding to five different values of β, namely, 15°, 30°, 45°, 60°, and 75°. Four different values of the horizontal seismic acceleration coefficient k_{h}, namely, 0.1, 0.2, 0.3, and 0.35, are considered. The problem parameters and adopted values are summarized in Table 1.

3. Adaptive Finite Element Limit Analysis
In the last two decades, the FELA methods are routinely used to analyze a variety of geotechnical engineering stability problems, which combine the limit theorems of classical plasticity with finite elements to give the UB and the LB on the collapse load [38]. These techniques do not require the loadsettlement curve and assumptions about the collapse mode to determine the limit load, which have been widely used in stability analysis of strip footings [39–41], tunnels [42–46], slopes [14, 47–49], and anchors [50, 51].
There are two distinct solutions produced: (1) the UB solution and (2) the LB solution. The proper FE techniques can be used to construct kinematically admissible velocity fields for UB solution and statically admissible stress fields for LB solution. In this paper, the discontinuous FE formulations proposed by Krabbenhoft et al. [52] and Lyamin et al. [53] are introduced for the discrete UB and LB problems, respectively. This will produce two nonlinear optimization models, which can be expressed as a same form [54, 55]:where α is a load multiplier; b^{T} is the discrete equilibriumtype operator; σ is the vector of discrete stresses σ_{i}; n is the total number of discrete stresses; p and p_{0} are the vectors of unknown and prescribed nodal forces, respectively; and F is the yield function.
To solve the discrete models presented in equation (2), the authors develop a general nonlinear optimization algorithm, which is a modified version of the feasible arc interior point algorithm (FAIPA) proposed by Herskovits et al. [56]. A distinct feature of the FAIPA algorithm used in this paper has been introduced by Zhang et al. in [57]; which is that a feasible arc is constructed for each iteration of the algorithm to avoid the violation of inequality constraints in search of a step length t, which is shown in Figure 2. From Figure 2, it can be observed that a secondorder feasible arc is constructed at the current iterate point x with a feasible direction d and a restoring direction , which takes the following form:
Since it has a curvature close to the active inequality constraint f, a large allowable step length t can be achieved, providing the “linesearching” proceeds along this feasible arc.
Moreover, the adaptive remeshing procedure proposed by Zhang et al. [58] is incorporated into the aforementioned FELA program to reduce the calculation errors, which is applied in both the UB and LB solutions. For simplicity, details of the adaptive remeshing procedure are not given here but can be found in Zhang et al. [58]. Because the adaptive mesh strategy is used, for all our calculations, the relative errors (REs) between the UB and LB measured by equation (4) are found to be within 3% or better:
Figure 3 shows a typical adaptive finite element mesh employed in the upper and lower bound for the case with c_{u}/(γB) = 5, λ = 1, H/B = 2, β = 45°, and k_{h} = 0.1. This figure also illustrates the stress boundary conditions for the LB analyses (the normal stress σ_{n} and the shear stress τ are labelled) and the velocity boundary conditions for the UB analyses (the horizontal velocity u and the vertical velocity are prescribed). This study adopts three iterations of adaptive mesh refinements for all numerical simulations, and similar meshes are applied for the UB and LB analyses. A typical final adaptive mesh has 9365 triangular elements and 14145 stressvelocity discontinuities, as shown in Figure 3. The size of the calculation domain is chosen to be so large that all the plasticzones at failure develop within the domain.
4. Comparisons with Previous Studies
4.1. Comparison of Static Bearing Capacity Factor N_{c}
The static bearing capacity factor N_{c} obtained from the present study for rough strip footings on cohesive slopes is compared with the solutions of (1) Vesic [2] by using the semiempirical solution, (2) Kusakabe et al. [11] on the basis of the upper bound method, (3) Georgiadis [12] by using finite element method (FEM), and (4) Leshchinsky [15] on the basis of the DLO approach. A comparison of these results is given in Figure 4 for the case with k_{h} = 0, λ = 0, and c_{u}/(γB) = 1. As seen, the AFELA results presented in this study and the results of Kusakabe et al. [11], Georgiadis [12], and Leshchinsky [15] are in excellent agreement, while Vesic’s solution provides smaller values of N_{c}.
For the case of a footing placed at a distance from the slope, a further comparison is made in Figure 5. All results presented in Figure 5 are for k_{h} = 0, c_{u}/(γB) = 5, and β = 30°. The values of N_{c} obtained by Georgiadis [12] based on the FEM and Leshchinsky and Xie [16] by using the DLO approach lie between the present upper and lower bound solutions. The present results agree reasonably well with the UB solution of Kusakabe et al. [11]. The limit equilibrium method reported by Meyerhof [5] and Castelli and Motta [4] provides a greater value of N_{c} since assumed failure mechanisms are used.
4.2. Comparison of Seismic Bearing Capacity Factor N_{ce}
The present results for seismic bearing capacity of strip footings placed adjacent to cohesive slopes are compared with the FEM conducted by Cinicioglu and Erkli [36]. Figure 6 shows the variations of the seismic bearing capacity factor N_{ce} with the horizontal seismic acceleration coefficient k_{h} for the case of λ = 0 and c_{u}/(γB) = 5. It can be observed that the present results agree reasonably well with the predictions of Cinicioglu and Erkli [36].
A final comparison of the results of this study is made for the case of β = 30°, k_{h} = 0.2, and c_{u}/(γB) = 5, as shown in Figure 7. As seen, the present values for N_{ce} lie in between the results obtained by the other researchers. The limit equilibrium method of Castelli and Motta [4] and the FEM of Cinicioglu and Erkli [36] are found to be greater than the present upper bound solution, and the values of N_{ce} obtained from the lower bound limit analysis of Farzaneh et al. [22] become slightly smaller than the present lower bound solution.
5. Results and Discussion
The upper and lower bound results for the seismic bearing capacity factor N_{ce} have been computed. Since the adaptive mesh strategy is employed for all the numerical simulations, in general, the relative errors are found to within 3%, showing a good accuracy for the present results. Therefore, average value of the upper and lower bound N_{ce} will be employed in the following discussions. The results calculated by AFELA for all the problem parameters considered are summarized in Tables 1–3 and depicted graphically in Figures 8–11. Note that the cases with an infeasible solution are indicated by “—” in Tables 1–3, indicating that the slope failure due to gravity occurs. In the following sections, the influences of the normalized footing distance λ, the dimensionless strength ratio c_{u}/(γB), the slope angle β, the slope height H/B, and the horizontal seismic acceleration coefficient k_{h} on N_{ce} will be discussed in detail.


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Figure 8 shows the variation of the seismic bearing capacity factor N_{ce} with λ for different combinations of k_{h} and β, where the values of N_{ce} are for the cases with c_{u}/(γB) = 1 and H/B = 1. As seen in Figure 8, it can be found that, for all magnitudes of k_{h} and β, the value of N_{ce} increases with the increase of λ up to a certain critical value of λ_{cr} beyond which the normalized footing distance does not influence the seismic bearing capacity. For example, considering the cases of β = 45°, the magnitude of N_{ce} reaches the constant values of 4.45 at λ = 3 for k_{h} = 0.1, 3.67 at λ = 2 for k_{h} = 0.2, 2.98 at λ = 1.5 for k_{h} = 0.3, and 2.69 at λ = 1.5 for k_{h} = 0.35, respectively. Interestingly, the value of λ_{cr} is generally smaller for greater value of k_{h}. The reason for this may be that the areas of the failure zone decrease with an increase in the value of k_{h}. It should be noted that the rate of increase of N_{ce} with λ is found to be more extensive for smaller values of k_{h}, when λ ≤ λ_{cr}.
The obtained values of the seismic bearing capacity factor N_{ce} with different combinations of c_{u}/(γB) and k_{h} are shown in Figure 9 for the cases of λ = 0, H/B = 1, and β = 15°, 45°, 60°, and 75°. The results indicate that for all the cases, as the value of c_{u}/(γB) increases, the seismic bearing capacity factor increases continuously and the rate of increase in N_{ce} decreases. It should be mentioned that the value of N_{ce} is affected slightly by the dimensionless strength ratio, when c_{u}/(γB) ≥ 5. Considering the cases with k_{h} = 0.2, for example, by increasing the magnitude of c_{u}/(γB) from 5 to 7.5 leads to increase in the seismic bearing capacity N_{ce} by less than 0.3% for β = 15° and 1% for β = 75°. On the other hand, the effect of c_{u}/(γB) on N_{ce} is found to be more prominent at smaller values of k_{h} and higher values of β.
The variations of the seismic bearing capacity factor N_{ce} with β for different combinations of k_{h} and c_{u}/(γB) are illustrated in Figures 10 and 11 for λ = 0 and 1, respectively. For the cases of strip footings at the crest of slopes (λ = 0), it can be seen that N_{ce} decreases linearly with an increase in the value of β, and this trend is predominant at the smaller magnitudes of k_{h}. On the contrary, for the case of λ = 1 shown in Figure 11, the existence of a certain critical value of β_{cr} can be seen beyond which the value of N_{ce} has a linear trend. The reason for this can be attributed to the fact that the value of λ_{cr} is smaller than 1, when β ≤ β_{cr}. It is noteworthy that the magnitude of β_{cr} is generally smaller for lower value of k_{h}.
Figure 12 shows the variation of the seismic bearing capacity N_{ce} with k_{h} for different combinations of β, c_{u}/(γB), and H/B, where the value of N_{ce} are for the case of λ = 0. It can be seen that for all the cases, the value of N_{ce} reduces linearly with increasing k_{h}. This trend is similar to the previous studies conducted by Kumar and Rao [28] Farzaneh et al., [22] and Cinicioglu and Erkli [36]. On the other hand, the effect of H/B on N_{ce} is also presented in Figure 12. When β ≥ 45° (Figure 12(b)), it can be found that the value of H/B does not influence the seismic bearing capacity factor. This can be attributed that the failure modes are not affected by H/B. However, from Tables 1–3, the value of H/B is found to affect the overall slope stability, especially for the case with higher value of k_{h} and lower value of c_{u}/(γB).
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6. Failure Mechanisms
Figure 13 illustrates the plots of power dissipations showing the influence of the normalized footing distance λ for the cases of c_{u}/(γB) = 1 and 7.5. As shown in Figure 13(a) for c_{u}/(γB) = 1 and λ = 0, a nonplastic triangular wedge below the footing base can be seen, and a slip line extends to the free surface. When the value of λ increases to 1 (Figure 13(b)), the slip line tends to extend the toe of the slope. For the case of c_{u}/(γB) = 1 and λ = 4 (Figure 13(c)), the failure mechanism is found to be not affected by the slope, which is consistent with the phenomenon illustrated in Figure 8. On the other hand, as shown in Figures 13(d), 13(e), and 13(f), the cases with higher value of c_{u}/(γB), the depth of failure zone is smaller than the cases with lower value of c_{u}/(γB). The only exception to this observation occurs for the cases with the greater value of λ = 4 (Figures 13(c) and 13(f)), where the failure modes are not unchanged. This indicates that the value of c_{u}/(γB) does not affect the seismic bearing capacity, when the normalized footing distance is larger enough.
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Several plots of power dissipations showing the effect of β and H/B are shown in Figure 14 for the cases with c_{u}/(γB) = 1, λ = 1, and k_{h} = 0.1. For the cases with small slope angles (Figures 14(a), 14(d), and 14(g)), it can be seen that the collapse mechanisms remain unchanged and the failure surface only extends to the free surface. As seen in Figures 14(b), 14(e), and 14(h), for the cases of moderate slope angles, similar failure modes, showing that the failure surface extends to the toe of the slopes, can be observed. However, for the cases with larger values of β, the failure mechanisms are significantly affected by H/B. As illustrated in Figure 14(c) for β = 75° and H/B = 1, the collapse mechanism is similar to the case of β = 45° and H/B = 1, but the depth of the failure surface is smaller. For the case of β = 75° and H/B = 2 (Figure 14(f)), the rigid triangular wedge disappeared, and only a slip line can be seen, which extends from the right of the footing base to the toe of the slope. For the case of β = 75° and H/B = 2, as shown in Figure 14(i), a larger slip surface noticeably toward the ground surface can be observed, indicating that the overall slope failure due to the gravity occurs (Table 4).
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For comparison, Figure 15 shows the plots of power dissipations for the cases with the greater value of c_{u}/(γB) = 5. The failure patterns remain unchanged and become independent of H/B, which are consistent with those of Figure 12. To show the effect of the horizontal seismic acceleration coefficient k_{h}, the upper bound failure mechanisms are presented in Figure 16 for the cases of c_{u}/(γB) = 2.5, λ = 1 and H/B = 2. It clearly can be seen that the depth of the failure zone decreases with an increase in the value of k_{h}. This can be used to explain the phenomenon shown in Figure 8 that the value of λ_{cr} is generally smaller for greater value of k_{h}.
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As summarized, the failure mechanisms may be classified into four types, which are shown in Table 5.

7. Conclusions
An adaptive finite element limit analysis (AFELA) program with incorporation of the pseudostatic approach has been used to investigate the seismic bearing capacity of strip footings on cohesive soil slopes. Based on the AFELA program, the upper and lower bounds for the seismic bearing capacity factor N_{ce} have been computed, showing a good accuracy for the results. To facilitate engineer use, the present results are presented in design tables and charts, and the influence of parameters, including λ, c_{u}/(γB), β, k_{h}, and H/B, on N_{ce} has also been examined. The following conclusions can be made.(1)In general, the value of N_{ce} increases with the increase of λ up to a certain critical value of λ_{cr} beyond which the normalized footing distance does not influence the seismic bearing capacity, and the value of λ_{cr} is smaller for greater value of k_{h}.(2)As the value of c_{u}/(γB) increases, the value of N_{ce} increases continuously and the rate of increase in N_{ce} decreases. When c_{u}/(γB) ≥ 5, the value of N_{ce} is affected slightly by c_{u}/(γB). And the effect of c_{u}/(γB) on N_{ce} is found to be more prominent at smaller values of k_{h} and higher values of β.(3)For the cases of strip footings at the crest of slopes (λ = 0), the magnitude of N_{ce} decreases linearly with an increase in the value of β, and this trend is predominant at the smaller magnitudes of k_{h}. For the case of a footing at a distance from the slope (λ ≥ 0), the existence of a certain critical value of β_{cr} can be seen beyond which the value of N_{ce} has a linear trend.(4)The value of N_{ce} reduces linearly with increasing k_{h}. The value of H/B does not influence N_{ce} for the case with higher value of k_{h} and lower value of c_{u}/(γB) but affects the overall slope stability.(5)Two types of failure mechanisms are discussed, including overall slope failure due to gravity and foundation failure. The depth of the failure zone decreases with an increase in the value of k_{h}.
Note that the soil properties are complex for the natural slopes; this study is limited to slopes for homogeneous soil properties. On the other hand, the effect of footing roughness has not been considered in this study, which will be investigated in the future work.
Notation
b^{T}:  Discrete equilibriumtype operator 
B:  Width of the strip footing 
c_{u}:  Undrained shear stress 
F:  Yield function 
H:  Slope height 
k_{h}:  Horizontal acceleration coefficient 
N_{ce}:  Seismic bearing capacity factor 
n:  Total number of discrete stresses 
p:  Vectors of unknown 
p_{0}:  Prescribed nodal forces 
u:  Horizontal velocity 
:  Vertical velocity 
q_{u}:  Ultimate seismic bearing capacity 
α:  Load multiplier 
β:  Slope angle 
γ:  Soil unit weight 
σ:  Vector of discrete stresses 
σ_{i}:  Discrete stresses 
σ_{n}:  Normal stresses 
τ:  Shear stresses 
λ:  Normalized footing distance to the crest. 
Data Availability
The data in Tables 1–3 used to support the findings of this study are included within the article. In addition, the data used in Figures 3–6 were from Refs. [2, 4, 5, 11, 12, 15, 16, 22, 36],and they are cited at relevant places within the text. The remaining data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to acknowledge the financial support of the China Postdoctoral Science Foundation (140050006) and the National Natural Science Foundation of China (Nos. 51478178 and 51608540), which made the work presented in this paper possible.
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Copyright © 2019 Weihua Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.