Advances in Civil Engineering
Volume 2019 (2019), Article ID 4548202, 16 pages
https://doi.org/10.1155/2019/4548202

Research Article

Seismic Bearing Capacity of Strip Footings on Cohesive Soil Slopes by Using Adaptive Finite Element Limit Analysis

Weihua Luo,1,2 Minghua Zhao,³ Yao Xiao,⁴ Rui Zhang,⁵ and Wenzhe Peng⁴

¹Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China
²Construction Development Corporation in Hunan Province, Changsha 410000, China
³Professor, Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China
⁴Ph.D., Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China
⁵Assistant Professor, Ph.D., Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China

Correspondence should be addressed to Wenzhe Peng; wzpeng@hnu.edu.cn

Received 17 April 2019; Revised 29 June 2019; Accepted 28 July 2019; Published 28 August 2019

Academic Editor: Sanjay Nimbalkar

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.

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 (Nce) 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], slip-line 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 Nγ 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 cohesive-frictional 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 cu, following the Tresca yield criterion with an associated flow rule. Based on the pseudostatic approach, a same horizontal seismic acceleration coefficient kh 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].

Figure 1: Schematic diagram of the model.

The seismic bearing capacity of strip footings on cohesive slopes can be represented by the dimensionless seismic bearing capacity factor Nce, which is defined as

where qu is the ultimate seismic bearing capacity.

In this paper, the magnitude of cu/(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 kh, namely, 0.1, 0.2, 0.3, and 0.35, are considered. The problem parameters and adopted values are summarized in Table 1.

Table 1: The problems parameters and adopted values.

Parameters
Descriptions
Values considered

cu/(γB)
Dimensionless strength ratio
0.625–7.5

H/B
Normalized slope height
1–4

Λ
Normalized footing distance
0–4

Β
Slope angle
15°–75°

kh
Horizontal seismic acceleration coefficient
0.1–0.35

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 load-settlement 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; bT is the discrete equilibrium-type operator; σ is the vector of discrete stresses σi; n is the total number of discrete stresses; p and p₀ 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 second-order feasible arc is constructed at the current iterate point x with a feasible direction d and a restoring direction , which takes the following form:

Figure 2: Illustration of the feasible arc (from Zhang et al. [57]).

Since it has a curvature close to the active inequality constraint f, a large allowable step length t can be achieved, providing the “line-searching” 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 cu/(γB) = 5, λ = 1, H/B = 2, β = 45°, and kh = 0.1. This figure also illustrates the stress boundary conditions for the LB analyses (the normal stress σn 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 stress-velocity discontinuities, as shown in Figure 3. The size of the calculation domain is chosen to be so large that all the plastic-zones at failure develop within the domain.

Figure 3: Finite element mesh showing boundary conditions for numerical limit analysis (cu/(γB) = 5, λ = 1, H/B = 2, β = 45°, and kh = 0.1).

4. Comparisons with Previous Studies

4.1. Comparison of Static Bearing Capacity Factor Nc

The static bearing capacity factor Nc 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 kh = 0, λ = 0, and cu/(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 Nc.

Figure 4: Comparison of variation in Nc with slope angle β for λ = 0 and cu/(γB) = 1.

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 kh = 0, cu/(γB) = 5, and β = 30°. The values of Nc 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 Nc since assumed failure mechanisms are used.

Figure 5: Comparison of variation in Nc with λ for β = 30°.

4.2. Comparison of Seismic Bearing Capacity Factor Nce

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 Nce with the horizontal seismic acceleration coefficient kh for the case of λ = 0 and cu/(γB) = 5. It can be observed that the present results agree reasonably well with the predictions of Cinicioglu and Erkli [36].

Figure 6: Comparison of variation in Nce with kh for λ = 0 and cu/(γB) = 5.

A final comparison of the results of this study is made for the case of β = 30°, kh = 0.2, and cu/(γB) = 5, as shown in Figure 7. As seen, the present values for Nce 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 Nce obtained from the lower bound limit analysis of Farzaneh et al. [22] become slightly smaller than the present lower bound solution.

Figure 7: Comparison of variation in Nce with λ for β = 30°, kh = 0.2, and cu/(γB) = 5.

5. Results and Discussion

The upper and lower bound results for the seismic bearing capacity factor Nce 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 Nce 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 cu/(γB), the slope angle β, the slope height H/B, and the horizontal seismic acceleration coefficient kh on Nce will be discussed in detail.

Table 2: Seismic bearing capacity factor Nce for strip footings on cohesive soil slopes with H/B = 1.

β (°)
cu/(γB)
kh
λ
kh
λ

0
0.5
1
1.5
2
3
4
0
0.5
1
1.5
2
3
4

15
0.625
0.1
3.81
4.13
4.37
4.41
4.41
4.41
4.41
0.3
2.74
2.97
2.98
2.98
2.98
2.98
2.98

1
3.92
4.22
4.43
4.45
4.45
4.45
4.45
2.77
2.99
2.99
2.99
2.99
2.99
2.99

1.5
3.97
4.26
4.46
4.47
4.47
4.47
4.47
2.79
3.01
3.01
3.01
3.01
3.01
3.01

2.5
4.01
4.29
4.48
4.48
4.48
4.48
4.48
2.80
3.01
3.01
3.01
3.01
3.01
3.01

5
4.04
4.31
4.49
4.49
4.49
4.49
4.49
2.81
3.02
3.02
3.02
3.02
3.02
3.02

7.5
4.05
4.32
4.50
4.50
4.50
4.50
4.50
2.81
3.02
3.02
3.02
3.02
3.02
3.02

0.625
0.2
3.20
3.49
3.62
3.62
3.62
3.62
3.62
0.35
—
—
—
—
—
—
—

1
3.30
3.58
3.66
3.67
3.67
3.67
3.67
2.49
2.69
2.69
2.69
2.69
2.69
2.69

1.5
3.34
3.62
3.69
3.69
3.69
3.69
3.69
2.52
2.70
2.70
2.70
2.70
2.70
2.70

2.5
3.38
3.65
3.71
3.71
3.71
3.71
3.71
2.54
2.71
2.71
2.71
2.71
2.71
2.71

5
3.40
3.68
3.72
3.72
3.72
3.72
3.72
2.55
2.72
2.72
2.72
2.72
2.72
2.72

7.5
3.41
3.68
3.72
3.72
3.72
3.72
3.72
2.56
2.72
2.72
2.72
2.72
2.72
2.72

30
0.625
0.1
3.12
3.46
3.76
4.04
4.28
4.41
4.41
0.3
—
—
—
—
—
—
—

1
3.37
3.82
4.17
4.43
4.45
4.45
4.45
2.45
2.84
2.98
2.98
2.98
2.98
2.98

1.5
3.46
3.93
4.28
4.47
4.47
4.47
4.47
2.50
2.92
3.00
3.00
3.00
3.00
3.00

2.5
3.54
4.00
4.34
4.48
4.49
4.49
4.49
2.54
2.96
3.01
3.01
3.01
3.01
3.01

5
3.58
4.04
4.38
4.50
4.50
4.50
4.50
2.56
2.99
3.02
3.02
3.02
3.02
3.02

7.5
3.60
4.06
4.39
4.50
4.50
4.50
4.50
2.57
3.00
3.02
3.02
3.02
3.02
3.02

0.625
0.2
2.67
2.92
3.14
3.33
3.50
3.62
3.62
0.35
—
—
—
—
—
—
—

1
2.89
3.32
3.62
3.66
3.66
3.66
3.66
2.25
2.63
2.69
2.69
2.69
2.69
2.69

1.5
2.96
3.41
3.69
3.69
3.69
3.69
3.69
2.30
2.68
2.70
2.70
2.70
2.70
2.70

2.5
3.02
3.47
3.71
3.71
3.71
3.71
3.71
2.33
2.71
2.71
2.71
2.71
2.71
2.71

5
3.05
3.51
3.72
3.72
3.72
3.72
3.72
2.35
2.72
2.72
2.72
2.72
2.72
2.72

7.5
3.07
3.52
3.72
3.72
3.72
3.72
3.72
2.36
2.72
2.72
2.72
2.72
2.72
2.72

45
0.625
0.1
2.46
2.89
3.28
3.64
3.95
4.41
4.41
0.3
—
—
—
—
—
—
—

1
2.81
3.30
3.71
4.05
4.34
4.45
4.45
2.12
2.50
2.78
2.98
2.98
2.98
2.98

1.5
2.96
3.50
3.91
4.24
4.47
4.47
4.47
2.20
2.69
2.99
3.00
3.00
3.00
3.00

2.5
3.06
3.63
4.04
4.38
4.49
4.49
4.49
2.26
2.76
3.01
3.01
3.01
3.01
3.01

5
3.12
3.71
4.13
4.45
4.50
4.50
4.50
2.30
2.84
3.02
3.02
3.02
3.02
3.02

7.5
3.14
3.74
4.15
4.48
4.50
4.50
4.50
2.31
2.85
3.02
3.02
3.02
3.02
3.02

0.625
0.2
2.13
2.43
2.73
2.99
3.23
3.60
3.60
0.35
—
—
—
—
—
—
—

1
2.44
2.88
3.22
3.51
3.67
3.67
3.67
1.97
2.33
2.58
2.69
2.69
2.69
2.69

1.5
2.56
3.08
3.44
3.69
3.69
3.69
3.69
2.04
2.51
2.70
2.70
2.70
2.70
2.70

2.5
2.64
3.20
3.58
3.71
3.71
3.71
3.71
2.09
2.59
2.71
2.71
2.71
2.71
2.71

5
2.69
3.27
3.66
3.72
3.72
3.72
3.72
2.12
2.63
2.72
2.72
2.72
2.72
2.72

7.5
2.71
3.29
3.68
3.72
3.72
3.72
3.72
2.14
2.65
2.72
2.72
2.72
2.72
2.72

60
0.625
0.1
1.95
2.44
2.92
3.34
3.70
4.28
4.41
0.3
—
—
—
—
—
—
—

1
2.29
2.85
3.34
3.75
4.10
4.45
4.45
1.75
2.15
2.48
2.77
2.98
2.98
2.98

1.5
2.45
3.07
3.55
3.96
4.28
4.47
4.47
1.87
2.37
2.73
2.99
3.00
3.00
3.00

2.5
2.57
3.22
3.72
4.10
4.42
4.48
4.49
1.95
2.52
2.90
3.01
3.01
3.01
3.01

5
2.66
3.34
3.82
4.21
4.49
4.50
4.50
2.00
2.61
3.00
3.02
3.02
3.02
3.02

7.5
2.68
3.37
3.85
4.24
4.50
4.50
4.50
2.02
2.64
3.01
3.02
3.02
3.02
3.02

0.625
0.2
1.67
2.04
2.41
2.74
3.03
3.47
3.52
0.35
—
—
—
—
—
—
—

1
2.00
2.48
2.89
3.24
3.53
3.66
3.66
1.63
2.01
2.30
2.56
2.69
2.69
2.69

1.5
2.15
2.70
3.13
3.48
3.69
3.69
3.69
1.75
2.22
2.55
2.70
2.70
2.70
2.70

2.5
2.25
2.86
3.30
3.64
3.71
3.71
3.71
1.82
2.37
2.70
2.71
2.71
2.71
2.71

5
2.32
2.97
3.41
3.72
3.72
3.72
3.72
1.86
2.45
2.72
2.72
2.72
2.72
2.72

7.5
2.34
3.00
3.45
3.72
3.72
3.72
3.72
1.88
2.47
2.72
2.72
2.72
2.72
2.72

75
0.625
0.1
1.48
2.02
2.58
3.07
3.50
4.13
4.41
0.3
—
—
—
—
—
—
—

1
1.80
2.44
3.02
3.50
3.89
4.45
4.45
1.39
1.82
2.22
2.56
2.84
2.98
2.98

1.5
1.98
2.67
3.25
3.71
4.09
4.47
4.47
1.53
2.05
2.47
2.83
3.00
3.00
3.00

2.5
2.10
2.83
3.41
3.87
3.23
4.49
4.49
1.63
2.21
2.66
2.99
3.01
3.01
3.01

5
2.19
2.95
3.54
3.98
4.33
4.50
4.50
1.69
2.33
2.78
3.02
3.02
3.02
3.02

7.5
2.22
2.99
3.57
4.01
4.36
4.50
4.50
1.71
2.37
2.82
3.02
3.02
3.02
3.02

0.625
0.2
1.27
1.68
2.11
2.50
2.83
3.34
3.36
0.35
—
—
—
—
—
—
—

1
1.58
2.11
2.60
3.00
3.34
3.66
3.66
1.30
1.70
2.06
2.36
2.61
2.69
2.69

1.5
1.74
2.34
2.84
3.25
3.58
3.69
3.69
1.44
1.92
2.31
2.63
2.70
2.70
2.70

2.5
1.85
2.51
3.02
3.43
3.71
3.71
3.71
1.53
2.08
2.49
2.71
2.71
2.71
2.71

5
1.93
2.63
3.15
3.55
3.72
3.72
3.72
1.58
2.20
2.61
2.72
2.72
2.72
2.72

7.5
1.95
2.67
3.19
3.58
3.72
3.72
3.72
1.60
2.24
2.65
2.72
2.72
2.72
2.72

Table 3: Seismic bearing capacity factor Nce for strip footings on cohesive soil slopes with H/B = 2.

β (°)
cu/(γB)
kh
λ
kh
λ

0
0.5
1
1.5
2
3
4
0
0.5
1
1.5
2
3
4

15
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
3.82
4.22
4.43
4.45
4.45
4.45
4.45
—
—
—
—
—
—
—

1.5
3.88
4.26
4.46
4.47
4.47
4.47
4.47
2.72
2.99
2.99
2.99
2.99
2.99
2.99

2.5
3.93
4.29
4.48
4.48
4.48
4.48
4.48
2.75
3.01
3.01
3.01
3.01
3.01
3.01

5
3.96
4.27
4.49
4.50
4.50
4.50
4.50
2.77
3.02
3.02
3.02
3.02
3.02
3.02

7.5
3.97
4.32
4.50
4.50
4.50
4.50
4.50
2.77
3.02
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
3.22
3.58
3.66
3.66
3.66
3.66
3.66
—
—
—
—
—
—
—

1.5
3.27
3.63
3.69
3.69
3.69
3.69
3.69
2.48
2.70
2.70
2.70
2.70
2.70
2.70

2.5
3.31
3.65
3.70
3.70
3.70
3.70
3.70
2.50
2.71
2.71
2.71
2.71
2.71
2.71

5
3.34
3.67
3.72
3.72
3.72
3.72
3.72
2.52
2.72
2.72
2.72
2.72
2.72
2.72

7.5
3.34
3.68
3.72
3.72
3.72
3.72
3.72
2.52
2.72
2.72
2.72
2.72
2.72
2.72

30
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
3.37
3.82
4.16
4.36
4.45
4.45
4.45
2.44
2.85
2.98
2.98
2.98
2.98
2.98

1.5
3.47
3.93
4.28
4.47
4.47
4.47
4.47
2.50
2.92
2.99
2.99
2.99
2.99
2.99

2.5
3.54
4.00
4.34
4.49
4.50
4.50
4.50
2.54
2.96
3.01
3.01
3.01
3.01
3.01

5
3.58
4.04
4.38
4.50
4.50
4.50
4.50
2.56
2.99
3.02
3.02
3.02
3.02
3.02

7.5
3.59
4.05
4.38
4.50
4.50
4.50
4.50
2.57
3.00
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
2.89
3.32
3.59
3.66
3.66
3.66
3.66
—
—
—
—
—
—
—

1.5
2.96
3.41
3.69
3.69
3.69
3.69
3.69
2.12
2.44
2.44
2.44
2.44
2.44
2.44

2.5
3.02
3.47
3.71
3.71
3.71
3.71
3.71
2.15
2.44
2.44
2.44
2.44
2.44
2.44

5
3.05
3.51
3.72
3.72
3.72
3.72
3.72
2.17
2.44
2.44
2.44
2.44
2.44
2.44

7.5
3.07
3.52
3.72
3.72
3.72
3.72
3.72
2.17
2.45
2.45
2.45
2.45
2.45
2.45

45
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
2.81
3.18
3.42
3.66
3.90
4.33
4.45
2.12
2.41
2.49
2.57
2.66
2.85
2.98

1.5
2.96
2.50
3.88
4.16
4.40
4.47
4.47
2.20
2.69
2.99
2.99
2.99
2.99
2.99

2.5
3.06
3.63
4.04
4.37
4.49
4.49
4.49
2.26
2.78
3.01
3.01
3.01
3.01
3.01

5
3.12
3.71
4.13
4.46
4.50
4.50
4.50
2.30
2.84
3.02
3.02
3.02
3.02
3.02

7.5
3.14
3.74
4.15
4.48
4.50
4.50
4.50
2.31
2.85
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
2.44
2.79
2.94
3.10
3.26
3.58
3.66
1.96
2.24
2.29
2.35
2.35
2.35
2.35

1.5
2.56
3.08
3.44
3.67
3.69
3.69
3.69
2.04
2.51
2.70
2.70
2.70
2.70
2.70

2.5
2.64
3.20
3.59
3.71
3.71
3.71
3.71
2.09
2.59
2.71
2.71
2.71
2.71
2.71

5
2.69
3.27
3.66
3.72
3.72
3.72
3.72
2.12
2.64
2.72
2.72
2.72
2.72
2.72

7.5
2.71
3.29
3.68
3.72
3.72
3.72
3.72
2.14
2.65
2.72
2.72
2.72
2.72
2.72

60
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
2.25
2.52
2.77
3.05
3.34
3.90
4.37
1.74
1.93
2.00
2.12
2.24
2.52
2.52

1.5
2.45
2.98
3.31
3.62
3.91
4.42
4.47
1.88
2.35
2.60
2.79
2.97
2.99
2.99

2.5
2.58
3.20
3.63
3.97
4.26
4.49
4.49
1.95
2.52
2.89
3.01
3.01
3.01
3.01

5
2.66
3.33
3.80
4.16
4.46
4.50
4.50
2.00
2.61
3.00
3.02
3.02
3.02
3.02

7.5
2.68
3.37
3.84
4.21
4.49
4.50
4.50
2.02
2.64
3.01
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
1.99
2.21
2.37
2.57
2.77
3.20
3.56
1.63
1.79
1.83
1.83
1.83
0.00
0.00

1.5
2.15
2.66
2.95
3.19
3.43
3.69
3.69
1.75
2.21
2.44
2.60
2.60
2.60
2.60

2.5
2.25
2.86
3.26
3.57
3.71
3.71
3.71
1.82
2.37
2.69
2.71
2.71
2.71
2.71

5
2.32
2.97
3.41
3.72
3.72
3.72
3.72
1.86
2.45
2.72
2.72
2.72
2.72
2.72

7.5
2.34
3.00
3.45
3.73
3.73
3.73
3.73
1.88
2.47
2.72
2.72
2.72
2.72
2.72

75
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
1.74
1.91
2.16
2.47
2.82
3.47
4.04
1.38
1.48
1.53
1.65
1.82
2.18
2.18

1.5
1.96
2.41
2.75
3.10
3.43
4.05
4.47
1.53
1.96
2.16
2.37
2.58
2.99
3.00

2.5
2.10
2.73
3.15
3.51
3.85
4.42
4.48
1.63
2.20
2.55
2.82
3.01
3.01
3.01

5
2.19
2.90
3.39
3.77
4.10
4.50
4.50
1.69
2.33
2.75
3.02
3.02
3.02
3.02

7.5
2.22
2.95
3.46
3.85
4.18
4.50
4.50
1.71
2.37
2.80
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
1.55
1.69
1.83
2.04
2.30
2.81
3.26
—
—
—
—
—
—
—

1.5
1.74
2.18
2.44
2.73
2.99
3.51
3.69
1.44
1.85
2.03
2.21
2.40
2.70
2.70

2.5
1.85
2.46
2.85
3.16
3.45
3.70
3.70
1.53
2.08
2.41
2.66
2.71
2.71
2.71

5
1.93
2.61
3.07
3.43
3.71
3.72
3.72
1.58
2.20
2.59
2.72
2.72
2.72
2.72

7.5
1.95
2.66
3.13
3.50
3.72
3.72
3.72
1.60
2.24
2.64
2.72
2.72
2.72
2.72

(a)

(b)

(c)

(d)

Figure 8: Variation of Nce with λ for cu/(γB) = 1 and H/B = 1: (a) β = 15°; (b) β = 45°; (c) β = 60°; (d) β = 75°.

(a)

(b)

(c)

(d)

Figure 9: Variation of Nce with cu/(γB) for λ = 0 and H/B = 1: (a) β = 15°; (b) β = 45°; (c) β = 60°; (d) β = 75°.

(a)

(b)

Figure 10: Variation of Nce with β for λ = 0 and H/B = 1: (a) cu/(γB) = 1; (b) cu/(γB) = 5.

(a)

(b)

Figure 11: Variation of Nce with β for λ = 1 and H/B = 1: (a) cu/(γB) = 1; (b) cu/(γB) = 5.

Figure 8 shows the variation of the seismic bearing capacity factor Nce with λ for different combinations of kh and β, where the values of Nce are for the cases with cu/(γB) = 1 and H/B = 1. As seen in Figure 8, it can be found that, for all magnitudes of kh and β, the value of Nce increases with the increase of λ up to a certain critical value of λcr beyond which the normalized footing distance does not influence the seismic bearing capacity. For example, considering the cases of β = 45°, the magnitude of Nce reaches the constant values of 4.45 at λ = 3 for kh = 0.1, 3.67 at λ = 2 for kh = 0.2, 2.98 at λ = 1.5 for kh = 0.3, and 2.69 at λ = 1.5 for kh = 0.35, respectively. Interestingly, the value of λcr is generally smaller for greater value of kh. The reason for this may be that the areas of the failure zone decrease with an increase in the value of kh. It should be noted that the rate of increase of Nce with λ is found to be more extensive for smaller values of kh, when λ ≤ λcr.

The obtained values of the seismic bearing capacity factor Nce with different combinations of cu/(γB) and kh 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 cu/(γB) increases, the seismic bearing capacity factor increases continuously and the rate of increase in Nce decreases. It should be mentioned that the value of Nce is affected slightly by the dimensionless strength ratio, when cu/(γB) ≥ 5. Considering the cases with kh = 0.2, for example, by increasing the magnitude of cu/(γB) from 5 to 7.5 leads to increase in the seismic bearing capacity Nce by less than 0.3% for β = 15° and 1% for β = 75°. On the other hand, the effect of cu/(γB) on Nce is found to be more prominent at smaller values of kh and higher values of β.

The variations of the seismic bearing capacity factor Nce with β for different combinations of kh and cu/(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 Nce decreases linearly with an increase in the value of β, and this trend is predominant at the smaller magnitudes of kh. On the contrary, for the case of λ = 1 shown in Figure 11, the existence of a certain critical value of βcr can be seen beyond which the value of Nce has a linear trend. The reason for this can be attributed to the fact that the value of λcr is smaller than 1, when β ≤ βcr. It is noteworthy that the magnitude of βcr is generally smaller for lower value of kh.

Figure 12 shows the variation of the seismic bearing capacity Nce with kh for different combinations of β, cu/(γB), and H/B, where the value of Nce are for the case of λ = 0. It can be seen that for all the cases, the value of Nce reduces linearly with increasing kh. 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 Nce 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 kh and lower value of cu/(γB).

(a)

(b)

Figure 12: Variation of Nce with kh for λ = 0 and H/B = 1: (a) β = 15°; (b) β = 45°–75°.

6. Failure Mechanisms

Figure 13 illustrates the plots of power dissipations showing the influence of the normalized footing distance λ for the cases of cu/(γB) = 1 and 7.5. As shown in Figure 13(a) for cu/(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 cu/(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 cu/(γB), the depth of failure zone is smaller than the cases with lower value of cu/(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 cu/(γB) does not affect the seismic bearing capacity, when the normalized footing distance is larger enough.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 13: Plots of power dissipations for H/B = 2, β = 45° and kh = 0.1: (a) λ = 0, cu/(γB) = 1; (b) λ = 1, cu/(γB) = 1; (c) λ = 4, cu/(γB) = 1; (d) λ = 0, cu/(γB) = 7.5; (e) λ = 1, cu/(γB) = 7.5; (f) λ = 4, cu/(γB) = 7.5.

Several plots of power dissipations showing the effect of β and H/B are shown in Figure 14 for the cases with cu/(γB) = 1, λ = 1, and kh = 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).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 14: Plots of power dissipations for λ = 1, cu/(γB) = 1and kh = 0.1: (a) H/B = 1, β = 15°; (b) H/B = 1, β = 45°; (c) H/B = 1, β = 75°; (d) H/B = 2, β = 15°; (e) H/B = 2, β = 45°; (f) H/B = 2, β = 75°; (g) H/B = 4, β = 15°; (h) H/B = 4, β = 45°; (i) H/B = 4, β = 75°.

Table 4: Seismic bearing capacity factor Nce for strip footings on cohesive soil slopes with H/B = 4.

β (°)
cu/(γB)
kh
λ
kh
λ

0
0.5
1
1.5
2
3
4
0
0.5
1
1.5
2
3
4

15
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
3.92
4.22
4.27
4.20
4.13
4.02
3.95
—
—
—
—
—
—
—

1.5
3.97
4.26
4.46
4.47
4.47
4.47
4.47
2.77
2.99
3.00
3.00
3.00
3.00
3.00

2.5
4.01
4.29
4.48
4.49
4.49
4.49
4.49
2.79
3.01
3.01
3.01
3.01
3.01
3.01

5
4.04
4.31
4.49
4.50
4.50
4.50
4.50
2.81
3.02
3.02
3.02
3.02
3.02
3.02

7.5
4.05
4.32
4.50
4.50
4.50
4.50
4.50
2.82
3.02
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
3.34
3.62
3.69
3.69
3.69
3.69
3.69
—
—
—
—
—
—
—

2.5
3.37
3.65
3.71
3.71
3.71
3.71
3.71
2.54
2.71
2.71
2.71
2.71
2.71
2.71

5
3.40
3.67
3.72
3.72
3.72
3.72
3.72
2.56
2.72
2.72
2.72
2.72
2.72
2.72

7.5
3.40
3.68
3.72
3.72
3.72
3.72
3.72
2.56
2.72
2.72
2.72
2.72
2.72
2.72

30
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
3.37
3.48
3.39
3.30
3.18
2.94
2.77
—
—
—
—
—
—
—

1.5
3.46
3.93
4.28
4.47
4.47
4.47
4.47
2.50
2.92
2.99
2.99
2.99
2.99
2.99

2.5
3.54
4.00
4.34
4.48
4.48
4.48
4.48
2.54
2.96
3.01
3.01
3.01
3.01
3.01

5
3.58
4.05
4.38
4.50
4.50
4.50
4.50
2.57
2.99
3.02
3.02
3.02
3.02
3.02

7.5
3.60
4.06
4.39
4.50
4.50
4.50
4.50
2.57
3.00
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
2.97
3.41
3.69
3.69
3.69
3.69
3.69
—
—
—
—
—
—
—

2.5
3.02
3.47
3.71
3.71
3.71
3.71
3.71
2.33
2.71
2.71
2.71
2.71
2.71
2.71

5
3.05
3.51
3.72
3.72
3.72
3.72
3.72
2.35
2.72
2.72
2.72
2.72
2.72
2.72

7.5
3.06
3.52
3.72
3.72
3.72
3.72
3.72
2.36
2.72
2.72
2.72
2.72
2.72
2.72

45
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
2.66
2.51
2.35
2.25
2.14
2.09
2.03
—
—
—
—
—
—
—

1.5
2.96
3.49
3.88
4.02
4.11
4.31
4.46
2.20
2.69
2.97
2.92
2.87
2.79
2.79

2.5
3.06
3.63
4.04
4.37
4.48
4.48
4.48
2.26
2.78
3.01
3.01
3.01
3.01
3.01

5
3.12
3.71
4.13
4.46
4.50
4.50
4.50
2.30
2.84
3.02
3.02
3.02
3.02
3.02

7.5
3.14
3.74
4.15
4.48
4.50
4.50
4.50
2.31
2.86
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
2.56
3.08
3.42
3.47
3.49
3.56
3.66
—
—
—
—
—
—
—

2.5
2.64
3.20
3.58
3.71
3.71
3.71
3.71
2.04
2.51
2.70
2.71
2.71
2.71
2.71

5
2.69
3.27
3.66
3.72
3.72
3.72
3.72
2.09
2.59
2.71
2.71
2.71
2.71
2.71

7.5
2.71
3.29
3.68
3.72
3.72
3.72
3.72
2.13
2.63
2.72
2.72
2.72
2.72
2.72

60
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
1.61
1.46
1.32
1.24
1.18
1.21
1.21
—
—
—
—
—
—
—

1.5
2.45
2.97
3.17
3.19
3.26
3.49
3.79
—
—
—
—
—
—
—

2.5
2.58
3.21
3.63
3.97
4.23
4.48
4.48
1.95
2.52
2.89
3.01
3.01
3.01
3.01

5
2.66
3.33
3.80
4.16
4.46
4.50
4.50
2.00
2.61
3.00
3.02
3.02
3.02
3.02

7.5
2.68
3.37
3.84
4.21
4.49
4.50
4.50
2.02
2.65
3.02
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
2.15
2.66
2.80
2.76
2.76
2.82
2.99
—
—
—
—
—
—
—

2.5
2.25
2.86
3.26
3.57
3.71
3.71
3.71
1.82
2.37
2.69
2.71
2.71
2.71
2.71

5
2.32
2.97
3.41
3.72
3.72
3.72
3.72
1.86
2.45
2.72
2.72
2.72
2.72
2.72

7.5
2.34
3.00
3.45
3.73
3.73
3.73
3.73
1.88
2.47
2.72
2.72
2.72
2.72
2.72

75
0.625
0.1
—
—
—
—
—
—
—
0.3
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
1.96
2.41
2.44
2.40
2.41
2.62
2.98
—
—
—
—
—
—
—

2.5
2.10
2.73
3.14
3.43
3.61
3.95
4.31
1.63
2.20
2.55
2.79
2.92
3.01
3.01

5
2.19
2.90
3.39
3.77
4.08
4.50
4.50
1.69
2.33
2.75
3.01
3.02
3.02
3.02

7.5
2.22
2.95
3.46
3.85
4.17
4.50
4.50
1.71
2.37
2.80
3.02
3.02
3.02
3.02

0.625
0.2
—
—
—
—
—
—
—
0.35
—
—
—
—
—
—
—

1
—
—
—
—
—
—
—
—
—
—
—
—
—
—

1.5
1.74
2.15
2.13
2.04
1.99
2.03
2.22
—
—
—
—
—
—
—

2.5
1.85
2.46
2.85
3.11
3.27
3.52
3.71
1.53
2.08
2.41
2.63
2.71
2.71
2.71

5
1.93
2.61
3.07
3.42
3.70
3.72
3.72
1.58
2.20
2.59
2.72
2.72
2.72
2.72

7.5
1.95
2.66
3.13
3.50
3.73
3.73
3.73
1.60
2.24
2.64
2.72
2.72
2.72
2.72

For comparison, Figure 15 shows the plots of power dissipations for the cases with the greater value of cu/(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 kh, the upper bound failure mechanisms are presented in Figure 16 for the cases of cu/(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 kh. This can be used to explain the phenomenon shown in Figure 8 that the value of λcr is generally smaller for greater value of kh.

(a)

(b)

(c)

Figure 15: Plots of power dissipations for β = 45°, λ = 1, cu/(γB) = 5 and kh = 0.1: (a) H/B = 1; (b) H/B = 2; (c) H/B = 4.

(a)

(b)

(c)

(d)

(e)

Figure 16: Plots of power dissipations for β = 45°, λ = 1, cu/(γB) = 2.5 and H/B = 2: (a) kh = 0; (b) kh = 0.1; (c) kh = 0.2. (d) kh = 0.3; (e) kh = 0.35.

As summarized, the failure mechanisms may be classified into four types, which are shown in Table 5.

Table 5: The types of slope failure.

Failure modes
Descriptions
Cases

Type 1
Bearing capacity failure for the slip line developed within the slope surface
H/B = 2, β = 15°, kh = 0.1, λ = 0 and cu/(γB) = 1 (Figure 12(a))

Type 2
Bearing capacity failure for the slip line extend to the toe of the slope
H/B = 2, β = 45°, kh = 0.1, λ = 1 and cu/(γB) = 1 (Figure 13(e))

Type 3
Bearing capacity failure without influence of the slope
H/B = 2, β = 45°, kh = 0.1, λ = 4 and cu/(γB) = 7.5 (Figure 13(f))

Type 4
Overall slope failure
H/B = 4, β = 75°, kh = 0.1, λ = 1 and cu/(γB) = 1 (Figure 14(i))

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 Nce 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, cu/(γB), β, kh, and H/B, on Nce has also been examined. The following conclusions can be made.

(1)In general, the value of Nce increases with the increase of λ up to a certain critical value of λcr beyond which the normalized footing distance does not influence the seismic bearing capacity, and the value of λcr is smaller for greater value of kh.(2)As the value of cu/(γB) increases, the value of Nce increases continuously and the rate of increase in Nce decreases. When cu/(γB) ≥ 5, the value of Nce is affected slightly by cu/(γB). And the effect of cu/(γB) on Nce is found to be more prominent at smaller values of kh and higher values of β.(3)For the cases of strip footings at the crest of slopes (λ = 0), the magnitude of Nce decreases linearly with an increase in the value of β, and this trend is predominant at the smaller magnitudes of kh. For the case of a footing at a distance from the slope (λ ≥ 0), the existence of a certain critical value of βcr can be seen beyond which the value of Nce has a linear trend.(4)The value of Nce reduces linearly with increasing kh. The value of H/B does not influence Nce for the case with higher value of kh and lower value of cu/(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 kh.

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

bT: | Discrete equilibrium-type operator
B: | Width of the strip footing
cu: | Undrained shear stress
F: | Yield function
H: | Slope height
kh: | Horizontal acceleration coefficient
Nce: | Seismic bearing capacity factor
n: | Total number of discrete stresses
p: | Vectors of unknown
p₀: | Prescribed nodal forces
u: | Horizontal velocity
: | Vertical velocity
qu: | Ultimate seismic bearing capacity
α: | Load multiplier
β: | Slope angle
γ: | Soil unit weight
σ: | Vector of discrete stresses
σi: | Discrete stresses
σn: | 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.

References

B. Hansen, “A general formula for bearing capacity,” Danish Geotechnical Institute, Bulletin, vol. 11, pp. 38–46, 1961.

A. S. Vesic, “Bearing capacity of shallow foundations,” in Foundation Engineering Handbook, H. F. Winterkorn and H. Y. Fang, Eds., Van Nostrand Reinhold, New York, NY, USA, 1975.

A. S. Azzouz and M. M. Baligh, “Loaded areas on cohesive slopes,” Journal of Geotechnical Engineering, vol. 109, no. 5, pp. 724–729, 1983.

F. Castelli and E. Motta, “Bearing capacity of strip footings near slopes,” Geotechnical and Geological Engineering, vol. 28, no. 2, pp. 187–198, 2010.

G. G. Meyerhof, “The ultimate bearing capacity of foundations on slopes,” in Proceedings of the Fourth International Conference on Soil Mechanics and Foundation Engineering, vol. 1, pp. 384–386, London, UK, August 1957.

K. Narita and H. Yamaguchi, “Bearing capacity analysis of foundations on slopes by use of log-spiral sliding surfaces,” Soils and Foundations, vol. 30, no. 3, pp. 144–152, 1990.

J. Graham, M. Andrews, and D. H. Shields, “Stress characteristics for shallow footings in cohesionless slopes,” Canadian Geotechnical Journal, vol. 26, no. 4, pp. 748–755, 1988.

V. V. Sokolovskii, Statics of Granular Media, Butterworth Scientific Publications, London, UK, 1960.

E. H. Davis and J. R. Booker, “Some adaptations of classical plasticity theory for soil stability problems,” in Proceedings of the Symposium Role of Plasticity in Soil Mechanics, A. C. Palmer, Ed., pp. 24–41, Cambridge, UK, September 1973.

K. Georgiadis, “An upper bound solution for the undrained bearing capacity of strip footings at the top of a slope,” Géotechnique, vol. 60, no. 10, pp. 801–806, 2010.

O. Kusakabe, T. Kimura, and H. Yamaguchi, “Bearing capacity of slopes under strip loads on the top surfaces,” Soils and Foundations, vol. 21, no. 4, pp. 29–40, 1981.

K. Georgiadis, “Undrained bearing capacity of strip footings on slopes,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 136, no. 5, pp. 677–685, 2010.

K. Georgiadis, “The influence of load inclination on the undrained bearing capacity of strip footings on slopes,” Computers and Geotechnics, vol. 37, no. 3, pp. 311–322, 2010.

J. S. Shiau, R. S. Merifield, A. V. Lyamin, and S. W. Sloan, “Undrained stability of footings on slopes,” International Journal of Geomechanics, vol. 11, no. 5, pp. 381–390, 2011.

B. Leshchinsky, “Bearing capacity of footings placed adjacent to c′-ϕ′ slopes,” Journal of Geotechnical & Geoenvironmental Engineering, vol. 141, no. 6, Article ID 04015022, 2015.

B. Leshchinsky and Y. Xie, “Bearing capacity for spread footings placed near c′-ϕ′ slopes,” Journal of Geotechnical & Geoenvironmental Engineering, vol. 143, no. 1, Article ID 06016020, 2017.

H. Zhou, G. Zheng, X. Yang, T. Li, and P. Yang, “Ultimate seismic bearing capacities and failure mechanisms for strip footings placed adjacent to slopes,” Canadian Geotechnical Journal, 2018.

M. Budhu and A. Al-Karni, “Seismic bearing capacity of soils,” Géotechnique, vol. 43, no. 1, pp. 181–187, 1993.

D. Choudhury and K. S. Subba Rao, “Seismic bearing capacity of shallow strip footings embedded in slope,” International Journal of Geomechanics, vol. 6, no. 3, pp. 176–184, 2006.

J. Kumar and N. Kumar, “Seismic bearing capacity of rough footings on slopes using limit equilibrium,” Géotechnique, vol. 53, no. 3, pp. 363–369, 2003.

S. K. Sarma and I. S. Iossifelis, “Seismic bearing capacity factors of shallow strip footings,” Géotechnique, vol. 40, no. 2, pp. 265–273, 1990.

O. Farzaneh, J. Mofidi, and F. Askari, “Seismic bearing capacity of strip footings near cohesive slopes using lower bound limit analysis,” in Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris, France, September 2013.

F. Askari and O. Farzaneh, “Upper-bound solution for seismic bearing capacity of shallow foundations near slopes,” Géotechnique, vol. 53, no. 8, pp. 697–702, 2003.

L. Dormieux and A. Pecker, “Seismic bearing capacity of foundation on cohesionless soil,” Journal of Geotechnical Engineering, vol. 121, no. 3, pp. 300–303, 1995.

K. Georgiadis and E. Chrysouli, “Seismic bearing capacity of strip footings on clay slopes,” in Proceedings of the 15th European Conference on Soil Mechanics and Geotechnical Engineering, pp. 723–728, Athens, Greece, September 2011.

J. Kumar and P. Ghosh, “Seismic bearing capacity for embedded footings on sloping ground,” Géotechnique, vol. 56, no. 2, pp. 133–140, 2006.

K. Yamamoto, “Seismic bearing capacity of shallow foundations near slopes using the upper-bound method,” International Journal of Geotechnical Engineering, vol. 4, no. 2, pp. 255–267, 2010.

J. Kumar and V. B. K. M. Rao, “Seismic bearing capacity of foundations on slopes,” Géotechnique, vol. 53, no. 3, pp. 347–361, 2003.

C. Jiang, J.-L. He, L. Liu, and B.-W. Sun, “Effect of loading direction and slope on laterally loaded pile in sloping ground,” Advances in Civil Engineering, vol. 2018, Article ID 7569578, p. 12, 2018.

J. S. Shiau, A. V. Lyamin, and S. W. Sloan, “Application of pseudo-static limit analysis in geotechnical earthquake design,” in Proceedings of the 6th European Conference on Numerical Methods in Geotechnical Engineering, pp. 249–255, Taylor & Francis, Graz, Austria, September 2006.

D. Raj, Y. Singh, and S. K. Shukla, “Seismic bearing capacity of strip foundation embedded in c-ϕ soil slope,” International Journal of Geomechanics, vol. 18, no. 7, Article ID 04018076, 2018.

J. Kumar and D. Chakraborty, “Seismic bearing capacity of foundations on cohesionless slopes,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 139, no. 11, pp. 1986–1993, 2013.

D. Chakraborty and J. Kumar, “Seismic bearing capacity of shallow embedded foundations on a sloping ground surface,” International Journal of Geomechanics, vol. 15, no. 1, Article ID 04014035, 2015.

D. Chakraborty and Y. Mahesh, “Seismic bearing capacity factors for strip footings on an embankment by using lower-bound limit analysis,” International Journal of Geomechanics, vol. 16, no. 3, Article ID 06015008, 2016.

H. Zhou, G. Zheng, X. Yin, R. Jia, and X. Yang, “The bearing capacity and failure mechanism of a vertically loaded strip footing placed on the top of slopes,” Computers and Geotechnics, vol. 94, pp. 12–21, 2018.

O. Cinicioglu and A. Erkli, “Seismic bearing capacity of surficial foundations on sloping cohesive ground,” Soil Dynamics and Earthquake Engineering, vol. 111, pp. 53–64, 2018.

R. Richards Jr. and X. Shi, “Seismic lateral pressures in soils with cohesion,” Journal of Geotechnical Engineering, vol. 120, no. 7, pp. 1230–1251, 1994.

S. W. Sloan, “Geotechnical stability analysis,” Géotechnique, vol. 63, no. 7, pp. 531–571, 2013.

J. S. Shiau, A. V. Lyamin, and S. W. Sloan, “Bearing capacity of a sand layer on clay by finite element limit analysis,” Canadian Geotechnical Journal, vol. 40, no. 5, pp. 900–915, 2003.

Y. Xiao, M. Zhao, and H. Zhao, “Undrained stability of strip footing above voids in two-layered clays by finite element limit analysis,” Computers and Geotechnics, vol. 97, pp. 124–133, 2018.

Y. Xiao, M. Zhao, H. Zhao, and R. Zhang, “Finite element limit analysis of the bearing capacity of strip footing on a rock mass with voids,” International Journal of Geomechanics, vol. 18, no. 9, Article ID 04018108, 2018.

J. P. Sahoo and J. Kumar, “Required lining pressure for the stability of twin circular tunnels in soils,” International Journal of Geomechanics, vol. 18, no. 7, Article ID 04018069, 2018.

D. W. Wilson, A. J. Abbo, S. W. Sloan, and A. V. Lyamin, “Undrained stability of dual circular tunnels,” International Journal of Geomechanics, vol. 14, no. 1, pp. 69–79, 2014.

Y. Xiao, M. Zhao, R. Zhang, H. Zhao, and W. Peng, “Stability of two circular tunnels at different depths in cohesive-frictional soils subjected to surcharge loading,” Computers and Geotechnics, vol. 112, pp. 23–34, 2019.

Y. Xiao, M. Zhao, R. Zhang, H. Zhao, and G. Wu, “Stability of dual square tunnels in rock masses subjected to surcharge loading,” Tunnelling and Underground Space Technology, vol. 92, Article ID 103037, 2019.

R. Zhang, Y. Xiao, M. Zhao, and H. Zhao, “Stability of dual circular tunnels in a rock mass subjected to surcharge loading,” Computers and Geotechnics, vol. 108, pp. 257–268, 2019.

K. Lim, M. J. Cassidy, A. J. Li, and A. V. Lyamin, “Mean parametric Monte Carlo study of fill slopes,” International Journal of Geomechanics, vol. 17, no. 4, Article ID 04016105, 2017.

Z. G. Qian, A. J. Li, R. S. Merifield, and A. V. Lyamin, “Slope stability charts for two-layered purely cohesive soils based on finite-element limit analysis methods,” International Journal of Geomechanics, vol. 15, no. 3, Article ID 06014022, 2015.

Y. Xiao, M. Zhao, R. Zhang, H. Zhao, and G. Wu, “Undrained bearing capacity of strip footings placed adjacent to two-layered slopes,” International Journal of Geomechanics, vol. 19, no. 8, Article ID 06019014, 2019.

J. Kumar and V. N. Khatri, “Effect of footing roughness on lower bound Nγ values,” International Journal of Geomechanics, vol. 8, no. 3, pp. 176–187, 2008.

J. Kumar and J. P. Sahoo, “Upper bound solution for pullout capacity of vertical anchors in sand using finite elements and limit analysis,” International Journal of Geomechanics, vol. 12, no. 3, pp. 333–337, 2011.

K. Krabbenhoft, A. V. Lyamin, M. Hjiaj, and S. W. Sloan, “A new discontinuous upper bound limit analysis formulation,” International Journal for Numerical Methods in Engineering, vol. 63, no. 7, pp. 1069–1088, 2005.

A. V. Lyamin, K. Krabbenhøft, A. J. Abbo, and S. W. Sloan, “General approach to modelling discontinuities in limit analysis,” in Proceedings of the 11th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG 2005), vol. 1, pp. 95–102, Torino, Italy, June 2005.

H. Ciria, J. Peraire, and J. Bonet, “Mesh adaptive computation of upper and lower bounds in limit analysis,” International Journal for Numerical Methods in Engineering, vol. 75, no. 8, pp. 899–944, 2008.

J. J. Muñoz, J. Bonet, A. Huerta, and J. Peraire, “Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading,” International Journal for Numerical Methods in Engineering, vol. 77, no. 4, pp. 471–501, 2009.

J. Herskovits, P. Mappa, E. Goulart, and C. M. Mota Soares, “Mathematical programming models and algorithms for engineering design optimization,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 30–33, pp. 3244–3268, 2005.

R. Zhang, G. Chen, J. Zou, L. Zhao, and C. Jiang, “Study on roof collapse of deep circular cavities in jointed rock masses using adaptive finite element limit analysis,” Computers and Geotechnics, vol. 111, pp. 42–55, 2019.

R. Zhang, L. Li, L. Zhao, and G. Tang, “An adaptive remeshing procedure for discontinuous finite element limit analysis,” International Journal for Numerical Methods in Engineering, vol. 116, no. 5, pp. 1–21, 2018.