Upper Bound Analysis of Two-Layered Slopes Subjected to Seismic Excitations Using the Layer-Wise Summation Method

Abstract

Due to natural sedimentation and artificial filling, slopes exhibit heterogeneity in the form of multi-layer soils, namely, layered slopes. Compared with homogenous slopes, the failure mechanism of layered slopes is more complex owing to the different shear strengths of each soil layer. Therefore, it is of great importance to gain insight into the stability of layered slopes. In this study, the upper bound theorem of limit analysis incorporated with a pseudo-static approach is utilized to investigate the seismic stability of two kinds of two-layered slopes: one with a stiff lower soil layer and the other with a weak lower soil layer. Three failure patterns, namely face failure, toe failure and base failure, are taken into account. A depth coefficient (Δ) is introduced to describe the distribution of two soil layers. The layer-wise summation method is adopted to calculate the safety factor and yield acceleration coefficient more conveniently. Based on Newmark’s method, the earthquake-induced horizontal displacement is estimated. The calculated results are validated by comparisons with published literature and the numerical method in terms of safety factor, critical failure surface and yield acceleration coefficient. The results show that the depth coefficient has a significant influence on the failure mechanism of two-layered slopes by determining whether the stability of upper-layered soil is dominant in the overall slope stability or not. Inaccurately identifying the failure patterns will overestimate the seismic performance of two-layered slopes in the aspects of safety factor and yield acceleration coefficient, leading to an underestimation of earthquake-induced horizontal displacement.

1. Introduction

The upper bound theorem of limit analysis is widely applied to the problems of stability analysis, i.e., slope stability [1], retaining wall stability [2], tunnel face stability [3] and embankment stability [4]. Generally, the Mohr–Coulomb failure criterion, represented by cohesion and internal friction angle of soils, is employed in the stability analysis of earth slopes [5,6,7]. The log-spiral rotational failure mechanism is proved to be the critical failure mechanism in slope stability [8]. Based on this failure mechanism, extensive research has recently been conducted on slope stability problems under different conditions and useful conclusions are drawn, providing guidance to the practical design of slopes. For instance, two- and three-dimensional stability charts and displacement charts of uniform slopes with the presence of seismic excitation and pore water pressure are established in order to quickly evaluate the stability of slopes for engineering interests [9,10,11,12,13]. Meanwhile, much attention has been paid to the failure pattern of slopes to further investigate the scale of landslides. Three primary failure patterns include face failure, toe failure and base failure in two- and three-dimensional slope stability analysis [14]. Toe failure commonly occurs in most cases. Base failure is triggered in mild slopes with low internal friction angle. Strong seismic excitation and large pore water pressure increase the likelihood of base failure. Face failure takes place when the slope is subjected to a large surcharge or when the three-dimensional effect is considered. Moreover, a rotational–translational combined failure is established when slopes involve a weak interlayer with lower shear strengths because landslides could occur along this interlayer [15]. In addition, the geometry of slopes has a significant influence on failure patterns. For instance, Rao et al. [16] and Zhang and Yang [17] explored the different failure patterns of two-staged slopes subjected to seismic excitations in two and three dimensions, respectively.

The aforementioned slope stability analyses are all aimed at homogenous slopes. However, owing to the natural geological action and artificial filling, slopes often exhibit heterogeneity and anisotropy [18], such as linear distributions of shear strengths along the depth or layered slopes composed of multi-layer soils. Therefore, it is of great importance to investigate the stability of slopes with heterogeneous and anisotropic soil. Within the framework of limit analysis, the same conclusion was drawn that heterogeneity and anisotropy of cohesion has a significant influence on slope stability and that a decrease in the heterogeneity coefficient and an increase in the anisotropy coefficient make slopes unstable [19,20]. In contrast, the anisotropy of the internal friction angle affects slope stability little [21,22]. For slopes composed of multi-layer soils, owing to the different shear strengths of each soil layer, the failure surface of each soil layer should be different. Therefore, the failure mechanism is composed of a series of failure surfaces instead of a single failure surface. Currently, numerous works have been conducted on the stability of layered slopes [23,24,25,26,27]. Li and Jiang [28] proposed three failure mechanisms to evaluate the stability of two-layered slopes. They found that face failure (failure mechanism A in their study) could occur for steeply two-layered slopes subjected to the large surcharger near the slope crest, and the rotational–translational failure mechanism (failure mechanism C in their study) is more critical than the rotational mechanism for gently two-layered slopes. Based on the combined failure mechanism, Deng et al. [29] adopted the limit equilibrium stress method to establish the stability charts for two-layered slopes for facilitating the design of slopes. Yao and Wang [30] established a horizontal layer slope failure mechanism for multi-layered slopes by combining a discrete algorithm with the upper bound theorem of limit analysis. Recently, a kinematic approach incorporated with the discretization technique has been developed to address this issue more conveniently than the traditional kinematic approach does. For instance, Chen et al. [31] calculated the earthquake-induced horizontal displacement of two-layered slopes with tension cutoff using the discretized failure mechanism. On the other side, a numerical method is an alternative way to conduct the stability analysis of multi-layered slopes. Based on the finite element method, Chatterjee and Krishna [32] found that translational failure occurred in the coarse-grained soil layer, while rotational failure took place in the fine-grained soil layer. However, few studies focus on the influence of the distribution of soil layers on the stability of the multi-layered slope, i.e., safety factor, failure mechanism and earthquake-induced horizontal displacement.

The objective of this study is to investigate the influence of the distribution of soil layers on the seismic stability of two-layered slopes in the application of the upper bound theorem of limit analysis. Two kinds of two-layered slopes are considered: one with a stiff lower soil layer and the other with a weak lower soil layer. To better describe the distribution of two soil layers, a depth coefficient (Δ) is introduced and defined as the ratio of the height of upper-layered soil to the height of the slope. Next, the pseudo-static approach is utilized to characterize the seismic forces acting on the slope. Both horizontal and vertical seismic forces are taken into consideration. A layer-wise summation method [33] is adopted to calculate external work rates and internal energy dissipation more efficiently. Subsequently, the safety factor (FS) and yield acceleration coefficient (k_y) are obtained by establishing an energy balance equation. Once the yield acceleration coefficient is determined, the earthquake-induced horizontal displacement (U_x) is calculated based on Newmark’s method [34]. Afterward, the calculated results are compared and verified with the published literature and upper bound solutions of the finite element limit analysis method, including k_y, FS and critical failure surfaces. Parametric studies including slope geometry and seismic excitation on the FS and failure pattern of selected slopes are conducted with different values of Δ. Finally, the influence of Δ on the seismic performance (i.e., k_y and U_x) of selected slopes is investigated, and some interesting conclusions are drawn.

2. Methodology

2.1. Upper Bound Solutions of FS for Two-Layered Slopes

Within the framework of limit analysis, the upper bound theorem is commonly applied to the stability analysis of slopes for finding rigorous upper bound values to the limit forces or loads causing failure or collapse of slopes. The application of the upper bound theorem requires that the soil mass should be perfectly plastic and obey the Mohr–Coulomb yield criterion. The deformation is governed by the associated flow rule that the angle between the tangent line at each point on the failure surface and the velocity vector is the internal friction angle. In other words, the internal friction angle controls the shape of the failure surface. The log-spiral rotational failure mechanism is proved to be a more critical failure mechanism than the translational failure mechanism. But for layered slopes, due to the different internal frictional angles of each soil layer, the failure mechanism is composed of a series of log-spiral failure surfaces instead of a single log-spiral failure surface. To comprehensively investigate the failure mechanism of two-layered slopes, three failure patterns, namely face failure, toe failure and base failure, are taken into consideration in this study, as illustrated in Figure 1. Herein, face failure is assumed to occur within the upper soil layer, which is verified by the finite element limit analysis method in Section 3.

2.1.1. Base Failure and Toe Failure

Since base failure can degenerate into toe failure, the calculation of upper bound solutions for base failure is presented in detail. Figure 2 shows how base failure is made up of two log-spiral rotational failure surfaces due to different internal friction angles in two soil layers. To describe the distribution of two soil layers, a depth coefficient (Δ) is introduced, which is defined as the ratio of the height of the upper soil layer (h) to the total height of the slope (H). In this study, the external work rates are represented by soil weight and seismic forces. Both horizontal and vertical seismic forces are considered. The internal energy dissipation is represented by soil cohesion along the failure surface. Then, the energy balance equation is established by equating the external work rates to the internal energy dissipation, which can be expressed as

$$W_{\gamma }+W_{kv}+W_{kh}=D$$

(1)

where W_γ is the work rate of the soil weight; W_kh and W_kv are the work rates of the horizontal and vertical seismic forces, respectively; and D is the internal dissipation of the soil cohesion. To account for the calculation of seismic forces, the pseudo-static approach is employed to treat seismic forces as a uniformly distributed inertial force, acting on the center of gravity of sliding soil mass [35]. This approximate but user-friendly approach is favored by engineers and is commonly applied in the assessment of seismic stability. Next, the calculation of the external work rates and the internal energy dissipation is elaborated by using the layer-wise summation method (LSM) [33] as follows.

Compared with the classical limit analysis, the LSM can better address the problems of highly nonlinear features of soil property, especially for layered soil. The core of the LSM is, firstly, to horizontally divide the sliding soil mass into several layers and, secondly, to calculate the external work rates and internal energy dissipation of each soil layer and sum them up. As shown in Figure 2, the sliding soil mass as one rigid body rotates around point O with angular velocity ω in the polar coordinate system. For the convenience of calculation, a cartesian coordinate system is established by setting the toe of the slope (point E) as the origin (E(0,0)). Point F is the intersection of the failure surface and the bottom of the slope (F(t,0)). When EF = 0, base failure degenerates into toe failure. Hence, t is set as an independent variable to distinguish two different failure patterns, and the value of t is negative. CD is the interface of the upper and lower soil layer. According to the LSM, the failure mechanism composed of two log-spiral failure surfaces is discretized by dividing the polar angles at the interval of δθ (0.1°). Then, the rigid sliding soil mass is divided into horizontal soil layers. The location of the ith soil layer (block mnn′m′) can be expressed in the polar coordinate system as

$$\theta =\{\begin{array}{l}{\theta }_0+\left(i-1\right)\delta \theta , {\theta }_0\le \theta <{\theta }_{\mathrm{D}}\\ {\theta }_{\mathrm{D}}+\left(i-1\right)\delta \theta , {\theta }_{\mathrm{D}}\le \theta \le {\theta }_{\mathrm{h}}\end{array}$$

(2)

$$r=\{\begin{array}{l}{r}_0{e}^{\left(\theta -{\theta }_0\right)\tan {\phi }_1}, {\theta }_0\le \theta <{\theta }_{\mathrm{D}}\\ r_{\mathrm{D}}{e}^{\left(\theta -{\theta }_{\mathrm{D}}\right)\tan {\phi }_2}, {\theta }_{\mathrm{D}}\le \theta \le {\theta }_{\mathrm{h}}\end{array}$$

(3)

$$r_{\mathrm{D}}=r_0{e}^{\left({\theta }_{\mathrm{D}}-{\theta }_0\right)\tan {\phi }_1}$$

(4)

where θ₀, θ_D and θ_h are the angles of points A, D and F on the failure surface with the horizontal direction, respectively. r₀ and r_D are the length of OA and OD, respectively; φ₁ and φ₂ are the internal friction angles of upper and lower soil layers, respectively. According to the geometrical features, θ_D can be deduced by a given Δ as

$$\Delta =\frac{h}{H}=\frac{e^{\left({\theta }_{\mathrm{D}}-{\theta }_0\right)\tan {\phi }_1}\cdot \sin {\theta }_{\mathrm{D}}-\sin {\theta }_0}{e^{\left({\theta }_{\mathrm{D}}-{\theta }_0\right)\tan {\phi }_1}\cdot e^{\left({\theta }_{\mathrm{h}}-{\theta }_{\mathrm{D}}\right)\tan {\phi }_2}\cdot \sin {\theta }_{\mathrm{h}}-\sin {\theta }_0}$$

(5)

For the convenience of calculation, the discrete horizontal soil layer is further divided into two triangle blocks (block mnn′ and block mm′n′). The gravity centers of two triangle blocks should first be determined due to the fact that the soil weight and seismic forces act on the gravity center. According to the geometrical features, the coordinates of gravity centers of block mnn′ and block mm′n′ can be obtained by coordinate transformation as, respectively,

$$\{\begin{array}{l}{x}_{G_{i1}}=\frac{2r_{\mathrm{h}}\cos \left(\pi -{\theta }_{\mathrm{h}}\right)+r_n\cos {\theta }_n+r_{n^{\prime }}\cos {\theta }_{n^{\prime }}+\left[r_{\mathrm{h}}\sin \left(\pi -{\theta }_{\mathrm{h}}\right)-r_n\sin {\theta }_n\right]\cot \beta }{3}+t\\ y_{G_{i1}}=\frac{3r_{\mathrm{h}}\sin \left(\pi -{\theta }_{\mathrm{h}}\right)-2r_n\sin {\theta }_n-r_{n^{\prime }}\sin {\theta }_{n^{\prime }}}{3}\end{array}$$

(6)

$$\{\begin{array}{l}{x}_{G_{i2}}=\frac{r_{\mathrm{h}}\cos \left(\pi -{\theta }_{\mathrm{h}}\right)+r_{n^{\prime }}\cos {\theta }_{n^{\prime }}+\left[2r_{\mathrm{h}}\sin \left(\pi -{\theta }_{\mathrm{h}}\right)-r_n\sin {\theta }_n-r_{n^{\prime }}\sin {\theta }_{n^{\prime }}\right]\cot \beta }{3}+t\\ y_{G_{i2}}=\frac{3r_{\mathrm{h}}\sin \left(\pi -{\theta }_{\mathrm{h}}\right)-r_n\sin {\theta }_n-2r_{n^{\prime }}\sin {\theta }_{n^{\prime }}}{3}\end{array}$$

(7)

where (x_Gi1,y_Gi1) and (x_Gi2,y_Gi2) are the coordinates of gravity center of block mnn′ and block mm′n′ in cartesian coordinate system, respectively; r_h, r_n and r_n′ are the length of OF, On and On′, respectively; and θ_n and θ_n′ are the angles of point n and point n′ of the ith soil layer with the horizontal direction, respectively (θ_n′ = θ_n + δθ).

Secondly, the total work rates of sliding soil mass duo to soil weight and seismic forces can be expressed by the sum of work rates of the ith soil layer, which are calculated by the dot product of soil gravity, seismic forces and velocity at the gravity center, respectively.

$$W_{\gamma }={\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(x_{G_{i1}}-x_O\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(x_{G_{i2}}-x_O\right)}$$

(8)

$$W_{kh}=k_{\mathrm{h}}\left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(y_O-y_{G_{i1}}\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(y_O-y_{G_{i2}}\right)}\right]$$

(9)

$$W_{kv}=k_{\mathrm{v}}{W}_{\gamma }$$

(10)

$$\{\begin{array}{l}{x}_O=r_{\mathrm{h}}\cos \left(\pi -{\theta }_{\mathrm{h}}\right)+t\\ y_O=r_{\mathrm{h}}\sin \left(\pi -{\theta }_{\mathrm{h}}\right)\end{array}$$

(11)

where γ_i is the soil unit weight of the ith soil layer; A_i1 and A_i2 are the areas of block mnn′ and block mm′n′, respectively; (x_O,y_O) is the coordinates of point O in the cartesian coordinate system; and k_h and k_v are the horizontal and vertical acceleration coefficient, respectively. Herein, k_v = λk_h. λ is the vertical-to-horizontal acceleration coefficient ratio, generally ranging from −0.5 to 0.5.

Owing to the sliding soil mass as one rigid block, the internal energy dissipation only occurs along the failure surface, which can be derived as

$$\begin{array}{ll}D& ={\displaystyle \sum [c_i\omega \sqrt{{\left(r_n\cos {\theta }_n\right)}^2+{\left(r_n\sin {\theta }_n\right)}^2}}\\ & \times \sqrt{{\left(r_n\cos {\theta }_n-r_{n^{\prime }}\cos {\theta }_{n^{\prime }}\right)}^2+{\left(r_n\sin {\theta }_n-r_{n^{\prime }}\sin {\theta }_{n^{\prime }}\right)}^2}\cos {\phi }_i]\end{array}$$

(12)

where c_i and φ_i are the cohesion and internal friction angle of the ith soil layer, respectively.

2.1.2. Face Failure

In this study, face failure is assumed to only occur within the upper soil layer, as depicted in Figure 3. Hence, the two-layered slope can be treated as a homogenous slope with upper-layered soil. The total height of the slope (H) is replaced by the height of the upper soil layer (h) in the calculation. The classical upper bound theorem of limit analysis is utilized to establish the energy balance equation. The work rates of the soil weight and seismic forces can be expressed in the polar coordinate system as [11,36]

$$W_{\gamma }={\gamma }_1\omega r_0^3\left(f_1-f_2-f_3\right)$$

(13)

$$W_{kh}=k_{\mathrm{h}}{\gamma }_1\omega r_0^3\left(f_4-f_5-f_6\right)$$

(14)

$$W_{kv}=k_{\mathrm{v}}{W}_{\gamma }$$

(15)

where γ₁ is the soil unit weight of the upper soil layer, and f₁~f₆ are non-dimensional functions, which can be found in Appendix A in detail. The internal energy dissipation along the failure surface D can be calculated as

$$D=\frac{c_1\omega r_0^2}{2\tan {\phi }_1}\left[e^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_1}-1\right]$$

(16)

where c₁ and φ₁ are the cohesion and internal friction angle of upper soil layer, respectively.

2.1.3. Safety Factor

As an indicator of the state of the slope, the safety factor (FS) is favored by engineers and commonly used in practice. FS can be calculated by the strength reduction method.

$$\{\begin{array}{l}{c}^{\prime }=c/\mathrm{FS}\\ {\phi }^{\prime }=\arctan \left(\tan \phi /\mathrm{FS}\right)\end{array}$$

(17)

where c and φ are the actual cohesion and internal friction angle of soil mass, respectively; c′ and φ′ are the cohesion and internal friction angle of soil mass in the limit equilibrium state, respectively. Substituting Equation (17) into Equation (1) and rearranging, a variable η, as a function of three independent variables, θ₀, θ_h and t (t for base failure only), is introduced to describe the stability of the slope as

$$\eta =\frac{D}{W_{\gamma }+W_{kv}+W_{kh}}=f\left({\theta }_0,{\theta }_{\mathrm{h}},t,\mathrm{FS}\right)$$

(18)

When the minimum of η is equal to 1.0, the slope is considered to be in the state of limit equilibrium, and the value of FS represents the safety factor of slopes. Owing to the implicit relationship between FS and η, an iteration method is adopted to determine the value of FS and the detailed procedure is described as follows.

Step 1. Input the information of a two-layered slope, including slope geometry (β and H), soil properties (c_i and φ_i of each soil layer), the distribution of two soil layers (Δ) and earthquake intensity (k_h and λ).

Step 2. Set FS = 1.0 and search the minimum of η = f(θ₀, θ_h, t, FS). The search domain of independent variables θ₀ and θ_h range from π/6 to (π − β) and t ranges from −H to 0. The constraint condition is θ₀ < θ_h. In the process of minimizing η, θ₀ and θ_h are varied with an increment of 0.01° and t is varied with a minimum of 0.001. The minimization procedure does not terminate until the difference between adjacent calculated results is less than 10⁻⁶ [14].

Step 3. If |η − 1| ≤ 10⁻³, which means the value of FS is an upper bound of the safety factor of the slope, stop the iteration. If not, set FS = $\sqrt{\mathrm{FS}}$ and take the updated FS back to Step 2 for the next iteration.

2.2. Yield Acceleration Coefficient and Seismic Permanent Displacements

Apart from FS, Newmark first proposed the basic elements of a procedure, namely Newmark’s method, to estimate the earthquake-induced displacement of an embankment in the evaluation of seismic stability [34], which is widely adopted by scholars [37,38,39,40,41]. To calculate the earthquake-induced displacement, the yield acceleration (k_yg) or yield acceleration coefficient (k_y) of a specific slope should be first determined. By equating the sum of external work rates of the soil weight and seismic forces to the internal energy dissipation, k_y can be deduced by rewriting Equation (1) as

$$k_{\mathrm{y}}=\frac{\left\{\begin{array}{l}{\displaystyle \sum [c_i\omega \cos {\phi }_i\sqrt{{\left(r_n\cos {\theta }_n\right)}^2+{\left(r_n\sin {\theta }_n\right)}^2}}\\ \hspace{1em}\hspace{1em}\times \sqrt{{\left(r_n\cos {\theta }_n-r_{n^{\prime }}\cos {\theta }_{n^{\prime }}\right)}^2+{\left(r_n\sin {\theta }_n-r_{n^{\prime }}\sin {\theta }_{n^{\prime }}\right)}^2}]\\ \hspace{1em}\hspace{1em} -\left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(x_{G_{i1}}-x_O\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(x_{G_{i2}}-x_O\right)}\right]\end{array}\right\}}{\left\{\begin{array}{l}{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(y_O-y_{G_{i1}}\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(y_O-y_{G_{i2}}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\lambda \left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(x_{G_{i1}}-x_O\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(x_{G_{i2}}-x_O\right)}\right]\end{array}\right\}}$$

(19)

for base failure and toe failure and

$$k_{\mathrm{y}}=\frac{\frac{c_1\omega r_0^2}{2\tan {\phi }_1}\left[e^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_1}-1\right]-{\gamma }_1\omega r_0^3\left(f_1-f_2-f_3\right)}{\lambda {\gamma }_1\omega r_0^3\left(f_1-f_2-f_3\right)+{\gamma }_1\omega r_0^3\left(f_4-f_5-f_6\right)}$$

(20)

for face failure. Therefore, k_y can be obtained by minimizing Equation (19) or Equation (20) with three independent variables: θ₀, θ_h and t (t for base failure only). The minimization procedure is the same as that of searching the minimum of η, while the value of FS is always set to 1.0, indicating the limit equilibrium state of the slope.

Once the external acceleration exceeds the yield acceleration, the soil mass starts to slide, causing its angular acceleration around point O to be not zero. Due to the inertial force, an additional moment should be considered in the energy balance equation. Equation (1) can be rewritten as

$$W_{\gamma }+W_{kv}+W_{kh}=D+W_M$$

(21)

where W_M is the work rate induced by the inertial forces. For base failure and toe failure, the formula of W_M can be expressed as [31]

$$W_M=\omega I_M\ddot{\theta }$$

(22)

$$\begin{array}{l}{I}_M={\displaystyle \sum \{\left[\frac{1}{36}{\gamma }_i{A}_{i1}\left(a^2+b^2+e^2\right)+{\gamma }_i{A}_{i1}{L}_{i1}\right]}\\ \hspace{1em} \hspace{1em}\hspace{1em} +\left[\frac{1}{36}{\gamma }_i{A}_{i2}\left(c^2+d^2+e^2\right)+{\gamma }_i{A}_{i1}{L}_{i2}\right]\}\end{array}$$

(23)

where I_M is the inertia moment of the sliding soil mass rotating around point O; $\ddot{\theta }$ is the angular acceleration of the sliding soil mass; L_i1 and L_i2 are the length of OG_i1 and OG_i2, respectively; a, b, c, d and e represent the length of each side of triangle block mnn′ and triangle block mnn′, respectively, as shown in Figure 2.

For face failure [11], it can be expressed as

$$W_M=\frac{G}{g}{l}^2\ddot{\theta }$$

(24)

$$l=\frac{{\gamma }_1{r}_0^3}{G}\sqrt{\left(f_1-f_2-f_3\right)+\left(f_4-f_5-f_6\right)}$$

(25)

$$G=\frac{1}{2}{\gamma }_1{r}_0^2\left[\frac{e^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_1}-1}{2\tan {\phi }_1}-\frac{L}{r_0}\sin {\theta }_0-\frac{h}{r_0}\frac{\sin \left(\beta +{\theta }_{\mathrm{h}}\right)e^{\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_1}}{\sin \beta }\right]$$

(26)

where G is the weight of sliding soil mass; g is gravity acceleration; l is the length of OG, as shown in Figure 3. The expressions of h/r₀ and L/r₀ are provided in Appendix A. Based on Newmark’s method, the horizontal displacement is accumulated when the external acceleration coefficient exceeds k_y. By substituting Equations (8)–(18) and Equations (22)–(26) into Equation (21), the expression of $\ddot{\theta }$ can be obtained after rearrangement.

For base failure and toe failure, we obtain the following:

$$\begin{array}{l}\ddot{\theta }=\frac{k-k_{\mathrm{y}}}{I_M}\{\left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(y_O-y_{G_{i1}}\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(y_O-y_{G_{i2}}\right)}\right]\\ +\lambda \left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(x_{G_{i1}}-x_O\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(x_{G_{i2}}-x_O\right)}\right]\}\end{array}$$

(27)

For face failure, we obtain the following:

$$\ddot{\theta }=\left(k-k_{\mathrm{y}}\right)\frac{{\gamma }_1{r}_0^3}{\frac{G}{g}{l}^2}\left(f_4-f_5-f_6\right)$$

(28)

where k is the actual horizontal acceleration coefficient of an earthquake which varies with time, while k_y is presumed to keep constant for a specific slope. The earthquake-induced horizontal displacement (U_x) can be calculated by double integrating $\ddot{\theta }$ over time interval dt, and displacement coefficient (C) is determined simultaneously.

$$u_{\mathrm{x}}=r_{\mathrm{h}}\sin {\theta }_{\mathrm{h}}{\displaystyle {\int }_t{\displaystyle {\int }_t\ddot{\theta }\mathrm{d}t\mathrm{d}t}}=C{\displaystyle {\int }_t{\displaystyle {\int }_t\left(k-k_{\mathrm{y}}\right)\mathrm{d}t\mathrm{d}t}}$$

(29)

For base failure and toe failure, we obtain the following [31]:

$$\begin{array}{l}C=\frac{r_{\mathrm{h}}\sin {\theta }_{\mathrm{h}}}{I_M}\{\left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(y_O-y_{G_{i1}}\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(y_O-y_{G_{i2}}\right)}\right]\\ \hspace{1em}\hspace{1em} +\lambda \left[{\displaystyle \sum {\gamma }_i{A}_{i1}\cdot \omega \cdot \left(x_{G_{i1}}-x_O\right)}+{\displaystyle \sum {\gamma }_i{A}_{i2}\cdot \omega \cdot \left(x_{G_{i2}}-x_O\right)}\right]\}\end{array}$$

(30)

For face failure, we obtain the following [11]:

$$C=r_{\mathrm{h}}\sin {\theta }_{\mathrm{h}}\frac{{\gamma }_1{r}_0^3}{\frac{G}{g}{l}^2}\left(f_4-f_5-f_6\right)$$

(31)

3. Results

3.1. Verification

As previously mentioned, the formulas of safety factor (FS) and the yield acceleration coefficient (k_y) are derived to evaluate the seismic performance of a two-layered slope. Herein, the results of published literature and the upper bound solutions calculated by the finite element limit analysis method (FELAM) are utilized to verify the correctness and effectiveness of the proposed method. Firstly, Figure 4 presents the comparisons of k_y for a homogenous slope and a two-layered slope. Different from the classical limit analysis, Chen et al. [31] employed the discretization technique to calculate the work rates and internal energy dissipation more conveniently, especially for heterogeneous slopes. Despite the different calculated methods, k_y of the proposed method matches well with the results of Chen et al. [31] with little difference. To exhibit the flexibility of the seismic stability analysis of two-layered slopes, the failure mechanism and FS of a 26.6° two-layered slope with a stiff lower soil layer (Slope 1) are calculated by using the proposed method and FELAM, respectively. The soil property is shown in

Table 1. The soil property of a two-layered slope (Adapted from Li and Jiang [28]).

Case	Layer	γ (kN/m3)	c (kPa)	φ (°)
Slope 1 with a stiff lower soil layer	#1	17.50	18.0	12.0
	#2	19.85	25.0	21.5
Slope 2 with a weak lower soil layer	#1	19.85	25.0	21.5
	#2	17.50	18.0	12.0

. As illustrated in Figure 5, both the failure mechanism and FS in this paper are in line with the solutions of FELAM under different values of the depth coefficient (Δ). Moreover, Δ does have an influence on the failure mechanism of the two-layered slope with a stiff soil layer, indicating that the occurrence of face failure is controlled by Δ. Overall, it can be inferred from the comparisons that the proposed method is fully validated for seismic stability analysis of two-layered slopes.

3.2. Safety Factor and Failure Pattern

To investigate the depth coefficient (Δ) on the seismic stability of the two-layered slope, both the safety factor (FS) and the failure pattern of the two-layered slopes possessing different slope inclination (β) and soil strength (i.e., cohesion (c) and internal friction angle (φ)) are calculated subjected to different Δ ranging from 0 to 1.4. The soil property of the two kinds of two-layered slopes is shown in Table 1: Slope 1 with a stiff lower soil layer and Slope 2 with a weak lower soil layer. Figure 6a,b show the negative correlation between β and FS for both Slope 1 and Slope 2, which is acknowledged by scholars [14,17]. Note that the range of Δ triggering face failure or base failure becomes narrow as β increases, indicating that gently layered slopes are more susceptible to the distribution of soil layers than steeply layered slopes are. The reason may be that Δ controls whether the stability of upper-layered soil is dominant in the overall slope stability or not. In addition, Figure 6c,d depict the different failure patterns with the same value of Δ for Slope 1 and Slope 2. It can be observed that FS will be overestimated if face failure or base failure are mistaken for toe failure, which demonstrates that the stability of two-layered slopes could be overestimated if the failure pattern is not accurately recognized, especially for the gently layered slopes (β ≤ 26.6°) (Figure 6a,b). Interestingly, the failure patterns with different methods exhibit distinct differences. Figure 7 compares the failure patterns obtained from the proposed method and FELAM. For Slope 1, the result of FELAM shows that toe failure along with face failure occurs when the two-layered slope becomes unstable, while only face failure takes place in the proposed method. For Slope 2, despite the base failure occurring in both methods, the results of FELAM show that rotational failure (base failure) and translational failure (near the toe of the slope) simultaneously take place. In addition, the difference in failure patterns leads to the difference in FS. This can be attributed to the deficiency in considering only one failure pattern in the stability analysis of the proposed method.

As expected, the presence of an earthquake weakens the slope stability in Figure 8 and Figure 9. However, the failure pattern is hardly affected by the horizontal acceleration coefficient (k_h) and the vertical-to-horizontal acceleration coefficient ratio (λ). The occurrence of face failure for Slope 1 and base failure for Slope 2 only depends on the value of Δ. For example, as shown in Figure 8a and Figure 9a, face failure occurs when Δ ranges from 0.6 to 1.0, and toe failure takes place when Δ ≤ 0.6 and Δ ≥ 1.0 regardless of the value of k_h and λ. In addition, Figure 9 also indicates that the downward vertical seismic excitation (λ = 0.5) has the greatest weakening effect on slope stability.

3.3. Earthquake-Induced Horizontal Displacement

To explore the depth coefficient (Δ) on yield acceleration coefficient (k_y) and earthquake-induced horizontal displacement (U_x) of the two-layered slope, the slope subjected to a specific earthquake is studied. Figure 10 depicts the acceleration–time records of the San Fernando earthquake, and its peak gravity acceleration (PGA) is 1.22 g. The slope inclination of two kinds of two-layered slopes is 45°, and the soil property is shown in Table 1. Based on Equations (19), (20) and (29)–(31), k_y, C (displacement coefficient) and U_x of Slope 1 and Slope 2 are presented in Figure 11 and Figure 12, respectively. Herein, the vertical acceleration is not included. For Slope 1, an increase in Δ leads to a decrease in k_y and C (Figure 11a,b), indicating the weakened seismic performance; consequently, U_x becomes larger. However, k_y, C and U_x remain constant when Δ exceeds 1.0 owing to the fact that the dominant role of the stability of the upper soil layer in the overall slope stability treats the two-layered slope as a homogenous slope composed of upper-layered soil, with the lower soil layer regarded as a firm stratum. In addition, U_x increases rapidly when face failure occurs, meaning that the seismic performance will be overestimated if face failure is treated as toe failure. For Slope 2, the opposite variations of k_y, C and U_x can be observed owing to the increasing soil strength of the overall slope when Δ ranges from 0.0 to 1.4. Similarly, mistaking base failure for toe failure will overestimate the seismic stability of the slope. Figure 11d and Figure 12d describe the detailed displacement–time curves under the cases of two different failure patterns with the same value of Δ. The displacement difference of different failure patterns is presented in

Table 2. Earthquake-induced horizontal displacements of different failure patterns for Slope 1.

Depth Coefficient, Δ	Horizontal Displacement, Ux (cm)		Difference (UxF − UxT)/UxF × 100%
	Face Failure	Toe Failure
0.7	50.74	38.50	24.12%
0.8	97.18	63.01	35.16%
0.9	167.93	112.33	33.11%

Note: U_xF and U_xT are the horizontal displacements of face failure and toe failure, respectively.

and

Table 3. Earthquake-induced horizontal displacements of different failure patterns for Slope 2.

Depth Coefficient, Δ	Horizontal Displacement, Ux (cm)		Difference (UxB − UxT)/UxB × 100%
	Base Failure	Toe Failure
1.0	56.42	49.79	11.74%
1.1	35.26	18.81	46.66%

Note: U_xT and U_xB are the horizontal displacements of toe failure and base failure, respectively.

. For Slope 1, compared with U_x under face failure, U_x under toe failure is reduced by 24.12%, 35.16% and 33.11% for Δ = 0.7~0.9, respectively. For Slope 2, the reductions in U_x under toe failure are, respectively, 11.74% and 46.66% for Δ = 1.0 and Δ = 1.1 if base failure is replaced by toe failure. Therefore, accurately identifying the failure patterns of two-layered slopes under a specific value of Δ is of great importance to evaluate slope stability.

4. Conclusions

Based on the upper bound theorem of limit analysis, this study mainly investigates the influence of the distribution of two soil layers on the seismic stability of two-layered slopes in the aspects of failure pattern, safety factor (FS), yield acceleration coefficient (k_y) and earthquake-induced horizontal displacement (U_x). Two kinds of two-layered slopes are considered: one with a stiff lower soil layer (Slope 1) and the other with a weak lower soil layer (Slope 2). A pseudo-static approach is employed to describe the presence of seismic forces. The upper bound theorem of limit analysis incorporated with a layer-wise summation method is employed to calculate the external work rates and internal energy dissipation. The calculated results are in line with the solutions of the published literature and finite element limit analysis. Emphasis is placed on the influence of the depth coefficient (Δ) on FS, k_y, U_x and critical failure surfaces of two-layered slopes. Some major conclusions are drawn as follows:

(1)

The distribution of two soil layers significantly affects the stability of two-layered slopes, especially for the failure pattern. The failure pattern of gently two-layered slopes is more susceptible to the variation of Δ than that of steeply two-layered slopes is. The range of Δ triggering face failure for Slope 1 and base failure for Slope 2 narrows with an increase in slope inclination. It can be attributed to the fact that Δ determines whether the stability of the upper soil layer dominates the overall slope stability.

(2)

The presence of an earthquake induces the reduction in FS of two-layered slopes. The horizontal acceleration coefficient (k_h) and the vertical-to-horizontal acceleration coefficient ratio (λ) have little influence on the failure patterns, which are only controlled by Δ.

(3)

Inaccurate recognition of failure pattern will overestimate the seismic performance (i.e., FS, k_y and U_x) of two-layered slopes with some range of Δ. For example, the earthquake-induced horizontal displacement will be underestimated by 24%~35% for Slope 1 and 11%~46% for Slope 2, respectively, if face failure or base failure are mistaken for toe failure. In practical engineering, much attention should be paid to the distribution of two soil layers to accurately identify the failure pattern of two-layered slopes.