Seismic Stability for 3D Two-Step Slope Governed by Non-Linearity in Soils Using Modified Pseudo-Dynamic Approach

Abstract

Seismic events are an active factor in causing slope instability, and the existing pseudo-dynamic method ignores some practical engineering conditions. In this paper, the seismic stability of three-dimensional (3D) two-step slopes is evaluated utilizing the modified pseudo-dynamic method based on the kinematical approach of limit analysis, in which the nonlinear characteristics of soil are considered using the generalized tangent technique. A 3D horn-like rotational failure mechanism is established to solve internal energy dissipation and external force work, in which the seismic work is considered in addition to the soil weight work and obtained by the layer-wise summation method. Based on the force-increase technique, the analytical expression for the safety factor (FS) of 3D two-step slopes is derived more readily. To verify the reliability of the new expression, the present results are compared with already posted solutions and the original pseudo-dynamic solutions. Ultimately, the sensitivity discussions are carried out to investigate the impacts of various factors on slope stability. This has some significance for the design and safety of 3D two-step slopes.

1. Introduction

Stability analysis for slopes has invariably been one of the most classic and hottest subjects in geotechnical engineering. New and numerous innovative approaches have been employed, and feasible engineering conditions are considered in the investigation of slope stability. Most of these research works are directed to slopes in the two-dimensional (2D) plane-strain situation. Nevertheless, slope instability and failure frequently exhibit distinct three-dimensional (3D) properties. Therefore, it is necessary to analyze the 3D stability for slopes in order to obtain more reasonable results. The stability analysis of 3D slopes can normally be carried out by the following three measures: (1) limit equilibrium method (LEM), (2) finite element method (FEM), and (3) limit analysis method (LAM). The existing 3D LEM is generally promoted on the basis of the 2D method; that is, the 2D slices are enlarged to the columns under the 3D space [1,2]. The assumptions of internal forces must be introduced in the analysis so as to ascertain the positive stress distribution on the sliding surface, yet the results obtained from this calculation and the actual stress state often have a large difference [3]. In numerical simulations based on the FEM, the strength reduction method is commonly utilized for calculations [4,5,6]. However, after discounting the slope strength, the stress state inside the slope will be changed so that the location of the most critical sliding surface is somewhat different from the actual situation. Recently, the LAM has been widely implemented in the stability assessment of slopes, which avoids the unreasonable assumptions on internal forces in the LEM and the tedious calculations in the FEM [7,8,9,10].

The primary challenge of applying the LAM to stability evaluation for 3D slopes is to construct a 3D kinematical admissible failure mechanism, and there has been a large amount of literature written on this field of research. The failure mechanisms adopted in the investigation of 3D limit analysis for slope stability can be summarized into two categories: (1) the block sliding mechanism, and (2) the integral sliding mechanism. The multi-block 3D sliding mechanism and the improved multi-block 3D sliding mechanism [11] belong to the first category, and the 3D horn-like rotational mechanism [12] and spherical mechanism belong to the second category. The application of the above two categories of sliding mechanisms in the stability assessment for 3D slopes can obtain more satisfactory results. However, the integral sliding mechanism can ensure calculation accuracy while greatly simplifying the calculation process.

Relevant studies have shown that slope geometry has a significant effect on slope stability. Therefore, in practical excavation engineering, the single-step is often modified into the double-step by adding a step platform to improve slope stability. In addition, earthquakes are also a factor that cannot be ignored in the stability assessment for 3D slopes [13]. The seismic stability of the 3D steep slope was investigated on the basis of the pseudo-static method. Nevertheless, the pseudo-static method evaluates the seismic effect in a highly simplified way, and it does not account for the influence of seismic excitation features on slope stability. A simplified pseudo-dynamic approach was proposed by Steedman and Zeng [14,15], which considers the temporal effects of seismicity but overlooks critical factors such as boundary conditions and material damping. On the basis of the existing methods, Bellezza [16,17] presented a modified pseudo-dynamic method, which has been frequently utilized in seismic stability analysis of geotechnical structures.

Furthermore, the linear Mohr-Coulomb (M-C) criterion, which has previously commonly been used in the stability assessment of soil slopes, cannot reflect the mechanical properties of soil material. Numerous experimental data show that the nonlinear envelope can better depict the shear strength of soils, but it is difficult to be applied in the stability evaluation of soil slopes based on the kinematic method of limit analysis. The generalized tangential technique proposed by Yang and Yin [8] succinctly solves this difficulty and is broadly employed in stability problems of geotechnical structures.

In this paper, the seismic stability of 3D two-step slopes is evaluated using the modified pseudo-dynamic approach based on the kinematical approach of limit analysis, in which the nonlinear characteristic of the soil is considered using the generalized tangent technique. The 3D horn-like rotational failure mechanism is discretized to solve the internal dissipation and the external work, in which the seismic work is considered in addition to the soil weight work and obtained by the layer-wise summation method. Based on the force-increase technique, the safety factor of 3D two-step slopes can be derived more readily. Eventually, the paper verifies the reliability of the results by comparing with already posted solutions and the original pseudo-dynamic solutions, and analyzes and discusses the effects of parameters.

2. Generalized Tangential Technique and Modified Pseudo-Dynamic Method

2.1. Generalized Tangential Technique

In conventional stability analysis for slopes, the linear M-C yield criterion is widely used to describe the shear strength of geotechnical materials. However, the relationship between shear strength and positive stress during actual material failure is often nonlinear [18,19]. The corresponding nonlinear strength envelope is shown in Figure 1, and the analytical equation in the σ_n-τ stress space is

$$\tau =c_0{\left(1+{\sigma }_n/{\sigma }_t\right)}^{1/m}$$

(1)

where ${\sigma }_n$ and $\tau$ are the normal stress and the shear stress on the failure surface, correspondingly. $c_0$ is the initial cohesion, ${\sigma }_t$ is the axial tensile strength at failure, and $m$ is the parameter evaluating the nonlinear feature of the envelope ($m\ge 1$). It can be seen that for $m=1$, Equation (1) transforms into the classical linear M-C criterion. In addition, ${\phi }_t$ can be introduced as an intermediate variable defined as $\tan {\phi }_t=\mathrm{d}\tau /\mathrm{d}{\sigma }_n$, then ${\sigma }_n$ and $\tau$ can be written as the function of ${\phi }_t$

$${\sigma }_n={\sigma }_t{\left(\frac{m{\sigma }_t\tan {\phi }_t}{c_0}\right)}^{\frac{m}{1-m}}-{\sigma }_t$$

(2)

$$\tau =c_0{\left(\frac{m{\sigma }_t\tan {\phi }_t}{c_0}\right)}^{\frac{1}{1-m}}$$

(3)

Nonetheless, the upper limit analysis of the bearing capacity or stability of geotechnical structures on rock under the nonlinear failure criterion has been one of the difficult problems in geotechnical engineering. In this paper, the generalized tangent technique [8] is adopted to obtain the upper limit solution for safety factor FS of 3D slopes subjected to non-linearity in soils. As shown in Figure 1, L is the tangential straight line to any point P on the nonlinear envelope, and its expression can be given as

$$\tau =c_t+{\sigma }_n\tan {\phi }_t$$

(4)

where $c_t$ is the intercept of that tangent line on the axis, bringing Equation (2) and Equation (3) into the Equation (4) whose analytic expression can be deduced

$$c_t=\frac{m-1}{m}{c}_0{\left(\frac{m{\sigma }_t\tan {\phi }_t}{c_0}\right)}^{\frac{1}{1-m}}+{\sigma }_t\tan {\phi }_t$$

(5)

The difficulty of using the nonlinear strength criterion directly in geotechnical engineering problems is avoided by the generalized tangent technique, and instead, the line tangent to the nonlinear envelope is used to describe the shear strength of the soils. Obviously, when the normal stresses are the same, the strength represented by the tangent line is greater than or equal to the strength of the nonlinear envelope. Therefore, the minimum upper limit solution of the safety factor for the 3D slope can be obtained by means of the generalized tangent line technique.

2.2. The Modified Pseudo-Dynamic Method

Seismicity can significantly increase the inducing forces of slope failure and reduce the shear strength of soils, which is one of the main factors inducing slope instability and landslide. Numerous new attempts and advances have been made in the aspect of seismic stability for geotechnical structures. Whereas the pseudo-static method, which is usually used to deliberate the effects of seismicity on 3D slope stability, neglects the fluctuation characteristics of seismic waves that vary with time, the existing pseudo-dynamic method only considers the time effect, ignoring some practical engineering conditions. Therefore, a new pseudo-dynamic method [16,17], which is improved on the basis of the original pseudo-dynamic method, is adopted in this paper. The present study puts forward a kinematic analytical solution for assessing a seismic 3D two-step slope subjected to the non-linearity in soils, which couples the modified pseudo-dynamic method with the upper limit theorem of limit analysis.

Bellezza [16] came up with the modified pseudo-dynamic method based on the assumptions of a visco-elastic medium and Kelvin–Voigt material, which surmounted the imperfections of the existing pseudo-dynamic method, namely: (1) The existing method cannot satisfy the zero-stress boundary condition because it only considers the upward propagation of the incident wave throughout the linear elastic soil medium [16,20]; (2) The acceleration value derived from the existing pseudo-dynamic method is considered to be augmented linearly toward the ground. Hence, an assumed amplification factor is required when using this method [21], which is regarded as equal to 1 in most relevant studies; (3) The existing method does not consider the energy dissipation or damping generated by the real material.

When k_v ≤ 0.5 k_h, Gazetas et al. [22] concluded that the effect of vertical seismic acceleration is almost negligible for the stability of 3D slopes based on the research of horizontal and vertical seismic waves. In order to optimize the program design process and computational speed, the dominant horizontal seismic motivator is taken into account with the least possible influence on the seismic effect in this paper. The horizontal seismic acceleration based on the improved pseudo-dynamic can be given as

$$a_h\left(z,t\right)=\frac{k_{h}g}{C_{\mathrm{S}}^2+S_{\mathrm{S}}^2}\left[\left(C_{\mathrm{S}}{C}_{SZ}+S_{\mathrm{S}}{S}_{SZ}\right)\cos \left(\omega t\right)+\left(S_{\mathrm{S}}{C}_{SZ}+C_{\mathrm{S}}{S}_{SZ}\right)\sin \left(\omega t\right)\right]$$

(6)

In which,

$$C_{sz}=\cos \left(y_{S1}\frac{H-z}{H}\right)\cos h\left(y_{S2}\frac{H-z}{H}\right)$$

(7)

$$S_{sz}=-\sin \left(y_{S1}\frac{H-z}{H}\right)\sin h\left(y_{S2}\frac{H-z}{H}\right)$$

(8)

$$C_S=\cos \left(y_{S1}\right)\cos h\left(y_{S2}\right)$$

(9)

$$S_S=-\sin \left(y_{S1}\right)\sin h\left(y_{S2}\right)$$

(10)

$$y_{s1}=\frac{\omega H}{V_s}\sqrt{\frac{\sqrt{1+4D^2}+1}{2\left(1+4D^2\right)}}$$

(11)

$$y_{s2}=-\frac{\omega H}{V_s}\sqrt{\frac{\sqrt{1+4D^2}-1}{2\left(1+4D^2\right)}}$$

(12)

where k_h and g are horizontal seismic coefficient and gravitational acceleration, respectively, and y_s1 and $y_{s2}$ are defined as a function of the normalized frequency wH/V_s and damping ratio D, in which H is the height of slope, and V_s is the velocity of S-wave propagating. The remaining parameters: w = 2π/T is angular frequency, where T is the seismic period; t is the arbitrary moment in the period T; z is the value corresponding to the vertical axis.

2.3. Force-Increase Technique

The resistance and the induction act together on the slope structure, and the slope structure will be destabilized or collapsed as the induction is critical to the resistance. In the stability analysis of slopes, two methods are frequently used: (1) lowering the resistance to make the slope critical, namely, the strength reduction method (SRM) [22,23,24]; (2) raising the induction to obtain the slope critical as well, namely, the force-increase technique (FIT) [25,26] used in this paper.

The SRM has universally been utilized to measure the safety factor FS of slopes, but what is derived from the SRM is an implicit equation containing FS. Thus, the force-increasing technique is used in this paper to facilitate the program design and subsequent solution. According to the force-increase principle, FS, namely, the factor of safety, can be customarily defined as

$$FS=\frac{{\gamma }_{cr}}{\gamma }$$

(13)

where $\gamma$ and ${\gamma }_{cr}$ are the respective initial unit weight of soil and the unit weight of soil that increases when the slope is in critical failure. It is well established that the external work can be written as the linear expression on the external force. According to this, FS can be correspondingly expressed as the ratio of the critical external work to the real external work

$$FS=\frac{W_{cr}}{W}$$

(14)

where $W_{cr}$ is the critical external work, and W is the real external work. On the basis of the kinematical approach of limit analysis, $W_{cr}=D$ (internal energy dissipation), so FS can be further written as the ratio between internal energy dissipation and real external force work, i.e.,

$$FS=\frac{D}{W}$$

(15)

As we can see from Equation (15), the equation on FS derived by the force increase technique is the explicit function, which greatly facilitates solutions of complex integral expressions obtained from both real external force work and internal energy dissipation in 3D conditions.

3. Kinematical Analysis Model of 3D Soil Slopes with Two Steps

The stability evaluation problems for geological engineering are frequently greatly simplified by adopting the kinematical approach of the LAM [27]. The kinematical approach can be formulated as follows: for any kinematically admissible plastic strain rate fields and velocity fields, if the internal energy dissipation is equated with the external force work, then the calculated solutions must be higher or identical to what actually occurs. Additionally, since the associated flow law must be satisfied in the kinematically admissible velocity field, which corresponds to the M-C criterion, there is a relationship for the principal strain rates, namely,

$${\dot{\epsilon }}_1+{\dot{\epsilon }}_2+{\dot{\epsilon }}_3+\left({\dot{\epsilon }}_1-{\dot{\epsilon }}_2-{\dot{\epsilon }}_3\right)\sin {\phi }_t=0$$

(16)

where ${\phi }_t$ is the slope of the tangent line at any point on the nonlinear envelope aforementioned in the section on generalized tangent technique, namely, the angle of internal friction of the soil. Applying the Equation (16) to the velocity interrupted curved surface formed at the time of slope failure, the following equation can be obtained

$$v_n=v_t\tan {\phi }_t$$

(17)

in which $v_n$ is the normal component of the velocity along the interrupted curved surface, and $v_t$ is the tangential component of the velocity along the interrupted curved surface. Furthermore, the 3D horn-like rotational mechanism, which has been shown to satisfy this condition well [12], is employed in this paper for the establishment of the failure mechanism for 3D two-step slopes.

3.1. Description of the 3D Failure Mechanism in Two-Step Slopes

Constructing the failure mechanism is one of the most crucial procedures in the stability analysis of geotechnical structures using the kinematic method of limit analysis. The 3D two-step slopes subjected to seismicity are analyzed in the framework of the 3D horn-like rotational failure mechanism in this paper.

In Figure 2, it can be observed that the exterior for the 3D two-step slope consists of the top surface for the slope AB, slope surface above the step BC with inclination ${\beta }_1$, slope surface below the step CD with inclination ${\beta }_2$, and step plane DE. The overall slope height is H. The height of slope above the horizontal step surface is ${\xi }_{1}H$, and the height of slope below the horizontal step surface is ${\xi }_{2}H$, for which ${\xi }_1$ and ${\xi }_2$ are the coefficients of depth. Apparently,

$${\xi }_1+{\xi }_2=1$$

(18)

The 3D horn-like rotational failure mechanism is implemented on the 3D two-step slope by crossing the peak and foot for the slope, and the symmetry plane for this failure mechanism is perfectly decided by two logarithmic spirals, whose analytic expressions can be given separately as

$$r=r_0{e}^{\left(\theta -{\theta }_0\right)\tan {\phi }_t}$$

(19)

and

$$r^{\prime }=r_0^{\prime }{e}^{-\left(\theta -{\theta }_0\right)\tan {\phi }_t}$$

(20)

in which r is the curve AE, r′ is the curve A′E′, and r₀ = OA, r′₀ = OA, and ${\theta }_0$ is the angle between the horizontal axis and the line OA, as shown in Figure 2. The 3D horn-like rotational failure mechanism is the curvilinear cone with vertex angle $2{\phi }_t$ shaped by the circle with progressively increasing radius in continuity around the center of rotation O. The length between point O and the center of the circle, and the radius of the circle can be expressed separately as

$$r_m=\left(r-r^{\prime }\right)/2=r_0{f}_1$$

(21)

and

$$R=\left(r-r^{\prime }\right)/2=r_0{f}_2$$

(22)

in which $f_1$ and $f_2$ can be written as

$$f_1=\frac{1}{2}\left[e^{\left(\theta -{\theta }_0\right)\tan {\phi }_t}+\frac{r_0^{\prime }}{r_0}{e}^{-\left(\theta -{\theta }_0\right)\tan {\phi }_t}\right]$$

(23)

$$f_2=\frac{1}{2}\left[e^{\left(\theta -{\theta }_0\right)\tan {\phi }_t}-\frac{r_0^{\prime }}{r_0}{e}^{-\left(\theta -{\theta }_0\right)\tan {\phi }_t}\right]$$

(24)

As illustrated in Figure 3, the 3D slopes can be extended to the case of 2D by embedding the block of width b with the plane-strain property in the 3D failure mechanism for two-step slopes, in accordance with Michalowski and Drescher [12]. The width b of the embedded block can be calculated as

$$b=B-{B^{\prime }}_{max}$$

(25)

where B and B’_max are the width for the entire two-step slope and the maximum width for the 3D section of the two-step slope, correspondingly. The 3D two-step slope is transformed into the 2D condition with the ratio of overall width B to height H of the two-step slope tending to infinity.

3.2. Internal Energy Dissipation

The internal energy dissipation inside the 3D two-step slope can be divided into two portions: (1) the dissipation generated along the non-continuous surface S_t where sliding is about to occur D_t; (2) the dissipation generated by the internal volume deformation within the slope D_V. In accordance with the upper limit theorem of limit analysis, D_t and D_V can be, separately, given as

$$D_t={\displaystyle {\iint }_{S_t}{c}_{t}v\cos {\phi }_t\mathrm{d}{S}_t}$$

(26)

and

$$D_V={\displaystyle {\iiint }_V{c}_t\cos {\phi }_t\left({\dot{\epsilon }}_1-{\dot{\epsilon }}_2-{\dot{\epsilon }}_3\right)}\mathrm{d}V$$

(27)

where S_t is the velocity non-continuous surface, and V is the volume of deformation occurring within the slope. v, namely, the velocity in the failure mechanism of 3D stepped slope, can be calculated by the following Equation (28) deduced from the local Cartesian coordinate systems founded on any circular cross-section of the 3D horn-like rotational failure mechanism with the center of the circular cross-section as the origin, as shown in Figure 2.

$$v=\left(r_m+y\right)\omega$$

(28)

in which y is the coordinate value corresponding to the y-axis in the local coordinate system. The rest of the parameters involved in Equations (26) and (27) are explicitly stated in the previous sections and will not be repeated here. According to the Gaussian dispersion theorem and after the rigorous derivation, the total internal energy dissipation for the sliding mechanism can be ultimately written as

$$\begin{array}{ll}D& =D_t+D_V\\ & =c_t\cot {\phi }_t{\displaystyle {\iint }_{S_r}v}\cdot n\mathrm{d}{S}_r\end{array}$$

(29)

where S_r is the remaining surface of the kinematically admissible failure mechanism after removing the velocity discontinuity surface. n is the outward unit normal vector of the surface S_r, and v is the corresponding velocity vector.

In general, the overall internal energy dissipation of the failure mechanism relatively easily deduced from the Equation (29) can be classified as four portions along the profile of the two-step slope: D_AB, D_BC, D_CD, and D_DE are the corresponding energy dissipation rates on the planes represented by AB, BC, CD, and DE, respectively, and their 3D portions can be shown as

$$D_{\mathrm{AB}-3\mathrm{D}}=-2\omega c_t\cot {\phi }_t{\displaystyle {\int }_{{\theta }_0}^{{\theta }_{\mathrm{B}}}{\displaystyle {\int }_0^{x_1^{*}}{r}_0^2}} {\sin }^2 {\theta }_0\frac{\cos \theta }{{\sin }^3\theta }\mathrm{d}x\mathrm{d}\theta$$

(30)

$$D_{\mathrm{BC}-3\mathrm{D}}=-2\omega c_t\cot {\phi }_t{\displaystyle {\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}{\displaystyle {\int }_0^{x_2^{*}}{r}_0^2}} {\sin }^2 \left({\theta }_{\mathrm{B}}+{\beta }_1\right)\frac{{\sin }^2{\theta }_0}{{\sin }^2{\theta }_{\mathrm{B}}}\frac{\cos \left(\theta +{\beta }_1\right)}{{\sin }^3\left(\theta +{\beta }_1\right)}\mathrm{d}x\mathrm{d}\theta$$

(31)

$$D_{\mathrm{CD}-3\mathrm{D}}=-2\omega c_t\cot {\phi }_t{\displaystyle {\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}{\displaystyle {\int }_0^{x_3^{*}}{\left(r_0\sin {\theta }_0+{\xi }_{1}H\right)}^2}}\frac{\cos \theta }{{\sin }^3\theta }\mathrm{d}x\mathrm{d}\theta$$

(32)

$$D_{\mathrm{DE}-3\mathrm{D}}=-2\omega c_t\cot {\phi }_t{\displaystyle {\int }_{{\theta }_{\mathrm{D}}}^{{\theta }_{\mathrm{h}}}{\displaystyle {\int }_0^{x_4^{*}}{r}_0^2}}{e}^{2\left({\theta }_{\mathrm{h}}-{\theta }_0\right)\tan {\phi }_{\mathrm{t}}}{\sin }^2\left({\theta }_{\mathrm{h}}+{\beta }_2\right)\frac{\cos \left(\theta +{\beta }_2\right)}{{\sin }^3\left(\theta +{\beta }_2\right)}\mathrm{d}x\mathrm{d}\theta$$

(33)

The upper limit of integration $x_i^{*}$ (i = 1, 2, 3, 4) of the second integral of the above double integration Equations can be written as $x_i^{*}=\sqrt{R^2-d_i^2}$, in which, according to Figure 2, the expressions of $d_i$ can be given as, separately

$$d_1=r_0\frac{\sin {\theta }_0}{\sin \theta }-r_m=r_0{f}_3$$

(34)

$$d_2=r_0\frac{\sin {\theta }_0\sin \left({\theta }_B+{\beta }_1\right)}{\sin {\theta }_B\sin \left(\theta +{\beta }_1\right)}-r_m=r_0{f}_4$$

(35)

$$d_3=\frac{r_0\sin {\theta }_0+{\xi }_{1}H}{\sin \theta }-r_m=r_0{f}_5$$

(36)

$$d_4=r_0{e}^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\frac{\sin \left({\theta }_h+{\beta }_2\right)}{\sin \left(\theta +{\beta }_2\right)}-r_m=r_0{f}_6$$

(37)

where $f_3$, $f_4$, $f_5$, and $f_6$ are dimensionless expressions are written as

$$f_3=\frac{\sin {\theta }_0}{\sin \theta }-f_1$$

(38)

$$f_4=\frac{\sin {\theta }_0\sin \left({\theta }_{\mathrm{B}}+{\beta }_1\right)}{\sin {\theta }_{\mathrm{B}}\sin \left(\theta +{\beta }_1\right)}-f_1$$

(39)

$$f_5=\frac{\sin {\theta }_0+{\xi }_1{\eta }_1}{\sin \theta }-f_1$$

(40)

$$f_6=e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\frac{\sin \left({\theta }_{\mathrm{h}}+{\beta }_2\right)}{\sin \left(\theta +{\beta }_2\right)}-f_1$$

(41)

In accordance with the geometric features, the angles ${\theta }_B$, ${\theta }_C$, and ${\theta }_D$ in Figure 2 can be deduced as

$${\theta }_{\mathrm{B}}=\arccos \frac{\cos {\theta }_0-{\eta }_2}{\sqrt{{\sin }^2{\theta }_0+{\left(\cos {\theta }_0-{\eta }_2\right)}^2}}$$

(42)

$${\theta }_{\mathrm{C}}=\arccos \frac{\cos {\theta }_0-{\eta }_2-{\xi }_1\cot {\beta }_1{\eta }_1}{\sqrt{{\left(\sin {\theta }_0+{\xi }_1{\eta }_1\right)}^2+{\left(\cos {\theta }_0-{\eta }_2-{\xi }_1\cot {\beta }_1{\eta }_1\right)}^2}}$$

(43)

$${\theta }_{\mathrm{D}}=\arccos \frac{e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\cos {\theta }_h+{\xi }_2\cot {\beta }_2{\eta }_1}{\sqrt{{\left(e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\sin {\theta }_h-{\xi }_2{\eta }_1\right)}^2+{\left(e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\cos {\theta }_h+{\xi }_2\cot {\beta }_2{\eta }_1\right)}^2}}$$

(44)

in which ${\eta }_1=H/r_0$ and ${\eta }_2=L/r_0$ are two dimensionless expressions, shown as

$${\eta }_1=e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\sin {\theta }_h-\sin {\theta }_0$$

(45)

$${\eta }_2=\cos {\theta }_0-e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\cos {\theta }_h-\left({\xi }_1\cot {\beta }_1+{\xi }_2\cot {\beta }_2+{\lambda }_a\right){\eta }_1$$

(46)

$L$ is the horizontal line AB’s length at the top of the slope, as illustrated in Figure 2. Consequently, the overall internal dissipation for two-step slopes after adding the plane insert section can be expressed in the following form

$$\begin{array}{ll}D& =D_{\mathrm{AB}-3\mathrm{D}}+D_{\mathrm{BC}-3\mathrm{D}}+D_{\mathrm{CD}-3\mathrm{D}}+D_{\mathrm{DE}-3\mathrm{D}}+D_{\mathrm{insert}}\\ & =\omega c_t\cot {\phi }_t{r}_0^3\left(g_1+g_2\right)\end{array}$$

(47)

where g₁ and g₂ are dimensionless expressions for the internal dissipation of 3D section and 2D plane-strain section, respectively, given as

$$\begin{array}{ll}{g}_1=& -2{\sin }^2{\theta }_0{\displaystyle {\int }_{{\theta }_0}^{{\theta }_{\mathrm{B}}}\frac{\cos \theta }{{\sin }^3\theta }}\sqrt{f_2^2-f_3^2}\mathrm{d}\theta \\ & -2{\sin }^2\left({\theta }_{\mathrm{B}}+{\beta }_1\right)\frac{{\sin }^2{\theta }_0}{{\sin }^2{\theta }_{\mathrm{B}}}{\displaystyle {\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}\frac{\cos \left(\theta +{\beta }_1\right)}{{\sin }^3\left(\theta +{\beta }_1\right)}}\sqrt{f_2^2-f_4^2}\mathrm{d}\theta \\ & -2{\left(\sin {\theta }_0+{\xi }_1{\eta }_1\right)}^2{\displaystyle {\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}\frac{\cos \theta }{{\sin }^3\theta }}\sqrt{f_2^2-f_5^2}\mathrm{d}\theta \\ & -2e^{2\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}{\sin }^2\left({\theta }_{\mathrm{h}}+{\beta }_2\right){\displaystyle {\int }_{{\theta }_{\mathrm{D}}}^{{\theta }_h}\frac{\cos \left(\theta +{\beta }_2\right)}{{\sin }^3\left(\theta +{\beta }_2\right)}}\sqrt{f_2^2-f_6^2}\mathrm{d}\theta \end{array}$$

(48)

$$\begin{array}{ll}{g}_2=& \frac{1}{2}\frac{b}{H}{\eta }_1\{\frac{{\sin }^2{\theta }_0}{{\sin }^2{\theta }_{\mathrm{B}}}-1+\frac{{\sin }^2{\theta }_0}{{\sin }^2{\theta }_{\mathrm{B}}}\left[\frac{{\sin }^2\left({\theta }_{\mathrm{B}}+{\beta }_1\right)}{{\sin }^2\left({\theta }_{\mathrm{C}}+{\beta }_1\right)}-1\right]\\ & +{\left(\sin {\theta }_0+{\xi }_1{\eta }_1\right)}^2\left[\frac{1}{{\sin }^2{\theta }_{\mathrm{D}}}-\frac{1}{{\sin }^2{\theta }_{\mathrm{C}}}\right]\\ & +e^{2\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\left[1-\frac{{\sin }^2\left({\theta }_h+{\beta }_2\right)}{{\sin }^2\left({\theta }_{\mathrm{D}}+{\beta }_2\right)}\right]\}\end{array}$$

(49)

3.3. External Force Work

The stability for two-step slopes is affected by the gravitational force of the soil and the horizontal seismic force, in which the effect of vertical seismic force is almost negligible at k_v ≤ 0.5 k_h, based on the conclusion of Gazetas et al. The soil weight work in the two wings of the two-step slope, namely, the 3D section, can be shown as

$$\begin{array}{ll}{W}_{\gamma -3\mathrm{D}}=2\omega \gamma & [{\displaystyle {\int }_{{\theta }_0}^{{\theta }_{\mathrm{B}}}{{\displaystyle {\int }_{d_1}^R{\displaystyle {\int }_0^{x^{*}}\left(r_m+y\right)}}}^2}\cos \theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta \\ & +{\displaystyle {\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}{{\displaystyle {\int }_{d_2}^R{\displaystyle {\int }_0^{x^{*}}\left(r_m+y\right)}}}^2}\cos \theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta \\ & +{\displaystyle {\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}{{\displaystyle {\int }_{d_3}^R{\displaystyle {\int }_0^{x^{*}}\left(r_m+y\right)}}}^2}\cos \theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta \\ & +{\displaystyle {\int }_{{\theta }_{\mathrm{D}}}^{{\theta }_h}{{\displaystyle {\int }_{d_4}^R{\displaystyle {\int }_0^{x^{*}}\left(r_m+y\right)}}}^2}\cos \theta \mathrm{d}x\mathrm{d}y\mathrm{d}\theta ]\end{array}$$

(50)

where γ is the unit weight of soil. The upper limit of integration $x^{*}$ of the third integral for the above triple integration Equations can be written as $x^{*}=\sqrt{R^2-y^2}$. Hence, the overall soil weight work after adding the intermediate 2D plane-strain section can be given as

$$\begin{array}{ll}{W}_{\gamma }& =W_{\gamma -3\mathrm{D}}+W_{\gamma -insert}\\ & =\omega \gamma r_0^4\left(g_3+g_4\right)\end{array}$$

(51)

where g₃ and g₄ are dimensionless expressions of the soil weight work for 3D section and 2D plane-strain section, respectively, shown as

$$\begin{array}{ll}{g}_3=& 2{\displaystyle {\int }_{{\theta }_0}^{{\theta }_{\mathrm{B}}}\left[\left(\frac{1}{8}{f}_2^2{f}_3-\frac{1}{4}{f}_3^3-\frac{2}{3}{f}_1{f}_3^2-\frac{1}{2}{f}_1^2{f}_3+\frac{2}{3}{f}_1{f}_2^2\right)\sqrt{f_2^2-f_3^2}+\left(\frac{1}{8}{f}_2^4+\frac{1}{2}{f}_1^2{f}_2^2\right)\arccos \frac{f_3}{f_2}\right]}\cos \theta \mathrm{d}\theta \\ & +2{\displaystyle {\int }_{{\theta }_{\mathrm{B}}}^{{\theta }_{\mathrm{C}}}\left[\left(\frac{1}{8}{f}_2^2{f}_4-\frac{1}{4}{f}_4^3-\frac{2}{3}{f}_1{f}_4^2-\frac{1}{2}{f}_1^2{f}_4+\frac{2}{3}{f}_1{f}_2^2\right)\sqrt{f_2^2-f_4^2}+\left(\frac{1}{8}{f}_2^4+\frac{1}{2}{f}_1^2{f}_2^2\right)\arccos \frac{f_4}{f_2}\right]}\cos \theta \mathrm{d}\theta \\ & +2{\displaystyle {\int }_{{\theta }_{\mathrm{C}}}^{{\theta }_{\mathrm{D}}}\left[\left(\frac{1}{8}{f}_2^2{f}_5-\frac{1}{4}{f}_5^3-\frac{2}{3}{f}_1{f}_5^2-\frac{1}{2}{f}_1^2{f}_5+\frac{2}{3}{f}_1{f}_2^2\right)\sqrt{f_2^2-f_5^2}+\left(\frac{1}{8}{f}_2^4+\frac{1}{2}{f}_1^2{f}_2^2\right)\arccos \frac{f_5}{f_2}\right]}\cos \theta \mathrm{d}\theta \\ & +2{\displaystyle {\int }_{{\theta }_{\mathrm{D}}}^{{\theta }_h}\left[\left(\frac{1}{8}{f}_2^2{f}_6-\frac{1}{4}{f}_6^3-\frac{2}{3}{f}_1{f}_6^2-\frac{1}{2}{f}_1^2{f}_6+\frac{2}{3}{f}_1{f}_2^2\right)\sqrt{f_2^2-f_6^2}+\left(\frac{1}{8}{f}_2^4+\frac{1}{2}{f}_1^2{f}_2^2\right)\arccos \frac{f_6}{f_2}\right]}\cos \theta \mathrm{d}\theta \end{array}$$

(52)

$$g_4=\frac{b}{H}{\eta }_1\left(f_7-f_8-f_9-f_{10}-f_{11}\right)$$

(53)

in which $f_7$, $f_8$, $f_9$, $f_{10}$, and $f_{11}$ are dimensionless expressions are written as

$$f_7=\frac{\left[\left(3\tan {\phi }_t\cos {\theta }_h+\sin {\theta }_h\right)e^{3\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}-3\tan {\phi }_t\cos {\theta }_0-\sin {\theta }_0\right]}{3\left(1+9{\tan }^2{\phi }_t\right)}$$

(54)

$$f_8=\frac{1}{3}{\eta }_2\sin {\theta }_0\left(\cos {\theta }_0-\frac{1}{2}{\eta }_2\right)$$

(55)

$$f_9=\frac{1}{3}{\xi }_1{\eta }_1\left(\cos {\theta }_0-{\eta }_2+\sin {\theta }_0\cot {\beta }_1\right)\left(\cos {\theta }_0-{\eta }_2-\frac{1}{2}{\xi }_1{\eta }_1\cot {\beta }_1\right)$$

(56)

$$f_{10}=\frac{1}{3}{\lambda }_a{\eta }_1\left(\sin {\theta }_0+{\xi }_1{\eta }_1\right)\left(\cos {\theta }_0-{\eta }_2-{\xi }_1{\eta }_1\cot {\beta }_1-\frac{1}{2}{\lambda }_a{\eta }_1\right)$$

(57)

$$f_{11}=\frac{1}{3}{\xi }_2{\eta }_1{e}^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\frac{\sin \left({\theta }_h+{\beta }_2\right)}{\sin {\beta }_2}\left[e^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\cos {\theta }_h+\frac{1}{2}{\xi }_2{\eta }_1\cot {\beta }_2\right]$$

(58)

In this paper, the effects of seismic force on the stability of two-step slopes are evaluated based on the modified pseudo-dynamic approach. Since the seismic acceleration keeps consistent on the same horizontal plane, the seismic force work can be solved using the layer-wise summation method.

As illustrated in Figure 4a, the planar right-angle coordinate system x-E-z is founded with E as the origin. P is the intersection of velocity discontinuity surface and symmetry plane for the 3D failure mechanism at arbitrary height, while P′ is the intersection of slope surface and symmetry plane for the 3D failure mechanism at the corresponding level. The polar angles of P and P′ are denoted as $\alpha$ and ${\alpha }^{\prime }$, respectively. Hence, the value on the z-axis corresponding to the horizontal surface PP’ can be derived as

$$z\left(\alpha \right)=r_0{e}^{\left({\theta }_h-{\theta }_0\right)\tan {\phi }_t}\sin {\theta }_h-r_p\sin \alpha$$

(59)

in which r_P is the distance from point P to point O, and can be written as

$$r_p=r_0{e}^{\left(\alpha -{\theta }_0\right)\tan {\phi }_t}$$

(60)

Bringing Equation (59) into Equation (6), the horizontal seismic acceleration a_h of arbitrary height in the two-step slope can be obtained.

In accordance with the layer-wise summation method, the seismic force work in the volume unit dV formed by the vertical increment dz and the horizontal surface PP′ within the slope failure mechanism, as illustrated in Figure 4b, can be given as

$$\mathrm{d}{W}_s=\frac{a_h}{g}\gamma v\sin \theta \mathrm{d}V$$

(61)

in which v is the velocity vector, and dV is the volume unit. According to the geometric relations, the volume units corresponding to the 3D section of two wings and the intermediate 2D plane-strain section inserted in the two-step slope failure mechanism can be deduced as, separately,

$$\mathrm{d}{V}_{3\mathrm{D}}=l\left(\theta \right)\frac{\sin \alpha }{{\sin }^2\theta }{r}_p\mathrm{d}\theta \mathrm{d}z$$

(62)

$$\mathrm{d}{V}_{insert}=b\frac{\sin \alpha }{{\sin }^2\theta }{r}_p\mathrm{d}\theta \mathrm{d}z$$

(63)

$l\left(\theta \right)$ in Equation (62) represents the width of the 3D portion for the stepped slope at any polar angle $\theta \left({\theta }_0\le \theta \le {\theta }_h\right)$, which can be derived as

$$l\left(\theta \right)=2\sqrt{R^2-d^2}$$

(64)

$$d=r_p\frac{\sin \alpha }{\sin \theta }-r_m=r_0{f}_{12}$$

(65)

where $f_{12}$ is the dimensionless expression, shown as

$$f_{12}=e^{\left(\alpha -{\theta }_0\right)\tan {\phi }_t}\frac{\sin \alpha }{\sin \theta }-f_1$$

(66)

In addition, the vertical increment dz can be further written as

$$\mathrm{d}z=\frac{\cos \left(\alpha -{\phi }_t\right)}{\cos {\phi }_t}{r}_p\mathrm{d}\alpha$$

(67)

Hence, the seismic force work in the 3D section of two wings and the intermediate 2D plane-strain section can be listed as the following double integral expressions, respectively

$$W_{s-3D}=\omega \gamma {\displaystyle {\int }_{{\theta }_0}^{{\theta }_h}{\displaystyle {\int }_{\alpha }^{{\alpha }^{\prime }}l(\theta )}}{r}_p^3\frac{{\sin }^2\alpha }{{\sin }^2\theta }\frac{\cos \left(\alpha -{\phi }_t\right)}{\cos {\phi }_t}\frac{a_h}{g}\mathrm{d}\alpha \mathrm{d}\theta$$

(68)

$$W_{s-insert}=\omega \gamma {\displaystyle {\int }_{{\theta }_0}^{{\theta }_h}{\displaystyle {\int }_{\alpha }^{{\alpha }^{\prime }}b{r}_p^3}}\frac{{\sin }^2\alpha }{{\sin }^2\theta }\frac{\cos \left(\alpha -{\phi }_t\right)}{\cos {\phi }_t}\frac{a_h}{g}\mathrm{d}\alpha \mathrm{d}\theta$$

(69)

The upper limit of integration ${\alpha }^{\prime }$ for the second integral in the above two equations is the function of $\alpha$, whose segmental function expressions are deduced as

$${\alpha }^{\prime }=\left\{\begin{array}{cc}\arccos \frac{\frac{r_0\sin {\theta }_0}{\tan {\theta }_B}-\frac{r_p\sin \alpha -r_0\sin {\theta }_0}{\tan {\beta }_1}}{\sqrt{{\left(\frac{r_0\sin {\theta }_0}{\tan {\theta }_B}-\frac{r_p\sin \alpha -r_0\sin {\theta }_0}{\tan {\beta }_1}\right)}^2+r_p^2{\sin }^2\alpha }}\hfill & \hfill {\theta }_0\le \alpha \le {\theta }_F\\ \arccos \frac{\frac{r_0\sin {\theta }_0+{\xi }_{1}H}{\tan {\theta }_D}-\frac{r_p\sin \alpha -r_0\sin {\theta }_0-{\xi }_{1}H}{\tan {\beta }_2}}{\sqrt{{\left(\frac{r_0\sin {\theta }_0+{\xi }_{1}H}{\tan {\theta }_D}-\frac{r_p\sin \alpha -r_0\sin {\theta }_0-{\xi }_{1}H}{\tan {\beta }_2}\right)}^2+r_p^2{\sin }^2\alpha }}\hfill & \hfill {\theta }_F<\alpha \le {\theta }_h\end{array}\right.$$

(70)

After simplification to Equations (68) and (69), the total seismic force work in the two-step slope failure mechanism can be shown as

$$\begin{array}{cc}{W}_s& =W_{s-3D}+W_{s-\mathrm{insert}}\\ & =\omega \gamma k_h{r}_0^4\left(g_5+g_6\right)\end{array}$$

(71)

in which g₅ and g₆ are dimensionless expressions of the seismic force work for the 3D section and the 2D plane-strain section, respectively, given as

$$g_5=2{\displaystyle {\int }_{{\theta }_0}^{{\theta }_h}{\displaystyle {\int }_{\alpha }^{{\alpha }^{\prime }}{e}^{3\left(\alpha -{\theta }_0\right)\tan {\phi }_t}}}\frac{{\sin }^2\alpha }{{\sin }^2\theta }\frac{\cos \left(\alpha -{\phi }_t\right)}{\cos {\phi }_t}\frac{a_h}{k_{h}g}\sqrt{f_2^2-f_{12}^2}\mathrm{d}\alpha \mathrm{d}\theta$$

(72)

$$g_6=\frac{b}{H}{\eta }_1{\displaystyle {\int }_{{\theta }_0}^{{\theta }_h}{\displaystyle {\int }_{\alpha }^{{\alpha }^{\prime }}{e}^{3\left(\alpha -{\theta }_0\right)\tan {\phi }_t}}}\frac{{\sin }^2\alpha }{{\sin }^2\theta }\frac{\cos \left(\alpha -{\phi }_t\right)}{\cos {\phi }_t}\frac{a_h}{k_{h}g}\mathrm{d}\alpha \mathrm{d}\theta$$

(73)

The overall external force work is equal to the sum of the seismic force work and the soil weight work, which is

$$W=W_{\gamma }+W_s$$

(74)

3.4. Factor of Safety

In order to make the expression of the final derived safety factor more concise, two dimensionless parameters are introduced here, namely, the cohesion coefficient ${\lambda }_c$ and the step-width coefficient ${\lambda }_a$, which are defined as follows

$${\lambda }_{ct}=c_t/\left(\gamma H\right)$$

(75)

$${\lambda }_a=a/H$$

(76)

In accordance with Equation (3), and combining with the external force work and the internal energy dissipation, the factor of safety FS can be derived as

$$FS=\frac{{\lambda }_{ct}{\kappa }_1\cot {\phi }_t\left(g_1+g_2\right)}{g_3+g_4+k_h\left(g_5+g_6\right)}$$

(77)

Furthermore, the geometric parameters must meet the following conditions in order for the failure mechanism to be kinematically admissible.

$$\{\begin{array}{l}0<{\theta }_0<{\theta }_{\mathrm{B}}<{\theta }_{\mathrm{C}}\le {\theta }_{\mathrm{D}}<{\theta }_h<\pi \hfill \\ 0<{\phi }_t<2/\pi \hfill \\ 0<r_0^{\prime }/r_0<1\hfill \\ 0<{\xi }_1<1\hfill \\ \begin{array}{l}0 < \mathrm{t} < \mathrm{T}\\ 0<b/H+B_{\max }^{\prime }/H\le B/H\end{array}\hfill \end{array}$$

(78)

After specifying the remaining parameters, an optimization program can be utilized to discover the minimum over total feasible kinematical solutions by optimizing the objective value FS in Equation (77) regarding the five variables ${\theta }_0$, ${\theta }_h$, ${\phi }_t$, $r_0^{\prime }/r_0$, and $t$.

4. Results and Discussions

4.1. Comparisons

Michalowski and Drescher [12] investigated the stability for 3D single-step slopes without considering the soil’s nonlinear features and seismic effects, and came up with the stability number $N_s$ ($N_s=\gamma H_{cr}/c_t$), from which the critical height for 3D single-step slopes can be derived. Where the slope with the critical height is about to collapse, the safety factor is FS = 1 at this time. To demonstrate the applicability of the current approach, the cohesion coefficient ${\lambda }_c$ deduced from the stability number $N_s$ ($N_s=1/{\lambda }_c$) is introduced in Equation (77) to calculate the safety factor FS, corresponding to ${\lambda }_a=0.0$, $m=1$, $k_h=0.0$, where the double steps effect, non-linearity effect, and seismicity effect are eliminated. The detailed data are shown in

Table 1. Comparisons of both the present outcomes and the solutions of Michalowski and Drescher [12], corresponding to ${\lambda }_a=0.0$, $m=1$, $k_h=0.0$.

B/H	Results	${\mathit{\beta }}_1={\mathit{\beta }}_2$
		$45°$	$60°$	$75°$	$90°$
1.0	Ns by Michalowski and Drescher	54.850	23.835	14.701	11.028
	FS by the present method	0.9893	0.9823	0.9993	0.9849
1.5	Ns by Michalowski and Drescher	46.845	20.773	12.976	8.935
	FS by the present method	0.9922	0.9743	0.9754	0.9924
2.0	Ns by Michalowski and Drescher	42.732	19.103	12.109	8.604
	FS by the present method	0.9994	0.9919	0.9796	0.9598
3.0	Ns by Michalowski and Drescher	39.956	17.873	11.184	7.974
	FS by the present method	1.0019	1.0010	0.9955	0.9925
5.0	Ns by Michalowski and Drescher	37.994	17.063	10.628	7.266
	FS by the present method	1.0063	1.0033	0.9986	0.9939
10.0	Ns by Michalowski and Drescher	36.703	16.527	10.265	6.944
	FS by the present method	1.0085	1.0026	1.0012	1.0004

, and it is evident that the result is pretty near to one, which verifies the effectiveness of the present method in this scenario.

In the research of seismic effects, the fundamental distinction between the modified pseudo-dynamic method and the existing pseudo-dynamic method lies in the various assumptions on seismic acceleration. The horizontal seismic acceleration in the original pseudo-dynamic analysis is written as

$$a_h=\left[1+\frac{z}{H}\left(f-1\right)\right]\cdot k_{h}g\cdot \sin \omega \left(t-\frac{z}{V_s}\right)$$

(79)

where $f$ is the amplification factor of the seismic wave. According to the above equation, after eliminating the time and position variables and setting $f=1.0$, the horizontal seismic acceleration will be simplified to $k_{h}g$, namely, the seismic acceleration in the pseudo-static analysis. Similarly, the seismic acceleration in the modified pseudo-dynamic analysis represented by Equation (6) will also be simplified to $k_{h}g$ by removing the time and position variables and setting the damping ratio $D=0.0$. Thus, with the same input parameters set, the results of the modified pseudo-dynamic method and the original pseudo-dynamic method should be theoretically consistent, which can demonstrate the modified pseudo-dynamic method’s efficiency in the seismic analysis of 3D slopes.

Table 2. Comparisons between the modified pseudo-dynamic solutions and the original pseudo-dynamic solutions, corresponding to $H=15 \mathrm{m}$, ${\beta }_1={\beta }_2=45°$, ${\lambda }_a=0.0$, $m=1.6$, $c_0=25 \mathrm{kPa}$, ${\sigma }_t=30 \mathrm{kPa}$, $T=0.1 \mathrm{s}$, $V_s=\infty$.

B/H	Modified Pseudo-Dynamic Method			Original Pseudo-Dynamic Method
	kh = 0.2	kh = 0.5	kh = 0.8	kh = 0.2	kh = 0.5	kh = 0.8
1.5	1.844	1.747	1.693	1.855	1.757	1.688
3.0	1.594	1.527	1.437	1.589	1.517	1.429
5.0	1.501	1.440	1.353	1.502	1.428	1.358
10.0	1.445	1.378	1.308	1.445	1.381	1.322
2D	1.412	1.330	1.276	1.423	1.333	1.277

shows the outcomes of the two pseudo-dynamic approaches, and they are strikingly similar, which verifies the applicability of the proposed method once more. The fundamental input parameters are as follows: $H=15 \mathrm{m}$, ${\beta }_1={\beta }_2=45°$, ${\lambda }_a=0.0$, $m=1.6$, $c_0=25 \mathrm{kPa}$, ${\sigma }_t=30 \mathrm{kPa}$, $T=0.1 \mathrm{s}$, $V_s=\infty$.

4.2. Parametric Effects Discussions

In accordance with the Equation (77), nonlinear properties of soil medium, geometric parameters of two-step slopes, and seismic parameters in the modified pseudo-dynamic analysis are all factors that affect the safety factor FS for 3D two-step slopes, the effects of which are depicted in Figure 5, Figure 6, Figure 7 and Figure 8, respectively, with the corresponding basic parameters: $B/H=3$, $H=15 \mathrm{m}$, ${\xi }_1=0.4$, ${\beta }_1=45°$, ${\beta }_2=60°$, ${\lambda }_a=0.1$, $m=1.6$, $c_0=25 \mathrm{kPa}$, ${\sigma }_t=30 \mathrm{kPa}$, $T=0.1 \mathrm{s}$, $D=0.0$, $V_s=150 \mathrm{m}/\mathrm{s}$, $k_h=0.2$.

As shown in Figure 5, the nonlinear coefficient $m$ and axial tensile strength ${\sigma }_t$ are inversely correlated with the safety factor FS; i.e., the larger the nonlinear coefficient and axial tensile strength lead to the smaller FS. Nonetheless, the initial cohesion c₀ is positively connected with FS, and the magnitude of the effect is stronger, as the FS varies from 0.4 to 2.3 corresponding to the initial cohesion value ranging from 12.5 to 32.5 kPa.

Then, Figure 6 illustrates the interactions between the nonlinear parameters. It can be observed that the trend of FS remains constant with ${\sigma }_t$ and $c_0$, but when $m$ is small, the slope stability changes dramatically. Moreover, when nonlinear parameters $m$, ${\sigma }_t$ and $c_0$ are fixed, FS decreases as B/H rises. It means that the stability of 3D slopes is better compared with 2D slopes at the same other parameters. This also demonstrates that treating the 3D slope simply as the 2D case in real engineering is overly conservative.

The effects of geometric parameters on slope stability are shown in Figure 7. It is obvious that the depth coefficient ${\xi }_1$ and step-width coefficient ${\lambda }_a$ are positively correlated with FS, while the inclined angles ${\beta }_1$ and ${\beta }_2$ are negatively correlated with FS. It is worth noting that the FS variation range corresponding to ${\beta }_2$ is larger compared with ${\beta }_1$ in the same range of 45~75°, indicating that ${\beta }_2$ has a stronger effect on slope stability. This has some significance for the design of two-step slopes. Furthermore, for the geometric parameters ${\xi }_1$, ${\lambda }_a$, ${\beta }_1$, and ${\beta }_2$, when the value of B/H is changed from 2 to 4, there is a significant drop in FS. With a rising B/H, the FS falls slowly and gravitates to 2D solutions.

In Figure 8, the influence of pseudo-dynamic parameters varying with B/H on slope stability is described. The variation pattern of FS with B/H is consistent with the foregoing conclusions; although, FS variations with the modified pseudo-dynamic parameters are all minor, not surpassing 5%.

By comparing the results obtained by the original pseudo-dynamic method and the previous relevant research literature, the reliability of the proposed method has been verified, repeatedly. The results obtained from the sensitivity tests of geometric parameters, pseudo-dynamic parameters, and nonlinear parameters of soil have considerable significance for the design and safety evaluation of three-dimensional stepped slopes in current practical projects.

The design mode of a two-order slope can resist the negative impacts of seismic events on slope stability to a certain extent. In addition, the greater the step width, the greater the depth of the step level, and the smaller the inclination angle ${\beta }_1$, the better the slope stability.

5. Conclusions

The present paper puts forward kinematic analytical solutions for assessing seismic 3D two-step slopes subjected to non-linearity in soils that couples the modified pseudo-dynamic method with the upper bound theorem. The 3D horn-like rotational failure mechanism is employed to solve internal energy dissipation and external force work, in which seismic work is considered in addition to soil weight work and obtained by the layer-wise summation method. Based on the force-increase technique, the safety factor of 3D two-step slopes can be derived more readily, and the results of this derivation can be applied to geotechnical engineering.

To verify the reliability of the approach in this paper, the results of Michalowski and Drescher [12] are extrapolated for the calculation of FS, which are extremely close to one. Additionally, the modified pseudo-dynamic solutions are compared to the original pseudo-dynamic solutions in order to evaluate the modified pseudo-dynamic method’s efficacy in the seismic stability analysis of 3D two-step slopes.

In this paper, the sensitivity tests are performed to investigate the impacts of various factors on slope stability. Among nonlinear parameters, the initial cohesion $c_0$ has a larger magnitude of influence on FS. When the nonlinear coefficient $m$ is small, FS varies more with the initial cohesion $c_0$ and axial tensile strength ${\sigma }_t$. Among the geometric parameters, ${\beta }_2$ has a stronger effect on slope stability compared with ${\beta }_1$. This has some significance for the design of two-step slopes. When the value of B/H is changed from 2 to 4, there is a significant drop in FS. With a rising B/H, the FS falls slowly and gravitate to 2D solutions. It is significant to mention that FS varies slightly with the pseudo-dynamic parameters. The above conclusions have a certain meaning for the design, construction, and stability evaluation for 3D two-step slopes subjected the seismic effects.

In the future, the authors will further study the stability of the 3D two-step slope in the following aspects: (1) The support methods of the 3D slope affected by seismic effects, such as prestressed anchor cable, anti-sinking pile reinforcements, etc.; (2) Special conditions of slope medium and complex engineering conditions, for instance, non-uniformity of soils, infiltration of surface water, etc.