On the seismic stability of soil slopes containing dual weak layers: true failure load assessment by finite-element limit-analysis

Abstract

Seismic stability analyses of soil slopes in the presence of weak interlayers are rather challenging within the framework of plasticity theory, due to the construction of kinematically admissible velocity fields and statically allowable stress fields at limit state. Finite-element limit-analysis procedures including finite-element upper-bound (FEUB) and finite-element lower-bound (FELB) approach are introduced in this study, retaining the merits of FEM and limit analysis theory to tackle above issues. Incorporating modified pseudo-dynamic approach, seismic slope stability analyses are transformed to linear programming models, in terms of lower- and upper-bound formulations. Pseudo-static and modified pseudo-dynamic solutions of the factor of safety (FoS) are sought through optimization with an interior-point algorithm. An appealing merit of the proposed procedure is that both lower and upper bounds are searched, aiding to better estimate the true solution of FoS. Limit equilibrium and Abaqus are applied to validate FEUB and FELB results. Effects of dual weak interlayers’ position and dimension on seismic slope stability are investigated. Critical failure surface and velocity field are plotted by post-processing, demonstrating a rotational-translational failure mechanism. Based on less than 5% difference between lower- and upper-bound solutions, the proposed procedure is capable of providing a reliable guidance for slope design and assessment.

Appendices

Appendix A: Finite-element upper-bound analysis

$$A_{11}^{e} = \frac{1}{{2A^{e} }}\left[ {\begin{array}{*{20}c} {b_{1}^{e} } & 0 & {b_{2}^{e} } & 0 & {b_{3}^{e} } & 0 \\ 0 & {c_{1}^{e} } & 0 & {c_{2}^{e} } & 0 & {c_{3}^{e} } \\ {c_{1}^{e} } & {b_{1}^{e} } & {c_{2}^{e} } & {b_{2}^{e} } & {c_{3}^{e} } & {b_{3}^{e} } \\ \end{array} } \right]$$

(A1)

$$A_{12}^{e} = \left[ {\begin{array}{*{20}c} {M_{1}^{{}} } & {M_{2}^{{}} } & \cdots & {M_{k}^{{}} } & \cdots & {M_{p}^{{}} } \\ {N_{1}^{{}} } & {N_{2}^{{}} } & \cdots & {N_{k}^{{}} } & \cdots & {N_{p}^{{}} } \\ {R_{1}^{{}} } & {R_{2}^{{}} } & \cdots & {R_{k}^{{}} } & \cdots & {R_{p}^{{}} } \\ \end{array} } \right]$$

(A2)

$$A_{23}^{d} = \left[ {\begin{array}{*{20}c} {T_{1}^{d} } & 0 \\ 0 & {T_{1}^{d} } \\ \end{array} } \right],\;\;T_{1}^{d} = \left[ {\begin{array}{*{20}c} {\sin \theta^{d} } & { - \cos \theta^{d} } & { - \sin \theta^{d} } & {\cos \theta^{d} } \\ { - \cos \theta^{d} } & { - \sin \theta^{d} } & {\cos \theta^{d} } & {\sin \theta^{d} } \\ \end{array} } \right]$$

(A3)

$$A_{24}^{d} = \left[ {\begin{array}{*{20}c} {T_{2}^{d} } & 0 \\ 0 & {T_{2}^{d} } \\ \end{array} } \right],\;\;T_{2}^{d} = \left[ {\begin{array}{*{20}c} {\tan \varphi } & {\tan \varphi } \\ 1 & { - 1} \\ \end{array} } \right]$$

(A4)

$$A_{3}^{b} = \left[ {\begin{array}{*{20}c} {T_{3}^{b} } & 0 \\ 0 & {T_{3}^{b} } \\ \end{array} } \right],\;\;T_{3}^{b} = \left[ {\begin{array}{*{20}c} {\cos \theta^{b} } & {\sin \theta^{b} } \\ { - \sin \theta^{b} } & {\cos \theta^{b} } \\ \end{array} } \right]$$

(A5)

$$B_{3}^{b} = \left[ {\begin{array}{*{20}c} {\overline{u}_{s1}^{b} } & {\overline{v}_{n1}^{b} } \\ \end{array} \;\;\;\;\begin{array}{*{20}c} {\overline{u}_{s2}^{b} } & {\overline{v}_{n2}^{b} } \\ \end{array} } \right]^{T}$$

(A6)

where $A^{e}$ is the area of a random element e (Fig. 1a), $b_{1}^{e} = y_{2}^{e} - y_{3}^{e}$, $b_{2}^{e} = y_{3}^{e} - y_{1}^{e}$, $b_{3}^{e} = y_{1}^{e} - y_{2}^{e}$, $c_{1}^{e} = - x_{2}^{e} + x_{3}^{e}$,$c_{2}^{e} = - x_{3}^{e} + x_{1}^{e}$, $c_{3}^{e} = - x_{1}^{e} + x_{2}^{e}$, with ($x_{1}^{e}$, $y_{1}^{e}$), ($x_{2}^{e}$, $y_{2}^{e}$), and ($x_{3}^{e}$, $y_{3}^{e}$) being the coordinates of three nodes in element e. For an external linearization of the MC failure criterion with p planes, $M_{k}^{{}} = \cos (2k\pi /p) + \sin \varphi$,$N_{k}^{{}} = - \cos (2k\pi /p) + \sin \varphi$,$R_{k}^{{}} = 2\sin (2k\pi /p)$, $k = 1,\;2, \cdots ,\;p$. $\theta^{d}$ is the angle of velocity discontinuity (e.g., edge d) inclined to x-axis as shown in Fig. 1b. $\theta^{b}$ is the angle of a random boundary (e.g., boundary b) with respect to x-axis as shown in Fig. 1c. $\overline{u}_{si}^{b}$ and $\overline{v}_{ni}^{b}$ denote the prescribed tangential and normal nodal velocity at node i (i = 1, 2), respectively.

$$C_{i1}^{e} = 2A^{e} c\cos \varphi [\underbrace {{\begin{array}{*{20}c} 1 & 1 & \cdots & 1 \\ \end{array} }}_{p}]$$

(A7)

$$C_{i2}^{d} = \frac{1}{2}cl^{d} [\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} ]$$

(A8)

$$C_{e1}^{e} = \frac{{A^{e} }}{3}\left[ {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {\begin{array}{*{20}c} \gamma & 0 \\ \end{array} } & \gamma & 0 & \gamma \\ \end{array} } \\ \end{array} } \right]$$

(A9)

$$C_{e2}^{q} = \frac{{l^{q} }}{2}\left[ {\begin{array}{*{20}c} 0 & {\overline{q}_{n} } & 0 & {\overline{q}_{n} } \\ \end{array} } \right]$$

(A10)

$$C_{e3}^{e} = \frac{{A^{e} \cdot \gamma }}{3}\left[ {\begin{array}{*{20}c} {k_{h} (t,y)} & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {k_{v} (t,y)} & {k_{h} (t,y)} \\ \end{array} } & {k_{v} (t,y)} & {k_{h} (t,y)} & {k_{v} (t,y)} \\ \end{array} } \\ \end{array} } \right]$$

(A11)

where $l^{d}$ denotes the length of a velocity discontinuity (e.g., d), $l^{q}$ is the length of an edge (e.g., q) where traction force $\overline{q}_{n}$ acts.

Appendix B: Finite-element lower-bound analysis

$$A_{1}^{e} = \frac{1}{{2A^{e} }}\left[ {\begin{array}{*{20}c} {b_{1} } & 0 & {c_{1} } & {b_{2} } & 0 & {c_{2} } & {b_{3} } & 0 & {c_{3} } \\ 0 & {c_{1} } & {b_{1} } & 0 & {c_{2} } & {b_{2} } & 0 & {c_{3} } & {b_{3} } \\ \end{array} } \right]$$

(B1)

$$B_{1}^{e} = \left[ {\begin{array}{*{20}c} {X_{x}^{e} } & {X_{y}^{e} } \\ \end{array} } \right]^{T}$$

(B2)

$$A_{2}^{d} = \left[ {\begin{array}{*{20}c} {T^{d} } & { - T^{d} } & 0 & 0 \\ 0 & 0 & {T^{d} } & { - T^{d} } \\ \end{array} } \right],\;\;\;T^{d} = \left[ {\begin{array}{*{20}c} {\sin {}^{2}\theta^{d} } & {\cos {}^{2}\theta^{d} } & { - \sin 2\theta^{d} } \\ { - \frac{1}{2}\sin 2\theta^{d} } & {\frac{1}{2}\sin 2\theta^{d} } & {\cos 2\theta^{d} } \\ \end{array} } \right]$$

(B3)

$$A_{3}^{b} = \left[ {\begin{array}{*{20}c} {T^{b} } & 0 \\ 0 & {T^{b} } \\ \end{array} } \right],\;\;\;T^{b} = \left[ {\begin{array}{*{20}c} {\sin {}^{2}\theta^{b} } & {\cos {}^{2}\theta^{b} } & { - \sin 2\theta^{b} } \\ { - \frac{1}{2}\sin 2\theta^{b} } & {\frac{1}{2}\sin 2\theta^{b} } & {\cos 2\theta^{b} } \\ \end{array} } \right]$$

(B4)

$$B_{3}^{b} = \left[ {\begin{array}{*{20}c} {q_{n1}^{b} } & {t_{s1}^{b} } & {q_{n2}^{b} } & {t_{s2}^{b} } \\ \end{array} } \right]^{T}$$

(B5)

$$A_{4}^{i} = \left[ {\begin{array}{*{20}c} {m_{1} } & {m_{2} } & \cdots & {m_{k} } & \cdots & {m_{p} } \\ {n_{1} } & {n_{2} } & \cdots & {n_{k} } & \cdots & {n_{p} } \\ {r_{1} } & {r_{2} } & \cdots & {r_{k} } & \cdots & {r_{p} } \\ \end{array} } \right]^{T}$$

(B6)

$$B_{4}^{i} = 2c\cos \varphi \cos (\pi /p)[\underbrace {{\begin{array}{*{20}c} 1 & 1 & \cdots & 1 \\ \end{array} }}_{p}]^{T}$$

(B7)

where $A^{e}$ is the area of a random element e (Fig. 2a), $b_{1}^{e} = y_{2}^{e} - y_{3}^{e}$, $b_{2}^{e} = y_{3}^{e} - y_{1}^{e}$, $b_{3}^{e} = y_{1}^{e} - y_{2}^{e}$, $c_{1}^{e} = - x_{2}^{e} + x_{3}^{e}$,$c_{2}^{e} = - x_{3}^{e} + x_{1}^{e}$, $c_{3}^{e} = - x_{1}^{e} + x_{2}^{e}$, with ($x_{1}^{e}$, $y_{1}^{e}$), ($x_{2}^{e}$, $y_{2}^{e}$), and ($x_{3}^{e}$, $y_{3}^{e}$) being the coordinates of three nodes in element e. $X_{x}^{e} ,\;X_{y}^{e}$ are body stress components in x- and y-direction. $\theta^{d}$ is the angle of interface d with respect to x-axis (Fig. 2b). $\theta^{b}$ is the angle of edge b inclined to x-axis (Fig. 2c). $q_{n1}^{b} ,\;t_{s1}^{b}$ ($q_{n2}^{b} ,\;t_{s2}^{b}$) represent nodal normal and shear stresses at the node 1 (node 2) of edge b. For an internal linearization of the Mohr-Coulomb criterion with p planes, $m_{k} = \cos (2k\pi /p) + \sin \varphi \cos (\pi /p)$,$n_{k} = - \cos (2k\pi /p) + \sin \varphi \cos (\pi /p)$, $r_{k} = 2\sin (2k\pi /p)$,$(k = 1,2, \cdots ,p)$.