Stability analysis of layered slopes in unsaturated soils

Abstract

This study presents stability analyses of layered soil slopes in unsaturated conditions and uses a limit equilibrium method to determine the factor of safety involving suction stress of unsaturated soil. One-dimensional steady infiltration and evaporation conditions are considered in the stability analyses. An example of a two-layered slope in clay and silt is selected to verify the used method by comparing with the results of other methods. Parametric analyses are conducted to explore the influences of the matric suction on the stability of layered soil slopes. The obtained results show that larger suction stress provided in unsaturated clay dominates the stability of the layered slopes. Therefore, the location and thickness of the clay layer have significant influences on slope stability. As the water level decreases, the factor of safety reduces and then increases gradually in most cases. Infiltration/evaporation can obviously affect the stability of unsaturated layered slopes, but their influences depend on the soil property and thickness of the lower soil layer.

Graphical abstract

Keywords

slope stability / suction stress / unsaturated soil / layered slope / limit equilibrium

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Appendix

The soil properties of the layered slopes could affect the location of the potential sliding surface, which yields to the complexity of the calculation on the moment of the self-weight gravity, M_G,i. Here, a two-layered soil slope is considered as the typical example, as shown in Fig. A1. The intersection between the sliding surface and the horizontal interface of the layered soils is Point F, at which is the angle β_F. Point E is the projection of the crest Point X on the sliding surface and the corresponding angle is β_E. The total moment M_G,i can be divided into five parts, i.e., M_G,1-1, M_G,1-2, M_G,2-1, M_G,2-2, and M_G,2-3.

Fig.A1 Different calculations on the moment of the self-weight gravity when (a) β_F > β_E and (b) β_F ≤ β_E.

The moments M_G,1-1 and M_G,1-2 for the first soil layer can be expressed as:

(A1)\begin{array}{}\text{(A1)}& \begin{array}{rl}{M}_{\mathrm{G},1\text{-}1}={\gamma }_1{\displaystyle {\int }_{{\beta }_{\text{C}}}^{{\beta }_{\text{D}}}}& (A_1{\text{e}}^{-{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\theta +A_1{\text{e}}^{-{\psi }_{\text{m,}1}\beta }\mathrm{c}\mathrm{o}\mathrm{s}\beta )\\ & A_1^2{\text{e}}^{-2{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},1}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta ,\end{array}\end{array}

(A2)\begin{array}{}\text{(A2)}& \begin{array}{rl}{M}_{\mathrm{G},1\text{-}2}={\gamma }_1{\displaystyle {\int }_{{\beta }_{\text{D}}}^{{\beta }_{\text{F}}}}& (H_1-y_{\text{c}}+A_1{\text{e}}^{-{\psi }_{\text{m},1}\beta }\mathrm{c}\mathrm{o}\mathrm{s}\beta )\\ & A_1^2{\text{e}}^{-2{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},1}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta .\end{array}\end{array}

The relationship between Points E and F can result in two cases of the calculations on the moment of the upper-layer, as below:

Case 1: β_F > β_E

(A3)\begin{array}{}\text{(A3)}& \begin{array}{rl}{M}_{\mathrm{G},2\text{-}3}=& {\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{D}}}^{{\beta }_{\text{E}}}(}{A}_1{\text{e}}^{-{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\theta -H_1+y_{\text{c}})\\ & A_1^2{\text{e}}^{-2{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},1}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta ,\end{array}\end{array}

(A4)\begin{array}{}\text{(A4)}& M_{\mathrm{G},2\text{-}4}={\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{E}}}^{{\beta }_{\text{F}}}(}H-H_1)A_1^2{\text{e}}^{-2{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},1}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta ,\end{array}

(A5)\begin{array}{}\text{(A5)}& \begin{array}{rl}{M}_{\mathrm{G},2\text{-}5}=& {\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{F}}}^{{\beta }_{\text{B}}}(}H-y_{\text{c}}+A_2{\text{e}}^{-{\psi }_{\text{m},2}\beta }\mathrm{c}\mathrm{o}\mathrm{s}\beta )\\ & A_2^2{\text{e}}^{-2{\psi }_{\text{m},2}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},2}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta .\end{array}\end{array}

Case 2: β_F ≤ β_E

(A6)\begin{array}{}\text{(A6)}& \begin{array}{rl}{M}_{\mathrm{G},2\text{-}3}=& {\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{D}}}^{{\beta }_{\text{F}}}(}{A}_1{\text{e}}^{-{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\theta -H_1+y_c)\\ & A_1^2{\text{e}}^{-2{\psi }_{\text{m},1}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},1}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta ,\end{array}\end{array}

(A7)\begin{array}{}\text{(A7)}& \begin{array}{rl}{M}_{\mathrm{G},2\text{-}4}=& {\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{F}}}^{{\beta }_{\text{E}}}(}{A}_2{\text{e}}^{-{\psi }_{\text{m},2}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\theta +A_2{\text{e}}^{-{\psi }_{\text{m},2}\beta }\mathrm{c}\mathrm{o}\mathrm{s}\beta )\\ & A_2^2{\text{e}}^{-2{\psi }_{\text{m},2}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},2}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta ,\end{array}\end{array}

(A8)\begin{array}{}\text{(A8)}& \begin{array}{rl}{M}_{\mathrm{G},2\text{-}5}=& {\gamma }_2{\displaystyle {\int }_{{\beta }_{\text{E}}}^{{\beta }_{\text{B}}}(}H-y_{\text{c}}+A_2{\text{e}}^{-{\psi }_{\text{m},2}\beta }\mathrm{c}\mathrm{o}\mathrm{s}\beta )\\ & A_2^2{\text{e}}^{-2{\psi }_{\text{m},2}\beta }\mathrm{s}\mathrm{i}\mathrm{n}\beta \left|\mathrm{c}\mathrm{o}\mathrm{s}\beta -{\psi }_{\text{m},2}\mathrm{s}\mathrm{i}\mathrm{n}\beta \right|\mathrm{d}\beta .\end{array}\end{array}

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant Nos. 52078185 and 51878248).

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