Non-associated plasticity analysis of slope stability under steady unsaturated flow conditions

Abstract

This study presents an analytical framework of limit analysis for the ultimate load of unsaturated soil slopes with the non-associated plasticity. The method of characteristics combined with boundary value problems is applied to build the slip line field that satisfies the stress boundary and the velocity boundary simultaneously. According to the proposed approach, a value of the ultimate load of slopes can be directly addressed, which is within the narrow range determined by the upper bound (UB) and the lower bound (LB) solutions. The non-associated Mohr-Coulomb criterion is presented to investigate the effect of the matric suction and the dilation on slope stability. The comparison with the existing methods is conducted to validate the presented solutions. The results indicate that the effect of the flow condition and the dilation on the ultimate load of four types of soil slopes can be estimated accurately by the proposed approach.

Introduction

Many soil slopes are unsaturated in nature, and soil strength can be enhanced by the matric suction, which consequently contributes to slope stability (Fredlund and Rahardjo, 1993). To evaluate the characteristics of unsaturated soil, great efforts have been made to find a representation of the shear strength. Fredlund et al. (1978) proposed an extended Mohr-Coulomb (MC) criterion with the linear shear strength to simulate unsaturated soil. Then, Vanapalli et al., 1996, Fredlund et al., 1996 proposed the nonlinear shear strength of unsaturated soil based on the soil–water characteristic curves. Furthermore, Lu et al. (2010) predicted an effective MC model that can directly introduce the suction stress into the shear strength without new assumptions and parameters. Based on the MC criterions mentioned above, the effect of the matric suction on slope stability has been investigated widely (Lu and Godt, 2008, Vahedifard et al., 2016, Li and Yang, 2018, Yuan and Du, 2018).

Limit analysis is an attractive choice to estimate the stability of unsaturated soil slopes. It usually generates the upper bound (UB) and lower bound (LB) solutions by the statically admissible field and the kinematically admissible field (Michalowski, 1995, Zhou and Wang, 2017, Qin and Chian, 2017, Li and Jiang, 2020), respectively. Compared with the limit equilibrium method, the bound theorems of classical plasticity produce more meaningful solutions considering the constitutive relationship of soil. Moreover, unlike the numerical simulation, limit analysis is easier to operate and has a lower computational cost.

Chen (1975) made such comments: “The slip line solution is not necessarily the true solution. If the associated flow rule is employed and the resulting stress–strain rate equations can be integrated to yield a kinematically admissible velocity field, the slip line solution is an UB solution. If, in addition, the slip line stress field can be extended over the entire half-space of the soil domain such that the equilibrium equations, the stress boundary conditions and the yield condition are satisfied, the slip line solution is also a LB solution and is hence the actual solution.” Therefore, if the slip line field of unsaturated soil slopes that satisfies the stress boundary and the velocity boundary simultaneously can be found out, the accurate solution will be directly predicted, not the narrow range determined by the conventional limit analysis.

In terms of limit analysis, the plastic flow rule plays an important role (Mizuno and Chen, 1983). The conventional limit analysis usually applies the associated flow rule to estimate slope stability. In actual, soil, when modelled as a rigid-perfectly plastic solid, does not obey the associated flow rule, and the ultimate load is no longer unique (Drescher and Detournay, 1993). Meanwhile, slope stability involving the non-associated plasticity is difficult to resolve, especially when the friction angle is much larger than the dilatancy angle (Chen et al., 2019). To overcome the difficulty, Davis and Lee (1968) introduced the modified shear strength of MC model to convert the non-associated plasticity into the equivalently associated plasticity, and this approach has been widely adopted to evaluate slope stability with the non-associated flow rule (Yang and Huang, 2009, Oberhollenzer et al., 2018, Wang et al., 2019).

Although many studies mentioned above have made great contributions to the stability assessment of unsaturated soil slopes and the non-associated plasticity problem, the ultimate load of unsaturated soil slopes with the non-associated plasticity has not been addressed. The aim of this study is to search for a slip line field of unsaturated soil slopes with the non-associated plasticity using the method of characteristics combined with boundary value problems. The slip line field should satisfy the statically admissible stress field and the kinematically admissible velocity field simultaneously. Thus, the accurate solution could be directly predicted, which is more effective than the UB and LB solutions. Although the kinematically admissible velocity field is not strictly fulfilled here, the presented solution is between the UB and LB solutions. This solution can be considered to be near the accurate one. Meanwhile, the effective MC criterion proposed by Lu et al. (2010) is modified to analyze the suction stress of unsaturated soil, and the Davis approach (Davis and Lee, 1968) is applied to account for the non-associated flow rule. Then, the effect of the matric suction and the dilation on slope stability can be investigated, and the results are compared with the existing methods.