The discrete technique combined with the upper bound approach is applied to analyze the stability of soil slopes with multiple layers in this paper. Based on the geometrical relationship derived from the normality requirement, an algorithm is proposed to determine the coordinates of discrete points on the failure surface. Specifically, these points are generated from the initial point to the terminal point. After completion of the discrete sliding surface, the sliding block is sliced by the vertical slice technique. Equations with integral form for rates of internal energy dissipation and external work are then derived. The scheme for calculation of the safety factor is presented by means of the strength reduction method and the optimization algorithm. Comparison analysis based on four examples under various conditions demonstrates the validity of the proposed method in addressing stability problem of slopes with multiple layers.

Slope stability assessment is one of the most important issues for geotechnical engineers. In the framework of a nonlinear failure criterion, four different types of safety factors are presented and their relationships are investigated for soil slope in this study. The variational method is incorporated into the kinematic approach of limit analysis to assess the stability of soil slope with known geometric boundary, and seismic effects are considered. The seismic loads are simplified as external forces acting on the slope. A rotational failure mechanism is used to describe the sliding mode of slopes. The differential equations for the sliding surface and corresponding stress distribution are derived using the variational method and then are employed to generate the sliding surface via the fourth-order Runge-Kutta approach. To avoid the computational complexity, the energy-work balance equation of the kinematic approach, instead of the static equilibrium equation, is used to judge whether the state of the slope is critical. The safety factor is designed as the minimum factor that brings the slope in the limit state. Computational schemes are proposed to calculate the values of safety factors. The validity of the proposed approach is demonstrated through comparison with previous works. Finally, parametric study is conducted to further reveal the relationships among the four types of safety factors.

Based on the nonlinear failure criterion and modified tangential technique, the upper bound solutions of the critical supporting pressure on the deep tunnel face were obtained under pore water pressure conditions. The influence of parameters on the critical supporting pressure and collapse range was investigated according to the unlimited block failure mechanism. It was found that the upper bound solutions of the critical supporting pressure increase with the growth of the nonlinear coefficient and pore water pressure coefficient. The collapse range of the tunnel face scales out with the increase in the nonlinear coefficient and shrinks with an increasing pore water pressure coefficient. Moreover, with the increase in the nonlinear coefficient, the impact strength on critical supporting pressure and collapse range declines gradually. According to the calculated results, both the pore water pressure and nonlinear criterion factors have negative impacts on the stability of the tunnel face. Thus, more attention should be paid to these parameters to ensure face stability in deep tunnel construction.

Based on the nonlinear failure criterion and modified tangential technique, the upper bound solutions of supporting pressure on the deep tunnel face were obtained under pore water pressure conditions. The influence of parameters on the supporting pressure and collapse range was investigated according to the unlimited block failure mechanism. It is found that the upper bound solutions of supporting pressure increase with the growth of the nonlinear coefficient and pore water pressure coefficient. The collapse range of the tunnel face scales out with the increase of the nonlinear coefficient and shrinks when increasing the pore water pressure coefficient. Moreover, with the increase of the nonlinear coefficient, the impact strength on supporting pressure and collapse range declines gradually. According to the calculating results, it is manifest that both the pore water pressure and nonlinear criterion factors have negative impacts on the stability of the tunnel face. Thus, more attention should be paid to these parameters to ensure face stability in deep tunnel construction.