The direct application of the Hoek–Brown failure criterion to practical slope engineering is still an urgent problem. The slope geometries and earthquake effect need to be considered in the determination of linear Mohr–Coulomb (MC) strength parameters from the Hoek–Brown criteria for slope stability analysis. This study adopted the tangential method to construct a three-dimensional (3D) rotational failure mechanism using the Hoek–Brown failure criterion for homogeneous rock slopes undergoing earthquake. The quasi-static method was employed to treat the seismic action as an external seismic force in the work–energy equation of the limit analysis theory. Based on the numerical optimization, the least upper-bound solutions and equivalent MC strength parameters were derived with respect to different strength parameters and seismic loads. The influences of nonlinear strengths, geometric parameters and earthquake load on the equivalent MC strength parameters were thoroughly investigated. The results suggested that the nonlinear parameters have different influences on the equivalent MC parameters for general steep slopes and vertical slopes. The effects of nonlinear parameters on the equivalent MC parameters become obvious for vertical slopes. The disturbance factor D affects the equivalent MC parameters only for very steep slopes in fractured rock masses. Additionally, the effect of slope inclination on the equivalent MC parameters becomes obvious for slopes in fractured hard rock masses. The 3D effect of the rock slope on the equivalent MC parameters was found to be slight. Moreover, the impact of earthquakes on the approximate MC parameters becomes weaker for steeper rock slopes. The tables of approximate MC strength parameters were given for various slopes with different nonlinear strength parameters. The presented tables can provide certain references for practical slope engineering.

This paper presents a new algorithm for assessing the reliability of three-dimensional (3D) slope stability considering the spatial variability of soil based on the Particle Swarm Optimization (PSO) algorithm. First, a 3D random field is generated using the Karhunen–Loève (K-L) expansion method. Then, the simplified Bishop method of limit equilibrium is coupled with the PSO algorithm to calculate safety factors of the slope. Finally, the failure probability of the slope is determined using the Monte Carlo Simulation method. After validating the rationality of the proposed method through a typical case study, this paper offers an in-depth examination of how soil spatial variability affects the stability of 3D slopes. It is observed that, given identical soil correlation lengths, slope geometric parameters, and failure surface widths, the failure probability is positively correlated with soil spatial variability parameters, while the mean safety factor demonstrates an inverse relationship with these variability parameters. Additionally, the failure probability tends to increase as the soil correlation lengths increase, and it also escalates with the expansion of the failure surface width. In contrast, the mean safety factor exhibits an upward trend with the augmentation of the horizontal correlation length, while it diminishes progressively as the vertical correlation length grows, and it also shows a decline with the widening of the failure surface width. The proposed algorithm significantly improves computational efficiency while ensuring accuracy, making it suitable for the reliability analysis of three-dimensional slopes.

An analytical solution is presented for scattering nonplanar SH waves emanating from a fault source using a lined tunnel. The lined tunnel is assumed to be an annular elastic solid in half-space. A simplified circular arc fault model is employed to represent the wave source. By means of the separation of variables method, all wave fields are given in terms of the wave function series with unknown coefficients. Taking advantage of the method of images, the zero-stress boundary condition on the horizontal ground surface is satisfied automatically. By applying Graf’s addition formula, a system of equations for seeking the unknowns is derived by taking advantage of the boundary conditions. The problem of wave scattering is finally solved after seeking solutions for the system of equations through standard matrix techniques. The effects of fault distance, fault curvature, and fault orientation are revealed with numerical results. It is found that the plane waves provide a good approximation to the fault-induced cylindrical waves when the source-receiver distance or fault radius of curvature is larger enough. Fault-induced topographic effects are strongly affected by source orientation.