Stability analysis of intermittently jointed rock slopes based on the stepped failure mode

In practical engineering, due to the noncontinuity characteristics of joints in rock slopes, in addition to plane failure, stepped sliding failure may occur for intermittently jointed rock slopes. Especially for intermittently bedding jointed rock slopes, the correlation and difference in strength parameters between joints and rock bridges, along with the various failure modes and intermittency of rock bridges, contribute to the complexity of stepped failure modes and the unpredictability of failure regions. Based on the upper-bound limit analysis method and multi-sliders step-path failure mode, considering the shear and tensile failure of rock bridges and the weakened relationship between the strength parameters of rock bridges and jointed surfaces, by introducing the modified M-C failure criterion and the formula for calculating the energy consumption of tensile failure of rock bridges, two failure mechanisms are constructed to obtain the safety factor (F_s) of intermittently jointed rock slopes. The sequential quadratic programming method is used to obtain the optimal upper-bound solution for F_s. The influence of multiple key parameters (slope height H, horizontal distance L, Slope angle β, shear strength parameters of the rock bridge φ_r and c_r, Dimensionless parameter u, weakening coefficients of the internal friction angle and cohesion between the rock bridges and joint surfaces K_φ and K_c) on the stability analysis of intermittently jointed rock slopes under the shear failure mode of rock bridges as well as under the tensile failure mode is also explored. The reliability of the failure mechanisms is verified by comparative analysis with theoretical results, numerical results, and landslide cases, and the variation rules of F_s with each key parameter are obtained. The results show that F_s varies linearly with φ_r and c_r of the rock bridge and with K_φ and K_c, whereas F_s changes nonlinearly with H and L. In particular, with the increase in K_φ and K_c, F_s increases by approximately 52.78% and 171.02% on average, respectively. For rock bridge tensile failure, F_s shows a nonlinearly positive correlation with φ_r, c_r, K_φ and K_c. In particular, with the increase in K_φ and K_c, F_s increases by approximately 13% and 61.69% on average, respectively. F_s decreases rapidly with increasing slope gradient β and decreasing dimensionless parameter μ. When K_φ and K_c are both less than 1.0, the stepped sliding surface occurs more easily than the plane failure surface, especially in the case of tensile failure of the rock bridge. In addition, rock slopes with higher strength parameters, taller heights, and greater weakening coefficients are prone to rock bridge tension failure with lower F_s, and more attention should be given to the occurrence of such accidents in actual engineering.