Exploration of face failure curve under earthquakes using calculus of variations

A solution of face collapse failure under earthquakes is obtained in the realm of plasticity theory with the help of the calculus of variations. A simplicity face collapse failure model is introduced. Based on the Hoek-Brown failure criterion and the upper bound theorem, an exact function of face failure curves under earthquakes is presented. The extreme collapse curves derived by the Euler-Lagrange equation are solved in a nonlinear PDF. And then the exact extreme curves of face collapse failure are got when B=0.5. It is shown that the most remote distances (z_max ) are consistent with the numerical results, but the area of collapse zone is too large. Also, a sensitivity analysis to some direction angles () of the face detachment and several face pressures () is presented. The resulting expressions are so simple and rational to show the characteristics of face collapse failure, and can be used to make comparisons of the empirical and numerical analyses.