Lower-Bound Axisymmetric Formulation for Geomechanics Problems Using Nonlinear Optimization

A lower-bound limit analysis formulation, by using two-dimensional finite elements, the three-dimensional Mohr-Coulomb yield criterion, and nonlinear optimization, has been given to deal with an axisymmetric geomechanics stability problem. The optimization was performed using an interior point method based on the logarithmic barrier function. The yield surface was smoothened (1) by removing the tip singularity at the apex of the pyramid in the meridian plane and (2) by eliminating the stress discontinuities at the corners of the yield hexagon in the $\pi \text{-plane}$. The circumferential stress (${\sigma }_{\theta }$) need not be assumed. With the proposed methodology, for a circular footing, the bearing-capacity factors $N_c$, $N_q$, and $N_{\gamma }$ for different values of $\varphi$ have been computed. For $\varphi =0$, the variation of $N_c$ with changes in the factor $m$, which accounts for a linear increase of cohesion with depth, has been evaluated. Failure patterns for a few cases have also been drawn. The results from the formulation provide a good match with the solutions available from the literature.