基于二阶锥规划有限元增量加载法的条形浅基础极限承载力分析

Ultimate bearing capacity analysis of shallow strip footing based on second-order cone programming optimized incremental loading finite element method

地基极限承载力分析是土力学研究中的一个经典课题。基于Hellinger-Reissner混合变分原理和有限元方法,将岩土体弹塑性问题构造成基于有限元框架的二阶锥规划(second-order cone programming,SOCP)问题,进而提出一种基于二阶锥规划理论的增量有限元法,即FEM-SOCP法。将岩土体弹塑性问题构造成二阶锥规划的数学优化问题,可以避免采用传统弹塑性计算中复杂的应力点积分等算法和屈服面棱角的平滑处理。此外,对于二阶锥规划问题,可以采用具有原始-对偶内点求解法的标准数学规划求解器MOSEK进行求解。将增量加载FEM-SOCP法应用于经典的基底粗糙的条形浅基础地基极限承载力分析中,分别考虑了关联和非关联塑性条件下的Mohr-Coulomb屈服准则。数值结果表明:所提出的增量加载FEM-SOCP法获得的地基承载力系数及地基承载力与传统FEM计算结果基本一致,而与常规有限元计算结果相比,基于增量加载的FEM-SOCP法所获得的屈服区更加平滑。

Ultimate bearing capacity analysis of the strip footing is a classical topic in soil mechanics. Based on the Hellinger-Reissner mixed variational principle and finite element method, a geotechnical elasto-plastic problem involving the Mohr-Coulomb model can be casted into a second-order cone programming(SOCP) problem in the finite element framework. A second-order cone programming based incremental finite element method named FEM-SOCP is then proposed. The proposed approach can avoid the complex stress integration algorithms and the smoothing treatment of the yield surface edges and corners, which are often necessary in the traditional elasto-plastic calculations. For the casted SOCP problem, the commercial solver MOSEK with the primal-dual interior point method can be used. The FEM-SOCP method is applied to the ultimate bearing capacity analysis of a shallow strip footing, with the associated and non-associated Mohr-Coulomb yield criteria considered, respectively. The numerical results show that the bearing capacity coefficients and bearing capacity obtained by the incremental loading FEM-SOCP method is consistent with the traditional FEM solution, but the yield zone attained is generally smoother compared to that obtained from conventional FEM.