A Simplified Kinematic Method for 3D Limit Analysis
The kinematic (upper bound) method of limit analysis is a powerful technique for evaluating rigorous bounds on limit loads that are often very close to the true limit load. While generalized computational techniques for two-dimensional (e.g., plane strain) problems are well established, methods applicable to three-dimensional problems are relatively underdeveloped and underutilized, due in large part to the cumbersome nature of the calculations for analytical solutions and the large computation times required for numerical approaches. This paper proposes a simple formulation for three-dimensional limit analysis that considers material obeying the Mohr-Coulomb yield condition and collapse mechanisms consisting of sliding rigid blocks separated by planar velocity discontinuities. A key advantage of the approach is its reliance on a minimal number of unknowns, can dramatically reduce processing time. The paper focuses specifically on tetrahedral blocks, although extension to alternative geometries is straightforward. For an arbitrary but fixed arrangement of blocks, the procedure for computing the unknown block velocities that yield the least upper bound is expressed as a second-order cone programming problem that can be easily solved using widely available optimization codes. The paper concludes with a simple example and remarks regarding extensions of the work.
Yuantong Gu, Hong Guan, Emilie Sauret, Suvash Saha, Haifei Zhan, Rodney Persky
J.P. Hambleton and S. W. Sloan, "A Simplified Kinematic Method for 3D Limit Analysis", Applied Mechanics and Materials, Vol. 846, pp. 342-347, 2016