Lower Bound Limit Analysis Using Nonlinear Programming

This paper describes a general numerical formulation of the lower bound theorem of classical plasticity. The method is based on simplex finite elements and nonlinear programming and is applicable to problems in one, two and three dimensions. The use of linear finite elements guarantees that the computed collapse loads are, within computational accuracy, rigorous lower bounds on the true collapse loads. The formulation is able to deal with complicated loading, which may include both surface tractions and body forces with prescribed or unknown distributions. The method can also model complex geometries, inhomogeneous material properties, and a wide variety of perfectly plastic yield functions. To ensure that the computed solution is statically admissible for cases involving semi–infinite domains, special “extension elements” have been developed for extending the stress field. The final statement of the lower bound optimisation problem involves a linear objective function (the collapse load), linear equality constraints, and convex inequality constraints. This means that the optimisation problem belongs to the family of “convex programs” and we can thus exploit the advantages of convex programming when looking for a suitable solution technique. A two–stage quasi Newton procedure, applied directly to the corresponding Kuhn–Tucker optimality conditions, is shown to solve the lower bound optimisation problem very efficiently. A number of examples are given to illustrate the utility of the procedure for large scale applications in geotechnical engineering. Scott W. Sloan and Andrei V. Lyamin 2