Limit analysis in plane strain: The lower bound approach.

A robust algorithm is developed to solve plane strain limit analysis problems by means of the lower bound approach. The theoretical foundation of this algorithm is the duality theorem which connects the classical lower bound and upper bound formulations by equating the least upper bound and the greatest lower bound. Using a suitable linearized yield function and the finite element equilibrium equations of a continuum, the lower found formulation leads to a linear programming problem. The constraint equations of the problem include the equilibrium equations, which allow a suitable form of discontinuity; the traction boundary conditions; and the linearized yield conditions. To solve the bounded form linear programming, a null-space basis algorithm has been developed, which is more effective than the traditional simplex algorithm in many aspects. Adopting the rectangular norm for this algorithm, a new set of variables is introduced by way of a linear transformation, which has the advantage of greatly reducing the number of constraint equations. This technique circumvents the major drawback of the lower bound formulations and solution methods facing other investigators, who had to solve greater number of constraint equations and who had to introduce additional slack or surplus variables. The algorithm developed is then applied to two classes of problems, namely punch indentation and metal extrusion. Since each of these applications involves extensive variations of the physical and geometrical parameters involved, they can be investigated thoroughly. Thus, some new important results are obtained for these two problems, e.g. it is possible to do a complete study of the dimensional effects on collapse pressure for the problem of indentation into a block with finite dimensions.PhDApplied SciencesMechanical engineeringMechanicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128492/2/9023525.pd