ABSTRACT
In this chapter, solutions are presented to some two-dimensional and axisymmetric problems. Ideally, the two formulations described in Chapter 6 should be used, namely, the Eulerian and the updated Lagrangian formulations. However, even for simple problems, use of the updated Lagrangian formulation requires a lot of computation. Therefore, only the Eulerian formulation will be used, with the current configuration as the control volume. (The only exception to this is the one-dimensional [1-D] problem of Section 8.2 where the total Lagrangian formulation is used.) The three governing equations need to be solved: (i) equilibrium equation, (ii) stress-strain (or strain rate) relation and (iii) strain-displacement (or strain rate-velocity) relation. A simplified approach shall be used to solve these problems. Especially, if the material is assumed to be perfectly plastic, then it becomes possible to find the stress field using only the equilibrium equation and the yield criterion. After that, the strain and displacement fields can be obtained by using the other two governing equations. Before some two-dimensional (plane stress/strain) or axisymmetric problems are solved, it is instructive to start with a 1-D problem. Therefore, first, the problem of symmetric beam bending of a perfectly plastic material shall be discussed.
