Solution of nonlinear problems of elastoplasticity by finite elementmethod.

By using the finite element and direct stiffness method, piecewise linear load deflection relationship, and step-by-step computational algorithm, a general procedure for the solution of problems of elastoplasticity is developed. The change of material properties at each step is expressed in the general differential form of Prandtl-Reuss equations of the theory of plastic deformations, with special treatment of the plane strain case. The effect of initial stresses is also considered when the rotations at each step are not negligible, although the strains are assumed small compared to unity. Application of the principle of virtual work to define the equilibrium of an element subject to initial and additional stresses yields a geometrical stiffness matrix which satisfies macroscopic equilibrium requirements. Computational difficulties arising from the repeated solution of large numbers of linear simultaneous equations are overcome by using an iteration method with an over-relaxation factor. The accuracy of the propagated solution is improved by the application of half-step procedure, which is a special case of the Runge-Kutta method.