A KINEMATICALLY EXACT FINITE-ELEMENT FORMULATION OF PLANAR ELASTIC-PLASTIC FRAMES

A finite element formulation of finite deformation static analysis of plane elastic-plastic frames subjected to static loads is presented, i n which the only function to be interpolated is the rotation of the ce ntroid axis of the beam. One of the advantages of such a formulation i s that the problem of the field-consistency does not arise. Exact non- linear kinematic relationships of the finite-strain beam theory are us ed, which assume the Bernoulli hypothesis of plane cross-sections. Fin ite displacements and rotations as well as finite extensional and bend ing strains are accounted for. The effects of shear strains and non-co nservative loads are at present neglected, yet they can simply be inco rporated in the formulation. Because the potential energy of internal forces does not exist with elastic-plastic material, the principle of virtual work is introduced as the basis of the finite element formulat ion. A generalized principle of virtual work is proposed in which the displacements, rotation, extensional and bending strains, and the Lagr angian multipliers are independent variables. By exploiting the specia l structure of the equations of the problem, the displacements, the st rains and the multipliers are eliminated from the generalized principl e of virtual work. A novel principle is obtained in which the rotation becomes the only function to be approximated in its finite element im plementation. It is shown that (N-1)-point numerical integration must be employed in conjunction with N-node interpolation polynomials for t he rotation, and the Lobatto rule is recommended. Regarding the integr ation over the cross-section, it is demonstrated by numerical examples that, due to discontinuous integrands, no integration order defined a s 'computationally efficient yet accurate enough' could be suggested. The theoretical findings and a nice performance of the derived finite elements are illustrated by numerical examples.