Elastica-plastica theory of Euler-Bernoulli beams subjected to concentrated loads

Elastica theory plays an important role in the study of large displacements of slender rods. However, the majority of the studies are still limited to elastic material only. In this paper, the traditional elastica theory is extended to an elastica-plastica theory. For the Euler-Bernoulli beams subjected to concentrated loads, the Legendre elliptic integral solution to elastic rods and the plastica solution to elastic-perfectly plastic rods are used. The deformations of elastic-perfectly plastic rods are classified into several cases, and the classification criteria are given. Furthermore, general solving methods that are applicable to various loading conditions are proposed. For an initial value problem, the highly efficient and accurate solution is completely analytical, while for a boundary value problem, the solution adopts a combination of the shooting method and the quasi-Newton method. The accuracy of the current elastica-plastica theory is verified by the corresponding two-dimensional FE simulations.