Finite element formulation of finite planar deformation of curved elastic beams

Abstract

The objective of this paper is to present a finite element formulation of finite deformation analysis of an arbitrarily curved, extensible, shear-flexible, elastic planar beam. The formulation is based on a modified Hu-Washizu variational principle in which exact non-linear kinematic equations of Reissner [ J. appl. Math. Phys. (ZAMP) 23, 795–804 (1972)] are taken into account. In such a case a new variational principle can be derived which is expressed in terms of only one function, the rotation of the cross-section of the beam. Thus only the rotation in the interior of an element needs to be approximated in its finite element implementation. The Euler-Lagrange equations of this principle are, among others, exact kinematic and equilibrium conditions for the beam. The solution capabilities are illustrated with numerical examples. Several finite elements of different order are examined. Excellent convergence of the solution of non-linear equilibrium equations by the Newton method and high accuracy of the solution for all elements considered is demonstrated. The results indicate that the accuracy of the present elements is not notably influenced by the length of the element and the order of numerical integration. Relatively large load increments are allowed. In some cases the results are insensitive to the number of load steps. These finite elements do not exhibit any kind of locking and describe equally precise extensible and inextensible beams as well as shear-stiff or shear-flexible ones. A beam subjected to a variety of loads and extremely deformed, may be modeled with only one element, but still with a very high precision.