Non-linear quadrature element analysis of planar frames based on geometrically exact beam theory

This paper presents a total Lagrangian quadrature element formulation for planar frames undergoing large displacements and rotations. The geometrically exact beam theory, first proposed by Reissner and later extended by Simo and Vu-Quoc, is used as the basis for the formulation. Quadrature element analysis starts with evaluation of the integrals involved in the weak form description of the problem. Neither the placement of nodes nor the number of nodes in a quadrature element is fixed, being adjustable according to convergence need. As a result, not only a member can be modeled with one quadrature element but the total number of degrees of freedom is minimized as well. Several examples of planar frames are given and comparison with analytical and finite element results is made to illustrate the high computational efficiency and accuracy of the weak form quadrature element method (QEM).