Implementation of consistent updated Lagrangian formulation for static analysis of geometrically nonlinear beam

摘要

In this paper, the consistent updated Lagrangian (CUL) formulation used for solving large-displacement non-linear problems is compared with the total Lagrangian (TL) and the updated Lagrangian (UL) formulations. This comparison indicates that these formulations use different reference configurations for developing stiffness and internal load vectors but yield identical terms. The equivalence of the three Lagrangian formulations was confirmed by developing the governing equilibrium equations for the CUL formulation using the continuum mechanics approach. For a two-noded, two-dimensional, large displacement-small strain Euler-Bernoulli beam element, the CUL finite element formulation of the equilibrium equations was developed to illustrate the procedure used to obtain the stiffness matrix and internal load vectors. All three finite element Lagrangian formulations were implemented and tested for static analysis of two-dimensional, Euler-Bernoulli straight cantilever beam subjected to a transverse tip load at the free end. They all gave identical results, which were comparable to those obtained using a co-rotational formulation.