Total Lagrangian formulation of 2D bar element using vectorial kinematical description

Abstract

Based on the total Lagrangian kinematical description, a geometrically nonlinear bar element is developed for large-displacement and post-buckling analyses of plane truss structures. Firstly, the vectorial form is used to explicitly describe the element kinematics and element strain in terms of element nodal displacements, thus eliminating the need of shape functions as required in a standard finite element formulation. Secondly, the virtual displacement principle is employed to represent the element equilibrium in the integral form. Due to the nonlinear nature of the system, the directional derivative operator is used to linearize the virtual displacement equation, resulting in an incremental form of equilibrium equations. Thirdly, the generalized displacement control method is adopted in obtaining the incremental solution of problems. Finally, several structures exhibiting different types of critical points are analyzed to verify the element accuracy and assess the ability of solution algorithm to trace nonlinear responses. In this paper, the MATLAB programming language is used for coding and in-house software development.