Treatment of Nonlinear Multi Freedom Boundary Constraints in Finite Element Analysis of Frame System using Lagrange Multiplier Method

Abstract

This paper is concerned with the treatment of nonlinear multi-freedom and multi-point boundary condition in finite element analysis of frame system. The treatment of boundary constraints is required to produce modified system of equation based on master stiffness equations considering nonlinear multi freedom constraints. The nonlinear constraints considerably increases the difficulty in constructing and solving the modified system of equations. Generally, the operation of imposing multi-freedom constraints can be developed using master-slave elimination, penalty augmentation or Lagrange multiplier adjunction methods. The master-slave method is useful only for simple cases but exhibits serious shortcomings for treating arbitrary constraints. The penalty method has difficulty in selecting appropriate weight values that balance solution accuracy with the violation of constraint conditions. In present work the Lagrange multiplier adjunction methods is employed and endowed with possibility of substitution and works particularly well for nonlinear constraints. The incremental-iterative algorithm based on Crisfield arc-length method is proposed to solve the nonlinear modified system of equation. Based on the presented algorithm, the paper proposed calculation procedure and established programs for determining internal forces and displacements of frames having nonlinear multi-freedom constrains condition. The numerical test examples are presented to investigate load-displacement and load-internal relationship of system having nonlinear multi freedom constraints. The calculation results show the efficiency and convergence of proposed algorithm.