Sparse self-stress matrices for the finite element force method

摘要

A basic problem in the finite element force method is that of obtaining a sparse and banded selfstress matrix and a sparse and banded structure flexibility matrix. Traditionally the self-stress matrix is obtained through the application of algebraic procedures to the equilibrium matrix. The self-stress matrix for an indeterminate structure is not unique, and it is possible to obtain another self-stress matrix from an existing one through algebraic operations and grouping of redundants. The purpose of this paper is to describe and test an algorithm, called REDUC, which combines the vectors of the self-stress matrix obtained from the LU procedure of the force method. The rows of the transpose of this matrix are combined by using a special form of the Gaussian elimination technique. A plane frame example is presented to demonstrate the algorithm at work. The algorithm REDUC is applied to a plane truss and physical interpretation of the resulting self-stress matrix highlights the grouping of redundants, improved sparsity and bandwidth. Improvements in the conditioning and bandwidth of the structure flexibility matrix are also observed. The algorithm yields results similar to those of the turn-back LU procedure, but requires less computation time and programming effort.