Consistent matrix formulations for structural analysis using finite-element techniques.

Abstract

The discrete-element stiffness matrix technique for formulation of linear structural problems in engineering mechanics is examined to develop techniques that give exact or closely approximate solutions for static load-displacement, elastic stability, and dynamic response problems. An exact relationship is derived for determining the coordinate load matrix equivalent to a general distributed load function. Use of the load matrix in static load-displacement problems results in an exact solution for the coordinate displacements consistent with the theoretical basis used for constructing the stiffness matrix. Explicit expressions are derived for a finite-displacement matrix for beam and plate elements for use in formulating the general elastic-stability problem. The approach discussed provides closely approximate buckling loads that are upper bounds to the precise solution. The dynamic problem, including elastic-stability considerations, is formulated using a consistent mass matrix approach. This approach provides closely approximate natural frequencies that are upper bounds to the exact solution. The data required for stiffness, load, finite-displacement, and mass matrices for a system composed of Timoshenko beam elements with linearly varying properties are provided. An example cantilever stepped beam problem is solved to illustrate the techniques involved and the exact or closely approximate nature of the solutions obtained.