Algebraic Equations of Linear Elasticity

and a Second-generation Force-based Solution Method

Abstract

It is shown how the differential equations of linear elasticity can be replaced by a completely equivalent set of governing algebraic equations. Those general algebraic equations can be reduced to a set of equilibrium equations for the displacements that is an algebraic version of Navier’s differential equations. The governing algebraic equations can also be used to define the compatibility equations in terms of a compatibility matrix that is the counterpart of a differential compatibility operator matrix found in the theory of linear elasticity. Remarkably, this algebraic compatibility matrix can be obtained from homogeneous solutions of the equilibrium matrix, thus enabling the stresses (or corresponding generalized forces) to be obtained directly, without introducing displacements. Unlike earlier force-based methods this second-generation force-based approach can be used effectively at all levels of engineering analysis, including basic engineering courses. As an example, the forces in a statically indeterminate truss are obtained using simple extensions of the statically determinate truss methods found in a first course in engineering statics. This example also shows that when second generation force-based methods are implemented in software such as MATLAB^®, solutions can be obtained without any programming, making the methods ideal tools for learning how to solve equilibrium problems in an engineering curriculum.