Algebraic Equations of Linear Elasticity and Their Solutions

A detailed derivation is given of the algebraic equations of elasticity that correspond to their differential equation counterparts. These algebraic equations are then reduced to two forms—one based on forces and another based on displacements. The new displacement approach is shown to be like the standard finite element method, where a stiffness matrix is used to solve for the displacements in statically indeterminate problems, but where the stiffness matrix is formed from equilibrium and inverse flexibility matrices. The force approach is based on equilibrium and compatibility equations and flexibility matrices where the compatibility equations are obtained from homogeneous solutions of the equilibrium equations. The reduced force-based and displacement-based methods are described in detail for a simple statically indeterminate problem using the symbolic algebra capabilities of MATLAB to demonstrate a highly efficient approach to solving problems at all technical levels in an engineering curriculum.