Chapter 10 - The Principle of Minimum Potential Energy for Two-Dimensional and Three-Dimensional Elements

Abstract

This chapter covers the implementation of the principle of minimum potential energy on two-dimensional (2D) and three-dimensional (3D) elements. Interpolation functions of nodal displacements are derived for linear triangular elements, quadratic triangular elements, bilinear rectangular elements, tetrahedral solid elements, eight-node rectangular solid elements, and plate bending elements. The next topic is dedicated to isoparametric elements and Lagrange polynomials. Using interpolation functions, strain-displacements formulae, Hooke’s law, and principle of the minimum potential energy, the derivation of the stiffness matrices for widely used 2D and 3D elements is described. Using such stiffness matrices, analytic procedures for the solution of plane stress problems in Cartesian and in polar coordinates are provided and the methodology of how to compose the matrix equation consisting of the global stiffness matrix and the boundary conditions submatrix is described. Finally, step-by-step instructions describing the sequence of ANSYS commands to be used for the solution of a typical plane stress problem containing curved boundaries are presented.