On the energy theorems of elastic pin-jointed frames

This paper shows, using only elementary mathematics, that, for a pin-jointed structure with non-linear elastic members, the classical minimum principles can be derived from one fundamental inequality. This is constructed from: (1) Young's inequality, which is applied to the force-extension law, which has been expanded to include terms for thermal expansion or lack of fit; (2) the principle of virtual work, which is first demonstrated using a compact version of an existing proof. The method clearly shows why the theorems of minimum total potential energy and minimum complementary energy are extremum principles and not just stationary ones. The energy theorems of Castigliano and Engesser are also derived from the same inequality. The case of Hookean behaviour is particularly simple, as Young's inequality reduces to the sum of the squares of the error in satisfying the force-extension law. The stiffness method for linear elastic pin-jointed frames is established without the need to use rotation matrices to generate the set of equations for the full frame. Matlab code for the analysis of a three-dimensional pin-jointed frame is included in an appendix.