2004 Volume 52 Issue 606 Pages 316-327
The theorems of Castigliano are revisited. Based on the Galerkin equation for the equilibrium equations, we obtain the generalized principle of virtual work with the virtual displacements on the displacement boundary, from which we get the theorem of minimum generalized potential energy for the linearly elastic material. From the minimum theorem, we have the generalized first theorem of Castigliano which can be reduced to the conventional first theorem of Castigliano for the case where body forces are absent. Then, based on the Galerkin equation for the strain-displacement relations, we obtain the generalized principle of complementary virtual work with the virtual stresses on the stress boundary, from which we get the theorem of minimum generalized complementary energy. From the minimum theorem, we have the generalized second theorem of Castigliano which can be reduced to the conventional second theorem of Castigliano for the case where the displacement boundaries are fixed. The obtained theorems are applied to analyze the extention of bars, demonstrating the effectiveness of the generalized theorems of Castigliano as well as that of the minimum theorems of generalized potential energy and the generalized complementary energy.
