Castigliano's Method for Warped Thin Curved Beams

Abstract

This presentation illustrates the use of Castigliano's theorem to solve problems of thin curved beams of arbitrary cross section loaded out of plane. This theorem, the result of the application of the principle of minimum complementary energy, is treated extensively in textbooks on strength of materials and elasticity. Solutions of statically determinate and indeterminate problems of thin uniform curved beams of any cross section, loaded out of plane (warping included), were previously developed by the author. Solutions of several simultaneous equations are required to solve such problems whereas Castigliano's Theorem requires mostly integration. To apply the theorem, the strain energy of structure is expressed in terms of point forces and distributed loads. Strain energy may be utilized, since for linearly elastic material the value of the complementary energy is identical to that of the strain energy. Two problems, one statically determinate and one statically indeterminate, are solved to illustrate the method.