On the modified Castigliano"s theorem

Abstract

This paper gives the modified Castigliano"s theorem, which is more convenient and more extensive for applications than the classical Castigliano"s theorem.

The modifications to the classical Castigliano"s theorem are in following two respects, the first respect is that the expression of the partial derivative with respect to the concentrated load p of the complementary energy density in the clasical Castigliano"s theorem is replaced by the expressions with the products of the external loads and influence functions, the modification brings us greatest simplicity and greatest convenience for calculations under various loads; the second respect is that the expressions with the products of the inhomogeneous boundary displacements and the influence functions are introduced into the classical Castigliano's theorem, the modification provides the theoretical fundamental for solving the problems of various boundary conditions.

We show also the method of how to apply the modified Castigliano's theorem to solve the problems of the surface structural mechanics.

Finally, as a calculated example of the applications of the modified Castigliano's theorem we solved the equation of the deflection surface of the rectangular plate with two adjacent edges built-in and other two adjacent edges free.