Involutory transformations and variational principles with multi-varoables in thin plate bending problems

In this paper, the generalized variational principles of plate bending problems are established from their minimum potential energy principle and minimum complementary energy principle through the elimination of their constraints by means of the method of Lagrange multipliers. The involutory transformations are also introduced in order to reduce the order of differentiations for the variables in the variation. Furthermore, these involutory transformations become in fact the additional constraints in the variation, and additional Lagrange multipliers may be used in order to remove these additional constraints. Thus, various multi-variable variational principles are obtained for the plate bending problems. However, it is observed that, not all the constraints of variation can be removed simply by the ordinary method of linear Lagrange multipliers. In such cases, the method of high-order Lagrange multipliers are used to remove those constraints left over by ordinary linear multiplier method. And consequently, some functionals of more general forms are obtained for the generalized variational principles of plate bending problems.