Cartesian Formulation of Membrane Theory

The equilibrium equations of the membrane theory in cartesian coordinates and their reduction to one differential equation for a stress function, following Pucher's approach, are considered. The problem of determining the stress resultants when boundary conditions of the displacement or elastic type are given is solved through the application of the principle of stationary complementary energy. Application of the direct methods of the calculus of variations allows the determination of the stress resultants without determining the displacements. When it is not desirable to use the direct methods of the calculus of variations, the problem of the membrane with boundary conditions of the displacement or elastic type is reduced to the solution of two differential equations with two boundary conditions. These are derived as the Euler equations and the natural boundary conditions respectively of the variational equation. The method introduces a Lagrange multiplier which is shown to be identical with a cartesian component of displacement.