governing equation is If and are constant, direct integration of the governing equation gives us where are constants. The slope of the deflection surface is For a circular plate, the requirement that the deflection and the slope of the deflection are finite at implies that . However, need not equal 0, as the limit of exists as you approach from the right. Clamped edges For a circular plate with clamped edges, we have and at the edge of the plate (radius ). Using these boundary conditions we get The in-plane displacements in the plate are The in-plane strains in the plate are The in-plane stresses in the plate are For a plate of thickness , the bending stiffness is and we have The moment resultants (bending moments) are The maximum radial stress is at and : where . The bending moments at the boundary and the center of the plate are Rectangular Kirchhoff-Love plates For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load. Sinusoidal load Let us assume that the load is of the form Here is the amplitude, is the width of the plate in the -direction, and is the width of the plate in the -direction. Since the plate is simply supported, the displacement along the edges of the plate is zero, the bending moment is zero at and , and is zero at and . If we apply these boundary conditions and solve the plate equation, we get the solution Where D is the flexural rigidity Analogous to flexural stiffness EI. We can calculate the stresses and strains in the plate once we know the displacement. For a more general load of the form where and are integers, we get the solution Navier solution Double trigonometric series equation We define a general load of the following form where is a Fourier coefficient given by . The classical rectangular plate equation for small deflections thus becomes: Simply-supported plate with general load We assume a solution of the following form The partial differentials of this function are given by Substituting these expressions in the plate equation, we have