Equating the two expressions, we have which can be rearranged to give The deflection of a simply-supported plate (of corner-origin) with general load is given by Simply-supported plate with uniformly-distributed load For a uniformly-distributed load, we have The corresponding Fourier coefficient is thus given by . Evaluating the double integral, we have , or alternatively in a piecewise format, we have The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by The bending moments per unit length in the plate are given by Lévy solution Another approach was proposed by Lévy in 1899. In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied. The goal is to find such that it satisfies the boundary conditions at and and, of course, the governing equation . Let us assume that For a plate that is simply-supported along and , the boundary conditions are and . Note that there is no variation in displacement along these edges meaning that and , thus reducing the moment boundary condition to an equivalent expression . Moments along edges Consider the case of pure moment loading. In that case and has to satisfy . Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as Plugging the expression for in the governing equation gives us or This is an ordinary differential equation which has the general solution where are constants that can be determined from the boundary conditions. Therefore, the displacement solution has the form Let us choose the coordinate system such that the boundaries of the plate are at and (same as before) and at (and not and ). Then the moment boundary conditions at the boundaries are where are known functions. The solution can be found by applying these boundary conditions. We can show that for the symmetrical case where and we have where Similarly, for the antisymmetrical case where we have We can superpose the symmetric and antisymmetric solutions to get more general solutions. Simply-supported plate with uniformly-distributed load For a uniformly-distributed load, we have The deflection of a simply-supported plate with centre with uniformly-distributed load is given by The bending moments per unit length in the plate are given by Uniform and symmetric moment load For the special case where the loading is symmetric and the