Bending of plates

Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.For a thin rectangular plate of thickness H {displaystyle H}  , Young's modulus E {displaystyle E}  , and Poisson's ratio ν {displaystyle u }  , we can define parameters in terms of the plate deflection, w {displaystyle w}  .The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829.Cylindrical coordinates are convenient for such problems. Here z {displaystyle z}   is the distance of a point from the midplane of the plate.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.For thick plates, we have to consider the effect of through-the-thickness shears onthe orientation of the normal to the mid-surface after deformation. Mindlin's theoryprovides one approach for find the deformation and stresses in such plates. Solutionsto Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations.Reissner-Stein theory for cantilever plates leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load q x ( y ) {displaystyle q_{x}(y)}   at x = a {displaystyle x=a}  .