moment is uniform, we have at , The resulting displacement is where The bending moments and shear forces corresponding to the displacement are The stresses are Cylindrical plate bending Cylindrical bending occurs when a rectangular plate that has dimensions , where and the thickness is small, is subjected to a uniform distributed load perpendicular to the plane of the plate. Such a plate takes the shape of the surface of a cylinder. Simply supported plate with axially fixed ends For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed . Cylindrical bending solutions can be found using the Navier and Levy techniques. Bending of thick Mindlin plates For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations. Governing equations The canonical governing equation for isotropic thick plates can be expressed as where is the applied transverse load, is the shear modulus, is the bending rigidity, is the plate thickness, , is the shear correction factor, is the Young's modulus, is the Poisson's ratio, and In Mindlin's theory, is the transverse displacement of the mid-surface of the plate and the quantities and are the rotations of the mid-surface normal about the and -axes, respectively. The canonical parameters for this theory are and . The shear correction factor usually has the value . The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations where is the displacement predicted for a Kirchhoff-Love plate, is a biharmonic function such that , is a function that satisfies the Laplace equation, , and Simply supported rectangular plates For simply supported plates, the Marcus moment sum vanishes, i.e., In that case the functions , , vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by Bending of Reissner-Stein cantilever plates Reissner-Stein theory for cantilever plates leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load at . and the boundary conditions at are Solution of this system of two ODEs gives where . The bending moments and shear forces corresponding to the displacement are The