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. Bending of Kirchhoff-Love plates Definitions For a thin rectangular plate of thickness , Young's modulus , and Poisson's ratio , we can define parameters in terms of the plate deflection, . The flexural rigidity is given by Moments The bending moments per unit length are given by The twisting moment per unit length is given by Forces The shear forces per unit length are given by Stresses The bending stresses are given by The shear stress is given by Strains The bending strains for small-deflection theory are given by The shear strain for small-deflection theory is given by For large-deflection plate theory, we consider the inclusion of membrane strains Deflections The deflections are given by Derivation In the Kirchhoff–Love plate theory for plates the governing equations are and In expanded form, and where is an applied transverse load per unit area, the thickness of the plate is , the stresses are , and The quantity has units of force per unit length. The quantity has units of moment per unit length. For isotropic, homogeneous, plates with Young's modulus and Poisson's ratio these equations reduce to where is the deflection of the mid-surface of the plate. Small deflection of thin rectangular plates This is governed by the Germain-Lagrange plate equation This equation was first derived by Lagrange in December 1811 in correcting the work of Germain who provided the basis of the theory. Large deflection of thin rectangular plates This is governed by the Föppl–von Kármán plate equations where is the stress function. Circular Kirchhoff-Love plates 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 is the distance of a point from the midplane of the plate. The governing equation in coordinate-free form is In cylindrical coordinates , For symmetrically loaded circular plates, , and we have Therefore, the