these stresses are In expanded form, where for plates of thickness . Using the stress-strain relations for the plates, we can show that the stresses and moments are related by At the top of the plate where , the stresses are Pure bending For an isotropic and homogeneous plate under pure bending, the governing equations reduce to Here we have assumed that the in-plane displacements do not vary with and . In index notation, and in direct notation which is known as the biharmonic equation. The bending moments are given by {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations for pure bending |- |For an isotropic, homogeneous plate under pure bending the governing equations are and the stress-strain relations are Then, and Differentiation gives and Plugging into the governing equations leads to Since the order of differentiation is irrelevant we have , , and . Hence In direct tensor notation, the governing equation of the plate is where we have assumed that the displacements are constant. |} Bending under transverse load If a distributed transverse load pointing along positive direction is applied to the plate, the governing equation is . Following the procedure shown in the previous section we get In rectangular Cartesian coordinates, the governing equation is and in cylindrical coordinates it takes the form Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates. {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations for transverse loading |- |For a transversely loaded plate without axial deformations, the governing equation has the form where is a distributed transverse load (per unit area). Substitution of the expressions for the derivatives of into the governing equation gives Noting that the bending stiffness is the quantity we can write the governing equation in the form In cylindrical coordinates , For symmetrically loaded circular plates, , and we have |} Cylindrical bending Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder. This type of bending is called cylindrical bending and represents the special situation where . In that case and and the governing equations become Dynamics of Kirchhoff-Love plates The dynamic theory of thin plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. Governing equations The governing equations for the dynamics of