|- |For the situation where the strains and rotations of the plate are small the virtual internal energy is given by where the thickness of the plate is and the stress resultants and stress moment resultants are defined as Integration by parts leads to The symmetry of the stress tensor implies that . Hence, Another integration by parts gives For the case where there are no prescribed external forces, the principle of virtual work implies that . The equilibrium equations for the plate are then given by If the plate is loaded by an external distributed load that is normal to the mid-surface and directed in the positive direction, the external virtual work due to the load is The principle of virtual work then leads to the equilibrium equations |} Boundary conditions The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work. In the absence of external forces on the boundary, the boundary conditions are Note that the quantity is an effective shear force. Constitutive relations The stress-strain relations for a linear elastic Kirchhoff plate are given by Since and do not appear in the equilibrium equations it is implicitly assumed that these quantities do not have any effect on the momentum balance and are neglected. The remaining stress-strain relations, in matrix form, can be written as Then, and The extensional stiffnesses are the quantities The bending stiffnesses (also called flexural rigidity) are the quantities The Kirchhoff-Love constitutive assumptions lead to zero shear forces. As a result, the equilibrium equations for the plate have to be used to determine the shear forces in thin Kirchhoff-Love plates. For isotropic plates, these equations lead to Alternatively, these shear forces can be expressed as where Small strains and moderate rotations If the rotations of the normals to the mid-surface are in the range of 10 to 15, the strain-displacement relations can be approximated as Then the kinematic assumptions of Kirchhoff-Love theory lead to the classical plate theory with von Kármán strains This theory is nonlinear because of the quadratic terms in the strain-displacement relations. If the strain-displacement relations take the von Karman form, the equilibrium equations can be expressed as Isotropic quasistatic Kirchhoff-Love plates For an isotropic and homogeneous plate, the stress-strain relations are where is Poisson's Ratio and is Young's Modulus. The moments corresponding to