Classic Plate Equation: Equilibrium

 Force Equilibrium The equilibrium equations describe how the plate carries external pressure loads with its internal stresses. To enforce equilibrium, consider the balance of forces and moments acting on a small section of plate. There are six (6) equilibrium equations, three for the forces and three for the moments, that need to be satisfied. The equations of force equilibrium are x direction: y direction: z direction: where Nx, Ny, Nxy, Nyx, Qxz, and Qyz are force resultants; px, py, and pz are distributed external forces applied on the plate. The equations of moment equilibrium are x direction: y direction: z direction: where Mx, My, Mxy, Myx, Nxy, and Nyx are moment resultants; and mx, my, and mz are distributed external moments applied on the plate. Note that all second and higher order terms are neglected.
 Further Simplicification To further simplify the problem, consider a plate subjected to transverse loads. In other words, only pz is the non-zero external force. All forces and moments in other directions are zero: px=py=mx=my=mz=0 The above six equations of equilibrium become Due to the lack of external force components other than pz, the shear stresses at any given point are paired as follows: This yields