The equilibrium equations describe how the plate carries pressure loads with its internal stresses.
Consider balancing of the forces and moments acting on a small section of a 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
where Nx, Ny, Nxy, Nyx, Qxz, and Qyz are force resultants, whereas px, py, and pz are distributed external forces applied on the plate.
The equations of moment equilibrium are
where Mx, My, Mxy, Myx, Nxy, and Nyx are moment resultants, and mx, my, and mz are distributed external moments applied on the plate.
Notice that all second and higher order terms are neglected.
In most practical cases, the external loading is limited to a distributed pressure. In other words, only pz is non-zero. All other forces and moments (px, py, mx, my and mz) are zero.
The 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: