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 N_x, N_y, N_xy, N_yx, Q_xz, and Q_yz are force resultants; p_x, p_y, and p_z are distributed external forces applied on the plate.

The equations of moment equilibrium are

x direction:

y direction:

z direction:

where M_x, M_y, M_xy, M_yx, N_xy, and N_yx are moment resultants; and m_x, m_y, and m_z are distributed external moments applied on the plate.

Note that all second and higher order terms are neglected.

To further simplify the problem, consider a plate subjected to transverse loads. In other words, only p_z is the non-zero external force. All forces and moments in other directions are zero: p_x=p_y=m_x=m_y=m_z=0

The above six equations of equilibrium become

Due to the lack of external force components other than p_z, the shear stresses at any given point are paired as follows:

This yields