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

x direction:

y direction:

z direction:

where N_x, N_y, N_xy, N_yx, Q_xz, and Q_yz are force resultants, whereas 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.

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 p_z is non-zero. All other forces and moments (p_x, p_y, m_x, m_y and m_z) are zero.

The 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