Force and moment resultants are convenient quantities for tracking the important stresses in plates. They are analogous to the moments and forces in statics theories, in that their influence is felt thoughout the plate (as opposed to just a local effect). Their convenience lies in the fact that they are only functions of x and y, whereas stresses are functions of x, y, and z.
Recall that the stress tensor has nine components at any given point. Each little portion of the direct stress acting on the cross section creates a moment about the neutral plane (z = 0). Summing these individual moments over the area of the crosssection is the definition of the moment resultants M_{x}, M_{y}, M_{xy}, and M_{yx},
where z is the coordinate pointing in the direction normal to the plate. Unlike other resultants that their subscripts indicate their action directions, the subscripts of moment resultants are the directions of stresses that cause the resultants. Hence, M_{x} is along y direction; M_{y} along x direction; M_{xy} along x direction; and M_{yx} along y direction.
Summing the shear forces on the crosssection is the definition of the transverse shear resultants Q_{x} and Q_{y},
There is one more set of force resultants that we need to define for completeness. The sum of all direct forces acting on the crosssection is known as N_{x}, N_{y}, and N_{xy},
N_{x}, N_{y}, N_{xy}, and N_{yx} are the total inplane normal and shear forces within the plate at some point (x, y), yet they do not play a role in (linear) plate theory since they do not cause a displacement w.
These force and moment resultants should be in equilibrium with all external forces and moments.
