Classic Plate Equation: Resultants
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 cross-section is the definition of the moment resultants Mx, My, Mxy, and Myx,
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, Mx is along y direction; My along -x direction; Mxy along -x direction; and Myx along y direction.
Summing the shear forces on the cross-section is the definition of the transverse shear resultants Qx and Qy,
There is one more set of force resultants that we need to define for completeness. The sum of all direct forces acting on the cross-section is known as Nx, Ny, and Nxy,
Nx, Ny, Nxy, and Nyx are the total in-plane 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.
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