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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. For a classical plate, the moment and force resultants can be expressed in terms of the normal and shear stresses in integral forms. They are

Perfect Bonding between Layers

For a laminated composite plate, the material properties can be quite different from layer by layer, and the stress distribution may be significantly different between layers. As a result, it is desirable to express the moment and force resultants in terms of the normal and shear stresses in each layer. To proceed, a few assumptions on how layers stack up need to be made.

In lamination theory, a laminate composed of two or more laminae (usually in several different directions) acts as an integral structural element. In other words, perfect bonding between the layers is assumed and the laminate is considered one-piece.

Perfect Bonding
1. The bonding itself is infinitesimally small (there is no flaw or gap between layers).
2. The bonding is non-shear-deformable (no lamina can slip relative to another).
3. The strength of bonding is as strong as it needs to be (the laminate acts as a single lamina with special integrated properties).
 

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 for the moment resultants Mx, My, Mxy, and Myx,

where z is the coordinate pointing in the direction normal to the plate. The subscript k indicates the kth layer from the top of the laminate and N is the total number of layers. Unlike other resultants whose subscripts indicate the action directions, the subscripts of the moment resultants are the directions of the stresses that cause the resultants. Hence, Mx is along the y direction; My is along the -x direction; Mxy is along the -x direction; and Myx is along the 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, the in-plane forces. The sum of all direct forces acting on the cross-section are known as Nx, Ny, Nxy, and Nyx.

Nx, Ny, Nxy, and Nyx are the total in-plane normal and shear forces acting within the plate at point (xy). However, they do not play a role in the (linear) plate theory since they do not cause an out-of-plane (transverse) displacement w.

These force and moment resultants should be in equilibrium with all external forces and moments.