The plate is assumed to be constructed by a homogeneous but not necessarily isotropic material and subjected to both transverse and in-plan loadings. Also, the Cartesian coordinate system is used. The goal is to develop the relations between the external loadings and the displacements. However, the relations between the resultants (forces N and moments M) and the strains (strains e and curvatures k) are of most interest in practice.
Replace the stresses in the force and moment resultants with strains via the constitutive equations, we have
By applying the summation and integration operations to their respective components, the force and moment resultants can be further simplified to
Combine the above equations we can write:
where A is called the extensional stiffness, B is called the coupling stiffness, and D is called the bending stiffness of the laminate. The components of these three stiffness matrices are defined as follows:
where tk is the thickness of the kth layer and is the distance from the mid-plan to the centroid of the kth layer. Forming these three stiffness matrices A, B, and D, is probably the most crucial step in the analysis of composite laminates.
In some situations, strains expressed in terms of resultants are more handy. The strain-resultant relations can be derived with appropriate matrix operations:
Note that A, B, D and A*, B*, D* are all symmetric matrices. Among them, A, B, and D are considered universal notations in the field of composites, i.e., the same notations appear in almost all literature of composite materials. A*, B*, and D*, on the other hand, are not.