Classical Lamination Theory

• Associated with each layer is the principal material coordinate system with an orientation denoted by q relative to the global or laminate coordinate system.

• For a unidirectional prepreg, the 1,2 axis would be parallel and transverse to the fibers, respectively. For a fabric the warp and fill directions would define the principal material system.
• The laminate may consist of lamina of several materials but each individual ply must obey linear stress-strain behavior.

• In the principal material coordinate system the stiffness relation for the kth ply is given by:

• The stiffness coefficients expressed in terms of the lamina engineering material properties are defined as follows:

• The stiffness matrix, however, must be expressed in the laminate coordinate system. Consequently, a straightforward transformation defined by the ply orientation, qk , where is the transformed ply stiffness in the laminate coordinate system .

• The laminate stiffness is obtained by appropriate averaging through the thickness.

• The laminate force and moment resultants are found through integration of stresses across the thickness of each lamina.
• Laminate strains are assumed to vary linearly through the thickness as a function of the laminate mid-plane strains, eo and curvatures, k .

• Substituting the assumed strain field into the transformed ply stiffness relations and performing integration yields the well known laminate stiffness relationship:

( i, j = 1, 2, 6 )