Resultant Laminate Forces and Moments

The resultant forces and moments acting on a laminate are obtained by integration of the stresses in each layer or lamina through the laminate thickness, for example, 

Sometimes the stiffness matrix for a lamina, [Q j], Is not constant through the thickness of the lamina. For example, if a temperature gradient or moisture gradient exists in the lamina and the lamina material properties are temperature dependent and/or moisture dependent, then [Qij]i( is a function of z and must be left inside the integral. In such cases, 

Twisting of a Two-Layered Antisymmetric Laminate under Tension (After Ashton, Hatpin, and Petit [4-3]) 

Some engineers have tried to characterize laminates with effective laminate stiffnesses, E, Ey, v y, and G y, and indeed such properties can be determined for a laminate by the usual measurements. However, it is crucial to recognize that with an effective laminate stiffness approach 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

---