8.1 Basic Equations

In Chap. 7, we derived the necessary formulas to calculate the strains and stresses through the thickness and the force and moment resultants given the strains and curvatures at a point ( x, y) on the reference surface. In this chapter, we will study the reverse process. Given the force and moment resultants, we want to calculate the stresses and strains through the thickness as well as the strains and curvatures on the reference surface. We also want to do this by computing the laminate stiffness matrix.

Figures 8.1 and 8.2 show the force and moment resultants, respectively. In the two figures, a small element of laminate surrounding a point ( x, y) on the geometric midplane is shown [1].

Figure 8.1: Schematic illustration of the force resultants on a composite laminate

Figure 8.2: Schematic illustration of the moment resultants on a composite laminate

The force resultants N _x, N _y, and N _xy can be shown to be related to the strains and curvatures at the reference surface by the following equation:

(8.1)

Similarly, the moment resultants M _x, M _y, and M _xy can also be shown to be related to the strains and curvatures at the reference surface by the following equation:

(8.2)

where the matrix components A _ij, B _ij, and D _ij are given as follows:

(8.3)

(8.4)

(8.5)

Equations (8.1) and (8.2) can be combined into one single...