STRUCTURAL ANALYSIS 177 On the other hand,an example of an unsymmetric arrangement of the same plies would be: 0°/0°/0°/0°/+45°/-45°/+45°/-45° These two cases are shown in Figure 6.4 where z denotes the coordinate in the thickness direction. 6.2.3.1 Laminate Stiffness Matrix.Consider now a laminate comprising n plies and denote the angle between the fiber direction in the kth ply and thex laminate axis by (with the convention defined in Fig.6.2).Subject only to the symmetry requirement,the ply orientation is arbitrary.It is assumed that,when the plies are molded into the laminate,a rigid bond (of infinitesimal thickness)is formed between adjacent plies.As a consequence of this assumption,it follows that under plane stress conditions the strains are the same at all points on a line through the thickness (i.e.,they are independent of z).Denoting these strains by ex,and y,it then follows from equation(6.7)that the stresses in the kth ply will be given by: x(k)=Qx(0x)Ex+Oxy(0x)Ey +Qxs(0x)Yxy y(k)=Qxy(0x)Ex +Qy()Ey +Qys(0x)Yy (6.10) Tx(k)=Qxs(0x)Ex+ys(0x)Ey+Qxs(0x)Yxy The laminate thickness is denoted by t and the thickness of the kth ply is h-h-1 with hi defined in Figure 6.5.Assuming all plies are of the same thickness (which is the usual situation),then the thickness of an individual ply is simply t/n.Now consider an element of the laminate with sides of unit length parallel to thex-and y-axes.The forces on this element will be denoted by NN 0 -45 0 +45 +45 -45 Mid-plane -45 +45 Mid-plane -45 0 +45 0 0 0 0 0 Fig.6.4 Symmetric (left)and non-symmetric (right)eight-ply laminates.STRUCTURAL ANALYSIS 177 On the other hand, an example of an unsymmetric arrangement of the same plies would be: O°lO°lO°lO° 1+45°1-45°1+45ol-45 ° These two cases are shown in Figure 6.4 where z denotes the coordinate in the thickness direction. 6.2.3. I Laminate Stiffness Matrix. Consider now a laminate comprising n plies and denote the angle between the fiber direction in the kth ply and the x laminate axis by Ok (with the convention defined in Fig. 6.2). Subject only to the symmetry requirement, the ply orientation is arbitrary. It is assumed that, when the plies are molded into the laminate, a rigid bond (of infinitesimal thickness) is formed between adjacent plies. As a consequence of this assumption, it follows that under plane stress conditions the strains are the same at all points on a line through the thickness (i.e., they are independent of z). Denoting these strains by ex, ey, and "Yxy, it then follows from equation (6. 7) that the stresses in the kth ply will be given by: O'x(k) = Q~( Ok)ex + axy( Ok)l?,y + Qxs( Ok)3'xy ~ry(k) = Qxy( ODex + ayy( Ok)f,y + Qys( Ok)'Yxy (6.10) "rxy(k) = Qxs( Ok)ex + ays( Ok)Sy + Qss( Ok)Yxy The laminate thickness is denoted by t and the thickness of the kth ply is hk -- hk-1 with hi defined in Figure 6.5. Assuming all plies are of the same thickness (which is the usual situation), then the thickness of an individual ply is simply t/n. Now consider an element of the laminate with sides of unit length parallel to the x- and y-axes. The forces on this element will be denoted by Nx, Ny, 0 0 *,kS Mid-plane - 45 -4.5 ",~.S 0 0 ~Z mw -45 *t.5 -45 * `k5 Mid-plane Fig. 6.4 Symmetric (left) and non-symmetric (right) eight-ply laminates