3.2 STIFFNESS MATRICES OF THIN LAMINATES 77 个y 个y Figure 3.13:Ply arrangements in orthotropic laminates.The ply's symmetry axes(dashed lines) must coincide with the laminate's orthotropy x,y axes. when the ply is a woven fabric and the ply's symmetry axes are aligned with the laminates orthotropy directions; when two adjacent unidirectional plies (oriented in different directions)are treated as a single layer and the symmetry axes of this layer are aligned with the laminate's orthotropy directions. For the orthotropic plies described above,the O16 and O26 elements of the ply stiffness matrix are zero (Eq.2.138): Q16=Q26=0. (3.35) With these values,Eq.(3.20)gives that the 16 and 26 elements of the [A],[B], and [D]matrices are zero: A6=A6=0B16=B26=0D16=D26=0 (3.36) Accordingly,there is no extension-shear,bending-twist,or extension-twist coupling in an orthotropic laminate (Table 3.5).On the other hand,when the laminate is not orthotropic,these couplings are present and result in unexpected deformations. We observe that the 16 and 26 elements of the [A],B],and [D]matrices are zero only in the x-y coordinate system,where x and y are the orthotropy directions (Table 3.4,page 76). For unsymmetrical orthotropic laminates the compliance matrix becomes(see Eq.3.23) 011 C12 0 P11 P12 07 12 C22 0 f21 f22 0 0 0 066 0 P66 B11 P21 0 (3.37) d11 812 0 P12 B22 0 0 0 B66 0 0 866」3.2 STIFFNESS MATRICES OF THIN LAMINATES 77 y x y x y x y x Figure 3.13: Ply arrangements in orthotropic laminates. The ply’s symmetry axes (dashed lines) must coincide with the laminate’s orthotropy x, y axes.
when the ply is a woven fabric and the ply’s symmetry axes are aligned with the laminates orthotropy directions;
when two adjacent unidirectional plies (oriented in different directions) are treated as a single layer and the symmetry axes of this layer are aligned with the laminate’s orthotropy directions. For the orthotropic plies described above, the Q16 and Q26 elements of the ply stiffness matrix are zero (Eq. 2.138): Q16 = Q26 = 0. (3.35) With these values, Eq. (3.20) gives that the 16 and 26 elements of the [A], [B], and [D] matrices are zero: A16 = A26 = 0 B16 = B26 = 0 D16 = D26 = 0. (3.36) Accordingly, there is no extension–shear, bending–twist, or extension–twist coupling in an orthotropic laminate (Table 3.5). On the other hand, when the laminate is not orthotropic, these couplings are present and result in unexpected deformations. We observe that the 16 and 26 elements of the [A], [B], and [D] matrices are zero only in the x–y coordinate system, where x and y are the orthotropy directions (Table 3.4, page 76). For unsymmetrical orthotropic laminates the compliance matrix becomes (see Eq. 3.23)          α11 α12 0 β11 β12 0 α12 α22 0 β21 β22 0 0 0 α66 0 0 β66 β11 β21 0 δ11 δ12 0 β12 β22 0 δ12 δ22 0 0 0 β66 0 0 δ66          . (3.37)