78 LAMINATED COMPOSITES When the layup is orthotropic and symmetrical,the elements of the compliance matrices are(see Eqs.3.29 and 3.30) a11 12 0 d山i 12 122 0 d 0 (3.38) 0 0 a66 0 0 d66 Isotropic laminate.We consider a laminate in which each ply is isotropic. (The material may be different in each ply.)Since in isotropic materials there is no preferred direction,the [O]matrix in the x-y coordinate system is the same as the [O]matrix in the xi-x2 coordinate system: [O=[Q (3.39) Consequently,the [A],[B],[D]matrices are independent of the coordinate directions.By introducing the elements of the [O]matrix given by Eq.(2.145)into Eq.(3.20),we obtain the following elements of the [A],[B],and [D]matrices: A 2=A1 A2 A66= 41-A2 2 A6=0 b6=0 B11 B2=B11 B12 86 B1-B2 (3.40) 2 B16=0 B26=0 D11 D2=D11 D12 D66= D1-D2 D26=0. 2 D16=0 In isotropic laminates there are no extension-shear,bending-twist,or extension-twist couplings(Table 3.5,page 76),but there may be in-plane-out- of-plane coupling. When the laminate consists of a single isotropic layer,the nonzero elements of the [A],B],and [D]matrices are (Egs.2.145 and 3.20) A1=42=A0 A12 =V AiSo A66 =1"Aiso (3.41) D11=D2=Diso D12 =v Diso D66 =1Diso, where Eh E 0=1-' D0=121-阿 (3.42) E is the Young modulus,v is the Poisson ratio,and h is the thickness.The preceding stiffnesses are identical to the stiffnesses of isotropic plates. 1 S.P.Timoshenko and S.Woinowsky-Krieger,Theory of Plates and Shells 2nd edition.McGraw-Hill, New York,1959,pp.5 and 81.78 LAMINATED COMPOSITES When the layup is orthotropic and symmetrical, the elements of the compliance matrices are (see Eqs. 3.29 and 3.30)    a11 a12 0 a12 a22 0 0 0 a66       d11 d12 0 d12 d22 0 0 0 d66    . (3.38) Isotropic laminate. We consider a laminate in which each ply is isotropic. (The material may be different in each ply.) Since in isotropic materials there is no preferred direction, the [Q] matrix in the x–y coordinate system is the same as the [Q] matrix in the x1–x2 coordinate system: [Q] = [Q] . (3.39) Consequently, the [A], [B], [D] matrices are independent of the coordinate directions. By introducing the elements of the [Q] matrix given by Eq. (2.145) into Eq. (3.20), we obtain the following elements of the [A], [B], and [D] matrices: A11 A22 = A11 A12 A66 = A11 − A12 2 A16 = 0 A26 = 0 B11 B22 = B11 B12 B66 = B11 − B12 2 B16 = 0 B26 = 0 D11 D22 = D11 D12 D66 = D11 − D12 2 D16 = 0 D26 = 0. (3.40) In isotropic laminates there are no extension–shear, bending–twist, or extension–twist couplings (Table 3.5, page 76), but there may be in-plane–out￾of-plane coupling. When the laminate consists of a single isotropic layer, the nonzero elements of the [A], [B], and [D] matrices are (Eqs. 2.145 and 3.20) A11 = A22 = Aiso A12 = ν Aiso A66 = 1−ν 2 Aiso D11 = D22 = Diso D12 = νDiso D66 = 1−ν 2 Diso, (3.41) where Aiso = Eh 1 − ν2 , Diso = Eh3 12(1 − ν2) , (3.42) E is the Young modulus, ν is the Poisson ratio, and h is the thickness. The preceding stiffnesses are identical to the stiffnesses of isotropic plates.1 1 S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. 2nd edition. McGraw-Hill, New York, 1959, pp. 5 and 81