3.2 STIFFNESS MATRICES OF THIN LAMINATES 81 The superscript o refers to Reference Plane 2.When the laminate is symmet- rical and the reference plane coincides with the midplane,the matrix [B]is zero. When the laminate is unsymmetrical,the matrix [B]is not zero. In general,there is no o value that results in a nonzero [B]matrix for an anisotropic composite laminate.In other words,for an unsymmetrical laminate there is no reference plane that is also a neutral plane. The compliance matrices [a],[B],and [8]refer to Reference Plane 1.The elements of these matrices for Reference Plane 2 are obtained by introducing Eq.(3.47)into Eq.(3.23).After algebraic manipulations,we obtain a喝=+Q(Bi+Bi)+Q2dj 号=+Q6j (3.48) 8号=8j The third of these equations shows that the bending compliance matrix [8]is independent of the choice of the reference surface. Curved laminates.The stiffness and compliance matrices derived in this chap- ter for flat laminates may be applied to thin curved laminates when the radius of curvature is large compared with the thickness. Numerical values of the stiffness and compliance matrices of selected lami- nates.Below,we present numerical values of the stiffness and compliance matrices of laminates with different lay-ups.The engineering constants used to calculate the laminate stiffnesses and compliances are listed in Table 3.6.While the properties in this table are not intended to depict a particular material,they are characteris- tic of many graphite-epoxy composites.Therefore,the properties in Table 3.6 are used in the examples in the book. 3.1 Example.Calculate the stiffness [A,[B],[D]and the compliance [a],[B],[8] matrices of a [Oo/45o]laminate made of graphite epoxy unidirectional plies.The ply properties are given in Table 3.6. Solution.The stiffness matrix of a unidirectional ply with the fibers in the 0- degree direction is []=[O].The stiffness matrix [O]is given by Eq.(2.147), Table 3.6.Properties of the material used in the examples [0 ±45 Longitudinal Young's modulus (GPa) E 148 16.39 Transverse Young's modulus (GPa) 9.65 16.39 Longitudinal shear modulus(GPa) G12 4.55 38.19 Longitudinal Poission's ratio 2 0.3 0.801 Thickness(mm) ho 0.1 0.23.2 STIFFNESS MATRICES OF THIN LAMINATES 81 The superscript  refers to Reference Plane 2. When the laminate is symmet￾rical and the reference plane coincides with the midplane, the matrix [B] is zero. When the laminate is unsymmetrical, the matrix [B] is not zero. In general, there is no  value that results in a nonzero [B] matrix for an anisotropic composite laminate. In other words, for an unsymmetrical laminate there is no reference plane that is also a neutral plane. The compliance matrices [α], [β], and [δ] refer to Reference Plane 1. The elements of these matrices for Reference Plane 2 are obtained by introducing Eq. (3.47) into Eq. (3.23). After algebraic manipulations, we obtain α i j = αi j +  (βi j + βji) + 2 δi j β i j = βi j + δi j (3.48) δ  i j = δi j . The third of these equations shows that the bending compliance matrix [δ] is independent of the choice of the reference surface. Curved laminates. The stiffness and compliance matrices derived in this chap￾ter for flat laminates may be applied to thin curved laminates when the radius of curvature is large compared with the thickness. Numerical values of the stiffness and compliance matrices of selected lami￾nates. Below, we present numerical values of the stiffness and compliance matrices of laminates with different lay-ups. The engineering constants used to calculate the laminate stiffnesses and compliances are listed in Table 3.6. While the properties in this table are not intended to depict a particular material, they are characteris￾tic of many graphite-epoxy composites. Therefore, the properties in Table 3.6 are used in the examples in the book. 3.1 Example. Calculate the stiffness [A], [B], [D] and the compliance [α], [β], [δ] matrices of a [010/4510] laminate made of graphite epoxy unidirectional plies. The ply properties are given in Table 3.6. Solution. The stiffness matrix of a unidirectional ply with the fibers in the 0- degree direction is [Q] 0 = [Q]. The stiffness matrix [Q] is given by Eq. (2.147), Table 3.6. Properties of the material used in the examples [0] ±45f Longitudinal Young’s modulus (GPa) E1 148 16.39 Transverse Young’s modulus (GPa) E2 9.65 16.39 Longitudinal shear modulus (GPa) G12 4.55 38.19 Longitudinal Poission’s ratio ν12 0.3 0.801 Thickness (mm) h0 0.1 0.2