8.1 Basic Equations 151 In order to be able to obtain the strains and curvatures at the reference surface in terms of the force and moment resultants,the inverse of(8.6)is written as follows [1]: e a11 a12116 611 612 b16 Nr es a12 a22 a26 b12 b22 b26 Ny a16 a26 a66 b16 b26 b66 Ncy 9 (8.7) b11 b12 b16 di d2 d16 M g b12 b22 b26 d12 d22 d26 My b16 b26 b66 dis d26 d66 where a11 a12 a16 b11 b12 b16 A11 A12 A16 B11 B12 B16]-1 a12 a22 a26 b12 b22 b26 A12 A22 A26 B12 B22 B26 a16 a26 a66 b16 b26 b66 A16 A26 A66 B16 B26 B66 b11 b12 b16 du di2 d16 B11 B12 B16 D11 D12D16 b12 b22 b26 d2 d22 d26 B12 B22 B26 D12 D22 D26 b16 b26 b66 d16 d26 d66 B16 B26B66 D16 D26D66 (8.8) Next,we consider the classification of laminates and their effect on the ABD matrix.Laminates are usually classified into the following five categories [1]: 1.Symmetric Laminates-A laminate is symmetric if for every layer to one side of the laminate reference surface with a specific thickness,specific material prop- erties,and specific fiber orientation,there is another layer the same distance on the opposite side of the reference surface with the same thickness,material prop- erties,and fiber orientation.If the laminate is not symmetric,then it is referred to as an unsymmetric laminate. For a symmetric laminate,all the components of the B matrix are identically zero.Therefore,we have the following decoupled system of equations: A11 A12 A16 A12 A22 A26 (8.9) Nry A16 A26 A66 .0 D12 D16 D12 D22 D26 (8.10) D16 D26 D66 Y 2.Balanced Laminates-A laminate is balanced if for every layer with a specific thickness,specific material properties,and specific fiber orientation,there is an- other layer with the same thickness,material properties,but opposite fiber ori- entation somewhere in the laminate.The other layer can be anywhere within the thickness.For balanced laminates,the stiffness matrix components A16 and A26 are always zero. 3.Symmetric Balanced Laminates-A laminate is a symmetric balanced laminate if it meets both the criterion of being symmetric and the criterion of being balanced. In this case,we have the following decoupled system of equations:8.1 Basic Equations 151 In order to be able to obtain the strains and curvatures at the reference surface in terms of the force and moment resultants, the inverse of (8.6) is written as follows [1]: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε0 x ε0 y γ0 xy κ0 x κ0 y κ0 xy ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ =                a11 a12 a16 b11 b12 b16 a12 a22 a26 b12 b22 b26 a16 a26 a66 b16 b26 b66 b11 b12 b16 d11 d12 d16 b12 b22 b26 d12 d22 d26 b16 b26 b66 d16 d26 d66                ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ Nx Ny Nxy Mx My Mxy ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (8.7) where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a11 a12 a16 b11 b12 b16 a12 a22 a26 b12 b22 b26 a16 a26 a66 b16 b26 b66 b11 b12 b16 d11 d12 d16 b12 b22 b26 d12 d22 d26 b16 b26 b66 d16 d26 d66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −1 (8.8) Next, we consider the classification of laminates and their effect on the ABD matrix. Laminates are usually classified into the following five categories [1]: 1. Symmetric Laminates – A laminate is symmetric if for every layer to one side of the laminate reference surface with a specific thickness, specific material prop￾erties, and specific fiber orientation, there is another layer the same distance on the opposite side of the reference surface with the same thickness, material prop￾erties, and fiber orientation. If the laminate is not symmetric, then it is referred to as an unsymmetric laminate. For a symmetric laminate, all the components of the B matrix are identically zero. Therefore, we have the following decoupled system of equations: ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ A11 A12 A16 A12 A22 A26 A16 A26 A66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ (8.9) ⎧ ⎪⎨ ⎪⎩ Mx My Mxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ D11 D12 D16 D12 D22 D26 D16 D26 D66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ κ0 x κ0 y κ0 XY ⎫ ⎪⎬ ⎪⎭ (8.10) 2. Balanced Laminates – A laminate is balanced if for every layer with a specific thickness, specific material properties, and specific fiber orientation, there is an￾other layer with the same thickness, material properties, but opposite fiber ori￾entation somewhere in the laminate. The other layer can be anywhere within the thickness. For balanced laminates, the stiffness matrix components A16 and A26 are always zero. 3. Symmetric Balanced Laminates – A laminate is a symmetric balanced laminate if it meets both the criterion of being symmetric and the criterion of being balanced. In this case, we have the following decoupled system of equations: