8 Laminate Analysis -Part II 8.1 Basic Equations In Chap.7,we derived the necessary formulas to calculate the strains and stresses through the thickness and the force and moment resultants given the strains and curvatures at a point (r,y)on the reference surface.In this chapter,we will study the reverse process.Given the force and moment resultants,we want to calculate the stresses and strains through the thickness as well as the strains and curvatures on the reference surface.We also want to do this by computing the laminate stiffness matrix. Figures 8.1 and 8.2 show the force and moment resultants,respectively.In the two figures,a small element of laminate surrounding a point (r,y)on the geometric midplane is shown [1]. The force resultants Nr,Ny,and Ny can be shown to be related to the strains and curvatures at the reference surface by the following equation: A11 A12 A16 e [B11 B12 B16 A12 A22 0 B12 B22 B26 (8.1) A16 A26 A66」 B16B26 B66 R%Y Fig.8.1.Schematic illustration of the force resultants on a composite laminate
8 Laminate Analysis – Part II 8.1 Basic Equations In Chap. 7, we derived the necessary formulas to calculate the strains and stresses through the thickness and the force and moment resultants given the strains and curvatures at a point (x, y) on the reference surface. In this chapter, we will study the reverse process. Given the force and moment resultants, we want to calculate the stresses and strains through the thickness as well as the strains and curvatures on the reference surface. We also want to do this by computing the laminate stiffness matrix. Figures 8.1 and 8.2 show the force and moment resultants, respectively. In the two figures, a small element of laminate surrounding a point (x, y) on the geometric midplane is shown [1]. The force resultants Nx, Ny, and Nxy can be shown to be related to the strains and curvatures at the reference surface by the following equation: ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ A11 A12 A16 A12 A22 A26 A16 A26 A66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ + ⎡ ⎢ ⎣ B11 B12 B16 B12 B22 B26 B16 B26 B66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ κ0 x κ0 y κ0 XY ⎫ ⎪⎬ ⎪⎭ (8.1) Fig. 8.1. Schematic illustration of the force resultants on a composite laminate
150 8 Laminate Analysis-Part II Fig.8.2.Schematic illustration of the moment resultants on a composite laminate Similarly,the moment resultants Mz,My,and Mry can also be shown to be related to the strains and curvatures at the reference surface by the following equation: B11 B12 B16 D12 D16 B12 B22 B26 (8.2) B16 B26 D26 D66 where the matrix components Ai,Bi,and Dii are given as follows: 4与=∑0.(k-k- (8.3) k=1 Bij= 2 O(绿-弟-） (8.4) Q(2-录-1） (8.5) Equations (8.1)and(8.2)can be combined into one single equation as follows: Nz A11 A12A16 B11B12 B16 c Ny A12 A22A26 B12 B22 B26 Ncy A16 A26A66B16 B26 B66 (8.6) M B11 B12 B16D11 D12 D16 My B12 B22B26D12 D22 D26 品 Mry B16B26B66D16D26D66 where the 6 x 6 matrix consisting of the components Ai,Bij,and Dij (i,j= 1,2,6)is called the laminate stiffness matrir,sometimes also called the ABD matrir. Note that the matrix components Aij,Bij,and Dij represent smeared or integrated properties of the laminate-this is because they are integrals(see [1])
150 8 Laminate Analysis – Part II Fig. 8.2. Schematic illustration of the moment resultants on a composite laminate Similarly, the moment resultants Mx, My, and Mxy can also be shown to be related to the strains and curvatures at the reference surface by the following equation: ⎧ ⎪⎨ ⎪⎩ Mx My Mxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ B11 B12 B16 B12 B22 B26 B16 B26 B66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ + ⎡ ⎢ ⎣ D11 D12 D16 D12 D22 D26 D16 D26 D66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ κ0 x κ0 y κ0 XY ⎫ ⎪⎬ ⎪⎭ (8.2) where the matrix components Aij , Bij , and Dij are given as follows: Aij = N k=1 Q¯ijk (zk − zk−1) (8.3) Bij = 1 2 N k=1 Q¯ijk z2 k − z2 k−1 (8.4) Dij = 1 3 N k=1 Q¯ijk z3 k − z3 k−1 (8.5) Equations (8.1) and (8.2) can be combined into one single equation as follows: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ Nx Ny Nxy Mx My Mxy ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 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 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε0 x ε0 y γ0 xy κ0 x κ0 y κ0 xy ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (8.6) where the 6 × 6 matrix consisting of the components Aij , Bij , and Dij (i,j = 1, 2, 6) is called the laminate stiffness matrix, sometimes also called the ABD matrix. Note that the matrix components Aij , Bij , and Dij represent smeared or integrated properties of the laminate – this is because they are integrals (see [1])
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 properties, and specific fiber orientation, there is another layer the same distance on the opposite side of the reference surface with the same thickness, material properties, 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 another layer with the same thickness, material properties, but opposite fiber orientation 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:
152 8 Laminate Analysis-Part II (8.11) Nv AcGYu (8.12) M [D11 D12 D16 My D12 D22 D26 0 (8.13) My) D16 D26 D66」 4.Cross-Ply Laminates-A laminate is a cross-ply laminate if every layer has its fibers oriented at either 0 or 90.In this case,the components A16,A26,Bi6, B26,Di6,and D26 are all zero. 8.2 MATLAB Functions Used The three MATLAB functions used in this chapter to calculate the [A],[B],and [D] matrices are: Amatrir(A,Qbar,z1,22)-This function calculates the [A]matrix for a laminate consisting of N layers where each layer k(k 1,2,3,...,N)has a transformed reduced stiffness matrix [There are four input arguments to this function. This function assembles the desired matrix after each layer's effect is included in a separate call to this function.The parameters zl and 22 are zk-1 and zk,respectively, for layer k.The function returns the 3 x 3 matrix [A]. Bmatrir(B,Qbar,21,22)-This function calculates the [B]matrix for a laminate consisting of N layers where each layer k(k=1,2,3,...,N)has a transformed reduced stiffness matrix [There are four input arguments to this function. This function assembles the desired matrix after each layer's effect is included in a separate call to this function.The parameters z1 and 22 are zk-1 and zk,respectively, for layer k.The function returns the 3 x 3 matrix [B]. Dmatrir(D,Qbar,z1,22)-This function calculates the [D]matrix for a laminate consisting of N layers where each layer k(k 1,2,3,...,N)has a transformed reduced stiffness matrix [There are four input arguments to this function. This function assembles the desired matrix after each layer's effect is included in a separate call to this function.The parameters z1 and 22 are zk-1 and zk,respectively, for layer k.The function returns the 3 x 3 matrix [D]. The following is a listing of the MATLAB source code for these functions: function y Amatrix(A,Qbar,z1,z2) %Amatrix This function returns the [A]matrix % after the layer k with stiffness [Qbar] % is assembled. % A -[A]matrix after layer k % is assembled. % Qbar -[Qbar]matrix for layer k % z1 -z(k-1)for layer k % z2 -z(k)for layer k
152 8 Laminate Analysis – Part II Nx Ny = A11 A12 A12 A22 ε0 x ε0 y (8.11) Nxy = A66γ0 xy (8.12) ⎧ ⎪⎨ ⎪⎩ Mx My Mxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ D11 D12 D16 D12 D22 D26 D16 D26 D66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ κ0 x κ0 y κ0 XY ⎫ ⎪⎬ ⎪⎭ (8.13) 4. Cross-Ply Laminates – A laminate is a cross-ply laminate if every layer has its fibers oriented at either 0◦ or 90◦. In this case, the components A16, A26, B16, B26, D16, and D26 are all zero. 8.2 MATLAB Functions Used The three MATLAB functions used in this chapter to calculate the [A], [B], and [D] matrices are: Amatrix(A, Qbar, z1, z2) – This function calculates the [A] matrix for a laminate consisting of N layers where each layer k (k = 1, 2, 3,... ,N) has a transformed reduced stiffness matrix ! Q¯"k . There are four input arguments to this function. This function assembles the desired matrix after each layer’s effect is included in a separate call to this function. The parameters z1 and z2 are zk−1 and zk, respectively, for layer k. The function returns the 3 × 3 matrix [A]. Bmatrix(B, Qbar, z1, z2) – This function calculates the [B] matrix for a laminate consisting of N layers where each layer k (k = 1, 2, 3,... ,N) has a transformed reduced stiffness matrix ! Q¯"k . There are four input arguments to this function. This function assembles the desired matrix after each layer’s effect is included in a separate call to this function. The parameters z1 and z2 are zk−1 and zk, respectively, for layer k. The function returns the 3 × 3 matrix [B]. Dmatrix(D, Qbar, z1, z2) – This function calculates the [D] matrix for a laminate consisting of N layers where each layer k (k = 1, 2, 3,... ,N) has a transformed reduced stiffness matrix ! Q¯"k . There are four input arguments to this function. This function assembles the desired matrix after each layer’s effect is included in a separate call to this function. The parameters z1 and z2 are zk−1 and zk, respectively, for layer k. The function returns the 3 × 3 matrix [D]. The following is a listing of the MATLAB source code for these functions: function y = Amatrix(A,Qbar,z1,z2) %Amatrix This function returns the [A] matrix % after the layer k with stiffness [Qbar] % is assembled. % A - [A] matrix after layer k % is assembled. % Qbar - [Qbar] matrix for layer k % z1 - z(k-1) for layer k % z2 - z(k) for layer k
8.2 MATLAB Functions Used 153 for1=1:3 for j=1 3 A(i,j)=A(i,j)+Qbar(i,j)*(z2-z1); end end y=A; function y Bmatrix(B,Qbar,z1,22) XBmatrix This function returns the [B]matrix % after the layer k with stiffness [Qbar] % is assembled. % B -[B]matrix after layer k % is assembled. % Qbar [Qbar]matrix for layer k % z1 -z(k-1)for layer k z2 -z(k)for layer k for i 1 3 for j=1:3 B(i,j)=B(i,j)+Qbar(1,j)*(z22-z1^2); end end y=B/2; function y Dmatrix(D,Qbar,z1,22) %Dmatrix This function returns the [D]matrix % after the layer k with stiffness [Qbar] % is assembled. % D [D]matrix after layer k % is assembled. % Qbar -[Qbar]matrix for layer k z1 -z(k-1)for layer k % z2 -z(k)for layer k for1=1:3 for j=1:3 D(1,j)=D(1,j)+Qbar(i,j)*(z2^3-z13); end end y=D/3; Example 8.1 Derive(8.3)and (8.4)in detail. Solution The derivation of (8.3)and(8.4)involves using (7.13a),(7.13b),and(7.13c)along with (7.12).Substitute the expression of obtained from (7.12)into (7.13a)to obtain:
8.2 MATLAB Functions Used 153 for i = 1 : 3 for j = 1 : 3 A(i,j) = A(i,j) + Qbar(i,j)*(z2-z1); end end y = A; function y = Bmatrix(B,Qbar,z1,z2) %Bmatrix This function returns the [B] matrix % after the layer k with stiffness [Qbar] % is assembled. % B - [B] matrix after layer k % is assembled. % Qbar - [Qbar] matrix for layer k % z1 - z(k-1) for layer k % z2 - z(k) for layer k for i = 1 : 3 for j = 1 : 3 B(i,j) = B(i,j) + Qbar(i,j)*(z2^2 -z1^2); end end y = B/2; function y = Dmatrix(D,Qbar,z1,z2) %Dmatrix This function returns the [D] matrix % after the layer k with stiffness [Qbar] % is assembled. % D - [D] matrix after layer k % is assembled. % Qbar - [Qbar] matrix for layer k % z1 - z(k-1) for layer k % z2 - z(k) for layer k for i = 1 : 3 for j = 1 : 3 D(i,j) = D(i,j) + Qbar(i,j)*(z2^3 -z1^3); end end y = D/3; Example 8.1 Derive (8.3) and (8.4) in detail. Solution The derivation of (8.3) and (8.4) involves using (7.13a), (7.13b), and (7.13c) along with (7.12). Substitute the expression of σx obtained from (7.12) into (7.13a) to obtain:
154 8 Laminate Analysis-Part II H/2 N:= [011(e9+z)+Q12(e9+zk9)+Q16(h2y+zk9)月dz (8.14) -H/2 Expanding (8.14),we obtain: H/2 H/2 H/2 H/2 N:=e2 Qidz+ Quzds+eQ12dz+ Q12zdz -H/2 -H/2 -H/2 -H/2 H/2 H/2 +7v Qio dz+ Q16 zdz (8.15) -H/2 -H/2 Next,we expand the first term of (8.15)as follows: H/2 Qndz= 1dz+…+ ++ 11dz(8.16) -H/2 艺0 k-1 zN-1 Recognizing that Qu is constant within each layer,it can be taken outside the integrals above leading to the following expression: H/2 01dz=Q11(1-z0)+011(2-z1）+…+011(-zk-1） -H/2 +…+Q11(2N-2N-1) (8.17) The above equation can be re-written as follows: H/2 N O1dz=∑O1(k-zk-1)=A1 (8.18) -H/2 k=1 Similarly,we can show that the other five integrals of (8.15)can be written as follows: H/2 N 012dz=02(k-zk-1）=A12 (8.19a) k=1 -H/2 H/2 N Q16d2=∑Q16(2k-zk-1)=A16 (8.19b) -H/2 k=1 H2 Quzd:= 1 ∑O(--)=B (8.19c) H/2 k=
154 8 Laminate Analysis – Part II Nx = H/ 2 −H/2 ! Q¯11 ε 0 x + zκ0 x + Q¯12 ε 0 y + zκ0 y + Q¯16 γ0 xy + zκ0 xy" dz (8.14) Expanding (8.14), we obtain: Nx = ε 0 x H/ 2 −H/2 Q¯11 dz + κ0 x H/ 2 −H/2 Q¯11 zdz + ε 0 y H/ 2 −H/2 Q¯12 dz + κ0 y H/ 2 −H/2 Q¯12 zdz +γ0 xy H/ 2 −H/2 Q¯16 dz + κ0 xy H/ 2 −H/2 Q¯16 zdz (8.15) Next, we expand the first term of (8.15) as follows: H/ 2 −H/2 Q¯11 dz = z1 z0 Q¯11dz + z2 z1 Q¯11dz + ··· + zk zk−1 Q¯11dz + ··· + zN zN−1 Q¯11dz (8.16) Recognizing that Q¯11 is constant within each layer, it can be taken outside the integrals above leading to the following expression: H/ 2 −H/2 Q¯11 dz = Q¯11 (z1 − z0) + Q¯11 (z2 − z1) + ··· + Q¯11 (zk − zk−1) + ··· + Q¯11 (zN − zN−1) (8.17) The above equation can be re-written as follows: H/ 2 −H/2 Q¯11 dz = N k=1 Q¯11 (zk − zk−1) = A11 (8.18) Similarly, we can show that the other five integrals of (8.15) can be written as follows: H/ 2 −H/2 Q¯12 dz = N k=1 Q¯12 (zk − zk−1) = A12 (8.19a) H/ 2 −H/2 Q¯16 dz = N k=1 Q¯16 (zk − zk−1) = A16 (8.19b) H/ 2 −H/2 Q¯11 zdz = 1 2 N k=1 Q¯11 z2 k − z2 k−1 = B11 (8.19c)
8.2 MATLAB Functions Used 155 H/2 -H/2 ah=20保-)= (8.19d) H/2 016dz= Q16(绿-录-1)=B16 (8.19e) 2 -H/2 k=1 Using the remaining two equations of the matrix(7.12),we obtain the general desired expressions as follows: N A=O(2k-2k-1) (8.20) k=1 0(绿-录-1)》 (8.21) k三1 MATLAB Example 8.2 Consider a graphite-reinforced polymer composite laminate with the elastic con- stants as given in Example 2.2.The laminate has total thickness of 0.500 mm and is stacked as a [0/90s laminate.The four layers are of equal thickness.Calculate the [A,[B],and [D]matrices for this laminate. Solution This example is solved using MATLAB.First,the reduced stiffness matrix [Q]for a typical layer using the MATLAB function ReducedStiffness as follows: >>Q=ReducedStiffness(155.0,12.10,0.248,4.40) 0= 155.7478 3.0153 3.0153 12.1584 0 0 0 4.4000 Next,the transformed reduced stiffness matrix is calculated for each layer using the MATLAB function Qbar as follows: >Qbar1 Qbar(Q,0) Qbar1 155.7478 3.0153 0 3.0153 12.1584 0 0 4.4000
8.2 MATLAB Functions Used 155 H/ 2 −H/2 Q¯12 zdz = 1 2 N k=1 Q¯12 z2 k − z2 k−1 = B12 (8.19d) H/ 2 −H/2 Q¯16 zdz = 1 2 N k=1 Q¯16 z2 k − z2 k−1 = B16 (8.19e) Using the remaining two equations of the matrix (7.12), we obtain the general desired expressions as follows: Aij = N k=1 Q¯ij (zk − zk−1) (8.20) Bij = 1 2 N k=1 Q¯ij z2 k − z2 k−1 (8.21) MATLAB Example 8.2 Consider a graphite-reinforced polymer composite laminate with the elastic constants as given in Example 2.2. The laminate has total thickness of 0.500 mm and is stacked as a [0/90]S laminate. The four layers are of equal thickness. Calculate the [A], [B], and [D] matrices for this laminate. Solution This example is solved using MATLAB. First, the reduced stiffness matrix [Q] for a typical layer using the MATLAB function ReducedStiffness as follows: >> Q = ReducedStiffness(155.0, 12.10, 0.248, 4.40) Q = 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000 Next, the transformed reduced stiffness matrix ! Q¯" is calculated for each layer using the MATLAB function Qbar as follows: >> Qbar1 = Qbar(Q,0) Qbar1 = 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000
156 8 Laminate Analysis-Part II >Qbar2 Qbar(Q,90) Qbar2 12.1584 3.0153 -0.0000 3.0153 155.7478 0.0000 -0.0000 0.0000 4.4000 >Qbar3 =Qbar(Q,90) Qbar3 12.1584 3.0153 -0.0000 3.0153 155.7478 0.0000 -0.0000 0.0000 4.4000 >Qbar4 Qbar(Q,0) Qbar4 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000 Next,the distances zk(1,2,3,4,5)are calculated as follows: >>z1=-0.250 z1= -0.2500 >>z2=-0.125 z2= -0.1250 >z3=0 z3= 0 >z4=0.125 z4= 0.1250
156 8 Laminate Analysis – Part II >> Qbar2 = Qbar(Q,90) Qbar2 = 12.1584 3.0153 -0.0000 3.0153 155.7478 0.0000 -0.0000 0.0000 4.4000 >> Qbar3 = Qbar(Q,90) Qbar3 = 12.1584 3.0153 -0.0000 3.0153 155.7478 0.0000 -0.0000 0.0000 4.4000 >> Qbar4 = Qbar(Q,0) Qbar4 = 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000 Next, the distances zk(k = 1, 2, 3, 4, 5) are calculated as follows: >> z1 = -0.250 z1 = -0.2500 >> z2 = -0.125 z2 = -0.1250 >> z3 = 0 z3 = 0 >> z4 = 0.125 z4 = 0.1250
8.2 MATLAB Functions Used 157 >>z5=0.250 z5= 0.2500 Next,the [A]matrix is calculated using four calls to the MATLAB function Amatrir as follows: >>A=zeros(3,3) A= 0 0 0 0 0 0 0 0 0 >A Amatrix(A,Qbar1,z1,z2) A= 19.4685 0.3769 0 0.3769 1.5198 0 0 0 0.5500 >A Amatrix(A,Qbar2,z2,z3) A= 20.9883 0.7538 -0.0000 0.7538 20.9883 0.0000 -0.0000 0.0000 1.1000 >>A Amatrix(A,Qbar3,z3,z4) A= 22.5081 1.1307 -0.0000 1.1307 40.4567 0.0000 -0.0000 0.0000 1.6500 >A Amatrix(A,Qbar4,z4,z5) A= 41.9765 1.5076 -0.0000 1.5076 41.9765 0.0000 -0.0000 0.0000 2.2000 Next,the [B]matrix is calculated using four calls to the MATLAB function Bmatrir as follows (make sure to divide the final result by 2 since this step is not performed by the Bmatrir function):
8.2 MATLAB Functions Used 157 >> z5 = 0.250 z5 = 0.2500 Next, the [A] matrix is calculated using four calls to the MATLAB function Amatrix as follows: >> A = zeros(3,3) A = 0 00 0 00 0 00 >> A = Amatrix(A,Qbar1,z1,z2) A = 19.4685 0.3769 0 0.3769 1.5198 0 0 0 0.5500 >> A = Amatrix(A,Qbar2,z2,z3) A = 20.9883 0.7538 -0.0000 0.7538 20.9883 0.0000 -0.0000 0.0000 1.1000 >> A = Amatrix(A,Qbar3,z3,z4) A = 22.5081 1.1307 -0.0000 1.1307 40.4567 0.0000 -0.0000 0.0000 1.6500 >> A = Amatrix(A,Qbar4,z4,z5) A = 41.9765 1.5076 -0.0000 1.5076 41.9765 0.0000 -0.0000 0.0000 2.2000 Next, the [B] matrix is calculated using four calls to the MATLAB function Bmatrix as follows (make sure to divide the final result by 2 since this step is not performed by the Bmatrix function):
158 8 Laminate Analysis-Part II >>B=zeros(3,3) B= 0 0 0 0 0 0 0 0 0 >B Bmatrix(B,Qbar1,z1,z2) B= -7.3007 -0.1413 0 -0.1413 -0.5699 0 0 0 -0.2063 >>B Bmatrix(B,Qbar2,z2,z3) B= -7.4907 -0.1885 0.0000 -0.1885 -3.0035 -0.0000 0.0000 -0.0000 -0.2750 >>B =Bmatrix(B,Qbar3,z3,z4) B= -7.3007 -0.1413 0 -0.1413 -0.5699 0 0 0 -0.2063 >>B =Bmatrix(B,Qbar4,z4,z5) B= 1.0e-015* 0 0 0 0 -0.1110 0 0 0 0 >>B=B/2 B= 1.0e-016*
158 8 Laminate Analysis – Part II >> B = zeros(3,3) B = 000 000 000 >> B = Bmatrix(B,Qbar1,z1, z2) B = -7.3007 -0.1413 0 -0.1413 -0.5699 0 0 0 -0.2063 >> B = Bmatrix(B,Qbar2,z2, z3) B = -7.4907 -0.1885 0.0000 -0.1885 -3.0035 -0.0000 0.0000 -0.0000 -0.2750 >> B = Bmatrix(B,Qbar3,z3, z4) B = -7.3007 -0.1413 0 -0.1413 -0.5699 0 0 0 -0.2063 >> B = Bmatrix(B,Qbar4,z4, z5) B = 1.0e-015 * 0 00 0 -0.1110 0 0 00 >> B = B/2 B = 1.0e-016 *