62 5 Global Coordinate System n sin(theta*pi/180); T=[m*mn知2*m知；n*nm*m-2*m知；-m*nm*nm*m-n*n]; Tinv [m*m n*n -2*m*nn*n m*m 2*m*n m*n -m*n m*m-n*n] y=Tinv*S+T; function y Qbar(Q,theta) %Qbar This function returns the transformed reduced stiffness matrix "Qbar"given the reduced 名 stiffness matrix Q and the orientation angle "theta". There are two arguments representing Q and "theta" The size of the matrix is 3 x 3. % The angle "theta"must be given in degrees. m=cos(theta*pi/180); n sin(theta*pi/180); T=[m*mnn2*m知；n*nm*m-2*m*知；-m*nm*nm*m-n*n]; Tinv [m*m n*n -2*m*nn*n m*m 2*m*n m*n -m*n m*m-n+n] y Tinv*Q*T; Example 5.1 Using(5.11),derive explicit expressions for the elements Sij in terms of S and8(use m and n forθ). Solution Multiply the three matrices in (5.11)as follows: 511 512 5161 「m2 n2 -2mn S11 S12 7 0 512 522 526 n2 m2 2mn S12 S22 0 516 526 566 mn -mn m2-n2 0 0 S66 23 (5.15) n2 2mn n2 m2 -2mn -mn mn m2-n2 The above multiplication can be performed either manually or using a com- puter algebra system like MAPLE or MATHEMATICA or the MATLAB Sym- bolic Math Toolbox.Therefore,we obtain the following expression: 511=S11m4+(2S12+S66)n2m2+S22n4 (5.16a) 512=(S11+S22-S66)n2m2+S12(n4+m4) (5.16b) 516=(2S11-2S12-S66)nm3-(2S22-2S12-S66)n3m (5.16c) 522=S11n4+(2512+S66)n2m2+S22m4 (5.16d) 526=(2S11-2S12-S66)n3m-(2S22-2512-S66)nm3 (5.16e 566=2(2S11+252-4S12-S66)n2m2+S66(n4+m4) (5.16f)
62 5 Global Coordinate System n = sin(theta*pi/180); T = [m*m n*n 2*m*n ; n*n m*m -2*m*n ; -m*n m*n m*m-n*n]; Tinv = [m*m n*n -2*m*n ; n*n m*m 2*m*n ; m*n -m*n m*m-n*n]; y = Tinv*S*T; function y = Qbar(Q,theta) %Qbar This function returns the transformed reduced % stiffness matrix "Qbar" given the reduced % stiffness matrix Q and the orientation % angle "theta". % There are two arguments representing Q and "theta" % The size of the matrix is 3 x 3. % The angle "theta" must be given in degrees. m = cos(theta*pi/180); n = sin(theta*pi/180); T = [m*m n*n 2*m*n ; n*n m*m -2*m*n ; -m*n m*n m*m-n*n]; Tinv = [m*m n*n -2*m*n ; n*n m*m 2*m*n ; m*n -m*n m*m-n*n]; y = Tinv*Q*T; Example 5.1 Using (5.11), derive explicit expressions for the elements S¯ij in terms of Sij and θ (use m and n for θ). Solution Multiply the three matrices in (5.11) as follows: ⎡ ⎢ ⎣ S¯11 S¯12 S¯16 S¯12 S¯22 S¯26 S¯16 S¯26 S¯66 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ m2 n2 −2mn n2 m2 2mn mn −mn m2 − n2 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ S11 S12 0 S12 S22 0 0 0 S66 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ m2 n2 2mn n2 m2 −2mn −mn mn m2 − n2 ⎤ ⎥ ⎦ (5.15) The above multiplication can be performed either manually or using a computer algebra system like MAPLE or MATHEMATICA or the MATLAB Symbolic Math Toolbox. Therefore, we obtain the following expression: S¯11 = S11m4 + (2S12 + S66)n2m2 + S22n4 (5.16a) S¯12 = (S11 + S22 − S66)n2m2 + S12(n4 + m4) (5.16b) S¯16 = (2S11 − 2S12 − S66)nm3 − (2S22 − 2S12 − S66)n3m (5.16c) S¯22 = S11n4 + (2S12 + S66)n2m2 + S22m4 (5.16d) S¯26 = (2S11 − 2S12 − S66)n3m − (2S22 − 2S12 − S66)nm3 (5.16e) S¯66 = 2(2S11 + 2S22 − 4S12 − S66)n2m2 + S66(n4 + m4) (5.16f)
5.2 MATLAB Functions Used 63 MATLAB Example 5.2 Consider a graphite-reinforced polymer composite lamina with the elastic con- stants as given in Example 2.2.Use MATLAB to plot the values of the six elements Sij of the transformed reduced compliance matrix [S]as a function of the orientation angle0 in the range-π/2≤0≤π/2. Solution This example is solved using MATLAB.First,the reduced 3 x 3 compliance matrix is obtained as follows using the MATLAB function ReducedCompliance of Chap.4. >>S=ReducedComp1 iance(155.0,12.10,0.248,4.40) S= 0.0065 -0.0016 0 -0.0016 0.0826 0 0 00.2273 Next,the transformed reduced compliance matrix [S]is calculated at each value of9 between-g0°andg0°in increments of 10°using the MATLAB function Sbar. >S1=Sbar(s,-90) S1= 0.0826 -0.0016 -0.0000 -0.0016 0.0065 0.0000 -0.0000 0.0000 0.2273 >S2=Sbar(s,-80) S2= 0.0909 -0.0122 -0.0452 -0.0122 0.0193 0.0712 -0.0226 0.0356 0.2061 >S3=Sbar(s,-70) S3= 0.1111 -0.0390 -0.0647 -0.0390 0.0528 0.1137 -0.0323 0.0568 0.1524
5.2 MATLAB Functions Used 63 MATLAB Example 5.2 Consider a graphite-reinforced polymer composite lamina with the elastic constants as given in Example 2.2. Use MATLAB to plot the values of the six elements S¯ij of the transformed reduced compliance matrix [S¯] as a function of the orientation angle θ in the range −π/2 ≤ θ ≤ π/2. Solution This example is solved using MATLAB. First, the reduced 3 × 3 compliance matrix is obtained as follows using the MATLAB function ReducedCompliance of Chap. 4. >> S = ReducedCompliance(155.0, 12.10, 0.248, 4.40) S = 0.0065 -0.0016 0 -0.0016 0.0826 0 0 0 0.2273 Next, the transformed reduced compliance matrix [S¯] is calculated at each value of θ between −90◦ and 90◦ in increments of 10◦ using the MATLAB function Sbar . >> S1 = Sbar(S, -90) S1 = 0.0826 -0.0016 -0.0000 -0.0016 0.0065 0.0000 -0.0000 0.0000 0.2273 >> S2 = Sbar(S, -80) S2 = 0.0909 -0.0122 -0.0452 -0.0122 0.0193 0.0712 -0.0226 0.0356 0.2061 >> S3 = Sbar(S, -70) S3 = 0.1111 -0.0390 -0.0647 -0.0390 0.0528 0.1137 -0.0323 0.0568 0.1524
64 5 Global Coordinate System >S4=Sbar(S,-60) S4= 0.1315 -0.0695 -0.0454 -0.0695 0.0934 0.1114 -0.0227 0.0557 0.0914 >S5=Sbar(S,-50) S5= 0.1390 -0.0894 0.0065 -0.0894 0.1258 0.0685 0.0033 0.0342 0.0516 >S6=Sbar(S,-40) S6= 0.1258 -0.0894 0.0685 -0.0894 0.1390 0.0065 0.0342 0.0033 0.0516 >s7=sbar(s,-30) S7= 0.0934 -0.0695 0.1114 -0.0695 0.1315 -0.0454 0.0557 -0.0227 0.0914 >s8=Sbar(s,-20) S8=- 0.0528 -0.0390 0.1137 -0.0390 0.1111 -0.0647 0.0568 -0.0323 0.1524 >>S9=Sbar(S,-10) S9= 0.0193 -0.0122 0.0712 -0.0122 0.0909 -0.0452 0.0356 -0.0226 0.2061
64 5 Global Coordinate System >> S4 = Sbar(S, -60) S4 = 0.1315 -0.0695 -0.0454 -0.0695 0.0934 0.1114 -0.0227 0.0557 0.0914 >> S5 = Sbar(S, -50) S5 = 0.1390 -0.0894 0.0065 -0.0894 0.1258 0.0685 0.0033 0.0342 0.0516 >> S6 = Sbar(S, -40) S6 = 0.1258 -0.0894 0.0685 -0.0894 0.1390 0.0065 0.0342 0.0033 0.0516 >> S7 = Sbar(S, -30) S7 = 0.0934 -0.0695 0.1114 -0.0695 0.1315 -0.0454 0.0557 -0.0227 0.0914 >> S8 = Sbar(S, -20) S8 = 0.0528 -0.0390 0.1137 -0.0390 0.1111 -0.0647 0.0568 -0.0323 0.1524 >> S9 = Sbar(S, -10) S9 = 0.0193 -0.0122 0.0712 -0.0122 0.0909 -0.0452 0.0356 -0.0226 0.2061