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 com￾puter algebra system like MAPLE or MATHEMATICA or the MATLAB Sym￾bolic 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)