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: