Calculation of matrices A, B, D of a laminate

 

    As an example, we consider the calculation of matrices A, B and D of a lamionater. The laminate considered is constituted of 4 layers of a  unidirectional material : EL = 38 GPa, ET = 9 GPa, υLT = 0.32, GLT = 3.6 GPa. The thicknesses of the layers are 1, 1,5, 1,5 and 1 mm, respectively starting from the bottom layer. The orientations of the layers are couches: 30, –15, 15 and –30°, respectively.

 

    We give herafter an example of the command file of Scilab and the results obtained for the matrices.

 

File.sce

 

n=4

data=[38 9 0.32 3.6 1 -30

      38 9 0.32 3.6 1.5 15

      38 9 0.32 3.6 1.5 -15

      38 9 0.32 3.6 1 30]

A=[0 0 0;0 0 0;0 0 0]; B=[0 0 0;0 0 0;0 0 0]; D=[0 0 0;0 0 0;0 0 0];

 et=0; e(1)=0;

for k=1:n

    e(k)=data(k,5);et=et+e(k);

end

disp(e,et );

zk=-et/2+0.5*e(1);z(1)=zk;

for k=2:n

    zk=zk+0.5*(e(k-1)+e(k));z(k)=zk;

end

for k=1:n ;

    EL=data(k,1);ET=data(k,2);nuLT=data(k,3);GLT=data(k,4);

    Q(k,1)=EL/(1-(ET/EL)*nuLT^2); Q(k,3)=ET*Q(k,1)/EL;

    Q(k,2)=nuLT*Q(k,3);Q(k,4)=GLT;

    teta=data(k,6);ci=cos(teta*%pi/180); si=sin(teta*%pi/180);

    Qp(k,1)=Q(k,1)*ci^4+Q(k,3)*si^4+2*(Q(k,2)+2*Q(k,4))*(ci*si)^2;

    Qp(k,2)=(Q(k,1)+Q(k,3)-4*Q(k,4))*(ci*si)^2+Q(k,2)*(ci^4+si^4);

    Qp(k,3)=(Q(k,1)-Q(k,2)-2*Q(k,4))*si*ci^3+(Q(k,2)-Q(k,3)+2*Q(k,4))*si^3*ci;

    Qp(k,4)=Q(k,1)*si^4+Q(k,3)*ci^4+2*(Q(k,2)+2*Q(k,4))*(ci*si)^2;

    Qp(k,5)=(Q(k,1)-Q(k,2)-2*Q(k,4))*ci*si^3+(Q(k,2)-Q(k,3)+2*Q(k,4))*ci^3*si;

    Qp(k,6)=(Q(k,1)+Q(k,3)-2*(Q(k,2)+Q(k,4)))*(ci*si)^2+Q(k,4)*(ci^4+si^4);

    A(1,1)=A(1,1)+Qp(k,1)*e(k);A(1,2)=A(1,2)+Qp(k,2)*e(k);

    A(1,3)=A(1,3)+Qp(k,3)*e(k); A(2,2)=A(2,2)+Qp(k,4)*e(k);

    A(2,3)=A(2,3)+Qp(k,5)*e(k); A(3,3)=A(3,3)+Qp(k,6)*e(k);

    A(2,1)=A(1,2); A(3,1)=A(1,3); A(3,2)=A(2,3);

    B(1,1)=B(1,1)+Qp(k,1)*e(k)*z(k); B(1,2)=B(1,2)+Qp(k,2)*e(k)*z(k);

    B(1,3)=B(1,3)+Qp(k,3)*e(k)*z(k); B(3,3)=B(3,3)+Qp(k,6)*e(k)*z(k);

    B(2,2)=B(2,2)+Qp(k,6)*e(k)*z(k);B(2,3)=B(2,3)+Qp(k,5)*e(k)*z(k);

    B(2,1)=B(1,2); B(3,1)=B(1,3); B(3,2)=B(2,3);

    D(1,1)=D(1,1)+Qp(k,1)*(e(k)*z(k)^2+(1/12)*e(k)^3);

    D(1,2)=D(1,2)+Qp(k,2)*(e(k)*z(k)^2+(1/12)*e(k)^3);

    D(1,3)=D(1,3)+Qp(k,3)*(e(k)*z(k)^2+(1/12)*e(k)^3);

    D(2,2)=D(2,2)+Qp(k,4)*(e(k)*z(k)^2+(1/12)*e(k)^3);

    D(2,3)=D(2,3)+Qp(k,5)*(e(k)*z(k)^2+(1/12)*e(k)^3);

    D(3,3)=D(3,3)+Qp(k,6)*(e(k)*z(k)^2+(1/12)*e(k)^3); D(2,1)=D(1,2);

    D(3,1)=D(1,3); D(3,2)=D(2,3)

end

disp("matrix A  x10^6 N/m");disp(A);disp("matrix B  x10^3 N");disp(B);disp("matrix D  Nm");disp(D);

 

 

Results

 

matrix  A  x10^6 N/m  

 

    158.2153     30.432002    0.        

    30.432002    51.277564    0.        

    0.           0.           33.674084 

 

 matrix   B  x10^3 N  

 

    0.           0.           22.658835 

    0.           0.           12.101159 

    22.658835    12.101159    0.        

 

 matrix   D  Nm  

 

    293.92554    77.332524    0.        

    77.332524    114.65288    0.        

    0.           0.           84.086861