1. Contents
Introduction ............................................................................................................................... 1
Case Studied.............................................................................................................................. 1
Procedure of MATLAB code................................................................................................... 2
Results ....................................................................................................................................... 2
Appendix ................................................................................................................................... 4
function to calculate A,B and D matrices for one lamina ......................................................... 4
Main code.................................................................................................................................. 4
Reference................................................................................................................................... 6

2. 1
Introduction
Most engineers have considerable training and experience in the design of simple
structural components, such as bars, shafts and beams, using isotropic materials. Laminate
moduli can be used to take advantage of this knowledge for the design of composite
structures. Using a laminate moduli, a laminate is thought of as a homogeneous plate with
apparent properties ( , etc). However, the laminate moduli can be computed only after
selecting the laminate configuration. To simplify the design process, plots of apparent moduli
for various laminate configurations can be produced beforehand. These are called carpet
plots, and they can be produced using the equations or directly using experimental data.
Case Studied
A laminate made of Epoxy fibers and isophthalic polyester matrix composed of eight laminas
is studied here. The orientations of the laminas are .
Material properties from table 1.1 are as follows:
Some symbols used in the code:

3. 2
Procedure of MATLAB code
1) A function called ''laminate'' is made to calculate the matrices for one
lamina. Its input is of the lamina and other parameters are connected to the main
code via "global" command.
2) Define the material properties of the laminate as mentioned above.
3) A 3 for-loops are made to construct the carpet plots. The first one is on ( . The
second one is on . The third loop is on ( . This third loop calculates the A,B and
D matrices for the whole laminate one value of orientation .
4) Change and calculate the for one and plot one
curve.
5) Change and plot a curve every change.
Results

4. 3

5. 4
Appendix
function to calculate A,B and D matrices for one lamina
% This function calculates A,B and D matrices for one lamina as a
function
% of the orientation of the lamina (theta)
function [Ai,Bi,Di]=laminate(theta)
global E1 E2 G12 nu_12 t z_i z_o
nu_21=nu_12*E2/E1;
delta=1-(nu_12*nu_21);
Q11=E1/delta;
Q12=nu_12*E2/delta;
Q22=E2/delta;
Q=[Q11,Q12,0;Q12,Q22,0;0,0,G12];
R=[1,0,0;0,1,0;0,0,2];
C=cosd(theta); S=sind(theta);
T=[C*C,S*S,2*S*C;S*S,C*C,-2*S*C;-S*C,S*C,C*C-S*S];
Q_bar=TQ*R*T/R;
Ai=Q_bar*t;
Bi=Q_bar*0.5*(z_i^2-z_o^2);
Di=(1/3)*Q_bar*(z_i^3-z_o^3);
Main code
% This is the main code for composite materials project for 8 laminas
% IT is made by Hossam Hassan AL-Kaleiby Abo Al-ela. Sec.(1),B.N.(26)
% Material: E-glass fibers and Isophthalic Ployester (Table 1.1)
% The laminate is symmetric.
clear;clc;close all
%% Properties of the material
global E1 E2 G12 nu_12 t z_i z_o
E1=39.9; % In GPa
E2=11.3; % In GPa
G12=3.3; % In GPa
nu_12=0.3;
tt=2; % Thickness of the plate
(assumed)
n=8; % No. of layers
% th=[0 90 45 -45 -45 45 90 0];
th=[-45 0 45 90 90 45 0 -45]; % Orientation of the laminas in
deg.
%% Calculations of A,B and D matrices for the plate consisted of 8
layers:
alfa=0:0.1:0.8; % Ratio of the 0 angle layer thick. to
tt
gamma=0:0.01:1; % Ratio of the +-45 angle layer thick.
to tt
for k=1:numel(alfa)
for j=1:numel(gamma)
beta=1-alfa(k)-gamma(j); % Ratio of the 90 angle layer thick.
to tt
if beta>=0
A=zeros(3,3);

6. 5
B=zeros(3,3);
D=zeros(3,3);
for i=1:numel(th)
if th(i)==0
t=alfa(k)*tt/2;
elseif th(i)==45 || th(i)==-45
t=gamma(j)*tt/2;
elseif th(i)==90
t=beta*tt/2;
end
z(i)=(2*i-n)*t/2;
z_i=z(i);
z_o=z_i-t;
theta=th(i);
[Ai,Bi,Di]=laminate(theta);
A=A+Ai;
B=B+Bi;
D=D+Di;
end
A11=A(1,1); A12=A(1,2); A22=A(2,2); A66=A(3,3);
D11=D(1,1); D12=D(1,2); D22=D(2,2); D66=D(3,3);
Exx(j)=(A11*A22-A12*A12)/(tt*A22);
nuu_xy(j)=A12/A22;
Gxxyy(j)=A66/tt;
Gxxyy_b(j)=D66*12/tt^3;
Exx_b(j)=12*(D11*D22-D12*D12)/(tt^3*D22);
end
end
Ex(k,:)=Exx; % Laminate inplane modulus(GPa)
nu_xy(k,:)=nuu_xy; % Laminate Poisson ratio
Gxy(k,:)=Gxxyy; % Laminate Shear modulus(GPa)
Gxy_b(k,:)=Gxxyy_b; % Laminate bending shear modulus(GPa)
Ex_b(k,:)=Exx_b; % Laminate inplane bending modulus(GPa)
end
%% Output carpet plots
figure
plot(gamma,Ex,'LineWidth',2);grid minor ;
xlabel('gamma','fontsize',18)
ylabel('E_x (GPa)','fontsize',15)
title('Carpet plot No.1')
figure
plot(gamma,nu_xy,'LineWidth',2);grid minor;
xlabel('gamma','fontsize',18)
ylabel('nu_x_y','fontsize',18)
title('Carpet plot No.2','fontsize',18)
figure
plot(gamma,Gxy, gamma,Gxy_b,'LineWidth',2); grid minor
xlabel('gamma','fontsize',18)
ylabel('Shear modulus (GPa)','fontsize',18)
title('Carpet plot No.3','fontsize',18)
legend('G_x_y','G_x_y^b')
figure
plot(gamma,Ex_b,'LineWidth',2);grid minor
xlabel('gamma','fontsize',18)
ylabel('E_x^b (GPa)','fontsize',18)
title('Carpet plot No.4','fontsize',18)
% figure
% plot(gamma,Gxy,'LineWidth',2); grid minor
% xlabel('gamma','fontsize',18)
% ylabel('Shear modulus (G_x_y) (GPa)','fontsize',18)

7. 6
% title('Shear modulus due to inplane load','fontsize',18)
Reference
Ever J.Barbero, "Introduction To Composite Materials Design".