Question

Solve with matlab

Determine the bending stiffness for the following laminates using the properties shown below. • [0]4 • [0/30]s • [0,+45], Layer properties aligned with the principal axes • E1 = 128 GPa • E2 = 11 GPa • G12 = 4.5 GPa • 12 = .25 %3D Layer Dimensions • b =.10 m • h= .0025 m

Answer 1

Things will be more clear if you read this first.

1] Stiffness mabix_in principal mestnenial ones (1,2) - To 02Q66 đ Ther, G4 = E , = 6, 1-212V21 Qiz V₂ El Qe = Gia - 1-2/22 2] Transformed stiffness mamix, L Qzy = Qxx Quy Qxes Qyx Quy Qus [ Qsx Qsy Qss. Reets Transformation mabix I = m² n² amn - _m2 -2mn - L-mn mn m?n? where m = cose, n = sino Rela ber" Qx, y & Q4, 2 2Qxs P22Qo Q2 Q22 0 1 Lo o 2 Q66d Qua Quy 2 Qys Qsx Qsy 2 Qss J from this, we need Quay Qan trony Qua I Qsa Quey Quy Qsy Qres Qus Qss

Dats Then Bending stiffness mabix. [p] = + @bey [he-bere K 1 = 2 K -3 KEL K-5 K36 the atk-thickness of each louper For [olu tx=2.5mm ho = -5mm (-2x tk) K-1 ..h1 = -2.5mm K:200 ---h2 = 0mm kisge --ch,= 2.5mm Ik hoo I-14 = 5mm (2x tk) For [0/0]s tx = 2.5mm bo=-5 mm (-2xtk) Kl 09 h = -2.5mm K:2 30° t h2=omm Tha = 2.5mm kho hu=5mm (2x tk) For [0, +45], tu = 2.5mm Tho - 7.5mm (0-3xtx) t-hi = -5mm to h₂=-2.5mm 1 hg = Omm -hy = 2.5mm h5=5mm Cha=705mm (3xtr) K3 309 +450 -450

MATLAB Part :-

Function to get transformed stiffness matrix Q_xy from Q_12 :-

Create separate file with filename same as function name(transform_stiffness.m) Don't run this file.

Function to obtain transformed stiffness matrix Q_xy in loading axes (X,Y) from stiffness matrix in principle material axes Q_12 function R_xy - transform_stiffness (Q_12, theta) m = cosd (theta); n = sind (theta); Transformation matrix, I T = {m^ 2 n ^2 2*m*n; n^2 m^2 -2*m*n; -m*n min mº2-n2]; Transformed stiffness matrix (Qxx Qxy 2*Qxs Qyx Qyy 2*Qys Qax Qay 2*Q33] Q_xy = inv(T)*Q_12*T; We want [Qxx Qxy Qxs Qyx QyY Qys Qax Qey 293] Q_xy - [Q_XY (1:3, 1:2), 0.5*Q_xy (1:3,3)]; end

Text :-

% Function to obtain transformed stiffness matrix Q_xy in loading
% axes(x,y) from stiffness matrix in principle material axes Q_12
function Q_xy = transform_stiffness(Q_12,theta)
    m = cosd(theta);        n = sind(theta);

    % Transformation matrix, T
    T = [m^2   n^2   2*m*n;
         n^2   m^2  -2*m*n;
         -m*n  m*n   m^2-n^2];

     % Transformed stiffness matrix [Qxx Qxy 2*Qxs
     %                               Qyx Qyy 2*Qys
     %                               Qsx Qsy 2*Qss]
     Q_xy = inv(T)*Q_12*T;

     % We want [Qxx Qxy Qxs
     %          Qyx Qyy Qys
     %          Qsx Qsy Qss]
     Q_xy = [Q_xy(1:3,1:2), 0.5*Q_xy(1:3,3)];
end

Script file :-

Create separate file with any name of your choice, and run this

clc; clear; # # $########### El = 128; E2 = 11; G12 = 4.5; GPa v12 = 0.25; & calculation for v21 v21 = (E2/E1) *v12; Layer Thickness tk = 2.5; **************** ******* #************* Sti Enne Matrix 912 incipal axe == ########### ==## ### Q_12 = [ E1/(1-v12*v21) v21*E1/(1-v12*v21) v21*E1/(1-v12*v21) E2/(1-v12*v21) 0; 2*G12]; ########################### Calculation for [0] 4 4 layers $################### ################################## thetas = [0 0 0 0]; angles for all layers D1 = zeros (3,3); $ Initializing bending stiffness matrix h = -2*tk:tk:2*tk; $hk of layers [ho, hi,...hk] -2tk to 2tk in steps of tk For k = 1:4 For all layers theta = thetas (k); Q_xy = transform_stiffness (Q_12, theta); Transformed stiffness matrix from developed function D1 = D1 + (1/3)*Q_xy* (h(k+1)^3 - h(k)^3); Summation end fprintf('For [0]4 laminate, bending stiffness matrix in GPa-mm 3:\n') disp (num2 3tr (round (D1,2))) Display matrix with elements rounded to 2 decimal places

=== tion for [0/30 ***** **** thetas = [0 30 30 0]; D2 = zeros (3,3); Initializing bending stiffness matrix h = -2*tk: tk:2*tk; hk of layers for k = 1:4 theta = thetas (k): Q_xy = transform_stiffness (Q_12, theta); D2 = D2 + (1/3) *Q_xy* (h(x+1)^3 - h(k)*3); end fprintf('For (0/30] 3 laminate, bending stiffness matrix in GPa-mm-3:\n') disp (num2str (round (D2, 2))) *#####################*************######## #### Calculation for [0, +-45] 3, 6layers ************************************************** ******* thetas = [0 45 - 45 -45 45 0]; D3 = zeros (3,3); Initializing bending stiffness matrix h = -3*tk:tk:3*tk; hk of layers, -3tk to 3tk in steps of tk for k = 1:6 theta = thetas (k); Q_xy = transform_stiffness (Q_12, theta); D3 = D3 + (1/3) *Q_xy* (h (k+1)^3 - h (k)*3); end Eprint('For [0/+-45] 3 laminate, bending stiffness matrix in Gea-mm-3:\n') disp (num2str (round (D3, 2)))

text :-

clc; clear;
%######################################################
%                    Layer Properties                  %
%######################################################
E1 = 128;       % GPa
E2 = 11;        % GPa
G12 = 4.5;      % GPa
v12 = 0.25;
% calculation for v21
v21 = (E2/E1)*v12;

% Layer Thickness
tk = 2.5;           % mm

%##################################################################
%        Stifnness Matrix Q12 (Principal axes)              %
%##################################################################
Q_12 = [  E1/(1-v12*v21)     v21*E1/(1-v12*v21)     0;
         v21*E1/(1-v12*v21)    E2/(1-v12*v21)       0;
                0                   0              2*G12];

%##################################################################
%                   Calculation for [0]4  , 4 layers               %
%##################################################################
thetas = [0 0 0 0];        % angles for all layers
D1 = zeros(3,3);           % Initializing bending stiffness matrix
h = -2*tk:tk:2*tk;        % hk of layers [h0,h1,...hk] -2tk to 2tk in steps of tk
for k = 1:4             % For all layers
    theta = thetas(k);
    Q_xy = transform_stiffness(Q_12,theta);     % Transformed stiffness matrix from developed function
    D1 = D1 + (1/3)*Q_xy*(h(k+1)^3 - h(k)^3);       % Summation
end
fprintf('For [0]4 laminate, bending stiffness matrix in GPa-mm^3:\n')
disp(num2str(round(D1,2)))      % Display matrix with elements rounded to 2 decimal places


%##################################################################
%                   Calculation for [0/30]s  , 4layers             %
%##################################################################
thetas = [0 30 30 0];
D2 = zeros(3,3);           % Initializing bending stiffness matrix
h = -2*tk:tk:2*tk;        % hk of layers
for k = 1:4
    theta = thetas(k);
    Q_xy = transform_stiffness(Q_12,theta);
    D2 = D2 + (1/3)*Q_xy*(h(k+1)^3 - h(k)^3);
end
fprintf('For [0/30]s laminate, bending stiffness matrix in GPa-mm^3:\n')
disp(num2str(round(D2,2)))

%##################################################################
%                   Calculation for [0,+-45]s , 6layers             %
%##################################################################
thetas = [0 45 -45 -45 45 0];
D3 = zeros(3,3);           % Initializing bending stiffness matrix
h = -3*tk:tk:3*tk;        % hk of layers, -3tk to 3tk in steps of tk
for k = 1:6
    theta = thetas(k);
    Q_xy = transform_stiffness(Q_12,theta);
    D3 = D3 + (1/3)*Q_xy*(h(k+1)^3 - h(k)^3);
end
fprintf('For [0/+-45]s laminate, bending stiffness matrix in GPa-mm^3:\n')
disp(num2str(round(D3,2)))

Results :-

For [0]4 laminate, bending stiffne 33 matrix in GPa-mm-3: 10724.27 230.4 230.4 921.62 375 For [0/30] 3 laminate, bending stiffness matrix in GPa-mm3: 10190.94 457.4 396.35 457.4 1000.96 134.24 396.35 134.24 601.99 For [0/+-45] 3 laminate, bending stiffne33 matrix in GPa-mm 3: 28871.81 3198.88 1838 3198.88 5590.51 1838 1838 1838 3686.89