%................................................................
% MATLAB codes for Finite Element Analysis
% problem21.m
% free vibrations of laminated plates
% See reference:
% K. M. Liew, Journal of Sound and Vibration,
% Solving the vibration of thick symmetric laminates
% by Reissner/Mindlin plate theory and the p-Ritz method, Vol.198,
% Number 3, Pages 343-360, 1996
% antonio ferreira 2008
% clear memory
clear all;colordef white;clf
% materials
thickness=0.001;h=thickness;kapa=pi*pi/12;
rho=1;I=thickness^3/12;
% symbolic computation
syms phi pi
% liew material
e2=1;e1=40*e2;g23=0.5*e2;g13=0.6*e2;g12=g13;
miu12=0.25;miu21=miu12*e2/e1;factor=1-miu12*miu21;
% angles for laminate
alfas=[0,pi/2,0];% 3 layers
% upper and lower coordinates
z(1)=-(h/2);z(2)=-(h/2)+h/3;z(3)=-z(2);z(4)=-z(1);
% [Q] in 0o orientation
qbarra(1,1,1)=e1/factor;
qbarra(1,2,1)=miu21*e1/factor;
qbarra(2,1,1)=miu12*e2/factor;
qbarra(2,2,1)=e2/factor;
qbarra(3,3,1)=g12;
qbarra(4,4,1)=kapa*g23;
qbarra(5,5,1)=kapa*g13;
% transformation matrix
T=[cos(phi)^2,sin(phi)^2,-sin(2*phi),0,0;sin(phi)^2,cos(phi)^2,sin(2*phi),0,0;sin(phi)*cos(phi),-sin(phi)*cos(phi),cos(phi)^2-sin(phi)^2,0,0;0,0,0,cos(phi),sin(phi);0,0,0,-sin(phi),cos(phi)];
% [Q] in structural axes
qBarra=T*qbarra*T.';
for s=1:size(alfas,2)
for i=1:5
for j=1:5
QQbarra(i,j,s)=subs(qBarra(i,j,1),phi,alfas(s));
end
end
Qbarra=double(QQbarra);
end
Q=Qbarra;
%______________________________________________
Astiff(5,5)=0;Bstiff(5,5)=0;Fstiff(5,5)=0;Istiff(5,5)=0;
for k=1:size(alfas,2)
for i=1:3
for j=1:3
Astiff(i,j)=Astiff(i,j)+Q(i,j,k)*(z(k+1)-z(k));
Bstiff(i,j)=Bstiff(i,j)+Q(i,j,k)*(z(k+1)^2-z(k)^2)/2;
Fstiff(i,j)=Fstiff(i,j)+Q(i,j,k)*(z(k+1)^3-z(k)^3)/3;
end
end
for i=4:5
for j=4:5
Istiff(i,j)=Istiff(i,j)+Q(i,j,k)*(z(k+1)-z(k));
end
end
end
pi=double(pi); % come back to numeric computation
% constitutive matrices
CMembranaMembrana=Astiff(1:3,1:3);
CMembranaFlexao0=Bstiff(1:3,1:3);
CFlexao0Flexao0=Fstiff(1:3,1:3);
CCorte0Corte0=Istiff(4:5,4:5);
% load
P = -1;
%Mesh generation
L = 1;
numberElementsX=10;
numberElementsY=10;
numberElements=numberElementsX*numberElementsY;
[nodeCoordinates, elementNodes] = rectangularMesh(2*L,L,numberElementsX,numberElementsY);
xx=nodeCoordinates(:,1);
yy=nodeCoordinates(:,2);
drawingMesh(nodeCoordinates,elementNodes,'Q4','k-');
axis off
numberNodes=size(xx,1);
% GDof: global number of degrees of freedom
GDof=5*numberNodes;
% stiffness and mass matrices
stiffness=formStiffnessMatrixMindlinQ45laminated5dof(GDof,numberElements,elementNodes,numberNodes,nodeCoordinates,CMembranaMembrana,CMembranaFlexao0,CFlexao0Flexao0,CCorte0Corte0);
[mass]=formMassMatrixMindlinQ4laminated5dof(GDof,numberElements,elementNodes,numberNodes,nodeCoordinates,rho,thickness,I);
% boundary conditions
[prescribedDof,activeDof,fixedNodeW]=EssentialBC5dof('cccc',GDof,xx,yy,nodeCoordinates,numberNodes);
% eigenproblem: free vibrations
numberOfModes=12;
[V,D] = eig(stiffness(activeDof,activeDof),mass(activeDof,activeDof));
% Liew, p-Ritz
D0=e2*h^3/12/(1-miu12*miu21);
D = diag(sqrt(D)*L*L/pi/pi*sqrt(rho*h/D0));
[D,ii] = sort(D); ii = ii(1:numberOfModes);
VV = V(:,ii);
activeDofW=setdiff([1:numberNodes]',[fixedNodeW]);
NNN=size(activeDofW);
VVV(1:numberNodes,1:12)=0;
for i=1:numberOfModes
VVV(activeDofW,i)=VV(1:NNN,i);
end
NN=numberNodes;N=sqrt(NN);
x=linspace(-L,L,numberElementsX+1);
y=linspace(-L,L,numberElementsY+1);
% drawing Eigenmodes
drawEigenmodes2D(x,y,VVV,NN,N,D);
Unable to convert expression containing symbolic variables into double array. Apply 'subs' function first to substitute values for variables.
Error in problem21 (line 44)
Qbarra=double(QQbarra)
Error in sym/double (line 868)
Xstr = mupadmex('symobj::double', S.s, 0);

1 Kommentar

Zuhdi
Zuhdi am 20 Jan. 2023
Hi, sorry for asking but have you solve the problem?

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am 20 Mär. 2022

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