University of Florida/Egm4507/s13 Team 7 Report 7
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Problem 7.1
[edit | edit source]On my honor, I have neither given nor received unauthorized aid in doing this assignment.
Given
[edit | edit source]Node 1 coordinate: (0,0)
Node 2 coordinate: (3.46,2)
Node 3 coordinate: (4.87,0.586)
Element length:
Young's modulus:
Cross sectional area:
Find
[edit | edit source]Verify the dimensions of and
Solve and plot the 2-bar truss problem. Compare the deformed shape to the undeformed shape.
Solution
[edit | edit source]The matrix is a 1x6 because there are 6 degrees of freedom due to the moments at each node induced by completely rigid nodes.
If the matrix has 6 degrees of freedom, the matrix must be constructed as a 6x6.
Truss Problem
[edit | edit source]The untransformed displacement matrix:
Element 1:
Element 2:
The rotation matrices are measured from the angle from the horizontal
Transforming (robots in disguise):
Element 2:
The rotation matrices are measured from the angle from the horizontal
Transforming (robots in disguise):
Find the global stiffness matrix by combining the element stiffnesses:
The transformation matrix for each element is the same one used to transform d tilde.
Eliminate rows and columns according to 0 displacements. There are no zero forces.
Solve for unknowns.
Plot based off of unknowns.
Sorry guys, this problem required a massive amount of typing and mediawiki coding during my finals week. I just couldn't devote the time to pull it off in a day and a half to work on it.
Our FEA exam was Monday night. I had finals the week before. I had an exam on Tuesday morning. This was due Wednesday afternoon. I had to leverage my exams over this report.
Please penalize me heavily for this problem, as opposed to the rest of my group members. -Joshua Plicque
%undeformed 2 element frame/truss system
Element1x1=[0 .692 1.384 2.076 2.768 3.46];
Element1y1=[0 0.4 .8 1.2 1.6 2];
Element2x1=[3.46 3.813 4.166 4.519 4.87];
Element2y1=[2 1.647 1.294 .941 0.586];
plot(Element1x1,Element1y1,'g',Element2x1,Element2y1,'r')
axis([-.5,5.2,-0.2,2.2]);
Problem 7.2
[edit | edit source]On my honor, I have neither given nor received unauthorized aid in doing this assignment.
Given
[edit | edit source]
Find
[edit | edit source]Provide plots of results and animation of deformed shape as function of time.
Solution
[edit | edit source]clear
clc
%GIVENS
E=5;
A=1/2;
rho=2;
ep=[E A];
f=zeros(12,1);
ff=zeros(12,1);
f(8)=-5;
%ELEMENT DOF
Edof=[1 1 2 3 4;
2 1 2 5 6;
3 3 4 5 6;
4 5 6 9 10;
5 5 6 7 8;
6 3 4 9 10;
7 3 4 7 8;
8 7 8 9 10;
9 9 10 11 12;
10 7 8 11 12];
%COORDINATES
ex=[0 1
0 1
1 1
1 2
1 2
1 2
1 2
2 2
2 3
2 3];
ey=[0 0
0 1
0 1
1 1
1 0
0 1
0 0
0 1
1 0
0 0];
%STIFFNESS AND MASS MATRIX
K=zeros(12);
M=zeros(12);
for i=1:10
E=ep(1);A=ep(2);
xt=ex(i,2)-ex(i,1);
yt=ey(i,2)-ey(i,1);
L=sqrt(xt^2+yt^2);
l=xt/L; m=yt/L;
ke=E*A/L*[l^2 l*m -l^2 -l*m;
l*m m^2 -l*m -m^2;
-l^2 -l*m l^2 l*m;
-l*m -m^2 l*m m^2;];
m=L*A*rho;
me=[m/2 0 0 0;
0 m/2 0 0;
0 0 m/2 0;
0 0 0 m/2];
edoft=Edof(i,2:5);
K(edoft,edoft)=K(edoft,edoft)+ke;
M(edoft,edoft)=M(edoft,edoft)+me;
end
bc=[1 0; 2 0 ; 12 0];
elm=[1:12]';
elm(bc(:,1))=[];
d0=zeros(12,1);
d0(elm)=K(elm,elm)\f(elm);
v0=zeros(12,1);
ndof=size(K,1);
freedof=[1:ndof]';
fixdof=[1 2 11 12]';
freedof(fixdof(:))=[];
Kred=K(freedof,freedof);
Mred=M(freedof,freedof);
Ks=Mred^(-1/2)*Kred*Mred^(-1/2);
[X1,D1]=eig(Ks);
for j=1:size(X1)
d=sqrt(X1(:,j)'*X1(:,j));
X3(:,j)=X1(:,j)/d;
end
[D1,i]=sort(diag(D1));
X4=X3(:,i);
eigenval=D1;
T=ceil(2*pi/eigenval(1));
ntimes=[0.1:0.1:T];
nhist=[4];
ip=[T/100 T 1/4 1/2 10*T 2 ntimes nhist];
[Dsnap,D,V,A]=step2(K,[],M,d0,v0,ip,ff,bc);
t=0:T/100:T
figure(1),plot(t,D(1,:),'-')
figure(2)
filename = 'r7p2.gif';
for jj = 1:100
Edb=extract(Edof,Dsnap(:,jj));
xd=(ex+Edb(:,[1 3]))';
yd=(ey+Edb(:,[2 4]))';
s1=['-' , 'k'];
axis manual
plot(xd,yd,s1)
set(gca, 'ylim', [-1 2], 'xlim', [0 4]);
drawnow
frame = getframe(1);
im = frame2im(frame);
[imind,cm] = rgb2ind(im,256);
if jj == 1;
imwrite(imind,cm,filename,'gif', 'Loopcount',inf);
else
imwrite(imind,cm,filename,'gif','WriteMode','append');
end
end
![](http://upload.wikimedia.org/wikiversity/en/0/00/R7P2_gif.gif)
![](http://upload.wikimedia.org/wikiversity/en/8/88/R7p2_plot.png)
References
[edit | edit source]Contributing Team Members
[edit | edit source]Problem Assignments | ||
---|---|---|
Problem # | Solved & Typed by | Reviewed by |
1 | Spencer Herran,Joshua Plicque, Kristin Howe | All |
2 | Matthew Gidel | All |
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.