University of Florida/Egm4507/s13 Team 7 Report 7

Problem 7.1

On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given

Node 1 coordinate: (0,0)

Node 2 coordinate: (3.46,2)

Node 3 coordinate: (4.87,0.586)

$P=100N$

Element length:

$L^{(1)}=4$

$L^{(2)}=2$

Young's modulus:

$E^{(1)}=3$

$E^{(2)}=5$

Cross sectional area:

$A^{(1)}=1$

$A^{(2)}=2$

Find

Verify the dimensions of ${\widetilde {K}}_{ij}$ and ${\widetilde {d}}_{j}$

Solve and plot the 2-bar truss problem. Compare the deformed shape to the undeformed shape.

Solution

The ${\widetilde {d}}_{j}$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 ${\widetilde {d}}_{j}$ matrix has 6 degrees of freedom, the ${\widetilde {K}}_{ij}$ matrix must be constructed as a 6x6.

Truss Problem

The untransformed displacement matrix: ${\begin{Bmatrix}d_{1}\\d_{2}\\d_{3}\\d_{4}\\d_{5}\\d_{6}\\\end{Bmatrix}}$

Element 1:
${\begin{Bmatrix}d_{1}=0\\d_{2}=0\\d_{3}=?\\d_{4}=?\\d_{5}=?\\d_{6}=0\\\end{Bmatrix}}$
Element 2:
${\begin{Bmatrix}d_{4}=?\\d_{5}=?\\d_{6}=0\\d_{7}=0\\d_{8}=0\\d_{9}=0\\\end{Bmatrix}}$

The rotation matrices are measured from the angle from the horizontal $\theta _{1}=30,\theta _{2}=210$
$R={\begin{Bmatrix}cos\theta &sin\theta \\-sin\theta &cos\theta \\\end{Bmatrix}}$
$R_{1}={\begin{Bmatrix}0.866&0.5\\-0.5&0.866\\\end{Bmatrix}}$
$R_{2}={\begin{Bmatrix}-0.866&-0.5\\0.5&-0.866\\\end{Bmatrix}}$

Transforming (robots in disguise): ${\begin{Bmatrix}{\widetilde {d}}_{1}\\{\widetilde {d}}_{2}\\{\widetilde {d}}_{3}\\{\widetilde {d}}_{4}\\{\widetilde {d}}_{5}\\{\widetilde {d}}_{6}\\\end{Bmatrix}}=$ ${\begin{Bmatrix}0.866&0.5&0&0&0&0\\-0.5&0.866&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-0.866&-0.5&0\\0&0&0&0.5&-0.866&0\\0&0&0&0&0&1\\\end{Bmatrix}}=$ ${\begin{Bmatrix}0\\0\\d_{3}\\-0.866(d_{4})+-0.5(d_{5})\\0.5(d_{4})-0.866(d_{5})\\0\\\end{Bmatrix}}$

Element 2: The rotation matrices are measured from the angle from the horizontal $\theta _{1}=210,\theta _{2}=120$
$R={\begin{Bmatrix}cos\theta &sin\theta \\-sin\theta &cos\theta \\\end{Bmatrix}}$
$R_{1}={\begin{Bmatrix}-0.866&-0.5\\0.5&-0.866\\\end{Bmatrix}}$
$R_{2}={\begin{Bmatrix}-0.5&0.866\\-0.866&-0.5\\\end{Bmatrix}}$

Transforming (robots in disguise): ${\begin{Bmatrix}{\widetilde {d}}_{4}\\{\widetilde {d}}_{5}\\{\widetilde {d}}_{6}\\{\widetilde {d}}_{7}\\{\widetilde {d}}_{8}\\{\widetilde {d}}_{9}\\\end{Bmatrix}}=$ ${\begin{Bmatrix}-0.866&-0.5&0&0&0&0\\0.5&-0.866&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-0.5&0.866&0\\0&0&0&0.866&-0.5&0\\0&0&0&0&0&1\\\end{Bmatrix}}=$ ${\begin{Bmatrix}-0.866(d_{4})+-0.5(d_{5})\\0.5(d_{4})+-0.866(d_{5})\\0\\0\\0\\0\\\end{Bmatrix}}$

Find the global stiffness matrix by combining the element stiffnesses: $\mathbf {k} ^{(e)}={\widetilde {\mathbf {T} }}^{{(e)}T}\,{\widetilde {\mathbf {k} }}^{(e)}\,{\widetilde {\mathbf {T} }}^{(e)}$

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

On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given

Figure 2.1: 10 member truss with L=1

$E=5,A=1/2,L=1,\rho =2,F=0.5$

Find

Provide plots of results and animation of deformed shape as function of time.

Solution

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

Figure 2.2: Animation of deformed shape

Figure 2.3: Plot of vertical displacement of node 2

Contributing Team Members

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.