University of Florida/Eml4507/s13.team3.andersondavyr7

Problem 7.1

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

Given

The free body diagram to the right was given for the problem.

File:Last.JPG

Plane Truss From Professor's Notes

Find

Solve 2-element frame system using same data for 2-bar truss system.
Solve 2-element truss system.

Solution

Frame Problem

A basic form is shown below:

$\left[\begin{array}{cccccc}{a}_1& 0& 0& -a_1& 0& 0\\ 0& 12a_2& 6L{a}_2& 0& -12a_2& 6L{a}_2\\ 0& 6L{a}_2& 4L^2{a}_2& 0& -6L{a}_2& 2L^2{a}_2\\ -a_1& 0& 0& a_1& 0& 0& \\ 0& -12a_2& -6L{a}_2& 0& 12a_2& -6L{a}_2\\ 0& 6L{a}_2& 4L^2{a}_2& 0& -6L{a}_2& 4L^2{a}_2\end{array}\right]\left[\begin{array}{c}{u}_1\\ v_1\\ t_1\\ u_2\\ v_2\\ t_2\end{array}\right]=\left[\begin{array}{c}{f}_{x1}\\ f_{y1}\\ c_{c1}\\ f_{x2}\\ f_{y2}\\ c_{c2}\end{array}\right]$

Equations derived from stiffness equations:

$a_1=\frac{E*A}{L}$

$a_2=\frac{E*I}{L^3}$

Globle stiffness equation is shown below:

$\left[\begin{array}{ccccccccc}{a}_1^{(1)}& 0& 0& -a_1^{(1)}& 0& 0& 0& 0& 0\\ 0& 12a_2^{(1)}& 6L{a}_2^{(1)}& 0& -12a_2^{(1)}& 6L{a}_2^{(1)}& 0& 0& 0\\ 0& 6L{a}_2^{(1)}& 4L^2{a}_2^{(1)}& 0& -6L{a}_2^{(1)}& 2L^2{a}_2^{(1)}& 0& 0& 0\\ -a_1^{(1)}& 0& 0& a_1^{(1)}+a_1^{(2)}& 0& 0& -a_1^{(2)}& 0& 0\\ 0& -12^{(1)}{a}_2& -6L{a}_2^{(1)}& 0& 12a_2^{(2)}-12a_2^{(1)}& 6L{a}_2^{(2)}-6L{a}_2^{(1)}& 0^{(2)}& -12a_2^{(2)}& 6L{a}_2^{(2)}\\ 0& 6L{a}_2^{(1)}& 2L^2{a}_2^{(1)}& 0& 6L{a}_2^{(2)}-6L{a}_2^{(1)}& 4L^2{a}_2^{(1)}+4L^2{a}_2^{(2)}& 0^{(2)}& -6L{a}_2^{(2)}& 2L^2{a}_2^{(2)}\\ 0& 0& 0& -a_1^{(2)}& 0& 0& a_1^{(2)}& 0& 0\\ 0& 0& 0& 0& -12a_2^{(2)}& -6L{a}_2^{(2)}& 0& -12a_2^{(2)}& -6L{a}_2^{(2)}\\ 0& 0& 0& 0& 6L{a}_2^{(2)}& 2L^2{a}_2^{(2)}& 0& -6L{a}_2^{(2)}& 4L^2{a}_2^{(2)}\end{array}\right]$

Q matrix is shown below:

$Q=\left[\begin{array}{c}{u}_1\\ v_1\\ t_1\\ u_2\\ v_2\\ t_2\\ u_3\\ v_3\\ t_3\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ u_2\\ v_2\\ t_2\\ 0\\ 0\\ t_3\end{array}\right]$

Force matrix is show below:

$F=\left[\begin{array}{c}{f}_{x1}\\ f_{y1}\\ c_{c1}\\ f_{x2}\\ f_{y2}\\ c_{c2}\\ f_{x3}\\ f_{y3}\\ c_{c3}\end{array}\right]=\left[\begin{array}{c}{f}_{x1}\\ f_{y1}\\ c_{c1}\\ 0\\ P\\ 0\\ f_{x3}\\ f_{y3}\\ 0\end{array}\right]$

Matrix for the system is shown below:

$\left[\begin{array}{ccccccccc}{a}_1^{(1)}& 0& 0& -a_1^{(1)}& 0& 0& 0& 0& 0\\ 0& 12a_2^{(1)}& 6L{a}_2^{(1)}& 0& -12a_2^{(1)}& 6L{a}_2^{(1)}& 0& 0& 0\\ 0& 6L{a}_2^{(1)}& 4L^2{a}_2^{(1)}& 0& -6L{a}_2^{(1)}& 2L^2{a}_2^{(1)}& 0& 0& 0\\ -a_1^{(1)}& 0& 0& a_1^{(1)}+a_1^{(2)}& 0& 0& -a_1^{(2)}& 0& 0\\ 0& -12^{(1)}{a}_2& -6L{a}_2^{(1)}& 0& 12a_2^{(2)}-12a_2^{(1)}& 6L{a}_2^{(2)}-6L{a}_2^{(1)}& 0^{(2)}& -12a_2^{(2)}& 6L{a}_2^{(2)}\\ 0& 6L{a}_2^{(1)}& 2L^2{a}_2^{(1)}& 0& 6L{a}_2^{(2)}-6L{a}_2^{(1)}& 4L^2{a}_2^{(1)}+4L^2{a}_2^{(2)}& 0^{(2)}& -6L{a}_2^{(2)}& 2L^2{a}_2^{(2)}\\ 0& 0& 0& -a_1^{(2)}& 0& 0& a_1^{(2)}& 0& 0\\ 0& 0& 0& 0& -12a_2^{(2)}& -6L{a}_2^{(2)}& 0& -12a_2^{(2)}& -6L{a}_2^{(2)}\\ 0& 0& 0& 0& 6L{a}_2^{(2)}& 2L^2{a}_2^{(2)}& 0& -6L{a}_2^{(2)}& 4L^2{a}_2^{(2)}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ u_2\\ v_2\\ t_2\\ 0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}{f}_{x1}\\ f_{y1}\\ c_{c1}\\ 0\\ P\\ 0\\ f_{x3}\\ f_{y3}\\ 0\end{array}\right]$

The reduced stiffness is
$k=\left[\begin{array}{cccc}5.75& 0& 0& 0\\ 0& 0.622& 0.619& 0.625\\ 0& 0.619& -0.818& 0.417\\ 0& 0.625& 0.417& 0.833\end{array}\right]$

To solve for the displacements
$\left[\begin{array}{c}0\\ 6\\ 0\\ 0\end{array}\right]====\left[\begin{array}{cccc}5.75& 0& 0& 0\\ 0& 0.622& 0.619& 0.625\\ 0& 0.619& -0.818& 0.417\\ 0& 0.625& 0.417& 0.833\end{array}\right]\left[\begin{array}{c}{u}_2\\ v_2\\ t_2\\ t_3\end{array}\right]$

After matrix math.
$\left[\begin{array}{c}{u}_2\\ v_2\\ t_2\\ t_3\end{array}\right]====\left[\begin{array}{c}0\\ 24.5\\ 7.32\\ -22.08\end{array}\right]$

After solving for the forces the result is
$\left[\begin{array}{c}f{x}_1\\ f{y}_1\\ C_1\\ f{x}_3\\ f{y}_3\end{array}\right]=\left[\begin{array}{c}0\\ -594\\ -4118\\ 0\\ -6.1\end{array}\right]$

Below is a figure of the new deformed frame plotted with the undeformed frame.

File:71.png

Deformed frame in red plotted with the undeformed frame in black.

Truss Problem

File:Truss undeformed plot.PNG

Undeformed frame in red plotted.

File:Deformed truss plot.PNG

Deformed frame in blue plotted.