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:

$[\begin{matrix} a_{1} & 0 & 0 & - a_{1} & 0 & 0 \\ 0 & 12 a_{2} & 6 L a_{2} & 0 & - 12 a_{2} & 6 L a_{2} \\ 0 & 6 L a_{2} & 4 L^{2} a_{2} & 0 & - 6 L a_{2} & 2 L^{2} a_{2} \\ - a_{1} & 0 & 0 & a_{1} & 0 & 0 \\ 0 & - 12 a_{2} & - 6 L a_{2} & 0 & 12 a_{2} & - 6 L a_{2} \\ 0 & 6 L a_{2} & 4 L^{2} a_{2} & 0 & - 6 L a_{2} & 4 L^{2} a_{2} \end{matrix}] [\begin{matrix} u_{1} \\ v_{1} \\ t_{1} \\ u_{2} \\ v_{2} \\ t_{2} \end{matrix}] = [\begin{matrix} f_{x 1} \\ f_{y 1} \\ c_{c 1} \\ f_{x 2} \\ f_{y 2} \\ c_{c 2} \end{matrix}]$

Equations derived from stiffness equations:

$a_{1} = \frac{E * A}{L}$

$a_{2} = \frac{E * I}{L^{3}}$

Globle stiffness equation is shown below:

$[\begin{matrix} a_{1}^{(1)} & 0 & 0 & - a_{1}^{(1)} & 0 & 0 & 0 & 0 & 0 \\ 0 & 12 a_{2}^{(1)} & 6 L a_{2}^{(1)} & 0 & - 12 a_{2}^{(1)} & 6 L a_{2}^{(1)} & 0 & 0 & 0 \\ 0 & 6 L a_{2}^{(1)} & 4 L^{2} a_{2}^{(1)} & 0 & - 6 L a_{2}^{(1)} & 2 L^{2} a_{2}^{(1)} & 0 & 0 & 0 \\ - a_{1}^{(1)} & 0 & 0 & a_{1}^{(1)} + a_{1}^{(2)} & 0 & 0 & - a_{1}^{(2)} & 0 & 0 \\ 0 & - 1 2^{(1)} a_{2} & - 6 L a_{2}^{(1)} & 0 & 12 a_{2}^{(2)} - 12 a_{2}^{(1)} & 6 L a_{2}^{(2)} - 6 L a_{2}^{(1)} & 0^{(2)} & - 12 a_{2}^{(2)} & 6 L a_{2}^{(2)} \\ 0 & 6 L a_{2}^{(1)} & 2 L^{2} a_{2}^{(1)} & 0 & 6 L a_{2}^{(2)} - 6 L a_{2}^{(1)} & 4 L^{2} a_{2}^{(1)} + 4 L^{2} a_{2}^{(2)} & 0^{(2)} & - 6 L a_{2}^{(2)} & 2 L^{2} a_{2}^{(2)} \\ 0 & 0 & 0 & - a_{1}^{(2)} & 0 & 0 & a_{1}^{(2)} & 0 & 0 \\ 0 & 0 & 0 & 0 & - 12 a_{2}^{(2)} & - 6 L a_{2}^{(2)} & 0 & - 12 a_{2}^{(2)} & - 6 L a_{2}^{(2)} \\ 0 & 0 & 0 & 0 & 6 L a_{2}^{(2)} & 2 L^{2} a_{2}^{(2)} & 0 & - 6 L a_{2}^{(2)} & 4 L^{2} a_{2}^{(2)} \end{matrix}]$

Q matrix is shown below:

$Q = [\begin{matrix} u_{1} \\ v_{1} \\ t_{1} \\ u_{2} \\ v_{2} \\ t_{2} \\ u_{3} \\ v_{3} \\ t_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \\ u_{2} \\ v_{2} \\ t_{2} \\ 0 \\ 0 \\ t_{3} \end{matrix}]$

Force matrix is show below:

$F = [\begin{matrix} f_{x 1} \\ f_{y 1} \\ c_{c 1} \\ f_{x 2} \\ f_{y 2} \\ c_{c 2} \\ f_{x 3} \\ f_{y 3} \\ c_{c 3} \end{matrix}] = [\begin{matrix} f_{x 1} \\ f_{y 1} \\ c_{c 1} \\ 0 \\ P \\ 0 \\ f_{x 3} \\ f_{y 3} \\ 0 \end{matrix}]$

Matrix for the system is shown below:

$[\begin{matrix} a_{1}^{(1)} & 0 & 0 & - a_{1}^{(1)} & 0 & 0 & 0 & 0 & 0 \\ 0 & 12 a_{2}^{(1)} & 6 L a_{2}^{(1)} & 0 & - 12 a_{2}^{(1)} & 6 L a_{2}^{(1)} & 0 & 0 & 0 \\ 0 & 6 L a_{2}^{(1)} & 4 L^{2} a_{2}^{(1)} & 0 & - 6 L a_{2}^{(1)} & 2 L^{2} a_{2}^{(1)} & 0 & 0 & 0 \\ - a_{1}^{(1)} & 0 & 0 & a_{1}^{(1)} + a_{1}^{(2)} & 0 & 0 & - a_{1}^{(2)} & 0 & 0 \\ 0 & - 1 2^{(1)} a_{2} & - 6 L a_{2}^{(1)} & 0 & 12 a_{2}^{(2)} - 12 a_{2}^{(1)} & 6 L a_{2}^{(2)} - 6 L a_{2}^{(1)} & 0^{(2)} & - 12 a_{2}^{(2)} & 6 L a_{2}^{(2)} \\ 0 & 6 L a_{2}^{(1)} & 2 L^{2} a_{2}^{(1)} & 0 & 6 L a_{2}^{(2)} - 6 L a_{2}^{(1)} & 4 L^{2} a_{2}^{(1)} + 4 L^{2} a_{2}^{(2)} & 0^{(2)} & - 6 L a_{2}^{(2)} & 2 L^{2} a_{2}^{(2)} \\ 0 & 0 & 0 & - a_{1}^{(2)} & 0 & 0 & a_{1}^{(2)} & 0 & 0 \\ 0 & 0 & 0 & 0 & - 12 a_{2}^{(2)} & - 6 L a_{2}^{(2)} & 0 & - 12 a_{2}^{(2)} & - 6 L a_{2}^{(2)} \\ 0 & 0 & 0 & 0 & 6 L a_{2}^{(2)} & 2 L^{2} a_{2}^{(2)} & 0 & - 6 L a_{2}^{(2)} & 4 L^{2} a_{2}^{(2)} \end{matrix}] [\begin{matrix} 0 \\ 0 \\ 0 \\ u_{2} \\ v_{2} \\ t_{2} \\ 0 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} f_{x 1} \\ f_{y 1} \\ c_{c 1} \\ 0 \\ P \\ 0 \\ f_{x 3} \\ f_{y 3} \\ 0 \end{matrix}]$

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

To solve for the displacements
$[\begin{matrix} 0 \\ 6 \\ 0 \\ 0 \end{matrix}] = = = = [\begin{matrix} 5.75 & 0 & 0 & 0 \\ 0 & 0.622 & 0.619 & 0.625 \\ 0 & 0.619 & - 0.818 & 0.417 \\ 0 & 0.625 & 0.417 & 0.833 \end{matrix}] [\begin{matrix} u_{2} \\ v_{2} \\ t_{2} \\ t_{3} \end{matrix}]$

After matrix math.
$[\begin{matrix} u_{2} \\ v_{2} \\ t_{2} \\ t_{3} \end{matrix}] = = = = [\begin{matrix} 0 \\ 24.5 \\ 7.32 \\ - 22.08 \end{matrix}]$

After solving for the forces the result is
$[\begin{matrix} f x_{1} \\ f y_{1} \\ C_{1} \\ f x_{3} \\ f y_{3} \end{matrix}] = [\begin{matrix} 0 \\ - 594 \\ - 4118 \\ 0 \\ - 6.1 \end{matrix}]$

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.

Template:CourseCat

University of Florida/Eml4507/s13.team3.andersondavyr7

Contents

Problem 7.1

Given

Find

Solution

Frame Problem

Truss Problem

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University of Florida/Eml4507/s13.team3.andersondavyr7

Problem 7.1

Given

Find

Solution

Frame Problem

Truss Problem

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