University of Florida/Eml4507/s13.team3.andersondavyr7

Frame Problem

A basic form is shown below:

${\displaystyle {\begin{bmatrix}a_{1}&0&0&-a_{1}&0&0\\0&12a_{2}&6La_{2}&0&-12a_{2}&6La_{2}\\0&6La_{2}&4L^{2}a_{2}&0&-6La_{2}&2L^{2}a_{2}\\-a_{1}&0&0&a_{1}&0&0&\\0&-12a_{2}&-6La_{2}&0&12a_{2}&-6La_{2}\\0&6La_{2}&4L^{2}a_{2}&0&-6La_{2}&4L^{2}a_{2}\\\end{bmatrix}}{\begin{bmatrix}u_{1}\\v_{1}\\t_{1}\\u_{2}\\v_{2}\\t_{2}\\\end{bmatrix}}={\begin{bmatrix}f_{x1}\\f_{y1}\\c_{c1}\\f_{x2}\\f_{y2}\\c_{c2}\\\end{bmatrix}}}$

Equations derived from stiffness equations:

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

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

Globle stiffness equation is shown below:

${\displaystyle {\begin{bmatrix}a_{1}^{(1)}&0&0&-a_{1}^{(1)}&0&0&0&0&0\\0&12a_{2}^{(1)}&6La_{2}^{(1)}&0&-12a_{2}^{(1)}&6La_{2}^{(1)}&0&0&0\\0&6La_{2}^{(1)}&4L^{2}a_{2}^{(1)}&0&-6La_{2}^{(1)}&2L^{2}a_{2}^{(1)}&0&0&0\\-a_{1}^{(1)}&0&0&a_{1}^{(1)}+a_{1}^{(2)}&0&0&-a_{1}^{(2)}&0&0\\0&-12^{(1)}a_{2}&-6La_{2}^{(1)}&0&12a_{2}^{(2)}-12a_{2}^{(1)}&6La_{2}^{(2)}-6La_{2}^{(1)}&0^{(2)}&-12a_{2}^{(2)}&6La_{2}^{(2)}\\0&6La_{2}^{(1)}&2L^{2}a_{2}^{(1)}&0&6La_{2}^{(2)}-6La_{2}^{(1)}&4L^{2}a_{2}^{(1)}+4L^{2}a_{2}^{(2)}&0^{(2)}&-6La_{2}^{(2)}&2L^{2}a_{2}^{(2)}\\0&0&0&-a_{1}^{(2)}&0&0&a_{1}^{(2)}&0&0\\0&0&0&0&-12a_{2}^{(2)}&-6La_{2}^{(2)}&0&-12a_{2}^{(2)}&-6La_{2}^{(2)}\\0&0&0&0&6La_{2}^{(2)}&2L^{2}a_{2}^{(2)}&0&-6La_{2}^{(2)}&4L^{2}a_{2}^{(2)}\\\end{bmatrix}}}$

Q matrix is shown below:

${\displaystyle Q={\begin{bmatrix}u_{1}\\v_{1}\\t_{1}\\u_{2}\\v_{2}\\t_{2}\\u_{3}\\v_{3}\\t_{3}\\\end{bmatrix}}={\begin{bmatrix}0\\0\\0\\u_{2}\\v_{2}\\t_{2}\\0\\0\\t_{3}\\\end{bmatrix}}}$

Force matrix is show below:

${\displaystyle F={\begin{bmatrix}f_{x1}\\f_{y1}\\c_{c1}\\f_{x2}\\f_{y2}\\c_{c2}\\f_{x3}\\f_{y3}\\c_{c3}\\\end{bmatrix}}={\begin{bmatrix}f_{x1}\\f_{y1}\\c_{c1}\\0\\P\\0\\f_{x3}\\f_{y3}\\0\\\end{bmatrix}}}$

Matrix for the system is shown below:

${\displaystyle {\begin{bmatrix}a_{1}^{(1)}&0&0&-a_{1}^{(1)}&0&0&0&0&0\\0&12a_{2}^{(1)}&6La_{2}^{(1)}&0&-12a_{2}^{(1)}&6La_{2}^{(1)}&0&0&0\\0&6La_{2}^{(1)}&4L^{2}a_{2}^{(1)}&0&-6La_{2}^{(1)}&2L^{2}a_{2}^{(1)}&0&0&0\\-a_{1}^{(1)}&0&0&a_{1}^{(1)}+a_{1}^{(2)}&0&0&-a_{1}^{(2)}&0&0\\0&-12^{(1)}a_{2}&-6La_{2}^{(1)}&0&12a_{2}^{(2)}-12a_{2}^{(1)}&6La_{2}^{(2)}-6La_{2}^{(1)}&0^{(2)}&-12a_{2}^{(2)}&6La_{2}^{(2)}\\0&6La_{2}^{(1)}&2L^{2}a_{2}^{(1)}&0&6La_{2}^{(2)}-6La_{2}^{(1)}&4L^{2}a_{2}^{(1)}+4L^{2}a_{2}^{(2)}&0^{(2)}&-6La_{2}^{(2)}&2L^{2}a_{2}^{(2)}\\0&0&0&-a_{1}^{(2)}&0&0&a_{1}^{(2)}&0&0\\0&0&0&0&-12a_{2}^{(2)}&-6La_{2}^{(2)}&0&-12a_{2}^{(2)}&-6La_{2}^{(2)}\\0&0&0&0&6La_{2}^{(2)}&2L^{2}a_{2}^{(2)}&0&-6La_{2}^{(2)}&4L^{2}a_{2}^{(2)}\\\end{bmatrix}}{\begin{bmatrix}0\\0\\0\\u_{2}\\v_{2}\\t_{2}\\0\\0\\0\\\end{bmatrix}}={\begin{bmatrix}f_{x1}\\f_{y1}\\c_{c1}\\0\\P\\0\\f_{x3}\\f_{y3}\\0\\\end{bmatrix}}}$

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

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

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

After solving for the forces the result is
${\displaystyle {\begin{bmatrix}fx_{1}\\fy_{1}\\C_{1}\\fx_{3}\\fy_{3}\\\end{bmatrix}}={\begin{bmatrix}0\\-594\\-4118\\0\\-6.1\\\end{bmatrix}}}$

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

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

Truss Problem

Undeformed frame in red plotted.

Deformed frame in blue plotted.