University of Florida/Eml4507/s13.team4ever.Wulterkens.R3

Given Equations

Given the two following equations, verify that they simplify to the bar element stiffness matrix ${\overline{k}}^{(e)}$.

$\mathrm{𝟏})$ ${\overline{k}}^{(e)}=T^{(e)T}{\hat{k}}^{(e)}{T}^{(e)}$

$\mathrm{𝟐})$ ${\overline{k}}^{(e)}={\tilde{T}}^{(e)T}{\tilde{k}}^{(e)}{\tilde{T}}^{(e)}$

Solution 1

Original Equation

${\overline{k}}^{(e)}=T^{(e)T}{\hat{k}}^{(e)}{T}^{(e)}$

Given Equations

${\hat{k}}^{(e)}=k^{(e)}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]=\left[\begin{array}{cc}{k}^{(e)}& -k^{(e)}\\ -k^{(e)}& k^{(e)}\end{array}\right]$

$T^{(e)}=\left[\begin{array}{cccc}{l}^{(e)}& m^{(e)}& 0& 0\\ 0& 0& l^{(e)}& m^{(e)}\end{array}\right]$

Plugging into original Equation

${\overline{k}}^{(e)}=\left[\begin{array}{cc}{l}^{(e)}& 0\\ m^{(e)}& 0\\ 0& l^{(e)}\\ 0& m^{(e)}\end{array}\right]\left[\begin{array}{cc}{k}^{(e)}& -k^{(e)}\\ -k^{(e)}& k^{(e)}\end{array}\right]\left[\begin{array}{cccc}{l}^{(e)}& m^{(e)}& 0& 0\\ 0& 0& l^{(e)}& m^{(e)}\end{array}\right]$

Solving

Multiple the first two equations. ${\overline{k}}^{(e)}=\left[\begin{array}{cc}{l}^{(e)}{k}^{(e)}& -l^{(e)}{k}^{(e)}\\ m^{(e)}{k}^{(e)}& -m^{(e)}{k}^{(e)}\\ -l^{(e)}{k}^{(e)}& l^{(e)}{k}^{(e)}\\ -m^{(e)}{k}^{(e)}& m^{(e)}{k}^{(e)}\end{array}\right]\left[\begin{array}{cccc}{l}^{(e)}& m^{(e)}& 0& 0\\ 0& 0& l^{(e)}& m^{(e)}\end{array}\right]$

Pull out the K term. ${\overline{k}}^{(e)}=k^{(e)}\left[\begin{array}{cc}{l}^{(e)}& -l^{(e)}\\ m^{(e)}& -m^{(e)}\\ -l^{(e)}& l^{(e)}\\ -m^{(e)}& m^{(e)}\end{array}\right]\left[\begin{array}{cccc}{l}^{(e)}& m^{(e)}& 0& 0\\ 0& 0& l^{(e)}& m^{(e)}\end{array}\right]$

Multiple the last two matrices to get the following equation: ${\overline{k}}^{(e)}=k^{(e)}\left[\begin{array}{cccc}{l}^{(e)2}& l^{(e)}{m}^{(e)}& -l^{(e)2}& -l^{(e)}{m}^{(e)}\\ l^{(e)}{m}^{(e)}& m^{(e)2}& -l^{(e)}{m}^{(e)}& -m^{(e)2}\\ -l^{(e)2}& -l^{(e)}{m}^{(e)}& l^{(e)2}& l^{(e)}{m}^{(e)}\\ -l^{(e)}{m}^{(e)}& -m^{(e)2}& l^{(e)}{m}^{(e)}& m^{(e)2}\end{array}\right]$

This shows that the original equation can be simplified down to the bar element stiffness matrix.

Solution 2

Original Equation

$k^{(e)}={\tilde{T}}^{(e)T}{\tilde{k}}^{(e)}{\tilde{T}}^{(e)}$

Provided equations

${\tilde{T}}^{(e)}=\left[\begin{array}{cccc}cos(\varphi )& sin(\varphi )& 0& 0\\ -sin(\varphi )& cos(\varphi )& 0& 0\\ 0& 0& cos(\varphi )& sin(\varphi )\\ 0& 0& -sin(\varphi )& cos(\varphi )\end{array}\right]$

${\tilde{k}}^{(e)}=\frac{EA}{L}\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 0& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

Plugging into Original Equation

$k^{(e)}=\left[\begin{array}{cccc}cos(\varphi )& -sin(\varphi )& 0& 0\\ sin(\varphi )& cos(\varphi )& 0& 0\\ 0& 0& cos(\varphi )& -sin(\varphi )\\ 0& 0& sin(\varphi )& cos(\varphi )\end{array}\right]\frac{EA}{L}\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 0& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{cccc}cos(\varphi )& sin(\varphi )& 0& 0\\ -sin(\varphi )& cos(\varphi )& 0& 0\\ 0& 0& cos(\varphi )& sin(\varphi )\\ 0& 0& -sin(\varphi )& cos(\varphi )\end{array}\right]$

Solving

Begin by multiplying the first two matrices together. $k^{(e)}=\frac{EA}{L}\left[\begin{array}{cccc}cos(\varphi )& 0& -cos(\varphi )& 0\\ sin(\varphi )& 0& -sin(\varphi )& 0\\ -cos(\varphi )& 0& cos(\varphi )& 0\\ -sin(\varphi )& 0& sin(\varphi )& 0\end{array}\right]\left[\begin{array}{cccc}cos(\varphi )& sin(\varphi )& 0& 0\\ -sin(\varphi )& cos(\varphi )& 0& 0\\ 0& 0& cos(\varphi )& sin(\varphi )\\ 0& 0& -sin(\varphi )& cos(\varphi )\end{array}\right]$

Next, multiply the last two together. $k^{(e)}=\frac{EA}{L}\left[\begin{array}{cccc}cos(\varphi {)}^2& cos(\varphi )sin(\varphi )& -cos(\varphi {)}^2& -cos(\varphi )sin(\varphi )\\ cos(\varphi )sin(\varphi )& sin(\varphi {)}^2& -cos(\varphi )sin(\varphi )& -sin(\varphi {)}^2\\ -cos(\varphi {)}^2& -cos(\varphi )sin(\varphi )& cos(\varphi {)}^2& cos(\varphi )sin(\varphi )\\ cos(\varphi )sin(\varphi )& -sin(\varphi {)}^2& -cos(\varphi )sin(\varphi )& sin(\varphi {)}^2\end{array}\right]$

This verifies our original expectation that the equation would simplify down to the bar element stiffness matrix.