User:Eml4500.f08.team.Bengtson/teamHW7b

November 21st, 2008 Meeting 37
$$[\varepsilon ] = \frac{[du]}{[dx]} = \frac{L}{L} = 1$$

$$[\sigma] = [E] = \frac{F}{L^2}$$

$$ [\mathbb{A}]=L^2$$

$$[\mathbb{I}]=L^4$$

$$ \frac{EA}{L}=[\tilde{k}_{11}] = \frac{(\frac{F}{L^2})L^2}{L} = \frac{F}{L} $$

$$[\tilde{k}_{11}\tilde{d}_1] = [\tilde{k}_{11}][\tilde{d}_1] = F$$

$$ [\tilde{k}_{23}\tilde{d}_3] = \frac{6[E][I]}{L^2} = \frac{(\frac{F}{L^2})L^4}{L^2} = F $$

$$ \begin{Bmatrix} \tilde{d}_1\\ \tilde{d}_2\\ \tilde{d}_3\\ \tilde{d}_4\\ \tilde{d}_5\\ \tilde{d}_6 \end{Bmatrix} =\begin{bmatrix} R& & 0 & 0 &0  &0 \\ & & 0 & 0 & 0 &0 \\ 0 & 0 & 1 &R  &  &0 \\ 0& 0 & 0 & &  &0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{Bmatrix} d_1\\ d_2\\ d_3\\ d_4\\ d_5\\ d_6 \end{Bmatrix}$$ See MATLAB portion for two bar system. Derivative of $$\hat{K}$$$$ \frac{\partial^2 }{\partial x^2}[(EI)\frac{\partial^2v }{\partial x^2}] - f_t(x) = m(x)\ddot{v}$$