User:Eml4500.f08.team.dwyer.jd/HW7 Nov 19 (Meeting 36)

Frame Elements & Dimensional Analysis


 $$ \tilde {\mathbf{k}} _ {6 x 6} ^ {(e)}\ \tilde {\mathbf{d}} _ {6 x 1} ^ {(e)}\ =\ \tilde {\mathbf{f}} _ {6 x 1} ^ {(e)}$$

$$ \tilde {\mathbf{d}} ^ {(e)} = \begin{Bmatrix} d_1^{(e)} \\ \vdots \\ d_6^{(e)} \end{Bmatrix} = \tilde {\mathbf{f}} ^ {(e)} = \begin{Bmatrix} f_1^{(e)} \\ \vdots \\ f_6^{(e)} \end{Bmatrix} $$

Note: $$ \tilde {f} _{3}^ {(e)}\ =\ f_3^{(e)}\, \ \tilde {f} _{6}^ {(e)}\ =\ f_6^{(e)} $$

moments (about $$  \tilde {z}\ =\ z) $$

$$\tilde{\mathbf{k}}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^{3}} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^{3}} & \frac{6EI}{L^2}\\ & & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ & &  & \frac{EA}{L} & 0 & 0\\ & sym & &  & \frac{12EI}{L^{3}} & \frac{-6EI}{L^2}\\ & &  &  &  & \frac{4EI}{L} \end{bmatrix}$$

$$\tilde{\mathbf{k}}^{(e)}_{6x6}\tilde{\mathbf{d}}^{(e)}_{6x1}=\tilde{\mathbf{f}}^{(e)}_{6x1}$$

$$\tilde{\mathbf{d}}^{(e)}=\begin{Bmatrix} \tilde{\mathbf{d}}^{(e)}_1\\ \tilde{\mathbf{d}}^{(e)}_2\\ \tilde{\mathbf{d}}^{(e)}_3\\ \tilde{\mathbf{d}}^{(e)}_4\\ \tilde{\mathbf{d}}^{(e)}_5\\ \tilde{\mathbf{d}}^{(e)}_6 \end{Bmatrix},\tilde{\mathbf{f}}^{(e)}=\begin{Bmatrix} \tilde{\mathbf{f}}^{(e)}_1\\ \tilde{\mathbf{f}}^{(e)}_2\\ \tilde{\mathbf{f}}^{(e)}_3\\ \tilde{\mathbf{f}}^{(e)}_4\\ \tilde{\mathbf{f}}^{(e)}_5\\ \tilde{\mathbf{f}}^{(e)}_6 \end{Bmatrix}$$

Dimensional Analysis
 $$\begin{bmatrix} \tilde{d}_1 \end{bmatrix}=L=\begin{bmatrix} \tilde{d}_i \end{bmatrix}\rightarrow i=1,2,4,5$$ $$\begin{bmatrix} \tilde{d}_1 \end{bmatrix}=1$$ $$\sigma =E\varepsilon\rightarrow[\sigma] =[E][\varepsilon] $$