User:EML4500.f08.FEABBQ.Hamdan/HW-7

notes form November-17-08


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

$${\tilde{d}}^{(e)}_{6x1}=\begin{Bmatrix} {\tilde{d}}^{(e)}_{1} \\ {\tilde{d}}^{(e)}_{2}\\ {\tilde{d}}^{(e)}_{3}\\ {\tilde{d}}^{(e)}_{4}\\ {\tilde{d}}^{(e)}_{5}\\ {\tilde{d}}^{(e)}_{6} \end{Bmatrix} $$   ,    $${\tilde{f}}^{(e)}_{6x1}=\begin{Bmatrix} {\tilde{f}}^{(e)}_{1} \\ {\tilde{f}}^{(e)}_{2}\\ {\tilde{f}}^{(e)}_{3}\\ {\tilde{f}}^{(e)}_{4}\\ {\tilde{f}}^{(e)}_{5}\\ {\tilde{f}}^{(e)}_{6} \end{Bmatrix}$$

$$\tilde{z}=z $$ ( Rotation about Z-axis)

Moments about the Z-axis are equal in both coordinates: $$\tilde{f}^{(e)}_{3}=f^{(e)}_{3}, \tilde{f}^{(e)}_{6}=f^{(e)}_{6}$$

$$

\tilde{k}=\begin{bmatrix} \frac{EA}{L} &0 &0  &\frac{-EA}{L}  &0  &0 \\ 0 &\frac{12EI}{L^{2}} &\frac{6EI}{L^2} &0 &\frac{-12EI}{L^3}  &\frac{6EI}{L^2} \\ 0&0 &\frac{4EI}{L}  &0  & \frac{-6EI}{L^2} &\frac{2EI}{L} \\ \frac{-EA}{L}& 0 & 0 & \frac{EI}{L}&  0& 0\\ 0&\frac{-12EI}{L^3} &\frac{-6EI}{L^2}  & 0 &  \frac{12EI}{L^3}& \frac{-6EI}{L^2}\\ 0&\frac{6EI}{L^2} & \frac{2EI}{L} &  0& \frac{-6EI}{L^2} &\frac{4EI}{L} \end{bmatrix}_{(6X6)}$$


 * Dimension Analysis:

For Displacement the dimension $$\begin{bmatrix} \tilde{d_1} \end{bmatrix}=L= \begin{bmatrix} \tilde{d_i} \end{bmatrix} \rightarrow   i= 1,2,4,5 $$

$$L\rightarrow$$ Lenght

$$\begin{bmatrix} \tilde{d_i} \end{bmatrix} \rightarrow$$ Dimension of $$\tilde{d}_i$$

The rotation has a dimension of 1 $$\begin{bmatrix} \tilde{d_3} \end{bmatrix}=1= \begin{bmatrix} \tilde{d_6} \end{bmatrix} $$



$$AB=R\cdot \theta$$

$$AB\rightarrow$$ Arc Length

$$\theta \rightarrow$$ Angle

$$ \theta =\frac{AB}{R}$$

$$\left[\theta \right]=\frac{\left[AB \right]}{\left[R \right]}=\frac{L}{L}=1$$

Notes from December 01-08
Computing $$U(\tilde{x}), V(\tilde{x)}$$ :



Local system: $$U(\tilde{x})=U(\tilde{x})\vec{\tilde{i}}+V(\tilde{x})\vec{\tilde{j}}$$

Global System:$$ U(\tilde{x})=U_{x}(\tilde{x})\vec+V_{y}(\tilde{x})\vec $$

Computing $$U(\tilde{x}), V(\tilde{x)}$$ using
 * $$V(\tilde{x)}=N_{2}(\tilde{x})\tilde{d_2}+N_{3}(\tilde{x})\tilde{d_3} +N_{5}(\tilde{x})\tilde{d_5}+N_{6}(\tilde{x})\tilde{d_6}$$
 * $$U(\tilde{x)}=N_{1}(\tilde{x})\tilde{d_1}+N_{4}(\tilde{x})\tilde{d_4}$$

Compute $$U_x(\tilde{x}), V_y(\tilde{x)}$$ from $$U(\tilde{x}), V(\tilde{x)}$$

$$\begin{Bmatrix} U_x(\tilde{x)}\\ U_y(\tilde{x)} \end{Bmatrix}= R^T \begin{Bmatrix} U(\tilde{x})\\V(\tilde{x})

\end{Bmatrix}$$

$$\begin{Bmatrix} U(\tilde{x})\\V(\tilde{x})

\end{Bmatrix}=\begin{bmatrix} N_1 &0 & 0 &N_4  &0  & 0\\ 0 &N_2 &N_3  & 0 & N_5 & N_6 \end{bmatrix}\begin{Bmatrix} \tilde{d_1^{(e)}}\\ \tilde{d_2^{(e)}}\\ \tilde{d_3^{(e)}}\\ \tilde{d_4^{(e)}}\\ \tilde{d_5^{(e)}}\\ \tilde{d_6^{(e)}} \end{Bmatrix}$$

let: $$ \begin{bmatrix} N_1 &0 & 0 &N_4  &0  & 0\\ 0 &N_2 &N_3  & 0 & N_5 & N_6 \end{bmatrix} =\mathbb{N(\tilde{\mathrm{x})}}$$

$$\begin{Bmatrix} U_x(\tilde{x)}\\ U_y(\tilde{x)} \end{Bmatrix}=R^T \mathbb{N(\tilde{\mathrm{x})}}\tilde{T}^{(e)}d^{(e)}$$