User:Eml4500.f08.RAMROD.F/HW7

Mtg. 36 November 19 2008


$$\tilde{K}_{6x6}^{(e)}\tilde{d}_{6x1}^{(e)}=\tilde{F}_{6x1}^{(e)} $$

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

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

Note: $$\tilde{F}_{3}^{(e)}=F_{3}^{(e)}$$ and $$\tilde{F}_{6}^{(e)}=F_{6}^{(e)}$$ are moments around the z-axis.

$$\tilde{K}_{6x6}^{(e)}=\begin{bmatrix} \frac{EA}{L} & 0 & 0 & \frac{-EA}{L} & 0 & 0\\ & \frac{12EI}{L^3} & \frac{6EI}{L^2} & 0 & \frac{-12EI}{L^3} & \frac{-6EI}{L^2}\\ & & \frac{4EI}{L} & 0 & \frac{-6EI}{L^2} & \frac{2EI}{L}\\ & &  & \frac{EA}{L} & 0 & 0\\ & SYM. & &  & \frac{12EI}{L^3} & \frac{-6EI}{L^2}\\ & &  &  &  & \frac{4EI}{L} \end{bmatrix}$$

Dimensional Analysis
$$\left[\tilde{d}_1 \right]=L=\left[\tilde{d}_i \right] \; \; i=1,2,4,5$$

$$ \left[\tilde{d}_3 \right]=L=\left[\tilde{d}_6 \right]$$

Let's take an arc of a circle of radius r



The arc length $$\bar{AB}=R\theta $$ where $$\bar{AB}$$ is the arc length and $$\theta$$ is the angle measure.

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

$$\left[\theta \right]=\frac{\left[\bar{AB} \right]}{\left[R \right]}= \frac{L}{L}=1$$ no dimension

$$ \sigma =E\varepsilon \Rightarrow \left[\sigma \right]=\left[E \right]\left[\varepsilon  \right]$$