User:Eml4500.f08.jamama.jan/Meeting12

Mtg.12:

$$ \mathbf{q}_{2 \times 1}^{(e)} = \mathbf{T}_{2 \times 4}^{(e)} * \mathbf{d}_{4 \times 1}^{(e)} \qquad \qquad \qquad \qquad \qquad \qquad (1) $$

Derivation of element F-D with respect to global coordinate sstem (p. 6-1)

$$ \mathbf{k}_{4 \times 4}^{(e)} * \mathbf{d}_{4 \times 1}^{(e)} = \mathbf{f}_{4 \times 1}^{(e)} $$



(p. 4-5) $$ \ \ k^{(e)} \begin{bmatrix}

1 & -1 \\ -1 & 1

\end{bmatrix}

\begin{Bmatrix}

q_1^{(e)} \\ q_2^{(e)}

\end{Bmatrix}=

\begin{Bmatrix}

P_1^{(e)} \\ P_2^{(e)}

\end{Bmatrix} \qquad \qquad (2)$$

$$ q_1^{(e)}\ $$ = axial displacement of element $$ \mathbf{e}\ $$ at local node

$$ P_1^{(e)}\ $$ = axial force of element $$ \mathbf{e}\ $$ at local node

Global: Derive eq. $$ (1) $$ from eq. $$(2)$$ $$ \rightarrow $$ equation $$(2)$$ was already derived in Mtg.4

Want to find relation between $$ \begin{matrix} \mathbf{q}_{2 \times 1}^{(e)} & and & \mathbf{d}_{4 \times 1}^{(e)} \\ \mathbf{P}_{2 \times 1}^{(e)} & and & \mathbf{f}_{4 \times 1}^{(e)} \end{matrix} $$

These relations can be expressed in the form: $$ \mathbf{q}_{2 \times 1}^{(e)} = \mathbf{T}_{2 \times 4}^{(e)} * \mathbf{d}_{4 \times 1}^{(e)} $$

Consider the displacement vector of local node, denoted by $$ \mathbf{d}_{(1)}^{(e)} $$



$$ \Rightarrow $$ $$ \vec {\mathbf{d}}_{(1)}^{(e)} =  \mathbf{d}_1^{(e)}\vec {i} +  \mathbf{d}_2^{(e)} \vec {j}$$



$$\Rightarrow \ q_{1}^{(e)} $$ = axial displacement of node is the orthogonal projection of the dispalcement. Vector $$ \vec {\mathbf{d}}_{1}^{(e)}$$ of node on the axis of $$ \tilde{i} $$ of element $$ \mathbf{e} $$

$$ q_{1}^{(e)}= \vec {\mathbf{d}}_{(1)}^{(e)}* \vec{\tilde{i}} \Rightarrow (\mathbf{d}_1^{(e)} \vec {i} +  \mathbf{d}_2^{(e)} \vec {j})* \vec{\tilde{i}}= \mathbf{d}_1^{(e)}( \vec{i} * \vec{\tilde{i}} ) + \mathbf{d}_2^{(e)} (\vec {j}* \vec{\tilde{i}}) \Rightarrow $$

$$ \Rightarrow \begin{matrix}

\vec{i} * \vec{\tilde{i}} & = & cos(\theta^{(e)}) & = & l^{(e)} \\ \vec{j} * \vec{\tilde{i}} & = & sin(\theta^{(e)}) & = & m^{(e)}

\end{matrix} \Rightarrow \ $$ $$q_{1}^{(e)}= l^{(e)} \mathbf{d}_1^{(e)} + m^{(e)}\mathbf{d}_2^{(e)} = {\llcorner l^{(e)} m^{(e)} \lrcorner}_{1 \times 2}

{\begin{Bmatrix}

d_{1}^{(e)} \\ d_{2}^{(e)}

\end{Bmatrix}}_{2 \times 1}$$

Similarly for node, $$ q_{2}^{(e)}= {\llcorner l^{(e)} m^{(e)} \lrcorner}_{1 \times 2} {\begin{Bmatrix}

d_{3}^{(e)} \\ d_{4}^{(e)}

\end{Bmatrix}}_{2 \times 1}$$

$$ \begin{matrix}

\begin{Bmatrix}

q_{1}^{(e)} \\ q_{2}^{(e)} \end{Bmatrix} & = &

\begin{bmatrix}

l^{(e)} & m^{(e)} & 0 & 0 \\ 0 & 0 & l^{(e)} & m^{(e)} \end{bmatrix} &

\begin{Bmatrix}

d_{1}^{(e)} \\ d_{2}^{(e)} \\ d_{3}^{(e)} \\ d_{4}^{(e)} \end{Bmatrix} \\

& & & \\

\mathbf{q}_{2 \times 1}^{(e)} & = & \mathbf{T}_{2 \times 4}^{(e)} & \mathbf{d}_{4 \times 1}^{(e)} \end{matrix}

$$