User:Eml4500.f08.qwiki.bishop/Lecture 12

=Lecture 12 - Monday, September 22, 2008=

For a more thorough understanding of the Finite Element Method, it is wise to derive the element force displacement with respect to the global coordinate system.

Recall from Page 6-1, $$k^{(e)}d^{(e)}=f^{(e)}$$ (Equation 1)



Therefore, the element force displacement matrix can be written as follows:

$$\mathbf{k^{(e)}}\begin{bmatrix} 1 & -1 \\ -1 & 1\\ \end{bmatrix}\begin{pmatrix} q_{1}^{(e)}\\ q_{2}^{(e)}\\ \end{pmatrix}=\begin{pmatrix} P_{1}^{(e)}\\ P_{2}^{(e)}\\ \end{pmatrix} $$

$$q^{(e)}_{i}$$=axial displacement of element e at local node $$i$$

$$P^{(e)}_{i}$$=axial force of element e at local node $$i$$

The overall goal is to derive equation 1 from equation 2 (already derived in Meeting 4). We want to find the relationship between: The relationships can be expressed in the form: $$q^{(e)}_{2x1}=T^{(e)}_{2x4}d^{(e)}_{4x1}$$
 * $$q^{(e)}_{2x1}$$ and $$d^{(e)}_{4x1}$$
 * $$P^{(e)}_{2x1}$$ and $$f^{(e)}_{4x1}$$

Consider the displacement vector of local node i, denoted by $$d^{(e)}_{i}$$:




 * $$\vec{d^{(e)}_{[i]}}=d^{(e)}_{1}\vec{i}+d^{(e)}_{2}\vec{j}$$


 * $$q^{(e)}_{i}$$=axial displacement of node [1] is the orthogonal projection of the displacement vector $$\vec{d_{[1]}^{(e)}}$$ of node [1] on the $$\tilde{x}$$ axis of element e

Therefore, for node [1], $$q_1^{(e)}$$ can be derived using the following steps.

$$q_{1}^{(e)}=d_{[1]}^{(e)}*\vec{\tilde{i}}$$ $$q_{1}^{(e)}=(d_{1}^{(e)}\vec{i}+d_{2}^{(e)}\vec{j})\vec{\tilde{i}}$$ $$q_{1}^{(e)}=d_{1}^{(e)}(\vec{i}*\vec{\tilde{i}})+d_{2}^{(e)}(\vec{j}*\vec{\tilde{i}})$$ $$q_{1}^{(e)}=l^{(e)}d_{1}^{(e)}+m^{(e)}d_{2}^{(e)}$$ $$q_{1}^{(e)}=\begin{bmatrix} l^{(e)} & m^{(e)}\\ \end{bmatrix}\begin{pmatrix} d_{1}^{(e)}\\ d_{2}^{(e)}\\ \end{pmatrix}$$

The values for $$q_2^{(e)}$$ and $$q_2^{(e)}$$ can now be substituted into the following matrix:

$$\begin{pmatrix} q_{1}^{(e)}\\ q_{2}^{(e)}\\ \end{pmatrix}=\begin{bmatrix} l^{(e)} & m^{(e)} & 0 & 0\\ 0 & 0 & l^{(e)} & m^{(e)}\\ \end{bmatrix}\begin{pmatrix} d_{1}^{(e)}\\ d_{2}^{(e)}\\ d_{3}^{(e)}\\ d_{4}^{(e)}\\ \end{pmatrix}$$