User:Eml4500.f08.delta 6.krueger/HW Prove FD-rel

Prove Element FD-rel

Start with: Eqn 1$$\tilde{f}^{(e)}_{6x1} = \tilde{k}_{6x6} ^{(e)} \tilde{d}_{6x1}^{(e)}$$

Then plug the two following equations into Eqn 1:

$$\tilde{f}^{(e)} = \tilde{T}^{(e)} f^{(e)}   $$ and $$ \tilde{d}^{(e)} = \tilde{T}^{(e)}  d^{(e)} $$

The following equation is found:

$$\tilde{k}^{(e)} \tilde{T}^{(e)} d^{(e)}= \tilde{T}^{(e)}  f^{(e)}$$

The transpose of the $$\tilde{T}$$ is taken to move the matrix from the right side to the left and the following equation is found: Eqn 2$$f^{(e)} = \begin{bmatrix} \tilde{T}^{(e)^{T}} \tilde{k}^{(e)}  \tilde{T}^{(e)}*d^{(e)} \end{bmatrix}$$

Now recall the following equation:

$$k_{6x6}^{(e)}=\tilde{T}_{6x6}^{(e)T}\tilde{k}_{6x6}^{(e)}\tilde{T}_{6x6}^{(e)}$$

This equation is substituted into Eqn 2 to get:

$$f_{6x1}^{(e)}=k_{6x6}^{(e)}d_{6x1}^{(e)}$$