User:Eml4500.f08.team.dwyer.jd/HW2

Lectures 6-1 through 6-3

= 2) Element picture =

= 3) Global FD at elem level=

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

k is the element stiffness matrix for elem (e = 1, 2).

d is element displacement matrix of elem e.

f is element force matix.

$$ k^{(e)} \begin{bmatrix} (l^{(e)})^2 & (l^{(e)}*m^{(e})) & -(l^{(e)})^2 & -(l^{(e)}*m^{(e})) \\ (l^{(e)}*m^{(e})) & (m^{(e)})^2 & -(l^{(e)}*m^{(e})) & -(m^{(e)})^2 \\ -(l^{(e)})^2 & -(l^{(e)}*m^{(e})) & (l^{(e)})^2 & (l^{(e)}*m^{(e})) \\ -(l^{(e)}*m^{(e})) & -(m^{(e)})^2 & (l^{(e)}*m^{(e})) & (m^{(e)})^2 \\ \end{bmatrix} $$

$$ l^{(e)}, m^{(e)} = $$ director cosines of ~x axis (goes from [1] to [2]) write global (x,y) coordinates.

Derivation of director cosines:

i~ = cosθ(e)i + sinθ(e)j

l(e) = i~ • i = (cosθ(e)i + sinθ(e)j) • i = cosθ(e)(i • i) + sinθ(e)(j • i), where i • i = 1 and j • i = 0, = cosθ(e)

m(e) = i~ • j = (cosθ(e)i + sinθ(e)j) • j = cosθ(e)(i • j) + sinθ(e)(j • j), where i • j = 0 and j • j = 1, = sinθ(e)