User:Eml4500.f08.jamama.justin/HW5

$$\hat{W}$$ = virtual axial displacement

W = Virtual displacement in global coordinate system corresponds to $$d^{(e)}$$ (4X1)

recall last equations:

$$q^{(e)}=T^{(e)}d^{(e)}$$  (3)

$$ \hat{w}=T^{(e)}w $$   (4)

Replace equations (3) and (4) into equation (2). (From page meeting 26, page 2)

$$(T^{(e)}w)\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$ for all $$\hat{w}$$ (4X1) (5)

Recall: $$\left(AB \right)^{T}=B^{T}A^{T}$$ (6)

pXqqXr

Recall:$$a\bullet b=a^{T}b$$ (7)

nx1nx1(1xnnx1)

(1x1 scalar)

Apply (7) and (6) in (5):

$$(T^{(e)}w)^{T}\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$

$$(T^{(e)T}w^{T})\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$

$$\Rightarrow w\bullet \left[\left( T^{(e)}\hat{k}^{(e)}T^{(e)}\right)d^{(e)}-T^{(e)T}P^{(e)} \right]=0$$ for all w (4x1)

$$\Rightarrow w\bullet \left[k^{(e)}d^{(e)}-f^{(e)} \right]= 0 $$ for all w.  $$\Rightarrow k^{(e)}d^{(e)}=f^{(e)}$$

So far, discrete case (non-continuous) matricies. Now, continuous case (Partial Differential Equations) motivational model problem: Elastic bar with varying A(x), E(x), subject to varying axial loads (distributed) + concentrated load + inertia force (dynamic). Axial loads being the dependent.