User:Eml4500.f08.FEABBQ.koby/HW5

=Rederivation of 3-D Matrix Relations= Axial F-D Relation

Now that a 3 dimensional system is being considered, there is now one more additional degree of freedom at each node. However, in any element there can still only be two forces and those forces always be in the direction of the element. This means that the relation of P to Q remains unchanged:

$$\begin{bmatrix} P^{(e)}_1\\ P^{e)}_2 \end{bmatrix} = k^{(e)}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} \begin{bmatrix} q^{(e)}_1\\ q^{e)}_2 \end{bmatrix}$$

Element DOF's and Element Forces in global XYZ coordinates (6x1 matrices) $$ \mathbf{d^{(e)}}=\begin{bmatrix} d^{(e)}_1 \\ d^{(e)}_2 \\ d^{(e)}_3 \\ d^{(e)}_4 \\ d^{(e)}_5 \\ d^{(e)}_6 \end{bmatrix} $$

$$ \mathbf{f^{(e)}}=\begin{bmatrix} f^{(e)}_1 \\ f^{(e)}_2 \\ f^{(e)}_3 \\ f^{(e)}_4 \\ f^{(e)}_5 \\ f^{(e)}_6 \end{bmatrix} $$

=Notes=

$$\mathbf{\hat{W}}_{2x1}$$ = Virtual axial displacement, corresponding to $$\mathbf{q^{(e)}_{1x1}}$$

$$\mathbf{W}_{4x1}$$ = Virtual displacement in the global coordinate system.

Replace equations (3) and (4) into equation (2)

$$\left( \mathbf{T^{(e)}w} \right)\bullet \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0$$ for all $$\mathbf{W}_{4x1}$$  (5)

Recall $$\mathbf{\left(AB \right)^{T}=B^TA^T}$$  (6)

Recall: $$\mathbf{a_{nx1} \cdot b_{nx1}=a^T_{1xn}b_{nx1}=S_{1x1}}$$ (7)

Apply (6) and (7) into equation (5):

$$\mathbf{\left( T^{(e)}w \right)^T} \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^TT^{(e)T}} \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^T}\cdot \left[ \mathbf{ \underbrace{ \left( T^{(e)T}\hat{k}^{(e)}T^{(e)} \right) }_{k^{(e)}} }d^{(e)} -\underbrace{ T^{(e)T}p^{(e)}}_{f^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^T}\cdot \left[ \mathbf{{k}^{(e)}d^{(e)} -f^{(e)}} \right]=0_{1x1}$$

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

The Continuous Case

So far we have only been analyzing discrete cases (with matrices). These cases are not continuous. The next phase of the class will look at the continuous case (with Partial Differential Equations, PDEs). There are many motivational problems such as an elastic bar with varying A(x), E(x), varying axial load (distributed and concentrating), and inertia forces (dynamics).