User:Eml4500.f08.RAMROD.A/MyContributionHW5

Virtual Displacement Terms
ŵ2x1 = virtual axial displacement, with corresponding q(e)2x1

w4x1 = virtual axial displacement in global coordinates system, with corresponding d(e)4x1

Equation (5)
$$ (T^{(e)}w)\cdot [\hat{K}^{(e)}(T^{(e)}d^{(e)})-P^{(e)}] = 0 $$   for all w4x1

We want to prove equation (5) using matrix identity equation (6):

Equation 6: $$ (\mathbf{AB})^T=\mathbf{B}^T\mathbf{A}^T $$

Homework Example to verify equation (6)
$$ A=\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6 \end{bmatrix}, B=\begin{bmatrix}7 & 8 & 9\\1 & 2 & 3\\4 & 5 & 6\end{bmatrix} , B^T = \begin{bmatrix}7 & 1 & 4\\ 8 & 2 & 5\\ 9 & 3 & 6\end{bmatrix} , A^T=\begin{bmatrix}1 & 4\\ 2 & 5\\ 3 & 6\end{bmatrix} $$

$$ AB=\begin{bmatrix}1\cdot7+2\cdot1+3\cdot4 & 1\cdot8+2\cdot2+3\cdot5 & 1\cdot9+2\cdot3+3\cdot6 \\4\cdot7+5\cdot1+6\cdot4 & 4\cdot8+5\cdot2+6\cdot5 & 4\cdot9+5\cdot3+6\cdot6 \end{bmatrix} =\begin{bmatrix}21 & 27 & 33\\57 & 72 & 87\end{bmatrix} $$

$$(AB)^T = \begin{bmatrix}21 & 57\\27 & 72\\ 33 & 87\end{bmatrix} $$

$$ B^TA^T = \begin{bmatrix}21 & 57\\27 & 72\\ 33 & 87\end{bmatrix} $$

This proves Equation (6): $$ (\mathbf{AB})^T=\mathbf{B}^T\mathbf{A}^T $$

We can recall that $$ \underline{a}\cdot\underline{b} = \underline{a}^T\underline{b} $$, where a and b are nx1 and aT is 1xn

It follows that from Eq(1):

$$ (T^{(e)}w)^T[\hat{k}^{(e)}(T^{(e)}d^{(e)})-P^{(e)}] = 0 $$, for all w,

and using Eq (6) above, that:

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

and that:

$$ w\cdot[(T^{(e)T}\hat{k}^{(e)}T^{(e)})d^{(e)}-(T^{(e)T}P^{(e)})] = 0 $$, for all w4x1

Also:

$$ w\cdot[\hat{k}^{(e)}d^{(e)}-f^{(e)}] = 0 $$, for all w.

All the above proves:

$$ k^{(e)}_{4x4} = T^{(e)T}_{4x2}\hat{k}^{(e)}_{4x2}T^{(e)}_{2x4} $$

Principal of Virtual Work
Equation 2:

$$ \hat{w}\cdot(\hat{k}^{(e)}q^{(e)}-P^{(e)}) = 0 $$, for all w

Equation 3:

$$ q^{(e)}_{2x1}=T^{(e)}_{2x4}d^{(e)}_{4x1} $$

and finally equation 4:

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

Problems of the Continuous Case (PDE's)
The new problems will contain varying parameters

Example of problem:

Elastic bar with varying cross section A(x), varying modulus E(x), subjective to varying axial load (distributed), a concentrated load and inertia force (dynamic).

note: load will be time dependent