User:Eml4500.f08.wiki1.handy

$${\hat{\mathbf{W}}}_{2x1}$$ is the virtual axial displacement which corresponds to $${\mathbf{q}}^{(e)}_{2x1}$$. On the other hand $${\mathbf{W}}_{4x1}$$ is the virtual displacement in the global coordinate system which corresponds to $${\mathbf{d}}^{(e)}_{4x1}$$. If we combine equations (2), (3) and (4) we get

$$(\mathbf{T^{(e)}}\mathbf{W})[\hat{\mathbf{k^{(e)}}}(\mathbf{T^{(e)}}\mathbf{d^{(e)}})-\mathbf{P^{(e)}}]=0$$ (5)

Which applies for all $${\mathbf{W}}_{4x1}$$. If we recall that

$$\mathbf{(A_{pxq}}\mathbf{B_{qxr})^T}=\mathbf{B^TA^T}$$ (6)

and

$$\mathbf{(a_{nx1}}\mathbf{b_{nx1})}=\mathbf{a_{1xn}^Tb_{nx1}}$$ (7)

we can manipulate (5) using (7). This gives us

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

Then by substituting in (6) we get

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

which can be simplified to

$$\mathbf{W[k^{(e)}d^{(e)}-f^{(e)}}]=0$$

and finally

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

Up until now we have dealt with the discrete case with matrices. From here forward we will consider the continuous case which involves partial differential equations. The model problem we will use consists of an elastic bar with varying area and modulus. It will be subjected to varying concentrated and axial loads, as well as inertial forces.