User:Eml4500.f08.wiki1.schaet/hw6

Using this general formula for stiffness matrix that works with non-constant E and A.

$$\mathbf{k}^{(i)}=\frac{1}{6L} \left( {2E_1A_1}+{(E_1A_2+E_2A_1)}+ {2E_2A_2}\right) \begin{bmatrix} 1 & -1\\ -1 & 1\end{bmatrix}$$

The matlab function of the axial forces and the function that calls it were modified to accomodate teh variances in E and A.

Boundary conditions:

Element 1

$$E_1^{(1)}=2$$ $$E_2^{(1)}=4$$ $$A_1^{(1)}=0.5$$ $$A_2^{(1)}=1.5$$

Element 2

$$E_1^{(2)}=3$$ $$E_2^{(2)}=7$$ $$A_1^{(2)}=1$$ $$A_2^{(2)}=3$$

Averages from before:

$$E_{avg}^{(1)}=3$$ $$A_{avg}^{(1)}=1$$

$$E_{avg}^{(2)}=5$$ $$A_{avg}^{(2)}=2$$

The plot of the undeformed beam, the deformed beam using the new equation, and the deformed beam using just the averages of E and A is as follows. The undeformed beam is the blue dotted line, the deformed beam using the averages is the solid red line, and the deformed beam using the varied E and A values is the solid green line. The matlab code that plots this image is as follows:  Two-Bar Plot

The matlab code used to attain the deformed beam using the varied E and A is as follows:  2 Bar Truss with varied E and A

The altered m-file, DeltaPlaneTrussElement.m, is as follows:  DeltaPlaneTrussElement.m Note: The deformed beam using the average values of E and A can be found in a previous hw.

Three and four bar eigenvalue problem revisited
The images for this section are as follows:

Three Bar Rectangular Truss
A for all elements is 1 E for all elements is 1 L for all elements is 1.

The following matlab code computed the results of this problem:

The results of this say that the 'Matrix is singular to working precision'. Essentially, this shows that the system is too unstable to compute results. If any load was applied at any node, the system would show NaN for the results. This is why a fourth bar is required.

AssumingThe eignevectors are found as follows: The plots of these eignevectors are as follows: image
 * code

Four Bar Rectangular Truss
The four bar rectangular truss system is as follows:
 * $$A^{(1)}=A^{(2)}=A^{(3)}=3$$
 * $$E^{(1)}=E^{(2)}=E^{(3)}=2$$
 * $$L^{(1)}=L^{(2)}=L^{(3)}=1$$ and $$L^{4}=\sqrt{2}$$

The following matlab code computed the results of this problem:
 * code

The following are the results:
 * code

The eignevectors are found as follows: The plots of these eignevectors are as follows: image
 * code