User:Eml4500.f08.delta 6.castrillon/Square 3-bar truss MATLAB simulation

The Three-Bar Square Truss
The following truss was assigned in homework 4, and an analysis of its behavior is performed.



The elements of this truss have the following properties:


 * $$A^{(1)}=A^{(2)}=A^{(3)}=1$$
 * $$E^{(1)}=E^{(2)}=E^{(3)}=1$$
 * $$L^{(1)}=L^{(2)}=L^{(3)}=1$$

The following is the MATLAB code for determining the global stiffness matrix, $$\mathbf{K}$$, and the dof values.

It is evident that the Finite Element Method is not enough to solve the behavior of this truss. It is evident that this truss is very inestable, and so there is an undetermined number of ways this truss will behave under any load. It is necessary that the engineer add another bar element for support of the structure.

The Eigenvalues and Eigenvectors
The following command yielded the eigenvalues (D) and eigenvectors (V) for the global stiffness matrix assembled above.

We can see that there are five zero-value eigenvalues. Below are the plots of these five eigenvalues of the three-bar square truss and the generic code for obtaining the plots below.

First Zero Eigenvalue



Second Zero Eigenvalue



Third Zero Eigenvalue



Fourth Zero Eigenvalue



Fifth Zero Eigenvalue



The Four-Bar Square Truss
The following truss is a more rigid version of the above three-bar square truss. Another bar element has been placed diagonally from the global node 1 to the global node 3.



This new truss system has slightly different properties:


 * $$A^{(1)}=A^{(2)}=A^{(3)}=3$$
 * $$E^{(1)}=E^{(2)}=E^{(3)}=2$$
 * $$L^{(1)}=L^{(2)}=L^{(3)}=1$$ and $$L^{4}=\sqrt{2}$$

The source code for assembling the global stiffness matrix, and determining the dof values is below:

It is now evident that by the addition of the fourth bar element the truss system is now stable.

The Eigenvalues and Eigenvectors
The eigenvalues (D) and eigenvectors (V) were determined with the code below. Also, the actual results are displayed.

We can see that there are four zero eigenvalues. The plots of these eigenvalues can be found below, as well as the code for the fourth zero eigenvalue, as an example.

First Zero Eigenvalue



Second Zero Eigenvalue



Third Zero Eigenvalue



Fourth Zero Eigenvalue