University of Florida/Egm4507/s13 Team 7 Report 7

Problem 7.1
On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
Node 1 coordinate: (0,0)

Node 2 coordinate: (3.46,2)

Node 3 coordinate: (4.87,0.586)

$$P=100 N$$

Element length:

$$L^{(1)}=4$$

$$L^{(2)}=2$$

Young's modulus:

$$E^{(1)}=3$$

$$E^{(2)}=5$$

Cross sectional area:

$$A^{(1)}=1$$

$$A^{(2)}=2$$

Find
Verify the dimensions of $$ \widetilde{K}_{ij}$$ and $$\widetilde{d}_{j}$$ Solve and plot the 2-bar truss problem. Compare the deformed shape to the undeformed shape.

Solution


The $$\widetilde{d}_{j}$$matrix is a 1x6 because there are 6 degrees of freedom due to the moments at each node induced by completely rigid nodes. If the $$\widetilde{d}_{j}$$ matrix has 6 degrees of freedom, the $$ \widetilde{K}_{ij}$$ matrix must be constructed as a 6x6.

Truss Problem
The untransformed displacement matrix: $$\begin{Bmatrix} d_1\\d_2\\d_3\\d_4\\d_5\\d_6\\ \end{Bmatrix}$$

Element 1: $$ \begin{Bmatrix} d_1 = 0\\d_2 = 0\\d_3 = ?\\d_4 = ?\\d_5 = ?\\d_6 = 0\\ \end{Bmatrix} $$ Element 2: $$ \begin{Bmatrix} d_4 = ?\\d_5 = ?\\d_6 = 0\\d_7 = 0\\d_8 = 0\\d_9 = 0\\ \end{Bmatrix} $$

The rotation matrices are measured from the angle from the horizontal $$\theta_1 = 30, \theta_2 = 210$$ $$R = \begin{Bmatrix} cos \theta & sin \theta\\ -sin \theta & cos \theta\\ \end{Bmatrix} $$ $$ R_1 = \begin{Bmatrix} 0.866 & 0.5\\ -0.5 & 0.866\\ \end{Bmatrix}$$ $$ R_2 = \begin{Bmatrix} -0.866& -0.5\\ 0.5 & -0.866\\ \end{Bmatrix}$$

Transforming (robots in disguise): $$ \begin{Bmatrix} \widetilde{d}_1\\ \widetilde{d}_2\\ \widetilde{d}_3\\ \widetilde{d}_4\\ \widetilde{d}_5\\ \widetilde{d}_6\\ \end{Bmatrix} = $$ $$ \begin{Bmatrix} 0.866 & 0.5 & 0 & 0 & 0 & 0\\ -0.5 & 0.866 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -0.866 & -0.5 & 0\\ 0 & 0 & 0 & 0.5 & -0.866 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{Bmatrix} = $$ $$ \begin{Bmatrix} 0\\ 0\\ d_3\\ -0.866(d_4) + -0.5(d_5)\\ 0.5(d_4) -0.866(d_5)\\ 0\\ \end{Bmatrix} $$

Element 2: The rotation matrices are measured from the angle from the horizontal $$\theta_1 = 210, \theta_2 = 120$$ $$R = \begin{Bmatrix} cos \theta & sin \theta\\ -sin \theta & cos \theta\\ \end{Bmatrix} $$ $$ R_1 = \begin{Bmatrix} -0.866& -0.5\\ 0.5 & -0.866\\ \end{Bmatrix}$$ $$ R_2 = \begin{Bmatrix} -0.5& 0.866\\ -0.866 & -0.5\\ \end{Bmatrix}$$

Transforming (robots in disguise): $$ \begin{Bmatrix} \widetilde{d}_4\\ \widetilde{d}_5\\ \widetilde{d}_6\\ \widetilde{d}_7\\ \widetilde{d}_8\\ \widetilde{d}_9\\ \end{Bmatrix} = $$ $$ \begin{Bmatrix} -0.866 & -0.5 & 0 & 0 & 0 & 0\\ 0.5 & -0.866 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -0.5 & 0.866 & 0\\ 0 & 0 & 0 & 0.866 & -0.5 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{Bmatrix} = $$ $$ \begin{Bmatrix} -0.866(d_4) + -0.5(d_5)\\ 0.5(d_4) + -0.866(d_5)\\ 0\\ 0\\ 0\\ 0\\ \end{Bmatrix} $$

Find the global stiffness matrix by combining the element stiffnesses: $$ \mathbf k^{(e)} = \widetilde{\mathbf T}^{{(e)}T} \, \widetilde{\mathbf k}^{(e)} \, \widetilde{\mathbf T}^{(e)} $$

The transformation matrix for each element is the same one used to transform d tilde.

Eliminate rows and columns according to 0 displacements. There are no zero forces.

Solve for unknowns.

Plot based off of unknowns.

'''Sorry guys, this problem required a massive amount of typing and mediawiki coding during my finals week. I just couldn't devote the time to pull it off in a day and a half to work on it.'''

'''Our FEA exam was Monday night. I had finals the week before. I had an exam on Tuesday morning. This was due Wednesday afternoon. I had to leverage my exams over this report.'''

'''Please penalize me heavily for this problem, as opposed to the rest of my group members. -Joshua Plicque'''

Problem 7.2
On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
$$E=5, A=1/2, L=1, \rho=2, F=0.5$$

Find
Provide plots of results and animation of deformed shape as function of time.

Contributing Team Members
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.