User:Eml4500.f08.wiki1.ambrosio/hw5

Part #3
It has already been shown that $$\mathbf{\tilde{d}^{(e)}} =\mathbf{\tilde{T}}^{(e)} \mathbf$$. We have already derived the transformation matrix $$\mathbf{\tilde{T}}^{(e)}$$, and we also know $$\mathbf$$ is the element DOF is the global coordinate. Therefore the axial displacement $$\mathbf{\tilde{d}^{(e)}} $$ can be written as

$$ \begin{align} \mathbf{\tilde{d}^{(e)}} & =\mathbf{\tilde{T}}^{(e)} \mathbf \\ \mathbf{q^{(e)}} & =\mathbf{\tilde{T}}^{(e)} \mathbf \\

\begin{pmatrix} q_1^{(e)} \\ q_2^{(e)} \end{pmatrix} & = \begin{pmatrix} l^{(e)} & m^{(e)} & n^{(e)}  & 0 & 0 & 0\\ 0 & 0 &0 & l^{(e)} & m^{(e)}  & n^{(e)} \end{pmatrix} \begin{pmatrix} d_1^{(e)}\\ d_2^{(e)}\\ d_3^{(e)}\\ d_4^{(e)}\\ d_5^{(e)}\\ d_6^{(e)} \end{pmatrix} \end{align} $$

Part #5
We have already shown that $$\mathbf{k^{(e)}} = \mathbf{\tilde{T}}^{(e)T} \mathbf{{\tilde{k}}^{(e)}} \mathbf{{\tilde{T}}^{(e)}} $$ .By using the transformation matrix we get the following.

$$\begin{align} \mathbf{k^{(e)}} & = \mathbf{\tilde{T}}^{(e)T} \mathbf{{\tilde{k}}^{(e)}} \mathbf{{\tilde{T}}^{(e)}} \\

\begin{pmatrix} k_1^{(e)} \\ k_2^{(e)} \end{pmatrix} & =

\begin{pmatrix} l^{(e)} & 0 \\ m^{(e)} & 0 \\ n^{(e)} & 0 \\ 0 & l^{(e)} \\ 0 & m^{(e)} \\ 0 & n^{(e)} \end{pmatrix}

\mathbf{{\tilde{k}}^{(e)}}

\begin{pmatrix} l^{(e)} & m^{(e)} & n^{(e)}  & 0 & 0 & 0\\ 0 & 0 &0 & l^{(e)} & m^{(e)}  & n^{(e)} \end{pmatrix}

\end{align} $$

Two Bar Truss
The following code was used to re-run the two bar truss system.

The following codes were used in the two bar truss code.

When the MATLAB code is ran, the following results are obtained.

Row 1 of the "results" represents element 1, while row 2 represents element 2. Column 1 is the strain, column 2 the stress, and column 3 is the axial force for each element.

Six Bar Truss Example
When the MATLAB code is ran, the following results are obtained.

As with the "results" matrix from the two bar truss code, the row indicates the element while column 1 is the strain, cloumn 2 is the stress, and column 3 is the axial force.

The figure below shows the undeformed truss (dashed blue line) and deformed truss (solid red line) with a weighting factor of 10000.



Six Bar Truss With Different Young's Modulus
When the MATLAB code is ran, the following results are obtained.

As with the "results" matrix from the two bar and six bar truss code, the row indicates the element while column 1 is the strain, cloumn 2 is the stress, and column 3 is the axial force.

The figure below shows the undeformed truss (dashed blue line), deformed truss where all elements have the same Young's Modulus (solid red line), and deformed truss where all elements have a different Young's Modulus (solid green line) with a weighting factor of 10000.