User:EML4500.f08.RAMROD.E/HW5

Six Bar Truss Code for Elements with Varying E This Part goes below Dan's Plot of 6 bar truss
Next, the six bar truss Matlab code used above, was modified to allow for different values of E for each element. This code uses all the same functions as above (i.e. PlaneTrussElement.m, PlaneTrussResults.m and NodalSoln.m). The modified code is shown below. The results from this situation, as well as the constant E case and the undeformed truss system are shown in the plot below. The varying E case is shown by the dotted nodes while the undeformed case is shown dashed. The element and node numbers are suppressed to avoid cluttering the diagram. The code used to create this plot is very similar to that shown above for the six bar truss with constant E, but repeated for the two deformation cases.

Continuation of Pergola's HW 5, Goes below Pergola's part
Using the Principle of Virtual Work, the global FD relationship is given by:

$$\displaystyle K*(d)=F$$

Also, we can represent the elemental matrix $$k^e$$ in terms of the axial matrix $$\hat{k}$$ and the transformation matrix, $$\displaystyle T$$. This is shown below:

$$k^e = \hat{k^e}(T)(T^T)$$

Lastly, the relationship between the element forces in global coordinates, $$\displaystyle f^e$$, and the axial forces, $$\displaystyle p^e$$, can be found using the transformation matrix, $$\displaystyle T$$.

$$\displaystyle f^e(T)=p^e$$

Or more detailed:

$$\begin{bmatrix}l^e & m^e & n^e & 0 & 0 & 0 \\ 0 & 0 & 0 & l^e & m^e & n^e\end{bmatrix} \begin{Bmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5\\ f_6\end{Bmatrix}=\begin{Bmatrix}p_1\\ p_2\end{Bmatrix}$$

Using the above derived equation for $$\displaystyle k^e$$, we can give the actual values for its elements. Multiplying out the right side of the equation gives us the following:

$$k^e = \hat{k^e}(T)(T^T)$$

$$k^e = \begin{bmatrix} 1 &-1 & 1\\ -1 & 1 & -1\\  1&  -1&1 \end{bmatrix}\begin{bmatrix} l^e & m^e & n^e & 0 & 0 & 0\\ 0 & 0 & 0 & l^e & m^e & n^e \end{bmatrix}\begin{bmatrix} l^e & 0\\ m^e & 0\\ n^e & 0\\ 0 & l^e\\ 0 & m^e\\ 0 & n^e \end{bmatrix}$$



This is the elemental stiffness matrix, $$\displaystyle k^e$$ for element e.