User:Eml4500.f08.delta 6.lopez/HW3

Three Bar Truss System
The three bar truss system is shown in the image below. The characteristics of each element is shown in the table that follows the image.





Convenient methods of numbering the local nodes are shown in the images below for each element. Element 1



Element 2



Element 3



$$ \sum{F_{x}}=0 $$ $$ \sum{F_{y}}=0 $$ $$ \sum{M_{A}}=0 $$

Question: What if we take the moment about a point B?



$$ \sum{M_{B}} = \vec{BA}\; X\; \vec{ F} = \vec{BA'}\; X\; \vec{ F} $$ The above equation stands only when A' lies on the line of action. We can derive this moment by defining the BA' vector, then substituting into the moment equation.

$$ \vec{BA'} = \vec{BA}+\vec{AA'} $$

$$ \vec{M_B} = {(\vec{BA}+\vec{AA'})}\; X\; \vec{F}= \vec{BA}\; X\; \vec{F} + \vec{AA'}\; X\; \vec{F} $$

where $$ \vec{AA'}\; X\; \vec{F} = 0 $$

The derivation of the moment about point B is quite simple:
 * NOTE

$$ \sum{\vec{F}_{i}} = 0 $$ $$ \sum{\vec{M}_{B_{i}}} = \sum{\vec{BA'}_{i}}\; X\; \vec{F}_{i} = \sum{\vec{BA}_{i}}\; X\; \sum{\vec{F}_{i}} $$ where $$ \sum{\vec{F}_{i}} = 0 $$ Therefore, the moment about point B is also equal to zero.

The 8x8 global stiffness matrix is found below, where the local stiffness matrices for each element correspond to the color-coded areas. The k values for the local stiffness matrix for element 3 can be found as shown below. Only two examples are shown, yet the values can be found under the homework section of this page. $$ k_{33} = k^{(1)}_{33} + k^{(2)}_{11} + k^{(3)}_{11} $$ $$ k_{34} = k^{(1)}_{34} + k^{(2)}_{12} + k^{(3)}_{12} $$

=HOMEWORK= The displacement vector for node 2 can be found by breaking it down into vector components. $$q^{(e)}_2= (d^{(e)}_3\vec{i}+d^{(e)}_4\vec{j}) \cdot \vec{\tilde{i}}$$ This is equivalent to $$q^{(e)}_2= d^{(e)}_{3}(\vec{i}\cdot \vec{\tilde{i}})+d^{(e)}_4(\vec{j} \cdot \vec{\tilde{i}})$$

q2(e) is shown below, after substituting in the director cosines.

$$q_2^{(e)}=l^{(e)}d_3^{(e)}+m^{(e)}d_4^{(e)}$$ where $$ l^{(e)} = cos\theta $$ $$ m^{(e)} = sin\theta $$

$$q_2^{(e)}=\begin{bmatrix}l^{(e)} & m^{(e)} \end{bmatrix}\begin{bmatrix}d_3^{(e)}\\ d_4^{(e)}\end{bmatrix}$$