User:Eml4500.f08.Ateam.philbin/HW3

Homework 3

Mtg. 17 10-05-08

Q: How to sum the moment about point B?



A: The moment is the same as any point along the line of action of the force. In this example, the point A' is used to simplify the cross product since the vector $$ \vec {BA'}$$ > is perpendicular to $$\vec F$$ >.

$$\sum M_B = \vec {AB} x  \vec F$$

$$\sum M_B = \vec {BA'} x  \vec F $$

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

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

$$ M_B = \vec {BA} x \vec F + \vec {AA'} x  \vec F  = 0$$

3-bar Problem



In this problem, there are three bars instead of two in the last example. The numbering of the elements and nodes is important to make the subsequent matrix manipulations as simple as possible. As a rule of thumb, one should use the overlapping numbering system. Put local node 1 on local node 2 of the previous element, if possible. This is shown in the following image.



The angle theta is determine by the vector going from local node 1 to 2 in each element. Therefore the angle from local node 1 to 2 in element three is pointing down and the corresponding angle is negative. By following these conventions when the global stiffness matrix is made, the three elements add into the same place in order to solve for the unknown forces. In this case, they all add into the columns and rows 3 and 4.



The finished matrix without numerical values is given in the following figure.

 \underline K= \begin{bmatrix} k_{11}^{(1)} & k_{12}^{(1)} & k_{13}^{(1)} & k_{14}^{(1)}&0&0&0&0\\ k_{21}^{(1)} &k_{22}^{(1)} &k_{23}^{(1)} &k_{24}^{(1)} &0 &0 &0 &0 \\ k_{31}^{(1)} &k_{32}^{(1)} &k_{33}^{(1)}+k_{11}^{(2)}+k_{11}^{(3)} &k_{34}^{(1)}+k_{12}^{(2)}+k_{12}^{(3)} &k_{13}^{(2)} &k_{14}^{(2)} &k_{13}^{(3)} &k_{14}^{(3)}\\ k_{41}^{(1)} &k_{42}^{(1)} &k_{43}^{(1)}+k_{21}^{(2)}+k_{21}^{(3)} &k_{44}^{(1)}+k_{22}^{(2)}+k_{22}^{(3)} &k_{23}^{(2)} &k_{24}^{(2)} &k_{23}^{(3)} &k_{24}^{(3)}\\ 0 &0 &k_{31}^{(2)} &k_{32}^{(2)} &k_{33}^{(2)} &k_{34}^{(1)} &0 &0\\ 0 &0 &k_{41}^{(2)} &k_{42}^{(2)} &k_{43}^{(2)} &k_{44}^{(1)} &0 &0\\ 0 &0 &k_{31}^{(3)} &k_{32}^{(3)} &0 &0 &k_{33}^{(3)} &k_{34}^{(3)}\\ 0 &0 &k_{41}^{(3)} &k_{42}^{(3)} &0 &0 &k_{43}^{(3)} &k_{44}^{(3)}\\ \end{bmatrix}