User:Eml4500.f08.RAMROD.A/MyContributionHW3

The 3 Bar Truss and the Moment Equation
A look at the 3 Bar Truss Setup



From Statics, We know that there are typically three useful equations:

$$\displaystyle \sum{Fx}=0 $$

$$\displaystyle \sum{Fy}=0 $$

$$\displaystyle \sum{M_A}=0 $$

Statically Indeterminate
For this truss setup, even the moment equation renders no useful information because all reactions pass through the same point, A, and the setup is statically indeterminate

Now, we ask the question: what if we tried to sum the moments about some arbitrary point, B?

We will need to look at another picture and vector math to find out answers:

Can we get Information Summing Moments about B?


A look at the vector algebra:

$$\displaystyle \sum{\overline{M_B}}= \overline{BA} \times \overline{F} $$ Thus, $$\displaystyle \sum{\overline{M_B}}= \overline{BA'} \times \overline{F} $$ for all A' on line of action of F (see image right) From Vector Algebra: $$\displaystyle \overline{BA'} = \overline{BA}+ \overline{AA'} $$ $$\displaystyle \overline{M_B} = (\overline{BA}+ \overline{AA'}) \times \overline{F} $$ $$\displaystyle \overline{M_B} = \overline{BA} \times \overline{F} + \overline{AA'} \times \overline{F} $$, With The latter cross product in the previous equation equal to zero, we are left with: $$\displaystyle \overline{M_B} = \overline{BA} \times \overline{F} + 0 $$ More Generally:

$$\displaystyle \sum_{i}^{}{B_i}=\sum_{i}^{}{\overline{BA_i} \times \overline}$$ Where Ai is any point on the line of action of F. Which Indicates: $$\displaystyle \sum{\overline{M_B}} = \overline{BA} \times \sum{\overline{F}} $$ We are already aware that: $$\displaystyle \sum{\overline{F}} = 0 $$ We can conclude that: $$\displaystyle \sum{\overline{M_B}} = \overline{BA} \times 0 = 0 $$ '''So, even if we take the moment from any arbitrary point, it will ultimately not provide any helpful information to solve the problem. The problem truly is statically indeterminate.

Three Truss Stiffness Matrix
Now we can solve for overall K by combining K(1), K(2) and K(3)

Matrix K $$\displaystyle K = \begin{bmatrix} k_{33}^{(1)}+k_{11}^{(2)}+k_{11}^{(3)} & k_{34}^{(1)}+k_{12}^{(2)}+k_{12}^{(3)}\\k_{43}^{(1)}+k_{21}^{(2)}+k_{21}^{(3)} & k_{44}^{(1)}+k_{22}^{(2)}+k_{22}^{(3)}\\k_{31}^{(2)}+k_{31}^{(3)} & k_{32}^{(2)}+k_{32}^{(3)}\\k_{41}^{(2)}+k_{41}^{(3)} & k_{42}^{(2)}+k_{42}^{(3)}\\k_{51}^{(3)} & k_{52}^{(3)}\\k_{61}^{(3)} & k_{62}^{(3)} \end{bmatrix} $$  =   $$\displaystyle K = \begin{bmatrix} k_{33} & k_{34}\\k_{43} & k_{44}\\k_{53} & k_{54}\\k_{63} & k_{64}\\k_{73} & k_{74}\\k_{83} & k_{84} \end{bmatrix} $$