User:Eml4500.f08.ateam.boggs.t/Homework Two

The Definition of a Moment
A force moment is a force applied to a system at a radial distance from the origin of that system. The SI units for a moment is the Newton-meter(N·m), the US Customary Units is the foot-pound(ft·lbf). The equation for a moment is:

$$ \overrightarrow{M_o} = \overrightarrow{F}\times\overrightarrow{r} $$

This is the definition of a moment about point O. A moment can also be referred to as the torque of a system.

A 2-Beam Truss System Solved Statically


$$\sum F_{x} = 0 = R_{1}cos(30) - R_{2}cos(-45)$$

$$\sum F_{y} = 0 = R_{1}sin(30) + R_{2}sin(-45) + P $$

$$ R_{1} = \frac{R_{2}cos(-45)}{cos(30)} = \frac{\sqrt{2}}{\sqrt{3}}R_{2} $$

$$ 0 = \frac{\sqrt{2}}{2\sqrt{3}}R_{2} - \frac{\sqrt{2}}{2}R_{2} + P $$

$$ 0 =\frac{\sqrt{2}(1 - \sqrt{3})}{2\sqrt{3}}R_{2} + P $$

$$ R_{2} = -\frac{2\sqrt{3}}{\sqrt{2}(1 - \sqrt{3})}P $$

$$ R_{1} = -\frac{2}{1 - \sqrt{3}}P $$

Element 1:
L(1) = 4 E(1) = 3 Ac(1) = 1 θ(1) = 30° - l(1) = cosθ(1) l(1) = cos30° = &radic; 3 /2 m(1) = sinθ(1) m(1) = sin30° = 1/2

Element 2:
L(2) = 2 E(2) = 5 Ac(2) = 2 θ(2) = -45° - l(2) = cosθ(1) l(2) = cos(-45°) = &radic; 2 /2 m(2) = sinθ(1) m(2) = sin(-45°) = &radic; 2 /2

The Stiffness Matrix:


K^{(e)}= k^{(e)} \cdot \begin{bmatrix} (l^{(e)})^2 & l^{(e)}m^{(e)} & -(l^{(e)})^2 & -l^{(e)}m^{(e)}\\ l^{(e)}m^{(e)} & (m^{(e)})^2 & -l^{(e)}m^{(e)} & -(m^{(e)})^2\\ -(l^{(e)})^2 & -l^{(e)}m^{(e)} & (l^{(e)})^2 & l^{(e)}m^{(e)}\\ -l^{(e)}m^{(e)} & -(m^{(e)})^2 & l^{(e)}m^{(e)} & (m^{(e)})^2 \end{bmatrix} $$



k^{(e)} = \frac{E^{(e)}A^{(e)}}{L^{(e)}} $$



k^{(1)} = \frac{(3)\cdot(1)}{4} = \frac{3}{4} $$



k^{(2)} = \frac{(5)\cdot(2)}{2} = 5 $$



k_{11}^{(1)} = (l^{(1)})^2 $$



k_{11}^{(1)} = (\frac{\sqrt{3}}{2})^2 $$



k_{11}^{(1)} = \frac{3}{4} $$



k_{12}^{(1)} = l^{(1)}m^{(1)} $$



k_{12}^{(1)} = (\frac{\sqrt{3}}{2})\cdot(\frac{1}{2}) = \frac{\sqrt{3}}{4} $$



k_{22}^{(1)} = (m^{(1)})^2 $$



k_{22}^{(1)} = (\frac{1}{2})^2 = \frac{1}{4} $$

Due to symmetry, where k12(1) = k21(1),



K^{(1)} = \frac{3}{4}\cdot \begin{bmatrix} \frac{3}{4} & \frac{\sqrt{3}}{4} & -\frac{3}{4} & -\frac{\sqrt{3}}{4}\\ \frac{\sqrt{3}}{4} & \frac{1}{4} & -\frac{\sqrt{3}}{4} & -\frac{1}{4}\\ -\frac{3}{4} & -\frac{\sqrt{3}}{4} & \frac{3}{4} & \frac{\sqrt{3}}{4}\\ -\frac{\sqrt{3}}{4} & -\frac{1}{4} & \frac{\sqrt{3}}{4} & \frac{1}{4} \end{bmatrix} $$



K^{(1)} = \begin{bmatrix} \frac{9}{16} & \frac{3\sqrt{3}}{16} & -\frac{9}{16} & -\frac{3\sqrt{3}}{16}\\ \frac{3\sqrt{3}}{16} & \frac{3}{16} & -\frac{3\sqrt{3}}{16} & -\frac{3}{16}\\ -\frac{9}{16} & -\frac{3\sqrt{3}}{16} & \frac{9}{16} & \frac{3\sqrt{3}}{16}\\ -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & \frac{3\sqrt{3}}{16} & \frac{3}{16} \end{bmatrix} $$

For Element 2,



k_{11}^{(2)} = (\frac{\sqrt{2}}{2})^2 $$



k_{11}^{(2)} = \frac{2}{4} = \frac{1}{2} $$



k_{12}^{(2)} = \frac{1}{2} $$



k_{22}^{(2)} = \frac{1}{2} $$



K^{(2)} = 5\cdot \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2}\\ -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$



K^{(2)} = \begin{bmatrix} \frac{5}{2} & \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2}\\ \frac{5}{2} & \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2}\\ -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2} & \frac{5}{2}\\ -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2} & \frac{5}{2} \end{bmatrix} $$