Question # $2$

$(40)$

Consider the truss in Figure, shown. All members are made from Douglas$-$fir wood with a modulus

of elasticity of $E=1\mathrm{.}90{10}^{6}l\frac{b}{i}{n}^2$ and a cross$-$sectional area of $8i{n}^2.$ Each member is $3$ ft length

and angle of each joint are $60.$ Using Finite Element Formulation $($Hand Calculation$)$ Determine

i$.$ Deflection of each joint.

ii$.$ Calculate average stress in member $2.$

Stiffness Matrix for Truss Element

$[K{]}^{(e)}=k\left[\begin{array}{ccccccccccccc}co{s}^2& sincos& -co{s}^2& -sincos& sin2& -sincos& -si{n}^2& -sincos& co{s}^2& sincos& -si{n}^2& sincos& si{n}^2\end{array}\right]$

$-sincos$

$-sincos$

$-co{s}^2$

$-sincos$

$-co{s}^2$

$sincos$

$k_i=\frac{A_i{E}_i}{L_i}$

Global Coordinate to Local Coordinate Transform Matrix

$\{[u_{ix}],[u_{iy}],[u_{jx}],[u_{jy}]\}=\left[\begin{array}{cccc}cos& sin& 0& 0\\ -sin& cos& 0& 0\\ 0& 0& cos& sin\\ 0& 0& -sin& cos\end{array}\right]\{[U_{ix}],[U_{iY}],[U_{jx}],[U_{jY}]\}$