Exercise $\backslash$#$3$ Select the lightest weight aluminum beam for a design

a$)$ The sketch depicts a continuous

tubular cross$-$section beam with

static loads P and Q applied at nodes

$1$ and $3$ as shown. Write the

expressions for the reaction forces in

terms of $\backslash (\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\},\backslash$mathrm$\{$P$\}\backslash )$ and Q $.$ Use the sign

convention shown in the inset.

$\backslash (\backslash$mathbf$\{$R $2$ X$\}=\backslash )$

$\backslash (\backslash$mathbf$\{$R $2$ Y$\}=\backslash )$

$\backslash ($ K S Y$=\backslash )$

Write the expression for the deflection of node $1$ relative to node $\backslash (2^\{*\}\backslash ),$ in of terms the above dimensions and and E$,($Young$\text{'}$s modulus$)$ and the Mol $("$I$")$

Deflection of node $1$: $=$ inches

Assuming a hollow rectangular cross$-$section beam with wall thickness $"$ t $"$ for all walls, write the expression for the maximum stress, in terms of $\backslash ($ P$,$ Q$,$ a$,$ b $\backslash )$ and the beam cross$-$section dimensions of $"$h$","$w$"$ and wall thickness $"$t$"($or use cross$-$section Area $"$A$"$ and Moment$-$of inertia "I IOI $").$

Maximum stress $\backslash (=\backslash )$

psi

Located at node:

b$)$ For $\backslash ($ a$=72\backslash )$ inches, $\backslash ($ b$=24\backslash )$ inches, $\backslash ($ P$=300\backslash$mathrm$\{$lbf$\}\backslash )$ and $\backslash ($ Q$=20,000\backslash$mathrm$\{$lbf$\}\backslash ),$ determine the lightest weight $\backslash (\backslash$underline$\{6061-\}\backslash )$

T$6$ aluminum tubular beam accounting for the following: The deflection of node $1$ relative to node $2$ must not exceed $"$a$"/480.$ A SF of at least $2\mathrm{.}0$ for yield stress is required for all locations along the beam. The tube's height$-$to$-$width ratio $(\backslash (\backslash$mathrm$\{$h$\}/\backslash$mathrm$\{$w$\}\backslash ))$ must not exceed $\backslash (4/1\backslash ).$ The wall thickness for the aluminum tube must be at least $0\mathrm{.}125$ inches. Your calculations should indicate that: