Problem $3$: Consider the structure shown below $($an $$Inverted King

Truss$)$ consisting of Beam AC$,$ Strut BD$,$ Rod ADC, Pin support A$,$

Roller support C$,$ and Applied load P$.$a$.$ Show that this structure is Statically Indeterminant $($i$.$e$.$ the

equilibrium equations alone are not sufficient to determine the loads

carried by the beam, strut, rod, pins, and rollers$)$

b$.$ Determine the loads in Beam AC$,$ Strut BD$,$ Rod ADC using

Castigliano$$s Theorem of Least Work. Assume strut BD to be the

Redundant Member. The truss has the following geometry, properties,

and loading

$$ Beam AC

$160-$mm$-$deep by $60-$mm$-$wide rectangle

$$ EAC $=200$ GPa $($steel$)$

$$ GAC $=79\mathrm{.}3$ GPa $($steel$)$

$$ IAC $=1$

$12$ b h$3=20\mathrm{.}48\backslash$times $10-6$m$4$

$$ AAC $=$ b h $=9\mathrm{.}6\backslash$times $10-3$

$$ Rod ADC

$15-$mm diameter

$$ EADC $=200$ GPa $($steel$)$

$$ AADC $=\backslash$pi

$4$ d$2=176\mathrm{.}7\backslash$times $10-6$m$2$

$$ Strut BD

$40$ mm by $40$ mm

$$ EBD $=12\mathrm{.}4$ GPa $($oak$)$

$$ ABD $=1\mathrm{.}6\backslash$times $10-3$m$2$