Q$.2$

$($a$)$ A free$-$standing rod comprised of an alloy square hollow section at top and a steel square solid section at bottom is subjected to axial loads of $150$ kN at point $C$ and $450$ kN at point $B$ as shown in Figure Q$\mathrm{.}2($a$).$ The dimension of the steel square solid section is $50$ mm $.$ The outer and inner dimension of the alloy square hollow section are $40$ mm and $20$ mm respectively. Given that the Young's moduli for steel and alloy are $210$ GPa and $105$ GPa respectively, and the Poisson's ratio for steel and alloy are $0\mathrm{.}30$ and $0\mathrm{.}20,$ respectively. Neglecting the bucking effect on the rod, calculate:

$($i$)$ the normal stress of each section of the rod, i$.$e$.$ section $(4$ marks$)$ AB and BC ;

$($ii$)$ the change in the total length of the composite rod; and

$(5$ marks$)$

$($iii$)$ the change in the lateral dimension of the steel section.

$(4$ marks$)$

Figure $0\mathrm{.}2($a$)$