Question $1$

$($a$)$ Derive the formula for the elastic displacement of a member subject to axial load, where the cross$-$sectional area of the member, and the load acting on the member, vary along the member's length. Provide a sketch where necessary and define all symbols used.

$[10$ marks$]$

$($b$)$ The column AB$,$ shown in Figure Q$1,$ is made from reinforced concrete with four $20$ mm diameter steel reinforcing bars. The concrete and steel act as a single composite body. Determine the average normal stress developed in the concrete and in the reinforcing bars if $P=350kN.$ Note: $E_{\text{s}\text{t}\text{e}\text{e}\text{l}}=200$GPa,$E_{\text{c}\text{o}\text{n}\text{c}\text{r}\text{e}\text{t}\text{e}}=30$GPa. $[19$ marks$]$

$($c$)$ Check the validity of the analysis if the yield stress of the steel and the concrete are $({}_Y{)}_{\text{s}\text{t}\text{e}\text{e}\text{l}}=180$MPa,$({}_Y{)}_{\text{c}\text{o}\text{n}\text{c}}=20$MPa $[4$ marks$]$

Figure Q$1$