Problem #$3$: $(30$ points$)$ An axially loaded bar ABCD consists of three segments: $(1)$ from $A$ to $B,$

$(2)$ from $B$ to $C,$ and $(3)$ from $C$ to $D,$ as shown in Figure $3.$ The bar subjected to three horizontal

forces: of $3$ kips applied at $B,4$kips applied at $C,$ and $2$ kips applied at $D,$ as shown. Segment $(1)$

from $A$ to $B$ is hollow with an outside diameter $d_o=1\mathrm{.}25in.$ and an inside diameter $d_i=\mathrm{.}875in.$

Segment $(2)$ from $B$ to $C$ is solid with a diameter $d_2=1\mathrm{.}25in.$ Segment $(3)$ from $C$ to $D$ is solid

with a diameter $d_3=\mathrm{.}875in.$ The lengths of the three segments are $L_1=10in.,L_2=15in.,$ and $L_3$

$=20$ in$.,$ respectively. The Young's moduli of the three segments are $E_1=1010,E_2=15{10}^6$

psi $,$ and $E_3=20{10}^6$ psi, respectively.

a$)$ Draw an appropriate free body diagram and find the reaction exerted by the wall at $A.$

b$)$ Take appropriate cuts, draw appropriate free body diagrams, and find the normal forces

$($denoted by $N_1,N_2,$ and $N_3$ in each segment, respectively$)$ and the normal stresses $($denoted

by ${}_1,{}_2,$ and ${}_3$ in each segment, respectively$)$ throughout the bar.

c$)$ Find the changes in length of each segment of the bar $($denoted by ${}_1,{}_2,$ and ${}_3,$

respectively$),$ and find the change in length $$ of the whole bar.