Problem $1$

The round bar shown in Figure $1$ has a uniform elastic modulus of E $.$ A force with magnitude $F_0$ is applied at point B with the sense shown, where there is a step in the diameter. The total length of the bar is $L_{\text{t}\text{o}\text{t}}.$ The diameters for each segment of the bar are: $d_{AB}$ and $d_{BC}.$ If the normal stress in segment $BC$ is ${}_{BC}$ and the normal stress in segment $AB$ is ${}_{AB},$ find the force $F_0$ and the lengths $L_{AB}$ and $L_{BC}.$

Figure $1$: Round bar with point loads

Take $E=220$GPa, $L_{\text{t}\text{o}\text{t}}=0\mathrm{.}9m,d_{AB}=50mm,$ and $d_{BC}=80mm.$

Take ${}_{BC}=-15$MPa and ${}_{AB}=50$MPa.