$2$ Stress on a "Composite" Bar

Consider a two$-$part bar. Its inner core is steel with diameter $100$ mm $.$ The outer portion is an aluminum

cylinder with inner diameter $100$ mm and outer diameter $150$ mm $.$ Both the steel core and aluminum cylinder

have the same length and always will have the same total displacement. The steel has Young's modulus

$E=205$GPa and coefficient of thermal expansion $=\frac{0\mathrm{.}000012}{}C($i$.$e$.,$ a $20C$ increase in temperature

causes a $1$ m bar of steel to extend in length by $(\frac{0\mathrm{.}000012}{}C)*(20C)*(1m)=0\mathrm{.}00024m,$ or $\{$:$240m).$

The aluminum has Young's modulus $E=70$GPa and coefficient of thermal expansion $=\frac{0\mathrm{.}000005}{}C.$

$2\mathrm{.}1$ Applied Load

The composite bar is subjected to an axial tensile load of $500$ kN $.$ Assuming the steel core and aluminum

cylinder displace the same total amount, calculate the axial stress in each portion $($steel core and aluminum

cylinder$).$

$2\mathrm{.}2$ Temperature Effect

Continuing on the previous part, the composite bar is now heated by $100C,$ but the total length is held

constant and the $500$ kN force still is applied. Calculate the resulting axial stress in each portion $($steel core

and aluminum cylinder$).$

HINT: This change in temperature will cause the steel and aluminum to expand, thus reducing the tensile

stress. A sufficiently large temperature change can even put the bar into compression since the length is

held constant!