Answer
$$\Delta T_{max} = 137.75^{\circ} F$$
Work Step by Step
1) Geometry
$\bullet$ Area of Steel Core:
$A_{st} = 1*1 = 1\space in^2$
$\bullet$ Area of Brass Shell:
$A_{br} = 1.5^2 - 1 = 1.25\space in^2$
2)Thermal Expansion
Since Brass expands more rapidly than Steel (as evident from $\alpha_{br}\gt\alpha_{st}$), as a result of an increase in temperature, the brass shell will experience compressive stress while steel will be under tensile stress.
Taking the stress in the steel core as limiting:
$\sigma_{st}=\frac{P_{int}}{A_{st}}\space \Rightarrow P_{int} = 8\space kip$ (Internal force generated due to temperature change)
$\sigma_{br} =\frac{P_{int}}{A_{br}} = -6.4\space ksi$ (compressive)
3)Deformation Compatibility
$\delta_{st} = \frac{\sigma_{st}}{E_{st}}*l + \alpha_{st} * \Delta T* l$
$\delta_{br} = \frac{\sigma_{br}}{E_{br}}*l + \alpha_{br} * \Delta T* l$
Since $\delta_{st} = \delta_{br}$ :
$\frac{\sigma_{st}}{E_{st}}*l + \alpha_{st} * \Delta T* l = \frac{\sigma_{br}}{E_{br}}*l + \alpha_{br} * \Delta T* l$
Solving for the unknown value, $\Delta T$:
$\boxed{\Delta T = 137.75^{\circ} F}\space\space\leftarrow\space ANS$
