# Chapter 2 - Problems - Page 91: 2.49

$$\Delta T_{max} = 137.75^{\circ} F$$

#### 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, $\Delta T$: $\boxed{\Delta T = 137.75^{\circ} F}\space\space\leftarrow\space ANS$

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