I'm working on a thermal expansion problem out of Beer & Johnston's Mechanics of Materials and I'm a bit confused by the underlying rationale of one of the steps in the solution process.
In the exercise text, $\alpha_s$ and $\alpha_b$ are given along with a $\sigma_{max}$ of the inner steel core that cannot be exceeded. The core is considered fully bonded to the shell and the goal is to derive the max $\Delta T$ that can be observed while remaining under $\sigma_{max}$. I understand the general solution process, but a sign change in the deformation equation has me confused, as the book does not give a specific rationale for its change. The equation the text gives is:
$$\delta = \alpha (\Delta T) L + \frac{PL}{EA}$$
I have verified the final solution with a solutions manual, and the solution process for this problem proceeds:
$$\begin{align} P &= A_s \sigma_{max} & (1) \\ \delta_b &= \alpha_b (\Delta T) L - \frac{PL}{E_b A_b} & (2) \\ \delta_s &= \alpha_s (\Delta T) L + \frac{PL}{E_s A_s} & (3) \\ & \text{solve for} \,\,\Delta T \,\, \text{below} \end{align}$$
The specific element in question is the sign change of the $\delta_P$ term in Equation (2). In particular,
- Why does this sign change occur?
- For what reason is it assigned to the brass shell and not the inner steel core?