Chapter 5
Q. 5.15
A thin cylindrical vessel is heated by a jacket from the outside such that the temperature distribution is as shown in Fig. 5.26. If E = 27 × 10^6 psi, α = 9.5 × 10^{-6} in. / in.°F, and μ = 0.28, determine (a) maximum thermal stress using Eq. 5.40 and (b) maximum thermal stress using Eq. 5.41.
\begin{array}{ll}\sigma_{\theta}=\sigma_{z}=\frac{-E \alpha T_{i}}{1-\mu}\left[\frac{2 r_{o}+r_{i}}{3\left(r_{o}+r_{i}\right)}\right] & \text { for inside surface } \\\sigma_{\theta}=\sigma_{z}=\frac{E \alpha T_{i}}{1-\mu}\left[\frac{r_{o}+2 r_{i}}{3\left(r_{o}+r_{i}\right)}\right] & \text { for outside surface }\end{array} (5.40)
\begin{array}{ll}\sigma_{\theta} & =\sigma_{z}=\frac{-E \alpha T_{i}}{2(1-\mu)} \quad \text { for inside surface } \\\sigma_{\theta} & =\sigma_{z}=\frac{E \alpha T_{i}}{2(1-\mu)} \quad \text { for outside surface }\end{array} (5.41)
Step-by-Step
Verified Solution
(a) T_{i}=400-700=-300^{\circ} F. Hence at inside surface
and at outside surface
(b) For inside surface
\begin{aligned}\sigma &=\frac{\left(-27 \times 10^{6}\right)\left(9.5 \times 10^{-6}\right)(-300)}{2(1-0.28)} \\&=53,400 psi\end{aligned}and for outside surface \sigma=-53,400 psi.