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## 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)

## Verified Solution

(a) $T_{i}=400-700=-300^{\circ} F$. Hence at inside surface

\begin{aligned}\sigma &=\frac{-\left(27 \times 10^{6}\right)\left(9.5 \times 10^{-6}\right)(-300)}{(1-0.28)}\left(\frac{2(13)+10}{3(13+10)}\right) \\&=55,800 psi\end{aligned}

and at outside surface

\begin{aligned}\sigma &=\frac{\left(27 \times 10^{6}\right)\left(\left(9.5 \times 10^{-6}\right)(-300)\right.}{(1-0.28)}\left(\frac{13+2 \times 10}{3(13+10)}\right) \\&=-51,000 psi\end{aligned}

(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.