Cylindrical shell: stresses, max shear and volume change

Data: $L = 4000$ mm, $d = 1000$ mm, $p = 2\ \text{N/mm}^2$, $t = 8$ mm. The change in volume needs $E$ and $\nu$; they are not given, so the usual values for steel are taken: $E = 2\times10^5\ \text{N/mm}^2$ and $\nu = 0.3$.

Stresses

$$\sigma_c = \frac{pd}{2t} = \frac{2(1000)}{2(8)} = 125.0\ \text{N/mm}^2, \qquad \sigma_l = \frac{pd}{4t} = 62.5\ \text{N/mm}^2$$

Maximum shear stress

$$\tau_{max} = \frac{\sigma_c-\sigma_l}{2} = \frac{125.0-62.5}{2} = 31.25\ \text{N/mm}^2$$

Change in volume

$$\varepsilon_v = \frac{pd}{4tE}(5-4\nu) = \frac{2(1000)}{4(8)(2\times10^5)}(5-1.2) = 1.1875\times10^{-3}$$ $$V = \frac{\pi}{4}(1000)^2(4000) = 3.1416\times10^{9}\ \text{mm}^3, \qquad \delta V = \varepsilon_v V = 3.7306\times10^{6}\ \text{mm}^3 = 3731\ \text{cm}^3$$

Answer: $\sigma_c = 125.0$ N/mm², $\sigma_l = 62.5$ N/mm², $\tau_{max} = 31.25$ N/mm², $\delta V \approx 3731$ cm³ (increase).