Thin vessel with extra water: stresses

Data: $d = 1000$ mm, $t = 8$ mm, $p = 10\ \text{N/mm}^2$ (the pressure raised above atmospheric), $E = 2\times10^5\ \text{N/mm}^2$, $\nu = 0.3$, $K_{water} = 2100\ \text{N/mm}^2$. (E, ν and K are needed only for volume change; the stresses depend only on $p$, $d$, $t$.)

Principal stresses

$$\sigma_1 = \sigma_c = \frac{pd}{2t} = \frac{10(1000)}{2(8)} = 625.0\ \text{N/mm}^2, \qquad \sigma_2 = \sigma_l = \frac{pd}{4t} = 312.5\ \text{N/mm}^2$$

The radial stress (maximum $-p = -10$ N/mm² on the inner surface) is small compared with these and is neglected in thin-shell theory.

Maximum shear stress

$$\tau_{max} = \frac{\sigma_1-\sigma_2}{2} = \frac{625.0-312.5}{2} = 156.25\ \text{N/mm}^2$$

(in the plane of the wall, on planes at $45^\circ$ to the axis). If the radial stress is taken as zero, the absolute maximum shear stress, $\sigma_1/2 = 312.5$ N/mm², occurs on planes at $45^\circ$ to the wall surface.

Answer: $\sigma_1 = 625.0$ N/mm², $\sigma_2 = 312.5$ N/mm², $\tau_{max} = 156.25$ N/mm².