Question

A thin-walled cylindrical tank is internally pressurized. If the hoop strain is thrice the axial strain, the Poisson's ratio of the material is \(\underline{\hspace{1cm}}\) (correct to one decimal place).

Hint:

For thin cylinders, hoop stress is twice axial stress. Use strain relations with Poisson's effect to relate material constants.

Answer 1

Correct Answer: 0.2

For a thin cylindrical pressure vessel:
Hoop stress: \(\sigma_h = \frac{pr}{t}\)
Axial stress: \(\sigma_a = \frac{pr}{2t}\)
Hoop strain:
\[ \epsilon_h = \frac{\sigma_h}{E} - \nu \frac{\sigma_a}{E} \] Axial strain:
\[ \epsilon_a = \frac{\sigma_a}{E} - \nu \frac{\sigma_h}{E} \] Given:
\[ \epsilon_h = 3\epsilon_a \] Substituting stresses (\(\sigma_h = 2\sigma_a\)):
\[ \frac{2\sigma_a}{E} - \nu\frac{\sigma_a}{E} = 3\left( \frac{\sigma_a}{E} - \nu\frac{2\sigma_a}{E} \right) \] Cancel \(\frac{\sigma_a}{E}\):
\[ 2 - \nu = 3(1 - 2\nu) \] \[ 2 - \nu = 3 - 6\nu \] \[ 5\nu = 1 \] \[ \nu = 0.2 \] Thus, the Poisson's ratio is \(0.2\).