Topic: Strength of Materials
A thin cylindrical shell of diameter (d), length (l) and thickness (t) is subjected to an internal pressure (p). The ratio of the longitudinal strain to hoop strain is
- A. $\frac{m-2}{2m-1}$
- B. $\frac{2m-1}{m-2}$
- C. $\frac{m-2}{2m+1}$
- D. $\frac{m+2}{2m+2}$
Answer: A. $\frac{m-2}{2m-1}$
Explanation: The longitudinal strain is given by $\epsilon_{l} = \frac{1}{E}[\sigma_{l} - \mu\sigma_{h}] = \frac{1}{E}[\frac{\sigma_{h}}{2} - \mu\sigma_{h}] = \frac{\sigma_{h}}{2E}[1 - 2\mu]$. The hoop strain is $\epsilon_{h} = \frac{1}{E}[\sigma_{h} - \mu\sigma_{l}] = \frac{1}{E}[\sigma_{h} - \frac{\mu\sigma_{h}}{2}] = \frac{\sigma_{h}}{2E}[2 - \mu]$. The required ratio is $\frac{\epsilon_{l}}{\epsilon_{h}} = \frac{(1 - 2\mu)}{(2 - \mu)}$. Using the relationship $\mu = \frac{1}{m}$, the ratio simplifies to $\frac{1 - 2(1/m)}{2 - (1/m)} = \frac{m-2}{2m-1}$.