Topic: Strength of Materials
Circumferential and longitudinal strains in the cylindrical boiler under internal steam pressure are $\epsilon_{1}$ and $\epsilon_{2}$ respectively. Change in the volume of the boiler cylinder per unit volume will be
- A. $\epsilon_{1}+2\epsilon_{2}$
- B. $\epsilon_{1}\epsilon_{2}^{2}$
- C. $2\epsilon_{1}+\epsilon_{2}$
- D. $\epsilon_{1}^{2}\epsilon_{2}$
Answer: C. $2\epsilon_{1}+\epsilon_{2}$
Explanation: Volume of a cylinder is $V=\pi r^{2}L$. Taking natural log on both sides: $log_{e}V=2~log_{e}r+log_{e}L$. Differentiating, we get $\frac{dV}{V}=\frac{2dr}{r}+\frac{dL}{L}$. Thus, volumetric strain $e_{v}=2\epsilon_{1}+\epsilon_{2}$, where $\epsilon_{1}$ is circumferential (hoop) strain and $\epsilon_{2}$ is longitudinal strain.