Question:
A thin cylindrical shell of diameter 'd', length 'l' and thickness 't' is subjected to an axial or internal pressure 'P'. What is the ratio of longitudinal strain to hoop strain in terms of Poisson's ratio (1/m)
A thin cylindrical shell of diameter 'd', length 'l' and thickness 't' is subjected to an axial or internal pressure 'P'. What is the ratio of longitudinal strain to hoop strain in terms of Poisson's ratio (1/m)
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Hoop stress is always twice the longitudinal stress in a thin cylinder.
Using this ratio ($\sigma_{\text{h}} = 2\sigma_{\text{l}}$) makes it easy to derive the strain expressions quickly during examinations.
Using this ratio ($\sigma_{\text{h}} = 2\sigma_{\text{l}}$) makes it easy to derive the strain expressions quickly during examinations.
Updated On: Jul 9, 2026
- (m-2) (2m+1)
- (2m-1) (m-2)
- (m-2) (2m-1)
- (2m+1) (m-2)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the ratio of longitudinal strain to hoop strain in a thin cylindrical pressure vessel subjected to internal pressure.
This ratio is to be expressed in terms of the material's Poisson's ratio, represented here as \(1/m\).
Thin shell theory applies when the wall thickness is small relative to the overall diameter of the cylinder.
Step 2: Key Formula or Approach:
The hoop (circumferential) stress \(\sigma_{\text{h}}\) in a thin cylinder is given by:
\[ \sigma_{\text{h}} = \frac{P d}{2 t} \]
The longitudinal stress \(\sigma_{\text{l}}\) in a thin cylinder is given by:
\[ \sigma_{\text{l}} = \frac{P d}{4 t} \]
Applying generalized Hooke's Law, the strains are:
\[ \epsilon_{\text{h}} = \frac{1}{E} \left( \sigma_{\text{h}} - \mu \sigma_{\text{l}} \right) \]
\[ \epsilon_{\text{l}} = \frac{1}{E} \left( \sigma_{\text{l}} - \mu \sigma_{\text{h}} \right) \]
where \(\mu = \frac{1}{m}\) is Poisson's ratio, and \(E\) is Young's Modulus.
Step 3: Detailed Explanation:
• Substitute the stress expressions into the strain equations:
\[ \epsilon_{\text{h}} = \frac{P d}{4 t E} \left( 2 - \mu \right) \]
\[ \epsilon_{\text{l}} = \frac{P d}{4 t E} \left( 1 - 2 \mu \right) \]
• Calculate the ratio of longitudinal strain to hoop strain:
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{1 - 2\mu}{2 - \mu} \]
• Substitute \(\mu = \frac{1}{m}\) into this ratio:
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{1 - \frac{2}{m}}{2 - \frac{1}{m}} \]
• Simplify the algebraic fraction by multiplying the numerator and denominator by \(m\):
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{m - 2}{2m - 1} \]
Step 4: Final Answer:
The ratio of longitudinal strain to hoop strain is \(\frac{m-2}{2m-1}\).
The question asks for the ratio of longitudinal strain to hoop strain in a thin cylindrical pressure vessel subjected to internal pressure.
This ratio is to be expressed in terms of the material's Poisson's ratio, represented here as \(1/m\).
Thin shell theory applies when the wall thickness is small relative to the overall diameter of the cylinder.
Step 2: Key Formula or Approach:
The hoop (circumferential) stress \(\sigma_{\text{h}}\) in a thin cylinder is given by:
\[ \sigma_{\text{h}} = \frac{P d}{2 t} \]
The longitudinal stress \(\sigma_{\text{l}}\) in a thin cylinder is given by:
\[ \sigma_{\text{l}} = \frac{P d}{4 t} \]
Applying generalized Hooke's Law, the strains are:
\[ \epsilon_{\text{h}} = \frac{1}{E} \left( \sigma_{\text{h}} - \mu \sigma_{\text{l}} \right) \]
\[ \epsilon_{\text{l}} = \frac{1}{E} \left( \sigma_{\text{l}} - \mu \sigma_{\text{h}} \right) \]
where \(\mu = \frac{1}{m}\) is Poisson's ratio, and \(E\) is Young's Modulus.
Step 3: Detailed Explanation:
• Substitute the stress expressions into the strain equations:
\[ \epsilon_{\text{h}} = \frac{P d}{4 t E} \left( 2 - \mu \right) \]
\[ \epsilon_{\text{l}} = \frac{P d}{4 t E} \left( 1 - 2 \mu \right) \]
• Calculate the ratio of longitudinal strain to hoop strain:
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{1 - 2\mu}{2 - \mu} \]
• Substitute \(\mu = \frac{1}{m}\) into this ratio:
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{1 - \frac{2}{m}}{2 - \frac{1}{m}} \]
• Simplify the algebraic fraction by multiplying the numerator and denominator by \(m\):
\[ \frac{\epsilon_{\text{l}}}{\epsilon_{\text{h}}} = \frac{m - 2}{2m - 1} \]
Step 4: Final Answer:
The ratio of longitudinal strain to hoop strain is \(\frac{m-2}{2m-1}\).
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