Question

A thin cylindrical shell subjected to an internal pressure resulted in hoop stress and longitudinal stress. If the radius and thickness of the shell are increased by 10%, then the increase in percentage of hoop stress and longitudinal stress respectively are

• A. \( 10 %, 5 % \)
• B. \( 0 %, 0 % \)
• C. \( 10 %, 10 % \)
• D. \( 5 %, 5 % \)

Hint:

Remember the formulas for hoop and longitudinal stresses in thin cylindrical shells. When both radius and thickness are scaled by the same factor, the stresses remain constant because the scaling factors cancel out.

Answer 1

The Correct Option is B

Step 1: Recall the formulas for Hoop Stress and Longitudinal Stress in a thin cylindrical shell.
For a thin cylindrical shell subjected to an internal pressure \( P \), with radius \( R \) and thickness \( t \):
The Hoop Stress (circumferential stress), \( \sigma_h \), is given by: \[ \sigma_h = \frac{PR}{t} \] The Longitudinal Stress (axial stress), \( \sigma_l \), is given by: \[ \sigma_l = \frac{PR}{2t} \]
Step 2: Analyze the effect of increasing radius and thickness by 10%.
Let the original radius be \( R \) and the original thickness be \( t \). The new radius, \( R' \), is increased by 10%: \[ R' = R + 0.10R = 1.1R \] The new thickness, \( t' \), is increased by 10 %: \[ t' = t + 0.10t = 1.1t \] The internal pressure \( P \) is assumed to remain constant.
Step 3: Calculate the new Hoop Stress (\( \sigma_h' \)).
Substitute \( R' \) and \( t' \) into the formula for hoop stress: \[ \sigma_h' = \frac{P R'}{t'} = \frac{P (1.1R)}{(1.1t)} \] \[ \sigma_h' = \frac{1.1PR}{1.1t} = \frac{PR}{t} \] Since \( \sigma_h = \frac{PR}{t} \), we have \( \sigma_h' = \sigma_h \).
Step 4: Calculate the new Longitudinal Stress (\( \sigma_l' \)).
Substitute \( R' \) and \( t' \) into the formula for longitudinal stress: \[ \sigma_l' = \frac{P R'}{2t'} = \frac{P (1.1R)}{2(1.1t)} \] \[ \sigma_l' = \frac{1.1PR}{2.2t} = \frac{PR}{2t} \] Since \( \sigma_l = \frac{PR}{2t} \), we have \( \sigma_l' = \sigma_l \).
Step 5: Determine the percentage increase in stresses.
Percentage increase in Hoop Stress: \[ \text{Percentage Increase} = \frac{\sigma_h' - \sigma_h}{\sigma_h} \times 100% = \frac{\sigma_h - \sigma_h}{\sigma_h} \times 100% = 0% \] Percentage increase in Longitudinal Stress: \[ \text{Percentage Increase} = \frac{\sigma_l' - \sigma_l}{\sigma_l} \times 100% = \frac{\sigma_l - \sigma_l}{\sigma_l} \times 100% = 0% \] Therefore, both the hoop stress and longitudinal stress remain unchanged. The increase in percentage for both is 0%. The final answer is $\boxed{\text{2}}$.