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A thick-walled hollow cylinder having outside and inside radii of 90 mm and 40 mm respectively is subjected to an external pressure of 800 MN/m2. The maximum circumferential stress in the cylinder will occur at a radius of
  • a)
    40 mm
  • b)
    60 mm
  • c)
    65 mm
  • d)
    90 mm
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A thick-walled hollow cylinder having outside and inside radii of 90 m...
The maximum circumferential stress in a thick cylinder occurs at the inner radius, i.e. 40 mm.
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Most Upvoted Answer
A thick-walled hollow cylinder having outside and inside radii of 90 m...
Given:
Outside radius (r1) = 90 mm = 0.09 m
Inside radius (r2) = 40 mm = 0.04 m
External pressure (P) = 800 MN/m^2 = 800 N/mm^2

To find:
The radius at which the maximum circumferential stress occurs.

Assumptions:
1. The cylinder is considered to be a thin-walled pressure vessel.
2. The cylinder is subjected to internal and external pressure only.

Solution:

1. Calculation of Stress:
The circumferential stress (σ) in a thin-walled cylinder is given by the formula:

σ = (P * r1) / (r1 - r2)

Substituting the given values:

σ = (800 * 0.09) / (0.09 - 0.04)
= 1440 N/mm^2

2. Calculation of Radius:
To find the radius at which the maximum circumferential stress occurs, we need to differentiate the stress equation with respect to the radius (r) and equate it to zero.

dσ/dr = 0

Differentiating the equation:

dσ/dr = (P * (r1 - r2) - P * r1) / (r1 - r2)^2

Setting the derivative equal to zero:

(P * (r1 - r2) - P * r1) / (r1 - r2)^2 = 0

Simplifying the equation:

(P * (r1 - r2) - P * r1) = 0
P * r1 - P * r2 - P * r1 = 0
- P * r2 = 0

Since P is a positive value, r2 must be zero for the equation to be satisfied.

Conclusion:
The maximum circumferential stress in the cylinder will occur at a radius of 40 mm (option A).
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A thick-walled hollow cylinder having outside and inside radii of 90 mm and 40 mm respectively is subjected to an external pressure of 800 MN/m2. The maximum circumferential stress in the cylinder will occur at a radius ofa)40 mmb)60 mmc)65 mmd)90 mmCorrect answer is option 'A'. Can you explain this answer?
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