In a thin cylindrical shell, the ratio of longitudinal stress to hoop stress is:
Try it first: pick an option, then read the explanation.
Correct answer: A. 0.5
Asked in SSC JE 2013; SSC JE 27/01/2018 (Evening)
Detailed Explanation
This is the previous ratio turned upside down: longitudinal stress pd/4t divided by hoop stress pd/2t.
\[ \frac{\sigma_l}{\sigma_h} = \frac{pd/4t}{pd/2t} = \frac{1}{2} = 0.5 \]
Why “0.5” Is Correct
The longitudinal stress is half the hoop stress, so the ratio is 0.5.
| Quantity | Thin cylinder (closed ends) | Thin sphere |
|---|---|---|
| Hoop stress | σh = pd/2t | — |
| Longitudinal stress | σl = pd/4t | — |
| Stress in any direction | — | pd/4t |
| Hoop strain | (pd/4tE)(2 − μ) | (pd/4tE)(1 − μ) |
| Longitudinal strain | (pd/4tE)(1 − 2μ) | — |
| Volumetric strain | (pd/4tE)(5 − 4μ) | (3pd/4tE)(1 − μ) |
Thin shell: wall thickness less than about 1/20 of the diameter, so stress is taken as uniform through the wall.
Why the Other Options Are Wrong
- B. 1.0: The two stresses are equal only in a thin sphere.
- C. 1.5: 1.5 does not follow from the formulas.
- D. 2.0: 2.0 is hoop to longitudinal, the inverse.
Key Point
σl/σh = 0.5 in a thin cylinder; = 1 in a thin sphere.
Key Concepts for Students
- A sphere needs half the wall thickness of a cylinder of the same diameter and pressure.
- Maximum in-plane shear in the wall = (σh − σl)/2 = pd/8t.
- Read the question carefully: hoop/longitudinal = 2, longitudinal/hoop = 0.5.
Practice more: this is question 7 in Thin Cylindrical and Spherical Shells MCQs – Part 1. Try the whole set and check every answer instantly.