Thin cylindrical shell under internal pressure: Qualitative deformation of diameter and length
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ADiameter decreases and length decreases
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BDiameter increases and length decreases
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CDiameter decreases and length increases
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DDiameter increases and length increases
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EDiameter unchanged and length increases
Answer
Correct Answer: Diameter increases and length increases
Explanation
Introduction / Context:A thin-walled cylinder (pressure vessel) subjected to internal pressure develops circumferential (hoop) and longitudinal stresses. These stresses cause elastic strains that change the vessel dimensions—important for design and safety checks.
Given Data / Assumptions:
- Thin shell: wall thickness t is small compared with diameter D (t ≪ D).
- Internal pressure p creates hoop stress sigma_h and longitudinal stress sigma_l.
- Material is linearly elastic within working range.
Concept / Approach:Hoop stress sigma_h = p * D / (2 * t) and longitudinal stress sigma_l = p * D / (4 * t). Both are tensile. Tensile strain circumferentially increases diameter; tensile strain longitudinally increases length. Poisson effects exist but the net effect under internal pressure is expansion in both directions for thin cylinders.
Step-by-Step Solution:
Compute sigma_h = p * D / (2 * t), tensile → causes circumferential expansion.Compute sigma_l = p * D / (4 * t), tensile → causes axial elongation.Strain relations (plain text): epsilon = sigma / E − nu * (sigma_perp / E).Net result: positive strain in both hoop and axial → D and L increase.Verification / Alternative check:Experimental observations on thin tubes under pressure exhibit increased diameter and gauge length. Thick-cylinder effects are negligible in thin-wall assumption.
Why Other Options Are Wrong:Any option indicating decrease in diameter or length contradicts tensile stress-induced expansion under internal pressure.
Common Pitfalls:Mixing up sign conventions or thinking Poisson contraction can outweigh direct tensile strain; in thin shells under typical pressures, direct tensile strains dominate.
Final Answer:Diameter increases and length increases