Thin cylindrical vessels under internal pressure: if two vessels have the same wall thickness but different diameters, which one can withstand the higher internal pressure before reaching the same hoop stress limit?

Introduction / Context:For thin-walled cylinders, the dominant stress from internal pressure is the circumferential (hoop) stress. Understanding how geometry influences allowable pressure is fundamental in preliminary sizing and material selection.

Given Data / Assumptions:

• Same material and same wall thickness t for both vessels.
• Thin-wall assumption applies (t ≪ d).
• Internal pressure only; no significant external loads.

Concept / Approach:The hoop stress for a thin cylinder is σ_h = p d / (2 t). For a given allowable stress and fixed t, the allowable pressure scales inversely with diameter: p_allow ∝ 1/d. Thus, reducing diameter increases the pressure that can be sustained before reaching the same stress limit.

Step-by-Step Solution:Write σ_h = p d / (2 t).For fixed σ_allow and t, solve p = 2 t σ_allow / d.Smaller d → larger p, hence the smaller diameter vessel withstands higher pressure.

Verification / Alternative check:Doubling diameter halves allowable pressure at unchanged thickness and allowable stress; this simple proportionality check confirms the conclusion.

Why Other Options Are Wrong:Larger dia vessel / larger dia long vessel: Both have higher hoop stress at the same pressure; thus lower allowable pressure for the same thickness.“Same strength irrespective of diameter”: Incorrect; the hoop-stress formula shows explicit dependence on d.

Common Pitfalls:Confusing longitudinal with hoop stress; applying thick-wall formulas to thin shells.

Final Answer:Smaller dia vessel.