Pressure vessel heads: for a cylindrical vessel closed with heads of equal thickness, which head shape can withstand the HIGHEST internal pressure for the same thickness and diameter?

Introduction / Context:
Pressure vessel end closures (heads) come in different shapes. For a given thickness and diameter, the geometry dictates membrane stresses and thus the pressure-carrying capability. Designers choose head shapes by balancing strength, fabrication cost, and available space. This question identifies the strongest shape for a given thickness.

Given Data / Assumptions:

• Thin-shell assumptions apply (membrane theory).
• Heads compared at equal thickness and identical diameter/knuckle proportions where applicable.
• Internal pressure loading; no external stiffening considered.

Concept / Approach:

Under internal pressure, membrane stresses are minimized when curvature is high and uniform. Hemispherical heads have constant principal curvatures and uniform membrane stress, giving the highest allowable pressure for the same thickness. Ellipsoidal (2:1) heads are next, then torispherical heads, while flat heads are weakest and require significantly more thickness or stays to withstand pressure.

Step-by-Step Solution:

Verification / Alternative check:

Design codes (e.g., thin-shell relations) rank hemispherical heads as requiring the least thickness for a given design pressure and diameter, confirming maximal strength per thickness.

Why Other Options Are Wrong:

Ellipsoidal and torispherical heads are strong but require more thickness than a hemisphere. Flat plates are the weakest and typically impractical at pressure without stays.

Common Pitfalls:

Ignoring knuckle stress concentrations in torispherical heads; assuming fabrication ease equals strength (hemispherical is best structurally but costlier to make).

Final Answer:

Hemispherical head