Internally pressurised cylindrical vessels: comparing closures of the same material and thickness, which head shape can withstand the highest internal pressure before reaching the same membrane stress limits?

Introduction / Context:
Pressure vessel ends (heads) convert internal pressure into membrane stresses. For a given material and thickness, geometry strongly influences the stress distribution. Identifying the most efficient head helps reduce weight or increase allowable design pressure.

Given Data / Assumptions:

• All heads have identical nominal thickness and are made from the same material.
• Elastic membrane theory applies; local discontinuities at knuckles are neglected for comparison.
• Internal pressure loading only; no external loads or vacuum buckling.

Concept / Approach:
Hemispherical heads develop uniform membrane stresses and have the most favorable curvature in both principal directions. As a result, for the same thickness, the allowable internal pressure is highest. Ellipsoidal (2:1) heads are next best, followed by torispherical or conical. Flat plates are worst because they carry bending in addition to membrane stresses.

Step-by-Step Solution:
Rank shapes by curvature efficiency: hemispherical > ellipsoidal > conical > flat.For equal thickness, the head with the lowest stress at a given pressure can withstand the highest pressure.Select hemispherical as best performer.

Verification / Alternative check:
Design codes use larger allowable pressure or smaller required thickness for hemispherical heads than for ellipsoidal/torispherical heads under the same conditions.

Why Other Options Are Wrong:
Ellipsoidal (2:1): efficient but not as strong as hemispherical for the same thickness.Conical: stress concentrations at the knuckle; lower pressure capacity.Flat plate: carries bending, lowest pressure capacity for a given thickness.

Common Pitfalls:
Comparing unequal thicknesses; ignoring required knuckle radii and code joint efficiencies.

Final Answer:
hemispherical