For a thin cylindrical shell with hemispherical ends designed for the same internal pressure, how does the required wall thickness of the cylindrical shell compare with that of the spherical (hemispherical) ends?

Introduction / Context:
Pressure vessels often use a cylindrical shell with hemispherical or elliptical heads. Because stress distributions differ between geometries, the required thickness for the same design pressure and material strength also differs. Knowing which section requires greater thickness guides economical and safe design.

Given Data / Assumptions:

• Thin-shell theory applies (t ≪ R).
• Same internal pressure p and material allowable stress.
• Joint efficiency and corrosion allowance not considered in the conceptual comparison.

Concept / Approach:
For a thin cylinder, the hoop (circumferential) stress is σ_cyl = p * R / t. For a thin sphere (hemispherical end), the membrane stress is σ_sph = p * R / (2 * t). To maintain the same allowable stress, the spherical head can be thinner for the same radius and pressure because the stress is half of that in a cylinder for the same thickness. Therefore, to achieve the same allowable stress, the cylindrical shell must be thicker than the spherical end.

Step-by-Step Solution:

Verification / Alternative check:
Design codes reflect this relationship in required thickness formulas; spheres are structurally more efficient pressure enclosures than cylinders.

Why Other Options Are Wrong:

• Equal or less-than statements contradict the membrane stress relationships.
• Option (d) introduces pressure dependence that does not change the fundamental geometric relationship under identical design conditions.

Common Pitfalls:
Ignoring the influence of nozzle reinforcements and joint efficiencies that may locally govern thickness; the geometric comparison still holds for the shell vs. head forms.

Final Answer:
More than