Pressure vessel heads and closures: identify the incorrect statement about flanged-and-dished, shallow-dished, flat heads, and stress considerations.

Introduction / Context:
Pressure vessel ends (heads) must safely transmit internal pressure to the shell while controlling stress concentrations. Common head types include flat heads, flanged-and-dished, shallow-dished, hemispherical, and variants. Recognizing which statements reflect real mechanical behavior is vital for safe design and code compliance.

Given Data / Assumptions:

• Thin to moderately thick shells designed under applicable codes (e.g., ASME) at typical process pressures.
• Standard geometries: flanged-and-dished (F&D), shallow-dished, flat heads.
• Comparisons are qualitative, not code formula derivations.

Concept / Approach:
Curved heads (dished/hemispherical) distribute membrane stresses efficiently and require less thickness than flat plates for the same diameter and pressure. Flat heads experience bending stresses and thus become impractically thick at large diameters or high pressures. Statements about crown radius relative to shell diameter reflect standard geometric definitions for dished heads.

Step-by-Step Solution:
Assess (a): For standard F&D heads, crown radius on the order of shell diameter is typical; statement is acceptable.Assess (b): Shallow dished heads indeed use larger crown radii than the shell diameter; acceptable.Assess (c): Flat heads being “most suitable” for large diameter at very high pressure is incorrect; they require extreme thickness and are rarely economical.Assess (d): Flanged-only head notionally reduces sharp discontinuity at the center; acceptable as qualitative stress reasoning.(e) is a true generalization: dished heads lower edge bending relative to flat plates.

Verification / Alternative check:
Code formulas show required thickness for flat heads grows rapidly with diameter and pressure, validating why curved heads are preferred for high pressure/large diameter applications.

Why Other Options Are Wrong:
(a), (b), (d), and (e) reflect accepted geometry/stress behavior; (c) contradicts fundamental bending vs. membrane stress efficiency.

Common Pitfalls:
Assuming flat covers are interchangeable with dished heads; overlooking nozzle-induced local stresses and the need for reinforcement regardless of head shape.

Final Answer:
Flat head covers are most suitable for larger vessels operating at very high pressure.