Thin spherical shells under internal pressure: which membrane stresses are significant for design?

Introduction / Context:
Spherical pressure vessels are used when uniform membrane stress and minimal weight per unit volume are desirable. Understanding which stresses govern design clarifies why spheres can be more material-efficient than cylinders for the same pressure rating.

Given Data / Assumptions:

• Thin shell assumption for a sphere (t << radius).
• Uniform internal pressure; no local discontinuities considered.
• Membrane theory validity (neglect bending except near boundaries).

Concept / Approach:
In a thin spherical shell, the membrane stress is equal in all circumferential directions and is the principal design driver. Radial stress within the wall is small compared with membrane stress and is typically neglected in thin-shell design, unlike in thick-walled vessels where radial gradients matter.

Step-by-Step Solution:
Apply membrane theory: σ_θ = σ_φ = p r / (2 t) uniformly over the sphere.Radial stress is of order p at the inner surface and decays through the thin wall; for thin shells, its magnitude relative to σ_θ is negligible for sizing.Hence, circumferential (membrane) stresses dominate the design basis.

Verification / Alternative check:
Comparing cylinder vs. sphere: cylinders have higher hoop stress for the same radius and thickness; spheres distribute membrane stress uniformly and reduce required thickness for equal allowable stress.

Why Other Options Are Wrong:
(a) Overstates radial stress relevance; (c) suggests both are large, which conflicts with thin-shell assumptions; (d) negates membrane theory; (e) “longitudinal” is a cylinder concept.

Common Pitfalls:
Confusing thick- and thin-wall behaviour; misapplying cylinder formulas to spheres.

Final Answer:
Circumferential (membrane) stresses dominate