Thin spherical pressure vessel – Membrane stress formula A thin spherical shell of diameter d and wall thickness t is subjected to an internal pressure p. What is the uniform membrane (hoop) stress developed in the shell material?

Introduction / Context:Thin-walled pressure vessels are analyzed using membrane theory. For a sphere, the stress is uniform in all tangential directions, making it a preferred shape for withstanding internal pressure.

Given Data / Assumptions:

• Spherical shell, diameter d, thickness t (t ≪ d).
• Internal gauge pressure p acts uniformly.
• Material behaves elastically and deformations are small.

Concept / Approach:Equilibrium of a hemispherical free body yields the tangential membrane stress σ. For a thin sphere, the resultant of membrane stresses around the great circle balances the internal pressure force on the hemisphere.

Step-by-Step Solution:Pressure force on hemisphere = p * projected area = p * (π d^2 / 4).Resultant resisting force = σ * (circumference) * thickness = σ * (π d) * t.Equate forces: σ * (π d) * t = p * (π d^2 / 4).Solve: σ = p d / (4 t).

Verification / Alternative check:Compare with thin cylinder (σ_hoop = p d / (2 t)); the sphere carries pressure more efficiently, showing half the hoop stress for the same d, t, p.

Why Other Options Are Wrong:(A) and (B) overestimate stress; (D) underestimates; “2 p d / t” is nonphysical for thin-shell equilibrium.

Common Pitfalls:Using the cylindrical formula for spheres; forgetting the projected area for pressure force; neglecting thin-wall criteria.

Final Answer:p d / (4 t)