Thin-walled spherical storage (Horton sphere): what is the expression for circumferential (hoop) membrane stress under internal pressure?\n\nUse symbols:\np = internal pressure\nR = internal radius of the sphere\nt = wall thickness (t ≪ R)\nσ = hoop (circumferential) membrane stress

Introduction / Context:
Horton spheres are spherical storage vessels used for pressurized liquids and gases (e.g., liquid ammonia). For thin shells under internal pressure, membrane theory provides simple expressions for the uniform biaxial stresses. Knowing the correct hoop stress relationship is essential for sizing wall thickness and checking code compliance.

Given Data / Assumptions:

• Thin-walled approximation applies (t ≪ R).
• Uniform internal pressure p.
• Neglect local discontinuities (nozzles, supports) for the basic formula.

Concept / Approach:
A spherical shell carries internal pressure with equal meridional and hoop membrane stresses because of symmetry. Free-body analysis of a hemispherical segment shows that the net pressure force equals the tensile ring force developed by the membrane stress over the cut section. Solving the force balance yields the stress relation.

Step-by-Step Solution:

Verification / Alternative check:
The spherical membrane stress is half that of a cylindrical shell (σ_cyl = p * R / t), consistent with the well-known result that spheres require less thickness for the same pressure.

Why Other Options Are Wrong:

• pR/t and 2pR/t overpredict stress (cylindrical or worse).
• (pR)/(4t) underpredicts stress; not supported by membrane equilibrium.

Common Pitfalls:
Applying the spherical formula to cylinders or ignoring joint efficiency and corrosion allowance required by design codes.

Final Answer:
σ = p * R / (2 * t)