Hoop Stress Calculator

Hoop stress, also called circumferential stress, is the stress experienced by the wall of a pressure vessel acting in the tangential (circumferential) direction. It arises due to internal pressure trying to expand the vessel radially outward. For thin-walled cylinders, hoop stress is the critical design stress because it is twice the longitudinal stress.

For a thin-walled cylindrical vessel, hoop stress is calculated as σ_h = (P × d) / (2 × t × η), where P is internal pressure, d is internal diameter, t is wall thickness, and η is the joint efficiency. For a spherical vessel, the formula is σ_h = (P × d) / (4 × t × η).

Circumferential stress (hoop stress) acts in the tangential direction around the circumference of the vessel and resists radial expansion. Longitudinal stress (axial stress) acts along the axis of the cylinder and resists axial splitting. For cylinders, hoop stress is always twice the longitudinal stress, making it the governing design parameter.

Longitudinal stress (σ_l) is the axial stress acting along the length of a cylindrical pressure vessel. It is calculated as σ_l = (P × d) / (4 × t × η). Note that for spherical vessels, hoop stress and longitudinal stress are equal because of symmetry, and both equal σ = (P × d) / (4 × t × η).

For a thin-walled spherical pressure vessel, hoop stress is σ_h = (P × d) / (4 × t × η). Because a sphere is symmetric, the stress is equal in all directions, and there is no separate longitudinal stress — both are equal. This also means spheres are inherently more efficient pressure vessels than cylinders.

Joint efficiency (η) accounts for the reduced strength of welded or riveted joints compared to the base material. A value of 1.0 means a seamless or fully efficient joint. Riveted joints typically range from 0.7 to 0.85. Lower joint efficiency increases the effective stress experienced at the joint, making it a critical design factor.

A pressure vessel is considered thin-walled when the ratio of internal diameter to wall thickness (d/t) is greater than 10, or equivalently when the radius is more than 5 times the wall thickness. Below this threshold, more complex thick-wall analysis (using the Lamé equations) is required for accurate stress prediction.

For cylindrical pressure vessels, hoop stress is exactly twice the longitudinal stress. This means the circumferential direction governs failure before the axial direction does. Designers must ensure the vessel material's yield or ultimate strength exceeds the hoop stress with an appropriate safety factor, which is why hoop stress is used as the primary design criterion.