Formula and Method for Hoop Stress in Pressure Vessels
Hoop stress — also called circumferential or tangential stress — is the stress that acts around the circumference of a cylindrical pressure vessel, pipe, or tank wall when internal pressure pushes outward against it. For a thin-walled cylinder (wall thickness small relative to diameter), the classic engineering formula is σh = P × D / (2 × t), where P is internal pressure, D is the inside diameter, and t is the wall thickness. This calculator also derives the longitudinal (axial) stress for a closed-end cylinder and, if you supply a material yield or allowable stress, a safety factor.
How the calculation works
Enter the internal pressure P, the inside diameter D, and the wall thickness t (D and t must use the same length unit — the ratio D/t is what matters, so millimeters, inches, or any other unit works as long as both fields match). The calculator computes hoop stress as σh = PD/(2t) and longitudinal stress as σl = PD/(4t), exactly half the hoop stress, which is the axial stress produced when internal pressure pushes against the circular end caps of a closed vessel. It also reports the diameter-to-thickness ratio D/t as a thin-wall validity check: the formula is accurate when D/t ≥ 20. If you enter a yield or allowable stress, the tool divides it by the hoop stress to give a safety factor.
Common mistakes
- Mixing diameter and radius: the formula σh = PD/(2t) uses diameter; if you only know the radius r, either double it first or use the equivalent form σh = Pr/t.
- Inconsistent length units: diameter and wall thickness must be entered in the same unit (both mm, both inches, etc.) — the formula's D/t ratio cancels the unit, but only if they match.
- Assuming thin-wall theory always applies: for thick-walled cylinders (D/t below about 20, such as gun barrels or high-pressure fittings), the simple formula understates peak stress at the inner wall — use the Lamé thick-wall equations instead.
- Confusing hoop and longitudinal stress: hoop stress is twice the longitudinal stress in a closed cylinder, which is why pressurized pipes typically fail along a longitudinal split rather than at the end caps.
Real-world applications
- Pressure vessel and boiler design uses hoop stress to size wall thickness against a material's allowable stress with an appropriate safety factor.
- Pipeline engineering applies the same relationship (often called Barlow's formula) to select pipe schedule and wall thickness for a given operating pressure.
- Compressed gas cylinders, scuba tanks, and hydraulic cylinders are all checked against hoop stress limits during design and periodic inspection.
- Storage tank and pipe manufacturers use the diameter-to-thickness ratio to decide whether thin-wall or thick-wall stress equations govern the design.