Calculate hoop stress, axial stress, and Von Mises stress in thin-walled pressure vessels. Supports cylinders and spheres with safety factor analysis.
Hoop stress is the circumferential stress in the wall of a pressure vessel — the primary stress that tends to split a cylinder lengthwise or burst a sphere. For thin-walled vessels (r/t ≥ 10), the classic formula σ = pr/t for cylinders (or σ = pr/2t for spheres) provides a quick and accurate estimate of this critical stress.
This Hoop Stress Calculator computes hoop stress, axial stress, and Von Mises equivalent stress for both cylindrical and spherical thin-walled pressure vessels. It also determines whether your design meets the safety margin by dividing the material yield strength by your chosen safety factor.
Pressure vessel design is safety-critical — boilers, gas cylinders, pipelines, and chemical reactors all rely on accurate stress analysis. This tool helps engineers, students, and technicians quickly verify wall thickness adequacy, find the maximum allowable pressure, or determine the minimum required thickness for a given service pressure. Check the example with realistic values before reporting.
Pressure vessel failure can be catastrophic. This calculator provides a rapid first-pass stress analysis, combining hoop, axial, and Von Mises stresses with material yield and safety factor. It replaces multiple manual calculations and provides visual stress-vs-allowable comparison bars for immediate pass/fail assessment. Keep these notes focused on your operational context.
Cylinder Hoop Stress: σ_hoop = p × r / t Cylinder Axial Stress: σ_axial = p × r / (2t) Sphere Hoop Stress: σ_hoop = p × r / (2t) Von Mises Equivalent: σ_vm = √(σ₁² + σ₂² − σ₁σ₂ + 3σ₃²) Where: p = internal pressure (MPa) r = inner radius (m) t = wall thickness (m)
Result: Hoop stress = 82.74 MPa
A cylinder with 1.379 MPa internal pressure, 0.3 m radius, and 5 mm wall thickness has a hoop stress of σ = (1.379 × 0.3) / 0.005 = 82.74 MPa.
Pressure vessels are closed containers designed to hold gases or liquids at pressures substantially different from ambient. They range from simple air receivers to nuclear reactor containment structures. The fundamental design requirement is that wall stresses remain safely below the material's strength, with appropriate safety factors to account for manufacturing variations, corrosion, fatigue, and unexpected overloads.
The thin-wall approximation assumes stress is uniform across the wall thickness, which simplifies calculations enormously. When r/t < 10, stress varies significantly from inner to outer surface, requiring Lamé's equations (thick-wall theory). Most industrial vessels are designed as thin-walled because it is more material-efficient.
The ASME Boiler and Pressure Vessel Code (BPVC) is the primary standard governing pressure vessel design in North America. Section VIII covers unfired pressure vessels, with Division 1 (design by rule) and Division 2 (design by analysis). These codes specify allowable stresses, required safety factors, weld inspection requirements, and hydrostatic testing procedures.
Hoop stress (circumferential stress) is the stress acting tangentially around the circumference of a pressure vessel. It is the dominant stress in cylinders and the one most likely to cause longitudinal splitting.
The thin-wall approximation is accurate when the ratio of inner radius to wall thickness (r/t) is 10 or greater. For thicker walls, Lamé equations (thick-wall theory) should be used.
In a cylinder, the hoop stress (pr/t) is exactly twice the axial stress (pr/2t). This is why cylinders tend to fail with longitudinal cracks — the hoop stress is the larger of the two.
ASME Boiler and Pressure Vessel Code typically uses a safety factor of 3.5–4 on ultimate tensile strength, or about 2/3 of yield. The exact factor depends on the code, service conditions, and inspection level.
Von Mises stress is an equivalent stress that combines all principal stresses into a single value for comparison against yield strength. Yielding occurs when Von Mises stress reaches the material yield strength (for ductile materials).
Not directly. In practice, you should subtract the corrosion allowance from the wall thickness before entering it, or add it to the minimum required thickness result.