Hoop Stress Calculator
Enter the internal pressure, diameter, and wall thickness of a thin-walled pressure vessel to get the hoop (circumferential) stress, longitudinal stress, radial stress, and safety factor instantly. Switch between metric and imperial units, choose a cylinder or sphere geometry, and see a full step-by-step breakdown of the calculation.
Formula
Worked example
A steel pressure vessel (cylinder) with inner diameter 500 mm, wall thickness 10 mm, and internal pressure 5 MPa: hoop stress = 5 x 500 / (2 x 10 x 1.0) = 125 MPa. Longitudinal stress = 62.5 MPa. With a yield strength of 250 MPa, the safety factor = 250 / 125 = 2.0. The d/t ratio is 50, confirming the thin-wall assumption is valid.
What is hoop stress?
Hoop stress (also called circumferential stress or tangential stress) is the internal tensile stress that develops in the wall of a pressure vessel in response to internal pressure. It acts tangentially, trying to "split" the vessel along a longitudinal seam. In a closed cylindrical vessel the hoop stress is twice the longitudinal (axial) stress, which is why longitudinal seam welds are more critical than circumferential ones. In a thin-walled sphere, hoop stress is equal in every direction because the geometry is symmetric. The formula was first formalized by Denis Poisson and is a foundation of pressure vessel engineering standards including ASME BPVC, EN 13445, and AS 1210.
How the thin-wall assumption works
The thin-wall (or membrane) theory treats the vessel wall as a simple membrane with no bending stiffness. It is valid when the wall thickness is small relative to the radius, specifically when the ratio of inner diameter to wall thickness (d/t) is greater than 20, or equivalently when t/r is less than 0.1. Under this assumption the stress is considered uniform through the wall thickness, which simplifies the calculation to a single formula. When d/t falls below about 20, the stress varies significantly across the wall thickness and the thick-wall (Lame) equations must be used instead. The difference in predicted hoop stress between thin- and thick-wall methods at d/t = 20 is roughly 5-10 percent, growing rapidly as the ratio decreases. This calculator flags the ratio so you can see at a glance whether the thin-wall assumption holds for your geometry.
Joint efficiency and design factors
The joint efficiency factor (eta, ranging from 0 to 1) accounts for the weakening effect of welds relative to the base metal. A seamless pipe or a vessel with fully radiographed (100% RT) welds uses eta = 1.0, meaning the weld is as strong as the parent metal. Spot-examined welds use eta = 0.85, and standard non-examined welds may use values as low as 0.60-0.70 depending on the applicable code. Dividing by eta in the stress formula is equivalent to treating the weld as the limiting cross-section. For critical pressure vessels always verify the appropriate joint efficiency from the applicable code (ASME BPVC Section VIII Division 1 Table UW-12, for example). The safety factor shown by this calculator is the ratio of material yield strength to hoop stress; most pressure vessel codes target a safety factor of at least 3.5 on tensile strength or 2.0 on yield strength for static loading.
Sphere vs cylinder - choosing the right geometry
A sphere is the most material-efficient shape for containing pressure. For the same internal pressure, diameter, and wall thickness, a sphere develops half the hoop stress of a cylinder. This is why high-pressure gas storage spheres and propane tanks are spherical: they use roughly half the material of an equivalent cylinder. Cylindrical vessels are preferred when length matters - pipelines, boilers, and horizontal storage tanks - because they are far cheaper to fabricate and can be made in any length. The penalty is the higher hoop stress, which means the cylinder wall must be designed for twice the circumferential force per unit area. In practice, cylindrical vessels often have domed or hemispherical end caps, which transition between the two regimes and are designed separately. This calculator handles the two pure geometries: a full cylinder with closed ends (giving both hoop and longitudinal stress) and a full sphere (where both principal stresses are equal).
Typical material yield strengths and Young's moduli
| Material | Yield strength (MPa) | Young's modulus (GPa) | Typical use |
|---|---|---|---|
| Carbon steel (A516 Gr.70) | 260 | 200 | Pressure vessels, boilers |
| Stainless steel 304 | 215 | 193 | Corrosive service, food and pharma |
| Stainless steel 316L | 170 | 193 | High corrosion resistance |
| Carbon steel API 5L X65 | 448 | 200 | Gas and liquid pipelines |
| Aluminium 6061-T6 | 276 | 69 | Lightweight vessels, aerospace |
| Titanium Grade 2 | 275 | 105 | Chemical and marine service |
| HDPE | 20-30 | 0.8-1.5 | Low-pressure pipes and tanks |
Reference values for common pressure vessel materials. Always confirm with certified material data sheets.
Frequently asked questions
What is the difference between hoop stress and longitudinal stress?
In a cylindrical pressure vessel, hoop stress (circumferential stress) acts around the circumference and is caused by the pressure trying to burst the cylinder radially outward. Longitudinal (axial) stress acts along the length of the vessel and is caused by the pressure acting on the closed end caps. For a thin-walled cylinder, hoop stress is exactly twice the longitudinal stress. This means the circumferential seam welds carry higher loads than the longitudinal ones, and design is usually governed by hoop stress.
When does the thin-wall assumption break down?
The thin-wall formula gives acceptably accurate results (within about 5% of the exact Lame solution) when the inside diameter is more than 20 times the wall thickness (d/t greater than 20). Below that ratio the stress distribution across the wall thickness becomes non-uniform and the simple membrane formula underestimates the peak stress at the inner surface. For thick-walled cylinders (d/t less than 10), the Lame equations, which account for the stress gradient through the wall, should be used instead.
How do I find the maximum allowable internal pressure from hoop stress?
Rearrange the hoop stress formula: P = (2 * sh * t * eta) / d for a cylinder, or P = (4 * sh * t * eta) / d for a sphere. Substitute the allowable stress (yield strength divided by the code safety factor) for sh. For example, if a cylindrical vessel has a wall of 10 mm, diameter 500 mm, joint efficiency 1.0, and allowable stress 125 MPa, the maximum pressure is 2 x 125 x 10 x 1.0 / 500 = 5 MPa.
Why is hoop stress also called circumferential or tangential stress?
All three names refer to the same stress component: the tension in the vessel wall that acts tangentially (along the circumference). "Hoop stress" is the most common engineering term, derived from the analogy of barrel hoops that resist the outward bulge of the staves. "Circumferential stress" emphasizes the direction. "Tangential stress" is mathematically precise - in cylindrical polar coordinates, it is the sigma-theta component.
What is Barlow's formula and how does it relate to hoop stress?
Barlow's formula (S = P*D / (2*t)) is simply the standard thin-wall hoop stress formula written using the outside diameter instead of the inside diameter. In pipeline engineering it is traditional to specify outside diameter (e.g. 12-inch pipe), so Barlow's formula is used with OD. For thick-walled pipes the difference between using OD and ID is significant, but for most pipelines (high d/t ratio) the two give nearly the same result. This calculator uses inner diameter, which is the more rigorous approach for pressure vessel design.
What safety factor is required for pressure vessels?
Required safety factors vary by code and application. ASME BPVC Section VIII Division 1 traditionally uses a safety factor of 3.5 on ultimate tensile strength, which for many steels corresponds to about 2.0-2.5 on yield strength. European standard EN 13445 uses partial safety factors on both loads and material strength. For pipelines, ASME B31.8 uses a design factor of 0.72 on SMYS for rural locations, reducing to 0.40 in high-density urban areas. Always apply the code appropriate to your jurisdiction and service conditions.
How does joint efficiency affect hoop stress?
Joint efficiency (eta) appears in the denominator of the hoop stress formula, so a lower eta directly increases the effective stress the weld must sustain. With eta = 0.85 (spot-examined weld), the wall must be about 18% thicker than for a seamless vessel to achieve the same safety factor. Pressure vessel codes specify which eta values are permitted for different weld types and inspection levels. Using a full-penetration butt weld with 100% radiography allows eta = 1.0 and minimizes required wall thickness.