Calculation
σt,max = pi · (ra²+ri²)/(ra²−ri²)
Hoop stress is not constant through a thick cylinder wall: it is highest at the inner surface. This calculator uses the radial stress distribution rather than a thin-wall approximation.
Lamé-Spannungsverteilung bei reinem Innendruck · σt(ri) = pi(ra²+ri²)/(ra²−ri²)
Calculate Maximum Hoop Stress in a Thick Cylinder
Hoop stress is not constant through a thick cylinder wall: it is highest at the inner surface. This calculator uses the radial stress distribution rather than a thin-wall approximation.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Maximum hoop stress σt,max — Circumferential tensile stress at the inner wall. Material checks also need radial and possibly axial stress, allowable stress and applicable code requirements.
- Internal pressure pi — Gauge pressure inside relative to the pressure-free outer face for this load case, not absolute atmospheric pressure. Check operating and peak pressures separately.
- Inner radius ri — Radius from cylinder axis to the pressurised inner surface. Divide inner diameter by two; 50 mm in the example.
- Outer radius ra — Radius from the axis to the free outer surface. Must strictly exceed ri; wall thickness is ra−ri.
Example
At pi = 10 MPa, ri = 50 mm and ra = 75 mm, (ra²+ri²)/(ra²−ri²) = (5625+2500)/(5625−2500) = 2.6. The inner-wall hoop stress is therefore 26 MPa.
Assumptions and limits
Long, straight, homogeneous, isotropic, linearly elastic cylinder of constant section; static internal gauge pressure and zero external gauge pressure. Only the undisturbed middle region away from end caps, openings and nozzles is considered. Not a complete pressure-equipment or strength verification.
Understand Hoop stress in a thick-walled cylinder
How strongly does internal pressure pull around the inner surface of a thick tube? Stress variation through the wall matters here.
What does this quantity describe?
Internal pressure pi acts on the cylindrical bore. In a thick wall, circumferential tensile stress decreases from the inside outward. Lamé's solution gives its maximum at inner radius ri: σt,max = pi(ra²+ri²)/(ra²−ri²), where ra is outer radius. Hoop stress acts around the circumference; by itself it is not a complete equivalent stress.
Formula and variables
σt,max = pi · (ra²+ri²)/(ra²−ri²)
Maximum inner hoop stress: σt,max = pi(ra²+ri²)/(ra²−ri²)Wall thickness: t = ra−riRadial stress at bore: σr(ri) = −pi
| Symbol / input | Meaning |
|---|---|
| Maximum hoop stress σt,max | Circumferential tensile stress at the inner wall. Material checks also need radial and possibly axial stress, allowable stress and applicable code requirements. |
| Internal pressure pi | Gauge pressure inside relative to the pressure-free outer face for this load case, not absolute atmospheric pressure. Check operating and peak pressures separately. |
| Inner radius ri | Radius from cylinder axis to the pressurised inner surface. Divide inner diameter by two; 50 mm in the example. |
| Outer radius ra | Radius from the axis to the free outer surface. Must strictly exceed ri; wall thickness is ra−ri. |
Choose the inputs correctly
pi is internal gauge pressure relative to the pressure-free outer face, for example the highest relevant operating pressure in MPa. ri and ra are radii, not diameters: halve drawing diameters. Wall thickness ra−ri must be positive. The example values 50 and 75 mm are arithmetic examples, not approved dimensions.
How to use the calculator
Enter internal pressure and both radii. Check the highest relevant pressure case separately. Evaluate inner-wall σt,max against allowable stress only after including radial and possibly axial stress and the applicable code checks.
Worked example
For pi = 10 MPa, ri = 50 mm and ra = 75 mm, the radii factor is (75²+50²)/(75²−50²) = 8125/3125 = 2.6. Thus σt,max = 26 MPa at the bore. The thin-wall estimate pi·ri/(ra−ri) gives only 20 MPa and is unsuitable here.
Understand the result and units
Maximum circumferential tension is at the inner face. Increasing ra while holding ri fixed lowers it. As the wall becomes thin, the formula approaches the boiler approximation; a thick wall cannot be treated as uniformly stressed.
The calculation uses pascals and metres internally. MPa pressure and mm radii use the shared converter; only the ratio of squared radii enters the factor.
Typical applications
Early plausibility checks for thick tubes, cylinders and pressure housings; comparing wall thicknesses before full code-based design.
Assumptions, limits and common mistakes
Homogeneous isotropic linearly elastic cylinder under static internal pressure with zero external gauge pressure, in its undisturbed middle region. End closures, holes, notches, thermal stress, fatigue, plasticity and manufacturing tolerances are excluded. Pressure equipment requires a full check against applicable rules.
Common mistake: Do not mix radii and diameters. ra must exceed ri. Hoop stress differs from radial and axial stress and cannot directly be read as an allowable pressure.
Frequently asked questions
What is “Hoop stress in a thick-walled cylinder” used for?
Early plausibility checks for thick tubes, cylinders and pressure housings; comparing wall thicknesses before full code-based design.
Where do the input values come from?
pi is internal gauge pressure relative to the pressure-free outer face, for example the highest relevant operating pressure in MPa. ri and ra are radii, not diameters: halve drawing diameters. Wall thickness ra−ri must be positive. The example values 50 and 75 mm are arithmetic examples, not approved dimensions.
What does the result not cover?
Homogeneous isotropic linearly elastic cylinder under static internal pressure with zero external gauge pressure, in its undisturbed middle region. End closures, holes, notches, thermal stress, fatigue, plasticity and manufacturing tolerances are excluded. Pressure equipment requires a full check against applicable rules.