# Hoop Stress Calculator

URL: https://speedycalculator.com/calculators/physics/hoop-stress/
Last updated: 2026-10-30

> The hoop stress calculator finds the hoop (circumferential), longitudinal and radial stresses in a thin-walled pressurised cylinder or sphere, plus the change in diameter and volume.

## Inputs

- Vessel shape
- Inner radius r (mm, cm, m, in, µm)
- Wall thickness t (mm, cm, m, in, µm)
- Internal pressure P (MPa, kPa, Pa, GPa, psi, ksi)
- Joint efficiency η
- Young's modulus E (GPa, MPa, Pa, psi, Mpsi)
- Poisson's ratio ν

## Results

- Hoop stress σ_h
- Longitudinal stress σ_l
- Radial stress at the inner wall
- Change in diameter
- Relative change in volume
- r/t ratio (thin wall if > 10)

## Formula

```
Cylinder: σ_h = P·r/(t·η),  σ_l = P·r/(2t·η) · Sphere: σ = P·r/(2t·η)
Δd/d = (σ_h − ν·σ_l)/E,  ΔV/V = Pr(5 − 4ν)/(2tE) (cylinder) or 3Pr(1 − ν)/(2tE) (sphere)
```

## Explanation

Pipes, boilers, scuba tanks and balloons hold pressure by tension in their walls. In a cylinder the hoop stress is twice the longitudinal stress, which is why overpressurised pipes split lengthwise.

### How to use the Hoop Stress Calculator

- **Vessel shape**
- **Inner radius r**
- **Wall thickness t**
- **Internal pressure P**
- **Joint efficiency η**
- **Young's modulus E**
- **Poisson's ratio ν**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose the vessel shape, enter the inner radius, wall thickness, internal pressure and joint efficiency (1 for seamless), and the material's E and ν for deformations. Valid for r/t > 10.

### Inputs explained

FieldWhat to enterVessel shapeChoose one: Cylinder, SphereInner radius rUnits: mm, cm, m, in, µmWall thickness tUnits: mm, cm, m, in, µmInternal pressure PUnits: MPa, kPa, Pa, GPa, psi, ksiJoint efficiency ηA numberYoung's modulus EUnits: GPa, MPa, Pa, psi, MpsiPoisson's ratio νA number

### What each result means

- **Longitudinal stress σ_l** = [Internal pressure P]×[Inner radius r]÷(2×[Wall thickness t]×[Joint efficiency η])
- **Radial stress at the inner wall** = -[Internal pressure P]
- **r/t ratio (thin wall if > 10)** = [Inner radius r]÷[Wall thickness t]

### Example values

How the main result changes when **Inner radius r** changes and everything else stays at the example values above:

Inner radius r (mm)Hoop stress σ_h (MPa)25050375755001006251257501501,000200

## Worked example

Inputs — Vessel shape: Cylinder; Inner radius r: 500 mm; Wall thickness t: 10 mm; Internal pressure P: 2 MPa; Joint efficiency η: 1; Young's modulus E: 200 GPa; Poisson's ratio ν: 0.3.

Results — **Hoop stress σ_h = 100 MPa**; Longitudinal stress σ_l = 50 MPa; Radial stress at the inner wall = -2 MPa; Change in diameter = 0.425 mm; Relative change in volume = 0.095 %; r/t ratio (thin wall if > 10) = 50.

## Key terms

- **Hoop stress**: Circumferential tension in a pressurised wall.

## FAQ

**Why do pipes burst lengthwise?**
The hoop stress is twice the longitudinal stress.

**When is the thin-wall formula valid?**
When the radius is more than about 10 times the wall thickness.

**What is joint efficiency?**
The relative strength of a welded or riveted seam (1 for seamless).

**Why are pressure vessels spherical?**
A sphere has half the wall stress of a cylinder of the same radius.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)