Calculate hoop stress in a pipe, cylinder or sphere from pressure, diameter and wall thickness, with longitudinal stress and a thick-wall (Lamé) check.

Internal minus external pressure; only nonnegative values are supported.

Changing basis reinterprets the entered diameter. Use the matching measurement.

Material yield comparison
Blank omits the yield-to-von-Mises ratio. This is not a design approval.

Hoop Stress Formula

For a thin-walled cylindrical pressure vessel or pipe, hoop stress is the circumferential stress caused by the pressure difference across the wall. The standard thin-wall relation is:

σₕ = (P rₘ) / (t)

If you prefer to work with mean diameter instead of mean radius, the equivalent form is:

σₕ = (P Dₘ) / (2 t)

Where:

Symbol Meaning Typical Units
σh Hoop stress (circumferential stress) Pa, kPa, MPa, psi
P Pressure difference across the wall, usually internal minus external pressure Pa, kPa, MPa, psi
rm Mean radius of the vessel wall m, cm, in, ft
Dm Mean diameter of the vessel wall m, cm, in, ft
t Wall thickness m, cm, in, ft

What the Calculator Does

Choose stress, pressure at a target stress, or wall thickness from a target stress. Select closed-end cylinder or sphere and identify whether the entered diameter is inner, mean or outer. The cylinder relation is shown below; spheres use σ = PDm/(4t). Mean radius rearrangements are background formulas, not a separate calculator mode.

Rearranged Forms

If you are solving for a different variable, these equivalent forms are useful:

P = (σₕ t) / (rₘ)
rₘ = (σₕ t) / (P)
t = (P rₘ) / (σₕ)

How to Calculate Hoop Stress

  1. Determine the net pressure. Use the pressure difference across the wall, not just internal pressure when significant external pressure exists.
  2. Use the mean radius. For thin walls, the mean radius is the radius measured at the midpoint of the wall thickness.
  3. Measure the wall thickness. Thickness must use the same length unit basis as the radius or diameter.
  4. Apply the formula. Multiply pressure by mean radius, then divide by thickness.

Mean Radius and Mean Diameter

This calculator consistently uses a mean-surface membrane approximation. It is not an exact thick-wall solution. The geometric relationships are:

rₘ = (rᵢ + rₒ) / (2)
Dₘ = (Dᵢ + Dₒ) / (2)

If outside diameter and thickness are known, the mean radius can also be written as:

rₘ = (Dₒ - t) / (2)

When This Equation Is Valid

The formula on this page is the thin-walled cylinder approximation. It is most appropriate when the wall is small compared with the vessel radius. A common rule of thumb is:

(t) / (rₘ) ≤ 0.1

Under this assumption, membrane stress is treated as approximately uniform through the thickness. Below a mean-diameter/thickness ratio of 20, the stress mode switches to the thick-wall Lamé stress at the bore. That screen alone does not validate a design; joints, radial stress, fatigue, corrosion, temperature, supports and other loads need separate analysis.

Units and Consistency

  • If pressure is entered in Pa and dimensions are in m, hoop stress is returned in Pa.
  • If pressure is entered in psi and dimensions are both in in, hoop stress is returned in psi.
  • Radius and thickness must use compatible length units before applying the equation.
  • Stress and pressure share the same dimensional unit, but they are not the same physical quantity.

Example

A cylindrical vessel has a pressure difference of 1.5 MPa, a mean radius of 0.25 m, and a wall thickness of 0.01 m.

σₕ = ((1.5 × 10⁶)(0.25)) / (0.01)
σₕ = 3.75 × 10⁷ Pa = 37.5 MPa

The wall therefore experiences a hoop stress of 37.5 MPa.

Hoop Stress vs. Longitudinal Stress

In a closed-end thin cylindrical vessel, the circumferential stress is larger than the axial stress. The longitudinal stress is:

σL = (P rₘ) / (2 t)

This means the hoop stress is typically twice the longitudinal stress for the same vessel geometry and pressure loading.

Common Mistakes

  • Using diameter in the radius equation without dividing by 2.
  • Mixing mm, m, in, or ft in the same calculation.
  • Using internal pressure instead of pressure difference when external pressure is not negligible.
  • Applying the thin-wall equation to a thick-walled vessel.
  • Using outside radius instead of mean radius when better accuracy is needed.

FAQ

What is hoop stress?
Hoop stress is the tangential tensile stress that acts around the circumference of a pressurized cylinder. It is the primary stress that tends to split a pipe or vessel along its length.

Why is mean radius used?
The selected approximation represents membrane stress at the mean surface. It does not include through-thickness stress variation or replace a thick-wall or design-code analysis.

Can hoop stress be negative?
A negative pressure difference can create compressive membrane stress, but this tool supports only nonnegative internal-minus-external pressure. External-pressure buckling requires a separate analysis.

Is this formula valid for spheres?
The cylinder formula is not the sphere formula. Select Sphere to use σ = PDm/(4t), with equal membrane stresses in both tangential directions. The same design limitations apply.

What if the wall is not thin?
For thick-walled cylinders, the hoop stress is not uniform through the wall, so the thin-wall equation should not be used for detailed design or high-accuracy analysis.