About the Hoop Stress Calculator
This hoop stress calculator finds the circumferential (hoop) stress and the longitudinal (axial) stress in a cylindrical pipe, tube, tank or pressure vessel under internal pressure. Enter the pressure, the diameter, whether that diameter is the inside or outside dimension, and the wall thickness. The calculator gives the thin-wall result σ = PD/2t that most hand calculations use, the exact thick-wall Lamé hoop stress at the bore, and the D/t ratio so you can see whether the thin-wall assumption is reasonable.
Add the allowable or yield stress of the material to get a factor of safety and the minimum wall thickness for that pressure. It suits mechanical and piping engineers, students working through mechanics of materials, and anyone sanity-checking a hydraulic line, air receiver or boiler shell. Work in psi and inches or in bar, millimetres and MPa.
Entering the outside diameter gives Barlow’s formula, which is slightly conservative and is widely used for pipe pressure ratings. This is a membrane-stress estimate for a straight, closed-end cylinder away from heads, nozzles and supports; design to ASME or EN pressure codes adds weld efficiency, corrosion allowance and code-specific formulas.
How to use the hoop stress calculator
- 1Choose imperial or metric units.
- 2Enter the internal pressure and the pipe or vessel diameter.
- 3Say whether the diameter is the inside or outside dimension.
- 4Enter the wall thickness and, optionally, the allowable stress.
- 5Compare the thin-wall and Lamé values and check the factor of safety.
Formula and method
Cutting a pressurised cylinder lengthwise, the pressure acting on the projected area (P × D × length) is resisted by two walls of thickness t, so the hoop stress is PD/2t. Cutting it crosswise, the pressure on the end area (πD²/4) is resisted by the wall ring (πDt), giving a longitudinal stress of PD/4t — half the hoop stress. These thin-wall results assume the stress is uniform through the wall, which is accurate when D/t is about 20 or more.
For thicker walls the Lamé solution gives the exact elastic hoop stress, which peaks at the inner surface: σ = P(ro² + ri²)/(ro² − ri²), with radial stress equal to −P there. Using the outside diameter in PD/2t is Barlow’s formula, a conservative approach common for pipe ratings. Metric inputs in bar are converted to MPa (1 bar = 0.1 MPa = 0.1 N/mm²) so stresses come out in MPa.
- σh, σl
- Hoop (circumferential) and longitudinal (axial) stress
- P
- Internal gauge pressure
- D
- Diameter (inside, or outside for Barlow)
- t
- Wall thickness
- ri, ro
- Inner and outer radius
- S
- Allowable or yield stress of the material
Worked examples
1,000 psi in a 10" ID vessel with a 1/2" wall
Hoop stress = 1,000 × 10 ÷ (2 × 0.5) = 10,000 psi and longitudinal stress = 5,000 psi. The exact Lamé value at the bore is about 10,524 psi, 5% higher, because D/t is only 20. Against a 20,000 psi allowable the factor of safety is 2.0 and the minimum wall would be 0.25 in.
Metric tank: 25 bar, 500 mm ID, 6 mm wall
25 bar is 2.5 MPa. Hoop stress = 2.5 × 500 ÷ 12 = 104.2 MPa and longitudinal stress = 52.1 MPa. With a 138 MPa allowable the factor of safety is 1.32, and the thin-wall minimum thickness is 2.5 × 500 ÷ 276 = 4.53 mm before any corrosion allowance or weld efficiency.
Barlow: 6" Sch 40 pipe (6.625" OD, 0.280" wall) at 1,500 psi
Using the outside diameter, Barlow’s hoop stress = 1,500 × 6.625 ÷ 0.56 = 17,746 psi. With a 35,000 psi yield strength (Grade B pipe) the factor of safety against yield is about 1.97.
Frequently asked questions
What is hoop stress?+
Hoop stress is the tensile stress acting around the circumference of a pipe or cylinder when it is pressurised from inside. It tries to split the cylinder along its length and is the largest stress in a thin-walled pressure vessel, twice the longitudinal stress.
What is the hoop stress formula?+
For a thin-walled cylinder σh = P × D ÷ (2 × t), where P is internal pressure, D the diameter and t the wall thickness. The longitudinal stress in a closed-end cylinder is P × D ÷ (4 × t). For a sphere the stress in every direction is P × D ÷ (4 × t).
When does the thin-wall formula stop being accurate?+
The thin-wall approximation is generally used when the diameter-to-thickness ratio is at least about 20 (radius-to-thickness about 10). Below that, use the Lamé thick-wall equations, which give a higher hoop stress at the inner surface.
Should I use inside or outside diameter?+
Mechanics textbooks use the inside or mean diameter. Barlow’s formula for pipe uses the outside diameter, which gives a slightly higher, conservative stress. Pressure vessel codes such as ASME Section VIII use their own forms with inside radius and a 0.6t term.
What is Barlow’s formula?+
Barlow’s formula, P = 2St ÷ D, relates the internal pressure P a pipe can hold to its allowable stress S, wall thickness t and outside diameter D. It is the thin-wall hoop stress formula rearranged and is widely used to estimate pipe pressure ratings.
Does this include corrosion allowance or weld efficiency?+
No. The calculator gives the membrane stress for the thickness you enter. Code design typically subtracts a corrosion allowance and manufacturing tolerance from the wall and divides by a joint efficiency for welded seams, so the required wall is larger.