Pressure Vessel Hoop Stress Calculator
Find the hoop and longitudinal stress in a thin-wall cylindrical pressure vessel from internal pressure, diameter, and wall thickness.
🛢️ What is Pressure Vessel Hoop Stress?
Hoop stress, also called circumferential stress, is the stress that develops around the circumference of a cylindrical pressure vessel as internal pressure pushes outward against the wall. For a thin-wall cylinder, where the wall thickness is small relative to the diameter, hoop stress is given by the simple membrane formula sigma_hoop = PD / 2t, and it is always exactly twice the longitudinal (axial) stress that acts along the length of the vessel.
Engineers rely on this formula constantly when sizing storage tanks, boiler drums, gas cylinders, pipelines, and process vessels, checking whether a proposed wall thickness keeps stress within the material's allowable limit at the design pressure, and sizing the required thickness for a target pressure rating. Because hoop stress governs (it is the larger of the two), it is the stress checked first in a thin-wall design.
A common point of confusion is when the thin-wall assumption stops applying. It holds well when the diameter-to-thickness ratio D/t is 20 or greater, a rule of thumb used across most design codes. Below that ratio, stress varies meaningfully through the wall thickness and the thin-wall formula understates the true peak stress at the inner surface, thick-wall (Lame) equations are needed instead.
This calculator computes both hoop and longitudinal stress from your pressure, diameter, and thickness, checks the D/t ratio automatically, and plots how hoop stress changes as wall thickness varies, so you can immediately see how much thicker (or thinner) the wall could be at your design pressure.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Compressed Air Receiver Tank
A tank with P = 2 MPa, D = 500 mm, t = 10 mm
Example 2 — High-Pressure Process Vessel
A vessel with P = 5 MPa, D = 1000 mm, t = 15 mm
Example 3 — Small Thick-Walled Tank (Invalid Thin-Wall Case)
A small tank with P = 1.5 MPa, D = 300 mm, t = 20 mm
❓ Frequently Asked Questions
🔗 Related Calculators
What is hoop stress in a pressure vessel?
Hoop stress (also called circumferential stress) is the stress acting around the circumference of a cylindrical pressure vessel, caused by internal pressure pushing outward on the vessel wall. For a thin-wall cylinder it is given by sigma_hoop = PD / 2t.
What is the formula for hoop stress?
sigma_hoop = P x D / (2 x t), where P is the internal pressure, D is the internal diameter, and t is the wall thickness. This applies to a thin-wall cylindrical pressure vessel under internal pressure.
What is longitudinal stress in a pressure vessel?
Longitudinal (axial) stress is the stress acting along the length of the cylinder, caused by pressure pushing against the end caps. It equals sigma_long = PD / 4t, exactly half the hoop stress, for a thin-wall cylinder under internal pressure alone.
Why is hoop stress twice the longitudinal stress?
The circumferential wall resists the internal pressure over a smaller effective area (per unit length) than the end caps resist it over the full cross-section, working through equilibrium of forces this ratio works out to exactly 2:1 for a thin-wall cylinder under uniform internal pressure.
What is the thin-wall assumption and when is it valid?
The thin-wall (membrane stress) assumption treats stress as uniform through the wall thickness, ignoring how stress varies across the wall. It is generally considered valid when the diameter-to-thickness ratio D/t is 20 or greater, below that, thick-wall (Lame) equations should be used instead.
What happens if D/t is less than 20?
Below a D/t of about 20, stress varies meaningfully through the wall thickness and the thin-wall formula underestimates the peak stress at the inner surface. Use thick-wall (Lame) cylinder equations, which account for radial stress variation, for an accurate result.
Does this formula account for welded joints?
No, this calculator computes the theoretical membrane stress only. Pressure vessel design codes such as ASME Section VIII apply a joint efficiency factor (less than 1.0) to account for weld quality, reducing the allowable stress or increasing the required thickness accordingly.
Should I use internal or external diameter in this formula?
Using the internal diameter gives the standard, slightly more conservative (higher) thin-wall hoop stress estimate. Some references use the mean diameter (average of internal and external) for a marginally more accurate result, the difference is small for a genuinely thin wall.
What units does this calculator use?
Internal pressure is entered in megapascals, diameter and wall thickness in millimeters, and both hoop and longitudinal stress results are shown in megapascals.
How is hoop stress different for a spherical pressure vessel?
A thin-wall sphere has a single membrane stress in every direction, sigma = PD / 4t, equal to the longitudinal stress of an equivalent cylinder. A cylindrical vessel has two different stresses, the higher hoop stress and the lower longitudinal stress, because it is only curved in one direction around its circumference.