The hoop stress calculator above gives the circumferential stress in the wall of a pipe or cylindrical pressure vessel, along with the longitudinal and radial stresses that act with it and the combined von Mises equivalent. It offers both the thin-wall membrane formula that most hand calculations use and the exact Lamé thick-wall solution, and it always shows you what the other model would have said.
Arb Digital builds free tools that are honest about their limits. This one solves the mechanics of a plain cylindrical shell. It is not a code calculation, and the section below on what a design code adds — joint efficiency, corrosion allowance, allowable stress from a certified table — explains why a passing number here is a starting point rather than a verdict.
What This Hoop Stress Calculator Does
It takes an internal pressure, an optional external pressure, a diameter, a wall thickness and a material yield strength, and returns the three principal stresses in the wall. Hoop stress acts circumferentially, trying to burst the cylinder open along a longitudinal seam. Longitudinal stress acts along the axis, trying to pull the ends off. Radial stress acts through the thickness and equals minus the internal pressure at the bore.
The diameter can be given as inside or outside, which matters more than it sounds. Pipe is normally specified by outside diameter and wall thickness, while vessel drawings frequently give the inside diameter, and using the wrong one with the wrong formula can shift the answer by several per cent on thick-walled sections.
The von Mises equivalent stress combines all three into the single number that yielding actually depends on, and the utilisation figure compares it against the allowable stress, taken here as yield divided by the design factor you enter. Anything approaching 100 per cent means the section is at its nominal limit before any code allowances have been applied.
How to Use It
- Enter gauge pressure, not absolute. The stress is driven by the difference across the wall, so a vessel at 10 bar gauge in atmosphere has a 10 bar differential, not 11.
- Say whether the diameter is inside or outside. The tool derives the other one from the thickness, so only one is ever entered.
- Start with the thin-wall model and check the ratio. If the diameter-to-thickness ratio comes out below about 20, switch to the thick-wall model, because the membrane assumption is no longer safe.
- Use the minimum wall, not the nominal. Pipe is manufactured with a mill tolerance, commonly minus 12.5 per cent, and the thin end of that tolerance is the one that governs.
- Read the von Mises figure for yielding, not the hoop stress alone. Yield depends on the combination of all three principal stresses, and hoop stress on its own is conservative in some geometries and not in others.
The Formulas: Membrane and Lamé
For a thin wall, the hoop stress is σh = P × Di ÷ (2t) and the longitudinal stress in a closed cylinder is σl = P × Di ÷ (4t), exactly half. These come from a force balance on a half-cylinder: the pressure acting on the projected area D×L is resisted by two wall sections of area t×L. The derivation and its assumptions are covered in MIT OpenCourseWare 2.001, Mechanics and Materials I, which treats thin-walled pressure vessels alongside the general stress–strain relationships.
For a thick wall the stress is not uniform, and the Lamé equations give the distribution. At the bore, with internal pressure Pi and external Po, the hoop stress is [Pi(ri² + ro²) − 2Poro²] ÷ (ro² − ri²), the radial stress is simply −Pi, and the longitudinal stress for closed ends is (Piri² − Poro²) ÷ (ro² − ri²).
Work the defaults. A pipe at 10 bar, which is 1.0 MPa, with a 200 mm bore and a 5 mm wall gives a thin-wall hoop stress of 1.0 × 200 ÷ 10 = 20.0 MPa and a longitudinal stress of 10.0 MPa. The Lamé solution for the same geometry, with ri = 100 and ro = 105, gives 20.51 MPa at the bore. The two agree to within about 2.5 per cent, which is what you should expect at a diameter-to-thickness ratio of about 40, and it is why the simple formula survives.
Thin Wall or Thick Wall: Where the Line Actually Is
The usual rule is that the membrane formula is acceptable when the diameter is at least twenty times the wall thickness, and that rule is about a five per cent error target rather than a physical transition. Nothing changes abruptly at 20; the error simply grows as the wall gets thicker relative to the bore.
At a ratio of 40 the discrepancy is around 2.5 per cent. At 20 it is roughly 5 per cent. At 10 it is about 10 per cent, and at a ratio of 5 the thin-wall formula understates the peak stress by more than 20 per cent — which is the wrong direction to be wrong in. This is why hydraulic cylinders, gun barrels, autoclaves and high-pressure tubing are always analysed with Lamé and never with the membrane formula.
The second reason the distinction matters is the radial stress. Thin-wall analysis ignores it entirely, treating the wall as a membrane in biaxial tension. In a thick wall the radial stress at the bore is a compression equal to the full internal pressure, and once the pressure becomes a significant fraction of the hoop stress that third principal stress changes the von Mises result appreciably. In a thin wall it is genuinely negligible; in a thick one it is not.
Why Hoop Stress Is Exactly Twice the Longitudinal Stress
This factor of two is not a coincidence or an approximation. It comes from geometry. The force trying to split the cylinder lengthwise acts on a rectangular projected area of diameter times length, and it is resisted by two wall strips each of thickness times length. The force trying to pull the end cap off acts on a circular area of πD²/4, and it is resisted by an annulus of circumference πD times thickness. Divide one by the other and the ratio is exactly two.
The practical consequences are everywhere once you look. Overpressured pipes split along a longitudinal seam, not around the circumference. Sausages and hot dogs crack lengthwise when cooked, for precisely this reason. A cylindrical vessel with a longitudinal weld has that weld in the highest-stressed direction, which is why longitudinal seams get more stringent inspection than circumferential ones.
It also explains why spherical vessels are more material-efficient. A sphere has no preferred direction, so the membrane stress is PD/(4t) in every direction — half the hoop stress of a cylinder of the same diameter and wall. A sphere holds the same pressure with half the wall, which is why large storage spheres exist despite being far more expensive to fabricate than a cylinder with dished ends.
Barlow's Formula and Which Diameter to Use
You will encounter three versions of the same equation using three different diameters, and they are not interchangeable. Barlow's formula, standard in the pipeline industry, uses the outside diameter: σ = P × Do ÷ (2t). The version used in much of pressure vessel work uses the inside diameter. A third uses the mean diameter, which is the most accurate of the three for moderately thick walls.
For a thin wall the differences are small, but they are systematic. The outside-diameter form always gives the highest stress and is therefore conservative for a burst check, which is exactly why the pipeline industry adopted it. The inside-diameter form gives the lowest. This calculator asks which diameter you are entering and computes the geometry consistently, using the inside diameter form for the thin-wall membrane result so that it lines up with the Lamé bore stress it is being compared against.
Where this bites is in reading somebody else's calculation. If two people get answers seven per cent apart on the same pipe, the diameter convention is usually the reason. State which one you used, every time.
What a Design Code Adds That This Does Not
A real vessel design does not stop at the membrane stress. ASME Boiler and Pressure Vessel Code Section VIII, Division 1, which governs vessels operating above 15 psig, wraps several further factors around it.
Allowable stress comes from a certified table for the specific material at the design temperature, not from yield divided by a factor you chose — and it falls sharply as temperature rises, because creep begins to govern above roughly 400 °C in carbon steel. Joint efficiency derates the wall according to the weld type and how much of it was radiographed, typically between 0.7 and 1.0. Corrosion allowance adds thickness that is expected to disappear over the vessel's life and must not be counted as structural.
Beyond that, the code deals with what this calculation cannot see: stress concentration at nozzles and openings, which needs reinforcement; the discontinuity stresses where a dished head meets the shell; external pressure and buckling, which is a stability problem rather than a strength one and is not governed by hoop stress at all; and fatigue where the pressure cycles. Treat the numbers here as the mechanics underneath a design, not as the design.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using absolute instead of gauge pressure — the wall responds to the differential across it, so a vessel at 10 bar gauge in atmosphere is loaded by 10 bar, not 11.
- Applying the thin-wall formula to a thick wall — below a diameter-to-thickness ratio of about 20 it understates the peak stress at the bore, and the error keeps growing as the wall thickens.
- Mixing up inside, outside and mean diameter — Barlow's formula uses the outside diameter and gives a higher, more conservative answer than the inside-diameter version of the same equation.
- Using nominal wall thickness — pipe carries a mill tolerance often as large as minus 12.5 per cent, and corrosion allowance has to come off as well before the structural wall is known.
- Treating the result as a code check — joint efficiency, certified allowable stress at temperature, nozzle reinforcement and external-pressure buckling are all outside this calculation.
Related Free Tools From Arb Digital
Once you have the stress, the stress and strain calculator converts it to strain and the Young's modulus calculator supplies the stiffness that governs how much the shell actually grows under pressure. If the pressure figure itself came in the wrong units, the pressure converter handles that, and the hydrostatic pressure calculator gives the head-driven pressure at the bottom of a tank. The pipe volume calculator covers contents and the material weight calculator the mass of the shell itself. For a vessel wall loaded in compression rather than tension, the column buckling calculator is the right stability check, and the full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
For a thin wall it is the pressure times the inside diameter divided by twice the wall thickness. For a thick wall the Lame equation gives the peak at the bore as the pressure times the sum of the squared radii, divided by the difference of the squared radii.
Because of geometry. The bursting force along the length acts on a rectangular projected area resisted by two wall strips, while the end force acts on a circle resisted by a full annulus. Dividing one by the other gives exactly two.
When the diameter is less than about twenty times the wall thickness. At that ratio the thin-wall formula is around five per cent low, and the error keeps growing, reaching more than twenty per cent at a ratio of five.
Both conventions exist. Barlow's formula uses the outside diameter and is conservative, which is why pipelines adopted it; pressure vessel work often uses the inside diameter. The important thing is to state which one your number came from.
It acts through the thickness and equals minus the internal pressure at the bore. In a thin wall it is tiny compared with the hoop stress so it is dropped, but in a thick wall it is a significant compression and changes the combined von Mises result.
No. A code design also needs a certified allowable stress at the design temperature, a weld joint efficiency, a corrosion allowance, nozzle reinforcement and checks for external pressure and fatigue. This gives the underlying mechanics only.
Because the hoop stress that acts circumferentially is twice the longitudinal stress, so the wall fails first on the plane the hoop stress pulls apart. That plane runs along the length of the pipe.
This tool is provided for educational and preliminary engineering use only. Pressure equipment must be designed, fabricated and inspected to the applicable code by a qualified engineer; nothing here substitutes for that.
