Hoop Stress Calculator

Pressure inside a pipe or vessel tries to split it lengthwise, and the wall’s answer is the hoop stress. Enter the internal pressure, the radius and the wall thickness, and this calculator returns the circumferential stress — and the longitudinal stress, exactly half as large.

Why the Wall Hurts Twice Around

Cut a pressurised cylinder mentally along its axis: the pressure pushes the two halves apart with force p × (projected area) = p × 2rL, resisted by the wall's two cut edges, 2 × σ × tL — the balance gives σ_hoop = pr ÷ t. Now cut across the axis: pressure on the end cap, p × πr², is resisted by the wall's whole circumference, σ × 2πrt — giving σ_long = pr ÷ (2t), exactly half. The factor of two is the most consequential integer in pressure engineering: vessels rupture along longitudinal seams because the hoop direction is loaded double; pipeline steels, boiler plates and rocket tanks are all sized by the hoop figure; and a sausage, splitting lengthwise on the grill, demonstrates the theorem at breakfast. The calculator evaluates both stresses and reports the wall's slenderness ratio, because the thin-wall formulas that produce them have a jurisdiction — r ÷ t of ten or more — that the answer itself should certify.

Worked Example: Ten Megapascal in a 200 mm Pipe

The defaults run a respectable pressure line: p = 10 MPa (100 bar), r = 100 mm, t = 10 mm. Hoop stress = 10 × 100 ÷ 10 = 100 MPa; longitudinal = 50 MPa. Against a typical pipeline steel's yield of 360–450 MPa, the hoop figure sits at a safety factor near 4 — the neighbourhood codes demand once welds, corrosion allowance and pressure surges have had their say (the thin-wall check passes: r ÷ t = 10, the jurisdiction's border, exactly where a careful designer switches to the thick-wall Lamé formulas the FAQs name). Halve the wall to 5 mm and the hoop stress doubles to 200 MPa — the linear t in the denominator is why corrosion, which thins walls, is pressure equipment's slow assassin, and why inspection measures remaining thickness with this division in mind.

Barlow's Formula and the Code Books

Rearranged to solve for pressure, σ = pr ÷ t becomes Barlow's formula, P = 2σt ÷ D — the working equation of pipeline and boiler codes for a century and a half, dressed by each code with its safety factors, weld efficiencies and temperature deratings. The physics this page computes is the code's skeleton: the ASME and EN documents are, in large part, this one equilibrium balance wrapped in hard-won cautionary factors after nineteenth-century boiler explosions taught the profession the price of optimism (the Hartford Steam Boiler inspection companies, and the codes that followed, were born from those wrecks). The calculator returns the unfactored stress — the honest physics figure — and the explainer names the wrapper, because mistaking one for the other, in either direction, is how vessels get mis-rated.

The Champagne Cork's Arithmetic

A champagne bottle at 6 bar with a 9 mm bore radius and 4 mm glass at the neck lives, by this page, at a hoop stress near 1.35 MPa in the glass — trivial against glass's strength, which is why bottles essentially never burst from pressure alone and the cork (held by wire against about 30 kg of push from that same pressure over the cork's area) is the designed weak point, released with a pop priced by the hydrostatic page. The exercise is worth the reader's thirty seconds precisely because it calibrates intuition: domestic pressures are kilopascal theatre next to the pipeline's megapascals, and this equation, entered with either, returns stress in the same honest units.

Arteries: The Body's Pressure Vessels

The aorta is a hoop-stress problem that evolution solved with elastin: at a pulse pressure near 16 kPa, a radius of 12 mm and a wall of 2 mm, this page returns a hoop stress around 96 kPa — small in engineering units, but cycled a hundred thousand times a day for a lifetime, which is why the wall is a living composite that repairs itself and why its failure mode, the aneurysm, is this equation's vicious circle made flesh: a bulge raises the local radius, the raised radius raises the stress at the same pressure, and the stress stretches the bulge further. Laplace's law (the same formula, in the physiology texts) is the aneurysm's growth chart, and surgeons intervene at diameters where the arithmetic turns ominous. One equilibrium balance thus sizes a pipeline, rates a rocket tank and times an operation — the reach of a correct force balance being bounded only by the number of pressurised cylinders the world contains, which is to say, unbounded for practical purposes.

Related Tools

More from the Physics Calculators hub: the Young's Modulus Calculator — the wall material's stiffness behind this stress — and the Hydrostatic Pressure Calculator, for pressures that come from depth rather than pumps.

Frequently Asked Questions

What changes for thick walls?

The stress stops being uniform through the wall: Lamé's formulas (1840s) give the hoop stress peaking at the inner surface, σ = p(r_o² + r_i²) ÷ (r_o² − r_i²) at the bore, falling toward the outside — which is why high-pressure vessels (gun barrels, hydraulic accumulators, deep submersibles' hulls viewed from outside-in) are analysed with Lamé and sometimes prestressed (autofrettage: deliberately yielding the bore so residual compression armours it). The thin-wall page's rule of thumb — trust it from r ÷ t ≥ 10, where its error against Lamé is a few percent — is printed in the widget's note with your own ratio computed, the calculator certifying its own licence on every run.

Why do rockets care so much about hoop stress?

Because a rocket is mostly pressure vessel: the tanks are the structure, and their mass is set by this equation inverted — t = pr ÷ σ_allowable — so every megapascal of tank pressure and every metre of radius must be paid in wall, while every improvement in the alloy's σ buys it back. The famous figure of merit, tank mass per contained energy, is this page's arithmetic wearing a spacesuit; the Falcon and Shuttle tanks' thin, balloon-like walls (some are literally pressure-stabilised, floppy when empty) are hoop stress optimised to the edge of the buckling page's separate terror.

What about spherical vessels?

Spheres halve the problem again: membrane stress in a sphere is pr ÷ (2t) in every direction — the longitudinal figure of the cylinder, with no weak direction — which is why storage spheres (Horton spheres) dot refineries and why deep-sea submersibles are spheres inside: geometry's most efficient answer to pressure. The mass penalty of cylinders is the price of manufacturability and plumbing; the calculator covers the cylindrical case in full and this FAQ gives the sphere's kinder quotient, which the reader can evaluate mentally from the cylinder's longitudinal output.

Does the formula work for external pressure — a submarine?

The stresses reverse sign (compression) and a worse failure mode takes command: buckling. A thin hull under external pressure collapses by instability at pressures far below the membrane stress's material limit — the Euler page's phenomenon in cylindrical dress — which is why submarine hulls are ribbed with frames that shorten the unsupported length, and why the hoop stress formula, while still computable, stops being the design driver. The calculator's domain is internal pressure stated plainly; the external case is flagged here because the difference between bursting and collapsing is the difference between the two pages of this wave.

How is a vessel proof-tested?

Hydrostatically: filled with water (nearly incompressible, so a failure weeps rather than explodes) and pumped to typically 1.3–1.5 times design pressure while inspectors watch for leaks and measure permanent set — the hoop stress at test pressure being this page's calculation at the elevated input. Every scuba tank's five-year hydro test, every boiler's commissioning, is Barlow's formula run as a physical argument. Pneumatic testing with gas is rarer and more feared, for the reason the energy arithmetic makes obvious: compressed gas stores the failure's shrapnel budget.

Why are LPG tanks and gas cylinders hydrotested with water, not air?

Energy: a vessel at test pressure stores energy in its contents' compression, and water, being nearly incompressible, stores almost none — a failure at 1.5× test pressure with water inside produces a crack and a puddle, where the same failure pneumatic releases the gas's stored expansion energy as a blast. The hoop stress at the test pressure is identical either way (this page's number does not care what the fluid is), which is precisely the point: the test audits the wall's stress at a pressure it will never see in service, while the choice of test fluid decides how forgiving the audit is if the wall fails it.

Who was Barlow?

Peter Barlow (1776–1862), English mathematician and engineer at the Royal Military Academy, Woolwich, whose name the pipeline form of the formula carries in American practice — though the equilibrium itself is older (Mariotte had the essence in the 1680s, reasoning about water pipes for Versailles' fountains). The formula's long, multi-author life is typical of engineering's foundations: discovered by natural philosophers for fountains, weaponised by boiler inspectors, and ending as a web calculator's division — each era re-deriving the same balance because pressure has never stopped asking the question.