Hoop Stress in a Pressure Vessel: Why Pipes Split Along Their Length

Watch a failed pressure pipe or an over-inflated cylinder and you will notice something consistent: it splits along its length, not around its circumference. A burst garden hose, a ruptured boiler, a failed compressed-gas cylinder — the tear runs lengthwise. That is not a coincidence. It is a direct, calculable consequence of how pressure stresses a cylindrical wall.

This article explains hoop stress and longitudinal stress in a thin-walled vessel, shows where the formulas come from, works a full numerical example, and explains why the circumferential direction is almost always the one that decides whether the wall holds.

Why this calculation matters

Pressure vessels are everywhere: gas cylinders, hydraulic accumulators, boilers, pipelines, aerosol cans, scuba tanks, and the cylindrical sections of chemical reactors. All of them contain a fluid pushing outward on every part of the wall, and the wall has to carry that push as membrane stress.

Getting the stress right is not academic. Pressure vessels store a large amount of energy, and a wall failure releases it suddenly. Design codes such as ASME Section VIII exist precisely because the consequences of an under-thick wall are severe. The thin-wall hoop stress formula is the first calculation in almost every vessel design, the basis for choosing wall thickness, and the sanity check you run before trusting any finite element model. If the simple formula and the simulation disagree, the simulation is the one you re-examine first.

The core formula

For a thin-walled cylinder, the wall is treated as a membrane: thin enough that the stress is essentially uniform through the thickness. The standard rule of thumb is that the formulas apply when the radius-to-thickness ratio r/t is greater than about 10. Below that, the through-wall stress variation matters and you need the thick-wall (Lamé) equations instead.

Two stresses act in the cylindrical wall. The hoop stress (also called circumferential stress) acts around the circumference. Cut the cylinder along a lengthwise plane and balance the pressure force on the projected area against the wall tension:

sigma_h = p * r / t

The longitudinal stress (also called axial stress) acts along the axis. Cut the cylinder across with a transverse plane; the pressure on the circular end face is carried by the full ring of wall material:

sigma_l = p * r / (2 * t)

Here p is the internal gauge pressure, r is the internal radius, and t is the wall thickness. The single most important fact in pressure vessel design follows immediately from comparing the two:

sigma_h = 2 * sigma_l

The hoop stress is exactly twice the longitudinal stress. The wall is twice as highly stressed in the circumferential direction as along the axis. That is why a failing cylinder tears lengthwise — the crack opens on the plane that the larger stress is trying to pull apart. It is also why a spherical vessel, which carries only the lower stress (p*r/2t) in every direction, is the most material-efficient shape for containing pressure.

A worked example

Consider a thin-walled cylindrical pressure vessel with an internal gauge pressure of 2 MPa, an inside diameter of 1.0 m (so the radius r = 0.5 m), and a wall thickness of 10 mm.

Step 1 — confirm the thin-wall assumption.

r / t = 0.5 / 0.010 = 50

Fifty is well above the threshold of 10, so the thin-wall membrane formulas apply with confidence.

Step 2 — hoop (circumferential) stress.

sigma_h = p * r / t
sigma_h = (2e6 * 0.5) / 0.01
sigma_h = 1e6 / 0.01 = 100 MPa

Step 3 — longitudinal (axial) stress.

sigma_l = p * r / (2 * t)
sigma_l = (2e6 * 0.5) / (2 * 0.01)
sigma_l = 1e6 / 0.02 = 50 MPa

The hoop stress is 100 MPa and the longitudinal stress is 50 MPa — exactly the 2:1 ratio the theory predicts. When you select a wall material and thickness for this vessel, the hoop stress is the value to compare against the allowable stress. Size the wall for the longitudinal stress alone and you would be optimistic by a factor of two, with the failure plane running straight down the side of the vessel.

Common mistakes

Using diameter where the formula wants radius. The thin-wall formula is p*r/t, with r the radius. If you substitute the diameter, every stress comes out twice too large. Some textbooks write the formula as p*D/(2t) — that is the same thing, but only if D really is the diameter. Be deliberate about which symbol you are using.

Applying thin-wall formulas to thick walls. When r/t drops below about 10, the stress is no longer uniform through the wall; it peaks at the inner surface. For thick cylinders you need the Lamé equations, and the membrane formula will underestimate the inner-wall stress.

Mixing gauge and absolute pressure. The stress is driven by the pressure difference across the wall. For a vessel open to the atmosphere outside, use gauge pressure. Plugging in absolute pressure adds a spurious atmosphere of loading.

Forgetting the longitudinal stress entirely. Hoop stress dominates, but the axial stress is still real. It matters for end caps, for longitudinal welds versus circumferential welds, and whenever an external axial load is superimposed on the pressure load.

Ignoring the ends and nozzles. The membrane formulas describe the smooth cylindrical body. Near end closures, nozzles, and supports, bending and local discontinuity stresses appear that the simple formulas do not capture.

Try the interactive NovaSolver calculator

The hand calculation is quick, but exploring how stress, safety factor, and required thickness respond to pressure and geometry is faster with a live tool. The Pressure Vessel Stress Calculator on NovaSolver computes the through-wall stress distribution using the Lamé equations, automatically classifies the vessel as thin- or thick-walled from the t/R ratio, and reports the peak hoop stress, the radial stress, the von Mises stress, a safety factor against the chosen material, and the ASME minimum wall thickness. You can switch between cylinder and sphere geometry and watch the stress distribution update.

Related calculators

• Pressure vessel design — for working through wall thickness selection and code allowances in a design context.
• Pressure vessel nozzle — for the local reinforcement needed where a nozzle penetrates the shell and disturbs the membrane field.
• Bolted flange joint — for the bolted, gasketed connections that seal vessel openings and pipework.

You can browse the full set in the mechanical calculators hub.

Closing note

Hoop stress is one of the cleanest results in mechanics of materials, and it carries an outsized practical lesson. A thin cylinder under internal pressure is stressed twice as hard around its circumference as along its length, so it splits lengthwise and so the hoop stress is the number you design the wall to. Confirm the thin-wall assumption with r/t, use radius not diameter, use gauge pressure, and treat the ends and nozzles as a separate problem. Get those right and the body of the vessel rarely surprises you.