Calculate the hoop stress σφ = p·r/t in the wall of a thin-walled cylindrical vessel or pipe under internal pressure. It is the larger of the two wall stresses – exactly twice the longitudinal stress σx = p·r/(2t) – and therefore governs the wall thickness. This is why cylindrical vessels and pipes burst lengthwise rather than around the circumference.
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Calculate the hoop stress σφ = p·r/t in the wall of a thin-walled cylindrical vessel or pipe under internal pressure. It is the larger of the two wall stresses – exactly twice the longitudinal stress σx = p·r/(2t) – and therefore governs the wall thickness. This is why cylindrical vessels and pipes burst lengthwise rather than around the circumference.
A cylindrical compressed-air receiver with mid-surface radius r = 500 mm and wall thickness t = 8 mm is under p = 10 bar = 1 N/mm² internal pressure. The hoop stress is σφ = 1 N/mm² · 500 mm / 8 mm = 62.5 N/mm², and the longitudinal stress only half of that, 31.25 N/mm². The thin-wall condition holds, since r = 500 mm exceeds 5·t = 40 mm. For structural steel S235 with a yield strength of 235 N/mm² this gives a safety factor against hoop stress of 235/62.5 ≈ 3.8.
Membrane theory of a thin-walled cylinder: the wall stresses are assumed constant across the wall thickness, and the radial stress perpendicular to the wall is neglected because it is of the same order as the pressure and therefore much smaller than the wall stresses. Valid for r > 5·t; for thicker walls the stress varies across the thickness and a thick-wall theory is required. Locally raised stresses at heads, nozzles, welds, flanges and supports, self-weight and contents, temperature effects and creep, buckling under external pressure, fatigue loading, and the allowances and safety factors of pressure-vessel codes are all excluded. The result is a nominal stress, not a verification to any code.
A cylindrical vessel under internal pressure is stretched in two directions at once – and one of the two stresses is twice the other. This calculator determines the governing hoop stress and can be rearranged for pressure, radius and wall thickness.
Two tensile stresses act in the wall of a thin-walled cylinder under internal pressure: the hoop stress σφ, also called circumferential or tangential stress, running around the cylinder, and the longitudinal stress σx along the cylinder axis. The Greek index φ (phi) denotes the circumferential direction and the Latin index x the axial one. Cutting the vessel perpendicular to its axis, force equilibrium gives σx = p·r/(2t); cutting it lengthwise gives σφ = p·r/t. Together these are known as the boiler formulas. Since symmetry means no shear stresses appear in either cut, σφ and σx are at the same time the principal stresses of the wall.
A sausage boiled too long always splits lengthwise, never around its circumference. That is exactly what the boiler formula says: the stress running around the sausage is twice the stress along its axis, so the skin tears there first. The same holds for every pipe and every cylindrical vessel.
σφ = p · r / t
Hoop stress: σφ = p·r/tLongitudinal stress: σx = p·r/(2t)Ratio: σφ = 2·σxWritten with the diameter: σφ = p·d/(2t)Required wall thickness: t = p·r/σallowThin-wall validity condition: r > 5·tFor comparison, the sphere: σ = p·r/(2t) in every direction| Symbol / input | Meaning |
|---|---|
| Hoop stress σφ | The tensile stress acting in the circumferential (tangential, i.e. around the cylinder) direction in the wall, trying to tear the vessel apart. The Greek index φ (phi) denotes the circumferential direction. It is the largest of the three wall stresses and therefore the quantity that governs wall thickness. Compare it with the material's allowable stress, i.e. yield strength divided by the required safety factor. Under external pressure the result turns negative and then denotes a compressive stress. |
| Internal pressure p | The gauge pressure inside the vessel, i.e. the pressure above ambient (not the absolute pressure). Source: the operating pressure from the plant data sheet, sensibly using the safety valve set pressure as the design pressure. If external pressure acts instead, enter p as negative – the wall then carries compressive stress, and a buckling risk arises that this formula does not cover. |
| Radius r of the mid-surface | The radius of the cylinder measured to the middle of the wall. For thin walls it makes little numerical difference whether the inner, mid or outer radius is used; for a clean calculation the mid-surface radius r = (inner diameter + wall thickness)/2 is correct. Note: do not enter the diameter – the boiler formula uses the radius, which is why the frequently quoted form σφ = p·d/(2t) with the diameter d gives the same value. |
| Wall thickness t | The load-bearing wall thickness of the cylinder. Only material that is actually available counts: allowances for corrosion and for the permitted undershoot of the nominal wall thickness must be subtracted first. For the formula to hold the wall must be thin – the literature gives r > 5·t as the criterion, i.e. radius greater than five wall thicknesses. |
p is the internal pressure as gauge pressure relative to ambient, not absolute; sensibly the design pressure is used, usually the safety valve set pressure. r is the radius of the mid-surface of the wall, not the diameter. t is the load-bearing wall thickness, i.e. the nominal thickness less the allowances for corrosion and manufacturing tolerance.
First check that the vessel qualifies as thin-walled at all: the radius must exceed five wall thicknesses. Then enter pressure, radius and load-bearing wall thickness. Compare the result with the material's allowable stress, i.e. yield strength divided by the required safety factor. Conversely the target can be switched to wall thickness: enter the allowable stress instead of the hoop stress and you get the theoretically required wall thickness – to which the allowances must then still be added.
A compressed-air receiver has a mid-surface radius r = 500 mm and a load-bearing wall thickness t = 8 mm under p = 10 bar internal pressure. Ten bar equal 1 N/mm², so σφ = 1 N/mm² · 500 mm / 8 mm = 62.5 N/mm². The longitudinal stress is only half of that, 31.25 N/mm². The thin-wall condition is clearly met, since r = 500 mm exceeds 5·t = 40 mm. Against the S235 yield strength of 235 N/mm² this gives a safety factor of about 3.8. For comparison: a sphere of the same radius and wall thickness would carry only 31.25 N/mm² in every direction – half as much as the cylinder.
The stress grows linearly with pressure and radius and falls inversely with wall thickness. A vessel of twice the size therefore needs twice the wall thickness at the same pressure – which is why large pressure vessels get heavy so quickly. The relation σφ = 2·σx is the real key insight: the circumferential direction is always the critical one, which is why pipes and vessels fail through longitudinal cracks. For design this means the longitudinal welds of a cylinder are more highly stressed than the circumferential ones and must be executed and inspected accordingly.
The calculation runs in pascal and metres internally. Pressure may be entered in bar, Pa, kPa, MPa or psi, stress in MPa (= N/mm²), kPa, GPa, psi or ksi, and radius and wall thickness in mm, cm, m or inches. Useful for hand calculations: 10 bar equal exactly 1 N/mm², and with r and t in millimetres the stress then comes out directly in N/mm².
Compressed-air and hydraulic accumulators, pipelines under internal pressure, silos and tanks, gas cylinders, heat-exchanger shells and boiler drums, cylinder tubes of hydraulic cylinders, inflated hoses and tyres, and checking an existing wall thickness in service.
Membrane theory of a thin-walled cylinder: the stresses are assumed constant across the wall thickness, and the radial stress perpendicular to the wall is neglected because it is only of the order of the pressure and therefore much smaller than the wall stresses. Valid for a radius greater than five wall thicknesses; beyond that a thick-wall theory is required. The markedly raised stresses at heads, nozzles, manholes, flanges and supports, self-weight and contents, temperature and creep, buckling under external pressure, fatigue loading, and all allowances and safety factors of pressure-vessel codes are not covered. The result is a nominal stress and does not replace a verification to any code.
Common mistake: The most common error is entering the diameter instead of the radius, which doubles the stress. Closely related is confusing the two usual forms: σφ = p·r/t with the radius and σφ = p·d/(2t) with the diameter mean the same thing. The second classic is confusion with the sphere formula σ = p·r/(2t): a sphere of the same dimensions carries only half the stress, which is why spherical vessels use material more efficiently. Third, the nominal wall thickness is often used instead of the load-bearing one, although corrosion and tolerance allowances do not carry load. And finally the formula must not stand alone under external pressure – there buckling, not stress, governs.
Compressed-air and hydraulic accumulators, pipelines under internal pressure, silos and tanks, gas cylinders, heat-exchanger shells and boiler drums, cylinder tubes of hydraulic cylinders, inflated hoses and tyres, and checking an existing wall thickness in service.
p is the internal pressure as gauge pressure relative to ambient, not absolute; sensibly the design pressure is used, usually the safety valve set pressure. r is the radius of the mid-surface of the wall, not the diameter. t is the load-bearing wall thickness, i.e. the nominal thickness less the allowances for corrosion and manufacturing tolerance.
Membrane theory of a thin-walled cylinder: the stresses are assumed constant across the wall thickness, and the radial stress perpendicular to the wall is neglected because it is only of the order of the pressure and therefore much smaller than the wall stresses. Valid for a radius greater than five wall thicknesses; beyond that a thick-wall theory is required. The markedly raised stresses at heads, nozzles, manholes, flanges and supports, self-weight and contents, temperature and creep, buckling under external pressure, fatigue loading, and all allowances and safety factors of pressure-vessel codes are not covered. The result is a nominal stress and does not replace a verification to any code.