Pressure Vessel Hoop Stress Calculator
Work out the hoop and longitudinal stress in a pressurised cylinder or sphere. Check thin-wall against the thick-wall Lamé equations, find the wall thickness or maximum pressure for an allowable stress, and estimate the burst pressure.
How to Use
- Choose what to work out: the Stresses in a vessel, the Wall Thickness a pressure needs, the Maximum Pressure a wall can take, or the Burst Pressure.
- Pick a cylinder or a sphere, then enter the inside diameter, the wall thickness and the internal pressure, each with its unit.
- For thickness and pressure, pick a material to fill in its allowable stress (or type your own) and the joint efficiency of the welded seam: 1.0 for seamless or fully radiographed, 0.85 spot-radiographed, 0.70 not radiographed.
- Read the hoop and longitudinal stress, the Lamé peak stress at the bore and the radius-to-thickness ratio; under 10 means the wall is thick and the Lamé figure is the one to use.
- Press a preset to load an air receiver, a thick hydraulic tube, a spherical tank, a thickness or pressure rating, or the burst pressure of a 2 in pipe.
Worked Example
An air receiver. A cylinder 500 mm inside diameter (r = 250 mm) with a 5 mm wall holds 10 bar, which is 1 MPa. The hoop stress is σh = 1 × 250 ÷ 5 = 50 MPa and the longitudinal stress half that, 25 MPa. r ÷ t = 50, so the thin-wall result is good: Lamé gives 50.5 MPa at the bore. Against SA-516 Grade 70’s 138.6 MPa allowable that is 36.1%, and Barlow’s formula with the 485 MPa tensile strength puts the burst at about 95.1 bar.
The wall it needs. A 1.5 m vessel at 15 bar with spot-radiographed seams (E = 0.85) needs t = 1.5 × 750 ÷ (138.6 × 0.85) = 9.549 mm; the ASME UG-27 form, which adds 0.6P to account for the radius growing through the wall, gives 9.623 mm. A real design adds a corrosion allowance and rounds up to a plate size.
The common mistake: the diameter in place of the radius. Putting 500 mm into σ = p r ÷ t instead of 250 mm gives 100 MPa for the air receiver, twice the true 50 MPa, and the thickness formula would then ask for twice the plate. Barlow’s formula, p = 2 S t ÷ D, uses the diameter because it already has the 2 in it; σ = p r ÷ t uses the radius.
Show Work
Formulas
Boiler Explosions and the First Pressure Vessel Code
The steam age was also the age of the boiler explosion. Thousands of boilers burst in the nineteenth century; the worst disaster on an American river, the steamboat Sultana in April 1865, killed more than a thousand people when her boilers exploded. The thin-wall formulas were known, but there was no agreed rule for the allowable stress, the riveted joints or the inspection.
Gabriel Lamé published the exact solution for a thick-walled cylinder in 1852, in his lectures on the mathematical theory of elasticity. In 1905 a boiler explosion destroyed the Grover Shoe Factory in Brockton, Massachusetts, killing 58 people; Massachusetts passed boiler rules in 1907, and the American Society of Mechanical Engineers approved its first Boiler Code in 1914. Its descendant, the ASME Boiler and Pressure Vessel Code, is still the reference in North America, alongside EN 13445 and PD 5500 in Europe and Britain.
The joint efficiency in these sums is a relic of riveted plates, where the holes weakened the seam; for welds it now depends on how much of the seam is radiographed.
About This Tool
This calculator works out the membrane stresses in a cylinder or sphere under internal pressure with the thin-wall formulas (using the inside radius), compares them with the thick-wall Lamé solution, and turns the hoop stress round to give the wall thickness or the maximum pressure for an allowable stress and joint efficiency. Barlow’s formula with the tensile strength gives a rough burst pressure. The ASME UG-27 forms are shown beside the results for comparison only.
The material presets fill in a simplified room-temperature allowable stress, the lower of the tensile strength ÷ 3.5 and two-thirds of the yield strength, from the specified minimums: SA-516 Grade 70 (485 MPa tensile, 260 MPa yield), 304 stainless (515 and 205 MPa), and typical values for 6061-T6 aluminium. This is not a design tool. It ignores nozzles, heads, external pressure, temperature, corrosion allowance, fatigue and testing, all of which a code such as ASME Section VIII requires; a pressure vessel must be designed and certified by a qualified engineer. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Hydraulic Cylinder & Pump Calculator, Stress, Strain & Young’s Modulus Calculator, and Ideal Gas Law Calculator.
Frequently Asked Questions
How do you calculate hoop stress?
For a thin-walled cylinder σh = p r ÷ t, with r the inside radius. A tank 500 mm across with a 5 mm wall at 10 bar (1 MPa) has σh = 1 × 250 ÷ 5 = 50 MPa, about 7,250 psi.
Why is hoop stress twice the longitudinal stress?
Along the length the pressure pushes on the end area πr², carried by the ring of wall 2πr t, so σl = p r ÷ 2t. Across a lengthwise cut it pushes on 2r per unit length, carried by only 2t, so σh = p r ÷ t. The 500 mm tank at 10 bar has 50 MPa hoop and 25 MPa longitudinal, which is why pipes and sausages split along their length.
When is a vessel thin-walled?
When the wall is less than a tenth of the radius (r ÷ t over 10). At r/t = 50 the thin formula is only 1.0% below the Lamé peak; a 20 mm bore tube with a 5 mm wall (r/t = 2) at 300 bar has a thin-wall hoop stress of 60 MPa but a real peak at the bore of 78 MPa, 30% more.
How thick does a pressure vessel wall need to be?
From the hoop stress, t = p r ÷ (S E), with S the allowable stress and E the joint efficiency. A 1.5 m vessel at 15 bar in SA-516 Grade 70 (S = 138.6 MPa) with spot-radiographed seams (E = 0.85) needs 1.5 × 750 ÷ (138.6 × 0.85) = 9.549 mm, before any corrosion allowance. The ASME UG-27 form gives 9.623 mm.
How do you estimate burst pressure?
Barlow’s formula uses the tensile strength and the outside diameter: p = 2 S t ÷ D. A 2 in schedule 40 steel pipe (2.375 in outside, 0.154 in wall) with a 60 ksi tensile strength bursts at about 2 × 60,000 × 0.154 ÷ 2.375 = 7,781 psi, 51.9 times a 150 psi working pressure.
How do I use the Pressure Vessel Hoop Stress Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Air receivers and tanks
A 500 mm receiver with a 5 mm wall at 10 bar runs at 50 MPa hoop stress, 36.1% of the 138.6 MPa allowable for SA-516 Grade 70.
Hydraulic tubes
A 20 mm bore stainless tube with a 5 mm wall at 300 bar peaks at 78 MPa at the bore by Lamé, against 60 MPa by the thin-wall formula.
Spherical tanks
A 10 m sphere with a 12 mm wall at 5 bar carries 104.2 MPa in every direction, half what a cylinder of the same size would see in hoop.
Sizing a shell
A 1.5 m shell for 15 bar needs 9.549 mm at 138.6 MPa allowable with a 0.85 joint efficiency.
Rating an existing vessel
A 24 in shell with a ⅜ in wall at a 20 ksi allowable can take 625 psi by pr/t (613.5 psi in the ASME UG-27 form).
Pipe burst margins
A 2 in schedule 40 pipe with 60 ksi tensile strength bursts at about 7,781 psi.
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