Instantly compute hoop stress, axial stress, and Von Mises stress for cylindrical or spherical vessels. Evaluate safety factor and minimum wall thickness.
Parameters
mm
mm
MPa
Live Values During Pressurisation
0.00
Internal pressure p (MPa)
0.0
Hoop stress σ_θ (MPa)
0.0
Axial stress σ_z (MPa)
0.0
Von Mises σ_vm (MPa)
—
Safety factor FS
Pressurisation Animation (stress visualisation)
The internal pressure p cycles from 0 to the set value P. The stresses follow thin-wall theory: σ_θ = p·r/t, σ_z = p·r/2t (sphere: σ = p·r/2t).
Thin-wall theory applies when the thickness-to-radius ratio is 1/10 or less. This tool still shows values outside that range, so re-evaluate with thick-wall theory (Lamé equations) when t/R>0.1.
What Is the Thin-Wall Pressure Vessel Stress Calculator?
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What is a "thin-wall pressure vessel"? Something like a boiler?
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Roughly speaking, it is a vessel made of thin plate that carries pressure from the inside. Boilers, gas tanks and air cylinders are typical examples. In this simulator, first choose the "Shape" — cylinder or sphere — and enter the internal pressure and diameter. The most basic stresses are calculated immediately.
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What is the difference between "hoop stress" and "axial stress"? Which one matters more?
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The circumferential tension you would see if you sliced the cylinder into rings is the "hoop stress"; the tension pulling the top and bottom lids of a drinks can apart is the "axial stress". In practice the hoop stress is usually the larger one, and failure tends to occur in that direction. Keep the "Shape" set to cylinder and raise the internal pressure: you can confirm that the hoop stress is exactly twice the axial stress.
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I see! But the results also show a "Von Mises equivalent stress". What is that used to judge?
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Good question. The vessel wall is in a "multiaxial stress state", pulled in both the hoop and axial directions at the same time. The Von Mises equivalent stress converts that into an equivalent "uniaxial tension", and is used to judge whether the material yields (deforms permanently). Change the "Material" from steel (SS400) to aluminium: the yield stress changes, and you can immediately see how the safety factor changes.
Frequently Asked Questions
This tool calculates σ = pD/2t and the other stresses with the inner diameter D. Using the outer diameter makes the stresses larger by (D+2t)/D (on the safe side); for thin walls the difference is small (4% at t/D = 0.02). Some code thickness formulas add a correction to the inner-diameter form (for example t = pR/(SE − 0.6p) in ASME Sec. VIII), so in design calculations follow the formula of the code you adopt.
In general design a safety factor of 2–3 against the material's yield stress is recommended. For high-pressure gas vessels and life-critical uses 4 or more, and for low-pressure, non-hazardous contents about 1.5 are typical guides. The minimum-thickness calculation in the current version uses a fixed design safety factor of 3.0 and does not accept an arbitrary value.
In a sphere the force from the internal pressure is spread evenly in all directions, so the membrane stress is uniform and equals half the hoop stress of a cylinder. In a cylinder the stresses differ between the circumferential (hoop) and axial directions, and the hoop stress is the largest, so for the same wall thickness a sphere withstands a higher pressure.
The Von Mises equivalent stress converts several principal stresses (hoop and axial) into a single equivalent stress. It is used to judge yielding: when it exceeds the yield stress, plastic deformation is assessed to begin. In design, a safety factor is applied to this value.
Real-World Applications
Boilers and heat exchangers:Essential for designing the shells of boilers that generate steam in power plants and factories, and of heat-exchange equipment. They are calculated rigorously, with the hoop stress at the centre, so that they do not fail at high temperature and pressure.
LP gas and high-pressure gas tanks:From domestic propane tanks to large industrial storage tanks, this is the basis of designing vessels that withstand the internal gas pressure. For light weight, designs with thin walls that still secure the safety factor are required.
Reaction vessels in chemical plants:High-pressure vessels (autoclaves) for carrying out chemical reactions. Because they often handle corrosive substances, material selection matters as well as the stress from internal pressure.
Hydraulic and pneumatic cylinders:The cylinder tubes used as actuators in construction and industrial machinery are thin-wall cylindrical vessels in the truest sense. Because the pressure is applied repeatedly, the design must also consider fatigue strength.
Common Misconceptions and Points to Note
There are several pitfalls that CAE beginners in particular tend to fall into when starting with this tool. First, do not take the definition of "thin-wall" lightly. It generally means a wall thickness (t) to inner diameter (D) ratio of about 1/20 or less (that is, t/D ≤ 0.05). For example, a vessel with an inner diameter of 1000 mm and a wall 60 mm thick is already in the "thick-wall" range. Because the tool's basic formulas consider only membrane stress, the radial stress that cannot be neglected in a thick-wall vessel is ignored, and the maximum stress at the inner surface is underestimated by a few to ten-odd percent (for an inner diameter of 1000 mm and a 60 mm wall, the value is about 6% below the inner-surface hoop stress from the Lamé equations). Always check this ratio in practice.
Next,mistakes in entering the "inner diameter". Drawings often give the "outer diameter". Entering the outer diameter makes the stresses larger by (D+2t)/D, that is, on the safe side. The danger is the opposite: underestimating the inner diameter, for example by subtracting the wall thickness only once instead of twice from the outer diameter. The tool always assumes the "inner diameter". Also,the interpretation of the safety factor needs care. The safety factor given by the tool is a pure margin against yielding under internal pressure. In real design there are many more factors to consider: thickness loss by corrosion (corrosion allowance), reduced strength at welds (joint efficiency), fatigue, creep and so on. Do not relax just because SS400 shows a safety factor of 3; treat it as "the basic skeleton — the actual design safety factor needs the joint efficiency of 0.7 applied, and so on", and refine it further.
Enter the inner diameter D (100–2000 mm) and wall thickness t (2–50 mm), for example D = 800 mm and t = 12 mm
Set the internal pressure P (MPa). Steam boilers are typically 1.0–2.5 MPa and high-pressure gas vessels 5–10 MPa
When an input changes, the hoop stress σ_θ, axial stress σ_z and Von Mises stress σ_vm are recalculated automatically
The safety factor FS is shown as the ratio of the yield stress (SS400 240 MPa, SUS304 210 MPa, A6061 100 MPa) to the Von Mises stress. To evaluate against a code allowable stress, use the value for the material and temperature in the code you adopt
Once the required minimum thickness t_min is shown, check that the design thickness exceeds it
Worked Example
For a carbon-steel (SS400) cylindrical vessel with D = 600 mm, t = 8 mm and P = 1.6 MPa: hoop stress σ_θ = 1.6×600/(2×8) = 60 MPa and axial stress σ_z = 1.6×600/(4×8) = 30 MPa. The Von Mises stress is σ_vm = √(60² + 30² − 60×30) ≈ 52.0 MPa, giving a safety factor FS of about 4.62 against the SS400 yield stress of 240 MPa. Because this tool uses yield stress/3 as the allowable stress for the minimum thickness, t_min = 1.6×600/(2×80) = 6.0 mm.
Practical Notes
Codes such as ASME Sec. VIII and JIS B 8265 add a corrosion allowance (usually 1–3 mm) to the wall thickness, so order plate thick enough to include the allowance on top of the calculated minimum
Re-evaluate with the allowable stress reduced by the weld joint efficiency (0.7–1.0 depending on the extent of radiographic examination). The longitudinal joint efficiency governs the hoop stress and the circumferential joint efficiency the axial stress
A spherical vessel has half the hoop stress of a cylinder, so for the same pressure the minimum thickness is smaller and the design is more efficient
Yield and allowable stresses fall as temperature rises. Evaluate with the values at the operating temperature read from the code's material tables