BIZOMAT MAKEOS · Strength
Spherical Shell under Internal Pressure
What stress does internal pressure cause in this spherical vessel or dished end?
Membrane stress of a thin sphere and the maximum (inner surface) stress of a thick sphere by Lamé’s equations, with the safety factor and the thickness needed for an allowable stress.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Inside diameter | mm | Inside diameter of the sphere or hemispherical head. |
| Wall thickness | mm | Minimum wall thickness. |
| Internal pressure (gauge) | MPa | Internal gauge pressure. |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification. |
| Required safety factor | Factor of safety required by your design code, customer specification or company standard. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Inside diameter
- What it is
- The diameter across the inside of the sphere or hemispherical head, measured through the centre from inner wall to inner wall.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Membrane stress and the thickness needed both rise in proportion to the diameter.
- Get it right
- Use the inside, not the outside, diameter; the tool adds the wall thickness itself for the thick-wall check.
What this means: Wall thickness
- What it is
- The thickness of the shell wall, using the thinnest point after forming, which is often at the crown of a pressed head.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Membrane stress is inversely proportional to the thickness.
- Get it right
- Deduct corrosion allowance and forming thinning from the nominal plate thickness before entering it.
What this means: Internal pressure (gauge)
- What it is
- The pressure inside the vessel above the surrounding atmosphere, as read on a normal pressure gauge, at the design or maximum operating condition.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 10000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Wall stress and required thickness rise in direct proportion to the pressure.
- Get it right
- Use gauge, not absolute, pressure, and convert bar to MPa: 1 bar = 0.1 MPa.
What this means: Yield strength
- What it is
- The stress at which the shell material starts to deform permanently, taken as the minimum specified value for the grade and thickness from the material certificate or standard.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safety factor rises in direct proportion; the required thickness falls in proportion.
- Get it right
- Use the minimum specified yield at the operating temperature if the vessel runs hot, since yield strength falls with temperature.
What this means: Required safety factor
- What it is
- The minimum ratio of yield strength to maximum wall stress that you need, set by your design code, customer specification or company standard. It drives the pass or fail verdict and the thickness needed.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thickness needed rises in direct proportion to the required factor.
- Get it right
- Pressure vessels are governed by a code; this screening tool does not replace a code design or certification.
Method and formulas
- Thin sphere: σ = p d ÷ 4t (half the hoop stress of a cylinder of the same diameter).
- Thick sphere (Lamé), inner surface: σt = p (ro³ + 2ri³) ÷ 2(ro³ − ri³).
- Thickness for the required factor (thin wall): t = p d × factor ÷ 4 Sy.
- Example: a 600 mm sphere 6 mm thick at 2 MPa has 50 MPa membrane stress.
Limitations
- Elastic stress away from nozzles, supports and welds.
- Pressure equipment must be designed to its code (AS 1210, ASME VIII).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check spherical tanks, floats and hemispherical heads
- Find the thickness for a spherical pressure part
- Compare spheres with cylinders
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Thin-wall membrane stress
- Thick-wall maximum stress (Lamé)
- Safety factor and thickness for the required factor
Interpreting the result
- A sphere needs about half the wall thickness of a cylinder of the same diameter and pressure.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 290–294. Topic reference; no source tables or recommended settings are reproduced.
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