BIZOMAT MAKEOS · Strength
Spherical Pressure Vessel Design (Thickness, Stress, Capacity, Hydrotest)
How thick must this spherical vessel be for its pressure, what stress and safety factor does the chosen plate give after corrosion, what does it hold and weigh, and is the hydrotest within limits?
Design of a spherical shell under internal pressure: thin-wall membrane stress and the thick-wall maximum stress at the bore (as the spherical shell tool) with the thickness reduced by the corrosion allowance, the safety factor on yield, the least thickness for the required factor plus allowance, capacity and shell mass, and the stress at the hydrotest pressure against your share of yield.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Inside diameter | mm | Inside diameter of the sphere. |
| Plate thickness | mm | Nominal plate thickness. |
| Corrosion allowance | mm | Corrosion allowance taken off the thickness for the stress check. |
| Design pressure | MPa | Internal design pressure. |
| Yield strength | MPa | Yield strength of the plate at temperature. |
| Safety factor on yield required | Safety factor on yield you require (code design uses allowable stresses instead; this is a screening check). | |
| Hydrotest pressure ÷ design pressure | Hydrotest pressure as a multiple of the design pressure (from your code). | |
| Largest hydrotest stress as a share of yield | % | Largest membrane stress during the hydrotest as a share of yield (your value). |
| Plate density | kg/m³ | Density of the plate. |
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 inside diameter of the spherical shell, which sets its stress, capacity and mass.
- 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 rises in direct proportion.
What this means: Plate thickness
- What it is
- The nominal plate thickness of the shell as ordered, before the corrosion allowance is taken off.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 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.
What this means: Corrosion allowance
- What it is
- The thickness allowed to be lost to corrosion over the life; the stress check uses the plate thickness minus this.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Design pressure
- What it is
- The internal design pressure of the vessel.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 500 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
- Stress rises in direct proportion.
- Get it right
- In MPa: 1 MPa = 10 bar.
What this means: Yield strength
- What it is
- The yield strength of the plate material at the design temperature, 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 3000 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.
What this means: Safety factor on yield required
- What it is
- The safety factor on yield the sphere must reach. This is a screening check; code design uses allowable stresses instead.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. 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.
What this means: Hydrotest pressure ÷ design pressure
- What it is
- The hydrostatic test pressure as a multiple of the design pressure, from the code you design to.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Largest hydrotest stress as a share of yield
- What it is
- The largest membrane stress allowed during the hydrostatic test, as a percentage of the yield strength.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 30% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Plate density
- What it is
- The density of the plate material, used to work out the shell mass.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 20000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Thin wall σ = p d_i ÷ 4t; thick wall maximum σ = p (r_o³ + 2r_i³) ÷ [2(r_o³ − r_i³)] (as the spherical shell tool), with t reduced by the corrosion allowance.
- Safety factor = S_y ÷ maximum stress; thickness needed = p d_i × factor ÷ (4 S_y) + allowance.
- Capacity = π d_i³ ÷ 6; shell mass = ρ π (d_o³ − d_i³) ÷ 6; hydrotest stress = membrane stress × hydrotest ratio.
- Example: 6 mm plate (5 mm after allowance) runs at 60.011 MPa, a factor of 4.1659; the sphere holds 113.1 L and weighs 54.341 kg.
Limitations
- Membrane region only; nozzles, supports and welds need code design.
- Pressure vessels must be designed and certified to the applicable code.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a spherical gas receiver or tank
- Compare a sphere with a cylinder for the same volume
- Check a sphere for a hydrotest
Who it is for
- Mechanical engineers
- Design engineers
- Inspection engineers
What you get
- Membrane and maximum stress, safety factor
- Thickness needed with allowance
- Capacity, shell mass and hydrotest stress
Interpreting the result
- A sphere needs about half the wall 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.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Spherical Pressure Vessel Design (Thickness, Stress, Capacity, Hydrotest) calculator