BIZOMAT MAKEOS · Strength
Circular Flat Plate under Pressure
What stress and deflection does uniform pressure cause in this round plate?
Maximum bending stress and centre deflection of a uniformly loaded circular flat plate with clamped or simply supported edges (small-deflection theory).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Edge support | choice | Clamped for a plate welded or bolted rigidly all round; simply supported for a plate resting on a gasket or ledge. |
| Radius to the edge support | mm | Radius at which the plate is supported (gasket or bolt-circle radius as appropriate). |
| Plate thickness | mm | Minimum plate thickness. |
| Uniform pressure | MPa | Pressure difference across the plate (1 MPa = 10 bar). |
| Elastic modulus | GPa | Elastic modulus from the material data sheet or an approved materials database (steel is typically about 200 GPa). |
| Poisson’s ratio | Poisson’s ratio of the material (about 0.3 for steel). | |
| Allowable bending stress | MPa | Allowable bending stress from your design code. |
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: Edge support
- What it is
- How the edge of the round plate is held: clamped (built in, unable to rotate) or simply supported (resting, free to rotate).
- Units and format
- Choose one of 2 options from the list: Clamped (built-in) edge; Simply supported edge.
- Example
- The form opens with “Clamped (built-in) edge” selected as a worked example. Choose the option that matches your case.
- Get it right
- Real bolted or gasketed edges lie between the two; the simply supported case gives the larger stress and deflection.
What this means: Radius to the edge support
- What it is
- The radius of the circle on which the plate is supported, from the centre to the support line such as the gasket or bolt circle.
- 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 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress rises with the square of the radius and deflection with its fourth power.
- Get it right
- Enter the radius, not the diameter.
What this means: Plate thickness
- What it is
- The thickness of the flat plate carrying the pressure.
- 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 12 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress falls with the square of the thickness and deflection with its cube.
- Get it right
- Use the minimum thickness after tolerance and any corrosion or machining allowance.
What this means: Uniform pressure
- What it is
- The pressure difference across the plate, spread uniformly over its face, between the two sides.
- 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 1 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress and deflection rise in proportion to the pressure.
- Get it right
- Enter MPa: 1 MPa is 10 bar. Use the difference between the two sides.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the plate material, from the material data sheet. It affects deflection, not stress.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Poisson’s ratio
- What it is
- Poisson’s ratio of the plate material: how much it contracts sideways when stretched, as a fraction, from the material data sheet.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It is a dimensionless ratio, not a percentage.
What this means: Allowable bending stress
- What it is
- The highest bending stress the plate is allowed to reach, from your design code. The calculated maximum stress is compared with it.
- 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 160 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Pressure vessel heads and bolted flanges must follow their own code rules.
Method and formulas
- Plate rigidity D = E t³ ÷ 12(1 − ν²).
- Clamped: σmax = 3pa² ÷ 4t² at the edge; δ = pa⁴ ÷ 64D.
- Simply supported: σmax = 3(3 + ν)pa² ÷ 8t² at the centre; δ = (5 + ν)pa⁴ ÷ 64(1 + ν)D.
- Example: a 12 mm clamped plate of 100 mm radius at 1 MPa has σmax = 52.083 MPa and δ = 0.049371 mm.
Limitations
- Small deflections only (δ less than about half the thickness); larger deflections need membrane theory.
- Real edges lie between clamped and simply supported; bolted flanges and pressure vessel heads must follow their codes.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check blind flanges, cylinder end caps and covers
- Size a window or inspection cover
- Compare clamped and simply supported edges
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Maximum bending stress and location
- Centre deflection
- Deflection-to-thickness ratio
Interpreting the result
- If unsure of the edge condition, check both cases and use the worse result.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 288–290. Topic reference; no source tables or recommended settings are reproduced.
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