BIZOMAT MAKEOS · Strength
Square and Rectangular Flat Plates
What stress and deflection does this square or rectangular plate have under load?
Maximum stress and deflection of square and rectangular flat plates with supported or fixed edges under uniform pressure or a central load, by the handbook’s plate formulas.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Plate and loading | choice | Plate shape, edge support and load. |
| Length (long side) | mm | Distance between supports (long side for rectangles). |
| Width (short side, rectangular) | mm | Short side of a rectangular plate. |
| Thickness | mm | Plate thickness. |
| Uniform pressure | kPa | Uniform pressure (1 kPa = 0.001 MPa; water depth 1 m ≈ 9.8 kPa). |
| Central load (central-load cases) | N | Concentrated load at the centre. |
| Radius of loaded area (central load) | mm | Radius of the area the central load is applied over. |
| Elastic modulus | GPa | Elastic modulus from the material data sheet (steel is typically about 200 GPa). |
| Allowable stress | MPa | Allowable 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: Plate and loading
- What it is
- The plate shape (square or rectangular), how its edges are held (simply supported or clamped fixed) and whether the load is a pressure spread over the whole face or a single load at the centre. Each choice selects a different handbook formula.
- Units and format
- Choose one of 7 options from the list: Square · edges supported top and bottom · uniform; Square · edges supported below only · uniform; Square · edges fixed · uniform; Square · edges fixed · central load; Square · edges supported · central load; Rectangular · edges supported · uniform; Rectangular · edges fixed · uniform.
- Example
- The form opens with “Rectangular · edges supported · uniform” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose “fixed” only when the edges are truly clamped against rotation, such as welded all round to a stiff frame; bolted or resting edges are closer to supported.
What this means: Length (long side)
- What it is
- The span of the plate between its edge supports. For a square plate it is the side length; for a rectangular plate it is the longer of the two sides.
- 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 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure between the lines of support, not the overall plate size including any edge that sits on the frame.
What this means: Width (short side, rectangular)
- What it is
- The shorter span between supports of a rectangular plate. It is only used by the two rectangular cases; square cases use the long side for both directions.
- 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.
- Get it right
- It must not be longer than the long side; if it is, swap the two values.
What this means: Thickness
- What it is
- The plate thickness through which it resists bending, measured at the thinnest part of the loaded panel.
- 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 8 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 thickness and deflection with the cube, so a small increase helps a lot.
- Get it right
- Use the actual or minimum thickness after any corrosion allowance or wear, not the nominal sheet size if it is likely to be thinner.
What this means: Uniform pressure
- What it is
- The pressure acting evenly over the whole face of the plate, such as fluid pressure, a stored bulk load or a distributed floor load. The tool multiplies it by the plate area to get the total load. Used by the uniform cases only.
- Units and format
- Enter a number in kPa: kilopascals. The form accepts 0 kPa to 1000000 kPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 kPa 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 direct proportion to the pressure.
- Get it right
- Use kilopascals: 1 kPa = 1 kN/m². Leave the central load aside for uniform cases; it is ignored there.
What this means: Central load (central-load cases)
- What it is
- A single concentrated force pressing on the middle of a square plate, such as a wheel, a foot of a machine or a jack. Used by the two central-load cases only.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5000 N 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 direct proportion to the central load.
- Get it right
- Enter the total force in newtons; a mass in kilograms must be multiplied by g (1 kg weighs about 9.81 N).
What this means: Radius of loaded area (central load)
- What it is
- The radius of the small circular patch over which the central load is spread, for example half the diameter of a pad or foot. Only the central-load cases use it.
- 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 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger loaded patch lowers the peak stress; it does not change the deflection.
- Get it right
- This is a radius, not a diameter. When it is small compared with the thickness, the tool replaces it with the handbook’s equivalent radius automatically.
What this means: Elastic modulus
- What it is
- The stiffness of the plate material in tension (Young’s modulus), from the material data sheet or standard. It sets how far the plate deflects but not how highly it is stressed.
- 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.
- Effect on the result
- Deflection is inversely proportional to the modulus; stress is unaffected.
- Get it right
- Enter gigapascals: 1 GPa = 1000 MPa. Typing a value in MPa here would make the plate a thousand times too stiff.
What this means: Allowable stress
- What it is
- The highest bending stress your design code or company standard permits in this plate, usually the yield strength divided by a safety factor. It is only used for the pass or fail check.
- 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
- Take it from the applicable design code for the material and loading, not the raw yield or tensile strength.
Method and formulas
- Square, uniform (W = p × area): supported S = 0.29W/t² (0.28 bottom only), d = 0.0443WL²/Et³; fixed S = 0.31W/t², d = 0.0138WL²/Et³.
- Square, central load: fixed S = 0.62W/t² ln(L/2r₀), d = 0.0568WL²/Et³; supported S = 0.62W/t² (ln(L/2r₀) + 0.577), d = 0.1266WL²/Et³ (r₀ replaced by √(1.6r₀² + t²) − 0.675t when r₀ < 1.7t).
- Rectangular supported: S = 0.75W ÷ t²(L/l + 1.61 l²/L²), d = 0.1422W ÷ Et³(L/l³ + 2.21/L²). Fixed: S = 0.5W ÷ t²(L/l + 0.623 l⁵/L⁵), d = 0.0284W ÷ Et³(L/l³ + 1.056 l²/L⁴).
- Example: a 1000 × 600 × 8 mm supported plate at 10 kPa reaches 31.302 MPa and deflects 1.2182 mm.
Limitations
- Small-deflection formulas based on ν = 0.3 (within about 3% for other metals).
- Deflections over about half the thickness need large-deflection (membrane) analysis.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check tank sides, access covers and floor plates
- Size a plate for a pressure or point load
- Compare supported and fixed edges
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Maximum stress
- Maximum deflection
- Total load
Interpreting the result
- Stiffeners usually save more weight than a thicker plate.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 288–290. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Deflection and stress of a flat panel
In Stiffening Sheet Metal by the Numbers
What it covers: Calculate a rectangular panel's deflection and bending stress under pressure for supported and clamped edges, see why the short side governs, and catch a calculation that used the long side.
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