Loaded Flat Plates – Stress and Deflection

This page provides engineering formulas and coefficients for estimating stress and deflection in thin flat plates subjected to transverse loading. Configurations include circular and rectangular plates, plates with central holes, uniform pressure loading and concentrated loads, with simply supported or clamped boundary conditions.

Contents

• Plate Theory and Assumptions
• Symbols and Units
• Circular Plates
• Rectangular Plates
• Circular Plates with Central Holes
• Circular Plates with Guided Central Holes
• Using the Plate Data

Plate Theory and Assumptions

The equations and coefficients on this page provide estimates of maximum stress and deflection for flat plates under a number of common loading and support conditions.

They are based on classical thin-plate bending assumptions. In particular:

• the plate is initially flat;
• the plate thickness is small compared with its other dimensions;
• deflection is small compared with the plate thickness;
• the material is homogeneous, isotropic and linearly elastic;
• Young's modulus and Poisson's ratio remain constant;
• the loading and boundary conditions correspond to the configuration shown;
• local stress concentrations and local contact effects are not represented by the simple plate equations.

The original RoyMech calculations were prepared principally for steel plates using a Poisson's ratio of approximately 0.3. Where coefficients or equations have been derived using this value, they should not automatically be applied to materials having substantially different Poisson's ratios.

The original notes also limited their use to plates having a thickness less than approximately 10% of the relevant plate span or diameter. This is a useful screening rule rather than a complete definition of thin-plate behaviour.

For the simply supported rectangular-plate cases on this page, the loaded plate edges are assumed to remain in contact with their supports and to be restrained from lifting away from them.

These equations are suitable for preliminary calculations and engineering estimates. For critical design, unusual geometry, large deflection, local loading or more accurate stress determination, a recognised plate reference such as Roark's Formulas for Stress and Strain, an appropriate analytical method or validated finite-element analysis should be used.

Symbols and Units

Plate Flexural Rigidity

For a homogeneous isotropic plate:

The cubic dependence on thickness is particularly important: plate bending stiffness changes rapidly with plate thickness.

Circular Plates

Uniform Load – Simply Supported Edge

Circular plate subjected to uniform transverse pressure with the outer edge simply supported.

Uniform Load – Clamped Edge

Circular plate subjected to uniform transverse pressure with the outer edge clamped.

Central Concentrated Load – Simply Supported Edge

Circular plate subjected to a concentrated transverse load at its centre, with the outer edge simply supported.

Central Concentrated Load – Clamped Edge

Circular plate subjected to a concentrated transverse load at its centre, with the outer edge clamped.

Rectangular Plates

The response of a rectangular plate depends strongly on its aspect ratio a/b as well as its loading and support conditions.

Uniform Load – Simply Supported Edges

Rectangular plate subjected to uniform transverse pressure with simply supported edges.

Uniform Load – Clamped Edges

Rectangular plate subjected to uniform transverse pressure with clamped edges.

Central Concentrated Load – Simply Supported Edges

Central Concentrated Load – Clamped Edges

The sign of a plate coefficient can identify the sense of the associated bending stress or moment according to the convention used in the accompanying equation. A negative coefficient should therefore not automatically be treated as an error.

Circular Plates with Central Holes

The following cases apply to annular plates. For these configurations the dimensions a and b and the ratio a/b are defined by the individual diagrams. Use the geometry shown with each equation rather than transferring the rectangular-plate definitions of a and b.

Concentrated Load at Hole – Simply Supported Outer Edge

Uniform Load over Ring – Simply Supported Outer Edge

Concentrated Load at Hole – Clamped Outer Edge

Uniform Load over Ring – Clamped Outer Edge

Circular Plates with Guided Central Holes

Uniform Distributed Load – Simply Supported Outer Edge

Uniform Distributed Load – Fixed Outer Edge

Using the Plate Data

A plate calculation is only meaningful when the geometry, loading and boundary conditions match the selected solution.

1. Identify whether the plate is circular, rectangular or annular.
2. Determine whether the load is distributed pressure or a concentrated force.
3. Identify the actual support condition.
4. Check that thin-plate and small-deflection assumptions are reasonable.
5. Use the equation and coefficient set corresponding to that exact configuration.
6. Use a consistent system of units throughout the calculation.
7. Compare the resulting stress with an appropriate material design criterion.
8. Check deflection against the functional requirements of the component.
9. Consider local stresses, fatigue, instability and other failure modes where relevant.

Small and Large Deflection

Classical small-deflection plate theory assumes that transverse deflection does not significantly alter the geometry of the plate or introduce important in-plane membrane forces.

As deflection becomes comparable with plate thickness, membrane action can become significant and the load-deflection relationship becomes nonlinear. The equations on this page should then be replaced by an appropriate large-deflection plate analysis.

Concentrated Loads

An ideal mathematical point load can produce highly localised stresses. Actual engineering loads are normally transferred through a finite contact area, washer, boss, bearing surface or attachment. Local bearing, bending, punching, yielding or contact stresses may therefore require separate assessment.

Stress Interpretation

The maximum bending stresses calculated by classical plate theory are nominal elastic stresses for the idealised plate model. Holes, welds, sharp changes in thickness, attachments and other geometric details can introduce local stress concentrations that are not represented by these simple solutions.

Related RoyMech References

• Component Loading – direct, shear, bending and torsional loading.
• Stress and Strain – elastic stress-strain relationships and material constants.
• Yield and Failure Criteria – Tresca, von Mises and principal-stress criteria.
• Mohr's Circle – principal stress and stress transformation.
• Stress-Strain Curve – tensile properties, yielding, ultimate tensile strength and ductility.
• Thick-Walled Cylinders – radial and hoop stresses in pressure-loaded cylinders.
• Beam Calculations – bending, shear and deflection of beam members.

External References and Calculation Resources

1. MITCalc – Plates – plate calculation software and engineering calculations.
2. Plate Theory – Wikipedia – general background on classical plate theory.
3. ExcelCalcs – Plates – engineering spreadsheets and plate calculations.
4. RoyMech / ExcelCalcs Plate Calculation – spreadsheet implementation of the plate equations historically associated with this RoyMech page.