Flat Plate Deflection & Stress Calculator

Find the maximum deflection and bending stress of a flat plate under uniform pressure or a central point load. Supports rectangular and circular plates with simply supported, clamped, or two-opposite-edges-fixed boundaries, using classical thin-plate theory and Roark coefficients. Pair the bending stress with our von Mises or Mohr's circle calculators for a full stress-state or yield check.

How Flat Plate Bending Works

A flat plate loaded perpendicular to its surface carries the load in two-way bending rather than the one-way bending of a beam. The stiffness of the plate is captured by the flexural rigidity D, which plays the role that EI plays for a beam. Because the plate spans in two directions, its deflection and stress depend on the aspect ratio, the edge support conditions, and how the load is distributed.

Here E is Young's modulus, t is the plate thickness, and ν is Poisson's ratio. The thickness enters as a cube, so doubling the plate thickness increases its bending stiffness eightfold (deflection drops to about one-eighth). Bending stress varies as 1/t², so doubling the thickness cuts the stress to about a quarter.

Rectangular Plate Formulas (Uniform Load)

For a rectangular plate under uniform pressure q, with short span b, the maximum deflection and bending stress are written with dimensionless coefficients α and β that depend on the long-to-short aspect ratio a/b and the edge conditions:

The coefficients are taken from Roark's Formulas for Stress and Strain (Table 11.4) for a Poisson's ratio of 0.3, and interpolated for the aspect ratio you enter. For simply supported edges the maximum stress is at the plate centre; for clamped edges it moves to the centre of the long edge, where the fixing moment is largest. As the aspect ratio grows past about 2, the plate behaves like a one-way strip spanning the short direction.

Two Opposite Edges Fixed

When only two opposite edges are built in and the other two are free, the plate spans one way between the fixed edges, behaving like a fixed-fixed strip. Using the span a between the fixed edges:

Because the unloaded edges are free, this case is treated as a one-way fixed-fixed strip spanning between the built-in edges. The peak moment is the fixed-end moment q a²/12 per unit width, giving the stress above, and the deflection uses the corresponding beam-strip value rather than a two-way plate coefficient. The free-edge width does not change the per-unit-width result, so it only matters when you enter a total load that needs spreading into a pressure.

Circular Plate Formulas (Uniform Load)

Circular plates of radius a under uniform pressure have exact closed-form solutions. For a simply supported edge:

For a clamped edge the deflection is smaller and the peak stress moves to the edge:

Clamping an edge roughly quarters the centre deflection compared with simple support, which is why fixed boundaries are so effective for plates.

Central Point Load

A concentrated load at the centre creates a theoretical stress singularity directly under the load, so the bending stress depends on how the load is spread. A standard tabulated approximation replaces the actual loaded radius e with an effective contact radius e′ that accounts for the plate thickness, keeping the result finite even for an ideal point load (e = 0):

For a circular plate of radius r carrying a central load P, the centre deflection and centre stress are:

For a rectangular plate with short span b, the deflection uses a tabulated coefficient k₁ and the centre stress uses a logarithmic term plus a coefficient (k₂ for simply supported, k₃ for clamped), with both coefficients interpolated by aspect ratio:

For a clamped rectangular plate the calculator also checks the fixing stress at the centre of the long edge (σ = k₂ P / t²) and reports whichever governs. These point-load results assume a central load or central contact patch; reduce the loaded radius toward zero for an ideal point load, or enter the real bearing radius for a finite patch load.

Pressure or Total Load

For a uniform load you can enter it either as a pressure q or as a total load P spread over the plate. In total-load mode the calculator converts to an equivalent pressure using q = P / area, where the area is the full rectangle or circle, and reports the pressure it used. This matches how loads are often quoted as a single force on a panel.

Units and Sign

In metric mode, enter pressure in kPa (or a force in N), dimensions and thickness in mm, and Young's modulus in MPa; results come back in mm and MPa. In imperial mode, enter pressure in psi (or a force in lbf), dimensions in inches, and modulus in psi; results come back in inches and psi. Deflection is reported as a positive magnitude in the direction of the load, and the bending stress is the peak surface stress at the governing location.

When These Formulas Apply

The calculator uses small-deflection (Kirchhoff) thin-plate theory, which assumes the plate is thin relative to its span and that deflections stay small, typically below about half the plate thickness. Beyond that range, in-plane membrane action begins to carry load and the real plate is stiffer than the linear result; a geometrically nonlinear analysis is then required. The closed-form cases also assume idealised edge supports and either a uniform pressure or a single central load. Real plates with off-centre or multiple loads, partial pressures, openings, stiffeners, or mixed edge conditions need specialist plate or shell finite element software.

From Bending Stress to Yield

This tool reports a peak bending stress (often at the centre or a supported edge). For a quick ductile check, treat that value as a normal stress in plane stress and use the von Mises stress calculator with the other components set to zero, or enter a biaxial state if you have both face stresses. For principal directions and maximum in-plane shear, use the principal stress & Mohr's circle calculator.

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