When thin-plate theory breaks down

Closed-form plate formulas are fast and transparent, and they fail in predictable ways. Knowing where helps you decide when an analytical estimate is enough.

Large deflection

Once the centre deflection exceeds about 20 % of the thickness, the mid-plane stretches and carries load as a membrane. The plate stiffens and the output becomes nonlinear. For a clamped circular plate, Timoshenko's energy solution gives:

w₀/h + 0.488 (w₀/h)³ = q a⁴ / (64 D h)

DiaphragmCalc solves this cubic and reports the deflection nonlinearity. For square plates it applies the same ratio of cubic to linear terms, which is an estimate. Example: a 1000 µm square, 8 µm thick silicon diaphragm at 1 bar has w₀/h = 2.2 and an estimated deflection nonlinearity of 43 %.

Thick plates

Kirchhoff plate theory ignores shear deformation. When a/h drops below about 10 the formulas under-predict deflection. Use Mindlin plate theory or FEA.

Things the formulas never include

• Anisotropy of single-crystal silicon (E varies from 130 to 188 GPa with direction).
• Residual stress from doping, oxides and bonding.
• Non-ideal clamping: sloped KOH sidewalls, glass frit or anodic bond compliance.
• Bossed, corrugated or multilayer diaphragms.
• Package-induced stress and thermal mismatch, often the largest error source in practice.

Use the analytical model to find a sensible geometry and the sensitivities of each parameter, then confirm with FEA and test.

Results are analytical first-order estimates and should be independently validated before use in safety-critical or production designs.