Square diaphragm deflection and stress

Most bulk-micromachined silicon pressure sensors use a square diaphragm, because anisotropic KOH or TMAH etching of (100) wafers produces square cavities bounded by {111} planes. The diaphragm behaves as a thin plate clamped on all four edges.

Formulas

For a clamped square plate of side a and thickness h under uniform pressure q, with flexural rigidity D = E h³ / 12(1 − ν²), small-deflection theory (Timoshenko & Woinowsky-Krieger, Table 35) gives:

w₀     = 0.00126 · q a⁴ / D          (centre deflection)
σ_edge = 0.3078 · q (a/h)²           (bending stress at mid-edge, normal to edge)
σ_centre ≈ 0.138 · q (a/h)²          (ν = 0.3)

The largest stress sits at the middle of each edge, on the surface, pointing normal to the edge. It is more than twice the centre stress, which is why piezoresistors go near the edge midpoints. Along the clamped edge the plate cannot curve, so the stress parallel to the edge is about ν·σ_edge.

Worked example

A 1000 µm × 1000 µm silicon diaphragm, 20 µm thick, E = 169 GPa, ν = 0.064, at 1 bar (100 kPa):

• D = 0.0001131 N·m
• Centre deflection w₀ = 1.11 µm, so w₀/h = 0.056: well inside the small-deflection range.
• Edge stress σ_edge = 77 MPa.
• At 3× overpressure the edge sees 231 MPa, a safety factor of 4.3 against a 1 GPa fracture stress.

Deflection scales with a⁴/h³ and stress with (a/h)². Doubling the side length multiplies deflection by 16 and stress by 4.

Limits

The 0.3078 and 0.00126 coefficients assume ideal clamping and an isotropic plate with ν ≈ 0.3. Real silicon diaphragms are anisotropic, the KOH sidewall slope softens the edge, and deflections above about 0.2·h add membrane stiffening. Treat these results as the starting point for FEA, not the answer.

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