Circular diaphragm deflection and stress

Circular diaphragms appear in DRIE-etched silicon sensors, metal-diaphragm transmitters and capacitive sensors. The clamped circular plate has exact closed-form solutions, which makes it a useful reference.

Formulas

For radius a, thickness h, uniform pressure q and D = E h³ / 12(1 − ν²):

w(r)     = q (a² − r²)² / 64D
w₀       = q a⁴ / 64D                 (centre)
σ_r(a)   = 3/4 · q (a/h)²             (radial stress at the edge)
σ_t(a)   = ν · σ_r(a)                 (tangential stress at the edge)
σ_r(0)   = σ_t(0) = 3(1+ν)/8 · q (a/h)²

The edge radial stress is the design-limiting value and the best place for a radially oriented piezoresistor.

Worked example

Silicon diaphragm, 1000 µm diameter (a = 500 µm), 15 µm thick, E = 169 GPa, ν = 0.064, full scale 1 bar:

• w₀ = 2.05 µm (w₀/h = 0.14)
• Edge radial stress = 83.3 MPa; tangential = 5.33 MPa
• First resonance in vacuum ≈ 240 kHz

Square or circular?

For the same span (side = diameter) and thickness, a square plate's edge stress is 0.3078/(0.75/4) ≈ 1.64 times the circular value, because the square has more unsupported area. The square also has a higher stress gradient at the edge midpoint, which suits small resistors. Circular diaphragms give lower peak stress for the same pressure and are easier to model exactly.

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