Kirchhoff-Love Circular Plate Flexural Rigidity and Diaphragm Deflection

Spot the (1 - nu^2) biaxial stiffening and 4x edge-support jump

Treating a circular pressure diaphragm like a 1D beam strip omits Poisson’s biaxial restraint (1 - nu^2) in plate flexural rigidity D = E * h^3 / (12 * (1 - nu^2)). Even bigger is boundary-condition sensitivity: under uniform differential pressure q_Pa on radius R_a, a clamped perimeter deflects w_clamp = q * R_a^4 / (64 * D), whereas a simply supported perimeter deflects w_simp = w_clamp * (5 + nu) / (1 + nu) — over 4x larger!

Check an aluminum sensor diaphragm: E = 70 GPa (B2), thickness h = 0.004 m (B3, 4 mm), Poisson’s ratio nu = 0.30 (B4), active radius R_a = 0.10 m (B5, 100 mm), and uniform pressure q = 50 kPa (B6, 50,000 Pa). Here D = 70e9 * 64e-9 / (12 * 0.91) = 4,480 / 10.92 = 410.2564 N*m, clamped center sag is 0.1904 mm, and simply supported sag jumps by 5.3 / 1.3 = 4.0769x to 0.7762 mm.

Plate Flexural Rigidity D (N*m) Formula

Guard thin-plate slenderness (h / (2 * R_a) > 0.10 requires thick-plate shear theory) and physical Poisson bounds (0 <= nu < 0.5).

=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 0.5, B5 <= 0), "Invalid Plate Input", IF(B3 / (2 * B5) > 0.1, "Thick Plate Regime", ROUND((B2 * 10^9) * (B3^3) / (12 * (1 - B4^2)), 4)))

Clamped Circular Diaphragm Deflection (mm) & Thickness Cube-Law Check

Because D scales with h^3, doubling diaphragm thickness in B3 from 0.004 m (4 mm) to 0.008 m (8 mm) multiplies D eightfold to 3,282.0513 N*m and cuts clamped center deflection eightfold from 0.1904 mm down to 0.0238 mm.

=LET(e_pa, B2 * 10^9, h_m, B3, nu_v, B4, r_m, B5, q_pa, B6 * 1000, IF(OR(e_pa <= 0, h_m <= 0, nu_v < 0, nu_v >= 0.5, r_m <= 0), "Invalid Plate Input", LET(d_rig, e_pa * (h_m^3) / (12 * (1 - nu_v^2)), ROUND(1000 * q_pa * (r_m^4) / (64 * d_rig), 4))))

Further reading

• Arte Sheets Lesson — Timoshenko Shear-Deformable Deep-Beam Deflection
• Arte Sheets Lesson — Lame Thick-Walled Pressure Vessel Hoop and Radial Stress