Mechanical & Structural Engineering
Mindlin-Reissner Thick Circular Plate Bending + Shear Deflection
Extend Kirchhoff thin-plate formulas to thick viewports and flange blinds using Mindlin-Reissner transverse shear deflection q*R^2 / (4*kappa*G*h) and exact (h/R)^2 scaling.
Predict the exact 36% transverse-shear boost on a thick circular viewport
When a high-pressure optical viewport, blind flange, or semiconductor chuck has thickness-to-radius ratio h / R >= 0.15, Kirchhoff-Love thin-plate theory w_KL = q * R^4 / (64 * D) misses substantial transverse shear deformation. Mindlin-Reissner plate theory adds w_shear = q * R^2 / (4 * kappa * G * h), where G = E / (2*(1 + nu)) and shear factor kappa = 5/6.
Algebraically, on a clamped circular plate, w_shear / w_KL simplifies to (16 / (5 * (1 - nu))) * (h / R)^2! Verify by hand for fused-quartz / ceramic properties E = 72 GPa (B2), thickness h = 0.030 m (B3, 30 mm), Poisson ratio nu = 0.20 (B4, so 16 / (5 * 0.80) = 4.0000!), active radius R = 0.100 m (B5, 100 mm, h/R = 0.30), pressure q = 2,400 kPa (B6, 2.4 MPa), and kappa = 5/6 (B7): shear-to-bending ratio is 4 * (0.30)^2 = 0.3600 (+36% extra deflection!).
| Thick-Plate Metric (h/R = 0.30, nu = 0.20) | Mindlin-Reissner Formula | Sample Output |
|---|---|---|
| Flexural Rigidity D & Shear G | E*h^3/(12*(1-nu^2)); G = E/(2*(1+nu)) | D = 168,750 N*m; G = 30 GPa |
| Kirchhoff Bending Term w_KL | 1000 * q_Pa * R^4 / (64 * D) | 0.0222 mm |
| Transverse Shear Term w_shear | 1000 * q_Pa * R^2 / (4 * kappa * G * h) | 0.0080 mm |
| Total Mindlin-Reissner w_MR | w_KL + w_shear (Shear/Bend = 0.3600) | 0.0302 mm (+36.0% over thin plate) |
Dimensionless Shear-to-Bending Amplification Ratio Formula
Verify the closed-form shear ratio (16 / (6 * kappa * (1 - nu))) * (h / R)^2 (which equals 4 * (h / R)^2 at kappa = 5/6, nu = 0.20) before inspecting micron/millimeter deflections.
=IF(OR(B3 <= 0, B4 < 0, B4 >= 0.5, B5 <= 0, B7 <= 0), "Invalid Plate Input", ROUND((16 / (6 * B7 * (1 - B4))) * ((B3 / B5)^2), 4))Total Mindlin-Reissner Clamped Center Deflection (mm) Formula
Sum Kirchhoff bending and Mindlin transverse shear in millimeters (or microns * 1000). Halving thickness h in B3 from 0.030 m to 0.015 m (h / R = 0.15) cuts the shear ratio fourfold from 0.3600 to 0.0900.
=LET(e_pa, B2 * 10^9, h_m, B3, nu_v, B4, r_m, B5, q_pa, B6 * 1000, kap, B7, IF(OR(e_pa <= 0, h_m <= 0, nu_v < 0, nu_v >= 0.5, r_m <= 0, kap <= 0), "Invalid Plate Input", LET(d_rig, e_pa * (h_m^3) / (12 * (1 - nu_v^2)), g_pa, e_pa / (2 * (1 + nu_v)), w_kl, (q_pa * (r_m^4)) / (64 * d_rig), w_sh, (q_pa * (r_m^2)) / (4 * kap * g_pa * h_m), ROUND(1000 * (w_kl + w_sh), 6))))Try the example
Use the original values or change them to test your understanding.