circular-plate

name	circular-plate
description	Circular plate analysis — simply supported/fixed/free boundary, uniform pressure, concentrated load, Kirchhoff theory, deflection/stress formulas, bimetallic plate, ASME pressure vessel heads.
metadata	{"priority":7,"promptSignals":{"phrases":["circular plate","round plate deflection","circular plate under pressure","annular plate","plate bending circular","pressure vessel head"],"minScore":3}}

Circular Plate Analysis — Complete Skill

Governing Equation (Kirchhoff Plate Theory)

Biharmonic equation for thin plate deflection w: ∇⁴w = q / D

Plate flexural rigidity: D = E t³ / [12(1 - ν²)] [N·m; E = Young's modulus; t = plate thickness; ν = Poisson's ratio]

Biharmonic in polar (axisymmetric): (d²/dr² + (1/r)d/dr)²w = q/D Or: d⁴w/dr⁴ + (2/r)d³w/dr³ - (1/r²)d²w/dr² + (1/r³)dw/dr = q/D

Solid Circular Plate — Uniform Pressure q

Simply Supported (at r = a)

Deflection: w(r) = (qa⁴ / 64D) × [(5+ν)/(1+ν) - 2(3+ν)/(1+ν) × (r/a)² + (r/a)⁴] [m] Maximum deflection (center): w_max = (5+ν)/(1+ν) × qa⁴ / (64D) [at r = 0] For ν = 0.3: w_max = 0.0671 × qa⁴ / D

Bending moments: M_r = (q a² / 16) × [(3+ν)(1 - (r/a)²)] M_θ = (q a² / 16) × [(3+ν) - (1+3ν)(r/a)²] Maximum stress (at r = a, top surface): σ_r_max = 6 M_r_max / t² = 3(3+ν) qa² / (8t²) [Pa]

Fixed Edge (clamped at r = a)

Deflection: w(r) = (qa⁴ / 64D) × [1 - 2(r/a)² + (r/a)⁴] [m; symmetric form] Maximum deflection (center): w_max = qa⁴ / (64D) [less deflection than SS; about 4× stiffer] For comparison: w_SS/w_fixed = (5+ν)/(1+ν) ≈ 4.0 for ν = 0.3

Maximum bending moment (at r = a, clamped edge): M_r(a) = -qa² / 8 [N; sagging at edge; maximum; causes tensile stress on top at edge]

Maximum radial stress: σ_r_max = 6 M_r(a) / t² = 3 qa² / (4t²) [at r = a, edge; on top face] For simply-supported: σ_r_max = 3(3+ν)qa² / (8t²) × (1/(SS)) = 0.1875(3.3)qa²/t² = 0.619 qa²/t² For clamped: σ_r_max = 0.75 qa²/t² (at edge)

Comparison (ν = 0.3): Simply supported (center): σ = 0.488 qa²/t² Clamped (edge): σ = 0.75 qa²/t² Clamped (center): σ = 0.488 - 0.25 = lower than SS center

Solid Circular Plate — Concentrated Load P at Center

Simply Supported

Deflection: w(r) = Pa² / (16π D) × [(3+ν)/(1+ν) - 4(r/a)² + 4(r/a)² ln(r/a)] [m; log singularity at r = 0 theoretical]

Maximum deflection (center, excluding singularity by treating load over small radius r₀): w_max ≈ (3+ν) P a² / (16π D (1+ν)) [m; dominant term]

Clamped

w_max = Pa² / (16πD) [for concentrated center load on clamped plate]

Annular Plate (Inner radius b, outer radius a)

Hollow plate with pressure; clamped outer, free inner: Additional constants determined from: M_r(b) = 0; V_r(b) = 0 (free inner); w(a) = 0; dw/dr(a) = 0 (clamped outer) Solution: superposition of four constants × four general homogeneous solutions

Approximate for small hole (b/a < 0.3): Treat as solid plate; stress concentration factor K_t ≈ 2–3 at hole edge under tension K_t = 2 for large plate with hole; K_t reduced for annular plate

Thermal Loading (Bimetallic Plate)

Circular disc with temperature gradient through thickness ΔT (T_bot - T_top): Curvature: κ = α × ΔT / t [1/m; α = CTE; bimetallic extension]

Deflection (clamped): zero (cannot deflect if fully clamped; stress builds) Deflection (free): w = κ r² / 2 [bowl shape]

Thermal bending moment (free plate): M_thermal = E α ΔT t² / [12(1 - ν)] [N·m/m] Results in uniform curvature; no reaction moment at free edge

Pressure Vessel Heads (ASME VIII Div 1)

Flat circular head (UG-34): Required thickness: t = d × √(C P / S_E) [in; d = inside diameter; C = factor (0.13–0.33); P = design pressure; S = allowable stress; E = weld efficiency] C = 0.13 for uniformly loaded, clamped (bolted flange); C = 0.33 for simple support

Hemispherical head: t = P × R / (2 × S × E - 0.2 × P) [membrane only; same as sphere] R = inside radius

2:1 Ellipsoidal head: t = P × D / (2 × S × E - 0.2 × P) [equivalent; factor same as cylinder] ASME preferred standard head; a/b = 2:1

Torispherical (ASME flanged and dished): t = 0.885 × P × L / (S × E - 0.1 × P) [L = crown radius = 1.0 D_i]

Stresses in flat head (bending control): σ_max = C₁ × P × (D/2)² / t² [governs thick flat heads; circular plate formula]

Deflection Limits

Pressure vessel flat heads: No explicit ASME deflection limit; but excessive deflection → flange leakage Design: w_max ≤ 0.01 × diameter (practical limit for gasketed joints)

Mechanical circular plates: Thin plate theory valid for: w < t/2 (large deflection theory needed above this) For w > t: membrane action dominates → significantly stiffer than plate theory predicts

Output

Provide: plate dimensions (a, t [mm]), boundary condition (simply supported/clamped), loading type (uniform q [Pa] or point P [N] or thermal ΔT), flexural rigidity D [N·m], maximum deflection w_max [mm] with location, maximum radial stress σ_r_max [MPa] with location (edge or center), maximum tangential stress σ_θ_max [MPa], comparison to material allowable, deflection ratio w/a, ASME VIII equation for pressure vessel head (if applicable), and thickness required per ASME (t_required [mm]).