rectangular-plate

name	rectangular-plate
description	Rectangular plate stress analysis — Kirchhoff thin plate bending (Navier and Lévy solutions), deflection w and moments Mx/My/Mxy under uniform and concentrated loads, Poisson's ratio effect, boundary conditions (simply supported, clamped, free edge), plate with hole (stress concentration), thermal bending, thick plate (Mindlin-Reissner, shear correction), yield-line theory (plastic collapse), AISC/ASME flat plate design, and large deflection (von Kármán equations).
metadata	{"priority":7,"promptSignals":{"phrases":["rectangular plate","plate bending","flat plate stress","Kirchhoff plate","plate deflection","Navier plate solution"],"minScore":3}}

Rectangular Plate Analysis — Complete Skill

Governing Equation — Kirchhoff (Thin) Plate

Biharmonic Equation

Equilibrium equation for thin plate bending: D × ∇⁴w = q(x,y) [D = flexural rigidity; w = deflection; q = distributed load [Pa]] ∇⁴w = ∂⁴w/∂x⁴ + 2∂⁴w/∂x²∂y² + ∂⁴w/∂y⁴

Flexural rigidity: D = E × h³ / [12 × (1 - ν²)] [N·m; h = plate thickness; ν = Poisson's ratio]

Moment-curvature relations: M_x = -D × (∂²w/∂x² + ν × ∂²w/∂y²) [N·m/m; bending moment per unit width in x-direction] M_y = -D × (∂²w/∂y² + ν × ∂²w/∂x²) [N·m/m; bending moment per unit width in y-direction] M_xy = D × (1-ν) × ∂²w/∂x∂y [N·m/m; twisting moment]

Bending stress from moments: σ_x = 12 × M_x × z / h³ = 6 × M_x / h² [Pa at z = ±h/2; maximum at surface] σ_y = 12 × M_y × z / h³ = 6 × M_y / h² σ_z = 0 (thin plate assumption); τ_xy from M_xy

Thin plate validity: h/a ≤ 0.1; h/w_max ≥ 5 (small deflection); if h/a > 0.1 → Mindlin theory

Navier Solution — SSSS Plate (All Edges Simply Supported)

Double Fourier Series

Fourier series expansion of load: q(x,y) = Σₘ Σₙ q_mn × sin(mπx/a) × sin(nπy/b) [m,n = 1,2,3,...; a,b = plate dimensions] q_mn = (4/(ab)) × ∫∫ q(x,y) × sin(mπx/a) × sin(nπy/b) dA

For uniform load q₀: q_mn = 16q₀/(π² × m × n) for odd m,n; q_mn = 0 for even m or n

Exact solution (SSSS): w(x,y) = Σₘ Σₙ w_mn × sin(mπx/a) × sin(nπy/b) w_mn = q_mn / [D × π⁴ × (m²/a² + n²/b²)²]

Maximum deflection (uniform load, center of plate): w_max = w(a/2, b/2) = Σ q_mn × sin(mπ/2) × sin(nπ/2) / [D × π⁴ × (m²/a² + n²/b²)²] First-term approximation (good to 1% for uniform load): w_max ≈ 0.0443 × q₀ × a⁴ / D [square plate a = b; from convergence of series; coefficient from tables]

Timoshenko coefficient table (SSSS, uniform q₀):

a/b ratio	α (w_max = α×q₀b⁴/D)	β (M_max = β×q₀b²)
1.0	0.0443	0.0479
1.2	0.0616	0.0627
1.5	0.0843	0.0843
2.0	0.1106	0.1017
∞ (strip)	0.1302	0.1250

Maximum moment (at plate center, square SSSS, uniform q): M_x,max = M_y,max = 0.0479 × q₀ × a² [a = b; by symmetry; at plate center] σ_max = 6 × M_max / h² = 6 × 0.0479 × q₀ × a² / h² = 0.287 × q₀ × (a/h)²

Lévy Method — Plates with One Pair of SSSS Edges

Opposite Edges Simply Supported

For plate with x = 0 and x = a simply supported (any BC on y = 0 and y = b): w(x,y) = Σₘ W_m(y) × sin(mπx/a) [automatically satisfies SS on x-edges]

W_m satisfies ODE: d⁴W_m/dy⁴ - 2(mπ/a)² × d²W_m/dy² + (mπ/a)⁴ × W_m = q_m(y)/D

General solution: W_m = (A_m cosh(α_m y) + B_m × y × sinh(α_m y) + C_m × sinh(α_m y) + D_m × y × cosh(α_m y)) + particular solution α_m = mπ/a

BC application: apply BC on y = 0 and y = b → 4 equations for 4 unknowns per m Typical: clamped edge (W_m = 0, dW_m/dy = 0); free edge (M_y = 0, V_y = 0)

Results for CCCC (clamped all edges, uniform load): w_max = α × q₀ × a⁴ / D [α = 0.0138 for square vs. 0.0443 for SSSS → clamped 3.2× stiffer] M_max at center: β × q₀ × a² with β = 0.0231 for square plate (at center) M_x at clamped edge (support): M_x_edge = -0.0513 × q₀ × a² (hogging; negative)

Results for SCSC (SS on x-edges, clamped on y-edges), a = b: w_max = 0.0209 × q₀ × a⁴ / D (intermediate between SSSS and CCCC)

Concentrated Load — Point Load P at Center (SSSS)

Navier solution converges slowly; use alternate: w_max = 0.1160 × P × a² / D [square plate; SSSS; load at center] M_max = 0.0368 × P [at load point; logarithmic singularity; bending stress diverges → use thick plate or local yield]

Practical note: Point load on thin plate → local stress concentration; use patch load or check punching shear

Large Deflection (von Kármán Equations)

When to Use

Large deflection criterion: w_max / h > 0.2–0.5 → membrane stresses become significant; linear theory non-conservative For square plate under uniform q: linear gives w_max = α×q×a⁴/D; valid while w_max < h/5

von Kármán equations (coupled nonlinear PDEs): D × ∇⁴w = q + h × (φ,yy × w,xx - 2φ,xy × w,xy + φ,xx × w,yy) [bending equation; φ = Airy stress function] ∇⁴φ = -E × (w,xy² - w,xx × w,yy) [compatibility; stretching of mid-plane]

Numerical solution: FEA (geometric nonlinear analysis); or Ritz/Galerkin approximate for simple geometries

Empirical large-deflection formula (SSSS square plate, uniform q): q × a⁴ / (E × h⁴) = C₁ × (w_max/h) + C₂ × (w_max/h)³ [C₁ = 49.9; C₂ = 84.7 for ν = 0.3] At w_max = h: linear term underestimates actual pressure by factor ~2.7

Thermal Bending

Temperature Gradient Through Thickness

Linear temperature gradient ΔT = T_top - T_bottom: Thermal moment: M_T = E × α_T × ΔT × h² / (12 × (1-ν)) = α_T × ΔT × D / h [per unit width] Equivalent load: q_T = ∇²M_T = 0 (if uniform thermal gradient → bending without load equivalent)

SSSS plate with thermal gradient: w_thermal = M_T × [a² + b²] / (D × π² × ...) [Navier series; similar to concentrated moment solution] In-plane thermal stresses (no bending gradient, uniform T change): σ = -E × α × ΔT / (1-ν) (fully restrained)

Mindlin-Reissner Thick Plate

When to use: h/a > 0.1; shear deformation significant Mindlin: adds ψ_x, ψ_y (independent rotation variables); shear correction factor κ = 5/6 Three equations instead of one biharmonic; natural BC on free edges differs from Kirchhoff

Effect on deflection: w_Mindlin = w_Kirchhoff × (1 + C × h²/a²) [shear correction; C ≈ 6(1-ν)/5 for SSSS] Significant when h/a > 0.2: up to 20% more deflection than Kirchhoff

Yield-Line Theory (Plastic Collapse)

Upper Bound Plastic Analysis

Yield-line: straight or curved lines of maximum moment (M_p) in plate at collapse Pattern assumed → virtual work equation → collapse load

Square plate SSSS under central point load: m_p = M_p × h (or σ_y × h²/4 for rectangular stress block) Yield-line pattern: 4 triangular panels meeting at center (yield line diagonal + central point) Virtual work: P × δ = Σ m_p × l_yl × θ_panel → P_collapse = 8 × m_p [for square plate, SSSS]

For uniform load SSSS square plate: q_collapse = 24 × m_p / a² [from corner-supported and edge yield lines]

CCCC square plate: q_collapse = (m_p + m_p') × ... → higher collapse load; edge moments add resistance

Standards and References

Standard	Scope
Timoshenko & Woinowsky-Krieger "Theory of Plates and Shells"	Classic reference; complete tables of deflection and moment coefficients
ASME VIII Div. 1 UG-34	Flat plate heads: t = d√(CP/SE) [flat plate design pressure]
AISC 360	Steel plate design including flexure
Roark's Formulas for Stress and Strain	Condensed plate tables for common cases
ASME BPVC Div. 2 App. O	Perforated plates; equivalent solid plate properties

Output

Provide: plate geometry (a × b [m]; h [mm]; a/b ratio), material (E [GPa]; ν; σ_y [MPa]), boundary conditions (SSSS/CCCC/SCSC/etc.), loading (uniform q₀ [Pa] or point P [N] and location), solution method (Navier/Lévy/Mindlin/FEA), maximum deflection w_max [mm] (Timoshenko coefficient α; position), moments M_x_max and M_y_max [N·m/m] (position; Timoshenko coefficient β), bending stress σ_max [MPa] = 6M_max/h² (at top/bottom surface; compare to σ_y and allowable), large-deflection check (w_max/h ratio; if > 0.2 → von Kármán nonlinear; q_nonlinear vs. q_linear [% difference]), thermal bending (if applicable: ΔT [°C]; thermal moment M_T; curvature), plasticity (collapse load P_c or q_c from yield-line; safety factor vs. design load), thick plate correction (Mindlin; h/a ratio; deflection correction [%]), and applicable reference (Timoshenko, ASME UG-34, Roark's).