| name | plate-shell |
| description | Thin plate and shell theory — Kirchhoff plate bending, Navier/Levy solutions, pressure vessel shells, cylindrical/spherical/conical shells, buckling, Roark tables. |
| metadata | {"priority":6,"promptSignals":{"phrases":["plate theory","thin plate","Kirchhoff","shell theory","plate bending","cylindrical shell","spherical shell","Roark"],"minScore":4}} |
Thin Plate & Shell Theory — Complete Skill
Kirchhoff Plate Theory Assumptions
- Plate is thin: h/L < 0.1 (h = thickness, L = span)
- Deflections small: w < h/5
- Mid-plane stress-free in bending (neutral surface)
- Normals remain normal after deformation (no shear deformation)
- Plane stress: σ_z = 0
Governing Equation
∇⁴w = q/D
Where:
- w(x,y) = deflection [m]
- q(x,y) = distributed load [N/m²]
- D = Eh³/(12(1-ν²)) = plate flexural rigidity [N·m]
- ∇⁴ = biharmonic operator = ∂⁴/∂x⁴ + 2∂⁴/∂x²∂y² + ∂⁴/∂y⁴
Plate Bending Moments
M_x = -D(∂²w/∂x² + ν∂²w/∂y²)
M_y = -D(∂²w/∂y² + ν∂²w/∂x²)
M_xy = -D(1-ν)∂²w/∂x∂y
Maximum stress: σ_x = 12M_x×z/h³, max at z = h/2 → σ_max = 6M_x/h²
Navier Solution (SSSS — All Edges Simply Supported)
Double Fourier series:
w(x,y) = ΣΣ (a_mn × sin(mπx/a) × sin(nπy/b))
a_mn = q_mn / [π⁴D(m²/a² + n²/b²)²]
For uniform load q₀:
q_mn = 16q₀/(π²mn) for m,n odd; = 0 otherwise
Max deflection (center, a = b = L, SSSS):
w_max ≈ 0.00406 × q×a⁴ / D (Roark)
Roark Key Results (Rectangular Plates)
| BC | Loading | w_max | M_max |
|---|
| SSSS | Uniform q | 0.00406 qa⁴/D | 0.0479 qa² (at center) |
| CCCC | Uniform q | 0.00126 qa⁴/D | 0.0231 qa² (at edges) |
| SSCC | Uniform q | 0.00220 qa⁴/D | varies |
| SCSС | Uniform q | 0.00192 qa⁴/D | — |
(a = short side; Roark Table 11.4 for complete tables)
Circular plate (radius R):
Clamped, uniform load: w_max = qR⁴/(64D) at center; σ_max = 3qR²/(4h²) at edge
Simply supported: w_max = qR⁴(5+ν)/(64D(1+ν))
Shell Theory — Membrane Analysis
For thin shells (R/t > 10), membrane theory neglects bending:
Meridional stress: σ_φ = pR/(2t) (for sphere)
Hoop stress: σ_θ = pR/t (for cylinder)
Cylinder under internal pressure p:
σ_hoop = pR/t, σ_long = pR/2t (σ_hoop = 2×σ_long)
Design: t = pR/(η×SE - 0.6p) [ASME VIII Div. 1, Eq. UG-27]
SE = allowable stress × weld joint efficiency E_joint
Sphere: σ = pR/2t (equal biaxial)
t = pR/(2SE - 0.2p) [ASME VIII]
Conical shell: σ_hoop = pR/(t×cosα), where α = half-angle, R = local radius
Cone-cylinder junction requires reinforcement ring (discontinuity analysis)
Bending of Cylindrical Shells (Discontinuity)
At edges/junctions: bending moment M and shear Q appear (boundary effects)
Characteristic length: β = [3(1-ν²)/(R²t²)]^(1/4)
Decay: exponential × sinusoidal, decays in distance ~3/β from edge
Edge effects significant within distance ≈ 4/β (about one "wavelength")
Edge loading of cylinder (at x=0):
w = (Q₀/2βD)e^(-βx)cos(βx) + (M₀/βD)e^(-βx)[cos(βx)-sin(βx)]/2
M_x = (Q₀/β)e^(-βx)sin(βx) + M₀e^(-βx)[cos(βx)+sin(βx)]
Buckling of Plates
Rectangular plate, uniaxial compression:
N_cr = k × π²D/b²
Buckling coefficient k:
SSSS: k = (m×b/a + a/(m×b))² → minimum k = 4 (for a/b = integer)
CCCC: k = 6.97
CSCS (load edges clamped): k ≈ 7.7
Shear buckling: τ_cr = k_s × π²D/b², k_s ≈ 5.35 (SSSS), 8.98 (CCCC)
Buckling of Cylindrical Shells
External pressure (long cylinder):
p_cr = 2E/(1-ν²) × (t/D)³ (Euler-like, very sensitive to imperfections)
Axial compression:
σ_cr = 0.6E(t/R) (classical — but real shells buckle at 20-60% of this!)
Knockdown factor γ = 0.5-0.7 for design (NASA SP-8007)
Torsion: q_cr = 0.272E(t/R)^(3/2)/(L/R)^(1/2)
Mindlin-Reissner (Thick Plates)
When h/L > 0.1: include transverse shear deformation
Modification: add 1.2/(Gh) term to flexibility (shear correction k_s = 5/6 for rect. section)
Software: ABAQUS S8R/S8R5 for thin, S8R for thick plate elements
Output
Provide: flexural rigidity D [N·m], max deflection w_max [mm], bending stress σ_max [MPa], shell hoop/meridional stress, buckling load.
