elastic-stability

name	elastic-stability
description	Elastic stability beyond columns — shell buckling, lateral-torsional buckling of beams, torsional-flexural buckling, combined loading interaction, imperfection sensitivity, AISC/AISI/Eurocode.
metadata	{"priority":7,"promptSignals":{"phrases":["lateral torsional buckling","LTB","torsional buckling","beam buckling","flexural torsional buckling","shell buckling","imperfection sensitivity"],"minScore":3}}

Elastic Stability — Complete Skill

Lateral-Torsional Buckling (LTB) of Beams

Elastic LTB Critical Moment (Doubly Symmetric I)

M_cr = (π/L_b) × √(E I_y × G J) × √(1 + (π E C_w / (G J L_b²)))

I_y = weak-axis moment of inertia J = St. Venant torsional constant = Σ b t³/3 (for open sections) C_w = warping torsional constant = I_y h_o²/4 (for I-sections; h_o = distance between flange centroids) G = shear modulus (79 GPa steel); E = 200 GPa; L_b = unbraced length

Compact-noncompact-slender classification: L_p = 1.76 r_y √(E/F_y) [fully plastic; no LTB] L_r = 1.95 r_ts (E/0.7F_y) √(J c / (S_x h_o) + √((J c)² / (S_x h_o)² + 6.76(0.7F_y/E)²))

r_ts = √(√(I_y C_w) / S_x) (effective radius for LTB)

AISC 360 Nominal Moment: L_b ≤ L_p: M_n = M_p = F_y Z_x (full plastic, no LTB) L_p < L_b ≤ L_r: M_n = C_b [M_p - (M_p - 0.7F_y S_x)(L_b - L_p)/(L_r - L_p)] ≤ M_p (linear interpolation) L_b > L_r: M_n = F_cr S_x ≤ M_p (elastic LTB)

C_b = moment gradient factor = 12.5 M_max / (2.5 M_max + 3 M_A + 4 M_B + 3 M_C) ≥ 1.0 C_b > 1.0 for non-uniform moments; C_b = 1.0 for uniform moment

Torsional-Flexural Buckling (Open Section Thin-Wall)

Critical for: channels, angles, tees, hat sections (no shear center at centroid)

Governing equation (3 coupled modes): Coupling between flexural (x,y) and torsional modes → interaction eigenvalue problem Simplified for doubly symmetric I: torsional mode = pure torsion or LTB (already covered)

For singly symmetric (channels, tees): critical mode from: [P_cr - P_eX][P_cr - P_eY][P_cr - P_ez] × r₀² - P_cr²(P_cr - P_eY) x₀² - P_cr²(P_cr - P_eX) y₀² = 0

P_eX, P_eY = Euler buckling loads about X, Y axes P_ez = π² E C_w / (K_z L)² + G J / r₀² (torsional mode) r₀ = √(r_x² + r_y² + x₀² + y₀²); (x₀, y₀) = shear center location

Shell Buckling (Cylindrical Shell Under Axial Compression)

Classical critical stress (Lorenz-Timoshenko): σ_cr = E/√3(1-ν²) × t/R (classical) = 0.605 E t/R (for ν = 0.3)

Knock-down factor γ (empirical — huge imperfection sensitivity): σ_cr,actual = γ × σ_cr,classical

γ = 1 - 0.901(1 - e^(-φ)) where φ = (1/16)√(R/t) For R/t = 100: γ ≈ 0.7; for R/t = 1000: γ ≈ 0.25

NASA SP-8007: recommended knock-down factors for aerospace shells Current NASA-STD-7001B: updated with stochastic imperfection approach

Shell buckling under combined loading: Axial + pressure: RS (interaction relationship from data) Axial + torsion: (σ_a/σ_cr,a)² + (τ/τ_cr)² ≤ 1 (approximately)

Imperfection Sensitivity

Geometric imperfections dramatically reduce actual buckling load for:

• Cylindrical shells (very sensitive; knock-down 0.2–0.8)
• Spherical shells (very sensitive; knock-down 0.2–0.5)
• Columns (moderate; Euler valid for small e)
• Flat plates (less sensitive; significant post-buckling strength)

Measurement: amplitude δ_0/R; wavelength as fraction of R Monte Carlo / stochastic FEM: sample imperfection fields → probabilistic buckling load distribution

Combined Loading Interaction

Column + bending (AISC H1-1): P_r/P_c + (8/9)(M_rx/M_cx + M_ry/M_cy) ≤ 1.0 (for P_r/P_c ≥ 0.2) P_r/(2P_c) + (M_rx/M_cx + M_ry/M_cy) ≤ 1.0 (for P_r/P_c < 0.2)

Beam-column buckling (Perry-Robertson): Used in BS 5950 and Eurocode 3 EN 1993-1-1 χ = 1 / (Φ + √(Φ² - λ̄²)) Φ = 0.5[1 + α(λ̄ - 0.2) + λ̄²] α = imperfection factor (0.13 curve a to 0.76 curve d depending on section) λ̄ = √(F_y/σ_cr) = relative slenderness

Torsion of Open Thin-Walled Sections

St. Venant (uniform) torsion: T = G J dφ/dz (dominant for stocky sections) Warping (non-uniform) torsion: T = -E C_w d³φ/dz³ (dominant for slender) Combined: GJ dφ/dz - E C_w d³φ/dz³ = m_z(z)

Solution involves hyperbolic functions; boundary conditions key Torsional parameter: λ_t = √(G J / E C_w) [1/m]

Output

Provide: M_n [kN·m] and φM_n (LRFD), C_b, L_p, L_r, L_b (actual), LTB failure mode (no LTB/inelastic/elastic), shell knock-down factor γ, shell critical stress [MPa], beam-column interaction ratio, combined loading unity check.