| name | column-buckling |
| description | Column buckling analysis — Euler, Johnson parabolic, AISC LRFD/ASD, effective length, slenderness ratio, combined axial+bending (beam-columns), local buckling, lateral-torsional buckling. |
| metadata | {"priority":7,"promptSignals":{"phrases":["buckling","column","Euler","slenderness","effective length","critical load","lateral buckling"],"minScore":4}} |
Column Buckling — Complete Skill
Effective Length
KL = effective length (K = effective length factor)
| End Conditions | K (theoretical) | K (design) |
|---|
| Pin-pin | 1.0 | 1.0 |
| Fixed-free (flagpole) | 2.0 | 2.1 |
| Fixed-pin | 0.7 | 0.8 |
| Fixed-fixed | 0.5 | 0.65 |
Slenderness Ratio
λ = KL/r where r = √(I/A) = radius of gyration
Check both axes: x-x and y-y (minor axis usually governs)
Euler Formula (elastic buckling — long columns)
P_cr = π²EI/(KL)² = π²EA/(KL/r)²
σ_cr = π²E/(KL/r)²
Validity: KL/r > λ_c (slenderness transition)
λ_c = π√(E/Sy) [where Euler is valid]
For steel (Sy=250MPa, E=200GPa): λ_c = π√(200000/250) = 89
Johnson Parabola (intermediate columns — Shigley)
Valid for λ < λ_c:
σ_cr = Sy - (Sy/(2π))²·(KL/r)²/E = Sy[1 - (KL/r)²·Sy/(4π²E)]
P_cr = σ_cr · A
Design equation (Shigley):
n = P_cr/P (n = 1.25-3.0 depending on loading uncertainty)
AISC LRFD Column Design (Specification E2-3)
Elastic: λ_c ≤ 1.5 → F_cr = (0.658^(λ_c²))·F_y
Inelastic: λ_c > 1.5 → F_cr = (0.877/λ_c²)·F_y
Where λ_c = (KL/rπ)·√(F_y/E)
φ_c = 0.90 (LRFD)
φ_c·P_n = φ_c·F_cr·A_g ≥ P_u (LRFD check)
ASD: P_n/Ω_c ≥ P_a (Ω_c = 1.67)
Local Buckling (Width-to-Thickness Limits)
Flange of W-shape: b/t ≤ 0.56√(E/F_y) (compact)
Web of W-shape: h/t_w ≤ 1.49√(E/F_y) (compact for uniform compression)
Tube/HSS: b/t ≤ 1.40√(E/F_y) (compact)
Round HSS: D/t ≤ 0.15E/F_y (compact)
If non-compact: use AISC reduced area A_eff
Beam-Column (Combined Axial + Bending)
AISC H1-1a (P_u/φP_n ≥ 0.2):
P_u/(φ_c·P_n) + 8/9·(M_ux/(φ_b·M_nx) + M_uy/(φ_b·M_ny)) ≤ 1.0
AISC H1-1b (P_u/φP_n < 0.2):
P_u/(2φ_c·P_n) + (M_ux/(φ_b·M_nx) + M_uy/(φ_b·M_ny)) ≤ 1.0
Amplification factor (P-delta): B₁ = C_m/(1-P_u/P_e1) ≥ 1.0
C_m = 0.6 - 0.4(M₁/M₂) [for members with no intermediate loads]
P_e1 = π²EI/(KL)² [Euler load for x-axis]
Lateral-Torsional Buckling (Beams)
For W-shapes with unbraced length L_b:
L_p = 1.76·r_y·√(E/F_y) [plastic, no LTB]
L_r = 1.95·r_ts·√(EJ/(S_x·h_o·F_y))·√(1+√(1+6.76(F_y·S_x·h_o/(EJ))²))
If L_b ≤ L_p: M_n = M_p = Z_x·F_y (full plastic moment)
If L_p < L_b ≤ L_r: M_n = M_p - (M_p-0.7F_y·S_x)·(L_b-L_p)/(L_r-L_p) [linear interpolation]
If L_b > L_r: M_n = F_cr·S_x [elastic LTB, F_cr = π²E/... from AISC Eq. F2-4]
Battened/Laced Columns
For built-up members: use modified slenderness (KL/r)_m = √((KL/r)₀² + (a/r_i)²)
a = spacing between connectors, r_i = minimum radius of gyration of component
Output
Provide: KL/r, governing axis, P_cr (Euler or Johnson), AISC F_cr, φP_n, utilization ratio, governing failure mode (global buckling, local buckling, or LTB).
