How to calculate the critical buckling load of a column: the axial load that triggers buckling of a slender column is given by Euler's formula P_cr = π²·E·I/(K·L)². Factor K depends on end conditions: K=1 pinned-pinned, K=0.5 fixed-fixed, K=0.7 fixed-pinned, K=2 cantilever. The formula applies only in the elastic regime (slenderness λ > λ_lim).

Formula and physical meaning

Euler's critical load equation, derived in 1744, applies to ideal slender columns: perfectly straight, linear-elastic material, slender enough to fail elastically. The effective length L_e = K·L accounts for the buckling mode shape. Buckling always occurs about the axis of minimum moment of inertia I_min: wide-flange sections (W, IPE, HEA) buckle about the weak axis unless laterally braced. Critical stress σ_cr = P_cr/A must not exceed yield f_y. AISC 360-22 §E3 and Eurocode EN 1993-1-1 §6.3 codify the check with χ-λ̄ buckling curves.

History and theory

Leonhard Euler published the formula in 1744 from the elastic-line ODE solution. Lagrange extended the concept. Engesser (1889) introduced the tangent-modulus formula for the inelastic regime. Tetmajer and Johnson developed empirical fits. Shanley (1947) resolved the inelastic paradox. Eurocode 3 was first published in 1992 and codifies 5 buckling curves a₀, a, b, c, d with imperfection factors α_imp from 0.13 to 0.76. AISC 360 in the US and CISC in Canada use equivalent formulations.

K factor by end conditions

• Pinned-pinned (base case): K = 1.0; AISC recommended 1.0
• Fixed-fixed: K = 0.5; AISC recommended 0.65
• Fixed-pinned: K = 0.7; AISC recommended 0.80
• Fixed-free (cantilever): K = 2.0; AISC recommended 2.10
• Fixed-fixed with sidesway: K = 1.0
• Fixed-pinned with sidesway: K = 2.0
• Unbraced frame: K from AISC alignment chart based on G_top, G_bottom rigidity ratios

Slenderness and buckling regime

Slenderness ratio λ = L_e/r where r = √(I/A) is the radius of gyration. Transition slenderness λ_lim = π·√(E/f_y) separates elastic (λ > λ_lim, Euler valid) from inelastic buckling (λ < λ_lim, yielding precedes buckling). For ASTM A36 (f_y ≈ 250 MPa): λ_lim ≈ 91. For ASTM A992 (f_y = 345 MPa): λ_lim ≈ 78. AISC 360-22 uses transition λ_e = 4.71·√(E/f_y). British Steel S275: λ_lim ≈ 86; S355: λ_lim ≈ 76.

Worked examples

Example 1 — IPE100 pinned column. L=3 m, IPE100 I_min = 1.59·10⁻⁶ m⁴ (3.82 in⁴), A = 1.03·10⁻³ m² (1.6 in²), steel A992. P_cr = 366 kN (82 kip). σ_cr = 355 MPa exceeds f_y=345 MPa → Euler invalid, inelastic regime. AISC 360 §E3 with F_e = 410 MPa: σ_cr = 0.658^(345/410)·345 = 246 MPa, P_n = 253 kN.

Example 2 — fixed-fixed pipe. D=100 mm (4 in), t=5 mm. A = 1,492 mm² (2.31 in²), I = 1.69·10⁶ mm⁴ (4.06 in⁴). L=4 m, K=0.5, L_e=2 m. P_cr = 876 kN (197 kip). r = 33.7 mm, λ = 59, inelastic.

Example 3 — cantilever mast. Pipe D=80 mm, t=4 mm. A=955 mm², I=6.95·10⁵ mm⁴. L=5 m, K=2, L_e=10 m. P_cr = 14.4 kN (3.2 kip). λ=372, well into elastic regime.

Example 4 — concrete column. 12×12 in (30×30 cm), A=0.09 m², I=6.75·10⁻⁴ m⁴. E_cm=31 GPa for C25/30. L=3 m fixed-fixed, K=0.5. P_cr=91.7 MN. Strength governs over global buckling for typical concrete columns.

Example 5 — truss diagonal. Angle L80×80×8 (3×3×5/16 in), I_min=73·10⁴ mm⁴, A=1,230 mm². L=2 m, K=1. P_cr=378 kN. λ=82, near transition.

Limits of Euler's hypothesis

1. Geometric imperfections: e_0/L of 1/300 to 1/1,000 cuts effective load 10-30%.
2. Residual stresses: differential cooling during rolling introduces self-stresses up to 0.3·f_y.
3. Load eccentricity: small alignment errors amplify bending and reduce margin.
4. Inelastic behavior: for λ < λ_lim material yields before buckling.
5. Local buckling: thin sections (AISC slender, EC3 Class 4) need effective area.
6. Lateral-torsional buckling: open sections under flexure too.
7. Second-order effects: P-delta in flexible structures.
8. Creep: timber and concrete reduce E over time.

Comparison with other formulas

Euler P_cr = π²·EI/L_e² applies for λ > λ_lim only. Intermediate slenderness (40 < λ < 110 for mild steel) uses Johnson's parabola σ_cr = f_y·[1 - (λ/λ_lim)²/4]. AISC §E3: σ_cr = 0.658^(f_y/F_e)·f_y when F_e ≥ 0.44 f_y, else 0.877·F_e (with F_e = π²·E/λ²). Eurocode 3 picks one of 5 curves based on section shape. CSA S16 in Canada uses similar but slightly different curve definitions.

Practical applications

1. Building columns: steel or RC, sized for SLS and ULS.
2. Trusses and bracing: compression members K=1 (in-plane) or K=2 (out-of-plane).
3. Antennas and telecom towers: cantilever K=2, wind dominated.
4. Aerospace: wing spars in compression during landing.
5. Engine connecting rods: limit dictating minimum section.
6. Earth retention struts: sized with K and f_y.
7. Ship structural framing: longitudinal stiffeners.
8. Tension structures: compressed posts when stiffness needed.
9. Silos and tanks: cylindrical shell buckling (more complex theory).
10. Wind turbine towers: combined local and global buckling.

Support

Questions: info@calcoi.com. Educational tool. For real projects consult a licensed PE and apply appropriate factors per AISC 360-22, EN 1993, or local code.

Updated July 2026