Euler Column Buckling Calculator

Buckling is the failure of a slender column that suddenly bows under axial compressive load, long before the material yields. The Euler critical load: P_cr = π²·E·I / (K·L)², where E is the elastic modulus, I the cross-section moment of inertia (weak axis), L the column length, and K the effective length factor from the end conditions. Its critical stress σ_cr = P_cr / A.

Circular cross-section d = 50 mm: I = π·d⁴/64 = 3.068×10⁻⁷ m⁴; A = 1,963.5 mm²; r = √(I/A) = 12.5 mm. With steel E = 200 GPa, L = 2 m, K = 1.0: P_cr = π² × 200×10⁹ × 3.068×10⁻⁷ / 2² = 151.4 kN. Critical stress σ_cr = 151.4 kN / 1,963.5 mm² = 77.1 MPa. Slenderness ratio λ = KL/r = 160, above the steel λ_cr limit of 125.7, so the Euler formula is valid.

K describes the column end conditions: K = 0.5 for fixed-fixed (both ends fixed), K = 0.7 for fixed-pinned, K = 1.0 for pinned-pinned (the standard case), and K = 2.0 for one free end and one fixed end (cantilever). A larger K means a longer effective length KL and a smaller critical load — a cantilever is 4 times weaker than pinned-pinned at the same length.

The slenderness ratio λ = KL/r, where r = √(I/A) is the radius of gyration. The Euler formula is only valid for slender columns, i.e., λ ≥ λ_cr, with λ_cr = √(2π²E/σ_y). Typical λ_cr values: steel (200 GPa, 250 MPa) = 125.7; aluminum 6061-T6 (69 GPa, 240 MPa) = 75.3; wood (12 GPa, 25 MPa) = 97.3; concrete (25 GPa, 30 MPa) = 128.3. If λ < λ_cr (short/intermediate column), the calculator shows a warning and Euler must not be used — use the Johnson formula or consult a structural engineer.

A column always buckles about the weak axis (smallest moment of inertia), so the calculator automatically uses the smallest I. Example: 80×120 mm steel cross-section, L = 3 m, K = 1.0: I_weak = 120×80³/12 = 5.12×10⁻⁶ m⁴, r = 23.1 mm, λ = 129.9 (> 125.7, Euler valid), P_cr = 1,122.9 kN, σ_cr = 117.0 MPa.

The I-beam mode uses a three-rectangle approximation (two flanges + one web) about the weak axis and ignores fillets and corner radii, so results are a conservative estimate within ±5–10% of standard profile tables. Example I200 (100×12 mm flanges, 8 mm web), steel, L = 4 m, K = 1.0: I_weak ≈ 2.008×10⁻⁶ m⁴, λ = 174.2, P_cr = 247.7 kN. For final design, use I values from the manufacturer's profile catalog.

Deliberate and honest: the safety factor depends on the applicable design code (e.g., AISC, SNI, Eurocode), load types, and failure consequences — not a universal constant. This calculator gives the theoretical critical load P_cr; you apply the safety factor per your project's standard (typically 1.67–2.0+ for steel columns). Never use P_cr as the allowable load.

No — and the calculator will warn you. Example: 300×300 mm concrete column, L = 3 m: r = 86.6 mm, λ = 34.6, far below the concrete λ_cr of 128.3, so Euler is not valid; the computed critical stress is 205.6 MPa anyway, exceeding the 30 MPa concrete compressive strength. Short/intermediate concrete columns are designed with reinforced-concrete column methods (interaction diagrams), not Euler.

Input: cross-section dimensions in millimeters (mm), column length in meters (m), elastic modulus in gigapascals (GPa), material strength in megapascals (MPa). Results: critical load in kilonewtons (kN), stress in MPa, area in mm², moment of inertia in m⁴ (scientific notation), radius of gyration in mm.

This calculator computes column buckling under axial compressive load (Euler formula). If you need support reactions, shear force, and bending moment on a transversely loaded beam, use our beam load calculator.

No. This is an educational and initial-estimate tool: Euler assumes a perfectly straight column, perfectly concentric load, and linear elastic material — ideal conditions that never exist in the field. Load eccentricity, initial imperfections, residual stresses, and local buckling can significantly reduce real capacity. Structural design must be verified by a licensed structural engineer per the applicable code.