EC3 Flexural Buckling (Buckling Curves)
Run the flexural buckling check of compression members using the equivalent-member method of EN 1993-1-1 (clause 6.3.1). Enter cross-section properties, buckling length, yield strength and buckling curve - the calculator returns the elastic critical load N_cr, the non-dimensional slenderness, the reduction factor χ and the buckling resistance N_b,Rd together with the utilisation N_Ed/N_b,Rd and a traffic-light rating, live with every input.
Calculation
χ buckling-curve method to EC3
Per EN 1993-1-1, 6.3.1 the resistance is obtained from the reduction factor χ = 1/(Φ + √(Φ² − λ_bar²)) ≤ 1.0 with Φ = 0.5·[1 + α·(λ_bar − 0.2) + λ_bar²]. The imperfection factor α (a0 = 0.13 … d = 0.76) accounts for initial bow and residual stresses. N_b,Rd = χ·A·f_y/γ_M1 with γ_M1 = 1.0.
Model: pure flexural buckling of straight, concentrically loaded compression members using the equivalent-member method of EN 1993-1-1 (6.3.1). Lateral-torsional buckling, torsional buckling, local buckling and the bending-and-buckling interaction of clause 6.3.3 are not included. The Euler load N_cr is textbook-based; the χ method, α values and γ_M1 are EC3 code values.
Results
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Formulas and fundamentals
Critical load and slenderness
The basis is the elastic critical (Euler) load:
The non-dimensional slenderness is:
It measures the susceptibility to stability failure independently of the absolute load level.
Reduction factor χ
The real resistance falls below the Euler load because of imperfections (initial bow, residual stresses). The equivalent-member method captures this via the reduction factor:
The imperfection factor α depends on the buckling curve: a0 = 0.13, a = 0.21, b = 0.34, c = 0.49, d = 0.76 (EN 1993-1-1, Table 6.1). For λ_bar ≤ 0.2 χ = 1.0 - buckling effects may be neglected.
Buckling resistance and check
The buckling resistance is:
The check requires N_Ed ≤ N_b,Rd, expressed as the utilisation η = N_Ed/N_b,Rd ≤ 1.0. The Euler critical load N_cr is a classical textbook result; the χ buckling-curve method with the factors α and γ_M1 is Eurocode 3 code content.
Worked example
A member of S235 (f_y = 235 N/mm²) with A = 1000 mm², I = 1·10⁶ mm⁴ and buckling length L_cr = 2000 mm is checked with buckling curve b (α = 0.34). The critical load is N_cr = π²·210000·10⁶/2000² = 518,154 N.
This gives the non-dimensional slenderness λ_bar = √(1000·235/518154) = 0.673. The auxiliary value is Φ = 0.5·[1 + 0.34·(0.673 − 0.2) + 0.673²] = 0.807 and the reduction factor χ = 1/(0.807 + √(0.807² − 0.673²)) = 0.799.
The buckling resistance is N_b,Rd = 0.799·1000·235/1.0 = 187,645 N ≈ 187.6 kN. A design compression force of N_Ed = 150 kN gives the utilisation η = 150/187.6 = 0.80 - the buckling check is satisfied.
Frequently asked questions
What does the non-dimensional slenderness λ_bar mean?
λ_bar = √(A·f_y/N_cr) compares the plastic limit load A·f_y with the elastic critical load N_cr. λ_bar < 0.2 means stocky (no buckling effect, χ = 1.0), λ_bar > 1 means slender (buckling governs, near-Euler resistance). It is the decisive input for the reduction factor χ.
How do I choose the correct buckling curve?
The assignment depends on cross-section shape, plate thickness, steel grade and buckling direction (EN 1993-1-1, Table 6.2). Hot-rolled I-sections buckle more unfavourably about the weak axis than the strong axis; welded and thick-walled sections sit on higher curves. A higher curve (more imperfection, larger α) yields a smaller χ and therefore a smaller resistance.
How do I determine the buckling length L_cr?
L_cr = β·L follows from the supports: pinned-pinned β = 1.0, fixed-free β = 2.0, fixed-fixed β = 0.5, fixed-pinned β ≈ 0.7. For frames and sway systems β is larger and must be derived from the overall system.
Why is γ_M1 = 1.0?
The partial factor for stability failure is set to γ_M1 = 1.0 in the German National Annex to EN 1993-1-1. The safety lies in the partial factors on the actions (N_Ed) and in the buckling curves themselves.
Which failure modes are not covered?
Only pure flexural buckling about the entered axis. Lateral-torsional buckling (sideways movement with twist), torsional buckling, local buckling of thin-walled sections and combined bending-and-buckling interaction (clause 6.3.3) are not included and must be checked separately.
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