C₁ Factor in M_cr

The elastic critical moment for LTB with moment gradient is:

M_cr = C₁ × (π² × E × I_z / L²) × √(I_w / I_z + L² × G × I_t / (π² × E × I_z))

The C₁ factor adjusts M_cr from the worst-case uniform moment (C₁ = 1.0) to account for the actual bending moment diagram. When the moment varies along the beam, the LTB resistance increases because the highest stresses are localized.

C₁ Factor Table — End Moments Only

For a beam segment with end moments M_A and M_B (ψ = M_B / M_A, where M_A is the larger end moment):

ψ = M_B / M_A C₁ (uniform section) C₁ (end restraint)
+1.00 1.00 1.00
+0.75 1.14 1.14
+0.50 1.31 1.31
+0.25 1.52 1.52
0.00 1.77 1.77
-0.25 2.06 2.06
-0.50 2.40 2.40
-0.75 2.82 2.82
-1.00 3.30 3.30

Positive ψ means double curvature (M_A and M_B produce tension on same side). Negative ψ means single curvature (opposite sides in tension). The worst case is ψ = +1.0 (uniform moment), giving C₁ = 1.0.

C₁ Factor Table — Transverse Loading

For simply supported beams with transverse loading (fork supports at ends):

Loading Type C₁ (fork supports)
Uniformly distributed load 1.12
Central point load 1.35
Two point loads at L/3 1.19
Two point loads at L/4 1.11
Triangular load (peak at midspan) 1.14

These values assume the load is applied at the shear center. If the load is applied on the top flange (destabilizing), C₁ should be reduced. If applied on the bottom flange (restoring), C₁ may be increased.

C₁ Factor Table — Combined End Moments and Transverse Load

For the common case of a simply supported beam with UDL and end moments (e.g., continuous beam):

End Moment Ratio ψ C₁ (with UDL)
+1.00 0.97
+0.50 1.08
0.00 1.23
-0.50 1.42
-1.00 1.64

Note that for ψ = +1.0 with UDL, C₁ = 0.97 (slightly less than 1.0), reflecting that the UDL adds to the already unfavorable uniform moment condition.

C₁ Equivalent Values vs AISC C_b

For designers familiar with AISC 360, the equivalent relationship is:

AISC C_b EN 1993 C₁ (equivalent) Moment Condition
1.00 1.00 Uniform moment
1.67 1.35 Central point load, S.S.
1.30 1.12 Uniform load, S.S.
2.27 1.77 Triangular moment (ψ=0)
2.40 1.77 Doubly symmetric I-section

The C₁ values are generally lower than C_b because EN 1993 applies the factor to M_cr rather than directly to the moment capacity. A direct numerical comparison depends on section geometry and slenderness.

C₁ Factor Application Example

For an IPE 300 beam (L = 5 m, S355) with uniformly distributed load:

Parameter Uniform Moment (ψ = +1.0) UDL (C₁ = 1.12)
M_cr 89.4 kN·m 89.4 × 1.12 = 100.1 kN·m
λ_LT 1.58 1.49
χ_LT 0.40 0.44
M_b,Rd 89.2 kN·m 98.1 kN·m
Improvement — +10%

While the C₁ = 1.12 improvement appears modest, for more severe moment gradients (point load or triangular moment), the improvement can exceed 50% over the uniform moment case.

C₁ for Cantilevers

Cantilevers have different C₁ factors because the support conditions and buckling modes differ:

Loading Type C₁
Cantilever — end point load 0.81
Cantilever — UDL 0.80
Cantilever — end moment 1.00

Cantilever C₁ values are below 1.0 in many cases because the buckling mode is less stable than a simply supported beam.

Frequently Asked Questions

What is the difference between C_b (AISC) and C₁ (EN 1993)?

C_b in AISC 360 is applied directly to the nominal moment capacity: M_n = C_b × M_p. C₁ in EN 1993 is applied to the elastic critical moment: M_cr = C₁ × M_cr,uniform. Both account for moment gradient effects, but they enter the calculation at different points. C_b values are typically higher than corresponding C₁ values because AISC uses it differently in the capacity equation.

What C₁ factor should I use for a simply supported beam with uniform load?

For a simply supported beam with uniformly distributed load and fork supports at both ends, use C₁ = 1.12 per EN 1993-1-1 guidance. This accounts for the parabolic moment diagram being more favorable than uniform moment.

Cb Values for Common Beam Loading Cases

For quick reference, pre-computed Cb values for simply-supported beams:

Loading Pattern Cb (approx.) Notes
Uniform load, no bracing points 1.14 Moment gradient from midspan moment
Central point load, no bracing 1.32 Higher gradient than uniform load
Equal end moments (double curvature) 1.00 M1/M2 = -1.0, uniform moment between braces
End moments, M1/M2 = 0.5 1.17 Moderate gradient
End moments, M1/M2 = 0 1.30 M1=0 at one brace point
End moments, M1/M2 = -0.5 1.10 Mild double curvature
Two point loads at L/3, no bracing 1.08 Nearly uniform moment in central third

For cantilevers, Cb = 1.0 unless the tip is laterally restrained:

Relationship Between EN 1993 and AISC Cb Factors

While EN 1993-1-1 Section 6.3.2.3 uses a modified moment factor approach (alpha_m or C1) rather than an explicit Cb, the concept is identical: both use the moment distribution between brace points to modify the elastic lateral-torsional buckling moment. For a linear moment gradient between brace points with end moment ratio psi = M_Ed,min / M_Ed,max:

EN 1993-1-1 Annex F (alternative method): C1 = 1.88 - 1.40*psi + 0.52*psi^2  (for kz=1.0, kw=1.0)
AISC 360-22 Equation F1-1: Cb = 12.5*Mmax / (2.5*Mmax + 3*MA + 4*MB + 3*MC)

Both formulas yield similar results (within ~5%) for common moment distributions. The AISC Cb equation is a curve fit to the exact C1 solution for a linear moment gradient with intermediate brace points.

EN 1993-1-1 Clause References for Lateral-Torsional Buckling

Related Pages

Educational reference only. C₁ values per EN 1993-1-1:2005 and ECCS Publication No. 119. Verify against actual support and loading conditions. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent verification.