The C1 Factor: Free Capacity in LTB Checks

Lateral-torsional buckling depends on how the compression flange is loaded along the unrestrained length. When the moment is constant, the whole flange is at full compression from end to end. When the moment peaks at one point and falls away, only part of the flange is highly stressed, and the beam is harder to buckle.

Where C1 enters

For a simply supported beam with the load applied at the shear centre and no warping restraint, the elastic critical moment takes the form:

$$M_{cr} = C_1\,\frac{\pi^2 E I_z}{L^2}\sqrt{\frac{I_w}{I_z} + \frac{L^2 G I_t}{\pi^2 E I_z}}$$

The part after $C_1$ is the critical moment for uniform bending. $C_1$ scales it for the actual moment diagram, so $C_1 = 1.0$ is the uniform moment case and every other shape gives a value above 1.

Values worth remembering

For a simply supported beam with free end rotation:

For end moments and continuous beams, take $C_1$ from a published table or a program that states its source, since several approximations exist.

What it does to the check

The non-dimensional slenderness is

$$\bar\lambda_{LT} = \sqrt{\frac{W_y f_y}{M_{cr}}}$$

so a larger $M_{cr}$ lowers the slenderness by the factor $1/\sqrt{C_1}$: about 0.94 for a uniformly loaded beam and 0.86 for a central point load. On the steep part of the buckling curve, that is a noticeable gain in the reduction factor $\chi_{LT}$.

EN 1993-1-1 §6.3.2.3 adds a second, separate allowance for rolled sections through the factor $f$, which uses a correction factor $k_c$ from Table 6.6: 1.0 for uniform moment, 0.94 for a uniformly distributed load and 0.90 for a central point load. Use it where your National Annex allows.

Two cautions

• $C_1$ assumes the load acts at the shear centre. A load on the top flange is destabilising and lowers $M_{cr}$; a load hung from the bottom flange raises it. Formulations with a $C_2$ term handle this.

• $C_1$ describes the moment shape between lateral restraints. Change where the restraints are and the segment, its length and its $C_1$ all change.

Takeaways

• Uniform moment is the worst case; most real beams are better.

• Use 1.13 for a uniformly loaded simply supported beam and 1.35 for a central point load.

• Mind the load height: top flange loading reduces the benefit.

• Check the §6.3.2.3 $k_c$ allowance separately, and only as your annex permits.

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