Lateral-torsional buckling — general method
Verification through M_cr, to EN 1993-1-1 clauses 6.3.2.2 and 6.3.2.3. Applies to any beam, including those without closely spaced lateral restraints.
Parameters of M_cr
▸ Fill in the data on the left and press Calculate
Tool information
What this calculator computes
The tool verifies a beam against lateral-torsional buckling by the general method, to EN 1993-1-1 clauses 6.3.2.2 and 6.3.2.3:
\(M_{cr} \rightarrow \bar{\lambda}_{LT} \rightarrow \chi_{LT} \rightarrow M_{b,Rd}\)
How it differs from the other LTB page
The bending page uses the simplified method of clause 6.3.2.4: the criterion on the compression-flange slenderness, \(\bar{\lambda}_f\). That one never goes through \(M_{cr}\) and applies to beams in buildings with closely spaced lateral restraints.
The method here applies to any beam, including those without intermediate restraints. It takes more input, but it is the only one available when the simplified method does not qualify.
$M_
The code requires the elastic critical moment but does not give its expression. The one used here is the classical form for doubly symmetric sections, from ENV 1993-1-1 Annex F:
\(M_{cr} = C_1 \frac{\pi^2 E I_z}{(k_z L)^2}\left[\sqrt{\left(\frac{k_z}{k_w}\right)^2 \frac{I_w}{I_z} + \frac{(k_z L)^2 G I_T}{\pi^2 E I_z} + (C_2 z_g)^2} - C_2 z_g\right]\)
The practical consequence: the result depends on a source outside the code. If you check against other software and get something different, the first question is which \(M_{cr}\) formula that one uses — not necessarily which of you is wrong.
The \(C_1\) factor: how much the diagram matters
\(M_{cr}\) is linear in \(C_1\). Uniform moment, the most unfavourable case, gives \(C_1 = 1.0\). A triangular diagram easily exceeds 1.7 — nearly double the critical capacity, from nothing but the shape of the diagram.
The page can derive it from the moments at five equidistant points:
\(C_1 = \sqrt{\frac{35 M_{max}^2}{M_{max}^2 + 9M_2^2 + 16M_3^2 + 9M_4^2}}\)
\(M_{max}\) is the true maximum over the span, which may fall between the sampling points. If you miss it, \(C_1\) comes out smaller — hence conservative, since \(C_1\) increases monotonically with \(M_{max}\). It is an error in the safe direction, but worth knowing which way it goes.
Two clauses, easily confused
Clause 6.3.2.2 "general case" and clause 6.3.2.3 "rolled sections or equivalent welded sections" are not variants of one another:
| General case (6.3.2.2) | Rolled sections (6.3.2.3) | |
|---|---|---|
| Curves | Table 6.4 | Table 6.5 |
| IPE with \(h/b \le 2\) | curve a (\(\alpha = 0.21\)) | curve b (\(\alpha = 0.34\)) |
| \(\bar{\lambda}_{LT,0}\) | 0.2 (fixed in the formula) | from the national annex (0.4) |
| \(\beta\) | 1.0 | from the national annex (0.75) |
| Extra cap | — | \(\chi_{LT} \le 1/\bar{\lambda}_{LT}^2\) |
| Factor \(k_c\) | not applicable | clause 6.3.2.3(2) |
Applying the wrong curve moves \(\chi_{LT}\) by several percent. The page always shows which curve it used.
The cap \(\chi_{LT} \le 1/\bar{\lambda}_{LT}^2\) is the most frequently forgotten one and governs on slender beams — the page flags when it governs rather than the formula.
\(k_c\) does not always help
\(f = 1 - 0.5(1-k_c)\left[1 - 2(\bar{\lambda}_{LT} - 0.8)^2\right], \quad f \le 1.0\)
The bracket turns negative once \(\bar{\lambda}_{LT}\) moves away from 0.8 by more than \(1/\sqrt{2} \approx 0.707\). Then \(f\) would exceed 1 and is capped, and \(k_c\) has no effect at all. So \(k_c\) is a correction for beams of moderate slenderness, not a general reduction.
Input data
- Beam — section, steel grade, section class, length between lateral restraints, \(M_{y,Ed}\).
- Method — general case or rolled sections, rolled or welded section, \(k_c\).
- \(C_1\) — directly, or from the moment diagram at five points plus the true maximum.
- \(M_{cr}\) parameters — \(k_z\), \(k_w\), \(z_g\) and \(C_2\).
- National annex — \(\gamma_{M0}\), \(\gamma_{M1}\), \(\gamma_{M2}\); \(\bar{\lambda}_{LT,0}\) and \(\beta\) come from there too.
Assumptions and limitations
Covered: doubly symmetric sections, major-axis bending, both clauses 6.3.2.2 and 6.3.2.3.
Not covered: monosymmetric and thin-walled sections; \(C_2\) and \(C_3\) are not selected automatically from the load type but entered; Class 4 uses \(W_{el}\) instead of \(W_{eff}\), which overestimates the resistance and is flagged as a note; \(k_z\) and \(k_w\) are not derived from the support conditions.
Related calculators
- LTB — simplified method — clause 6.3.2.4, for beams with close restraints
- Buckling — the same \(\bar{\lambda} \rightarrow \chi\) mechanism, in compression
- Bending and compression — where \(\chi_{LT}\) enters the interaction