Lateral-torsional buckling

What this calculator checks

The calculator verifies the overall stability of a beam in bending — loss of stability by lateral-torsional buckling — using the simplified equivalent compression flange method, EN 1993-1-1 §6.3.2.4.

The idea: if the compression flange, seen as a compression member in its own right, is slender enough, then the beam cannot buckle laterally, and the more laborious general check §6.3.2.1 becomes unnecessary. Condition (6.59):

\(\bar\lambda_f = \frac{k_c \, L_c}{i_{f,z} \, \lambda_1} \le \bar\lambda_{c0} \cdot \frac{M_{c,Rd}}{M_{y,Ed}}\)

When the condition is met, the beam develops its full bending capacity \(M_{c,Rd}\), with no reduction for instability.

Why lateral buckling occurs

A beam in bending has its compression flange on one face and its tension flange on the other. The compression flange tends to buckle sideways, exactly like a compression member — but it is held by the web and the tension flange, so buckling appears as a combination of lateral displacement and twist of the whole section.

That is why the spacing of the lateral restraints \(L_c\) matters: not the length of the beam, but the interval over which the compression flange is free to move sideways. A connected concrete slab or closely spaced purlins shorten \(L_c\) and can remove the problem entirely.

The quantities of the method

• \(i_{f,z}\) — the radius of gyration of the equivalent compression flange (the flange plus one sixth of the web) about the minor axis; \(i_{f,z} = \sqrt{I_{f,z}/A_{f,z}}\).
• \(k_c\) — a factor accounting for the shape of the moment diagram over \(L_c\). A constant moment is the most unfavourable (\(k_c = 1.0\)); a triangular or parabolic diagram is more favourable, with \(k_c < 1\).
• \(\bar\lambda_{c0} = \bar\lambda_{LT,0} + 0.1 = 0.5\) — the slenderness threshold below which instability does not reduce the capacity.
• \(M_{c,Rd}\) — the section moment resistance, with \(W_{pl}\) for classes 1–2 and \(W_{el}\) for class 3.

The calculator also returns the maximum admissible moment before the general check is required: \(M_{y,Ed,max} = (\bar\lambda_{c0}/\bar\lambda_f) \cdot M_{c,Rd}\).

Input data

• \(M_{y,Ed}\) — the design bending moment, in kNm.
• Profile and section — the dimensions from which \(i_{f,z}\) is computed.
• \(L_c\) — the distance between lateral restraints of the compression flange, in m.
• \(k_c\) — the moment distribution factor, chosen from the moment diagram.
• Cross-section class (1–3) — decides whether \(M_{c,Rd}\) uses \(W_{pl}\) or \(W_{el}\).
• Steel grade — \(f_y\) per Table 3.1.
• \(\gamma_{M1}\) — the recommended value is 1.0, but the National Annex takes precedence.

Assumptions and limitations

• The method is simplified and on the safe side: failing the condition does not mean the beam fails, but that the general check §6.3.2.1 with the full \(\chi_{LT}\) factor must be carried out.
• It applies to doubly symmetric I or H beams. Monosymmetric or open sections need separate treatment.
• Interaction with axial force is not treated — see bending and compression.
• Effective lateral restraints are assumed at the ends of the interval \(L_c\).

Related calculations

• Bending moment — the section capacity \(M_{c,Rd}\), the upper bound
• Flexural buckling — the equivalent for compression members
• Bending and compression — the \(N\)–\(M\) interaction
• Cross-section class — it decides \(M_{c,Rd}\)