Learn: LTB non-dimensional slenderness
A beam can deflect sideways and twist under bending. Lateral-torsional nondimensional slenderness compares its reference section moment with the elastic critical moment for that lateral-torsional mode.
What the formula is saying
Find W fy/Mcr and take its square root. The critical moment includes the chosen restraint and loading model; a lower Mcr means the beam is more sensitive to lateral-torsional buckling.
Read the symbols in plain language
- Wy
- Relevant section modulus
Relevant section modulus. Use the section modulus appropriate to the stated cross-section class.
mm³Cubic millimetres here describe a section modulus; they are a length-cubed unit.
- fy
- Yield strength
Specified yield stress of the relevant steel grade and thickness, before the material partial factor unless explicitly stated otherwise.
N/mm²One N/mm² equals one MPa.
- Mcr
- Elastic critical moment
Elastic critical moment. The two moment values must have identical units before division.
N·mmUse N·mm as the base unit shown here. W is mm³ and fy is N/mm², so Mcr must be entered in N·mm. A critical moment of 500 kN·m equals 500,000,000 N·mm. The resulting λLT has no unit.
- λ̄LT
- Result to find
LTB non-dimensional slenderness. The square root defines the nondimensional LTB parameter used in the later curve calculation.
ratio / no unit
Sort out the units first
W is mm³ and fy is N/mm², so Mcr must be entered in N·mm. A critical moment of 500 kN·m equals 500,000,000 N·mm. The resulting λLT has no unit.
Assumptions before calculating
Use the stated first-generation teaching equation with compatible section properties, material strengths and supplied partial factors. The required section class, buckling curve, National Annex values and design situation must be established separately.
This is a first-generation Eurocode teaching relationship or an explicitly simplified coefficient calculation. The numbers supplied here are exercise data, not a recommendation for any country. Check the adopted edition, relevant clause, National Annex, applicability conditions and all other limit states before any real design.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find λ̄LT and explain the result in the stated output unit.
- Wy · Relevant section modulus
- 800000 mm³
- fy · Yield strength
- 355 N/mm²
- Mcr · Elastic critical moment
- 500000000 N·mm
Calculate the reference section moment
Use the section modulus appropriate to the stated cross-section class.
(800000) × (355) = 284000000 N·mmCompare with the elastic critical moment
The two moment values must have identical units before division.
(284000000) ÷ (500000000) = 0.568Find lateral-torsional slenderness
The square root defines the nondimensional LTB parameter used in the later curve calculation.
√((0.568)) ≈ 0.7536577473
Does this worked answer make sense?
If the critical moment equals W fy, λLT is 1. A fourfold increase in Mcr halves λLT without changing the section’s reference yield moment.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- Wy · Relevant section modulus
- 600000 mm³
- fy · Yield strength
- 275 N/mm²
- Mcr · Elastic critical moment
- 300000000 N·mm
Calculate the reference section moment
Use the section modulus appropriate to the stated cross-section class.
(600000) × (275) = 165000000 N·mmCompare with the elastic critical moment
The two moment values must have identical units before division.
(165000000) ÷ (300000000) = 0.55Find lateral-torsional slenderness
The square root defines the nondimensional LTB parameter used in the later curve calculation.
√((0.55)) ≈ 0.7416198487
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- Wy · Relevant section modulus
- 900000 mm³
- fy · Yield strength
- 355 N/mm²
- Mcr · Elastic critical moment
- 600000000 N·mm
Find: Learn: LTB non-dimensional slenderness
A hint, not the answer
Find W fy/Mcr and take its square root. The critical moment includes the chosen restraint and loading model; a lower Mcr means the beam is more sensitive to lateral-torsional buckling.
W is mm³ and fy is N/mm², so Mcr must be entered in N·mm. A critical moment of 500 kN·m equals 500,000,000 N·mm. The resulting λLT has no unit.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Calculate the reference section moment
Use the section modulus appropriate to the stated cross-section class.
(900000) × (355) = 319500000 N·mmCompare with the elastic critical moment
The two moment values must have identical units before division.
(319500000) ÷ (600000000) = 0.5325Find lateral-torsional slenderness
The square root defines the nondimensional LTB parameter used in the later curve calculation.
√((0.5325)) ≈ 0.729725976
Avoid the common trap
Do not mix kN·m with N·mm. Do not use an axial critical force Ncr in the moment denominator or confuse λLT with the member’s geometric length-to-radius ratio.
When this method applies — and when it does not
This does not calculate Mcr, select the appropriate W for the section class, choose a buckling curve or apply the final reduction factor. Warping restraint, load height and moment distribution must already be represented in Mcr.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: LTB non-dimensional slenderness. First-generation EN 1993/EN 1994 teaching: cross-section resistance, stability, connections or composite action as relevant. Member classification and other limit states remain separate checks.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.