Lateral torsion buckling (LTB) is a very dangerous phenomenon, that can easily cause the collapse of a poorly designed beam. In civil engineering codes, the *critical bending moment* is crucial in the proper design of bent beams susceptible to LTB, as it allows for slenderness calculation. In “typical” cases everything is ok since code equations allow engineers to obtain the value of the *critical moment*. Those equations, however, require a lot of conditions to be met in order to work, and if at least one is not fulfilled… problems start. Today I will show you how to calculate critical moment in any situation you may encounter in your engineering work

**Critical bending moment equation and required conditions**

There are many equations for the critical moment which vary slightly in terms of parameters (some are more complicated/accurate than others). If you are interested in hand calculations of the critical moment I believe this is a nice guide. Note that most available equations follow the same set of rules that need to be followed in order to use the equation.

Required conditions for calculationcritical bending momentaccording to equation:

Beam must be symmetric in at least 2 planes– this is a huge drawback, forget L-sections, C-sections (even threw old code in my country stated that for C-sections you can calculate slenderness as for I-section and then reduce it by 25%), and many others including custom welded cross-sections.Beam must have a constant cross-section on its length– so no “optimized” beams with thinner flanges near end hinged supports.Beam must be straight (linear)– equations do not allow for curved beams like hopper rings in silos made of corrugated sheets.Beam must be bended in plane of it’s symmetry– this actually is important, as when you have bending in 2 directions you do not fulfill this requirement. You would be surprised how many beams are bended in both directions.Beam must be restraint in transverse movement and in cross-section plane rotations at its end– so many purlins do not fulfill this condition – it is not enough to screw the beam by it’s bottom flange – top flange have to be supported in transverse direction as well.It would be nice if the beam would have a relatively simple moment distribution along it’s length– this is often the case, but from time to time it might get problematic.

As you can see there are many limitations and many engineers are not aware of them. Each time your software make a design for you, you actually assume that all of the above is correct, and unfortunately some of those assumptions, when unfulfilled may have a drastic influence on capacity reduction due to *lateral torsional buckling*. Of course it is impossible to verify each beam in the design, but for the most important elements or those obviously not fulfilling the requirements given above this should be verified. If you cannot find an equation to calculate *critical moment* in your case doesn’t worry – there is a numerical way to solve this problem.

**Numerical method for ***critical bending moment* calculation

*critical bending moment*calculation

Most of the finite element programs have the possibility to calculate the critical moment. As long as your software uses plates and can do linear buckling you should be fine 🙂

**Actions to take:**

**Model the beam using plate elements:**define surfaces and apply corresponding thickness to each one**Support your beam in a realistic way:**remember that software now “sees” your beam as a 3D object, you can for instance support only one edge of the beam**Load your beam in a realistic way:**same as above, since model “sees” your beam in 3D you can actually choose at which part of the beam load is applied**Perform a linear analysis:**check for the maximal moment in the analyzed beam (sometimes I use a secondary simplified beam model so I do not have to integrate stresses from plates to derive bending moment in the beam).**Perform linear buckling analysis:**outcome would be a stability failure shape and*critical load multiplier.***Bending moment from linear analysis multiplied by***critical load multiplier*is the critical bending moment

Below I have recorded how to do this in RFEM software. I use it in my engineering office for beam static and simpler designs, while Femap and NX Nastran is used in mode demanding cases.

If you are interested in FEA analysis to be sure to see my free FEA course – you can get it below!

This article was created as Mathias asked me on how to calculate a critical moment in one of the elements he was designing. If you have a problem with design feel free to write me about it – I will try to help 🙂

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If you have any questions feel free to leave them below in the comments section.

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MathiasJune 2, 2016 at 1:28 pmSuper stuff, thanks for this really good tutorial!

After have run the analysis and got the critical moment for an optimized beam (*), I still have 2 more questions:

1)if I wish to calculate Mb,Rd can I continue use the general method of the Eurocode?

2)if yes, can the section modulus of the beam be integrated along the length for the formula of Mb,Rd?

(*) optimized beam : asymmetric beam with different sections along its length

Łukasz SkotnyJune 2, 2016 at 7:15 pmHi Mathias,

Thank you for kind words and good questions 🙂

1.Yes – this method provides the critical moment, you can then calculate slenderness as usual and then proceed with “normal” Eurocode procedure2.This is tricky – basically Eurocode in most places refers to members with constant cross section (hence need for numerical calculation of critical moment if the cross-section is not constant). I would say at this point we should divide capacity of a beam into 2 categories:A. Plastic capacity – this is the simpler part – in each cross section you have a certain moment and a certain moment of inertia so design for plastic capacity is easy and Eurocode provides guidelines for such calculations.

B. Stability capacity – whether or not the beam will loose stability. In Eurocode procedure you are using “buckling curves” that are assign to your cross-section (the is the parameter alpha in the procedure). Those refer to beams with constant cross section, and as such are not the best fit. Also the way capacity is calculated you use the reduction factor to plastic capacity (in order to take stability into account), and that would lead to situation in which beam would have “different” stability capacity in different cross sections… while there is only one stability capacity – a certain multiplier for the loads that when applied leads to stability failure (theoretically there are more since there is unlimited amount of eigenvalues but this is beyond point – in engineering interest is in the first one since it will cause collapse).

I would say that when you have a constant cross section (even a weird one) using critical moment and Eurocode procedure make sense, however if the cross section is not constant I would rather perform a “numerical” design which is also described in Eurocode. I will try to make a post about how to do that in near future 🙂

Hope that helps 🙂

MathiasJune 3, 2016 at 11:04 amThank you it helps a lot, I’m waiting for the next post to have the full answer to my question then 🙂

Łukasz SkotnyJune 4, 2016 at 4:43 pmHey Mathias,

The way I organize things here it will take few weeks to make an article. For now I would suggest calculating stability capacity using minimal cross-section this is on the safe side of course 🙂

Have a good one 🙂

MichaelNovember 22, 2016 at 1:32 pmThank you Lukasz

I from your video, LTB and stability issues is now clear to me.

Michael

Łukasz SkotnyNovember 24, 2016 at 8:08 amHi Michael,

I’m happy that you enjoyed the video 🙂