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29 May 2016

# Critical bending moment: How to calculate it with numerical analysis (in RFEM)

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 moments 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 calculation critical bending moment according to the 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 makes 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 a critical moment in your case doesn’t worry – there is a numerical way to solve this problem.

## Numerical method for 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 how to calculate a critical moment in one of the elements he was designing.

This is awesome! I’ve prepared a special free FEA course for my subscribers. You can get it below.

#### Author:Łukasz Skotny Ph.D.

I have over 10 years of practical FEA experience (I'm running my own Engineering Consultancy), and I've been an academic teacher for a decade. Here, I gladly share my engineering knowledge through courses, and on the blog!

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### Join the discussion

Zygmunt Kowalczyk - 2021-05-30 06:36:31

Hallo Łukasz,
but what if we have a two-span beam? Is it also possible to get critical bending moment in this way?

Łukasz Skotny Ph.D. - 2021-05-30 16:14:44

Of course - you can get a critical load for any system with compressive stress (somewhere). Of course in more complex load patterns it's more like critical load multiplier rather than a critical bending moment or critical bending force, but the "meaning" is the same (although it's more tricky to use that in code procedures).

BTW I'm afraid that your comment will disappear in some time - I'm redoing the website and I already migrated content to the new environment... I hope you will get to read that before this happens!

Dejan Matić - 2021-03-22 19:42:27

Łukasz Skotny Ph.D. - 2021-03-26 20:48:15

Thank you so much Dejan for sharing my work - this is a real help, and I really appreciate this!

All the best!
Ł

Andris Gaidelions - 2021-02-16 04:10:28

I was try few times to get the seme result as you, filany I found problem with support boundary conditions. I was design ass beam not 3D where alow rotation!

Łukasz Skotny Ph.D. - 2021-02-16 11:20:48

Hey Andris!

I'm glad that you managed to follow the example. I know how frustrating this may be, but hearing that you succeeded, at the end, makes me very happy!

Al the best!
Ł

Rubicel Garcia-Alena - 2019-07-09 22:55:01

Hello Łukasz,

Thank you for putting together your blog and the very interesting videos. I found both really helpful. I am also taking your free online course. I will go for the Premium version afterward.

I would like to have your comments on some tests I run based on one of your videos, specifically the one tilted “Critical bending moment: How to calculate it with numerical analysis (in RFEM)”.

I did the same project but this time I picked other beam sizes. I am more familiar with the Canadian codes that’s why I used a W310x21 – Steel 345W and a W200x19 – Steel 345W.
I run a Stability analysis to get the critical load factors. I compared the results to the Canadian Code, the American and the European.

For my tests the results are not that good compared to the ones you got in your video tutorial.
The Elastic Critical Moment for LTB for beam W310x21 is 26% higher than the one from de EC3 and the American code. It is 42% higher than the one from de Canadian Code.
The Elastic Critical Moment for LTB for beam W200x19 is 10% higher than the one from de EC3 and the American code. It is 25% higher than the one from de Canadian Code.
I used the same setting as you used for the video tutorial, except I changed then beam size and the material.

Have a great day,

Łukasz Skotny Ph.D. - 2019-07-10 12:23:32

Hey Rubicel

I'm not very familiar with US and Canada codes, nor US cross-section types... that being said:

Definitely check what the boundary conditions are assumed in US and Canadian codes... as there is the difference between the codes themselves!

10% difference is not the end of the world, but 26% is a difference for sure. Are you sure that you used the equations for the position where you apply the load (on the upper flange, lower flange or in the middle)? That is a common mistake - and something you need to select for the code procedure. BTW there is no "EC procedure for the critical moment" so when you say that the difference is between LTB and EC I'm not sure which equation you are referencing (there are no critical moment equations in any EC... and honestly, I'm not sure how it is in Canadian and US codes).

If what you are referencing to is the answers RFEM provides in its modules - those values are influenced by RFEM module settings. I haven't used US nor Canadian modules, but this is definitely the case for the EC module (after all it takes a critical moment from "somewhere", while EC does not provide info on how to calculate it!).

So make super sure, that the critical moment in LBA and in the modules (assuming that this is what you do) are actually calculating the same thing! It's possible to set up modules differently then your FEA model, and if you are using plates, then it's a separate model altogether, and you made several assumptions while creating that model, that should be reflected in RFEM modules settings as well.

Hope this helps

Jakub Beszczyński - 2018-05-30 11:10:37

Hi Łukasz! Thank You for this video and the whole blog. I have succesfully tested the LBA approach to find Mcr value for an I-beam. But as I was researching the realm of LBA-to-Mcr approach, I've stumbled upon a problem. For some examples, in my case a RHS beam, no eigenvalue could be found. Lowering the demanded convergence factor helped find eigenvalues, but the results were trash (obviously). Making the section more slender resulted in nice results again, and then stabilizing the slender section with a lateral support in the middle made it impossible to find eigenvalues yet again.
My take is that RHS's are really inherently stable, so this means just that torsional instability would not occur at reasonable load level.
Am I interpreting this correctly?

Thanks and have o good day!

Łukasz Skotny Ph.D. - 2018-05-30 11:20:54

Hey Jakub!

I'm really happy that you like the blog : )

I guess that by RHS you mean Rectangular Hollow Section (and not "Round Hollow Section"). If yes, then it would depend on the cross-section dimensions. If it is a square in cross-section there will be no lateral torsional buckling. The effect happens when you are bent in the "strong axis" and cross section deforms in the "weak axis". If both axes are the same, there is no effect. The same would go for a Circular pipe (it doesn't have a strong axis so there is no lateral torsional buckling).

However, if you have a rectangle and you apply bending in the strong axis it should show lateral torsional buckling as normal. Even though critical capacity is higher than say for an I-Section, there is still a "capacity". Meaning that you should be able to calculate critical moment. Maybe if you are "a bit rectangular" but very close to a square, the multiplier is so high that your LBA algorithm goes crazy - hard to guess really...

All the best
Ł

Jakub Beszczyński - 2018-05-30 13:49:33

Thanks for such a swift reply!
I mean a rectangular section and it is 120x220mm bent along stronger axis. As You say, It's probably the algorithm going crazy, like the infamous Robot Stiffness difference problem. I'll try it on a different FEM software.

Łukasz Skotny Ph.D. - 2018-05-31 04:01:18

Well - this is a rather slender beam I would say. almost 2:1 in height/width. I don't have my license with me so I can't do a quick check, but at a gut feeling, I think you should get "something" from the analysis.

Let us know how it goes!

All the best
Ł

Michael - 2016-11-22 13:32:14

Thank you Lukasz
I from your video, LTB and stability issues is now clear to me.
Michael

Łukasz Skotny Ph.D. - 2016-11-24 08:08:51

Hi Michael,

I'm happy that you enjoyed the video :)

Mathias - 2016-06-02 13:28:09

Super 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 Skotny Ph.D. - 2016-06-02 19:15:27

Hi 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 procedure
2. 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 :)

Mathias - 2016-06-03 11:04:58

Thank you it helps a lot, I'm waiting for the next post to have the full answer to my question then :)

Łukasz Skotny Ph.D. - 2016-06-04 16:43:05

Hey 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 :)