How to calculate critical bending moment with linear buckling

Linear buckling allows for calculation of critical moment in any case. It doesn't matter if your cross-section is symmetric or not or how it is supported! If you have ever struggled to design a beam due to lateral torsion buckling read on!

Critical bending moment equation and required conditions

There are many equations for critical moment which vary slightly in terms of parameters (some are more complicated / accurate than others). If you are interested in hand calculations of 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 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 don't worry - there is a numerical way to solve this problem.

Numerical method for critical bending moment calculation

Most of finite element programs have the possibility to calculate the critical moment. As long as your software uses plate or shell elements 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 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 be sure to see my free FEA course!

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