Linear vs nonlinear buckling explained!

Choosing between linear buckling and nonlinear buckling in given case can be difficult. Both approaches have positive and negative aspects. I explain here what are the differences, and when to use each analysis. I promise not to use complicated mathematics, nor highly theoretical approach. Instead, I will use a simple example to easily describe differences.

My first encounter with this problem

Linear vs nonlinear buckling dilemma got to me when I started my Ph.D. in shell stability. As a civil engineer, I wasn’t really trained in FEA during studies (apart from obvious programs for structural design). Modeling a shell for Abaqus or NX Nastran was definitely a challenge, but one I could quickly overcome. Problems started when I had to actually define an analysis type for my problem.

I quickly realized that there are far too many possibilities I have never heard about to choose from. Analysis called “buckling” obviously looked promising, and was my first bet 🙂 Later I realized that such approach was frown upon by some of the “scientific” crowd, so I have searched for an alternative. That way (and thanks to Cornelia Doerich Ph.D. thesis!) I managed to learn about nonlinear buckling! It was then when something “clicked” for me, and I simply never looked at any engineering problem in the same way as before. If you have no experience with nonlinear analysis, I would greatly advise you to take a look – you will be surprised how incredible it may be!

Linear vs nonlinear buckling: all you need to know

Below, you will see a short video, where I describe what are the differences in linear and nonlinear buckling on a simple shell example. If you have missed it, you can read on my blog about both linear buckling, and nonlinear buckling.

Happy watching 🙂

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  1. Joetox Goaline January 31, 2017 at 8:13 am - Reply

    My guess is the observed shape of the peak in the graph you show coordinating force with deflection will be less cornice-like and more like a bell if nonlinear material yielding is an influence to buckling.

    • Łukasz Skotny January 31, 2017 at 10:44 am - Reply

      Hey Joetox 🙂

      I would say it depends. I completely agree that this might happen in some cases, however many buckling phenomenons appear in a strictly elastic regime with stresses far smaller that yield. And in those cases I would say we are still looking at the chart as I have shown.

      My experience is that imperfections greatly reduce such “peaks” and “smooth things out”

      Have a great day!

  2. Andrew May 5, 2018 at 3:06 am - Reply

    Hi Łukasz,

    Great tutorial! I’m wondering what constraints you used on the uniformly loaded shell on 4 columns example?


    • Łukasz Skotny May 5, 2018 at 6:57 pm - Reply

      Hey Andrew!

      Ha! That is a tough question after all this time 😛 I don’t think this model “survived” to any of my archives as it was to simple to store.

      Knowing me it was:

      radial support on the top and bottom part, plus additional vertical support where the support actually is at the bottom. And then symmetry boundary conditions on the vertical edges.

      If you have any more questions feel free to write – I will try to help out 🙂
      Are you asking as you try to reproduce the outcomes? Any specific issues/

      All the best

  3. Harish April 2, 2019 at 5:12 pm - Reply

    Hello Łukasz,
    Great video but I have certain doubts regarding imperfection sensitivity factor that is given during non-linear buckling analysis. I usually take it as a certain percentage of the shell thickness ( 0.1 % to 100 %) i.e if the thickness of my structure is 10 mm then i will take 5 ( 50 % of thickness of the pressure vessel). Is this a correct way to do it ? can you please explain about imperfection sensitivity factors. Thanks in advance.

    • Łukasz Skotny April 3, 2019 at 6:27 am - Reply


      This is a really tricky question. I mostly do shell design according to EN and in EN 1993-1-6 they actually introduced the methods to calculate how big imperfections should be. I don’t use a “fixed percentage” imperfections in my work, but of course, this is very industry-specific. I can see a rationale behind doing so, ut 10% of thickness seems quite low… even though in relatively thick shells this actually may be reasonable I guess…

      I hope this helps a bit? If you have any follow up questions shoot!

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