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8 minutes read
3 April 2017

Materially nonlinear analysis – how does it work

8 minutes read

When people think about nonlinear analysis usually what first comes to mind is material nonlinearity. This is the most “visible” aspect of nonlinearity in calculations. I won’t dare to say that materially nonlinear analysis is the most important one (I think that there is no such thing), but without a doubt, this is an important part of engineering knowledge. Let’s learn how does this analysis works, and what it really means

Word of warning

There are so many aspects of material nonlinearity that to this day it makes my head spin! With all honesty, I can tell you that I have no experience with a lot of those aspects, as it will take years for me to gain experience in many fields (still working on it!).

I’m a civil engineer with a steel structures background so for me mostly material nonlinearity connects with yielding. This is what I will focus on here. If you happen to have vast experience with a different aspect, and you are willing to share knowledge let me know, it would be great to host you here 🙂

Below you can expect a simple language explanation without a complex theory of how material nonlinearity works. If you are interested in different nonlinear material models in FEA go here.

Let’s start with the steel stress-strain curve

Materially nonlinear analysis - stress-strain curve

I won’t go into a lot of detail here. I’m pretty sure that if you are into engineering you already saw this chart… hundreds of times! This is an engineering stress-strain curve for steel. It can also be drawn as true stress – true strain curve but this is not the topic for today. Fortunately, in the part, we will use the curve today this difference is insignificant.

The important part is that there is an elastic part (where stress is proportional to strains), plastic plateau (where plastic strain develops with almost no change in material stress), and then the strengthening and necking.

I will mostly focus today on 2 parts of this chart – the elastic part and a plastic plateau. This means that for the limited strains I could treat my steel as a bi-linear material with the following characteristic:

Materially nonlinear analysis - bi-linear material model

This is a typical bilinear elastoplastic model. Sometimes a work hardening is used (the plastic plateau line is not horizontal but inclines slightly). You can read more about it in the post I mentioned at the beginning.

The bilinear elastoplastic model works only to a certain maximal plastic strain (at which strengthening starts). I marked it as “max allowed plastic strain” on the left chart. For now, I assume that our model won’t reach such a high strain value. In real analysis, this is checked simply by displaying plastic strain values and checking the maximal values in the place of interest. As you see on the chart on the right for higher plastic strain values model will act as if the plastic plateau is infinite. This can have a negative influence on the accuracy of the outcomes so checking max plastic strain is a good idea.

 “Free” strain

Many materials have a nonlinear stress-strain curve. Most are stiffer first and gradually became less stiff as the stress increases. However, steel has an incredible trick up in its sleeve – it can yield.

Yielding means, that steel material will maintain the stress (so the capacity remains the same) but will increase the strain. This is a huge advantage. There are a lot of benefits to yielding (and some issues of course!) but here I will focus on aspects concerning plastic stress distribution.

Think about it this way:

Materially nonlinear analysis - 3 guys with a rock (cool right?)

3 guys keep a stone over their head. The middle one is too short to reach the stone, so he basically does nothing, while taller guys on the sides carry all the weight. Imagine that they represent a brittle material and the stone is too heavy. First, the 2 tall guys would be overwhelmed, and then the middle guy alone would be hopeless against the weight of the stone. However, as I wrote before:

Steel can maintain a certain stress level (yield), while increasing strains

In our example that would mean that both tall guys can squat while still holding the stone! In this way, the stone will go lower and finally, the guy in the middle will be able to help out… and now 3 guys are carrying the stone, not two!

Materially nonlinear analysis - 3 guys, 2 squats, 1 rock!

This is called plastic redistribution or plastic adjustment. There are limits to this of course:

  • Plastic strains can’t be too high – there is only so deep squat the guys can make after which they are done anyway
  • You can’t pull this trick all that often – if you expect the tall guys to squat and stand up all the time they will be really tired and fatigue kicks in (the low cycle one – dangerous stuff!)
  • Material isotropic/kinematic hardening – each cycle the stress-strain curve change a bit.

How materially nonlinear analysis works in an element

Analogies are very useful but let’s see how this really works in a steel element. The simplest example is a beam in bending:

Linear stress distribution in bended beam

As you can see the stress on the bottom and top edge of cross-section are highest, while in the middle stresses are gradually smaller. This is clearly an elastic state.

To explain how plasticity works we need to make a small experiment. Imagine I will cut a small section of the beam, somewhere in the middle. Bending means that the “vertical sides” of our cut-out section want to rotate like this (of course, deformations are completely out of scale, but we want to see them!):

Deformation of section cut-out in bended beam

Now I will cut this small piece into several horizontal layers. I will also measure strain in those layers. The strain is basically a difference between the “original undeformed shape” and “deformed shape when bending is applied” in relation to the original length. You can easily observe that the outer layers (1 and 7) deform a lot (so the strain is high). Strain decreases when we move toward the middle of the cross-section.

Strain distribution in bended beam

Of course, strain in each layer is proportional and changes linearly when cross-sections remain flat in bending (and usually that is the case). So if the max strain in layer 1 is 0.0035 then max strain in layer 2 is 0,0025 and in layer 4 max strain is 0,0005. This is a simple linear ramp.

We assumed that cross-sections will remain flat in bending. This means that almost vertical sides of section cut-out will remain straight lines. This is a reasonable assumption. Armed with that knowledge we know, that increasing strain in layer 1 by a factor of 2, will increase strain in each layer by the same factor. In other words strain in our cross-section looks like this:

Materially nonlinear analysis - strain distribution in a beam

So let’s go back to the bi-linear stress-strain curve for steel we adopted for this task. When the material is in the elastic zone stress increases proportionally to strain. But when we reach plastic plateau strain is increasing, but stress remains the same.

Materially nonlinear analysis - bi-linear elasto plastic model

There is a limit value for strain where yielding occurs. It is very easy to calculate as you can see above. Since we already know strains are changing in any cross-section let’s see how this influences stress distribution in the same cross-section:

Materially nonlinear analysis - essence of stress and strain distribution

As you can see at the beginning there was very high stress at the outer parts of the cross-section, while the middle section did nothing. This is a similar case as with the 3 guys keeping the stone above their heads. With increased strain (for our 3 friends that would be squatting) steel is still able to carry the stress. However, there is a benefit – the higher the strain the more of the cross-section actually carries the weight! Higher strain in the outer edges allows the inside of the cross-section to be strained as well. This way the stress appears there as well and helps with carrying of the bending. Just as the shorter guy in the middle did!

This is not a small effect – for a rectangle, you get an additional 50% of capacity due to bending this way! For an I-beam that will be more like 6-10%, but still something worth remembering 🙂

What to remember:

Material nonlinearity is fun! I hope that I managed to show you something worth your time reading this. To sum this up, the most important highlights:

  • Whenever strain in the material is so high that you reached yield, you should use nonlinear material model.
  • The easiest model to use is the bi-linear model. However, you need to check if maximal strains in your model do not exceed strains under which strengthening takes place.
  • Yielding allows transferring some portion of the overwhelming load to another part of the structure (aka “the middle guy”).
  • Plastic redistribution of loads is super useful, but problems like low cycle fatigue must be considered if the loads will be changing in time.

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If you have a spare 15 seconds write a comment with your thoughts on the matter or any questions you might have. I have a good history of replying to each and every comment.

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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    Comments (30)

    Umesh - 2023-05-12 14:53:32

    Wow! Great article. Understood the plastic redistribution very well.

    Łukasz Skotny Ph.D. - 2023-05-15 20:07:43

    Thanks Mate! I'm really glad that you liked it!
    All the best!

    Pranav - 2021-11-12 07:40:50

    Your way of explaining concepts is really creative !!

    about this post One thought is confusing me,

    (Lets keep aside fatigue as of now)
    what is a use of linear elastic analysis then. Why do we even use it. I meas we can carry non linear plastic every time and say component is safe. Because I have observed nearly in 90% cases the zone above yielding is very high in elastic analysis but plastic analysis shows very very less strain making it safe.

    In simple words shouldn't we consider component as a "FAILED" if it is yielding ?

    Łukasz Skotny Ph.D. - 2021-11-12 10:35:55

    Hey Pranav!

    First of all, thank you for your kind words - I'm glad that you like my style :)

    As for your question, there is no simple answer. While I don't like "putting things aside" in discussions, let's ignore the fatigue as you suggested (although this is a big thing).

    If you would consider "stress reached yield limit in linear elastic analysis" as a failure, then indeed LA would be conservative in most cases. Not all of course, as some materials are brittle, and all that.

    In such a case, doing the nonlinear analysis (assuming that you can allow plastic stress redistribution) would be a reasonable thing to do, and would lead to higher capacity.

    Still, Linear Analysis is easier to do, doesn't require super sophisticated software, etc. Sure, the outcomes may not be as cool, but they are easy to get and fast to compute. The biggest problem with LA is, that many engineers do not understand what it is and how it works... and that is an issue! This leads to a situation that in many cases people really overinterpret those outcomes, and that may lead to problems.

    All the best

    SONGKE - 2020-12-24 10:37:50

    Thank you for sharing your thought, Lukasz!

    I have been reading your posts for a while, it's always great that you can explain the things with funny sketches.

    Please keep going with you excellent work.

    If there is something to add, I would supplement as below:
    1) The bilinear elastoplastic analysis is also called "limit analysis", to define the maximum acceptable loads which the structure can withstand. Of course, the maximum allowable strain shall be pre-defined.

    Łukasz Skotny Ph.D. - 2020-12-24 18:35:41

    Hey Songke!

    I must admit that I've heard "limit analysis" somewhere, but I never dived deep into the subject! Thank you for letting me know - I really didn't know that :)

    And... I'm really glad that you like my work :)

    All the best!

    Deeksha S G - 2020-09-13 08:51:27

    Great! Every content you post, it's like i can explain it to anyone in simple words so that even they can never forget it.
    Thank you so much!!

    Łukasz Skotny Ph.D. - 2020-09-14 06:44:09

    Thank you for your kind words Deeksha! I'm really glad that you like my work :)

    All the best!

    Yaniv Ben-David - 2020-02-04 21:44:06

    Great stuff Lukasz!

    As always, the experience of following your approach is totally enjoyable!

    If I run a linear analysis (why the hell should I do that?!) and get an annoying-above-yielding stress concentration somewhere - I can run the FEA again, this time in a bilinear elastoplastic mode. The stress will be redistributed and lets say that the strain would be acceptable and so the total deformation.
    However, we would still get a plastic zone. What would be our approach from here?

    Łukasz Skotny Ph.D. - 2020-02-05 17:13:51

    Hey Yaniv!

    I'm glad that you like the text :)

    As to your question - check plastic strains. If you have the yielded (plastic) zone, plastic strains will appear. Those have "acceptable" limits depending on what material you use. Usually, codes describe what sort of plastic strains are acceptable - at least this is my experience with ferritic/austenitic steel.

    Just be careful with fatigue and effects like this. If yielding happens under cyclic loading this will be "fun". Still doable in FEA, but a much more complex problem :)

    All the best!

    Sounghoon - 2019-11-01 12:30:43

    Thank you for your post. I have some questions.

    1. Is this related to 'stress linearization' ? I sometimes use the function to review FEA result. I heard that stress concentration can be ignored for a static analysis if the material is a ductile like a steel. So I am wondering the plastic load redistribution is related to the linearization.

    2. How can I calculate the max allowed plastic strain?

    3. If a stress condition of a structure is on the plastic plateau, I think the structure has a permanent deformation even though the stress does not increase. Could I ignore the deformation?

    Łukasz Skotny Ph.D. - 2019-11-01 14:32:14

    Hey Sounghoon!

    I will try to briefly answer your questions, but they would require a post of its own each I guess...

    1. No, not really. Stress linearization is something you do if you want to calculate i.e. pressure vessels according to ASME VIII code. I'm not familiar with it, but I think it uses linear analysis as a basis, so the post is not connected to this really.

    In static analysis (so no fatigue etc.) stress from stress concentrations will be redistributed, sure. But how much will? Until you calculate nonlinear material FEA it's guessing more or less. I don't like using linear analysis for design.

    2. Max allowable plastic strain can be derived according to different codes i.e. DNV RP C280, EN 1993-1-6 or EN 1993-1-4 (for stainless steels). In essence, it will depend on the Young Modulus, yield stress, etc. Also, it is obviously limited by the max elongation at breaking from the tensile test (but usually allowable strain values are significantly smaller than those max values from lab tests...

    3. No, you cannot simply ignore the deformation! It will be there obviously. But also, there is a limit of how much structure can deform plastically. On one end, plastic strains will be high. On another too big zone of yielding may cause instabilities in your model (plastic collapse) that can cause failure regardless of strain values (simply put rigidity decreases when the yield area is high, and this alone can cause failure).

    Hope this helps!

    Javed - 2018-10-11 09:23:10

    nice explanation...but still confuse..please explain with taking example were stiffness comes in to picture.

    Łukasz Skotny Ph.D. - 2018-10-11 11:14:13

    Hey Javed!

    This is a very good question. Sadly, for now, I simply don't have time to dwell into such topics. Somewhere at the start of the new year, I should launch a nonlinear FEA course. It will include a lot of advanced topics so keep watching my blog so you won't miss it :)

    All the best

    Freddy de Jong - 2018-07-09 07:27:28

    Today I'm more and more concerned with non-linearity.
    I have recently refreshed my depth knowledge with various modules from Coursera (Georgia Tech University)
    Here you'll find the course topics https://sterkteberekening.nl/kennis-en-certificering/
    But your FEA-subjects are a very welcome addition to that.
    I've already followed your Free FEA essentials course.
    After the holiday, I'll definitely follow the Premium on-line course.
    Already looking forward to start with it.

    Łukasz Skotny Ph.D. - 2018-07-09 13:46:08

    Hey Freddy!

    I should start with the "oh it's great that you are interested in my course" (you know, the moneyz!) But instead, I will start with: I'm so super excited that you spend the time to learn! You would be surprised how few people actually learn anything new after University. They just do what they always did... I'm really excited for you, as I strongly believe that the constant learning is an actual superpower!

    ... and of course, I'm really happy that you are interested in the course - can't wait to have you as a student - it will be fun I'm sure :)

    Have a great one mate!

    Pramod Ravichandran - 2018-07-07 04:26:36

    Dear Lukasz,
    I must say this , everytime I read your content I am excited that I will learn something new today. Not that these words are new to me, the way of expressing complex engineering terms in simple terms is just simply great. I need to learn lot more from you, thanks for sharing the knowledge.

    Łukasz Skotny Ph.D. - 2018-07-07 05:35:04

    Thanks for kind words!

    I'm really happy that you like it :)

    All the best

    Don - 2018-04-03 06:49:30

    Excellent work. It is beneficial for structural engineers especially beginners to understand more about NL analysis and advantages in safe designs.
    Thank you very much for your invitation to participate.

    Łukasz Skotny Ph.D. - 2018-04-03 15:46:28

    Hey Don!

    Thank you so much for the kind words :)

    All the best

    P V Narayana Rao - 2018-01-23 05:07:46

    excellent .. thanks for sharing such a complex subject in a simple thought provoking way. Highly useful for freshers and refresher for those practicing....like to see many more from you... cheers ...

    Łukasz Skotny Ph.D. - 2018-01-23 12:37:23


    Thank you for kind words!

    All the best

    rob - 2018-01-19 17:18:47

    Amazing job! So difficult problem presented in suche a simple way. It is first article of your authorship that I read and I really liked it. Tomorrow I will catch up and read the rest of it. Have a nice weekend!

    Łukasz Skotny Ph.D. - 2018-01-19 19:56:40


    I'm really glad that you like it :)

    Have a great weekend as well!

    alam - 2017-09-19 09:45:56

    Thanks for this article,

    I really enjoyed to read the stress distribution (how middle guy works) vs strain (plateau).

    Łukasz Skotny Ph.D. - 2017-09-19 17:39:02

    Hey, Alam!

    I'm so glad that you like it :)

    All the best

    Anup - 2017-04-18 03:31:20


    Łukasz Skotny Ph.D. - 2017-04-18 05:54:17

    Hey Anup!

    I'm glad you like it :)
    Have a great day!

    Jos - 2017-04-04 15:17:05

    Hi Lukasz,

    Interesting post! If you want to know more about using non-linear FEA in (offshore) construction, give DNV RP C208 a read. It covers material models, maximum allowable plastic strains, etc.

    Keep these interesting posts up, you are succeeding in putting complicated stuff into simple and understandable example.

    Łukasz Skotny Ph.D. - 2017-04-04 18:30:37

    Hey Jos

    I have already read the C208 standard. I must admit it was a great read :) I really liked the fact that the code actually specified allowable plastic strains in few cases - something I found lacking in Eurocodes (or at least I couldn't find such information there, I admit I haven read them all).

    Thank you for this comment! Also I'm very happy that you like my posts - nice to hear kind words :)

    Have a great day!


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