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8 minutes read
26 March 2017

Geometrically nonlinear analysis – how does it work?

8 minutes read

Geometric nonlinearity is incredibly useful in structural analysis. It can help you check if your model is correct, or allow for great structural optimization. Today I will discuss the basics of what does geometrically nonlinear analysis does using not the theory but real-life examples!

What you will learn here

I will explain what nonlinear geometry actually does. I promise there is no complex theory here – I will only use simple real-life examples.

Of course, I have chosen examples that actually give different outcomes in linear and nonlinear approaches. There are a lot of cases when linear and nonlinear design gives practically the same result. However there are also many cases where the nonlinear approach is simply required, and I wish to showcase a few of them.

Always start with something positive: Laundry!

I really love simple real-life examples. When it comes to geometric nonlinearity the best one is a string you hang your laundry on.

Geometrically linear analysis

Take a look. A wet sweater is what… 5 kg? Assuming the string is 5m long that would produce a bending moment of 0.0625kNm. Seems small right? But the string is only 5mm in diameter (assuming you really have an awesome laundry string, mine is more like 3mm…). Section modulus of such cross-section is 0.0123cm^3. This means that in order to carry the weight of the sweater (as a beam) the string should withstand the stress of 0.0625kNm/1.23e-8 m^3 = 5081MPa… only 20 times more than steel.

When I look at my string I rather doubt the gigantic strength hidden within. And yet it works…

The above example is a typical linear approach to a problem. Perhaps you noticed that my example cheated a bit. I didn’t draw any deflection! In fact, it would be easy to calculate it as well. If my string would be made from steel (far from it in reality) that would be only… 20.2m.

Think how much my poor string would have to elongate to deform that much (5m string deflecting 20m?!?). This question brings us closer to understanding nonlinear geometry… where large deformations theory is being used.

When the string becomes nonlinear

You see, the string cannot elongate 20.2m as the linear approach suggest. But it will elongate some for sure! As you know this elongation will cause a tensile force in the string. In turn, this tensile force will stabilize the whole thing, reducing deflections!

Geometrically nonlinear analysis - basic example

In the large deformation case above the deflection is only 57mm (when I assume my string is made from steel). For nylon, it would be around 237mm and the force in such case would be only 0.26kN (opposed to 1.07kN for steel wire).

I can easily believe that my string will somehow manage to carry the equivalent of 26kg in tension.

But how does geometrically nonlinear analysis work?

Geometrically nonlinear analysis - how does membrane state works

Look at the schematic above. Assuming that I have cut the string in half in the middle there is a tensile force N (the one we just calculated) pulling the string to the “inside”. On the wall where the string is attached, there is a reaction force in the horizontal direction, also equal to N. Both those forces create a pair of forces, that can carry the “moment” from the load. This is a certain simplification, but it really well describes what is going on.

You can easily verify the above. In the linear case, we got the bending moment of 0.0625kNm. In a nonlinear case, steel wire deflected 57mm with the normal force of 1.07kN. 0.057m x 1.07kN = 0.061kNm. In the second example, nylon string deflected 237mm with the normal force of 0.26kN. 0.237m x 0.26kN = 0.0616kNm. Both outcomes aren’t perfect, but quite close, which simply shows how to think about this issue.

Membrane state of string observations:

  • The higher the deflection (i.e. when you have an initial deflection) the lower the tensile forces (this is why for the nylon string the force decreased).
  • Theoretically, everything should strive to deflect as much as possible to reduce loads.
  • There is a limit to how much you can deflect. Material has certain Young Modulus, and cross-section used to have a certain moment of inertia and cross section area. It will deform as much as it can, and then it will stop.
  • If the deformations are “small”, the beam (hard to call it a string anymore since it is very rigid!) will have a lot of bending moment and little to no tensile force. Just as in the linear case.
  • If the deformations are “large”, bending moment will be small and the tensile force will appear.
  • Of course, this effect is “continuous”, so if the deformations are “medium” there is a bit of bending and a bit of tension involved.

Look out! There is a catch!

At this point, someone could make a conclusion that nonlinear geometry is positive and by “ignoring” it, you are on the safe side. Unfortunately, this is not the case:

As always in static design – nothing is for free

Here, we “pay” for this nice nonlinear membrane effect (which helps with the capacity of the string) with tensile force in supports. Do you want to verify if this is true? Try hanging laundry on the same string but in a different setup. Instead of attaching both ends to the wall lay one end simply on the table. For a typical beam, this is not a problem (the table still carries the vertical force). But we all intuitively know how will it end for a string right?

This is the ultimate proof that there is a tensile horizontal force in both supports – after all this is what makes the whole system works!

Nature of geometrically nonlinear analysis

Nonlinear geometry is neither positive or negative… it simply is!

Difference between linear and geometrically nonlinear analysis

Imagine you have our laundry string between two columns. If you look from the perspective of the string linear geometry is “bad” or at least “conservative”. This is because the forces you will estimate in linear design will greatly overestimate the reality. This means that you can design the string safely, even if very conservative.

On another hand for columns linear approach is actually quite optimistic. It takes into account the vertical load that will be transferred from the string, but it completely omits the horizontal force. This means that there will be no bending in columns, and that will lead to a not conservative design in linear static.

As you can see nonlinear geometry “helps” the string and “hinders” the columns. Unfortunately, you can’t pick if you want to use it or not… it is always there!

Here a word of comment is needed. Assuming that following the linear static you would design the string to withstand the moment calculated as for the beam (in the linear case). Then we would no longer have a “string” but a “beam”. Such a case would lead to far smaller deflections and in turn the horizontal force on columns would be much smaller. Perhaps even negligible or not even there.



This is why the use of static design is possible:

If deformations of the model are small, effects of nonlinear geometry will be small as well. The only problem is that sometimes it is hard to guess if the deformations are “small enough” to ignore nonlinear geometry.

In this example, there is something else we should consider. If the columns would be heavily loaded in the vertical direction, deflection seen in the nonlinear approach would introduce an eccentricity in the columns. This would lead to additional bending (often called second-order bending or secondary bending) that in some cases may be very important. This is also a geometrically nonlinear effect.

To sum this up

The geometrically nonlinear analysis is a very useful tool in structural design. If you encounter elements that deflect a lot in their load-caring process this is the best approach. If the linear approach would be correct in your case, you will get the same outcomes from the geometrically nonlinear analysis. However, if the case should use this nonlinearity, then this is the only option for a correct solution.

It is also great in dealing with secondary bending, where deformations of the model cause loads perpendicular to deformation direction cause additional bending.

In structural design, there is also a nice debate about linear/nonlinear buckling, which is strongly connected to linear/nonlinear geometry in the analysis. This is the topic I described elsewhere as I focused here on stress design mostly.

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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 (24)

Iman - 2021-05-17 23:23:41

hi Łukasz,

thank you for such a great educational and easily understood post.

after reading the content on your website and other papers, it seems that a combination of running a linear buckling analysis, defining an initial imperfection based on that LBA and then running geometric non-linear analysis on the imperfect model should result in quite accurate magnified moments.

the Canadian Code provides a simplified equation in calculating the Cr (compressive resistance) of sections in flexural buckling which is Cr = φs AFy (1 + λ2n)–1/n

λ = KL/r (Fy/pi^2 x Es)^1/2

n = 1.34, except for welded H-shapes with flame-cut flange edges and hollow structural sections
manufactured in accordance with CSA G40.20, Class H (i.e., hot-formed or cold-formed
stress-relieved sections), where n = 2.24

there are also equations that calculate the magnification factor for the moments.

i'm just wondering, after extracting results from our non-linear analysis, which we should have magnified moments from, what is the compressive capacity that I should use in my D/C ratio?

is it the Cr = φs AFy (1 + λ2n)–1/n
or
Cr = φs AFy

I have a feeling that we can use the full compressive resistance of the section (Cr = φs AFy ) since we have already accounted for the buckling/instability of the section with our analysis, but then again since this number is going to be larger than the Euler buckling critical load, I can't help but doubt that I can use a capacity higher than the Euler buckling critical load since buckling was a concern in the first place!

I hope my question was clear enough but please do let me know if I can email you directly and clarify in hopes that you can shed some light on the matter.

cheers,
Iman

Reply
Łukasz Skotny Ph.D. - 2021-05-30 16:22:51

Hey Iman!

Let me start by saying that I never have even seen a Canadian code (not to mention reading/using one!).

But without a doubt, you cannot get a capacity due to compression higher than the Euler force for that element... it's just impossible! Furthermore, with beams, I would expect, that the real capacity of a member would be 2-3 times lower than the Eulers force of that element. Ok, maybe sometimes only 1.8 times lower or something, but definitely not like 1.1 of the critical force, etc.

But this is a HUGE topic, and I'm not a fan of imperfections from LBA, I should also mention that such imperfections usually are for a single member only etc. There is just too much to cover in a single reply... but please be careful!

All the best!
Ł

Reply
Omkar - 2020-09-23 04:45:54

Hi,
Can you explain the details of how you calculated the deflection considering nonlinearities?
Thank you!

Reply
Łukasz Skotny Ph.D. - 2020-09-23 10:57:36

Hey Omkar!

I think I will disappoint you here...

I know there are differential equations for this, but I don't play around with this stuff. I simply launched my FEA software, defined the model, defined that I want this to be a "nonlinear geometry case" and I pressed "calculate". This is the quickest way for me, and to be honest I strongly suspect that most cases of nonlinear geometry I have to deal with won't have any differential equations that will allow for easy calculation of deformations anyway :)

And of course, on real cases, you can simply measure the deformation (it's nonlinear in "nature" just by itself)

All the best!
Ł

Reply
MD ADIL - 2020-08-29 07:35:41

Thank you so much sir for sharing your knowledge and making this topic so interesting. I am glad that I came across your blog.

Reply
Łukasz Skotny Ph.D. - 2020-08-29 17:20:02

I'm glad that you like my work!

All the best!
Ł

Reply
Avnish - 2020-06-12 12:01:51

Very nicely explained through an interesting real life example. Keep writing such articles explaining complex concepts in your lucid amnner. Thank you.

Reply
Łukasz Skotny Ph.D. - 2020-06-14 18:42:39

Thank you Avnish! I'm glad that you like my style :)

Reply
Mohamed - 2020-05-09 12:54:57

Very useful and Great..Thank you

Reply
Łukasz Skotny Ph.D. - 2020-05-12 17:08:43

Thank you Mohamed! I'm really glad that you like the article :)

Reply
Yasaman - 2019-10-30 05:29:26

Hi Łukasz
As always, it was great. I like your posts and makes me to feel that I am reading an attractive story. well done and God bless you!!! I learn alot!

Reply
Łukasz Skotny Ph.D. - 2019-10-30 07:34:10

Thank you Yasaman!

You are most kind :) I'm really glad that you like my work, and that you learn from it!

All the best
Ł

Reply
Yaniv Ben-David - 2019-09-11 08:48:20

Hi Łukasz!

Thanks for the article. I find you whole blog really great!

If you don't mind, I have several comments:
1. You sketched the sweater in the middle of the string but treated it as a distributed load all along it.
In the real case, the string would be deflected in a more triangular way...
While it is not that important since the general idea is not affected, I think you should mention it in the text.
2. Your first string observation is quite misleading: "The higher the deflection the lower the tensile forces".
For the same case (i.e the same string shape and material) - larger deflections would be accompanied with larger tensile force. As you well explained, these deflections are the actual reason for this tensile force. At first, there is no tensile force so the mass moment causes the string to deflect. As it gets longer, a tensile force is created and becomes larger as well. This whole thing keeps going on up until a point by which the tensile force is large enough (with a large enough arm) to create a moment that equals to the mass moment.
Of course, for a weaker material the tensile force will be smaller for the same amount of deflection. This is why the string must elongate even more in order to equalize the mass moment.

What do you think?

Reply
Łukasz Skotny Ph.D. - 2019-09-11 09:01:17

Hey Yaniv!

First of all, thank you for reaching out! I'm really glad that you like my blog.

As for your comments:

1. Yea, this is what you get while sketching - those were one of the first drawings I did, and to be honest I didn't pay as much attention to details like that back then. I will be more careful in the future for sure. But I doubt this has any significance in the topic at hand, so I'm not convinced it requires some additional explanation in this case.

2. Well, I'm not sure if it is misleading. Maybe it is... but I came from a different angle. If the same string has some initial deflection then the overall deflection would be bigger, and that helps. Of course, your observation about the same string needing bigger load is 100% valid - I added a small note there indication initial deflection as an example to avid unnecessary confusion - thank you for catching that!

That all being said I really appreciate your effort in helping me to do a better job. I'm still "learning to teach", and such comments really help out! When I will write a book one day this will really help me!

All the best
Ł

Reply
Illimar Kalk - 2019-04-01 08:11:29

Hello Łukasz,

It's good to read, how you are explaining the FEA things.

Reply
Łukasz Skotny Ph.D. - 2019-04-01 10:07:36

I'm glad that you like it Illimar :)

All the best
Ł

Reply
Abhimanyu Singh - 2018-11-12 05:49:09

Great article Łukasz!

You always add to the clarity on the subject.

Reply
Łukasz Skotny Ph.D. - 2018-11-14 07:41:20

Hey!

I'm super glad that you like it!

All the best
Ł

Reply
Ahmed Rashed - 2017-12-31 08:47:20

Thank you for that great explanation and great article as usual :) . the question is what if I have member
pre-tensioned and loaded vertically , how can I know if i need nonlinear analysis ?

Reply
Łukasz Skotny Ph.D. - 2018-01-02 09:47:04

Hey Ahmed!

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

It's a bit hard to imagine your case from that description. Do you mean like a horizontal beam with pretension and loaded vertically?

In most cases, this depends on the geometry. For instance, if the beam is short and has a "big" cross-section there is no need for geometric nonlinearity. If it is long and has "small" cross-section you need to consider geometrically nonlinear effects in some cases (i.e. when it is not "simply supported" with sliding support at one end).

All in all, I would say if you are in doubt... use geometric nonlinearity. In worst case it won't have any effect : )

Have a great day!
Ł

Reply
Gautam nagaraj - 2017-04-01 07:23:04

You are a genius. You know the art of describing mechanics in very simple way. Indeed its very helpful for all engineers fresh out of college or experienced to reorganize there thought process and focus on basics

Reply
Łukasz Skotny Ph.D. - 2017-04-01 08:31:54

Hey Gautam!

Thank you so much for such kind words!
I'm very happy that you like the post - there are more coming for sure :)

Have a great day!
Łukasz

Reply
barmin - 2017-03-28 05:59:12

Hi Łukasz!
Thanks for another great blog entry - wish I had that kind of explanation while I was studying ;)

Reply
Łukasz Skotny Ph.D. - 2017-03-28 08:53:55

Hey Barmin!

Thank you for kind words :) I'm glad you like it :)

Have a great day!
Łukasz

Reply

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