Web under local loads – Hand calculations
It’s very hard to calculate the capacity of small sections of any structure under concentrated loads. Luckily EN 1993-1-5 gives us a decent solution for webs under concentrated loads!11 January 2021
I remember well when I first tried to understand buckling. For whatever reason, this was not natural for me. There were so many equations, that didn’t make any sense back then! But it’s way easier to understand stability thanks to FEA. To go there, we need to find a good answer to a question: what is Buckling Analysis?
Buckling Analysis is an FEA routine that can solve all the difficult buckling problems that cannot be solved by hand calculations. Linear Buckling (LBA) is the most common Buckling Analysis. The nonlinear approach, on the other hand, offers more robust solutions than Linear Buckling.
As with everything in engineering, there are different “levels” of complexity in Buckling Analysis. Definitely the more you know, the more you can calculate accurately… but let’s start at the beginning.
Firstly, I feel I should start with something pretty important:
Usually, when people say “Buckling Analysis” they have Linear Bifurcation Analysis (LBA) in mind.
While LBA is a great tool, and we will definitely talk more about it here, this is just the begining!
In truth, many problems require nonlinear approach to buckling, and I will discuss this here as well! Such approach is also a “buckling analysis”, but you may have heard “nonlinear buckling” used in this regard.
To be fair, I prefer to talk about “stability” rather than “buckling”. While this may be my weird quirk (I guess a Ph.D. in the field of stability does that to you!), I think it’s a bit more precise. I guess this comes from the fact, that when we discuss buckling, people instantly think about Eulers column, and the “buckling” of compressed straight members.
While in fact, stability covers a LOT more effects. I can comfortably call most of them “buckling” anyway, but I think stability simply fits better into the role. So before we go any further let’s take a look at some cool ways in which stuff can lose stability.
There is one thing we have to cover before we go into FEA buckling analysis and all the “technical” stuff. You simply needed to be aware of how things can actually fail due to stability! After all, it’s critical to know what you are trying to solve in FEA. It’s not only the “know your enemy” thing. The actual approach to modeling and analysis will differ in some cases, depending on the stability type you have to deal with. This is why I compiled a list of all the awesome ways something may fail due to stability:
All right, I know it’s a lot to take in. But truth be told, without understanding all that, doing FEA Buckling Analysis isn’t a great idea. It’s pretty easy to omit stuff if you are unaware of it! We will get back to this close to the end of this post.
To this day I don’t understand why they don’t teach this in kindergartens. This is the most “basic” and user-friendly analysis FEA can provide for buckling problems. While limited, and pretty inaccurate in some cases, Linear Bifurcation Analysis (LBA) is so easy to use, that it’s a default stability analysis in most FEA packages. It is so popular, that in some software this type of analysis is simply called “buckling”.
The idea is, that you input, support, and load the model, and then the analysis provides you with a critical load multiplier as an outcome.
Load multipliers as outcome:
If you are used to using design codes, it may seem weird to present a capacity as a “load multiplier”. But this actually makes a lot of sense. In essence you should read such outcomes as “how many times I should increase my load for the structure to reach failure”. Of course, load multiplier higher than 1.0 means that the load you applied doesn’t reach “analysis failure” while multiplier lower than 1.0 leads into doom!
You can read more about load multiplier as an outcome here!
The critical load multiplier you get from Linear Buckling is a pretty peculiar one.
The outcome should read: how many times do I have to increase the loads for my ideal model to fail due to elastic stability.
Engineers often read the above as a “capacity due to stability”, which is sadly completely wrong! The below graph shows it clearly:
It’s like with the Eulers column we started with. While critical force might be quite high, the “actual” capacity will be much lower. There are three main aspects we should look into:
We need to discuss those issues one by one before we move on. This is the critically important stuff (the pun!).
This is the part of FEA design that many engineers would gladly forget about! I’m pretty certain that if imperfections would be simple to implement, nobody would complain. Sadly, this is a rather complicated thing, and also seriously time-consuming. So it’s only natural that people want to ditch it!
Fun note: with my several years of experience in corporate training I’ve learned something curious. It is often the case, that the most difficult things in design are usually “overlooked” or plainly “ignored”. And when I ask about those topics, I usually hear things like “it’s impractical”, or “too little time”, and the old time classic “we always did it that way”.
I think that showing on examples how the ignored effect reduces capacity is a pretty important thing in such cases.
Since imperfections are also often ignored, let’s do a simple example, shall we?
Very basics of imperfections!
Before we go into more complex things, let’s take a look at how this really works. We will use our trusted Euler’s column for this ok?
Firstly, let’s take a look at how this works in LBA. We have a nice column, and we load it with the load P=1. It doesn’t really matter how much load “1” means. Notice that the column does not deform to the sides at all. It may shorten a bit due to Hooke’s law and all that, but this is a minuscule problem here.
All right, let’s increase the load to P=2. Again, nothing really happens. Side deformations are still zero.
Finally, we apply a “big enough load”. I denoted it as Pcr, but it is often called “Eulers Force” or simply “Critical Force”. With this much load, our column buckles! Funny enough, it doesn’t matter how big the side deformations are. Instantly, in a place called bifurcation point, our column “snaps” into the failure mode… and it’s done! Formally, the side deformations become infinite (this is what LBA would show), but of course the “real beam” would simply break at some point (LBA doesn’t see that though).
Of course, “infinite” looks really bad on charts in FEA software, so instead the shape is “normalized”. Often times you will see that the maximal horizontal deformations in LBA are equal to “1”. But it’s not 1m or 1mm or anything else. It’s just a way to show you the “shape”. Think about it this way: if this point moves “1”, and the other point moves only “0.5” then the second point moves half the distance of the first. How far they go is irrelevant. This is simply linearly scalable.
What did we learned from that?
Well there is only one conclusion to make. Since we are analyzing a “perfect model” horizontal deformations happen only at failure!
This is not the case in reality… so let’s take a look!
Now, let’s take a look at our wonderful column, but let’s bend it initially.
Usually “the bend”, is the way in which we bead with all the things that are uncertain (straight axis, constant material properties, eccentricities and all!). Taking all of those effects separately wouldn’t be practical. Instead, we will combine them into one value of “imperfection” I marked as U0 below:
Note how the reaction changes of the model:
This is what I would call a “limit load”, or the actual capacity of the beam. To be precise I should add, that if the stress level would rise to the yielding point, yielding should be taken into account to get a “proper” limit load. For now, let’s assume that the buckling happened in a purely elastic zone for simplicity.
Please note, that the “real capacity” I called limit load is smaller than the critical load predicted by LBA!
This means that imperfections along with second-order effects really reduce capacity. Sometimes by a very high margin. The maximal reduction between a “perfect” and “imperfect” model I saw in my own designs was 2.5 times smaller capacity!
I feel you don’t want to make such mistakes… so it’s better to include imperfections in your models if you are using FEA as the main design tool.
But there is a problem with LBA… even if you implement imperfections, LBA won’t take secondary effects into account! There will be no increase of deformations in the middle with the increase of the load. While taking secondary effects into account seems necessary… LBA won’t do tricks like this for you. This leads us to the second point of the list we are discussing!
The phenomenon I just described (with the increasing deformation due to imperfection) is actually nonlinear. Sure, you need to implement imperfections in the first place. Without them, the whole thing doesn’t happen in our example anyway. But I covered this already in the previous part. But even if you implement imperfections things aren’t great for LBA! Since this is a linear algorithm, it can’t take second-order geometry effects (like the increase of deformations I just described) into account. There is no setting that can change this! Simply put LBA uses “linear geometry”. For the second-order geometry effects to happen in analysis you need “nonlinear geometry”.
Small disclaimer for picky people:
I understand that “second-order geometry” and “nonlinear geometry” is not exactly the same thing. Some software (especially cheaper ones in civil engineering) only have “second-order geometry” implemented. It is often called “second-order analysis” in those programs. With it, you can’t really do “geometrically nonlinear analysis” unless it is implemented separately. “Full nonlinear geometry” allow for some more tricks. I.e. things can enter the membrane state during analysis. You can read more on nonlinear geometry or how it works here!
I suspect that this will die off at some point, as all the FEA packages will allow for nonlinear geometry. As of now, to get the increase in horizontal deformations under load you don’t need the “full nonlinear geometry”. All you need is a remnant of the past called “second-order geometry”. Think about it as “nonlinear geometry of the old times”.
Just to avoid angry stares, I will refer to this mostly as “second-order effects” in the rest of this post! Just be aware that it’s almost the same as “nonlinear geometry”.
All right! Let’s assume that the solver wants to include an increase of deformations (second-order effects). The only way to do it would be to apply load as I did. Divide it into smaller portions, and see how the model behaves with a smaller load. And then progressively add more and more load, noticing how the model deforms and taking that into account.
Sadly, LBA cannot do it… since it is linear!
This means that even if you would include imperfections, LBA won’t take into account this additional deformation from imperfection (often called “second-order deformation”). Not everything is susceptible to this. Most of the beam models won’t care that much since design rules cover this effect. But for instance, shells will cry in outrage!
For beam models, the problem is smaller. But it is not because you can “completely ignore” second-order effects. That would be too much even for beams! However, beams (and columns and other “members”) are pretty easy to predict. This means that code design procedures that you may use (like those in EN 1993-1-1) already include second-order effects “hidden” in the procedure itself. In essence, you can estimate the critical load without those, and factor for second-order effects later, thanks to the design procedure.
For shell models, things are grimmer. It’s almost impossible to predict how various shells will react to imperfections, and what will be the scope of “second-order effects”. It’s hard to make a blanket “safe side” assumptions about this. What you would get would lead to extremely conservative design in many cases (while being extremely needed in others!). This means, that you simply cannot use LBA in shell design. Well, you may use LBA, but such a procedure must be governed by insane expectations and assumptions. You know, like calibrating the design method for each particular case!
The final point on this shortlist of Linear Buckling limitations is a fun one. Here, we will wonder what on earth the capacity of an element really is!
This may come as a surprise, but this is actually quite a complex topic!
Firstly, let’s throw out of the window all the “weird stuff”. We won’t talk here about fatigue, nor dynamic effects, corrosion, and many other things that may cause collapse. Since things are complicated already, let’s agree that we will keep strictly to static design. Also, we won’t check additional stuff like maximal allowable deformations and the like. Those have no real impact on buckling.
That being said we are left with stability and plasticity:
Plastic failure: You can think about it as a “stress failure”. In essence, a part is loaded with so much, that the material fails to carry stresses it is subject to. You know, like a concrete brick in a hydraulic press. There is no stability influence. I don’t think I have to convince you that when you compress a concrete cube in a hydraulic press… it won’t buckle!
Stability failure: I myself refer to this as “elastic failure” but it may not be the best way to describe this phenomenon (hence I used stability instead). This is the “pure buckling failure”. Since in this case buckling happens in “purely elastic range” plastic capacity of the material is irrelevant (as long as it’s high enough to keep buckling within elastic range).
Most designing procedures (the analytical ones described in codes) use a single capacity metric (usually in percentages). In most cases it’s derived in a way, that you calculate the plastic capacity first, and then reduce it to take stability into account. In the end, you get one single value describing the capacity ratio of an element.
But in such an approach, it is pretty easy to lose a very important thing! Those two are not separate!
In fact plastic failure and stability failure can “cooperate” to cause failure sooner than any single one of them could!
This can definitely surprise you!
I often see people checking stresses (i.e. plastic analysis) and then checking stability. Usually, if both are ok, the outcome they see is “structure is ok”. Sadly, this may not be the case! As with everything, there are several possibilities here:
This is where the fun happens. Because you see… the pipe from the last example could fail let’s say under load 80. While its capacity due to stability could be 90, and plastic capacity could be 100. And this is when people get it wrong! It is often the case, that buckling and plastic capacity are somewhat close to each other (like 90 and 100 in this example). In situations like that… they will “cooperate” to destroy the model, and the final capacity will be lower than any of the two effects alone!
To put this in perspective let’s consider our 3 cases again, but with numbers:
If the stability and plastic capacity are close to each other it is not enough to check them both separately!
And this is another thing where LBA won’t help you. It cannot use nonlinear material (to calculate plastic capacity). This means it won’t take into account the “cooperation” of failure modes!
Ok! It’s high time for something positive right!
I just told you, that Linear Buckling (LBA) will provide you with some sort of an outcome… but it won’t include important effects like imperfections, second-order deformations, or the influence of yielding on buckling capacity.
So the only reasonable question is… why on earth even bother with LBA?!
Luckily there is a solid answer to such a question!
LBA shines when you have a defined design procedure!
This is mostly the case with beam models. In order to design a column or a beam, you need to work pretty hard. Most codes (like EN 1993-1-1) describe design procedures that allow you to take all the effects we just discussed into account. However, to use them, you need to know the member slenderness.
You can calculate it manually – it’s doable in many cases, but in some pretty time-consuming. But instead you can use LBA to calculate the critical load – the most troublesome component of calculating slenderness!
This is where this analysis shines! It can greatly simplify the work you need to put in to design a problematic element in your beam models. But… that is more or less “it”. If you are not into beam models and code-driven design… LBA is just a nice toy!
Sure, based on the outcome you can make “educated guesses” about capacity due to stability… but those will be guesses, as you don’t include all the factors I just described!
Teaching various companies about FEA and stability I often hear that they are using a “magical LBA limit”. You know something like “if critical load multiplier is higher than X we are fine”. I must admit that “X” varies greatly (from around 2 to even 50!), but everyone is convinced that their value works! Sadly, there is no single “blanket value” and I can show this based on a simple example.
Let’s look at an example, how LBA outcomes are “relative”:
Imagine a 2m heigh truss. Each compressed strout designed according to Eurocodes will have it’s capacity around twice lower than critical Eulers force. This means, that if the truss is “well designed” the critical load multiplier for those elements will be abound 2.0-3.0.
The same outcome (2.0-3.0) would raise some flags for a simple frame system if this load multiplier would be assosiated with a sway stability failure.
And the same outcome (2.0-3.0) would be a likely disaster in a design of silo shell, as the actuall capacity may be several times lower.
As you can see above, “the same outcome” from LBA in one situation is perfectly acceptable (truss diagonals) while leads to almost certain failure in other cases (shell buckling).
What I am trying to say here is, that there is no “magic critical load multiplier” that solves all of your problems! And in fact, LBA outcomes cannot be used as capacity measures! At least not without a lot of work beforehand. Let’s discuss this further, shall we?
Sure, I can imagine a procedure that would help you to learn more for very specific and highly repetitive models. But it would be up to you to figure this out. Below is the list of what I would do (if I would work in highly repetitive tasks of course) to generate enough knowledge to use LBA more wisely:
As you see, it’s completely doable to use LBA to judge the capacity of the models quickly. But there is a good reason why no of the codes I know of, allow using LBA directly as “capacity”. It simply doesn’t work this way.
I would love to tell you “Yup! it’s so simple. Just take the LBA outcome and you are fine”. Sadly, the most beautiful part of engineering is, that it is way more complicated than this. And it’s actually dangerous to you and your designs to make such shortcuts without understanding how things work!
Of course, there is a solution to this. It’s the 21st century after all!
But I will start with when not to use it ok?
When you don’t need a nonlinear approach:
Beam models! Yea, as I told you before, all the things “omitted” by LBA are “hidden” in the code-driven procedures of design. Since you will use codes in design of the beams I assume, you are absolutely fine with not doing the nonlinear analysis I will describe later. All the phenomena are already taken into account by the design code.
Ok, maybe not all of them… just read the code and make sure if you comply with all the assumptions. If that is the case, you are set!
Plate/Shell structures where buckling is not an issue! You know, not everything buckles (this may come as a surprise, I know!). But since this is a topic about buckling I guess your structure has to be verified anyway. Also, nonlinear analysis is useful even if the structure does not buckle, as it deals with the irritating stress concentrations for you! Yup, no more guessing how much stress is “allowable” beyond yield, and no more radically-safe-side-estimates! Nonlinear FEA is good for you when you design stuff!
All right, so let’s wonder how Nonlinear Buckling Analysis works. It’s actually dramatically simple! Let’s take a look at what the solver will do:
What you don’t see at the animation is, that the model had some imperfections applied to it. I automatically changed its geometry with the certain “imperfection shape” with the “imperfection amplitude” taken from the design code.
Notice, how the model response changes. Firstly it deforms more or less uniformly around the circumference. Then it selects one “dimple” and this one grows greatly. Afterward, a nearby dimple grows as well (that is after failure actually). This is purely a nonlinear solution, as linear one would show the same deformation all the time, only scaled with a higher and higher number (it could not change the character!).
And finally, notice that at some point stresses stop increasing when reaching 138.2MPa. This is because if yielding (in high temperatures, but it’s irrelevant). The model is set up in such a way, that at a certain stress (138.2MPa) no more stress can be carried, and plastic deformations start.
All of the 3 effects I just describe solve the 3 problems we had with LBA!
Benefits of Nonlinear Buckling:
This is why Nonlinear FEA is in high demand. You can simply design stuff accurately, without guessing and over-conservative assumptions!
Of course, there is a lot more to discuss in this topic, here I wanted to show you how the nonlinear analysis work, and what is the difference between linear and nonlinear buckling approach. If I managed to interest you in this subject, you can read more on nonlinear FEA in this post!
If you start to wonder if this will ever be over… this is the last part so take heart!
So far you have learned about the Linear and Nonlinear approaches to Buckling Analysis. In a sense, this is all I wanted to teach you here. But it’s only fair to point out some potential flaws in analysis, and what you need to pay special attention to!
Remember when I told you that your modeling will depend on the type of analysis you may want to check? Let’s start there!
Problems with beam models in Buckling Analysis!
Regardless if you want to verify stability with a linear or nonlinear approach, there are some universal issues with beam (member) models! This is the shortlist:
As you can see above, if you have a very important part to design, and you are not certain what stability issues it may have… it’s better to use the plate model. Sadly, the beam model may omit some pretty important things in the analysis!
And finally, at the very end… stability is very susceptible to change in boundary conditions!
Normally we don’t care if the support is blocked against torsion under our column or not. Or is the hinge in both directions or only in one? Or even if our shell is supported in the pinned way, or it can slide in one direction or another.
Practical experience shows, that such decisions don’t impact normal stress analysis at all. But stability cares about those! Just be sure, that you are using reasonable Boundary Conditions in your analysis. I know, I know, you always should pay attention to those. But in stability analysis accuracy in Boundary Conditions is even more important!
This is a very big and important topic. You can read all you should know about Boundary Conditions here!
Och wow… you are still here? It’s been the third day that I write this post and I feel like writing a book, not a blog! Thanks for reading through all of this. I really hope this was useful. Definitely let me know what you think about it below!
Since we covered so much, let’s try to summarize this a bit!
And… that’s all!
Let me know what you think in the comments below, and if you like the post, definitely share it with friends!
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