When I was doing my first civil engineering design I hardly thought about dynamics. Static analysis was “all there was” for me. And to some degree, it might have been even justified back then. Now, when I understand a bit more, I would like to take you on a trip! We will learn about the differences between statics and dynamics!

The main difference between static and dynamic analysis is TIME! If the load is applied so slowly, that inertia effects won’t play a role, all you need is static analysis. Dynamic analysis handles impacts and other “fast” happening situations, but also vibrations (which happen in time).

But of course, there is implicit and explicit, and all the exciting stuff! So let’s get rolling!

## Starting slow (deeeespaaaacito!)

This part will be short (and slow!) because it’s about static analysis.

The basic idea is, that the load you have applied to your structure just is there. Furthermore, it could have already been there those precious seconds after the Big Bang! In short, it doesn’t matter how this load “got there”, but it’s there now and it won’t change later. If you want a slightly more scientific description, this more or less means that the load is applied extremely slowly! So slow in fact, that the speed of the load application can be omitted!

Static analysis in a nutshell:

• It doesn’t matter how you apply the load. Solver assumes that this is happening extremely slowly. This means that the way you apply the load has no impact on structural behavior.
• The load is not changing in time – it is just there… and that’s that!
• Structural response to the static load CAN differ in time (you know, things like creep, relaxation, etc.). Engineers usually view such analysis as more specialistic, and not “simple static”. Still, structural response changing in time is an option in static design.
• The fact the load is not changing doesn’t mean that the structural response is linear! All sorts of fun things can happen! If the load is high enough it can cause buckling, yield and all sort of other cool things. This, however, doesn’t change the fact that the analysis is “static” in nature!

Without a doubt, the static design is really popular. In fact, in Poland when someone will do calculations of a structure they would say that they are going to do “static design” or simply “statics”. I know the same is true in several other countries as well.

This is mostly because static is much easier to calculate then dynamic, and require less sophisticated software to do so! This also means that people will prefer doing statics. Which in turn leads to something pretty interesting and that is…

You see, “back in the day” it was almost impossible to calculate impact, etc. Simply put software was to “weak” to do so. I assume you could do such things at Universities, etc. But in a typical structural office, it was out of reach – at least in civil engineering.

But of course, this doesn’t mean that impacts didn’t happen! There were stone crushers, things were thrown out of trucks on structures (in gravel plants and similar facilities) and myriads of other things (including even a car hitting the building you are designing).

But how people handled those if they couldn’t perform the dynamic analysis?

… well, they increased the static load!

In essence if something will impact our structure you may not have to calculate the actual impact. I know it would be super cool to do such things! But often you don’t have the software, and more importantly, the time to do such analysis.

This is where static equivalent of dynamic load comes in play.

The idea is simple: just increade the load with the “dynamic factor”! Then you can treat it as static load in your analysis.

I saw various “dynamic factors” in my career, starting from a humble 1.5 and going up to around 10. If I would have to give the most popular value it would definitely be 4.0. However, 2,0 would be close behind it.

Of course, the value of “dynamic factors” depends on the industry and what you are trying to do. Often times, those were estimated and then “passed along” for decades. The origins of many of the values are long forgotten. But it doesn’t mean that this approach doesn’t work! Far from it! I think that most civil engineering structures with impact loads were designed this way!

This approach is so popular, that many manufacturers of various technical equipment supply this information in their datasheet! For instance, you get the machine drawing (to know how to connect it to the structure, etc). Usually, this drawing contains machine weight (the real one) and the “static weight” you should use in design. If the machine can cause horizontal loads, the manufacturer should provide those as “static equivalent” as well. Cool huh!

You will also get a frequency of the machine, along with all of the above. And this nicely leads us to another part of this article!

## Dynamic or not dynamic – vibrations!

This is where things start to be a bit more interesting. I think it’s obvious that “vibrations” are a dynamic thing. But, you most likely won’t need all the fancy stuff to analyze vibrations! This is the realm of “linear dynamics”.

In essence, you can use modal analysis to predict vibration modes of your structure (as long as the structure itself behaves in a linear way). And what is interesting is, that loads in this analysis don’t change in time. In fact, the solver will “change” the loads you select into the mass of your model. It will simply ignore the rest of the loads! So there is no “load changing in time” component yet!

You may think about modal analysis as about Linear Buckling Analysis (LBA) of dynamics! It’s there, it does help, but it’s not the pinnacle of human achievement in the field!

Modal analysis

Modal analysis allows you to predict natural frequencies of your linear model. This way, you can check if you may have vibration problems. Of course, you don’t want the applied load frequency to be close to the one you’ve got for your structure in modal analysis. If that is the case, it’s better to be careful, since your structure can enter resonance, and this hurts!

Resonance is of course dangerous. It’s the situation where the amplitude of vibrations increases A LOT! I don’t want to say that it increases to “infinity” because of the damping. But still, it increases enough to destroy your structure if the source of vibrations isn’t shut off quickly enough!

Storytime!

When I was a student I was in a building during resonance once. And nothing really bad happened, so I live to tell the tale. In Wrocław, the civil engineering department is a 10 story building, and they were making a parking space behind it. Since they needed to compact sand they used those small “hoppers”. You know the small machines that basically “jump” up and down to compact the ground below them. It so happened that the frequency of the hopper “jumps” they used matched almost perfectly to the natural frequency of our building. So naturally, after some time the building started to shake!

I was in the lecture on the 1st floor so it wasn’t so bad. But people from the 10th run down in panic on the stairs (they were afraid to use elevators). Luckily someone realized what was going on! They run to the guys doing the parking space and ask them to take a break… and things stabilized!

But I also helped in the design of repairs of the steel structure that entered resonance! This time the source was some technology thing, and before they shut it down, many of the welds and bolts cracked. Luckily the crew was wise enough to run off and shut down the machine by killing the electricity from afar!

So yea… you may want to pay attention to natural vibrations, and modal analysis will help you here. Sadly, it’s pretty costly to change the natural frequencies of the structure once it is built! So it’s better to pay attention!

## High-end shaking!

Of course, modal analysis isn’t all you can do. It’s more or less the beginning I would say. Just as LBA in buckling, the modal analysis doesn’t paint a “full picture”.

In fact, you can do a forced response analysis, that would be an equivalent of nonlinear buckling (jut to drag the LBA analogy a bit more). This would be a type of more advanced analysis, and I won’t have anything against calling it “dynamics”!

In essence, you define the loads/accelerations that are applied to the structure. Those loads/accelerations change in time of course! Many programs have some “historical earthquakes” already implemented there. But you can let your imagination run wild as well (i.e. to fit your load to a certain machine, etc.). Then you run the analysis and see how your structure will react to such loading changing in time.

Of course, forced response analysis is more difficult and time-consuming to perform. So it has to have some advantages because no one would use it otherwise! This is the main one:

Forced response allows you to see, how your structure reacts to various frequencies at once. Modal analysis tells you what are the “clean” frequencies your structure will vibrate in (eigenmodes). And what if in reality, you will have more than one frequency present? You also learn how your model reacts to vibrations that have different frequencies than eigenmodes. The fact that frequency of the vibration doesn’t “hit” the eigenmode doesn’t mean that you can simply ignore it…

There is a trick here. If you have a load, that changes 20 times in 2 seconds… you could actually calculate 20 different linear static analyses (for each different load). The “animation” between those 20 static analysis outcomes may look as if you really did forced response analysis. I’ve seen people who claimed such a thing…

This is of course not what this is about! In “real” forced response analysis your structure will vibrate even after the load ends. Since you usually define just a few seconds of the load, it’s simple to check! Just see what is happening 3 seconds after the load ends. And it’s shouldn’t be “nothing”, unless you have some seriously crazy damping! You are expecting that the structure will still vibrate because of the load you applied before.

## Dynamic or not dynamic – fatigue!

So we explored vibrations so far, tackling the idea of loads changing in time in force response analysis.

Another interesting phenomenon that you can meet in your designs is fatigue! Here, the situation is literally opposite to the one I in modal analysis! The load changes in time, but we will do static analysis to handle it (in most cases at least)!

I’m mentioning it briefly here since I don’t think most people would classify fatigue as “dynamic”. Although fatigue is often associated with vibrations. But since I used “load changing in time” as my dynamic definition it’s only fair to mention it here!

The idea is, that repeating load cycles can cause cumulative damage to the material. So it’s not only about the stress that is higher than yield. Rather, that 130MPa of stress in S235 steel changing from tension to compression may cause failure over time as well!

The thing is, that usually, load cycles are “static in nature” and they don’t happen very fast. This means that there will be no “inertia effects” in your analysis. And in such a case, it’s “murky” to classify fatigue as a dynamic problem. Simply put you will solve static cases to see the maximal and minimal stress in any given place. Afterward, you perform the fatigue checks “outside of FEA” (with scripts or even hand calculations). Those static cases usually will be linear, unless low-cycle fatigue will be considered. In such a case, you have to include yielding you our analysis.

Of course, it can happen, that the cycles are “dynamic” in nature with vibrations. That would be a nice mix of forced response and fatigue case in such a situation!

## Time for dummies…

Definitely, we are getting closer and closer to the analysis of the actual dynamics. But before we start…

…there is one thing I should mention, and that is “dummy time”. You see some solvers (like Adina used in SOL 601 of NX Nastran) want you to define time… even in static analysis! It is seemingly stupid, but this is only a way of introducing the “load factor” into the analysis.

While it may look like your load “depends” on time it’s not the case! Time (in such a case “dummy time”) is only used as a “counter” or load multiplier if you prefer.

The idea is simple: define how the load changes in the “dummy time”. Usually, you want a linear dependence. You know if dummy time is “0” then the Load is “0”. Of course, you also define maximal dummy time “X” when the load has “full value”. Just be aware that in the case of dummy time “X” means… some measure of dummy time. It is not in seconds (nor any other time unit). Dummy time is just a measure of how much load is applied. In essence when dummy time is equal to “X/2” then 50% of the load is applied.

The best way to think about dummy time is as a load multiplier. In fact, there is only one difference. The load multiplier for the load you applied is 1.0. This means that the load multiplier of 0.5 always means that 50% of the applied load. With dummy time it’s not the case! You can actually say, that maximal load is applied when the dummy time is equal to whatever you like. Of course, it is still wise to use 1… just to avoid mistakes. But technically you can use a dummy time of 2346 to represent the “full value” of the load you applied. In such a case, at dummy time 2346/2=1173 you will get 50% of the load.

Whenever you play like this with dummy time never forget that the solver uses it to iterate the solution. So when you change the dummy time at max load from 1 to let’s say 100 that is not all! It would be wise to adjust “dummy time stepping” as well:

Using dummy time

Let’s assume we have a load of 1000kN. We want to apply it in 100 equal steps of 10kN each.

Firstly the “simple case”. Let’s say that when the full load is applied (1000kN) we have the dummy time = 1.0. In such a case, each step should be 1/100=0.01 units of dummy time to get our 10kN per step.

However, we may want to have the max dummy time = 300 units when the load is 1000kN. In that case, each increment of the analysis should be 300/100=3 units of dummy time. In essence, we still get 10kN in each step.

It is easy to forget changing the “dummy time stepping” in our analysis! This is why it’s best to stick with dummy time = 1.0 for the full load. Just to avoid weird mistakes!

Dummy time has its uses, even though they are not as “grand” as you may think. I consider it a “perk” of certain solvers. Some use it, some don’t but in the end, all work the same. It just a matter of understanding how your solver increments loads. There is only one “benefit” you get from dummy time. You don’t have to toy with “steps” in your analysis when you use it. Let’s imagine you want to do such a multi-step analysis:

• Step 1: apply 100% of the load
• Step 2: decrease of 50% of the load
• Step 3: increase the load to 75%
• Step 4: Decrease load to 0…

Normally you will have to set up “steps” in your analysis. Effectively there would be 4 analyses one after another, as described above. But with “dummy time” you can make a single analysis step. All you have to do is to say that the dependence between time and load is:

Of course, you must say that the analysis should be from Time “0” to Time “1”… and that is it. You don’t have to learn how to make steps or how to restart analysis with other loads. Perhaps this is a bit easier this way. That is one of the differences between SOL106 and SOL 601 in NX Nastran by the way.

Of course, it’s not all sunshine and rainbows. Dummy time can be “irritating” to understand. Especially if your solver needs it for nonlinear static and you don’t know about it. It took me some time to figure it out for the first time! And even now I forget to set “dummy time” on occasion when I do SOL 601 analysis in my NX Nastran.

There is one important thing to remember! The fact that you defined “time dependence” for your loads doesn’t mean automatically that you are making a dynamic analysis! There is a chance this is a static analysis with a “dummy time”. It is always worth checking that in your solvers manual. Sadly, in many cases “dummy time” is described as “time” in your software. It is super easy to get confused!

If you are unsure if you are using “dummy” or “real” time test it!

Set a time at max load to 0.0001 and do analysis. If you have a static analysis with “dummy time” it will work just fine. If you are really making a dynamic analysis applying load in 0.0001s will cause some funky effects! Most likely you will see impact waves in your model and stuff. It is actually quite possible that your dynamic analysis won’t converge with this setting without some “fighting” for it. Just remember, that if you want to apply a load in 100 steps, each of those steps should be 0.0001/100=0.000001 units of time! It’s easy to forget to change the incrementation settings in the solver!

## The “True” Dynamic Analysis

Finally, we got to the heart of this. I guess that if you would like to “oppose” static analysis with dynamic one – this is it! Sure, along the way we have discussed some interesting topics on vibrations, etc. Some of those analyses can easily be called “dynamic”. But the “real” dynamic analysis starts here!

The difference between static and dynamic analysis is simple. As I wrote at the beginning, static analysis means, that the load “is just there” and does not change in time (which means it was applied really slowly!). Dynamic analysis is precisely on the opposing side of the scale. Here, we wonder how the load is applied and how fast it happened. We take into account inertia effects and all the jazz.

If you never met the term “inertia effects” it is simply this:

On the left, you see a nonlinear static analysis. Rotation is applied to the handle… and the entire thing just rotates. Nothing fancy really. This is what would happen if you would apply the rotation to the handle very slowly. This is the static domain, the load is applied so slowly, that you can basically ignore inertia effects!

On the right – the same thing… but with a twist! This time I applied the rotation “fast”, which called for dynamic analysis. I actually had to set how fast the 90der rotation will happen (in seconds) during load definition. Notice, that at the beginning handle moves before the tip realizes that there is a movement to be made. Then the tip tries to “catch up” and stuff begins to shake!

Note how the rod vibrates, even after the rotation is done!

Inertia effects!

The faster you rotate the fishing rod, the bigger the vibrations you get at the end. This “additional” movement is caused by inertia.

This is why you can ignore “slow-motion” and treat it as static in your models! The speed is so low, that there are basically no inertia effects. There will be no vibration at all after the load is done. Most loads happen in this “slow” domain.

But in “high speeds” inertia effects take place, and you have to use dynamic analysis instead. Otherwise, you may be missing important aspects of your model response!

It’s not simple to say how fast is “too fast” for static. If in doubt, it is better to use dynamic analysis “just in case”. But if I would have to make a limit, I would say that if the load is applied in minutes, it would be a good ground to consider static analysis. Anything faster calls for dynamics.

Of course, dynamic analysis allows for a lot of other cool things. For instance, you can analyze an impact:

Notice how nicely all things come together here. Firstly, it’s plainly obvious that I didn’t use “static load equivalent”. It would be a possibility of course! Instead of the ball, I would model a load on the impact area. The dynamic factor would be an issue for sure! I don’t think I ever heard about the values for such a case… and this is why I’ve made the dynamic analysis instead! I simply didn’t have to guess the dynamic factor, I could analyze what would really happen at the impact!

Notice that after the ball bounced you can see the shell vibrating slightly due to impact. This is a really nice example of inertia effects! Of curse, you can set some “crazy” dumping into the dynamic problem. In such a case, the vibrations will be very small. Usually, however, you expect some “shaking” in the dynamic analysis even after the load disappears.

The vibrations above are not the same as in the case of modal or forced response analysis. There, you have a constant source of the vibrations. You know, something like rotating machinery, etc. Modal analysis (and forced response) requires a constant “existence” of the vibration source. Without the source, vibrations die out due to dumping. Here, vibrations are just a “side effect” of dynamic load. They are caused by the effects of inertia, and of course without the “constant source” they die out eventually.

## The great battle of dynamic solvers!

The fishing rod and shell impact examples were done in an implicit solver. This is usually how a “typical” dynamic solver is called. But most likely you have heard about explicit solvers as well. Those would fit to solve the above task as well.

But of course, there must be a difference between implicit and explicit solvers. Otherwise, no one would bother to implement both types!

In essence, the difference is in the “speed” of the phenomenon you wish to analyze. If things are happening in time longer than let’s say 1s (maybe even 0.1s) implicit solver is great. If things happen faster (100ms or less) most likely it will be better to use explicit solver.

In theory, both will work just fine for all problems. It’s just that implicit solvers will compute much faster when the analysis time is long, while explicit solvers excel at quick solutions of problems with really short periods of “analysis time”.

The implicit vs explicit battle is fascinating, and without a doubt requires a post of its own. You can read much more about it in this post! Here, I just wanted to mark that there are 2 possibilities for solving dynamic problems in FEA.

## Summary

I hope that you find this useful. While there is a lot of content here, let’s try to wrap this up a bit. The goal is to make it easier to digest and remember for later.

• Dynamic analysis involves time! Whenever the speed of things is of essence or loads change in time, dynamic analysis is your tool! But this means, that if things are happening really slowly… you can simply use static analysis instead!
• … but there is a problem with the above definition! It’s obvious that “impact analysis” requires a dynamic approach. But, there are some other effects where loads change in time. Those include:
• Vibrations! They are caused by loads that are changing in time in a constant manner. And since they are changing in time, they fit into the definition. There are two approaches to this. You can do the modal analysis to see the natural frequencies of your structure. But you can also perform a forced response. This will show you how your model will react to the given excitation. Both are fun, but without doubt, a forced response in a more advanced approach.
• Fatigue! This is where it gets really “murky”! In fatigue loads are time-dependent as well… but you usually solve those as static problems anyway. This is because changes in the loads may happen really slowly over long periods of time. Of curse, fatigue can also happen when the loads change quickly (in vibrations). It’s just something associated with time in analysis, so I decided to mention it here.
• Not everything dynamic needs to be solved that way! Often times, you will just increase the impact loads with a “dynamic factor” and then analyzed them in a static way. This way, you don’t have to run “fancy” analysis all the time. There is always an important question, however! Who should say what is the value of such a “dynamic factor” and who is responsible for that value?
• Even when you have “time” set up in your analysis this may not mean you are doing dynamic analysis! Sometimes nonlinear solvers (like Adina) may require you to set “dummy time” simply to iterate nonlinear static problems. This is just the solver set up, and such time has no physical meaning. Also, the effects of inertia won’t be considered in such an analysis, which may be important in your task! Read your solver manual to make sure, but you can also do a simple test. Set the time to be 0.000001 and then 1000 in a second analysis. Then iterate your solution in both. If the outcomes are the same, most likely this is a “dummy time”!
• 2 flavors of dynamic analysis! You can solve the “real” dynamic problems with implicit and explicit algorithms. You need to do this when things are happening “fast” in your analysis. Such analysis also includes the effects of inertia. Both implicit and explicit approaches are fine, and not a single one of them is “better”. But I should say that the explicit solver is a part of fewer FEA packages. Since not every FEA package even has one, the explicit solver is seen as a “more advanced” thing.

I really hope that you’ve enjoyed the post. I would love you to share your opinion (or questions) in the comments below!