Designing steel silos with analytical approach always yields very conservative results. Nonlinear Finite Element Method allow for great optimization of those structures (and other shell structures as well!). I will discuss this based on a silo I have recently designed.

Hero of today

I will describe advantages of the nonlinear numerical design of shells based on silos I have designed not so long ago. Below a picture of the structure:

Analytical approach od silos design – a brief history

History of designing silo with analytical equations is fascinating (at least for me :P). This is however not the main topic. Instead of going through all equations in EN 1993-4-1 or other similar codes I will simply give you a short history. If this doesn’t interest you, skip this part 🙂

All I wish to say here is, that up to this day we are using Timoshenko equation for critical buckling load of ideal uniformly supported/loaded shell. This is important, not only because the equation is almost 100 years old (equation for square area is somewhat older and still works), but also because this is the only closed analytical solution we have. In other words, if you want to design a shell that is supported on columns (like a silo) or not uniformly compressed from the top (a bit like a silo, I’m not picky) then… tough luck – there is no solution.

This leads to fabulous stories on how people tried to get around it. There were failures, some success, and shocks. I.e. in 1992 Samuelson and Eggwertz published a book Shell Stability Handbook. They proved that stress in shells does not spread at 45 deg angle but far steeper, meaning that concentrated stress… tends to remain concentrated with height. I can only imagine how bad a day in work I would have when I would learn that after designing several silos… I won’t even start describing the point when we realized how greatly shell capacity drops with imperfections.

Analytical approach to silos design – step by step

Generally, silos design is made in steps as shown below. If you are interested in post showing and describing all equations as well leave a mark in the comments.

  • Decide how accurately the shell must be made (there are strict rules for this step in EN 1993-1-6).
  • Calculate the imperfection reduction factor.
  • Think if “elephant foot buckling” (aka “plastic buckling”) is possible and if so update the imperfection reduction factor accordingly.
  • Use Timoshenko equation to calculate slenderness (regardless if you have uniformly supported/loaded shell or not). If your shell is supported on columns this is considered in imperfection reduction factor.
  • Calculate buckling reduction factor based on the slenderness and imperfection factor.
  • Stress in shell must be smaller than yield limit multiplied by buckling reduction factor.

Pretty reasonable right? There are few issues with this approach, but all in all, this is a doable process.

Note however that it still uses Timoshenko equation and adapts it for the purposes of discretely supported shells. This is done with imperfection reduction factor (its value is somewhat dependent on stress distribution along shell circumference). This is the place where conservative approach appears, as such adaptation must be made on the safe side. This means the equations are “ready” for a worst case, while your case is most likely not as bad 🙂

For our silos, analytical capacity ratio due to vertical compression near column support is 117%. Sounds bad, right?

Finite Element Method approach to silos design

This is why I like Eurocodes so much (even though I am aware of all of the flaws). They allow for numerical design. Plain and simple. Code EN 1993-1-6 clearly describe what analysis should be performed, and when you can assume the capacity is sufficient. Also, those are not rules saying how much overstress (stress above yield limit) you may have here and there. A full-blown nonlinear analysis with imperfections is required.

This is really great 🙂 Even through nonlinear analysis isn’t the easiest approach, the benefits are incredible! Again I don’t want to go into details about how to do what – if you are interested take my free FEA course!

The steps for FEA design are as follows:

  • Linear buckling. This is not an answer (far from it!) but it checks if the model works, where you can expect stability problems and let you roughly estimate where you are in terms of stability capacity.
  • Nonlinear buckling. This one is fun! You do not need a closed analytical solution for your case. You don’t need to adapt Timoshenko equation either. Solving nonlinear buckling gives you the answer you need.
  • Nonlinear buckling with imperfections. I mentioned this earlier, imperfections have a great impact on capacity. Of course, you never know which set will be the worse one, so you need to check at least several. Imperfections amplitudes are given in EN 1993-1-6.
  • Plastic collapse. The “second part” of the deal… no one claims that your shell will fail in an elastic manner. What about elephant foot buckling or plain plastic collapse. This is your answer.
  • Plastic collapse with nonlinear geometry. The first part of “interaction checking”. Now (after all the above steps) you know what is worse: elastic failure or plastic collapse. But surely they interact so you check how in here.
  • Plastic collapse with nonlinear geometry and imperfections… or nonlinear buckling with imperfections and plastic material. This is basically the same deal. Thing is that at this stage you know what will be worse and you will choose another set of imperfections based on that. Sometimes plastic collapse dominates (so you pick imperfections sets that further decrease capacity in this mode), sometimes elastic buckling failure dominates and imperfections change. After all, this is why you are doing all of the above: to understand how your particular case works, so you can act accordingly.

After all this work you look at the final stability path from GMNIA and you simply read the maximal load multiplier. Few factors later you have your designed capacity.

In the case of this silos capacity calculated numerically (according to EN 1993-1-6 and EN 1993-4-1) is 78%…

This means that the same capacity of the same silos under the same conditions is 50% higher when you calculate it with nonlinear finite element analysis. I think it is well worth the hassle, as material savings are significant!

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