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
Structural rigidity is a very underrated concept I’m afraid. You see, usually, which part is more rigid than others does not directly influence the code-driven design methods. “Magic” happens when you do static design, and this process is far less regulated than dimensioning itself. I often meet engineers who completely do not understand this concept. I think that this hinders their possibility to find the best solutions (or even good ones sometimes!).
If you are using FEA to analyze stresses and displacements of the structure, it is really important to understand what rigidity is all about. If I would have to ask my potential new employee one question to check their skill it would definitely be “what is structural rigidity and where does it influence the design?”. Let’s find an answer to that question – shall we?
I love talking about complex problems in a simplistic, caricatural way. I think this way it is easier to explain something (and in style!). But what is most important it will be far easier for you to actually memorize and recall what you learn this way! This is why many examples here (like my favorite wet sweater or guys carrying a rock) are so weird.
After more than 10 years of teaching at University, I strongly believe that we have developed far too many long complex words to describe things. Sure it makes us sound smart, but this is getting us nowhere. The best specialists I know can easily explain almost any concept using simple examples without any “uncommon” words, why do beginners often hide their uncertainty behind complex definitions… This is why I always refer to structural rigidity as Gummy Bears (I mean in all honesty… science without gummy bears?).
Normal version: Imagine a table with 4 legs.
University version: Imagine a table with 4 legs that have an infinitely rigid table top that can move only in vertical direction.
When I stand on a table my weight (100kg – yea… I know) is evenly distributed between 4 legs meaning that each “gets” 25kg. We got used to thinking that stress “appears” in the table legs, but let’s look a tiny bit closer. What really happens is that the leg shortens a bit (and becomes a bit wider according to Hooke’s law). This shortening means that particles of material that forms each leg got a bit closer to each other
Since they are closer, they start to repel each other – this repealing is stressful. This is also why legs became a tiny bit shorter. It is a very small movement (let’s say 1nm) so it is easy to forget about this nuance (it will be critical in a second!). Also, let’s assume that 25kg is the maximal capacity of the table legs. If the “leg shortening” is higher than 1nm the legs are destroyed. The reason for that is simple – higher shortening means that a higher force than 25kg was applied.
Now imagine we have the same table, but legs are made from gummy bears. The cross-section stays the same, we just used another material. What will that change?
For one, let’s assume that those are really “strong” gummy bears and that they can actually withstand 25kg each.
If that is the case, you can easily imagine that the only difference is the fact that the tabletop will move downward, as gummy bears deform under load. It is quite obvious, that gummy bears will “compress” far more than normal wooden legs (but we did the gummy math correctly and there will be no buckling!). The difference here is the Young Modulus (just so you know, this is not a story about Young).
Assuming that gummy bears follow Hooke’s law if the leg “shortens” 10cm under full 25kg of load, then it will shorten 5cm under half of the load (this is linear). How much force will “appear” in the gummy bear column with 1nm of shortening? Literally nothing, or at least nothing worth considering.
Let’s summarize what we know:
Now imagine that we have a normal table with 4 wooden legs and I stand on it again – each leg gets 25kg right? But as I go up I feel that the table shakes and generally is overloaded. A quick calculation later I realize that each leg can carry only 20kg instead of 25…
The easiest solution is to add another leg right? Luckily for me, I have a gummy bear leg, that I know can withstand 25kg… awesome right?
So I build this contraption together and I climb the table once more. You know already things are going south right?
The mechanism is pretty simple – tabletop carries the load to table legs (all 5) and they shorten under the load.
University add on: Since table top won’t deflect (is infinitely rigid) and can move only vertically deformations in each leg are identical.
Unfortunately, with very small shortening (around 1nm) all of the force goes again into 4 wooden legs. Even though we have added the 5th leg, and each of them has sufficient capacity now, the table will still fail. This is because the gummy bear leg is not rigid enough to actually “take” the load. Failure goes as follows:
Structural recap: We had a slab or anything really, that was supported by columns. Calculating columns capacity one by one and adding them shows that the total capacity is sufficient. Columns are evenly distributed and should be evenly loaded (like on the circumference of the circle). However, the structure still failed as we did not provide equal rigidity of the supports!
As you can see structural rigidity is incredibly important. Note that it is not taken into account during code design! In both examples (table and slab) you would get that each element’s code capacity is ok! Only correctly executed FEA could show you that something is off. This is one of the reasons why it is worth it to understand this concept.
There is one last thing that we should consider: you don’t have to use 2 different materials as I did in this example to “get” this effect.
In essence, rigidity is connected to the deformation of the parts but it is not limited to differences in Young modulus – that is simply the easiest example to use! Equally problematic would be 5 columns standing on foundations and one of the foundations starts to settle (going deeper into the ground as the ground gives in). The column does not get shorter, check it might be identified as the other columns… but still, it won’t take any load so everything acts just as I described.
It can also happen that something goes unstable (buckling). The same material, good “foundations” and still elements stopped transferring any force!
In truth, structural rigidity can be found everywhere, from weld design (that is a great topic BTW) to silo supports… and of course gummy bears!
With this all in mind let’s answer the question I started with: “what is structural rigidity and where it influences the design?”
Structural rigidity is a phenomenon responsible for force (or stress) redistribution in the model (among others). If some parts of the analyzed structure deforms more than others, then usually load is carried onto the more rigid elements, and may cause an overload of those parts. Small rigidity can be caused by various reasons like different material, boundary conditions, geometry of the structure or stability issues.
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