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7 minutes read
9 October 2017

Structural rigidity: Moment of inertia

7 minutes read

I got a question from Manoj about the basics of rigidity. Since the post about structural rigidity (gummy bears!) is so popular I figured it would be nice to share another story about rigidity I often use… cavemen!

The question may seem simple, but I’m sure there will be plenty of cool ideas in this post you may have never heard before! After all, who doesn’t like a good story, that explains stuff in a simple way?

Always tell a story…

I literally hated classes on strength of materials. They were boring, the stuff seemed unpractical, and there were differential equations and integration… and not much more!

For years I despised those courses. Until I really needed to use some of the knowledge from that area. Of course, I didn’t learn the strength of materials as a student (straight A’s BTW). Since I was thinking this is a “theoretical poop” I simply did the bare minimum to get an “A” without any effort to understand this stuff. Bah! I even learned some differential equations by heart, as I was too lazy to learn how to solve them!

So, when the need to use the knowledge came… it was time to really learn this stuff. Luckily I always had a good intuition to understand things in this funky-childish way. With this skill, I was able to tell stories to myself about things, which helped me understand this all. Later, I found to my surprise, that people usually don’t learn this way. So I started using those stories in the Uni courses I ran.

I strongly believe that until you cannot explain how something works in engineering to your grandma, you simply don’t understand that enough yourself! This is what the stories are for!

I knew most students will have serious problems with the strength of materials, so most of my classes start with the “all you need to know about the strength of materials in 45min or less”. This is the cavemen story!

Once upon a time in France…

Our cavemen tribe lives in France. This is an important fact since it is common knowledge that in France trees aren’t round! A typical cross-section of a French tree is rectangular as you know. This is important and will come in handy!

Trees in fight for better understanding of strength of materials!

Imagine our small primitive tribe encountered a very first engineering problem: they want to cross the river!

Since there is plenty of trees around all they need to do is to cut one, push it over the river, and cross on it.

Of course, cutting a thinner tree means less work… and this is how we found out a first optimization issue!

With nothing more to do, our small tribe decided to make some tests, just to see what will happen.

Cavemen engineering science!

The first observation is primitively simple. The thicker the tree the stronger it is. Surprisingly the outcomes aren’t all that consistent, but after few years cavemen can more or less “guess” if the cross-section is sufficient or not. First engineering tables are created:

Structural rigidity basice... or primitives :P

This is the simple cross-section area! The more you have it the better!

Accidents in service of development

Theoretically, the story could end here, if not for the fact that someone made something stupid. Instead of putting the rectangular cross-section “flat as always” someone put the tree in the upright position… and the capacity was off the charts!

The same cross-section, but placed in a different direction gave dramatically different results.

The first thing that is easy to notice is that the “flat as always” trunk deflects far more than the “upright” trunk under the same cavemen load (let’s call him Ugh!).

There is one very important conclusion to make here: deflection does not depend on the cross-section! There is “something else”!

Name your direction!

In order to look deeper into this fantastic discovery, we need to make few assumptions. We are a bit hopeless when it comes to defining tree “position”. Using “flat as always” and “upright” make a lot of sense for a French tree… but for an irregular shape, such description won’t work!

Also we cannot use “top”, “bottom”, “right” and “left” to describe this position. It is simple to prove:

This forced cavemen to use the axis. It was easy to experimentally check in which direction the tree bent the most, and in which direction the tree trunk deflected the least. Surprisingly, those directions were always perpendicular to each other.

Note: Load direction (and direction of deformation) is perpendicular to the axis! This means that if the load acts downward the deformation will be against the horizontal axis. Just as I have drawn above.

This may be a bit confusing at the beginning, but after a while it becomes natural 🙂

Thanks to this discovery, instead of going with “top” or “upright”, the direction of load (or direction of deflection if you like) will be defined against an axis. This is perfect, as the cross-section always has the “strongest” and the “weakest” direction. This doesn’t change if you rotate this cross-section! Also to avoid confusion we will always place the axis in a way that they cross in the center of gravity of our cross-section.

You could also notice, that the trunk in the upright position wobbles a bit when loaded. Maybe not wobble in a real sense, but at least seems unstable somehow. This is a stability issue: every cross-section wants to be bent against its weakest axis. If you bend it against the strong axis it will try to rotate to change where the load is applied. This is called lateral-torsional buckling, and it is a topic for a completely different story!

Where is the strength hidden in a tree?

With the axis defined, there is only one piece of the puzzle we are missing. That is: what is the difference between the strong and the weak axis?

Let’s take a look once more:

If we would look long enough it would definitely hit us! Wood is closer to the weak axis, and more “far away” from the strong axis. Basically, this is the only and the most important difference.

If the material of the cross-section is “far away” from the axis of this cross-section, the cross-section will be “stronger” or more rigid. If the material is close to an axis, that will be a relatively “weak” axis.

This phenomenon is described by a parameter called moment of inertia.

If you would think about it, this makes perfect sense. After all, this is why we make I-sections right?

The story goes on…

To be honest, there is more to this story, but we reached the point where I can simply answer Manoj’s question. I’m pretty certain that sooner or later I will get back to this!

To answer the original question

Flat-panel has a very small moment of inertia. Simply put, it has material very close to the axis against which you will bend it. A curved panel has a much higher moment of inertia on the same axis. This is why it behaves far more rigidly!

Of course, the moment of inertia influences far more than deflection during bending. I.e. if the section has a higher moment of inertia it is also less likely to buckle.

Overall people refer to the moment of inertia as to rigidity. To be fair, rigidity is a broader term, but usually, the moment of inertia is there somewhere!

I hope you have found this useful!

Want to learn more?

First of all, if you have a question related to FEA or structural steel feel free to leave it in the comments below, or send it via Linkedin. This way I will be able to answer your question, just as I did with the one I got from Manoj!

Learning about rigidity is a great step toward using FEA. This is because rigidity is one of the fundamental concepts you need to understand to master the Finite Element Method. If you like FEA, you can learn some useful things in my special free FEA course for my subscribers. You can get it below.

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 (28)

    Santosh - 2021-05-04 14:36:15

    Thank you for wonderful explanation. (Student from Nepal)

    Łukasz Skotny Ph.D. - 2021-05-07 08:52:24

    You are welcome Santosh! All the best!

    Supreeth - 2021-03-30 16:14:15

    Hello, thanks for the wonderful explanation and illustrations
    It makes sense, thank you, but I had a doubt - the longer the distance, the longer the bending moment right? Technically shouldn't it undergo more deflection? Thanks

    Łukasz Skotny Ph.D. - 2021-04-02 19:22:52

    Hey Supreeth!

    Absolutely, if you have a cantilever that is loaded at the end, the longer the cantilever (with the rest being the same) the more it will deform :)

    All the best

    Yaniv Ben-David - 2020-02-01 20:01:55

    Hi Lukasz,

    Thanks as always.
    My grandma gives you great wishes too...:)

    Łukasz Skotny Ph.D. - 2020-02-04 20:49:08

    Hey Yaniv!

    Thank you, man! All the best. Of course, tell your Grandma that I great her as well!

    All the best

    Freddy de Jong - 2018-09-21 09:01:23

    So much fun with this guy Łukasz :-)
    Easy learning classic difficulties.

    shared on my page

    Łukasz Skotny Ph.D. - 2018-09-21 09:53:00

    Hey Freddy!

    I'm super happy that you like it!

    All the best

    Z - 2018-05-04 14:50:31

    Hi, L,

    Everything is easier to remember when linked with something easily accessible.

    In a huge parking you can quickly find your car using designation letters, combined with numbers, but also colours, ducks and beavers, etc.

    When I was in University, I was going to see lectures in different shift just to see different teacher; I knew the name before joining, from some publications and his engagement with middle school folks...

    There, you could even make a joke, which he would pick up and build upon!

    As you mentioned elsewhere, lots of fun, and things go surprisingly smoothly.

    Aside that, the presentations he made were simply unforgettable due to the lively "picturing" of the matter, although it was followed by relatively "heavy" formulations.

    On the other hand, I am sure there is matter that cannot be closer understood without strong support from the pure mathematics. But, I still walk on the ground :)


    Łukasz Skotny Ph.D. - 2018-05-04 16:32:59

    Hey Z!

    This really had to be an awesome teacher! It's great that you cherish this memory, as such people should be remembered and praised!
    I too believe that there things that won't be understood without complex math. I just think that using complex FEA isn't one of those... or maybe I have to little experience and I'm still in the playground. Who knows :)

    All the best!

    Srikanth - 2017-10-23 16:23:24

    I guess the purpose in school is not to make you a great designer or make you understand to the extent you can explain stuff to your primary school kid (they ask so many whys!) or grandma if you prefer that way.

    Imagine the effort something like this would take to explain other concepts in engineering!

    We should try for mass moment of inertia...the forgotten one :-)

    Łukasz Skotny Ph.D. - 2017-10-23 18:10:13

    Hey Srikanth!

    I agree and disagree at the same time (it's my blog... I'm allowed to do that :P)

    I agree that using such stories is an effort. But also it takes a certain character to think this way, and also a certain character to like such approach. I'm sure not everyone like what I have described here, and not all teachers would agree that this is the optimal way. Still, this is my way and I enjoy the effort I put into this. Also, I got a lot of great feedback so far so maybe there is something to this.

    I'm certain not everything can be described with a caveman stories... but absolutely everything can be stripped down from those long many-syllables words and simply be told in an easy and understandable way. This is not impossible, far from it. After all, I can discuss shell stability on this level and this is far from simple topic :)

    Let me know what do you think :)

    All the best!

    Yarko - 2017-10-13 05:51:17

    Amazing article, funny and interesting! Keep doing this!

    Łukasz Skotny Ph.D. - 2017-10-13 05:53:34

    Hey Yarko!

    I'm so glad that you like it :)

    All the best

    Prabhakar Joshi - 2017-10-13 05:43:03

    But somehow our academicians don't teach this way, engineering has to be taught in a complicated way, only through partial differential equations.

    Łukasz Skotny Ph.D. - 2017-10-13 05:53:00

    Hey, Prabhakar!

    This is unfortunate, but this is also reality. I think that part of the reason is that all academic teachers are... well, academic. It happens relatively rarely that teachers at Universities have actual businesses and do design for a living. Such work changes how you look at things I would say, but it's hard to do at the same time as working at Uni, simply because of time constraints!

    All the best

    Manoj Kumar - 2017-10-12 10:35:12

    Great Lukasz!! Thanks a Lot, I asked you this question in LinkedIn. I didn't think this problem from Moment of Inertia point of view. Now I got an answer for 'Why Lays Potato chips are curved!!'.

    Łukasz Skotny Ph.D. - 2017-10-12 10:48:33

    Hey Manoj!

    Thank you for the comment, and the question :)

    I'm glad that I could help you out :)

    All the best

    Amirhossein - 2017-10-11 08:00:16

    "until you cannot explain how something works in engineering to your grandma, you simply don’t understand that enough yourself" so true!

    Łukasz Skotny Ph.D. - 2017-10-11 09:48:37

    Hey, Amirhossein!

    I'm glad that you like it ;)

    All the best

    Z - 2017-10-10 13:57:27

    "Story telling" is very important in engineering.
    Many a guy memorize many things, and apply that perfectly to the work at hand. But if they don't really have a good grasp of the nature of things, when it fails it will fail with a big bang.
    In today's world of gadgets and Google it is even more evident, and I would say important.
    Comes to mind my professor of Mechanics of Fluids, decades ago. After you passed written part, and say were about to finish your verbal "defense", a question comes like e.g. this one, "describe IN YOUR WORDS, NO FORMULAS what happens at the transition from laminar to turbulent flow. Some people got surprised that they were sent to repeat the exam!

    Łukasz Skotny Ph.D. - 2017-10-10 14:15:01

    Hey Z!

    I love such questions. This is a great example!

    I also agree that failures coming from not understanding stuff usually come with a big bang :)


    Michał Grotowski - 2017-10-10 12:17:35

    I still remember this story from a few years back! Surprisingly, recently I have stumbled upon a project in a way similar to the one discussed during the steel structures course - a roof that was supported on a brick wall from 1942. Was actually astonished at all the relevant issues I still remembered from back then.


    Łukasz Skotny Ph.D. - 2017-10-10 12:28:23

    Hey Michał!

    Who would thought :) Great to know you are doing fine!

    All the best!

    Niño Gem Ngo Lee - 2017-10-10 02:26:47

    Good day! I am new to FEA what is the purpose of the right hand rule and would you explain its significance in the result acquisition? thanks a lot

    Łukasz Skotny Ph.D. - 2017-10-10 07:44:32

    Hey Nino!

    Thank you for asking... to be honest, I'm not sure what rule do you have in mind. Is there a description somewhere, or could you tell me more?

    All the best

    Paul - 2017-10-23 19:45:21

    You go from +x to + y with the four fingers of your rigth hand and your thumb point to +z axe. It works not only for FEA but for any dextro rotatory cordinate system.

    Łukasz Skotny Ph.D. - 2017-10-23 19:54:00

    Hey Paul!

    Seems you know more about hands than I do : )

    I remember that there was one hand rule on physics that was pretty funky... no clue how to find what it was about though :)

    All the best and thanks for spreading knowledge : )


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      Your personal data administrator is Enterfea Łukasz Skotny, Skrzydlata 1/7, 54-129 Wrocław/POLAND, Email. By subscribing to the newsletter that includes marketing messages you consent to your personal data processing in accordance with this privacy policy