Hello,
This may get a little depressing, so reader be warned.
Yesterday while at work, I was listening to this podcast entitled
Geek Friday. This in itself is not entirely out of the ordinary, as I often listen to something while working, and I attempt to listen to Geek Friday every week. This week, however, the hosts started talking about a
BBC News article about Lego blocks and the theoretical tallest a stack of blocks can get before the growing weight of the top blocks simply crushes the bottom block. This is an absolutely fascinating question, so I went in search of the article (it was not too difficult, as they posted a link in the show notes). So I read the article and all was well. However something caught my eye (actually my co-workers eye, but it’s all the same, really). The first normal line of text reads thusly:
There has been a burning debate on the social news website Reddit.
A burning debate!? (would love to use an interrobang here, but alas, I do not know how)!? I just had to find out how exciting this was. (Don’t worry, it was lunch time, so I was using my time).
This, however, is what led me to my small existential crisis. If I had only known…
From the get go, just a little bit down the page, we have a fascinating response to the question by duggatron (it’s the first post with any substantial length). I read it, and I was like, hey cool, I understand most of that. And ooh, an FEA. I did that in senior design. For completeness, I will include the entirety of the post.
This may be a little bit too simplified to get an accurate result.
Taking into account the fact that it's hollow is going to make a big difference, both because the brick will have significantly less structure and because you will get stress concentrations at the corners of the brick.
Also, the compressive strength listed on that chart isn't to failure, just 10% strain. Using the flexural strength might be a more accurate metric for failure.
The cross sectional area of a lego brick is about 182mm2 if you account for the hollowness, so the stress per block is actually closer to 0.0001341MPa/block or 0.01944psi.
Since calculating the failure stresses by hand would take a while due to the complex shape, I just did it in CAD and used finite element analysis to figure out what force would lead to stresses that exceed the flexural strength.
The resulting FEA plot is here: (i.imgur.com/1QQF7.jpg) and here: (i.imgur.com/oOuFQ.jpg)
Based off this analysis, my estimate for the number of bricks is about 220,000 bricks resulting in a failure stress of 11.04ksi. That would be a tower around 2.112km high.
Edit: Some people have pointed out that some of the assumptions I've made are impractical.
First, people have mentioned the change in gravity as a function of height. The inaccuracy from this is pretty low. The acceleration due to gravity at the equator at sea level is 9.78049m/s2 vs. 9.77432m/s2 at 2000m source. That's a difference of ~0.063%, so it's a fairly safe assumption to ignore it. Remember, the Earth's diameter is over 12,750km at the equator. A 2km tower is insignificant by comparison.
Second, people have mentioned the fact that the buckling in the tower could limit how high you could actually build. My solution assumes that you could stabilize the tower in some way so only the weight of the bricks is taken into account. Another way of visualizing the problem as simply the number of bricks is to think of it as an inverted pyramid of 220,000 bricks (mentioned here). If the pyramid looked like this, only with the center filled in, the pyramid wouldn't have to be very tall. The perimeter increases by two bricks each layer, and the center of layer x is equal to the number of bricks on layer x-2. That means the pyramid would have 236 layers, or just over 2.3m high.
Pretty exciting stuff, right? I couldn’t be content with that, though, because for some very odd reason, I enjoy reading comments even though I know they make me angry most of the time due to the complete lack of tact most of the internet possesses. So, I kept reading down the page until I go to the next lengthy block of text; this one dealing more with equations and less with CAD drawings. It was interesting in its own right, and I understood what they were talking about, though they weren’t really dealing with the actual question at hand. Main equation used shown below.
The buckling force is F=(pi x pi)xExI/(K x L)2.
Then I continued down the page to get to the next block of text. This one is what really threw me off. Again, for completeness, I’ll provide the entirety of the comment here. (Warning, contains language; I cannot control the internet!).
Now, this is going to be lost in the sea of envelopes you just received from replies to you, and I'm perfectly fine with that. But I have to say:
I'm so fucking lost. I'm so. fucking. lost. I can't even imagine what you just spent the time to think out. It kinda hurts. The reason it hurts is because part of me understands. Part of me thinks "Okay, yeah, yeah. Just following basic formulas here. I don't remember the formulas, but this is just advanced applications." I understand the ideas, but not the math. And then I hit FEA. And then you just start piling other factors on top of it. And my brain just... hits a wall. It cannot compute any farther than this. You have made me hit critical load. And it scares me. Seriously, all I want to write in this comment is "NERD!" and be done with it. And move on. Pretend it didn't happen. Go on with my life.
But... your comment, these past two comments may have just changed my life. For the last two years I've thought, "Hey, if I go to college, I should get a degree in Engineering. I know I don't know much advanced math because of how my schooling played out in high school, but I've always been okay at it, and I liked physics before I had to un-enroll from that school. Maybe I can do that, and see where it takes me, and maybe if I'm lucky end up with a job up-state at Microsoft. That would really be amazing, and I'd really love to be a part of that environment."
And now... I'm afraid of that path. Not because of the work it might take to do it, but because I'm afraid that I might not be able to do it. That I might not even be able to start to comprehend the calculations that you just did. Even though my field of engineering wouldn't be the same yours. I'm afraid of looking at a problem in a book and feeling exactly what I just did looking at your work in something relatively insignificant. That feeling of helplessness. Inability. I don't know if I can do that. I don't know if I can learn more advanced math anymore, I don't know whether or not I'll look at it and think, "Okay, so this... this goes here... and then... and... maybe... hrm... okay, okay... I think I've got it." like most other things I've dealt with, or if instead it'll go, "...What. Wait. No. What? ... I can't even. What? Fuck. What? Fuck. Fuck. Fuck. ..." It scares the shit out of me. I feel so stupid. I feel so amazingly, fucking stupid. I know I'm supposed to be smarter than this, that I could do whatever I wanted if I tried to do whatever I wanted, if I developed a sense of motivation and forced myself and became eager to learn these things, but now I know that that's not true. I very well could try as hard as I wanted to, and not get any farther than where I am now. But that's setting a limit on myself, that's not saying "Oh, if only I did this a certain way things would be better. If I did [blank] I would end up with a better job. If I was [blank] [blank] I wouldn't have to be so concerned about wondering the next time I found a girl who liked me because she would just be there already. If I had [blank] I could do [blank], If I [blank] [blank] I would/could [blank] [blank] [blank]." That's forcing myself to realize that, no, that's now how it works. Sometimes you just can't do [blank] at all.
It's fucking terrifying. This is more terrifying than every near death, every what if, every if only, all rolled into one. All of those put together couldn't even hit the halfway mark of what I just experienced. My heart hurts. My brain hurt first, but then after I realized it, I could physically feel my heart just sink a little, and beat a little slower. The part of my chest where it's at is warmer than normal, I can sense where my heart is, I can just feel the pulse it's sending out, I wouldn't consider it a beat, I'd consider it a wave. It feels like a wave pool. And I can feel the defeat in it. The sadness. It feels like my mom died. Or my little sister. Or like I just got dumped again after 2 years in a relationship, that moment of realization after the initial shock, after the fight had ended and an hour later after all the anger had ebbed away, the feeling of utter disbelief that what I loved so much was gone, left with only myself and my thoughts for the months to follow...
I think the worst part is, tomorrow it won't be different. I'll revert right back to thinking "If I go to college I'll go for a degree in engineering." But I'll know, too. It won't just be that. I know my real thought will be "If I go to college I'll go for a degree in... business. That sounds good. More money there anyways." and from there I'll push into politics. And from there, well, we've seen how it typically ends up. I promise that's not how I intend it to be. That I'm not a right-wing anything, let alone an extreme one. I feel like Anakin. Maybe this is how people like Murdoch get started, with a feeling of failure, followed by pure hatred for ever experiencing it. A need to gain power so that nothing can ever provide that sensation again. So that he can instead exert it over them, not the other way around. Maybe he started out thinking "I can't wait. I'm going to run the best newspaper that ever existed. We'll expose all the wrongs, so that the people can turn them into rights. I might not be a hero, but I'll change the world." Maybe.
tl;dr You just gave a 19-year old an existential crisis
I found myself empathizing with this person, this 19 year old on the cusp of a major life change. My life doesn’t match his/hers completely as I have already completed college with an engineering degree, but the overall though process is still the same. After reading this person’s existential crisis, I had one of my own. Here I am with a bachelor’s degree in mechanical engineering from one of the top engineering schools in the nation, and I don’t really feel good enough. I feel like I just got by though school, not really learning anything practical. I know people say that most real knowledge comes with on-the-job training. That's not really the point though. You would think that after spending five years at an engineering institution, I would be able to comfortably calculate the required mass to crush a single Lego block, and right now, I don't think I could just come up with the equations and do it all by myself. I've got no back-of-the-envelope math skills to speak of.
So that's where I am on this Saturday evening: thinking I'm not really smart enough to cut it in the real world as I struggle to emulate this
Christmas tree made of books and utterly failing to do so. It seems fitting that I would be failing to do something engineering-like while thinking about how I'm not possibly a good engineer.
[Insert catchy sign off here]
