Triangular bracing - which orientation

[Captainowie]

One for the structural engineers out there...

I am building a structure to support my latest GBC module. I have vertical supports, and a horizontal cross-member. I am using a 12-long brick to form a 6-8-10 triangle to maintain the right angle between the support and the cross-member (thank you Pythagoras!). But which is the best way to do it?

In the red version (on the left), the 6 side is horizontal, and the 8 side is vertical. In the blue version, it's the other way around. Which one should I use? Or are they equivalent?

Under which circumstances should I use one over the other?

In this case, the brace will be under compression (holdling something up). Would the answer be any different if it was under tension (holding something down)?

Thanks

Owen.

P.S. The forces involved are not great, so I'm certain that either one would suffice. I'm just curious about whether one is better than the other. I'd settle for which one is more asthetically pleasing.

Edited April 19, 2014 by Captainowie

[JM1971]

I thinks its best to try and get all the angles to same for maximum strength, blue looks close.

Edited April 19, 2014 by JM1971

[DrJB]

You can get a bit sturdier design by using a technic 3×5 L liftarm, placed at the corner.

A side note, I know it is often (incorrectly) referred to as Pythagoras ... but really, it is called an Egyptian triangle, thanks to the Pharaohs and their pyramids.

The pharaohs came long before Pythagoras ...

Edited April 19, 2014 by DrJB

[Captainowie]

I thinks its best to try and get all the angles to same for maximum strength, blue looks close.

The angles are the same in both cases. The difference between them is the orientation - whether the small angle is on the horizontal or vertical brick.

You can get a bit sturdier design by using a technic 3×5 L liftarm, placed at the corner.

A side note, I know it is often (incorrectly) referred to as Pythagoras ... but really, it is called an Egyptian triangle, thanks to the Pharaohs and their pyramids.

The pharaohs came long before Pythagoras ...

Surely you mean "as well as" rather than "instead of" for the liftarm. I would wager that what I've got there would withstand a lot more pressure than if I replaced the diagonal brick with the liftarm.

Most things aren't named after the person who invented them. That doesn't make it incorrect to refer to things by their names.

[BEAVeR]

My guess would be that the blue one would be sturdier, since a lever produces a greater force towards the fulcrum. So, if you keep the problem simple and don't look at whether those beams are better in compression or in bending, a rule of thumb could be: the farther you point of support is from the fulcrum, the more load it can hold.

[OzShan]

It's a trade off between having Max bending in the upright or the cross beam. Which to you want to flex more? Is the load out at the tip or spread across?

[Lipko]

I think it depends on the type of load and how/where you apply it. Would it be a single force at the end (all the weight is supported by only small region at the end)?

Or is the weight well distributed all along the horizontal beam?

If the latter, I'd go for the red solution, as the diagonal beam is more vertical and puts less horizontal force (thus less bending torque) on the vertical beam.

Actually, for this reason I would go for a diagonal beam which is closer to vertical if the vertical beam is very long (so you can only attach the diagonal beam to the middle of the vertical beam).

If the load is applied at the end of the horizontal beam, I would go for a longer diagonal beam with an attaching point closer to the load application place. But this will introduce stability ("buckling") issues, so you would have to add another diagonal beam which connects the middle of the main diagonal to the corner of the vertical and horizontal beams.

(TBH, I don't do much mechanics stuff nowadays and I haven't put too much thought into the matter)

[drdesignz]

I do structural engineering work for a living. Basically, what you have is a column, beam, and brace, that creates a cantilevered frame. The strongest version depends on where to load is applied. Without really knowing the final version of your design, or how the column is being supported, I'd say attach the end of the brace to the beam closest to the center of the load. If the load is closer towards the end of the beam, you want the blue version. If it's closer to the center, red. That said, there isn't that much strength difference between the two versions. It's likely that either would work well for your application.

[DrJB]

The angles are the same in both cases. The difference between them is the orientation - whether the small angle is on the horizontal or vertical brick.

Surely you mean "as well as" rather than "instead of" for the liftarm. I would wager that what I've got there would withstand a lot more pressure than if I replaced the diagonal brick with the liftarm.

Most things aren't named after the person who invented them. That doesn't make it incorrect to refer to things by their names.

Not to dwell on this longer than needed ... but Pythagora's theorem only refers to the relationship between the lengths of a triangle (whether it has a right angle or not). It has got nothing to do with the perfect squares, which is really attributed to Fermat ... But, let us get back to Lego. shall we?

Edited April 19, 2014 by DrJB

[vmln8r]

In the corner, you could also do the ubiquitous technicbrick-plate-plate-technicbrick thing (the distance between the holes on the pair of technicbricks is three studs).

Not to dwell on this longer than needed ... but Pythagora's theorem only refers to the relationship between the lengths of a triangle (whether it has a right angle or not). It has got nothing to do with the perfect squares, which is really attributed to Fermat ... But, let us get back to Lego. shall we?

Huh? Why the but? Captainowie's point still stands, right?

[Captainowie]

I do structural engineering work for a living. Basically, what you have is a column, beam, and brace, that creates a cantilevered frame. The strongest version depends on where to load is applied. Without really knowing the final version of your design, or how the column is being supported, I'd say attach the end of the brace to the beam closest to the center of the load. If the load is closer towards the end of the beam, you want the blue version. If it's closer to the center, red. That said, there isn't that much strength difference between the two versions. It's likely that either would work well for your application.

Thanks, that's exactly the answer I was looking for.

[Captainowie]

Not to dwell on this longer than needed ... but Pythagora's theorem only refers to the relationship between the lengths of a triangle (whether it has a right angle or not). It has got nothing to do with the perfect squares, which is really attributed to Fermat ... But, let us get back to Lego. shall we?

Ok, I'm sorry for pursuing this, but I can't let it slide. What you've said is simply not true. The only relationship that I can think of between the lengths of an arbitrary triangle is that each side must be shorter than the sum of the other two (otherwise the shape wouldn't be able to close). Pythagoras' theorem applies ONLY to right-angled triangles.

You're right that nothing in the theorem restricts the solution to whole numbers, but as you said earlier, the existance of the 3-4-5 triangle has been known for thousands of years, and has nothing to do with Fermat. You're perhaps getting confused with Fermat's Last Theorem, which states that there are no whole-number solutions to equations of the form x^n + y^n = z^n for any n greater than two. We know there exist solutions for n = 2, because that's just Pythagoras' theorem. We also know there exist solutions for n = 1, because that's just adding numbers. But there is no perfect cube that is the sum of two other perfect cubes, and so on for higher powers.

Owen.

[DrJB]

Fair enough ... I 'misread' your original post as Pythagora's 'Triangle' .... which I often see mentioned. My point was much simpler than the ensuing 'debate'. I often see people refer to (3,4,5) as pythagora's triangle .. when in fact there is no such thing. (3,4,5) is called an Egyptian triangle, and it does satisfy Pythagora's. As for my excursion into Fermat ... I am not confused as I did not want to venture into 3D bracing as I'm not sure one can do that with Lego parts. Incidentally, there is another triangle (5,12,13) and it satisfies Pythagoras, though it is not called as such.

Lastly, this is not about letting it 'slide' as I see this as a 'healthy' debate ... and hopefully we both would walk away having learned something.

Cheers.

Edited April 20, 2014 by DrJB

[Captainowie]

I'm glad you agree that this is a 'healthy' debate - there aren't many places on the Internet where such things don't disintegrate into name-calling and worse.

I have not seen (3,4,5) called "Pythagoras' Triangle" - to my mind that triangle would be no more Pythagorean than any other right-angled triangle. However, triplets of numbers such as (3,4,5), (5,12,13), (7, 24, 25), and (infinitely) many more besides, as well as all their integer multiples, are known as Pythagorean Triads, or Pythagorean Triples.

To insist that the (3,4,5) triangle is properly called an Egyptian Triangle and not Pythagoras' Triangle on the basis that the Egyptians came before Pythagoras is to assert that it was discovered by the Egyptians. Certainly we know they used it, but what evidence is there that they didn't copy it from the Babylonians? Or the Arabs? Or the Chinese? (okay, maybe China is not very likely).

Incidentally, the sole reason I mentioned him in the first place is so that readers who maybe didn't like having to sit through that stuff in school can see that it's not so useless and irrelevant after all.

I'm not sure what you mean by 3D bracing - if you mean something like connecting the diagonally opposite vertices of a box, then the appropriate equation is a^2 + b^2 + c^2 = d^2 (where a, b, and c are side lengths of the box) which again has nothing to do with Fermat. But I'll bet you can do it with LEGO! Though it might not be particularly structurally sound, I'll grant you that.

Owen.

[Technyk32231]

If you're using Pythagorean triples, you have to add one to each number to get the stud length. For example, to make a 3,4,5 triangle, you would use a 4L, 5L, and 6L liftarm. Anyway, I'm pretty sure the orientation doesn't matter, as long as the longest side isn't one of the legs forming the right angle. I hope this makes sense, because I'm not really the best explainer.

[DrJB]

You proved my point ... I did not know of the (7, 24, 25) triplet.

[Erik Leppen]

Of the original two images I'd say the blue version is slightly better because the connection with the 6-stud brick is on the longer side, meaning the horizontal 16-brick has a little less play if the two-pinned connection between the two horizontal bricks bends.

But that's really the only difference. If you use two 2x4 plates covering the connection with the 6-stud brick, the two are equivalent, because they have the same triangle (6/8/10).

By the way, have people noticed that in bridges and other structures where a rectangle needs to be braced with a diagonal, most often the diagonal that is in tension is added, because the diagonal is longer than the side, and it's best to put the longest edges in tension. This is because when an edge in compression it matters how long it is - a long edge buckles sooner than a short edge - while for an edge in tension the length doesn't matter.

See image below for what I mean. The diagonals are in tension.

[Captainowie]

By the way, have people noticed that in bridges and other structures where a rectangle needs to be braced with a diagonal, most often the diagonal that is in tension is added, because the diagonal is longer than the side, and it's best to put the longest edges in tension. This is because when an edge in compression it matters how long it is - a long edge buckles sooner than a short edge - while for an edge in tension the length doesn't matter.

Surely that's more to do with the properties of the material concerned? i.e you put steel in tension, because it buckles under compression. On the other hand, if you were building in concrete, you'd put the braces under compression, because concrete crumbles under tension. That's why steel-reinfocred concrete is so strong - the concrete resists compression and the steel resists tension.

[Lipko]

Surely that's more to do with the properties of the material concerned? i.e you put steel in tension, because it buckles under compression. On the other hand, if you were building in concrete, you'd put the braces under compression, because concrete crumbles under tension. That's why steel-reinfocred concrete is so strong - the concrete resists compression and the steel resists tension.

I would be surprised if a concrete beam with the same width would be more stable than a steel beam (or maybe it would be more fair to compare beams with the same mass per metre). I believe the reason for using concrete in compressed elements is simply that it's cheaper than steel (or maybe it has also some better non-mechanical properties that's beneficial in certain situations).

[Blakbird]

Surely that's more to do with the properties of the material concerned? i.e you put steel in tension, because it buckles under compression. On the other hand, if you were building in concrete, you'd put the braces under compression, because concrete crumbles under tension. That's why steel-reinfocred concrete is so strong - the concrete resists compression and the steel resists tension.

You won't find any trusses made of concrete.