Physics of Triangles: Why Are They the Strongest?

[dreit]

I have always heard that triangles are the strongest geometric shapes. Can anyone explain why this is so?

 

[Studiot]

I think your confidant was probably referring to pin jointed trusses in the theory of structures.

A pin jointed triangle is the only polygon (regular or irregular) that can enclose area and support load without bracing. Bracing may be either corner stiffening or diagonal bracing.
In the case of diagonal bracing the structure is reduced to an assembly of simple triangles and is said to be triangulated.

 

[schip666!]

Triangles are strong because they are the only polygon in which the angles at the corners cannot change without an associated change in the length of an edge. All other polygons can change shape by just wiggling a couple angles someplace, their strength is dependent on the strength of the corner connection itself. With a triangle an edge must collapse in order to change the triangle's shape and edges can be made way stronger than connections.

 

[Studiot]

Triangles are strong because they are the only polygon in which the angles at the corners cannot change without an associated change in the length of an edge.

Are you sure? I suggest you draw a few.

 

[sophiecentaur]

Are you sure? I suggest you draw a few.

? which bit of the statement are you doubting? Or is he being sloppy in some detail?

 

[schip666!]

Are you sure? I suggest you draw a few.

Well, I <u>have</u> drawn a lot of triangles in my time and never been able to get any of them to collapse without cutting an edge... Maybe I was wrong about the "all other polygons" statement? I have not done an exhaustive search -- just mentally constructed a square and dismissed the rest -- hmm, maybe odd-number polygons have the un-collapsible feature? I'll have to go make a pentagon and see...

 

[schip666!]

Nope...a pentagon wiggles... onward to the hexagons!

 

[Studiot]

Triangles are strong because they are the only polygon in which the angles at the corners cannot change without an associated change in the length of an edge.

Nobody said the polygons have to be regular, so I have drawn one regular and one irregular example of where the angle changes substantially, from A to B, without change in the length of any side.

The first case is simple shear, the second is inversion of two sides.

 

[JaredJames]

Nobody said the polygons have to be regular,

No, he said any, regular or irregular. So I don't understand why you included this statement.

so I have drawn one regular and one irregular example of where the angle changes substantially, from A to B, without change in the length of any side.

Which is exactly what he said. Polygons can have angle changes without a change in side length. Followed by something along the lines of: triangles are the only shapes which you cannot change an angle without changing a side length.

 

[Dr Lots-o'watts]

What about arches and domes?

 

[JaredJames]

What about arches and domes?

I was thinking that, the stone arches support massive weights thanks to their shapes. If anything, I'd say more so than triangles (please correct if I'm wrong).

 

[Studiot]

Yes Jared, you are quite right, I didn't read it properly. Thank you for correcting me. Apologies to Schip.

A triangle is the simplest form of 'arch'.

 

[Dr Lots-o'watts]

Incidentally, here's a nice picture a friend took of the Quebec city bridges. One is made out of triangles and the other, arches.

 

[schip666!]

Nobody said the polygons have to be regular, so I have drawn one regular and one irregular example of where the angle changes substantially, from A to B, without change in the length of any side.

The first case is simple shear, the second is inversion of two sides.

Ah. I see. I was perhaps too glib in my specifications... What would be a better way to say it? The elements -- angles and sides -- of a triangle cannot change unilaterally? If an angle changes a side must also change? There must be a way to state it succinctly, but I'm somewhat dull just now.

In your rectangle case all four angles have to change at once if the sides remain the same. In your irregular example you have two final configurations, where two of the angles have changed, but the traverse between (in 2D) requires at least one more angle change at some other vertex. How could we work all that into the explanation without hiring a lawyer?

Maybe I should have said that given any three line segments you can make only ONE triangle, and let the consequences fall where they may?

 

[sophiecentaur]

@schipp
Perhaps you could say that for a triangle, if you change the length of one side, the length of another side must change. That's the only polygon for which that applies - in all other polygons the angles can change to accommodate the change in length of one side, withought another side having to change length.
The amazing thing was that we all (apart from studiot) just accepted your original, somewhat loose statement because it 'seemed' right.

No - I just read that and it's still not right.
Over to you. It must be something to do with angles too. Like, if you change one angle, at least one side must change length. Is that better?

 

[Studiot]

Schip is correct, even the case of ambiguity in the sine rule for triangles requires different side lengths. He has a better explanation than my original and lead to the reason that the pin jointed triangle is the only polygon that cannot become a mechanism.

Schip I hope you have also read post#12?

I don't follow the relevance of the (nice) photos of the Quebec bridges?

 

[schip666!]

@schipp
Over to you. It must be something to do with angles too. Like, if you change one angle, at least one side must change length. Is that better?

Oy...It's got to have something to do with it, yah...

The OP question was "why are triangles so strong?" and the answer has to do with using the strength of the sides in tension or compression, rather than relying on the angles to remain constant at the vertices. Studiot's square is the counter example because the sides and vertices are independent.

Well, all but one of you knew what I meant anyway...

 

[schip666!]

Yes, Apollo Gee graciously given and accepted. But we still haven't really nailed that danged definition...

And now I'm not sure that arches are the same as triangles because the load is distributed differently. Maybe as a simplification?

 

[Studiot]

The OP was really too vague, and had not elaborated since.

Strongest shape for what?

Cross section of a beam?

Tension structure like the Quebec suspension bridge?

Forming part of the tetrahedral structure of diamond?
Or forming part of the hexagonal structure of graphite?

Or...?

 

[Dr Lots-o'watts]

I don't follow the relevance of the (nice) photos of the Quebec bridges?

Just a photo I suddenly realized I had where triangles and arches were actually living side-by-side. Giving life to an interesting thread until the OP returns. I also live there.

 

[Archosaur]

I see the triangle debate is settled, no?

As far as domes/arches go, the strongest shape is the "catenary". It is the shape a piece of string makes when it is suspended between two points. (note: it's NOT a parabola, or any conic section, for that matter). Flip that curve upside down and you have the best possible load bearing arch. Why is that? Because at every point on the curve, force is directed directly along the curve. There is no sheer stress, only compression.

http://en.wikipedia.org/wiki/Catenary"

 

[JaredJames]

I see the triangle debate is settled, no?

As far as domes/arches go, the strongest shape is the "catenary". It is the shape a piece of string makes when it is suspended between two points. (note: it's NOT a parabola, or any conic section, for that matter). Flip that curve upside down and you have the best possible load bearing arch. Why is that? Because at every point on the curve, force is directed directly along the curve. There is no sheer stress, only compression.

http://en.wikipedia.org/wiki/Catenary"

Very interesting, did not know that.

 

[sophiecentaur]

Gaudi, the famous Catalan architect with buildings all over Barcelona used to develop his dome structures using simple but upside down versions using strings and weights suspended. All the forces were tension and he could measure the stresses easily and make modifications as he approached an optimum structural design. Making the dome, later, with struts (all in compression), he already knew the stresses involved and 'got it right' without the availability of fancy computer progs.

 

[schip666!]

I _knew_ there was a reason I liked Gaudi...
thx

 

[dreit]

Thanks for all of the responses reguarding the triangles, and I had no idea about the catenary part. The wikipedia page actually put the whole concept into perspective.

 

[Archosaur]

I'm glad you liked it. Yea, it's a cool shape that not enough people know about.