My Math Forum (Compressive - tensile) stresses

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 January 10th, 2018, 10:21 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 412 Thanks: 27 Math Focus: Number theory (Compressive - tensile) stresses What is the widest arch constructible from cubes, each of side S?
 January 11th, 2018, 03:14 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,644 Thanks: 2084 There's no maximum width.
January 11th, 2018, 06:31 AM   #3
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Quote:
 Originally Posted by skipjack There's no maximum width.
Perhaps Loren meant something like this:

What is the maximum length of a beam that can be constructed such that the sag in the middle of the beam is less than or equal to S. This will depend on the material properties of the beam, so assume the the cross-sectional area of the beam is a square of area S^2 and that the beam length is an integer multiple of S. The height of the beam (using the mid-point of the beam) is 3S/2.

I'll see if I get time to answer this one, but my engineering is rusty for sure!

 January 11th, 2018, 09:57 AM #4 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 412 Thanks: 27 Math Focus: Number theory I recall the book stacking problem, perhaps important for an arch of cubes: Book Stacking Problem -- from Wolfram MathWorld Also, does friction play a role in this situation where it might not in a standard arch?
January 12th, 2018, 12:52 AM   #5
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Quote:
 Originally Posted by Loren I recall the book stacking problem, perhaps important for an arch of cubes: Book Stacking Problem -- from Wolfram MathWorld Also, does friction play a role in this situation where it might not in a standard arch?
But that is not an arch. It is a corbel.

And a beam is not an arch, though I once designed a bridge with a beam curved to look like an arch.

skipjack is right there is not (theoretical) maximum width, but what he didn't say is that you have to be prepared to allow an increasing rise with increasing span.

So there is a maximum width for a given rise.

This span to rise ratio was a serious limitation in early bridge construction.

loren, you did not say what sort of arch (i.e. how flat).
Semicircular arches have the greatest rise, but you can make them flatter by using another profile.

January 13th, 2018, 11:46 AM   #6
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Quote:
 Originally Posted by studiot But that is not an arch. It is a corbel. And a beam is not an arch, though I once designed a bridge with a beam curved to look like an arch. skipjack is right there is not (theoretical) maximum width, but what he didn't say is that you have to be prepared to allow an increasing rise with increasing span. So there is a maximum width for a given rise. This span to rise ratio was a serious limitation in early bridge construction. ...loren, you did not say what sort of arch (i.e. how flat). Semicircular arches have the greatest rise, but you can make them flatter by using another profile...
Is the strongest arch a semicircle, or is it a parabola? A corbel is like a door with a lintel? Can a corbel, as opposed to an arch, be constructed simply with multiple, free standing components like cubes?

Quote:
 Benit13 ...assume the the cross-sectional area of the beam is a square of area S^2 and that the beam length is an integer multiple of S. The height of the beam (using the mid-point of the beam) is 3S/2.
If the "sag" is any greater than S, will the cubic structure fall apart? Thank you for your generalization.

January 13th, 2018, 03:09 PM   #7
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Quote:
 Originally Posted by Loren Is the strongest arch a semicircle, or is it a parabola? A corbel is like a door with a lintel? Can a corbel, as opposed to an arch, be constructed simply with multiple, free standing components like cubes? If the "sag" is any greater than S, will the cubic structure fall apart? Thank you for your generalization.
Let's tackle the corbel first.

A lintel has two supports and acts like a beam to span an opening (eg window or door) and support the wall above. So it is supported on either side.

A corbel is built out like the pieces in your drawing from one side only.
So would be used at the edge of a building to support an overhanging roof.
Usually the pieces are plank shaped to get maximum forward extension past the edge of the building.
If you had an opening you could build two corbels, one from either side, which just touched in the middle, but were each self supporting without the other.
Some bridges are built on a version of this principle.

Both beams, corbels and lintels ( I think you americans call them lintols ?) are subject to bending forces, arches are not (or should not be).

The blocks that go to make up an arch are called voussoirs and are specially shaped, they are not cubical, they are tapered to follow the radial lines of the curve of the arch.
The inside of the arch is called intrados or soffit, the outer surface is called the extrados.

All the blocks that make up the arch ring are held together in compression only, so the maximum allowable stress will be the crushing strength of the particular block. This maximum occurs along the intrados line of the arch.

The false arch I mentioned is a reinforced concrete beam with the bottom surface (soffit) curved concave downwards to resemble a masonry arch.
Being a road bridge, the deflection is imperceptible (by design).

 Tags compressive, stresses, tensile

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