Re: [math-fun] reinforced concrete
Great idea! But how accurate is this translation of the catenary idea into 2D? What you really want is to hang a parallel array of chains all parallel to the X axis, and then hang another array of chains all parallel to the Y axis, and then adjust the length of each chain until all X chains touch all Y chains at all intersection points. This way, there is no assumption about the strength of the cloth. In other words, the problem with putting a non-elastic cloth into a catenary-type shape is that the cloth doesn't have enough degrees of freedom to assume the best position. Thus, when the dried concrete shape is turned over, the stresses will not all be tangential. A rubber sheet is somewhat better, but the thickness of the rubber is inverse to what you want: the highest stress will have the thinnest rubber, but that is where you want the thickest concrete. At 02:25 PM 12/27/2014, Allan Wechsler wrote:
The technique I heard about in the 70s was to hang a cloth sheet from a support the same shape as the intended foundation, let it settle into a nice catenary-like shape, and gradually spray it with concrete. You have to do it gradually, in thin, even layers, so that the heavy spots don't drag the cloth down and distort the shape. After it dries, you invert it. Perfect vaulted dome, ideal for bearing its own weight down to the foundation.
Great idea
... already used by Gaudi Best, É. Catapulté de mon aPhone
Le 28 déc. 2014 à 00:59, "Henry Baker" <hbaker1@pipeline.com> a écrit :
Great idea!
But how accurate is this translation of the catenary idea into 2D?
What you really want is to hang a parallel array of chains all parallel to the X axis, and then hang another array of chains all parallel to the Y axis, and then adjust the length of each chain until all X chains touch all Y chains at all intersection points.
This way, there is no assumption about the strength of the cloth.
In other words, the problem with putting a non-elastic cloth into a catenary-type shape is that the cloth doesn't have enough degrees of freedom to assume the best position. Thus, when the dried concrete shape is turned over, the stresses will not all be tangential.
A rubber sheet is somewhat better, but the thickness of the rubber is inverse to what you want: the highest stress will have the thinnest rubber, but that is where you want the thickest concrete.
At 02:25 PM 12/27/2014, Allan Wechsler wrote:
The technique I heard about in the 70s was to hang a cloth sheet from a support the same shape as the intended foundation, let it settle into a nice catenary-like shape, and gradually spray it with concrete. You have to do it gradually, in thin, even layers, so that the heavy spots don't drag the cloth down and distort the shape. After it dries, you invert it. Perfect vaulted dome, ideal for bearing its own weight down to the foundation.
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I believe I saw the technique demonstrated at one of Arthur Loeb's "Design Science" conventions at Harvard. There seemed to be a good deal of art in it. A fairly heavy, sailcloth-like sheet was suspended from a polygonal frame, and adjusted by eye, by pulling slack up over the frame, or letting slack out where it "seemed to need it". A few weights were suspended from various random points of the sheet, marking where the thing was intended to bear load when hardened and inverted. The hardening was done by painting the sheet with a dilute glue, which was allowed to harden, then a thin plaster, and then slathered with a layer of more serious structural plaster. The shape hardened for about a day and a half, and then with great fanfare was inverted. Weights modelling the intended load were placed on it at the points where the original random weights had been hung. It looked very stable. Eric, the technique was not presented as original: it was apparently an idea that was well-known in certain architectural design circles, so it doesn't surprise me that Gaudi used it. On Sat, Dec 27, 2014 at 6:57 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Great idea!
But how accurate is this translation of the catenary idea into 2D?
What you really want is to hang a parallel array of chains all parallel to the X axis, and then hang another array of chains all parallel to the Y axis, and then adjust the length of each chain until all X chains touch all Y chains at all intersection points.
This way, there is no assumption about the strength of the cloth.
In other words, the problem with putting a non-elastic cloth into a catenary-type shape is that the cloth doesn't have enough degrees of freedom to assume the best position. Thus, when the dried concrete shape is turned over, the stresses will not all be tangential.
A rubber sheet is somewhat better, but the thickness of the rubber is inverse to what you want: the highest stress will have the thinnest rubber, but that is where you want the thickest concrete.
At 02:25 PM 12/27/2014, Allan Wechsler wrote:
The technique I heard about in the 70s was to hang a cloth sheet from a support the same shape as the intended foundation, let it settle into a nice catenary-like shape, and gradually spray it with concrete. You have to do it gradually, in thin, even layers, so that the heavy spots don't drag the cloth down and distort the shape. After it dries, you invert it. Perfect vaulted dome, ideal for bearing its own weight down to the foundation.
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Perhaps off the direct topic, but if bricks and mortar and such interest you, this is worth a watch: http://www.pbs.org/wgbh/nova/ancient/great-cathedral-mystery.html
participants (4)
-
Allan Wechsler -
David Wilson -
Eric Angelini -
Henry Baker