Tuesday, December 25, 2012

Proportionality, strength and buoyancy

I have written several times on this blog about how masonry structures are scaleable.  That is, a given masonry structure can be made larger or smaller and will still have adequate strength, so long as the proportions remain intact.  For example, if a round arch is ten feet in radius and has walls which are one foot thick, then the same design will work with one hundred foot radius and walls which are ten feet thick.

The example cited above is a good one to look at because a “thin-shelled structure” is defined as having a ratio of radius-to-wall thickness of ten-to-one.  This ratio of radius-to-wall thickness provides adequate strength for a masonry arch under earth’s gravity.

If we take this example one step further, and consider not just a semi-circular round arch (or barrel vault, or Roman arch) but we look at a complete sphere, such that the arch describes a completed circle, then we have a spherical structure which can have strength adequate to withstand the pressures at great depth, underwater.

Going further still, if we look at the example of a sphere whose wall thickness is one tenth of its radius, we see that such a sphere will always be buoyant in water, no matter what size it is.  A small hollow concrete sphere 2 inches in diameter, with concrete walls 0.1 inches thick, will have the same proportional buoyancy as a sphere 200 feet in diameter with concrete walls 10 feet thick.

Since the volume of a sphere is (4/3) * pi * r3; the density of concrete is around 2.4 g/cc; and the density of water is around 1.0 g/cc, it is a simple matter to show that the buoyant force acting on a hollow concrete sphere with wall thickness equal to one-tenth of its radius will always exceed the weight of the concrete.  Such a sphere will always float, no matter how big it is or how thick its walls get. (I can show the math, but spare the reader here.  It’s simple stuff).

Given that thicker concrete walls are stronger, a larger sphere (following the scaleability rule for masonry) can be made of great size, with great wall thickness, and can be sunk to great depth and can maintain structural integrity under the great forces found there.  Another feature of increasing the radius of such a hollow sphere is the exponential increase in volume.  As the radius increases linearly, the volume is increased as a function of the radius cubed.

These simple facts of proportionality, buoyancy, strength and volume regarding a submerged masonry sphere are really pretty interesting.  It indicates that a sphere can be used at great depth, if it is made large enough.  The increase in scale will simultaneously provide the economy of scale for tasks such as desalination, as described several times earlier on this blog (here, here and here).  Simple yet elegant.