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.

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 * r

^{3}; 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.

Where does the size of the sphere happer the efforts to submerge it? At some point it would be nearly impossible to submerge the sphere since the positive bouancy will be working against you....

ReplyDeleteThe work required to sink a buoyant object is the buoyant force times the distance submerged (work = force * distance). The larger the sphere, the greater the buoyant force. The buoyant force per unit volume is constant; there is always buoyant force "working against you." For example, if a sphere is submerged to make fresh water from salt water, then a larger sphere is harder to sink, but it makes more fresh water. The work/volume required to make fresh water like this is constant.

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