ESG7@DFVLROP1.BITNET (10/25/86)
I have been observing the debate between Robert Maas and John Hogg with
growing irritation. Maas proposed the idea of lighter-than-air vacuum
filled metal structures while John has been trying to argue against this
odd notion. One is reminded of the old saying that, "one should never
argue with a fool because others might be unable to tell the
difference". However I believe that pseudo-scientific arguments should
be slapped down forcefully. The notion of lighter-than-air metals based
on a vacuum containing structure can be rigorously disproven. The
geometry that can best withstand compression is a sphere. If a hollow
sphere will collapse under pressure then all other hollow geometries
will also fail. The equation for the classical buckling pressure of a
sphere can be found in the "Handbook of Engineering Mechanics" by W.
Fluegge. The equation is
pcr=2.0*E*((t/r)**2)/((3.0*(1.0-(nu**2)))**0.5)
with the assumption of t<<r , where E = Young's modulus,
nu=Poisson's ratio, t=wall thickness, r=radius
pcr=critical buckling pressure. The mass of a hollow sphere
with the assumption of t<<r is m=4*dnsmtl*pi*(t/r)*(r**3) where
dnsmtl=densitiy of the metal. By Archimedes principle for an object to
be bouyant, its mass must be less than the mass of the air that it
displaces. This leads immediately to the inequality
(t/r) < (1/3)*(dnsair/dnsmtl) where dnsair=air density. This
new inequality validates our original assumption of t<<r. We may now
derive form the buckling equation the inequality constraining a
lighter-than-air structure based on vacuum which is:
(P/E) < (2/(9*((3*(1-(nu**2)))**0.5)))*((dnsair/dnsmtl)**2)
where P=atmospheric pressure consistent with the density of air, dnsair.
We note the thickness ratio t/r has dropped from the equation. No
manipulating of the thickness will effect the outcome of the inequality.
We also note that the air density increases as a square against the
pressure. Under the perfect gas law p=dnsair*R*T where R is the gas
constant and T is the absolute temperature. The absolute temperature
only varies by about 25% in the first 10,000 meters, so as a first
approximation we can assume that pressure is proportional to density.
Based on this we see that the inequality is most likely to work at sea
level where density is highest and thereby benefitting from its being
squared. Steel has one of the highest strength-to-weight ratios, so it
would serve as a best case for this problem. We now plug in numbers for
steel at sea level and find that P/E = 4.9e-7 and
(2/(9*((3*(1-(nu**2)))**0.5)))*((dnsair/dnsmtl)**2) = 3.21e-9
The inequality fails by over two orders of magnitude. This conclusion
is an obvious one. I will not comment on how this reflects upon the
intelligence of Robert Maas who proposed this idea and then so staunchly
defended it against John Hogg's criticism. I hope that If Maas wishes
to continue to defend this idea, he would make this a private
discussion or move it over to the SF-LOVERS forum where ideas of similar
quality are often presented.
Gary Allen