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From: ESG7@DFVLROP1.BITNET
Newsgroups: sci.space
Subject: A mathematical refutation of vacuum filled lighter-than-air metal structures
Message-ID: <8610251522.AA12069@s1-b.arpa>
Date: Sat, 25-Oct-86 11:25:43 EST
Article-I.D.: s1-b.8610251522.AA12069
Posted: Sat Oct 25 11:25:43 1986
Date-Received: Sun, 26-Oct-86 02:15:12 EST
Sender: daemon@ucbvax.BERKELEY.EDU
Organization: The ARPA Internet
Lines: 95

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