APPENDIX I

By A. F. Zahm

As inventors frequently propose the construction of a vacuum balloon, to secure buoyancy without the use of gas, it may be desirable to estimate the strength of material required to resist crushing, say in a spherical balloon.

The unit stress in the wall of a thin, hollow, spherical balloon subject to uniform hydrostatic pressure, which is prevented from buckling, is given by equating the total stress on a diametral section of the shell to the total hydrostatic pressure across a diametral section of the sphere, thus:

2πrtS = πpr²

in which S may be the stress in pounds per square inch, p the resultant hydrostatic pressure in pounds per square inch, r the radius of the sphere, t the wall thickness.

The greatest allowable mass of the shell is found by equating it to the mass of the displaced air, thus:

4πr²tς₁ = 4πr³ς₂/3

in which ς₁ is the density of the wall material, ς₂ the density of the atmosphere outside.

Now, assuming p = 15, ς₁/ς₂ = 6,000, for steel and air, the equations give:

S = 3pς₁/2ς₂ = 45 × 6,000/2 = 135,000 pounds

per square inch as the stress in a steel vacuum balloon.

For aluminum ς₁ is less, but the permissible value of S is also less in about the same proportion.

The last equation shows that for a given material and atmospheric environment, the stress in the shell or wall of the spherical balloon is independent of the radius of the surface. It is also well known that the stress is less for the sphere than for any other surface. Hence, no surface can be constructed in which S will be less than 3pς₁/2ς₂. The argument is easily seen to apply to a partial vacuum balloon, since a balloon of one nth vacuum will float a cover of but one nth the mass and strength.

The above result was obtained on the assumption that the shell was prevented from buckling. As a matter of fact, it would buckle long before the crushing stress could be attained. We must conclude, therefore, that while a vacuum balloon has alluring features, the materials of engineering are not strong enough to favor such a structure. Perhaps it is nearer the truth to say that such a project is visionary, with the materials now available.

A like argument applies to the balloon reservoir in which it has been proposed to compress the surplus gas taken from a balloon hull on expansion of its contents by change of level or temperature. If a given mass of gas obeying Boyle’s law be pumped into a receiver of given shape and mass, the resultant stress in the receiver wall will be independent of the size. Hence the material of the proposed reservoir, if expanded to the size of the hull itself, will weigh the same, and suffer the same increment of unit stress, for a given mass increment of gas. Hence, instead of pumping the above-mentioned gas surplus from the hull into the reservoir, this latter may be discarded and its mass of material spread over the hull itself. This argument applies only if the shapes of hull and reservoir be equally effective, as, for example, if both be cylindrical.