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