Vacuum Airship Principles

Conventional airships fill a vessel with some gas having a lower density than the surrounding air.  This creates a volume which weighs less than the air it displaces, creating a net upward buoyant force.  A vacuum airship uses a similar principle; however instead of a lighter-than-air gas a rigid vessel is emptied of the air inside.  If the total weight of the displaced air of the interior is greater than the weight of the vacuum vessel then the vessel will experience a net upward force similar to that experienced by the gas filled balloon.

Recalling the buoyant force:

A vacuum vessel is a form of pressure vessel, one in which the forces experienced are compressive.  The critical pressure difference for a thin-walled pressure vessel, or the pressure difference that will cause the vessel to buckle, is given by:

In terms of the skin thickness, radius, Young's modulus, and Poisson ratio.  For a sphere the volume and surface area are given by:


Mass of the sphere is given by multiplying the surface area, thickness, and density.  By subtracting the mass of the sphere and the air it contains from the mass of the displaced air the total lift can be found.
 
Setting lift equal to zero gives the limiting case of neutral buoyancy.  Combining this with the ideal gas law, and assuming any gas within the vessel is at the same temperature as  the ambient atmosphere, gives a relation for the necessary pressure difference for any tank material choice in terms of the ambient temperature, itself a function of altitude:

Unfortunately at no temperature that exists within the Earth's atmosphere will the necessary pressure difference be one that can be achieved.  This means that no real material can construct a thin walled vacuum vessel capable of displacing more than its own mass.
This is not, as it might initially appear, a fatal flaw for the vacuum airship concept.  Rather it will be necessary to develop a pressure vessel that does not fail by buckling.  Several proposals have been put forward to overcome this difficulty, the most promising of which involve the use of either tensegrities or geodesics.  Such structures distribute loads among different members so that each member suffers only compressive or tensile loads, and thus is not prone to bending or buckling.