How about a 30km tall building? Using multilayered columns using rolled up or rippled graphene and nanotubes, in various patterned cross sections, it should be possible to make strong threads, ribbons and membranes, interwoven to make columns and arrange them into an extremely tall pyramid.
Super-tall structures for science and tourism
Think of a structure like the wood and bark of a tree, with the many tubular fine structures. Engineering can take the ideas nature gives us and optimise them using synthetic materials. Graphene and carbon nanotube will become routing architectural materials in due course. Many mesh designs and composites will be possible, and layering these to make threads, columns, cross members with various micro-structures will enable extremely strong columns to be made. If the outer layer is coated to withstand vacuum, then it will be possible to make the columns strong enough to withstand atmospheric pressure, but with an overall density the same as the surrounding air or less. Pressure is of course less of an issue higher up, so higher parts of the columns can therefore be lighter still.
We should be able to make zero weight structures in lower atmosphere, and still have atmospheric buoyancy supporting some of the weight as altitude increases. Once buoyancy fails, the structure will have to be supported by the structure below, limiting the final achievable height. Optimising the structures to give just enough strength at the various heights, with optimised mesh structure and maximal use of buoyancy, will enable the tallest possible structures. Very tall structures indeed could be made.
So, think of making such a structure, with three columns in a triangular cross-section meeting at 43 degrees at the top (this is the optimal angle for the strongest A frame in terms of load-bearing to weight ratio, though that is a simplistic calculation that ignores buoyancy effects, so it ‘needs more work’.
Making a wild guess, 30km tall structures may be feasible, but that is just a wild guess and I would welcome comments from any civil engineers or graphene architects. These would not be ideal for habitation, since most of the strength in the structure would be to support the upper parts of the structure itself and whatever platform loading is needed. The idea may be perfect for pressurised platforms at the top for scientific research, environmental monitoring, telescopes, space launches, tourism and so on. The extreme difference in temperature may have energy production uses too.
Getting the first 30km off the ground without needing any rocket fuel would greatly reduce space development costs, not to mention carbon and high altitude water emissions.
A simple addition to this would be to add balloons to the columns at various points to add extra buoyancy. I dare not try to calculate how much higher this would permit, but I suspect not all that much more since even with balloons, they cannot give much extra lift once the atmosphere is too thin.
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