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To my understanding the main problem with space elevators is that the strength to weight ratio is not high enough to support the cable's weight. Naturally the first thought that would come to mind is to use a cord with varying thickness. Ie. the cord gets thinner as it gets closer to earth. Why is the strength to weight ratio still a problem?

Quoted from wikipedia: Artsutanov's idea was introduced to the Russian-speaking public in an interview published in the Sunday supplement of Komsomolskaya Pravda in 1960,[10] but was not available in English until much later. He also proposed tapering the cable thickness so that the stress in the cable was constant. This gave a thinner cable at ground level that became thickest at the level of geostationary orbit.

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    $\begingroup$ I would guess that it's probably because even if you taper the cable, our current level of technology is not enough to manufacture a cable of sufficient strength and lightness to make a space elevator. $\endgroup$ – Phiteros Sep 10 '16 at 7:59
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    $\begingroup$ The cable taper has to be exponential for constant stress; for modern materials the characteristic length (strength-to-mass ratio divided by gravitational acceleration) is small relative to geostationary orbit altitude, which means the top of the cable would end up being too thick. $\endgroup$ – 2012rcampion Sep 10 '16 at 8:12
  • $\begingroup$ One problem is manufacturing technology. We have not been able to manufacture long "strings" without flaws or defects at the molecular/crystalline level. The longer the "string" the more flaws it will have & each flaw introduces weaknesses which reduces the overall strength. $\endgroup$ – Fred Sep 10 '16 at 14:19
  • $\begingroup$ I don't have enough background info to answer reliably, but isn't the issue of attaching an elevator cabin to the cable to do anything useful with it a slight problem, as well? $\endgroup$ – O. R. Mapper Sep 10 '16 at 15:31
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It's the exponential nature of the equation determining the thickness.

Let's discretize the problem to make it simpler to picture: you have a thread of very strong material, that has a self-breaking length of two kilometers. That means, if you hang more than two kilometers of this rope on iself, it will break under own weight.

Let's cut it in kilometer long pieces, so each piece can hold another time its weight. Then let's try to extend it.

First two segments are single strings, each a kilometer long, first one with 50% redundancy, the other at breaking limit.

The third kilometer must hold the weight of two lengths - so it must be two such pieces in parallel.

Fourth kilometer holds weight of four lengths (the initial two and the segment with two), so four ropes in parallel.

Fifth - eight times the thickness. Sixth - sixteen. Seventh - thirty-two. To add the twelth kilometer, the thickness must be 1024 times the initial.

It's not actually AS bad, as with altitude gravity starts dropping both with distance of the central body and with fraction of orbital velocity. But it's a long time and a huge stress before we get there and the thickness starts stabilizing, then dropping as it extends past GEO. Still, with current material engineering, the amount of material needed is huge - and in case of materials less durable than nanotubes the exponential explosion means there's not enough material on Earth to build it, as it quickly gets into continent-sized thicknesses..

One of earlier estimates for using nanotubes gave the thickness of 1km around GEO for 1 ton payload. There are newer ones that give more promising estimates, but that should give you a clue why this is a problem.

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    $\begingroup$ The good news is as you reduce the raw ressources, of planets earth, we now have a lesser gravity and need less materials ! The bad news is we are now having a very weird orbit pattern. $\endgroup$ – Antzi Sep 12 '16 at 8:06
  • $\begingroup$ @Antzi: Since the elevator is geostationary, it remains stationary relative to the gravitational anomalies, its orbital pattern unaffected - and at GEO altitude the discrepancies are all but gone; the problem is only valid for LEO - and once we have the elevator, who cares about LEO? $\endgroup$ – SF. Sep 12 '16 at 9:32

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