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Taper ratio is a central concept for space elevators and tethers. It tells how much larger the cross section of the tether has to be at one end of the tether, compared to the other. Being a dimensionless quantity, it is simply found by:

$$R_{taper}=e^{\frac{Y}{S}}$$

Where $S$ is the tensile strength of the material divided by its density, giving the unit $m^2/s^2$, and $Y$ is the acceleration experienced by the tether, integrated over its length. (again $m^2/s^2$).

This works nice and well for a tether, which is stretched, and that is also the most common application, as materials tends to have a greater tensile strength than compression strength. For instance, Zylon has a tensile strength of $5.8GPa$, which together with a density of $1560kg/m^3$ gives it an $S$ of $3.72Mm^2/s^2$.

By that reason, space elevators are supported from the top. However, by the exponential nature of the mechanics, supporting half the elevator ("half" is here referring to the acceleration-distance integrated quantity, not mass or distance.) from the bottom should give a taper ratio equivalent to the square root of a top-only design. That is only the case when the $S$ of the ground-supported structure is equal to the $S$ of the tether. That is not the case. For finding actual taper ratio improvements for an elevator with dual support, one has to know a reasonable value for $S_{ground}$. An updated formula for taper ratio is surprisingly then just:

$$R_{taper}=e^{\frac{Y}{S_{tether}+S_{ground}}}$$

(yes, you can simply add them!)

Data for compression strength of various materials is easily available, however, it is insufficient as the design of the structure is an important factor when it is no longer a tether.

What is a reasonable value for $S_{ground}$, in $m^2/s^2$, for a supporting, compressing strength limited structure?

(Note that this quantity is independent of surface gravity.)

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  • $\begingroup$ Supporting half the tower from the bottom requires a building 18,000 km high. I think you'll have to assume the entire elevator is supported from the top, any support from the bottom is small enough to be ignored. $\endgroup$ – Hobbes Apr 18 '16 at 18:35
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    $\begingroup$ @Hobbes This question is just to get an approximate value of the differences between $S$ for compression and stretching. "Halving the distance" (18000km) is an extremely wrong approximation in this case, as what counts is the acceleration-over-distance integral, not length. It is also worth noting that the exponential nature of this is on pair with that of the rocket equation, so the ratios may be counter intuitive. I am perfectly aware about that this is a minor effect, but S_tether-over-S_ground is still an interesting question. $\endgroup$ – SE - stop firing the good guys Apr 18 '16 at 18:40

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