These elevator are most likely constructed (in a very basic sense) as a taut line between earth and a geosynchronous orbital counter weight. The tether between the two is kept taut by the outward (from earth's perpective) force the weight exerts. Because of the lack of protection from space radiation, the elevator is unlikely to be used for biological transport of any kind, therefore it's a reasonable assumption that a space dock would be on the other end, where frigate-class space-ships too large for re=entry can refuel or restock easily.
The tether, due to the incredible force keeping it taut must be both felxible and incredibly strong. Either a complicated carbon-based weave of microtubes or something of that nature. Earth cannot have an orbital elevator as it stands as it's gravity is far too strong, however, should we colonise the Moon, it would be feasible to construct one there.
It is uncertain the effect this would have on the Moon's orbit and Earth's Solar orbit.
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