It simply seems improbable that a space elevator would stay in space. It couldn’t be sent into orbit, since it would orbit faster than its tether and be pulled into the atmosphere. It couldn’t be kept aloft by centripetal force, since gravity is still stronger. And it simply seems infeasible to attach rockets to the top and keep them firing all the time. So. How does a space elevator stay in space?
Geostationary orbit on carbon fiber tethers…
From those links you should be able to see that the elevator would be in orbit, but in a geostationary orbit, like the communication satellites that are up there now. It would not move with respect to the surface of the Earth.
In theory the tether would stretch in two directions from that location: away from the Earth and towards the Earth, so that the centre of gravity of the elevator would remain at the geostationary point.
When the elevator is long enough one end would touch the ground and the other end would stretch tens of thousands of kilometers past geostationary orbit, but the centre of gravity would still remain in the same place.
That’s the theory, anyway- there are lots of difficulties with making one, but the basic idea is sound.
First of all, you meant to say “centrifugal”, there, not “centripetal”. Centripetal force is the force (pointing towards the center) in the non-rotating coordinate system which causes an object to move in a circle, which in this case is gravity. Centrifugal force is the apparent force in the rotating coordinate system which points away from the center. Which one is appropriate to use depends on the coordinate system you’re in.
Second, the balance between the centrifugal force and gravity is precisely what an orbit is, so those two explanations are equivalent.
And third, an orbit (however you describe it) wouldn’t work near the Earth’s surface, but as you go higher, the centrifugal force gets stronger, while gravity gets weaker. Get high enough, and they balance (an orbit at this height is geosynchronous). The central part of the cable is supported in this way, and the higher and lower parts are supported by hanging from the central parts.
I smile whenever I read about a Space Elevator. Star Trek Voyager had an episode about one in Feb, 1997. Science Fiction often foreshadows the real thing long before its possible to do.
How the hell could you protect such a structure from sabotage?
Or storms? I can imagine what wind could do.
I think this is in understandable mistake on the OP’s part, as every single reference I’ve ever seen in my life to “centrifugal force” has always been followed by “no, you meant to say ‘centripetal’.” 
The Wikipedia article mention Tsiolkovky’s proposed “tower” in 1895. Just a leetle before ST-V. More recent proposals date back to the 1950s, when Roddenberry was writing for cowboys with horses instead of starships.
We’re a long way from feasibility of a space elevator. I’d like to see at least a few decades of experience with carbon nanotube structures before we try something so big and risky. Kim Stanley Robinson’s “Red Mars” talks about the consequences of a fallen space elevator on barely-inhabited Mars. Imagine the disaster if an Earth space elevator was taken down, and wrapped itself around the equatorial region a time or two.
An object, such as a space station, in Clarke (geosynchronous) orbit remains directly above one point on the equator. It orbits the earth in the same 24 hours the earth takes to rotate. (Note in passing that other orbits than equatorial are possible, but not relevant to this discussion.)
An idealized thought-experiment no-mass tether of sufficient tensile strength dould therefore connect two locations fixed with relation to each other. In practice, of course, you run into two issues: 1. Nothing material is mass-free; and 2. Almost nothing is of sufficient tensile strength.
Both these problems are difficult to deal with but not necessarily insuperable. The mass may be counteracted by continuing the tether on past Clarke orbit, where something massive fixed in a 24-hour ‘orbit’ (by its tether) has a net centrifugal effect, counteracting the pull of gravity on the portion of the tether below Clarke orbit. The tether itself can in theory be made of fullerene, the carbon form discovered a couple of decades ago. There may be other materials with sufficiently low weight and high tensile strength but only fillerene is known at the moment.
Any time anybody says that an invention was from science fiction, I’ll bet them Romney’s $10,000 that they’re wrong. If I lose I’ll pay the $10,000 (after I borrow it from him). If I’m right they pay $10,000 per decade that they’re off. I’d make Romney money on these bets.
Even in science fiction terms, a 1997 cite is pretty bad.
What’s interesting is that both books came out in 1979 and were completely independent. Neither had known the other was writing one.
Two noteworthy things I’ve pointed out in previous space elevator threads:
First, a space elevator doesn’t necessarily have to be the same thickness it’s whole length: if it tapers from the geosynch part down to the ground, a lower tensile strength is acceptable, since the thicker part can support the thinner part. This means that you could use the same steel used in bridge cables to build an orbital tower- IF the tower was a thousand miles thick at the upper end. 
The question becomes, “can you make it of a material that’s both strong enough and economical enough?”.
Second, a space elevator doesn’t necessarily have to be huge. It’s going to be long, but there’s no reason it couldn’t be a ribbon a few centimeters wide and a few microns thick. This is important because it means that the concept is scalable; that is, you can start with a smaller cheaper version and if it works then move up to a bigger one. Figuratively speaking, you can start out with a Curtiss biplane and work your way up to a 747, rather than inventing all or nothing.
A space elevator is only going to be used once if it stays up. I recommend the kind that goes up and down.
Nah, send one elevator per passenger, then put a dozen in each one coming back. Use the remainder as raw material for the spaceport.
I’m a big fan of KSR’s Mars trilogy but it’s easy to grossly overestimate the density of these proposed tethers esp. as they have yet to be invented. It’s not going to be anything like a steel structure collapsing.
Clarke’s The Fountain of Paradise is a very interesting look at the design and building of a space elevator. I’m afraid any such structure would be a terrorism magnet, however. Ideally, we’ll be technologically capable of building one at just about the same time as we’re mature enough not to have some people want to knock it down.
Meh, ANY infrastructure can be a terrorism target. Establish a secure perimeter around it and set up AA SAM batteries and you’re done. It would be a strategic asset so presumably it would be guarded by the military.
I don’t see that has ever been many successful attacks on heavily guarded US military installations.
I worry about the Chunnel, because if it was opened to the seabed it would be virtually irrepairable.