Ringworlds are a bit of a SciFi staple but as they're commonly portrayed, the large ones have a major problem... they'd, well... explode if they were built out of materials that actually existed. This, as you can probably imagine, makes habitation rather difficult...
The issue is that as the rings get bigger, the tensile strength needed to hold them together grows with the diameter (assuming the "artificial gravity" remains the same), at some point the ring would need to be spinning so fast for a decent surface gravity that it'd rip itself apart. As Gav and Dan demonstrate with this poor, innocent CD. Presumably this would result in greatly reduced house prices on the ring...
In fact, for an Earth-like surface gravity, an Earth-like day-night cycle and an average density about that of carbon, the tensile strength needed to hold the ring together is 350 times greater than the strongest material known to man, Graphene...
This got me thinking... what if, instead of having just the spinning ring, there was a slightly larger stationary ring weighing down on the rotating one? Given that a ring of this size would end up using at least one whole planet to build, there's got to be an awful lot of gravity trying to crush the whole thing down into a nice neat sphere (gravity is good like that).
Assume that there is a construction crew big enough for the job, that there's enough material to build both these rings (the outer ring can literally be made out of any old rubbish) and there's some way to build a friction-less bearing (super-maglev?) between the two that can handle the immense pressure between the inner and outer rings. We have plenty of energy (i.e. someone's sun) and a couple planets to play with but no new physics or material science.
So, assuming all that, is this way of building Ringworlds actually possible using current (albeit massively scaled up) science?
Update: A couple people have pointed out that there needs to be a way of keeping the stationary ring from touching the rotating ring while still balancing the huge forces involved. Superconducting Maglevs is the way to go, using the numbers from the chart above, there would be ~25 kilotonnes per square meter between the stationary ring's inner surface and the rotating ring's outer surface, don't get me wrong, this is a lot of pressure, but it's no more than the fuel injectors in a modern diesel engine and a whole order of magnitude less than a high end waterjet cutter. Using the lift formula from this wiki page on maglevs you'd need magnets that could generate 25 Teslas to balance the forces, a lot, but seeing as the current (non-exploding) magnet world record is over 40 Teslas, difficult but still doable.
Also, the inhabitants on the inside (along with the atmosphere) should not actually feel any effects from the mass of the rings as for some physics reason I don't fully understand, the gravity inside a hollow sphere (or ring, which is just a slice of sphere) is always zero when the mass is evenly distributed, so no worries about the air being sucked into the center, the only force it and everything else on the inside surface should feel is centrifugal (yes, I know) force trying to squash them flat against the inner surface. Apparently that only applies to a completely closed shell
I have no clue how much thicker and heavier the outer stationary ring needs to be to generate the same kind of forces as the inner ring, I have a hunch that you'd need more than one planet as the ring is more spread out so the gravity trying to crush the outer ring might not be as strong as if it was still in planet form.
P.S. the whole idea of building a counterbalanced Ringworld is a) it's cool, and b) the particular one in the chart above would have both Earth-like "gravity" (centrifugal-gravity?) for the inner-surface inhabitants along with a 24hr day-night cycle by default