Why can't there be a trap door under nuclear reactors in case of meltdown? This may be a naive question, but after the Fukushima Daiichi partial meltdown and studying the aftermath of Chernobyl it seems they could be helped by this idea.
In Chernobyl, the liquidators that cleaned up the disaster tunneled concrete under the reactor core and covered the whole complex in a big containment unit.  Why could they not have a 30 meter pit below a reactor filled with water with a trap door holding the reactor up?  This pit could be very thick reinforced concrete similar to a missile silo  If there were a meltdown or imminent meltdown and there were no other options the reactor core could be dropped into this pit by triggering the trap door and then perhaps have lead shot dumped on it or sealed in concrete. 
This seems very simple but there must be some reason why this is not practical.  I know that the core would still be hot and it would still be a problem but would this not be better than it being exposed to air?
 A: I don't have much more than a interested amateur's understanding of the construction and operation of nuclear reactors, but...
If nothing else you add the task of insuring that this large, complicated system fails safely. And does so while managing the huge weight involved, despite the fact that the core is hooked to a complicated system of pipes carrying water that is likely tritated and may be contaminated and that system is coupled to a second water loop that needs to be guaranteed to remain clean.
Your system needs to be arranged so that the core material does not fall into a dense pile, and has to insure that despite the large weight of the core material and the possibility of very high temperatures.
And all of that has to be fit into a containment structure that is already a huge project to build (and yes, they are contained from below as well as the 
sides and top ... else what happens to the nearby ground wanter).

A lot of engineering effort has gone into attempts to design "intrinsically safe" reactor starting decades ago when the first of the Generation III reactors went in. Alas, this has proved harder than expected.
A: By the time the core melts it is probably way beyond passive cooling. If you read up on the Chernobyl event they went to great lengths to prevent the core from melting its way into a water reservoir beneath the reactor (the bubbler pool) for fear it would result in a steam explosion. Three Russians in diving suits performed a suicide mission: entered the bubbler pool and opened the sluice gates to drain it. They died very soon after from radiation poisoning.
A: 
Why could they not have a 30 meter pit below a reactor filled with water with a trap door holding the reactor up?

If you read about the Molten-Salt Reactor, you will find that it has something rather similar to this.  In the image below, item number 13 is a freeze valve.  If the reactor overheats, the plug melts and the molten core flows down to the tanks 10 and 11.  This "escape route" would probably be passively cooled by something like atmospheric convection.  Of course, this was just one design and we can't speak for all systems that might be used.

In the generation of reactors being built today, AREVA has implemented something called a "core catcher".  It is basically a passively cooled sheet of metal at the bottom of the containment building that a molten core would fall on, and then be safely contained and cooled.
My latter example, however, is only a worst-case scenario because if at all possible, we would prefer to not harm the physical integrity of the fuel.  The rods of present-generation nuclear fuel hold in radioactive gases at a pressure significantly above sea-level atmosphere pressure.  If many of the rods break, then that is still a core failure, and will release lots of radioactivity.  Because of our reliance on the physical integrity of a small thickness of heavily irradiated metal cladding, we really can't move it except under controlled conditions and very slowly.
A: If the core ever touches air during its meltdown, it will get so hot that it will maintain a steady independent reaction that will continue until the fuel is spent. It is so hot that it melts through concrete, steel and other man-made obstacles. If it were to hit water, the steam that would result would be disastrous. It would put tons of radioactive particles into the atmosphere.
Modern reactors are equipped with several layers of concrete in case of a total meltdown.
