The problem with the original STP55N06L’s is that they were underrated for the original job, and putting any high load on them will cause lots of heat. They also have a very high Rds(on) value. FDP8870 or IRLB8743 or IRLB3034 have really low Rds(on) values (order or two of magnitude less than the nornal MosFET). I use all of these on various RAMPS boards with no issue, though I always add a small heatsink (usually just a strip of metal with a hole in it), just in case.
Also RAMPS is designed around REQUIRING a fan on it for cooling. If you haven’t got a fan on your RAMPS then you need one.
I would not put 2 STP55N06L’s in parallel as @Sebastian_Schmidt suggests, mainly because they’re getting harder and harder to source and two will cost you more than one of the others. Cheaper to buy one of the replacements I listed above. Feel free to buy any of the above solely on price since they are all quite suitable. Prices vary a lot from area to area, and supplier to supplier, so I can’t tell you what is cheap for you.
Also note: There were a bunch of fake STP55N06L’s at one point. They were relabelled devices that had an even higher Rds(on) and worse heat dissipation specs than a true STP55N06L. Made a bad problem worse.
FYI: The added heating from extra current causes a feedback effect that raises the resistance just enough to make the area even hotter if there is enough current. Too much current and it’s possible for the package to melt or even the solder joints.
If you really have lots of current (and given the size of the bed, quite possible), what @Sebastian_Schmidt says Re: FET on a heatsink and cables to the board. BUT, make sure those cables are damn good, as they will be carrying 10-14 Amps of current, and they will get warm if you use too thin a cable. You might look at the Gen7 Heated Bed external MOSFET board, as it has its own power inputs, etc. People have retrofitted this to a RAMPS board in the past no problems. And the FET the board recommends is one of the above.
@Sebastian_Schmidt Re: Method of driving the FETs with PWM: What you say is true but I don’t know of any of the RepRap electronics that do it this way. They all go the cheap route.
FWIW: When RAMPS was designed, heated beds were not expected to draw more than 10 Amps max. The reality now is that a standard 200mm x 200mm heated bed can easily draw 14A, and larger beds can draw much more. The screw connectors on your board are at most rated to 10A, possibly less depending on brand.
Q: Do you really know how much current your bed pulls?