Ever sat back and said, "Oh no, double-darn, aw shucks (or maybe something else?)!'' when you've broken off a bolt in a blind hole? After you get the anger out of your system, sit back for a moment and analyze the problem. There are a variety of methods to remove that offending broken bolt.
You may have a shoulder or stub sticking out-very often, the bolt will break off right at the head. If so, you may be able to grab the shank with a pair of locking pliers. But before you try turning it again, do a bit of preparation. For some reason, that bolt has attached itself to its surroundings, most likely due to corrosion. Something has to break that bond. A thermal or physical shock may do the trick, as may some type of penetrant. A physical shock is nothing more than giving the area around the broken bolt a series of sharp raps with the hammer. Just don't get carried away and break something. Thermal shock can be accomplished by heating and cooling; heating alone may loosen the part through expansion.
Penetrating oils really do help, but you've got to give them time to work. The longer the "soak'' time, the better your chance of a successful removal. Another trick that works surprisingly well is to heat the area just enough to allow canning paraffin or candle wax to melt and follow the threads. I'm not sure exactly how or why it works, but it does work, especially well on cast-iron parts.
If there's no stub sticking out, we have to improvise. Take the thickest flat washer you can locate with a center hole slightly smaller than the broken bolt's diameter. Lay this over the bolt and carefully weld it to the bolt's shank. This is where a MIG welder (i.e., "wire feed'') really comes in handy. (With a "stick'' welder, it's a bit more difficult, but not impossible.) Let the weld cool a bit, and then put an ice cube on top of the washer. We're trying to accomplish two things here: (1) We've got a place to grab onto with locking pliers, and (2) we're applying a bit more thermal shock. Again, heat the part, apply a lubricant, and give it a try.
Once, on a large bolt that I knew was really going to be stubborn, I sacrificed an impact socket and welded it directly to the bolt, but the welder may not solve every problem. Welding is a bit scary in some locations, and you may not have a welder handy. Or the part may not be accessible to a welder, or is broken off totally flush.
Our choice now is to drill a hole in the direct center and use a properly-sized "easy out.'' Don't use an ordinary drill bit. Get yourself a left-turn bit from an industrial supply house, the reason being that its rotation direction is the same as the removal direction. Naturally, you've got to have a reversible driver. I've seen broken bolts come right out by just the drilling action. No such luck? Then insert the "easy out,'' and again, with some heat and penetrating oil, attempt to remove the bolt. Don't play "gorilla'' and break off the easy out-they're harder than any drill bit known to mankind.
Still won't come out? Then it's back to the industrial supply house for another left-hand drill bit, slightly smaller than the bolt's diameter. Hopefully, your first hole was dead center in the bolt, because you're going to attempt to drill out everything but the threads. After the holes been drilled, you may be able to pick out any remaining pieces with a sharp, pointed tool. Most likely, though, your hole wasn't dead center and you damaged some of the threads. Hopefully, they'll clean up with a tap. If things are a bit on the loose side when reinstalling the bolt, you might want to use a chemical locking compound.