Grab the appropriate tool and try removing the bolt. If it only comes loose a bit and then gets to the point you feel it's going to break, give it another hit with the penetrating lube and turn it back in. You may have to do this cycle a bunch of times, with each cycle the bolt coming out further. Hey, this takes a lot less time than what you have to go through if that bolt breaks off.
Here is another trick my engine-builder father-in-law taught me that seems to work great on cast iron, such as engine blocks. Heat the bolt and the area around it just enough to allow canning paraffin or candle wax to melt and follow the threads. I am not sure why or how it works but it does.
Now, let's go back to that broken bolt. If there is 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. Place this over the bolt and carefully weld it to the bolt's shank. This is where a MIG-welder (i.e., "wire feed") comes in handy. (With a "stick" welder, it's a bit more difficult, but not impossible.) For obvious reasons, you want to be careful and not weld the washer to the material the bolt is screwed into. Now on top of the washer, weld on a nut. Let the weld cool a bit, and then put an ice cube on top of the washer/nut you welded on. We're trying to accomplish two things here: (1) We've got a place to grab onto with a wrench, 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 nearly impossible to remove, I sacrificed an impact socket and welded it directly to the washer. However, 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 of the bolt and use a properly-sized "easy out." This is a square-shanked pin that you drive into the hole and then turn, which hopefully will remove the bolt. 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 also have 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 be a "gorilla" and break off the "easy out," as they're harder than any drill bit known to mankind.
That bolt 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 now you're going to attempt to drill out everything but the threads. After the hole has 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.
If you really screw up the threads during removal, then: (1) If you have room, you can drill and tap the hole to the next over size; (2) Drill it over size and install a thread insert. There are fairly inexpensive kits available to do this now; (3) Use some Loctite Form-A-Thread. What you do is fill the cleaned hole with the mixed material, put a release agent on a new bolt, and screw it in place. After 30 minutes, remove the bolt and let the material continue to cure for another 90 minutes. Then retorque the bolt in place. While it's not up to super torque loads, I've used it with success.
Plan on breaking off lots of bolts? There are some really trick bolt-removing specialty tools available from most tool supply houses, the tool trucks, and even Sears has a neat tool. Myself, I would rather take a bit of time and use lots of penetrating lube and say a little prayer to the bolt gods before I loosen a bolt.
Oh, and my buddy with the YJ offered this bit of advice. Whenever you remove a bolt that is subject to corrosion of any kind, use lots of antiseize. Sure it's messy to start with, but in the long run if you have to remove that bolt again, you're free and clear.
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