Cracked and sinking concrete floor

[Cian]

You may also want to look at foam concrete leveling.

Not really a DIY project, but they have some pretty dense foams that they shoot out of setups similar to what they foam insulate with.

What he said. I had my front walkway approaching my front stoop raised by this method. Fairly common fix for what the OP described.

https://www.youtube.com/watch?v=c8aqUTXIpSo

[Snipe Hunter]

What he said. I had my front walkway approaching my front stoop raised by this method. Fairly common fix for what the OP described.

https://www.youtube.com/watch?v=c8aqUTXIpSo

That works well in your situation. But broken, sinking areas on a large slab is a different animal. I'm guessing you had to move one single piece of concrete, not pieces of concrete.

[MstrCarpenter]

I watched the video Cian linked to along with a few others. Some were serious hack jobs but they still used foam to support the slab and it worked well enough for them. I worked on a large machine pad at Sig Sauer that needed vibration isolation from the rest of the shop floor. The Engineers approved the use of Dow Great Stuff (pro gaps and cracks -fire rated) to seal gaps between the (basically) rubber blocks that created the  16' x 26' x 3' form to keep concrete from seeping out of or under the blocks. Even when poured in multiple lifts, that's a lot of outward pressure near the bottom. That got me checking the product specifications. Compressive strength is 9.3 P.S.I., or 1,339.2 Lb.s /per square foot. I don't think I'd use it on a driveway, but maybe in a corner of my shop. Hey, for $5 and a half-hour, if it eliminated a tripping hazard 'til I overloaded it and had to do it right, I think it would be money well spent.

[gear jammer]

Decided to do it right and bust out the old and pack in sand and repour. Drilled some half inch rebar holes and found some scrap concrete wire to use. I don't have a tamper so used a 5 gal bucket about 3/4 full to pack the sand. Concrete company said they would bring over some left over from other jobs. Did it Friday about mid morning. Cost was $125. 

Thanks for the replies.

[Snipe Hunter]

Cool deal. I've got some concrete work coming up, not looking forward to it.