I have used old style lag shields, going back 40 years ago. I found they pretty much always loosen up. Nowadays, I use Hilti epoxy for most of my concrete anchoring. I just wore out my second milwaukee rotohammer, probably drilled 20,000 holes with it before it died. I have set a LOT of anchors. Epoxy is the bomb. Drill the hole, in your case, rent a hilti gun from the rental yard, epoxy in threaded inserts.
In my fab building, I actually inset 1' x 1' x 3/4" plate on an 8' grid, flush in the concrete, each plate has 4 tapped holes that take 1" threads. this is much better than epoxying in an anchor, but a bit hard to do on an existing slab.