Installing cabinets over tile
Another issue that comes to mind is how secure does the toe base need to be? It sounds like you're installing cabinets with no toe-kick notch. This seems to be more prevalent in the west than east, where factory cabinets are made with built-in toe-kicks.
Here's an approach that could avoid the need to fasten the sleeper to the floor at all. The 2X4 sleeper is scribed level to the floor, "tie-backs" (2X4's that run back and get fastened to the wall/sill plate) are installed to hold it in place, the cabinets are set on top of the sleeper and fastened to the wall, and the screw through the cabinet base into the sleeper. The cabinets won't be going anywhere after everything is assembled (and the counter will help "unitize" everything as well). This really isn't much different than using adjustable feet.
Which begs the question: have you considered switching to adjustable feet instead of toe bases? When the cabinetmaker I installed for switched to adjustable feet, it was a happy day for me. I never once wished I was using the platform system or toe board system, and I made *much* better time on my installations. I've included a photo below of the adjustable feet I use.
All that said, a refrigerator end panel would have to be approached differently, since it might be all on its own at the end of a cabinet run.
In your picture, it seems you are using 4 inch (or so) tall legs to provide for the toe kick space. Do you somehow attach a "skin" or front to the legs themselves?
From contributor C:
Here is a sketch that sort of shows this detail.
I use loose bases exclusively made out of 3/4" cdx plywood. I use plywood so that in the event of a water leak, the cabinetry will still be on a sound foundation rather than a particleboard sponge. I cover these with a 1/4 finished skin as I detail out the job. My kicks are 4 sided with sleepers as needed so as to be universally applicable. I also make one long kick even in the event of several smaller cabinets - it just makes everything easier to lay out during installation.
The thought of levelers seems a novel idea, although I have not had any experience with them. I don't know that the use of levelers in conjunction with a loose toe would nullify the benefit of the leveler. With my method, I would level the base (with or without levelers) and simply screw it to the back wall, shimming at the wall as well to keep my kick straight. You can then slide your boxes right onto that secure base. A few screws to secure your boxes to the wall and you're done.
To install that free standing refer panel, I would use a metal angle bracket on the inside of the opening at the wall and run a bead of clear silicone around the bottom.
We glue 2x6 blocks to the deck (or drive pin them) in the inside corners of the toe kicks, shim, level, attach to blocks and walls. (FS100 glue is best.) We do not see much radiant heat here in the South, but I can't imagine it would get hotter in any one spot.
We have been using the leveling legs for quite some time now and have never looked back. The speed and accuracy that one can achieve with this system and a laser level is amazing. We cut our installation times down by a third and that means more money in my pocket! I've had the very same problem before installation over tile, and those feet worked out fine. We even had an island - we embedded the feet in Liquid Nail, which worked fine. I urge you to check into the feet. We use the Camar brand, but there are others.
We also build separate toe kicks and level them well. We just put a couple of screws through the back of the toe into the bottom plate of the wall to hold them in place. I would not recommend fastening to the floor if it has heating pipes in the concrete because if you penetrate one, you could have a real mess. If you have an island cabinet you could just glue a 2 X 4 to the floor to fasten to. Or use concrete fasteners that don't go in very far, though that would be taking a chance on hitting a pipe.
If you need to skin the toe kick after leveling, try the Quickscribe. It follows the floor as well as it follows walls.
From experience in my own house I would insulate (blue board) under any cabinets that have food in them. When I laid the pipe in my house, I did not put it under the cabinets, but there is still quite a bit of heat radiating into them. Not a big issue, but one to think about.
Silicone, liquid journeyman in a tube - just set it and forget it. Be sure to remove dust from all adjoining surfaces. I have used this method for years where other methods are not desirable for whatever reason
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