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In our true masonry fireplaces, the hearth support is usually a cantilevered concrete slab. The hearth itself is a separate slab with its top at the level of the finish floor, usually 3/4" above story zero.

This finish slab may vary in thickness, and will often be thicker than the typical finish floor. You want to see this slab in plan; put it on A Fireplace or A Cabs2. The concrete slab will be placed directly below the finish slab. It should go on A Chimney3.

Modeling the hearth structure tends to be a construction documents phase task. When you began modeling the chimney in schematics and/or design development, you might have placed the firebox object at project zero, which was OK then. Once you have the real hearth structure in place, however, you need to lower the firebox to meet the structural slab, not the finish hearth. Make sure you add this adjustment to the height of the firebox object, so the top stays put. The polygon wall surrounding the firebox object also needs to be stretched down to meet the slab. (The wall height can be stretched in section, not so much for the object. LAME.)

The structural slab usually extends all the way through the chimney stack. The core portion of the chimney below should be shortened to meet the bottom of the slab.

The other clashes between the hearth slabs and surrounding elements are solved with solid element operations. Both hearth slabs need to be subtracted from the main joist deck, and the concrete slab may need to be subtracted from the non-core wall below.

To be clear, don't move the joist deck to accommodate the hearth; you need it where it is to complete the ceiling of the story below. But the finish floor should be edited to go around the finish hearth. If there's a floor finish fill, it needs to go around too.

Flush hearth structure
Hearth flush with finish floor

A raised hearth is not much different. The concrete slab will usually be at the level of the joist deck. The finish hearth, the firebox parts, the smoke chamber, and the flue are all raised by whatever distance. You need another slab of core-type stuff, probably CMU, on top of the concrete slab. The front surface of the hearth will be some sort of veneer material, consisting of walls on the A Chimney3 layer.

Chimney hearth structure
Raised hearth

Chimney/Fireplace 1: Fireplace in Plan
Chimney/Fireplace 2: Chimney in Plan and 3D
Chimney/Fireplace 3: Flues
Chimney/Fireplace 5: Chimney Top JM11