Anybody can advise on is it possible to join the floor before the staircase and make the profile slanded follow the staircase as per the attached image. Thank you in advance =)
Unfortunately, stairs and floors still don't join in Revit.
1) Read about the "Cut Profile" tool. It is a 2D solution for sections. This will allow you to "cut" the edge of the slab.
You might need to do this anyway to fix the floor. Notice that in your section, the finish floor is getting inside the stairs.
2) Do the stairs with a "landing" at the starting level, that matches the floor. Not ideal, either, but, a workaround.
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