I'm wanting to install a pull-up bar in my basement, but the 2x10 joists for the floor above would be too low when factoring in how far the pull-up bar hangs down. To increase the mounting height of the pull-up bar, I was thinking of sistering a 2x6x4 (or 2x6x6, or longer if necessary) further up on the 2x10 using structural screws, and then mounting the pull-up bar hardware to the 2x6, which would give me the proper height to hang from.
When I've sistered beams together previously, I've always used the same dimensional lumber (2x8x8 to a 2x8x8, etc.) Would sistering a 2x6 to the 2x10 cause any issues as far as structural strength? Since my kitchen floor would be above, I obviously don't want to create an issue of having too much of a dead load when I'm hanging on it, that ends up sagging the floor or weakening it.
On a related note, the ceiling is a drop ceiling with removable panels. To mount the bar, I would need to remove at least 2 of the panels. Is there any solution to have the bar mounted, but not have those 2 panels permanently open? I thought of just measuring where the mounting bars would go up through the panel, and cutting out that section. The only problem I see is that I would then never be able to remove the panels without destroying them.