My house was built in the 1950's with a flat roof. In my garage there is a set of exposed ceiling joists that make a cross. Attached is a quick drawing of what it looks like. The problem is that the space perpendicular beams that sit on top of the header and the joists is a bunch of unused space. What makes it worse is that the joists are at an angle so it goes from a small amount of wasted space (about 6 inches) to 2 feet to rest above a door header and another supporting header. I know these were added as a part of the structure. I was hoping to replace these with something that would return that space to me so I can mount a ceiling bike rack.
Any Suggestions. I can get some pictures if further detail is need.