Jazzy47:
Is the 28" pipe pile and its cap pl. already in place, or do you have some control over what it looks like and how it is welded? You have to know much more about these conditions before you can come up with meaningful details. And, it is not a particularly good condition to try to take your reactions into the pipe walls, through the cap pl. and its welds to the pipe. And, you should know more about the mating of the top of the pipe wall and the cap pl., is it a good bearing surface or cut free hand with a torch? You might be better off taking that 1" pile cap pl. off, to improve your detailing options. I would want to cut the cap pl. back to a 28" OD for about a 14" long arc under your canti. beam and rework the welds btwn. the cap pl. and the pipe wall as PJP welds. The compressive reaction column/point might be a 3' +/- long channel, with a cap pl. and toes welded to the pipe. The tension reaction point might be an end pl. on the beam, welded to the beam, with particular attention to the web welding to pick up the reaction. This end pl. would extend 2 or 3' down the pipe wall, and it would have a wide vert. slit centered below the beam, with a nice circular termination under the beam bot. flg. This slot would be welded to the pipe wall to pick up the tension reaction. These details have to be sized and cleaned up depending upon what’s actually there.
It seems to me that the basic math here is pretty straight forward, and I think it goes something like this: I’m using a 27" ID as a backspan, so you have a 11.125' beam (10' + 13.5"/12). The compressive reaction on the pipe wall is about 247.22k {(50k)(11.125')/2.25'} and the tension reaction on the pipe wall, at the back span is about 197.22k {(50k)(8.875')/2.25'}, and the sum of the verts. even add up. Don’t forget the DL’s and the lateral stability of that canti. beam. You said that the pipe pile has already been checked, but I would want to be darn sure of that. Then, your primary problem is getting your canti. reactions into the pipe wall without causing it to buckle or yield.
A statement like... “I find that here I am generally encouraged to make vastly simplifying assumptions and call everything rigid, but I'd like to understand the more accurate analysis.” is a rather inexperienced comment. We do this all the time to even begin the make our problems analyzable and solvable. Your assumptions should be conservative, not overly so, and they should reasonably represent the way the structure will actually act. You will find that most of our structures act the way we force them to act by the nature of the way we detail them. And, of course, these details should provide a clean/smooth load path from one element to the next. When we don’t provide this smooth load path in our details is when structures perform poorly or fail.
RareBugRA’s kicker idea probably won’t work because it would induce a punching/buckling reaction in the pipe wall, some feet below the canti. bm. If you have the headroom, I might be tempted to look at a truss form about 4 or 5' high over the compressive reaction, and with a turnbuckle and rods for a sloped top chord out to the 50k load.
