1) The Code has been around a lot longer than these fancy computer programs - a Code built for slide rules. B31.3 had a paragraph that suggested including that axial force in the EXP calcs, if significant. CAESAR II used to have an ON/OFF for this item, too.
2) Axial stress is not bending stress so I would not expect those terms to be mixed in a bending stress number. Except for Code Stress, I consider these other stress columns to be "text book" calculations, not Code-defined calculations.
3) CAESAR II now used the SSI's in B31.3. I do not believe the current (Code) calculation overly conservative. If you apply any additional multiplier (in the configuration file) you may be overly conservative.
4) SSI's are provided for bends as well.
_________________________
Dave Diehl