Hi there!
Probably, the big differences regarding stresses on intersections are due to the Axial SIFs that are not taken into account by ASME B31.1. You may check if the over-stressed sections belong to header part - if yes, this is the reason...
The immediate solution that comes in my mind is to let the intersections to move - this means to let the pipe un-buried at intersections. Depending on the pipe layout & environment this approach might not always be acceptable.
Another way to act is to review what temperatures you use for analysis - you should use REALISTIC TEMPERATURES, meaning the ACTUAL Maximum & Minimum Temperatures reached by metallic pipe wall. In this regard, see Project Specs, Design Basis and, if nothing is provided there, liaise with Process Discipline - they should provide the temperatures for calculation - might be a tough issue, but this is the situation...
Another direction may be to assess accurate SIFs. Depending on branch/header sizes, the SIFs on the header part may be significantly lower than the default B31.3 SIFs that correspond mainly to the branch part. You may use FEA assessment for this scope - PRG FEA Package r FEA Tools (if you afford this...).
I hope this thoughts may be of some help.
Good luck!
Regards,
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Dorin Daniel Popescu
Lead Piping Stress Engineer