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See AutoPIPE Help Code Compliance Calculations for actual equations, example: ASME B31.1 2016:
Hoop (Circumferential) pressure =
Axial (Long) pressure =
Note: equation could be different depending on "Long Pressure [P/A/M/N]" setting
"Include Axial P case in sustain" setting =
It is true that P1 case does not directly apply cap force. It does that indirectly by apply pressure strain that accounts for it and for poisson ratio. In the case of a flexible joint, AutoPIPE does apply the cap force P x Pressure Area to properly get the extension at the joint.
Loads and Load Sets - AutoPIPE