Piping Codes are simplified.
In my opinion, once B31 sets wall thickness, safety regarding hoop stress is satisfied.
While an "exact" evaluation of Tresca stress (maximum shear stress) in the plane normal to radial must include hoop, longitudinal and shear stress, a simpler and more conservative calculation ignores the hoop term. That's why it is called longitudinal stress - the focus is on the (remaining) non-hoop terms: i.e., longitudinal and shear.
But don't go too far with your octahedral shearing stress calculations. This equivalent stress calculation may be the root of this stress evaluation but it is not used today. The Code defines the stress calculation and the allowable limit.
CAESAR II calculates many stresses. One is octahedral shearing stress and it includes hoop, longitudinal, radial and shear terms. The one called Code Stress is the one with direct Piping Code implications.
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Dave Diehl