So, in this situation, you are mixing and matching failure modes. Focus on the individual failures modes, one at a time, and do not get distracted by the method (elastic in your case).

For Protection Against Plastic Collapse, you need to focus on the Design Load Combinations. In this failure mode you will find the limits on P_m, P_L, and P_L+P_b - equations 5.2, 5.3, and 5.4. There is no use in applying operating loads for this failure mode - it is simply not permitted. Pay special attention to the General Note in Table 5.3.

(Caveat - with Design Load Combinations 6, 7, and 8, there is a parameter Ω_PP, which has the effect of being the operating pressure (to be used in combination with the wind and seismic occasional loads to form the indicated Design Load Combinations). However, great care was taken in the ASME Code Committees while creating this parameter so as to not confuse it with P_O).

Likewise, you use the design load combinations for Protection Against Local Failure and Protection Against Collapse From Buckling.

You use the operating loads for the failure modes of Protection Against Failure From Cyclic Loads: Fatigue and Ratcheting.

Quote (FPPE)

(why in Figure 5.1 does it not also consider Pm+Pb+Q?

There is no such quantity.

For all intents and purposes, please consider P_m as a hand-calc value. You should use your hand-calc to validate your FEA where you would categorize a stress as P_m - you never use FEA to calculate P_m.

In the context of the P+Q stresses, it matters not whether the membrane stress is general or local.

And again, the limit on P_L+P_b+Q is not 3*S, it it S_PS. Pay special attention to 5.5.6.1(c) - particularly the last two sentences:

Quote (ASME Section VIII, Division 2, Part 5 - 5.5.6.1(c) last two sentences)

In this case, the value of S_PS may vary with the specified cycle, or combination of cycles, being considered since the temperature extremes may be different in each case. Therefore, care shall be exercised to assure that the applicable value of S_PS for each cycle, or combination of cycles, is used (see 5.5.3).

Thanks TGS4 for your explanations, always precise and professional.

With your last message, you "took the earth out from under my feet" with this statement:

Quote (TGS4)

For all intents and purposes, please consider Pm as a hand-calc value. You should use your hand-calc to validate your FEA where you would categorize a stress as Pm - you never use FEA to calculate Pm.

Do you mean that the Pm stress you get from FEA may not be correct? Or rather, do you mean for example a cylindrical tank? But in the case, for example, of an Air Cooled API 661 with a flanged cover or in the case of a device with more complex shapes, how can the Pm value be obtained with hand-calc?
I don't want to ask stupid questions, but I've never asked myself this kind of problem.
Thank you in advance

Happy to help.

I will share some of my thoughts with respect to my contention that P_m should be a hand-calc. First, it is not my contention that the value of P_m is wrong - in fact if you re-read my 21 Jun 22 03:44 post, you will see that I said

Quote (TGS4)

You should use your hand-calc to validate your FEA where you would categorize a stress as P_m - you never use FEA to calculate P_m.

Validating your FEA is extremely important, but the work process ought to be one where you use hand-calcs to validate your FEA - and the hand-calc for P_m is ideally suited for this situation.

Quote (ASME Section VIII, Division 2, Part 5 - 5.2.2.2)

(a) General Primary Membrane Equivalent Stress (P_m)
(1) The general primary membrane equivalent stress (see Figure 5.1) is the equivalent stress, derived from the average value across the thickness of a section, of the general primary stresses produced by the design internal pressure and other specified mechanical loads but excluding all secondary and peak stresses.

Accordingly, P_m is calculated as F/A, M*c/I, and P*r/t or C*P*d^2/t^2 (for a flat head)- or any of the other hand-calc method in Part 4. If you have a complex shape that you can't perform a hand-calc, then you are into a situation where paragraph 5.2.1.2 comes into play:

Quote (ASME Section VIII, Division 2, Part 5 - 5.2.1.2)

For components with a complex geometry and/or complex loading, the categorization of stresses requires significant knowledge and judgment. This is especially true for three-dimensional stress fields. Application of the limit load or elastic–plastic analysis methods in 5.2.3 and 5.2.4, respectively, is recommended for cases where the categorization process may produce ambiguous results.

Basically, if you can't do a hand-calc to calculate P_m, then your geometry may be too complex for the elastic stress analysis method for demonstrating Protection Against Plastic Collapse.

Thank you very much TGS4, I hope to get to at least half of your knowledge.