PL is local membrane, so it is average through thickness (and Pb average linearized stress).
Otherwise they are local, that is you must take the maximum value of PL+Pb in the region under consideration (e.g. a vessel wall portion around a nozzle). However they are not peak stresses.
Sorry, I think that I used the wrong terminology. My question should have been:
For Pb+Pl < 1.5 Sm, is Pb an average or maximum stress?
For instance, Prex's example of a vessel wall portion around a nozzle: If the ID of the vessel wall portion around a nozzle is Pb = 2000 psi and the OD of the vessel around the same nozzle is Pb = 8000 psi and the average Pb from ID to OD is 4000 psi, (Prex, if I understand you correctly) then Pb = 8000 for the equation Pb+Pl<1.5Sm.
Can't follow your way of reasoning. This is mine:
1) You have a region like the portion of a vessel wall around a nozzle
2) You know the six components of stress at any location in this region and also the distribution of the same along the thickness (of course you don't really need to calculate everything, you'll know in advance where maximum stresses are).
3) At any location in the region you may now calculate PL+Pb: comes from the through thickness linear stress distribution (or averaged stress) that has the same resultant normal load and bending moment as the original distribution (see the formulae in a FAQ in this same forum)
4) As an alternative to 2) and 3), if you calculate the stresses with a thin shell formulation (FEA or by formula is the same), then you directly get the membrane load N (this is not psi, it is lb/in) and the bending moment M, and from them you have PL=N/t and Pb=6M/t2
5) Now determine the maximum value of PL+Pb in the region (considering that Pb has both plus and minus signs): this is the value to be compared to 1.5Sm
I think my question was answered when Prex said, "you'll know in advance where the maximum stresses are" and "at any location in the region you may now calculate Pl+Pb".
Pl+Pb should be calculated where the maximum stresses are and NOT an average thru the thickness of the vessel.
BTW, we are using FEA and the ID of the vessel near the nozzle has much less stress than the OD of the vessel near the nozzle.
Well, not really heaterguy.
The correct procedure is to get PL and Pb from a through thickness stress linearization (or averaging) procedure.
By taking the highest stress at inner or outer vessel faces you could be a lot too conservative, but also sometimes fairly under conservative: there are well known stress distributions through thickness having their highest stress lower than the corresponding linearized stress.
However, as I pointed out above, if your FEA is with thin shell elements, you don't need to linearize anyway, as with thin shells all the calculated stresses are linear through thickness. On the contrary what you don't get with FEA is the distinction between primary and secondary stresses, but that's a completely different story.
1) The ONLY location where one will see Pb is at the centre of a flat plate. So, in the context of a vessel-nozzle intersection, Pb does not exist.
2) So, that leaves you with Pl. If you are using shell elements, you can get the membrane value directly from the "mid-plane" or membrane stresses in the elements. If you are using solid elements, Pl will be an average. HOWEVER, these averages will not be the average of the stress intensity values. Rather, the stress intensity is calculated from the averages of the six component stresses.
3) The location of where to perform the Pl check is partly a matter of art. Strictly-speaking, this check should be performed everywhere. Practically speaking, you should check a number of locations to confirm that everywhere Pl<1.5Sm. Remember, you're not looking to prove that one location meets the requirement - you're looking to prove that everywhere meets the requirement (or proving the negative, that there isn't a location that exceeds the limit).
heaterguy, let us know whether or not you are dealing with shell elements or solid elements. This will better help us understand your question.
TGS4,
concerning your ...ONLY... statement I wouldn't be so sharp, though I agree that rarely Pb is of concern apart from the center region of flat closures.
An example of a different situation where Pb matters is the outer region of a flat cover, when this one may be considered built in to the cylinder: of course you can safely consider the head as simply supported (and the bending at the periphery becomes secondary) or consider the head as partially fixed (by using for example the formulae of art.4-2 and following) and you'll get a primary bending at the periphery that could well be higher than the center one.
Agreed. My main point for comment #1 is that Pb only occurs in flat head - in the two locations mentioned. In the context of the vessel-nozzle intersection, Pb doesn't exist. All bending in this location is secondary.
