Sec.VIII-1 shell formula origin 4

[Paulettea]

Dear All

I have a very basic question regarding shell thickness equation as per ASME BPVC Sec. VIII-Div.1.
I know from very elementary calculations that the stress in long cylindrical shells under pressure is given by:
S=PR/t
And therefore the minimum thickness is given by:
t=PR/S
where, S is the allowable stress.
However, in VIII-Div.1 UG-27 the minimum thickness is given by:
t=PR/(SE-0.6P).
I can understand that this formula wants to consider the nonlinear state of stress distribution especially when the pressure is high and hence the thickness is high. However, what I do not understand is the origin of that factor 0.6 by which P is multiplied. I tried very much to find an analytical method to derive that coefficient but failed. So, is this number some sort of empirical number or is it found by some trial and error procedure or are there any mathematical analysis behind it?
Please, help me with this or I cannot sleep at all.

Warm Regards

 

[TGS4]

I am greatly troubled by your trying to find an equivalent between the Code rules, empirically based on semi-thin shells, and then applying the linear-elastic DBA approaches. The two are barely remotely related, especially the limit on Pl+Pb (which is Spl, not just 1.5S - how old of a version of VIII-2 are you using?)

 

[r6155]

Simple. See ASME VIII Div.1

MANDATORY APPENDIX 1
SUPPLEMENTARY DESIGN FORMULAS
1-2 CYLINDRICAL SHELLS

t= R (exp(P/SE)-1)

Regards
r6155

 

[prex]

I'll only make some statements, in trying to come back to the original question.
In a thin cylinder, the circumferential stress is PR/t; the radial stress is -P on inside and 0 outside, so its average (membrane) value is -0.5P.
So, if we use the Tresca criterion for failure, the thickness would be t=PR/(SE-0.5P), and indeed there are other pressure vessel codes that use this formula (e.g. EN13445).
Now mandatory App.1 of Div.1 offers an alternative formula, t=R(exp(P/SE)-1), that is mandatory for thick cylinders, but may be used for all cases. This formula gives values of t that are consistently lower wrt those given by t=PR/(SE-0.6P), but also wrt those given by t=PR/(SE-0.5P)!
My answer to the original question is that the factor 0.6 (instead of 0.5) may be only justified based on historical grounds and has no logical foundation (again wrt 0.5)

prex
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[Paulettea]

Thank you TGS4 for your reply.

TGS4, Can you help me understand what I say below is right or not?

the following is my understanding I do not know if I have come to the correct conclusion or not so please help me.

[highlight #73D216]"In design by analysis when you obtain stress values in a shell if the stress at one point is greater that the allowable it does not mean that there is a plastic collapse. Because it simply shows at the point under consideration there may be some yielding. And yielding in one point does not mean the whole section fails. Because the tension in adjacent points may be far from yielding and can continue resisting the applied load.
Now, if the stress through the whole thickness of the shell exceeds yield then there will be failure because there is no point in that section left to resist the load. Therefore, if the average stress through the thickness exceeds yield there will be plastic collapse."[/highlight]

prex, I tried before to justify this 0.6 with Tresca and the coefficient of P became 1. Now as you say, if the average value is used the coefficient will be 0.5 which is closer to 0.6. However, I do not remember the reference but I saw somewhere mentioning that the stress criterion for Div.1 is not Tresca and it is Maximum Normal Stress. for Div.2 previously it was Tresca but now it is Maximum Distortion Energy.

Warm Regards.

 

[TGS4]

Your understanding is correct. However, demonstrating this with linear elastic analysis and pseudo-elastic stresses is rather difficult. Elastic-plastic analysis is much more robust in demonstrating it.

Regarding failure criteria, both VIII-1 and VIII-2 in DBR use maximum (component) stress, although VIII-2 does apply von Mises for combined loading. And, of course, DBA in VIII-2 uses von Mises.

The exponent formula is the exact formula (even Lame's work was an approximation) considering plastic collapse of a cylinder when implementing an elastic-perfectly-plastic material model.

 

[Paulettea]

Again many thanks for the insight you add here TGS4.

I'm so sorry to make this discussion so lengthy and take your precious time.

TGS4,as you and some other ones in here mentioned this -0.6P term is in the required thickness formula on an empirical basis to approximate the maximum stress through the thickness of a cylindrical shell under internal pressure. But I think it is not necessary to follow the maximum stress through the thickness of a shell section in the first place. The important factor to deal with is the average stress through the thickness of a shell which if followed gives the required thickness to be PR/SE rather than PR/(SE-0.6P) the latter is the thickness require if you want to prevent yielding in all point of a shell thickness. That is why I tried to relate the empirical shell thickness in Div.1 to DBA criterion in Div.2 for membrane stress.

Warm Regards.

 

[jamesl]

I think the paper by H.C. Boardman in 1943 “Formulas for the design of cylindrical and spherical shells to withstand uniform internal pressure” was the basis of the UG-27 equation. The paper started with classical equations, which had been used in Appendix 1-2 of Div 1 Code up to 2007 Edition. The paper re-wrote the equation to have the factor before t in terms of t/R. Then you will see the factor varies from 0.5 to 0.6 when t/R varies from 0 to 0.5. Factor of 0.6 was recommended to error on safe side.