## Sec.VIII-1 shell formula origin

## Sec.VIII-1 shell formula origin

(OP)

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

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

## RE: Sec.VIII-1 shell formula origin

So don't try to verify the formula that is given by the code, just use it.

You seem to be not experienced in the design, and I understand your frustration. If you have problem with the thickness you may use higher grade material if acceptable for the process, cost and availability.

Or you may offer the client alternative rule, ASME VII Div 2 which will give you less thickness but you will be forces to provide more specific calculations.

Unfortunately these are the short answers for your questions.

Hope it helps.

## RE: Sec.VIII-1 shell formula origin

Companion guide to the ASME Boiler and Pressure Vessel and Piping Codes, Fourth Edition - Volume 2, mentions in para 21.4.2.4## RE: Sec.VIII-1 shell formula origin

Starting with the Lame' equation, I came up with a 1.0 instead of a 0.6 term. That involved dropping a t-squared term, so assuming thickness is small relative to radius.

One of the references I found mentioned using the stress range instead of just the hoop stress to match the failure theory in question. Doing so affects that number ,but doesn't make it 0.6, either.

I found some references to pipe design that used a 0.4 factor in that place.

Sometimes, stuff like this is just the result of committee thinking. You've got 7 people that think it ought to be a 1.0 and 3 people that think it ought not be there at all, so the compromise is a 0.6.

## RE: Sec.VIII-1 shell formula origin

saplantifor your reply.As you rightly guessed I am not much experienced in pressure vessel design at least not as much as you are. Maybe that is the reason why I ask too many questions. Nonetheless, there are things regarding what you mentioned that I want to say.

Dear saplanti!I know very well that each code or standard has its own design formulae to be followed by the users of the code. If the user does not like the formulae that’s his or her own problem.When I want to design a pressure vessel I just follow the rules provided by the code nothing more. Even, most of the times I do not bother reading the code because I give these calculations to be done by software (most of the times PV Elite). So I agree with you that in practice, we just follow the rules.

However, I did not ask this question in order to change the design of a real vessel nor did I ask it because I thought it is wrong. In fact it is totally the other way around as I know definitely this formula is right and there is a reason behind every term in it and I am trying to understand that reason. Even if I never understand the true reason for this formula I won’t stop using it.

The only motive for me to ask this question is just curiosity and I think that is the motive for all knowledge.

Before asking this question I searched a lot in other sources and the only thing I found was “Guidebook for The Design of ASME Sec VIII - 4th Ed by James R. Farr & Maan H. Jawad” which is a very good book. In chapter 2 page 32 of this book it is mentioned that the reason that there is the term (-0.6P) in the denominator is to take into account the nonlinear state of stress distribution which is quite understandable. Unfortunately, it does not mention why the coefficient of P is (-0.6). Please note that this coefficient for spherical shell is (0.4).

Regarding what you said about the manufacturing tolerance, with all due respect, I do not agree that this term is related to that as there are specific parts of the code to address the issue of the manufacturing tolerances and acceptance criteria for them.

## RE: Sec.VIII-1 shell formula origin

Please note that those specific parts of the codes give you the acceptable tolerances in manufacture of the pressure vessels. The Lame equation is for perfect cylinder only. So the codes cover the tolerances and deviations by changing this equation slightly for the expectations. Please do not forget these equations are part of the rules for design.

Hope this help.

## RE: Sec.VIII-1 shell formula origin

## RE: Sec.VIII-1 shell formula origin

and this

are 2 different things.

Following code rules is not accomplished by using software like PV Elite. Following a code starts by reading and understanding a code, and doing (I guess) the first dozen of calculations by hand (i.e. paper or excel). Trusting PVE without knowing what it does, and why it does it, is very dangerous, to put it gentle.

## RE: Sec.VIII-1 shell formula origin

I think the minimum thickness will be under by 0.3 mm of the nominated thickness. If the plate come with the minimum thickness you will under design if you use the nominal thickness for your calculation. This is one of the examples for tolerances/deviations in the code.

## RE: Sec.VIII-1 shell formula origin

"I think the minimum thickness will be under by 0.3 mm of the nominated thickness. If the plate come with the minimum thickness you will under design if you use the nominal thickness for your calculation using Lame equation." not to cause confusion.

## RE: Sec.VIII-1 shell formula origin

I do not know exactly the criteria for Div. 1. But if i want to make a safe design in Div.2 part 5, I need to calculate the primary membrane stress. If the primary membrane stress is supposed to be calculated with very simple formulas again that 0.6 does not appear in calculations of average stress. And more importantly, you do not need to take the non-linearity into account since only the average stress over the thickness is important. The only reason that the non-linearity in the Lame equation may be important is in calculation of bending stresses. however, the allowable stress for bending and bending plus membrane stress is 1.5S. Even, if you want to use Part 4 of Div. 2 the governing formula for shell thickness is:

t=(D/2)*(exp(P/SE)-1)

Now if maclaurin series is used for the exponential term we have:

t=(D/2)*((1+(P/SE)/1!+(P/SE)^2/2!+(P/SE)^3/3!+(P/SE)^4/4!+...)-1)

if the term P/SE is much less than 1 (which is the case when the pressure is nit much high) then higher order terms can be neglected and we will have:

t=(D/2)*((1+P/SE)-1)=PD/2SE=PR/SE

which is again the basic formula without 0.6

Warm Regards

## RE: Sec.VIII-1 shell formula origin

You are definitely right. However, unfortunately I am not much experienced and in order to have a reliable understanding of the code I need time. And you see time is something that is not provided by managers at all. I remember once I told my manager that I need time to understand the code first and then start designing a vessel. then in response he said " then what were you doing at university?"

Warm Regards

## RE: Sec.VIII-1 shell formula origin

Fixed . Seiously, you have an issue with your manager. Dont take the task of doing PV analysis if you did not have sufficient training.

## RE: Sec.VIII-1 shell formula origin

Say goodbye your position and search an serius company. This is your responsability.

Regards

r6155

## RE: Sec.VIII-1 shell formula origin

## RE: Sec.VIII-1 shell formula origin

Note that with the Lame equations to find thickness, I had to iterate, it wasn't a direct formula, so there's some advantage to using that version.

## RE: Sec.VIII-1 shell formula origin

JStephenvery much for your bright answer.This correction with 0.6 a is very smart and clever method to get close to the Lame state of stress. Since, as you have correctly mentioned it is very easy to use.

However, what I can not fully understand is that why should we consider the stress variations through the thickness of a shell. As far as I know for a static design there is no need to take stress concentrations into account. Even stress variations are not important when we are measuring the membrane stress. Remember, if there is one point in the shell which has a stress higher than the yield, it does not mean that the whole section will collapse. In elasticity if a plastic failure is supposed to occur for a section in a component, the average stress of the section has to go higher than the yield. That is the reason why primary membrane stress is defined. Now even if you integrate the Lame equation over the thickness and then divide it by the thickness you will end up with the average stress formula which is again without the 0.6 term (PR/SE).

Warm Regards

## RE: Sec.VIII-1 shell formula origin

The first is from "Pressure Vessels Design and Practice - Somnath Chattopadhyay", second is from "

Process Equipment Design - Brawnell & Young". I attached one document and will attache another to support the information.

## RE: Sec.VIII-1 shell formula origin

Other than this there is another reference book but I cannot scan and attache the entire chapter. You need to search this book if you need more information on the Lame equation: "Mechanics of Materials_2nd edition, E.J. Hearn, Volume 1, City of Birmingham Polytechnic, UK". Chapter 10 Thick Cylinders give you the derivation and use of Lame equation. However I suggest the practicing engineer to have both Vol 1 and Vol 2. There are a lot of information in SI units.

I hope these will help you a bit in understanding why Div 1 is using this formula.

Regards

## RE: Sec.VIII-1 shell formula origin

saplantivery much.The second reference that you have sent says it very clearly.

"The range of the membrane equation has been extended by the

empirical modificationof adding the constant 0.6"Therefore, it is not an analytical constant to be derived by analysis.

Again thanks o lot for helping me. Now I can sleep.

Warm Regards

## RE: Sec.VIII-1 shell formula origin

## RE: Sec.VIII-1 shell formula origin

MANDATORY APPENDIX 1

SUPPLEMENTARY DESIGN FORMULAS

1-2 CYLINDRICAL SHELLS

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

Regards

r6155

## RE: Sec.VIII-1 shell formula origin

In a

thincylinder, 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

lowerwrt those given by t=PR/(SE-0.6P), butalsowrt 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

http://www.xcalcs.com : Online engineering calculations

https://www.megamag.it : Magnetic brakes and launchers for fun rides

https://www.levitans.com : Air bearing pads

## RE: Sec.VIII-1 shell formula origin

TGS4for 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.

"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."

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.

## RE: Sec.VIII-1 shell formula origin

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.

## RE: Sec.VIII-1 shell formula origin

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.

## RE: Sec.VIII-1 shell formula origin