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# Sec.VIII-1 shell formula origin2

## 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

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

This formula is accepted by that particular code to consider manufacturing tolerances given by the code. It is the same approach used by all other codes with similar or different formulation.

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.

Hope it helps.

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

Im not sure if the OP doesnt understand how to apply a Code, but rather might be interested (out of curiousity - it's better to ask and understand then to take everything for granted) how ASME, or whatever Code organisation, came up with the derivation of the formulae, and e.g. the exact coefficient like 0.6. Why not 0.55? The Companion guide to the ASME Boiler and Pressure Vessel and Piping Codes, Fourth Edition - Volume 2, mentions in para 21.4.2.4

#### Quote:

These equations are very straightforward and do not require much discussion. However, there are some pertinent issues that are discussed. These equations are based on thin wall theory, but do include provisions to account for the variation of stress through the wall of the vessel (called the Lame’ effect) that becomes significant for very thick cylinders. For example, the “0.6 P” term included in the cylinder equations is added in a simplistic way to account for variation in stress through the thickness. The limits that are provided for using these equations assure that the simplification of the Lame’ effect is not used for very thick shells where the effect is more accurately defined by the equations given in Appendix 1. Use of Equation (21.1) is limited to a pressure that does not exceed 0.385 SE or where the thickness does not exceed 1/2 of the inside radius.

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

I looked at it some, and didn't come up with any good answer, either.
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

(OP)
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.

#### Quote (saplanti)

This formula is accepted by that particular code to consider manufacturing tolerances given by the code. It is the same approach used by all other codes with similar or different formulation.

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

Paulettea,

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

Ha, very good question. Any code committee member here or whoever knows the history ? Only they can address why -06P is used.

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

This

#### Quote:

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).
and this

#### Quote:

So I agree with you that in practice, we just follow the rules.
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

And please note that the plate manufacturing minimum tolerance is part of the manufacturing tolerance/deviation. So the plate that you will buy will not be in the exact nominated thickness, the thickness will deviate from the nominated thickness in the width and length of the plate within the tolerances given by the code (please see the nominated plate material code and its reference).
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

The sentence in the second paragraph should be read as:
"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

(OP)

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

(OP)
Thank you XL83NL again with very useful comment.

#### Quote (XL83NL)

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.
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

#### Quote:

However, unfortunately I am not much experienced and in order to have a reliable understanding of the code I need time a mentor. And you see time is something that is not provided by managers at all, since they dont understand the risks involved in designing pressure vessels. 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?" I will give you time and hire an experienced PV engineer, because I dont want to see this company go bankrupt based on a stupid decision I made as an MBA to give an unexperienced engineer a task which requirs years of proper mentoring and training, which resulted in a bad design for whoch our company got convicted.

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

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

Regards
r6155

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

I apologize for my last two comments above, I wish I can delete them. I should not be writing after midnight. Of course everything is based on the Lame equation and some acceptance within the code. Now I have a doctor appointment, I will search my books again after I come back. I think it is related to the D/t ratio and stress distribution within the thickness. However the findings will not change the code rule at all.

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

Just as an exercise, I made up a spreadsheet to calculate thickness using the Lame formulas, and from that, calculated what that 0.6 factor ought to be to match the thickness. With low pressures, it comes out 0.5. As pressure and wall thickness increases, it gradually increases to 0.6 or a little higher depending on where you make the cutoff. At the low end, that 0.6P term is fairly negligible, so it doesn't make much difference what it is, and at the high end, that formula gives you results very close to the Lame formulas. So, I don't know the theoretical derivation for it, but it does appear to be an excellent (and much simplified) way to account for the variation of stress through the thickness as per the Lame formulas. That being the case, I would think it has no relation to vessel tolerances, plate tolerances, etc.

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

(OP)

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

Hi I am back again, and found some explanation for your interest:

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

Here is the attachment for the second reference (this shows you how they derived the formula):

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

(OP)
Thank you saplanti very much.

The second reference that you have sent says it very clearly.
"The range of the membrane equation has been extended by the empirical modification of 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

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?)

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

Simple. See ASME VIII Div.1

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

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
http://www.xcalcs.com : Online engineering calculations
https://www.megamag.it : Magnetic brakes and launchers for fun rides

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

(OP)

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

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.

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

(OP)
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.

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

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.

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