Stress linearization tool

[VonFea]

Hi,
I’m doing excel tool for stress linearization and I need some help. I have six stress components for all nodes in stress classification line (SCL). I know the formulas are found in FAQs (

http://www.eng-tips.com/faqs.cfm?fid=982),

but I have hard time to solve these integral equations. I would appreciate very much if someone could show me how to solve for example bending stress with real component values (attached). Example model is pipe with pressure and moment (My) load.
Is membrane stress same as equivalent stress calculated with average component value (green 31,99MPa)?

Reagards
VonFea

 

[prex]

You have five nodal values at equal intervals: number them from 0 to 4.
The average stress components are in this case: (s[sub]1[/sub]+s[sub]2[/sub]+s[sub]3[/sub]+(s[sub]0[/sub]+s[sub]4[/sub])/2)/4
So for example XX[sub]m[/sub]=35.18 (not far from the value in your table, but the table is incorrect).
The bending components are: 6(s[sub]1[/sub]-s[sub]3[/sub]+(s[sub]0[/sub]-s[sub]4[/sub])/2)/4
For example XX[sub]b[/sub]=5.481
Having all the m and b stress components you can follow that FAQ.

prex
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[TGS4]

I guess it all depends on the Code to which you are performing the linearization for.

If it is the ASME Code (ASME Section VIII, Division 2, Part 5), then the answer to your second question is that I can't tell from what you did. The procedure is to calculate the membrane and bending stresses at the component level, and then calculate equivalent membrane and membrane-plus-bending stresses from those component-level values.

Also, note that the membrane-plus-bending tensor should look like
σxx,m σxy,m σxz,m
σxy,m σyy,m± σyy,b σyz,m
σxz,m σyz,m σzz,m± σzz,b
Where X is through-thickness, Y is hoop, and Z is meridional

 

[VonFea]

Hi,
Thank you for your answers prex and TGS4. I got now two m+b value (38,13 and 25,85) but they are greater than total stress (linearization2.JPG). I know this is possible, but because I’m new with this I smell error. Total stress is calculated with equivalent stress formula in EN 13445-3 (previous post) and results are same in FEA (nodal von mises).

I’m planning to use this with ASME VIII div 2 and EN 13445-3. Linearization formulas are same although in EN 13445-3 there is a formula for more accurate radial linearization of bending stress (linearization2.JPG). If you can tell how to solve this equation it would be great. I know values at both end at SCL is almost always enough, but parabola would look better than straight line

TGS4: In this example I have used horizontal pipe where Z is through-thickness, Y is hoop, and X is meridional. I’m not sure what do you mean by:

Also, note that the membrane-plus-bending tensor should look like
σxx,m σxy,m σxz,m
σxy,m σyy,m ± σyy,b σyz,m
σxz,m σyz,m σzz,m ± σzz,b
Where X is through-thickness, Y is hoop, and Z is meridional

I know sigma is missing and orientation of coordinate is different but I don’t understand “plus minus” thing. Can you open this little more please? Perhaps like prex did with simple equations and s-values. That was simple enough for me

When I convert element values to nodal values in fea model should I use maximum or average nodal values?

Regards
VonFea

 

[VonFea]

Hi,
If I have more nodes in SCL what would be common form for these formulas?
(s[sub]1[/sub]+s[sub]2[/sub]+s[sub]3[/sub]+(s[sub]0[/sub]+s[sub]4[/sub])/2)/4 --> (s[sub]1[/sub]+s[sub]2[/sub]+s[sub]3[/sub]+…+s[sub]n-1[/sub]+(s[sub]0[/sub]+s[sub]n[/sub])/2)/n
Otherwise average of components except SCL first and last nodal values have been averaged. Is this right?
5-A.4.1.2 Step 1. … The membrane stress tensor is the tensor comprised of the average of each stress component along the stress classification line
Why the formula isn’t pure average? Why do I have to average first and last value (s[sub]0[/sub]+s[sub]4[/sub])/2?

6(s[sub]1[/sub]-s[sub]3[/sub]+(s[sub]0[/sub]-s[sub]4[/sub])/2)/4 --> 6(s[sub]1[/sub]-s[sub]n-1[/sub]+(s[sub]0[/sub]-s[sub]n[/sub])/2)/n or 6(s[sub]n/2-1[/sub]-s[sub]n/2+1[/sub]+(s[sub]0[/sub]-s[sub]n[/sub])/2)/n ???
Where this formula comes from and how do I apply it when I have more nodes?
How come there is no radial distance anywhere in the formula?

Regards
VonFea

 

[prex]

I'm very sorry...
The integral in the FAQ is correct, but the translation into a summation I did above is fully wrong.
Let's start with the membrane stress.
The translation of the integral into a summation is:
[Σ][σ][sub]i[/sub]dx[sub]i[/sub]/t where dx[sub]i[/sub] is the length of the SCL tributary of stress [σ][sub]i[/sub] (so this formula applies also to unequal subdivisions).
For equal subdivisions, for i=0 and i=n dx/t=1/2n and dx/t=1/n in the other cases. This gives the formula for membrane stress above.
For bending the translation into a summation is:
6[Σ][σ][sub]i[/sub](x[sub]i[/sub]-t/2)dx[sub]i[/sub]/t[sup]2[/sup]
For equal subdivisions (x[sub]i[/sub]-t/2)/t=i/n-1/2
So for 4 equal subdivisions:
[σ][sub]b[/sub]=6([σ][sub]0[/sub](-1/2)(1/8)+[σ][sub]1[/sub](-1/4)(1/4)+[σ][sub]3[/sub](1/4)(1/4)+[σ][sub]4[/sub](1/2)(1/8))=3(-[σ][sub]0[/sub]-[σ][sub]1[/sub]+[σ][sub]3[/sub]+[σ][sub]4[/sub])/8
This will hopefully give more reasonable results.
But be careful: you also made a mistake, it's not only me!
To calculate the m+b stress, you need to do [σ][sub]m[/sub] +/- [σ][sub]b[/sub] at the component level, and only then calculate the two equivalent m+b stresses.

prex
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[TGS4]

Thanks for your response, prex. Very good explanation. It's very important to recognize that the membrane and membrane-plus-bending equivalent stresses are calculated by the membrane and membrane-plus-bending component stresses.

VonFEA - I don't understand your confusion about what I wrote. For ASME Section VIII, Division 2, Part 5, perhaps the paragraph in 5-A.4.1.2(b) is more to your liking. It's the same thing. And, as prex indicates, membrane-plus-bending is both membrane + bending and membrane - bending, hence the ±. (If your membrane is negative, then the subtraction will provide a higher absolute value, and vice versa.)

 

[VonFea]

Hi,
Thank you for your reply. TGS4 – I didn't understand your answer because I didn't know that m+b summation must be made in component level. Now your answer makes perfect sense

Prex- Can you verify that this is what you meant.

Membrane:
Equal subdivisions: (n= 4 elements): dx/t = 1/(2n)=1/8 when i=0 and i=n=4
dx/t=1/n=1/4 when i=1-3

Σσ[sub]i[/sub]dx[sub]i[/sub]/t = 36,986MPa x 1/8 + 36,105MPa x 1/4 + 35,178MPa x 1/4 + 34,251MPa x 1/4 + 33,382 x 1/8 = 35,1795MPa (=XXm) is this right?

Then I calculate XYm, YYm, XZm, YZm, ZZm at the same way. Whit these six membrane component, I calculate equivalent stress and that is membrane stress. Right?

Bending: 6Σσ[sub]i[/sub](x[sub]i[/sub]-t/2)dx[sub]i[/sub]/t[sup]2[/sup]

Equal subdivisions: (x[sub]i[/sub]-t/2)/t=i/n-1/2 (what is that t?) Do you mean (x[sub]i[/sub]-t/2)=i/n-1/2 ?
Then i/n-1/2:

i=0 0/4-1/2=-1/2
i=1 1/4-1/2=-1/4
i=2 1/2-1/2=0
i=3 3/4-1/2=1/4
i=4 4/4-1/2=1/2

dxi/t[sup]2[/sup]

dxi=distance from origo relative to SCL (always positive?)
t=distance from origo (=number of elements from origo)

i=0 d[sub]x0[/sub]/t[sup]2[/sup]=(1/2)/2[sup]2[/sup]=1/8
i=1 (1/4)/1[sup]2[/sup]=1/4
i=2 0
i=3 (1/4)/1[sup]2[/sup]=1/4
i=4 (1/2)/2[sup]2[/sup]=1/8

σb=6(σ0(-1/2)(1/8)+σ1(-1/4)(1/4)+σ3(1/4)(1/4)+σ4(1/2)(1/8))=3(-σ0-σ1+σ3+σ4)/8

Then I calculate XYb, YYb, XZb, YZb, ZZb at the same way. Whit these six membrane component, I calculate equivalent stress and that is bending stress. Right?

Then I calculate like TGS4 showed: (although my coordinate orientation is different)
σxx,m σxy,m σxz,m
σxy,m σyy,m ± σyy,b σyz,m
σxz,m σyz,m σzz,m ± σzz,b

Have I understand this right?

Regards
VonFea

 

[prex]

t is thickness, in other words t=[Σ]dx[sub]i[/sub]
The numbers seem correct though.
I don't understand the last part.
You correctly say

Then I calculate XYm, YYm, XZm, YZm, ZZm at the same way. Whit these six membrane component, I calculate equivalent stress and that is membrane stress.

Then you should calculate XX[sub]m[/sub]±XX[sub]b[/sub], XY[sub]m[/sub]±XY[sub]b[/sub] and so on, to obtain two sets of six m+b stress components: from these you calculate two equivalent stresses and the larger is the m+b stress.

prex
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[VonFea]

Hi,

Also, note that the membrane-plus-bending tensor should look like
σxx,m σxy,m σxz,m
σxy,m σyy,m ± σyy,b σyz,m
σxz,m σyz,m σzz,m ± σzz,b
Where X is through-thickness, Y is hoop, and Z is meridional

I thought that TGS4 meant that only hoop and meridional m+b components need to be summed
In my case YYm±YYb and XXm±XXb and other components are membrane components. With these two sets of six components I calculate m+b equivalent stress.

Prex- So you are saying that I have to sum all components XXm±XXb, XYm±XYb, YYm±YYb, XZm±XZb, YZm±YZb, ZZm±ZZb

Regards
VonFea

 

[prex]

Right

prex
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[VonFea]

Hi,
Prex- So you mean that you are right and TGS4 is wrong? Or did I misunderstood what TGS4 meant?
Does TGS4 wan't to comment this?

Regards
VonFea

 

[TGS4]

If you want to comply with 5-A.4.1.2(b) then follow what I said. If you are designing to a different Code that does not have such mandatory requirements, then you are free to do what Prex indicates.

(BTW - I was one of the Code Committee members who pushed for 5-A.4.1.2(b) to be specifically written as it is. Annex 5.A is mandatory for ASME Section VIII, Division 2. It is based in WRC 429 recommendations.)

I have the greatest respect for Prex and their abilities/capabilities. However, in this specific case I have first-hand knowledge and insight (I won't claim that I wrote that specific article in the Code, but I provided significant input to ensure that it was written as it currently stands).

 

[VonFea]

Hi,
Thank you Prex and TGS4 for your exellent answers! I think I got this now.

EN 13445-3 calculation is made like Prex stated because the formula is σ[sub]ij,l[/sub]=σ[sub]ij,m[/sub]+σ[sub]ij,b[/sub] without any special requirements.

While in ASME there are requiremets: Bending stresses are calculated only for the local hoop and meridional (normal) component stresses, and not for the local component stress parallel to the SCL or in-plane stress. This is like TGS4 mentioned.

Regards
VonFea

 

[prex]

Let me add that what ASME states is an additional allowance for the design (or a reduction of the safety factor): this is evident if you think in term of elastic energy, as neglecting part of the actual stress tensor will neglect part of the stored energy and thus finally reduce the equivalent stress.
Of course this is of minor importance for SCL crossing shells far from discontinuities, as we know that shear stresses are negligible there, but at a gross structural discontinuity the influence of this choice might be more important.
I'm curious about the rationale that is behind this position: TGS4, as you supported it, could you enlighten this more?

prex
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[datsnl]

Hi,

Thanks for this excellent post for new commers like myself.

I just had a simple question, If there are 5 elements i = 0 to 5, what will be the Bending Stress formula?

I have attached excel file that I am having for 6 nodes, 5 elements.

Thanks
datsnl

Dharmit
Moonish Ent Pty Ltd

http://engineering.moonish.biz

 

[prex]

The answer is in the post above at 4:14.
[σ][sub]b[/sub]=6([σ][sub]0[/sub](-1/2)(1/10)+[σ][sub]1[/sub](-3/10)(1/5)+[σ][sub]2[/sub](-1/10)(1/5)+[σ][sub]3[/sub](1/10)(1/5)+[σ][sub]4[/sub](3/10)(1/5)+[σ][sub]5[/sub](1/2)(1/10))

prex
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[TGS4]

prex - the rationale was that it was the recommended approach from WRC 429. It will be a few weeks until I return to the office, but when I do I will forward to you the exact part of WRC 429. I recall that it was due to, in the "original" or "hand-calc" definitions of bending that only bending in the hoop and longitudinal directions had any physical meaning for pressure equipment. Besides, M+B is relevant only in Protection Against Ratcheting, as opposed to Protection Against Plastic Collapse.