Stress linearization tool
Stress linearization tool
(OP)
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
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





RE: Stress linearization tool
The average stress components are in this case: (s1+s2+s3+(s0+s4)/2)/4
So for example XXm=35.18 (not far from the value in your table, but the table is incorrect).
The bending components are: 6(s1-s3+(s0-s4)/2)/4
For example XXb=5.481
Having all the m and b stress components you can follow that FAQ.
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool
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
RE: Stress linearization tool
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
RE: Stress linearization tool
If I have more nodes in SCL what would be common form for these formulas?
(s1+s2+s3+(s0+s4)/2)/4 --> (s1+s2+s3+…+sn-1+(s0+sn)/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 (s0+s4)/2?
6(s1-s3+(s0-s4)/2)/4 --> 6(s1-sn-1+(s0-sn)/2)/n or 6(sn/2-1-sn/2+1+(s0-sn)/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
RE: Stress linearization tool
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:
Σσidxi/t where dxi is the length of the SCL tributary of stress σi (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:
6Σσi(xi-t/2)dxi/t2
For equal subdivisions (xi-t/2)/t=i/n-1/2
So for 4 equal subdivisions:
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
This will hopefully give more reasonable results.
But be careful: you
To calculate the m+b stress, you need to do σm +/- σb at the component level, and only then calculate the two equivalent m+b stresses.
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool
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.)
RE: Stress linearization tool
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
Σσidxi/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: 6Σσi(xi-t/2)dxi/t2
Equal subdivisions: (xi-t/2)/t=i/n-1/2 (what is that t?) Do you mean (xi-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/t2
dxi=distance from origo relative to SCL (always positive?)
t=distance from origo (=number of elements from origo)
i=0 dx0/t2=(1/2)/22=1/8
i=1 (1/4)/12=1/4
i=2 0
i=3 (1/4)/12=1/4
i=4 (1/2)/22=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
RE: Stress linearization tool
The numbers seem correct though.
I don't understand the last part.
You correctly say Then you should calculate XXm±XXb, XYm±XYb 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
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool
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
RE: Stress linearization tool
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool
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
RE: Stress linearization tool
(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).
RE: Stress linearization tool
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 σij,l=σij,m+σij,b 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
RE: Stress linearization tool
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
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool
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
RE: Stress linearization tool
σb=6(σ0(-1/2)(1/10)+σ1(-3/10)(1/5)+σ2(-1/10)(1/5)+σ3(1/10)(1/5)+σ4(3/10)(1/5)+σ5(1/2)(1/10))
prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads
RE: Stress linearization tool