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# B31.1 tm calculation - y coefficient

## B31.1 tm calculation - y coefficient

(OP)
Could someone please explain what the y coefficient is there for? It is explained in the code a coefficient having values as given in the table 104.1.2 (A). It seems to be a safety factor as it reduces the calculated value of tm.

### RE: B31.1 tm calculation - y coefficient

As I understand it, it is a temperature-dependent creep factor.  Note how the Y factor changes with temperature.  I will look in my references.  John B - comment?

Thanks!
Pete

### RE: B31.1 tm calculation - y coefficient

Hello,

You nailed it Pete, "Y" is a factor to compensate for the non-linear reduction in allowable stress at design temperatures above 900 degrees.

Regards, John.

### RE: B31.1 tm calculation - y coefficient

Factor y has a clear physical meaning.
If you take y=d/(d+Do) as required by B31.3 for thick pipes at low temperatures, you'll find that SE equals exactly the value for the maximum circumferential stress (at inner face) in cylinders under pressure: S=P(Do2+d2)/(Do2-d2).
For thin pipes y=0.5, for thick pipes y<0.5

0.5 is also the value to be used if one wanted to account for the average circumferential stress, not the peak value.
Hence y is there because it must be there.
Now an expert in the foundations of B31.1 should help, but I suppose that the use of 0.4 (e.g.for all non ferrous metals and cast iron) is a safety value to account for thick pipes.
I can't explain the increasing values at higher temperatures: if it is a safety factor, then it decreases the safety with increasing temperature! Would be curious to know the correct answer.

prex
motori@xcalcsREMOVE.com
http://www.xcalcs.com
Online tools for structural design

### RE: B31.1 tm calculation - y coefficient

The factor Y comes from the Boardman equation which is an empirical equation which matches the Lame equation solution for the circumferential stress on the inside diameter of a cylinder.  The circumferential stress on the inside of a cylinder under internal pressure is higher than the circumferential stress on the outside because of the effect of radial stress (=internal pressure on inside surface), poisson's effect, and the fact that meridional (axial) strain through the thickness of a cylinder must remain constant (increase in circumferential stress offsets axial strain caused by radial stress, both poisson's effects). The same equation is used in Section VIII Div 1 but the equation is recast based on inside diameter rather than outside diameter.  The empirical relation with y=0.4 becomes less accurate for thick walled cylinders, and the equation cited by Prex corrects this.  In the creep regime, the stress distribution through the wall becomes more constant, which is why the y factor is changed for temperatures in the creep regime. If you want more detail (or more long winded description), you can purchase the book I just published through ASME, Process Piping, The Complete Guide to ASME B31.3.

### RE: B31.1 tm calculation - y coefficient

Chuck, there is no effect of Poisson's ratio in the solution for stresses in a thick walled cylinder with capped ends: see Roark's Formulas for Stress and Strain.
Poisson's ratio appears only in the relationships for deformations.

Also, if at high temperature creep tends to flatten the stress distribution, this would justify taking y=0.5, not 0.7 as allowed by the code.

Can you give some more highlights on why y is allowed to go over 0.5, so that the resulting average circumferential stress will be slighly higher than the allowable stress?

prex
motori@xcalcsREMOVE.com
http://www.xcalcs.com
Online tools for structural design

### RE: B31.1 tm calculation - y coefficient

It is a Poisson's effect but it does not appear in the equation because Poisson's ratio cancels out (at least for isotropic material).  Mentioning the Poison's effect is part of a thumbnail sketch of why the stresses differ through the thickness.  You can look up a reference on Lame to get the full description.

Increasing Y decreases required thickness (which means the calculated stress is lower for the same thickness).  Thus, Y increases from 0.4 to a higher value in the creep regime.

If you are worried about why Y can be greater than 0.5, equilibrium is satisfied (more or less) with Y=1.0 (the pressure acts on the inside diameter, not midwall.

### RE: B31.1 tm calculation - y coefficient

Thought I would give a little more thorough explanation of the physical phenomenon.

The radial stress in a cylinder under internal pressure is equal to the internal gauge pressure on the inside surface and is compressive and is zero on the outside surface.

The meridional or axial strain in a cylinder has to be constant through the thickness, or it will cease to be a cylinder (e.g. if there is a bending strain through the thickness, the cylinder walls would have to roll inward or outward).

The radial compressive stress causes a meridional tensile strain due to Poisson's effect.  This meridional strain due to radial stress is proportional to internal pressure on the inside of the cylinder and is zero on the outside.

This difference in meridional strain due to radial stress cannot exist, so something else must happen to cancel it out.

The other thing that happens, is the circumferntial stress becomes nonuniform through the thickness.

The circumferential tension due to internal pressure on the inside is larger than the outside.  This larger circumferential stress causes a larger meridional compressive strain (inside relative to outside) which offsets the larger tensile strain caused by radial stress (again inside relative to outside).

Thus, Lame exists.

### RE: B31.1 tm calculation - y coefficient

And thanks to Chuck becht & prex.

Once again thank you very much.

Amit

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