Pipeline Moment of Inertia Equations 1

[JayKimYou]

Hello,

I have a question regarding two different equations used to calculate bending moment of inertia for pipes. Any help would be much appreciated.

The first equation I came across shows my moment of inertia as:

I = pi/64(D^4-d^4) where;

D= outer diameter of pipe
d= inner diameter of pipe

I have also come across another equation for moment of inertia in a paper regarding HDD installation loading and stress analysis. This equation is to find the pulling load within a curved section of HDD. They used moment of inertia to ultimately determine the Normal forces applied to the pipe. They specified I as "bending moment of inertia", however the equation they used was:

I = pi(D-t)^3 * t/8 where;

D = outer diameter of pipe
t = thickness of pipe

I was wondering if someone could please explain why they used a different equation, when most of the equations online and in textbooks show the former equation rather than the latter. I have only seen the latter equation once on these forums, but I would love an explanation of the differences in use. Thank you for your time.

 

[BigInch]

The first as you probalby know is just the value for the simple moment of inertia of a circular tube.

The 2nd equation appears to be derived from V*Q/I/t, relating normal force (vertical shear) to bending moment, where the bending moment is probably being related to axial pull load x some eccentricity factor of a circular pipe cross section. They are using a somewhat simplified calculation for the section modulus of a circular section where they use the average diameter, then multiply again by t/8 to approach a Q/I/t value of a tube. It is not clear from your description exactly how axial load induces shear in that HDD model, so it would help if you can show how pull load is being related to bending moment in the HDD calculation. I imagine that it is axial load x a certain eccentricity factor.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

 

[JayKimYou]

Thanks for the quick reply, I apologize, I can't see why shear is induced in the model either, if you have a few minutes, would you please take a quick scan through the paper:

http://storage.cloversites.com/jdha...talled by Horizontal Directional Drilling.pdf

Page 42 begins to describe how they found the equations for pulling the pipe through curved sections. I believe this paper will explain more than I can at the moment... Thanks again for your help.

 

[rconner]

I know there are at least two commonly used "moments of inertia" involved with various "bending" of pipes/tubes or rings. I'm not sure if either one is expressed correctly in your post. If e.g. a pipe or tube is bent longitudinally, I believe the applicable moment of inertia about its central axis is conventionally expressed as,

I(long.) = pi(D^4-d^4)/64

On the other hand, if a pipe is crushed e.g. as a ring loaded in opposite line bearing loads at the top and bottom, the effective moment of inertia to determine e.g. curved beam stress is essentially that of the rectangular pipe wall thickness cross-section i.e. I(ring) = bh^3/12, or per unit pipe length in effect,

t^3/12

Am curious could you point me to the specific thread, context or reference you find the "latter" expression in, and maybe can try to divine where the developer was coming from?

 

[rconner]

Thanks (you were reading my mind - your response to BI and my request crossed in cyberspace!) Will look at this.

 

[JayKimYou]

It took me a bit to find the forum page again,

http://www.eng-tips.com/viewthread.cfm?qid=167106

this is where I saw the equation appear once more. They are referring to the inertia of a ring... any explanation would be great. Thanks again for everything.

 

[rconner]

fyi

http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia

(approximation, ok for thin tubes)

 

[prex]

As d=D-2t you can write:
D⁴-d⁴=(D²-d²)(D²+d²)=(D-d)(D+d)(D²+d²)=2t(2D-2t)(D²+D²-4Dt+4t²)=8t(D-t)(D²-2Dt+2t²)
Now, for normal proportions of pipes, where t<<D you can state
D²-2Dt+2t²[&asymp;]D²-2Dt+t²
so the expression above becomes
8t(D-t)³
that leads to the second formula. So the two are almost exactly equivalent (for pipes).

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[JayKimYou]

Is there a minimum ratio of t/D to determine whether or not you can assume it is thin walled (i.e. 2t = t)? Is it critical to use the thin wall equation?

Finally, Regarding HDD, can I use either equation? The differences of inertia between the two is within a ~0.1 depending on the thickness. I don't think this will ultimately affect the outcome of stress analysis, however, I just want to use the most correct equation in my calculations.

Thanks everyone for your responses. It is very encouraging to know I can depend on this forum.

 

[BigInch]

BTW neither of those is the "thick walled" pipe equation.
Don't get all balled up in, which one is more correct. Engineering is not mathematics, nor is it accounting, or sub-atomic physics. In general you should always strive to use the simplest solution that gets you an answer that you can live with. If you use an approximate equation and the answer is difficult to live with, then (and only then) refine your analysis. You will always have to communicate your design to someone that doesn't understand as much as you do (you're the expert) and explaining the simplest solution will always be easiest for both of you. Let's see how using either one of those equations affects the solution and the answer in your case here.

You should use the first equation whenever you need to know the stress on the outside surface, or at the surface of the inside wall, as each of those stresses are different. If you just need the average stress in the wall, then you use the average stress formula. To calculate axial stress and pull load from the normal stress resulting from a bending moment, you have a choice. You can calculate the outside surface bending stress and the inside surface bending stress and take the average of those two, or you can use the I formula from the average diameter and calculate the average stress directly. As axial load is the summation of all stress in the wall, not just the outside stress, or just the inside stress alone, it is the average stress that you need. So, calculate the average stress the easiest way possible and that is by formula 2. Now, is the difference of 0.1 explained by calculating only the outside fiber stress and wrongly assuming that the outside stress value is uniform across the wall?

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

 

[chicopee]

JayKimYou, both equations are coorect, however, the second one is for thin wall tubes and is an approximate to the first equation. You will find in some of the literature that the second equation is also expressed as pie*t*D^3/8.

 

[prex]

JayKimYou, by a bit of inspection you'll find that the relative error in the 2nd formula is of the order of t²/D². So if the pipe is "really" thick (say D/t=10) the error will be 1%, and if the pipe is with "normal" proportions (say D/t=50) the error will be 0.04%.
I don't think you have to worry about which is the correct formula, but as both are really simple formulae, you have the choice.

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[rconner]

Actually, the first expression I(long.) = pi(D^4-d^4)/64 appears accurate for even a tube with an infinitessimally small bore I.e a solid rod. If you set d = 0 , this expression resolves to (long.) = pi(D^4)/64, that happens to be exactly the moment of inertia of a circular area or rod about its center.
I believe the "approximation", on the other hand, would give you a substantial error in such extreme thickness case, with a predicted I only about 50% of what it ought to be if you set t = D/2.

 

[BigInch]

The first is correct for all wall thicknesses, the second is correct to use when you need the median wall stress and has nothing to do with calculating moment of inertia of the section. It is not a formula for calculating "moment of inertia".

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek