Help a Structural Guy - Anchor Forces 2

[VAStrEngr]

Attached is a sketch depicting a simple straight run of pipe between two anchor blocks with an expansion loop. The sketch represents an above group fuel pipeline for which I would like to know the reaction forces at the anchors due to a thermal load variation of 100 F.

I have been given anchor forces by our supposed "fuel" engineer but am having trouble accepting them because results from a computer FEA analysis I ran on a structural engineering program indicate reactions almost 1/3 of what he was computing. Given a huge increase in the size of the anchors that would be needed if I used his values, I strongly desire some consideration of our discrepencies.

His method can be summarized as the following:

1. determine virtual expansion (delta) due to thermal growth of 100 degree using the equation

delta = e(thermal coeff) * 100F * L .....where L=x1 + l + x3 + l + x2 (as shown in attached sketch).

2. determine anchor force = P = 12*E*I*delta/ l^3 ....where l is equal to the length of one leg of the loop.


While I agree with the premise behind it (although it seems to be a very conservative approach), my biggest concerns are:

1. the delta calculation should not include the 2 l's (the lenght of the expansion loop) because these are not contributing to the reaction force that is being calculated.

2. The P equation is only taking into account 1 leg of the loop. The "spring constant" part of the equation (12EI/l^3) could be much lower because you have two legs (or springs) in series which by Hooke's Law =

1/k = 1/k1 + 1/k2


Basically, I need to have my ducks in a row before I approach the senior engineer with these assumptions (especially since I am not actually in the fuels group).

Since anchor forces are a pretty standard component of pipeline design, I am appealing to you the Eng Tips community to help educate me on how this gets done.

Thank you!

 

[ione]

Take a look here

http://www.minikins.co.uk/IMG/minikindesignbook4thedition.pdf

http://www.spiraxsarco.com/resource...m-distribution/pipe-expansion-and-support.asp

 

[VAStrEngr]

Very nice resources...thank you. I need to investigate more into the equations but from the first look through, it appears to confirm my initial assumptions.

 

[BigInch]

Do not include the vertical lengths in the horizontal expansion calculation.

Calculate the thermal expansion using the (longest) distance between 1 anchor and the first elbow in the "loop", consider that to be the same deflection of a cantilever comprized of one vertical leg fixed at the top.

Solving the cantilever deflection equation for an anchor reaction would give, P = 3 * [Δ] * EI / Lv^3
where Lv is the height of the loop.

So the thermal deflection gives a point load (anchor reaction) of P_thermal = 3 * [Δ] * EI / Lv^3

That would indicate a reaction of 3/12 ths of the fuel guy's answer, or right about 1/4th.

You do now have the moment at the upper elbow from considering that totally fixed, which restrains free expansion of the run below and that would equal M = P_moment * Lv

Moment at top elbow = P_moment * Lv
Deflection of a cantilever with a moment,
[Δ] = M*L^2/(2EI)
using M = P_moment * Lv
[Δ] = (P_moment*Lv) * Lv^2/(2EI)
[Δ] = P_moment * Lv^3/(2EI)
P_moment = [Δ]* 2EI / Lv^3
The Moment gives a point load (anchor reaction) of
P_moment = 2 * [Δ]* EI / Lv^3

Thermal deflection gives a point load (anchor reaction) of
P_thermal = 3 * [Δ] * EI / Lv^3

Final result must be approximately equal the sum of the two cases, or
5 * [Δ] EI / Lv^3

Since the upper elbow is not fixed and actually does rotate, the reaction force due to that moment must be less, so let's let it rotate a bit and you can reduce that 5 to say about 4.

4 [Δ] EI / Lv^3 is 1/3 of the fuel guy's 12 [Δ] EI / Lv^3, so my guess is that you're much closer to the right answer and that the fuel guy's solution is not correct.

I don't understand why the fuel guy is using 12. A totally fixed cantilever is only 3 [Δ] EI / Lv^3

Just show them the FE calculation and let them decide which one they want to use.

 

[ione]

I concur with you: leg’s length should not be entered when computing the length between the two anchors.

Further to forces and moments acting on anchors M.W. Kellogg method could be of help.

Being:
-) A and B the anchors,
-) L the distance between A and B [inch]
-) deltaL the linear thermal expansion = alpha*L*?deltaT [inch]
-) I the moment of inertia of your pipe [inch^4]

FA = - FB = 10^6*A1*(I*?deltaL/L^3) expressed in [lb]

MA = - MB = 10^5*A2*(I*?deltaL/L^2) expressed in [ft-lb]

A1 and A2 are reported on nomographs where you have to enter loops parameter (leg’s length and horizontal segment length).

Please note that what above only accounts for forces produced by thermal expansion, but the total force acting on anchors could be caused even by other things.

 

[chicopee]

Are you sure the loop is rectangular as shown in the sketch?

 

[VAStrEngr]

The sketch was merely an illustrate because my engineer brain cannot describe things with words as well as I can draw. However, the method I am looking for reassurance on should work for all cases, square/rectangular loops. Then the issue becomes sizing the loop to provide the most efficient flexibility vs cost of pipe. The references that ione provided give a pretty good breakdown of this.

Thanks for the feedback ione and biginch...excellent.