Hello,

I'm trying to understand the matrix formulation behind the elbow element. I'm not understanding how the flexibility factor is utilized to create a beam-like element.

I originally thought that the in-plane and out-plane flexibility factors were simply used to reduce the flexural rigidity by multiplying the respective constant by the reciprocal of the flexibility factor.

According to a similar source, a flexibility matrix is first created and then inverted to create a stiffness matrix. Is this how CAESAR 2 also calculates the stiffness matrix for elbows?

Source is here: https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/ans_arch/thy_el18.html

Through my research, I continually run into a source by Chen regarding flexibility analysis by stiffness matrix, but it seems to be out-of-print and very hard to find.

Does anyone have a copy of "L. H. Chen. "Piping Flexibility Analysis by Stiffness Matrix". ASME, Journal of Applied Mechanics. December 1959." they could send me?

I'm just a piping engineer trying to find the bottom of the rabbit hole that is beam-type pipe stress analysis.

Thank you for any help!

That's an interesting twist to developing a bend stiffness matrix.
The Codes provide a bend flexibility factor, not a bend "stiffness" factor. So to produce a bend stiffness matrix (that we can combine with other component stiffness matrices), we must invert the bend flexibility matrix. This bend flexibility matrix was developed by the Mare Island program MEC-21. You will see the bend flexibility factor included there.
There is an older post on this subject but I am unable to locate it through the search utility here. I suggest you Google some of these terms to find more (e.g., bend flexibility MEC 21).

I also found this: https://repository.lib.ncsu.edu/bitstream/handle/1840.20/27617/F1-2.pdf?sequence=1&isAllowed=y

It states:

"In order to properly account for the cross-section ovalization in pipe elbows for Class 2 piping, the ASME Code requires the flexibility factor, FF, be used to modify the corresponding straight pipe bending moment of inertia, I, for both in-plane and out-of-plane bending."

Does this mean I can simply take a beam-element and divide the moment of inertia by the flexibility factor to calculate the increased flexibility of an elbow? Or do I need to perform the entire construction and inversion of the flexibility matrix?

Thank you.

But wouldn't that approach adjust all six terms and not just in-plane and out-plane bending?

Dave,

I think you apply the flexibility factor to the local in-plane and out-of-plane bending moments of inertia, ignoring the torsional moment of inertia, before transformation to the global coordinate system.

I think. I honestly have no idea.

This reference for an elastic pipe element has a parameter for the flexibility factor: https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/ans_arch/thy_el16.html

It states:



The torsional moment is unchanged.

Is that all that is required to model an elbow? I still feel like I'm missing something.

It's hard to unscramble those eggs though. Torsion on one end of a 90 degree elbow is out-plane bending on the other end.

Yes--that's why I'm confused about that method. It seems to be missing something.

Is it possible that the out-of-plane flexibility is "absorbed" in the element and forces/moments aren't applied to the opposite node? One node (where forces applied) would then have increased displacements, but the opposing node of the same beam element would retain the same angular displacement (same torsional flexibility).

Attached is a hard copy scan of a bend flex matrix from Coade. There must be some other post where this came from.

Bob,

Thank you for the attachment! That makes things very clear.

Here is another source for the matrix in Fortran code: https://repository.lib.ncsu.edu/bitstream/handle/1840.20/27617/F1-2.pdf?sequence=1&isAllowed=y