On B31J Note 3: A Parametric FE Evaluation of the 1.3/h Bend k-Factor

Findings:

k=1.3/h without Figure 1-5 modification is good for bends attached to rigid elements (e.g. flanges) as close as 1.5OD from both ends.

k=1.3/h with at least 1-flange modification (c=h^1/6) starts to become necessary with rigid element attachments at 1 to 1.5OD from both ends.

k=1/3h with 2-flange modification (c=h^1/3) may still be non-conservative where flanges are attached directly to both ends of the bend.

Background

The 2020 version of ASME B31.3 and B31.1 removed their Appendix D that had been in used for decades. Users are now being referred to B31J for SIF and Flexibilities of bends and branch connections.

I was in the process of evaluating the new CAESAR II v13 which natively incorporates B31J. To my surprise, I could not pass the displacement stress check on the Example 1 calculation of the updated Appendix S of B31.3. My investigation revealed that it’s most likely due to the difference in flexibility factors of bends. It was initially perplexed since the formulae for bend SIF and k-factor in B31J appear to be unchanged compared to Appendix D (you’d have to look at B31J errata for the correct formulae for the SIFs though). Then, the kind folks at Hexagon pointed out that I missed the Note 3 of B31J. Oops!

The note basically states that if you have a 90 deg bend with the thickness matching the connected pipes, you “may” take k=1.3/h. I have since found that there is no consensus on this. Some pipe stress analysis program developers take this as mandatory if B31J flexibility is used, while some don’t. And I’m fairly certain that the updated Appendix S too does not make use of this note.

My interpretation of the note as written is that it’s optional. On the other hand, when the codes give you an alternative, it’s usually allowing you to use something less conservative, or easier, or both. This note is neither (yes, one can also argue that in some cases, higher stiffness can be less conservative).

So I went ahead and read multiple papers on this subject, until I came across what I believe was the source of this 1/3h factor. It’s a 1978 paper by Rodabaugh, Iskander and Moore. In the paper, they also suggested 1.1/h for 45 deg bends.

Since Rodabaugh et al. used FEA as a basis, I figured I should do my own too with my computer that I believe is many orders of magnitude more powerful than what they had back in 1978. And since their element count was no more than 900 by my estimate, I wondered if their model was too rigid. Perhaps, using many more elements that I can easily do now would prove that 1.65/h is indeed a reasonable estimate.

I went ahead and performed a parametric FEA on 294 specimens (seven sets of 42 specimens). Each set consists of LR bends of NPS3 to NPS12 (in S-10S, S-STD, S-80, and S-160), and NPS14 to NPS24 (in S-STD, S-80, and S-160). Six of the sets have 90 deg bends attached at both ends to pipes with lengths L (L = 0 OD, 1 OD, 1.5 OD, 2 OD, 3 OD, and 4 OD). The remaining one set has 45 deg bends attached at both ends to pipes of L=2 OD only. In all cases, the pipes are of the same thickness as the bends. The other ends of the pipes (ends A and B) were modeled as rigid to simulate relatively rigid bodies such as flanges. 0 OD essentially means that the bends are attached directly to flanges.

End A is modeled as fixed, while in-plane and out-plane moments are applied to end B in two separate load cases. Rotational displacements were measured at both ends of the bend, as well as point B. The in-plane (ki) and out-plane (ko) flexibility factors were calculated using the bend-end displacement readings, and then compared with the factors generated based on point B displacements as sanity check. (They match).

My analyses were performed using ANSYS 2021 R2. Each model was meshed into 27648 quadratic brick elements, with three through-thickness elements. For my result validation, I performed a mesh sensitivity study to determine the optimum number of elements. I also compared some of my calculated in-plane flexibility factors with those calculated using PRG software, FEBend.

Findings

The results of my analysis are shown in the log-log charts attached (my calculated results are shown as FEA ki and ko). Also included in the charts are the lines representing the 1.65/h, Rodabaugh’s recommendations, and others.

For the range of h I evaluated:

1.??????90 deg bends connected to 2OD, 3OD and 4OD-long pipes:

a.??????We can clearly see that the FEA calculated k’s are always lower than 1.65/h. If we take lower k as being conservative, 1.65/h is therefore non-conservative. 1.3/h as recommended in B31J Note 3 appear to be a much better rule to use for everyday pipe stress calculations.

b.??????Perhaps a more accurate equation to use is k=1.4/h, as shown by the trendline of my results.

c.??????The chart for 2OD also includes a line for k=1.3/h adjusted with 1-flange attachment. Some developers recommend the use of 2-flange modifier (c=h^1/3) if there are two rigid elements within two pipe diameters from the bend. From this chart, it’s clear that even 1-flange adjustment is not needed when attached pipes are 2OD or greater in length.

2.??????90 deg bends connected to 0 OD, 1 OD and 1.5 OD-long pipes:

a.??????With flanges attached directly to both ends of the bend, even 2-flange adjustment may severely overestimate the flexibility of the bend (non-conservative). c=h^2/3 modifier may provide a better estimate.

b.??????With flanges attached to both ends at a distance of 1OD, 1-flange adjustment (c=h^1/6) becomes necessary.

c.??????Starting at 1.5 OD-long pipe attachment, we can see that the calculated k’s are generally on the 1.3/h line. So, despite having both ends with flanges at 1.5OD away, reduction adjustment may not be necessary. Alternatively, this can be used as a starting point to consider only 1-flange adjustment (c=h^1/6).

3.??????45 deg bends connected to 2 OD-long pipes:

a.??????1.1/h appears to be a very good estimate. It may become more non-conservative at higher h (generally very thick walls).

b.?????1.65/h unmodified is definitely too non-conservative for 45 deg bends.

c.??????Modification to the k factor for 45 deg bends (e.g. 1.1/h) has not been explicitly addressed in the Code yet. PIP does suggest modifying 1.65/h with the factor for 2-flange attachment for all 45 deg bends. My chart shows that you can reasonably and conservatively estimate the k as 1.3/h with 1-flange attachment (c=h^1/6). 2-flange attachment adjustment (c=h^1/3) used with 1.3/h may be overly conservative, unless perhaps the rigid elements are closer than 2OD.

Conclusion

In all cases, the calculated ki and ko do appear to be virtually the same.

Please note that pressure stiffening effects was not considered in this evaluation. Also note that I have not yet addressed very high or very low h values. I’ve only investigated h of between 0.082 and 0.841. Rodabaugh’s paper does suggest that the recommended k equations may become non-conservative as h gets to about 1 or above.

In conclusion, I'm personally comfortable using 1.3/h for 90 deg bends in most cases. It is my humble opinion however that the note 3 wording could have been clearer. I wouldn't mind seeing it replacing 1.65/h altogether in the table. Now as pipe stress engineers, this is obviously a bit unpleasant especially when we're dealing with sensitive equipment. It is true that the point of pipe stress analysis is not to reflect reality at high degree of accuracy. We approximate, but we'd better err on the conservative side.

Further Work

I may expand this review later on 45 deg bends with more variety of pipe attachment lengths.