I am analyzing a slug flow case. We observe that an elbow gets slammed every 3-5 seconds and displaces less than 1" in one direction, and even less in the other two. The estimated rise in pressure due to the slugging has been given to me by the process department. I multiplied this by the cross-sectional area of the pipe to get an applied force. I first did a modal analysis, and found that one of the natural frequencies Caesar calculated matches the movement of the pipe very well.

Now to calculate the time the force is applied (and thus the impulse on the elbow). I have read quite a few articles discussing the subject that suggest using the time the pressure wave takes to traverse the longest distance between elbows in the system. Since this line is coming out of a furnace, I first tried using the total length of the helical coil. That produced displacements that were much larger than observed. I next tried a 14' span between elbows that was in the same direction as the largest component of the observed deflection. The displacement results were consistent with the observed deflection.

The dynamic stress report shows that this line failing B31.3 code (the stress exceeds the allowable). I have been working feverishly on this for four days, and I may no longer be thinking clearly, but I interpret these results to mean:

1. Failure due to secondary stress (fatigue) is more likely because the system exceeds code stress. The fact that it has not failed in over 3 million cycles (6 months) is because the stress it is experiencing is somewhere in the safety factor of the code allowables.
2. Inherent in the analysis I've done is the assumption of low-cycle fatigue. In other words, there is a fatigue factor that is 1 for anything under 7000 cycles, but should be applied in this case. If I go back and do that, the results will be even worse due to attenuated allowable stress.
3. I should put a snubber on that elbow and see if I can resolve the problem.
4. I want to go home and play Guitar Hero with my kids until my fingers are as numb as my brain.

1. Estimate for this dyn. case the highest peak stress (on a Bend ?).
2. Look/Find for your material a highfatique curve
3. Take the stressamplitude at from the right site ( biggest loadcycle)
4. If your calculated stress less then this value, then is it ok.

You have combined a few terms here - slug and pressure rise. I see these as two separate items - 1) a slug of liquid in a vapor line will cause a momentum imbalance at each elbow in sequence & 2) a sudden change in flow can cause a hammer or a pressure rise traveling from one leg to the next. I am not sure of what you are trying to model here.
But as far as I know, these events do not occur every 3-5 seconds.
(Another imbalance that is randomly repetitive would be the generation and collapse of vapor "bubbles" in a tall liquid column at the right temperatures and pressures.)
The reciprocating pump people deal with big cyclic vibration issues. As I understand it, they will try to keep the stress magnitude caused by these events to less than 6500 psi (zero-to-peak)- see API 618. But I'm not sure if they calculate stress the same way.

What you have is the dyn. system answer -- 0,2Hz until 0,33 Hz with a max displacement 1" - from a not exact describable event.
First step i woud start a harmonic analyses with this displacement at the node for a frequency range 0,2 until 0,33 Hz.
I mean after 3 million cycles do have not a primäry or secundary stress problem. Its a fatique problem.
A period test at the highest stress woud i made.

Your parenthetical comment nails the situation exactly. Perhaps I should not use the term slug flow. What we have is two-phase flow of a hydrocarbon that enters a furnace high above the convection section and exits at the bottom of the radiant section. The flow is about 5% vapor (by mass) at around 400 degrees F. The piping makes a flat turn under the furnace, comes out from under it and rises to connect with a tower inlet. The elbow where the flat turn is made is at the low point of the system, and it is slammed approximately every 3-5 seconds. I imagine that a vapor pocket forms briefly in the low piping, then bubbles up, allowing a slug of fluid to slam into the elbow. I was given a pressure rise produced by this event from a HySys model, which I multiplied by the cross-sectional area at the elbow to give a load. To form the impulse spectrum, I divided the key imbalance distances by the speed of sound in the fluid, then used 10% for the rise/fall times (this is a trapezoidal waveform). The first distance I tried yielded too much displacement. The second one gave me displacements at the elbow that were consistent with observation, and failed (exceeded B31.3 allowables).

If I understand correctly, Caesar takes my load and spectrum and produces a DLF to multiply the stress in the sustained load case. The code compares this value with the allowable for an occassional load. No fatigue has been applied, but if it fails with a fatigue factor of 1, it will certainly fail with a lower factor. In other words, the system already fails at less than 7,000 cycles (even though it has lasted longer in real life), so there's no need to check an SN diagram.

My View :
First you must to darifying the fail possibilities and what is you load Character.
Primary ?
The primary stress evaluation respects constant primary loads.
But you have ever changin loads.
Ok, conservativ you can say its a sustained case.
But your ever chaging load have not this sustained character.
If you had over 3 million load cycles without fail, then this show its not
a primary stress "problem".

Secundary ?
The secundäry stress evaluation respects the ratcheting effect.
That mean increasing strain after ever load cycle.
Its improbable without fail after 3 million load cycles.

Fatique ?
Most Likely is you case only a fatique problem.

Is this not a accetable way for you, then take a snubber.

Hi There,

I would do the following procedure:

1- Measure harmonic displacements at critical points, normally at elbows.
2- Run Modal analysis and identify the mode shape and frequency which match your problem.
3- Perform Harmonic analysis with given displacements with matched frequency. You may tweak damping ratio and friction factor multiplier in control parameteres for better and accurate modeling.
4- Apply fatigue curve and perform fatigue analysis.
5- if falied. Try with sway-brace assembly or snubber.