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nureg

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gtheler vor 6 Jahren
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@@ -1093,27 +1093,79 @@ A pretty nice list of steps, which definitely I would not have been able to tack

# Cumulative usage factors {#sec:usage}

__Terminamos FEM__
Strictly speaking, finite-elements are not needed anymore at this point of the analysis. But even though we are (or wannabe) FEM experts, we have to understand that if the objective of a work is to evaluate fatigue (or fracture mechanics or whatever), finite elements are just a mean and not and end. If we just mastered FEM and nothing else, our field of work would be highly reduced. We need to use all of our computational knowledge to perform actually engineering tasks and tell our bosses and clients whether the pipe would fail or not. This tip is induced in college but it is definitely reinforced afterward after working with real clients and bosses.

__Hay que aprender!__
Another comment I would like to add is that I had to learn fatigue practically from scratch when faced with this problem for the first time in my engineering career. I remembered some basics from college (like the general introduction from [@sec:fatigue]), but I lacked the skills to perform a real computation by myself. Luckily there still exist books, there are a lot of interesting online resources (not to mention Wikipedia) and, even more luckily, there are plenty of other fellow engineers that are more than eager to help. My second tip is: when faced to a new challenging problem, read, learn and ask for help to real people to see if you read and learned it right.

1025823
\medskip

Back and distantly, in\ [@sec:case] we said that people noticed there were some environmental factors that affected the fatigue resistance of materials. The basic ASME approach does not take care of these factors, and it is regarded as fatigue “in air.” We are interested in taking them into account, so we follow the US Nuclear Regulatory Commission guidelines to evaluate fatigue “in water.”

## In air (ASME’s basic approach) {#sec:in-air}

We already said in\ [@sec:fatigue] that fatigue analysis gives the limit number\ $N$ of cycles that a certain mechanical part can withstand when subject to a certain periodic load of stress amplitude\ $S_\text{alt}$. If the actual number of cycles\ $n$ the load is applied is smaller than the limit\ $N$, then the part is fatigue-resistant. In our case study there is a mixture of several periodic loads, each one expected to occur a certain number of times. ASME’s way to evaluate the resistance is to break up the stress history into partial stress amplitudes\ $S_{\text{alt},j}$ and compute individual usage factors\ $U_k$ for the\ $k$-th amplitude as

$$U_k = \frac{n_k}{N_k}$$

If the extrema of the partial stress amplitude correspond to different transients, then the following note in ASME-3224(5) should be followed:

> In determining $n_1$, $n_2$, $n_3$, $\dots$, $n_k$ consideration shall be given to the superposition of cycles of various origins which produce a total stress difference range greater than the stress difference ranges of the individual cycles. For example, if one type of stress cycle produces 1,000 cycles of a stress difference variation from zero to +60,000\ psi and another type of stress cycle produces 10,000 cycles of a stress difference variation from zero to −50,000\ psi, the two types of cycle to be considered are defined by the following parameters:
>
> (a) for type 1 cycle, $n_1 =$ 1,000 and $S_{\text{alt},1} = (60,000 + 50,000)/2 = 55,000$\ psi;
> (b) for type 2 cycle, $n_2 =$ 9,000 and $S_{\text{alt},2} = (50,000 + 0)/2 = 25,000$\ psi.

This cryptic paragraph can be better explained by using a clearer example. To avoid using actual sensitive data from a real power plant, let us use the same test case used by both the NRC (in its report NUREG/CR-6909) and EPRI (report 1025823) called “EAF SAmple PRoblem 2-Rev. \2 (10/21/2011)”.

![A low-alloy steel vessel nozzle (blue) welded to a stainless steel pipe (gray).](axi-inches-3d.png){#fig:axi-inches-3d width=60%}

It consists of a typical vessel (NB-3200) nozzle with attached piping (NB-3600) as illustrated in\ [@fig:axi-inches-3d]. These components are subject to four transients, that give rise to linearized stress histories (slightly modified according to NB-3216.2) which are given as individual stress values juxtaposed so as to span a time range of about 36,000 seconds ([@fig:nureg1]). As the time steps is not constant, each stress value has an integer index\ $i$ that uniquely identifies it:

| Transient number\ $k$ | Time range [s] | Index range | Number of cycles\ $n_k$ |
|:---------------------:|:--------------:|:-----------:|:-----------------------:|
| 1 | 0--3210 | 1--523 | 20 |
| 2 | 3210--6450 | 524--959 | 50 |
| 3 | 6450--9970 | 960--1595 | 20 |
| 4 | 9970--35971 | 1596--2215 | 100 |

A design-basis earthquake was assumed to occur briefly during one second at around\ $t=3470$\ seconds, and it is assigned a number of cycles\ $n_e=5$. The detailed stress history for one of the SCLs including both the principal and lineariased stresses, which are already offset following NB-3216.2 so as to have a maximum stress equal to zero, can be found as an appendix in NRC's NUREG/CR-6909 report.

::::: {#fig:nureg}
![Full time range of the juxtaposed transients spanning $\approx$ 36,000 seconds](nureg1.svg){#fig:nureg1 width=80%}

We already said in\ [@sec:fatigue] that fatigue analysis gives the limit number\ $N$ of cycles that a certain mechanical part can withstand when subject to a certain periodic load of stress amplitude\ $S_\text{alt}$. If the actual number of cycles\ $n$ the load is applied is smaller than the limit\ $N$, then the part is fatigue-resistant. In our case study there is a mixture of several periodic loads, each one expected to occur a certain number of times. One way to evaluate the resistance is to break up the stress history into partial stress amplitudes\ $S_{\text{alt},j}$ and compute individual usage factors\ $U_j$ for the\ $j$-th amplitude as
![Detail of transients showing the ids of some extrema](nureg2.svg){#fig:nureg2 width=80%}

$$U_j = \frac{n_j}{N_j}$$
![Detail of the earthquake](nureg3.svg){#fig:nureg3 width=80%}

If the extrema of the partial stress amplitude correspond to different transients, then the number of expected $n_j$ is the minimum of the cycles expected to occur.
Linearised stress\ $\text{MB}_{31}$ time history of the example problem from NRC and EPRI. The id of the extrema are shown in green (minimums) and red (maximums).
:::::

To compute the global usage factor, we first need to find all the combinations of local extrema pairs and then sort them in decreasing order of stress difference. For example, the largest stress amplitude is found between $i=447$ and $i=694$. The second one is 447-699. Then 699-1020, 699-899, etc. For each of these pairs, defined by the indexes\ $i_{1,j}$ and $i_{2,j}$, a partial usage factor\ $U_j$ should computed. The stress amplitude\ $S_{\text{alt},j}$ which should be used to enter into the $S$-$N$ curve is

## In air {#sec:in-air}
$$
S_{\text{alt},j} = \frac{1}{2} \cdot k_{e,j} \cdot \left| MB^\prime_{i_{1,j}} - MB^\prime_{i_{2,j}} \right| \cdot \frac{E_\text{SN}}{E(T_{\text{max}_j})}
$$
where $k_e$ is a plastic correction factor for large loads (NB-3228.5), $E_\text{SN}$ is the Young Modulus used to creat the $S$-$N$ curve (provided in the ASME fatigue curves) and\ $E(T_{\text{max}_j})$ is the material’s Young Modulus at the maximum temperature within the interval.

We now need to comply with ASME’s obscure note about the number of cycles to assign a proper value of\ $n_j$. Back to the largest pair 447-694, we see that 447 belongs to transient\ #1 which has $n_1=20$ and 694 belongs to the earthquake with\ $n_e=5$. Therefore\ $n_1=5$, and the cycles associated to each index are decreased in five. So $i=694$ disappears and the number of cycles associated to $i=447$ are decreased from 20 to 15. The second largest pair is now 447-699, with 15 (because we just spent 5 in the first pair) and 50 cycles respectively. Now $n_2=15$, point\ 447 disappears and 699 remains with 35 cycles. The next pair is 699-1020, with number of cycles 35 and 20 so $n_3=20$, point 1020 disappears and 699 remains with 15 cycles. And so on, down to the last pair.

::::: {#fig:cuf}
![Reference from NUREG/CR6909](cuf-nrc.png){#fig:cuf-nrc width=100%}

![Results reproduced by the author](cuf-seamplex.png){#fig:cuf-seamplex width=100%}

Tables of individual usage factors for the NRC/EPRI example. One table is from an most-powerful-country-with-almost-a-billion-dollar-year-budget agency and the other one is from a third-world engineering startup.
:::::

## In water
Why all these details? Not because you could perform fatigue evaluations just reading this section without resorting to ASME, fatigue books and even other colleagues. It is to show that even though these computation can be made “by hand” (i.e. using a calculator or, God forbids, a spreadsheet) when having to evaluate a few SCLs within several piping systems it is far (and I mean really far) better to automatise all these steps by writing a set of scripts. Not only will the time needed to process the information be reduced, but also the
introduction of human errors be minimised and repeatability of results will be assured. This is true in general, so here is another tip: learn to write scripts to post-process your FEM results (you will need to use a script-friendly FEM program) and you will gain considerable margins regarding time and quality.

## In water (NRC’s extension)

Environmentally assisted fatigue

1025823


fatigue mutiplier

$$F_\text{en} = \frac{N_\text{air}}{N_\text{water}}$$
@@ -1137,6 +1189,10 @@ errors and uncertainties: model parameters (is E what we think? is the material

do you trust your FEM program?

do you trust your engineering judgment?

would you put your signature in the report?

# Conclusions

Back in college, we all learned how to solve engineering problems. And once we graduated, we felt we could solve and fix the world (if you did not graduate yet, you will feel it shortly). But there is a real gap between the equations written in chalk on a blackboard (now probably in the form of beamer slide presentations) and actual real-life engineering problems. This chapter introduces a real case from the nuclear industry and starts by idealising the structure such that it has a known analytical solution that can be found in textbooks. Additional realism is added in stages allowing the engineer to develop an understanding of the more complex physics and a faith in the veracity of the FE results where theoretical solutions are not available. Even more, a brief insight into the world of evaluation of low-cycle fatigue using such results further illustrates the complexities of real-life engineering analysis.
@@ -1160,6 +1216,8 @@ Back in college, we all learned how to solve engineering problems. And once we g
* clone the [parametric tee repository](https://bitbucket.org/seamplex/tee), understand how the figures from\ [@sec:parametric] were built and expand them to cover “we might go on...” bullets
* think thermal-mechanical plus earthquakes as “bake, break and shake” problems
* the elastic problem is still linear after all
* keep in mind that FEM are a mean to get an engineering solution, not and end by themselves
* learn to write scripts to post-process FEM results (from a script-friendly FEM program)


# Referenced works

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