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nafems4.md

@@ -67,7 +67,7 @@ In each of the countries that have at least one nuclear power plant there exists
 
 How come that pipes are subject to fatigue? Well, on the one hand and without getting into many technical details, the most common nuclear reactor design uses liquid water as coolant and moderator. On the other hand, nuclear power plants cannot by-pass the thermodynamics of the Carnot cycle, and in order to maximise the efficiency of the conversion between the energy stored in the uranium nuclei into electricity they need to reach temperatures as high as possible. So, if we want to have liquid water in the core as hot as possible, we need to increase the pressure. The limiting temperature and pressure are given by the [critical point of water](https://en.wikipedia.org/wiki/Critical_point_(thermodynamics)), which is around 374ºC and 22\ MPa. It is therefore expected to have temperature and pressures near those values in many systems of the plant, especially in the primary circuit those that directly interact with it, such as pressure and inventory control system, decay power removal system, feedwater supply system, emergency core-cooling system, etc.
 
-Nuclear power plants are not always working at 100% power. They need to be maintained and refuelled, they may undergo operational transients, they might operates at a lower power due to load following conditions, etc. These transient cases involved changes both in temperatures and in pressures that the pipes are subject to, which in turn give rise to changes in the stresses within the pipes. As the transients are postulated to occur conservatively cyclically during a number of times during the life-time of the plant (plus its extension period), mechanical fatigue in these piping systems arise especially at the interfaces between materials with different thermal expansion coefficients.
+Nuclear power plants are not always working at 100% power. They need to be maintained and refuelled, they may undergo operational (and some incidental) transients, they might operates at a lower power due to load following conditions, etc. These transient cases involved changes both in temperatures and in pressures that the pipes are subject to, which in turn give rise to changes in the stresses within the pipes. As the transients are postulated to occur conservatively cyclically during a number of times during the life-time of the plant (plus its extension period), mechanical fatigue in these piping systems arise especially at the interfaces between materials with different thermal expansion coefficients.
 
 An important part of the analysis that almost always applies to nuclear power plants but usually also to other installations is the consideration of a possible seismic event. Given a postulated design earthquake, the civil structures that hold the pipes have a response spectra for each floor level. One has to combine these spectra with the natural oscillation modes and frequencies of the piping system using one of several available methodologies such as the “Square Root of Sum of Squares” or SRSS method. This way, an static-equivalent  distributed internal load can be computed and then applied as extra loads conservatively (see [@sec:kinds]) at the moment of highest mechanical demand.
 
@@ -340,7 +340,7 @@ avoid monolithic
 divert(0)
 
 
-# Piping in nuclear rectors
+# Piping in nuclear rectors {#sec:piping-nuclear}
 
 So we need to address the issue of fatigue in nuclear reactor pipes that
 
@@ -352,7 +352,7 @@ So we need to address the issue of fatigue in nuclear reactor pipes that
     b. heat transients, and
     c. seismic loads.
 
-As I wanted to illustrate in [@sec:five], it is very important to decide what kind of problem (actually problems) we should be dealing with. As a nuclear engineer, I learned (theoretically in college but practically after college) that there are some models that let you see some effects and some that let you see other effects (please [say “modelling” not “simulation.”](https://www.seamplex.com/blog/say-modeling-not-simulation.html)]). And even if, in principle, it is true that more complex models should let you see more stuff, they definitely might show you nothing at all if the model is so big and complex that it does not fit into a computer (say because it needs hundreds of gigabytes of RAM to run) or because it takes more time to compute than you may have before the final report is expected.
+As I wanted to illustrate in [@sec:five], it is very important to decide what kind of problem (actually problems) we should be dealing with. As a nuclear engineer, I learned (theoretically in college but practically after college) that there are some models that let you see some effects and some that let you see other effects (please [say “modelling” not “simulation.”](https://www.seamplex.com/blog/say-modeling-not-simulation.html)). And even if, in principle, it is true that more complex models should let you see more stuff, they definitely might show you nothing at all if the model is so big and complex that it does not fit into a computer (say because it needs hundreds of gigabytes of RAM to run) or because it takes more time to compute than you may have before the final report is expected.
 
 First of all, we should note that we need to solve
 
@@ -366,6 +366,8 @@ So for each time of the operational transient, the pipes are subject to
  b. a uniform internal temperature $T_i(t)$ that gives rise to a non-trivial time-dependent temperature distribution\ $T(\vec{x},t)$ in the bulk of the pipes, and
  c. internal distributed forces\ $\vec{f}=\rho \cdot \vec{a}$ at those times where the design earthquake is assumed to act.
 
+## Thermal transient {#sec:thermal}
+
 Let us invoke our imagination once again. Assume in one part of the transients the temperature of the water inside the pipes falls from say 300ºC down to 100ºC in a couple of minutes, stays at 100ºC for another couple of minutes and then gets back to 100ºC. The temperature within the bulk of the pipes change as times evolves. The internal wall of the pipes follow the transient temperature (it might be exactly equal or close to it through the [Newton’s law of cooling](https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling)). If the pipe was in a state of uniform temperature, the ramp in the internal wall will start cooling the bulk of the pipe creating a transient thermal gradient. Due to thermal inertia effects, the temperature can have a non-trivial dependence when the ramps start or end (think about it!). So we need to compute a real transient heat transfer problem with convective boundary conditions because any other usual tricks like computing a sequence of steady-state computations for different times would not be able to recover these non-trivial distributions.
 
 Remember the main issue of the fatigue analysis in these systems is to analyse what happens around the location of changes of piping classes where different materials (i.e. different expansion coefficients) are present, potentially causing high stresses due to differential thermal expansion (or contraction) under transient conditions. Therefore, even though we are dealing with pipes we cannot use beam or circular shell elements, because we need to take into account the three-dimensional effects of the temperature distribution along the pipe thickness. And even if it we could, there are some tees that connect pipes with different nominal diameters that have a non-trivial geometry, such as the weldolet-type junction shown in\ [@fig:weldolet-cad;@fig:weldolet-mesh]. In this case, there are a number of SCLs (Stress Classification Lines) that go through the pipe’s thickness at both sides of the material interface as illustrated in\ [@fig:weldolet-scls]. It is in these locations that fatigue is to be evaluated.
@@ -443,9 +445,9 @@ The material interface in the system from [@fig:real-life] is configured by an o
 :::::
 
 
-## Seismic loads
+## Seismic loads {#sec:seismic}
 
-Before considering the actual mechanical problem that will give us the stress tensor at the SCLs and besides needing to solve the transient thermal problem to get the temperature distributions, we need to address the loads that arise due to a postulated earthquake during a certain part of the operational transients. The full computation of a mechanical transient problem using the earthquake time-dependent displacements is off the table for two reasons. First, because the computation would take more time than we might have to deliver the report. And secondly and more importantly, because civil engineers do not compute earthquakes in the time domain but in the frequency domain. Time to revisit our [Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform) exercises from undergraduate math courses.
+Before considering the actual mechanical problem that will give us the stress tensor at the SCLs and besides needing to solve the transient thermal problem to get the temperature distributions, we need to address the loads that arise due to a postulated earthquake during a certain part of the operational transients. The full computation of a mechanical transient problem using the earthquake time-dependent displacements is off the table for two reasons. First, because the computation would take more time than we might have to deliver the report. And secondly and more importantly, because civil engineers do not compute earthquakes in the time domain but in the frequency domain using the [response spectrum method](https://en.wikipedia.org/wiki/Response_spectrum). Time to revisit our [Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform) exercises from undergraduate math courses.
 
 ### Earthquake spectra
 
@@ -499,7 +501,7 @@ The equivalent accelerations for the piping section of [@fig:modes] for the spec
 The ASME code says that these accelerations (depicted in [@fig:acceleration]) are to be applied twice. Once with the original sign and once with all the elements with the opposite sign during two seconds of the transient each time.
 
 
-## Linearity (not yet linearisation)
+## Linearity (not yet linearisation) {#sec:linearity}
 
 Even though we did not yet discuss it in detail, we want to solve an elastic problem subject to an internal pressure condition, with a non-uniform temperature distribution that leads to both thermal stresses and variations in the mechanical properties of the materials. And as if this was not enough, we want to add at some instants a statically-equivalent distributed load that comes from a design earthquake. This last point means that at the transient instant where the stresses (from the fatigue’s point of view) are maximum we have to add the distributed loads that we computed from the seismic spectra to the other thermal and pressure loads. But we have a linear elastic problem (well, we still do not have it but we will in\ [@sec:break]), so we might be tempted exploit the problem’s linearity and compute all the effects separately and them sum them up to obtain the whole combination. We may thus compute just the stresses due to the seismic loads and then add them up to the stresses of any instant of the transient, in particular to the one with the highest ones. After all, in linear problems the result of the sum of two cases is the results of the sum of the cases, right? Wrong.
 
@@ -941,15 +943,37 @@ Von\ Mises stress and 400x warped displacements for three values of\ $d_b$.
  * using distributed forces from earthquakes,
  * etc.
 
-Most of the time at college we would barely do what is needed to be approved and move on to another subject. If you have the time and consider a career related to finite-element analysis, please do not.
+Most of the time at college we would barely do what is needed to be approved and move on to another subject. If you have the time and consider a career related to finite-element analysis, please do not. Now clone the repository from <https://bitbucket.org/seamplex/tee> and start playing. If you are stuck, do not hesitate asking for help in [wasora’s mailing list](https://www.seamplex.com/lists.html). It will pay off later on.
 
-**Clone the repository with the input files here. Feel free to ask for help in our community mailing list.**
 
 ## Bake, shake and break {#sec:break}
 
+A fellow mechanical engineer who went to the same high school I did, who went to the same engineering school I did and who worked at the same company I did, but who earned a PhD in Norway once told me two interesting things about finite-elements graduate courses. First, that in Trondheim the classes were taught by faculty from the the mathematics department rather than from the mechanical engineering department. It made complete sense to me, as I always have thought finite elements mainly a maths subject. And even though engineers might know some maths, it is nothing compared to actual mathematicians. Secondly, that they called the thermal, natural oscillations and elastic problems as the rhyming trio “bake, shake and break” (they also had “wake” for fluids, but that is a different story) which are just the three problems listed in section\ [@sec:piping-nuclear] that we need to solve in our nuclear power plant.
+
+So here we are again with the case study where we have to compute the linearised stresses at certain SCLs located near the interface between two different kinds of steels during operational and incidental transients of the plant. The first part is then to “bake” the pipes, modeling a thermal transient with time-dependent temperature or convection (it depends on the system) boundary conditions. This steps gives a time and space-dependent temperature\ $T(x,y,z,t)$.
+
+Then we proceed to “shake” the pipes, i.e. to compute the natural frequencies and associated oscillations modes. Employing the floor response spectra and combining the individual contributions with the SRSS method discussed in section\ [@sec:seismic], we obtain a distributed load vector\ $\vec{f}(x,y,z)$ which is statically equivalent to the design earthquake.
+
+Finally we attempt to “break” the pipes successively solving many steady-state elastic problems for different times\ $t$ of the operational transient. Some remarks about this step:
+
+ 1. The material properties are temperature-dependent (we use data from ASME\ II part\ D).
+ 2. Thermal expansion is taken into account. The reference temperature (i.e. the temperature at which there is no expansion) is\ 20ºC that coincides with ASME’s decision of the reference temperature for the mean thermal expansion coefficients.
+ 3. The temperature distribution\ $T(x,y,t,z)$ for bullets 1 & 2 is the generalisation of the reduced-model procedure explained in\ [@sec:thermal].
+ 4. The internal faces of the pipes are subject to an uniform pressure\ $p(t)$ given by the definition of the transient
+ 5. There are mechanical supports throughout the pipe system. Depending on the type of the support (i.e. vertical, lateral, axial, full, etc.) one or more degrees of freedom (i.e. $u$, $v$ and/or $w$) are fixed to zero. The ends of the CAD models were chosen always to have axially-null displacements.
+ 6. The earthquake-equivalent volumetric force\ $\vec{f}(x,y,z)$ should only be applied at the time\ $t$ where the maximum stresses occur. Note that due to the discussion from\ [@sec:linearity] we cannot compute the stresses raised just by\ $\vec{f}(x,y,z)$ and then add them to the main stresses. The force has to be included into the “break” step. An educated guess of the time where the maximum stress occur is usually enough. Anyway, it might be necessary a trial and error scheme to find the sweet spot.
+ 7. According to ASME\ III, the seismic load has to be applied during two seconds with the two possible signs. That is to say, apply $\vec{f}(x,y,z)$ during two seconds and then  $-\vec{f}(x,y,z)$ during two further seconds when the main stresses are maximums.
+ 8. A number of stress classification lines have to be defined. The locations were previously accorded with the plant owner and/or the regulator.
+ 9. The stress linearisation has to be performed individually for each principal stress\ $\sigma_1$, $\sigma_2$ and $\sigma_3$ to fulfill the requirements ASME\ III\ NB-3126 (see [@sec:in-air] below).  
+ 10. This “break” step is linear.
+
+geometrical
+material
+
+
 # Fatigue
 
-## In air
+## In air {#sec:in-air}
 
 ## In water
 
@@ -979,4 +1003,4 @@ Back in college, we all learned how to solve engineering problems. But there is
  * learn this by heart: the complexity of a FEM problem is given mainly by the number of _nodes_, not by the number of elements
  * remember that welded materials with different thermal expansion coefficients may lead to fatigue under cyclic temperature changes
  * if you have time, try to get out of your comfort zone and do more than what other expect from you (like parametric computations)
- * clone the parametric tee repository, understand how the figures from\ [@sec:parametric] where built and expand them to cover “we might go on...” bullets
+ * clone the [parametric tee repository](https://bitbucket.org/seamplex/tee), understand how the figures from\ [@sec:parametric] where built and expand them to cover “we might go on...” bullets