seismic

nafems4.md

@@ -329,7 +329,7 @@ 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 “modeling” not “simulation.”](https://www.seamplex.com/blog/say-modeling-not-simulation.html)] And even if it is in principle 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 doe 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 “modeling” 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
 
@@ -337,12 +337,14 @@ First of all, we should note that we need to solve
  ii. a frequency analysis of the piping system to get the natural oscillation modes and use them to obtain the pseudo-accelerations created by the design earthquake, and finally
  iii. the elastic problem to obtain the stress tensor needed to compute the alternating stress to enter into the fatigue curve.
 
-So for each time of the transient, the pipes are subject to
+So for each time of the operational transient, the pipes are subject to
 
  a. an uniform internal pressure\ $p_i(t)$ that depends on time,
  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.
 
+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.
 
 dnl 33410 07-3-4D-29
@@ -420,10 +422,73 @@ The material interface in the system from [@fig:real-life] is configured by an o
 
 ## Seismic loads
 
-### Natural frequencies
+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.
 
 ### Earthquake spectra
 
+In case you are wondering, the answer is yes: all nuclear power plants are designed to withstand earthquakes. Of course, not all plants need the same level of reinforcements. Those built in large quiet plains will be, seismically speaking, cheaper than those located in more geologically active zones. Keep in mind that all the 54 Japanese nuclear power plants did structurally resist the 2011 earthquake, and all of the reactores were safely shut down. What actually happened in [Fukushima](http://www.world-nuclear.org/information-library/safety-and-security/safety-of-plants/fukushima-accident.aspx) is that one hour after the main shake, a 14-metre tsunami splashed on the coast, jumping over the 9-metre defenses and flooding the emergency Diesel generators that provided power to the pumps in charge of removing the remaining [decay power](https://en.wikipedia.org/wiki/Decay_heat) from the already-stopped reactor core.
+
+Back to our case study, the point is that each site where nuclear power plants are built must have a geological study where a postulated design-basis earthquake is to be defined. In other words, a theoretical earthquake which the plant ought to withstand needs to be specified. How? By giving a set of three spectra (one for each coordinate direction) giving acceleration as a function of the frequency for each level of the building. That is to say, once the earthquake hits the power plant, depending on soil-structure interactions the energy will shake the building foundations in a way that depends on the characteristics of the earthquake, the soil and the concrete structure. Afterwards, they way the oscillations travel upward and shake each of the mechanical components erected in each floor level depends on the design of the civil structure in a way which is fully determined by the floor response spectra like the ones depicted in\ [@fig:spectrum].
+
+![A sample spectrum for a certain floor level of a certain nuclear power plant.](spectrum.vector_format){#fig:spectrum}
+
+### Natural frequencies
+
+As the earthquake excites some frequencies more than others, it is mandatory to know which are the natural frequencies and modes of oscillations of our piping system. Mathematically, this requires the computation of an eigenvalue problem. Simply stated, we need to find all the non-trivial solutions of the equation
+
+$$
+K \phi_i = \lambda_i \cdot M \phi_i
+$$
+where $K$ is the usual finite-element stiffness matrix, $M$ is the mass matrix, $\lambda_i$ is the $i$-th natural frequency of the structure and $\phi_i$ is a vector containing the nodal displacement corresponding to the $i$-th mode of oscillation.
+
+Practically, these problems are solved using the same mechanical finite-element program one would use to solve a standard elastic problem, provided such program supports these kind of problems ([Fino](https://www.seamplex.com/fino/) does!). There are only two caveats to take into account:
+
+ 1. The computation of the natural frequencies is “load free”, i.e. there can be no surface nor volumetric loads, and
+ 2. The displacement boundary conditions ought to be homogeneous, i.e. only displacements equal to zero can be given. One may fix only one of the three degrees of freedom in certain surfaces and leave the others free though, as long as all the rigid body motions are removed as usual.
+
+A real continuous solid has infinite modes of oscillation. A discretized one has three times the number of nodes modes of oscillation. In any case, one is usually interested only in a few of them, namely those with the lower frequencies because they take most of the energy with them. Each mode has two associated parameters called modal mass and excitation parameter that reflect how “important” the mode is regarding the absorption of energy from an external oscillatory source. Usually a couple of dozens of modes are enough to take up more than 90% of the earthquake energy. Figure\ [-@fig:modes] shows the first six natural modes of a sample piping section.
+
+::::: {#fig:modes}
+![$i=1$](mode1.png){width=50%}
+![$i=2$](mode2.png){width=50%}
+
+![$i=3$](mode3.png){width=50%}
+![$i=4$](mode4.png){width=50%}
+
+![$i=5$](mode5.png){width=50%}
+![$i=6$](mode6.png){width=50%}
+
+First six natural oscillation modes for a piping section.
+:::::
+
+These first modes that take up most of the energy are then used to take into account the earthquake load. There are several ways of performing this computation, but the ASME\ III code states that the method known as SRSS (for Square Root of Sum of Squares) can be safely used. This method mixes the eigenvectors with the floor response spectra through the eigenvalues and gives an spatial (actually nodal) distribution of three accelerations (one for each direction) that, when multiplied by the density give a vector of a distributed force (in units of Newton per cubic millimeter for example) which is statically equivalent to the load coming from the postulated earthquake.
+
+::::: {#fig:acceleration}
+![$a_x$](ax.png){width=80%}
+
+![$a_y$](ay.png){width=80%}
+
+![$a_z$](az.png){width=80%}
+
+The equivalent accelerations for the piping section of [@fig:modes] for the spectra of\ [@fig:spectrum].
+:::::
+
+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 of displacements and stresses
+
+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 lead 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.
+
+no era que todo era lineal y podemos sumar todo?
+si y no
+
+podemos sumar los desplazamientos y los tensores de tensiones, pero ojo que las tensiones principales no son lineales a la suma (si a la multiplicacion)
+
+
+
+cantilever beam, principal stresses, linearity of von mises
+
 
 # The infinite pipe revisited after college
 
@@ -446,9 +511,6 @@ complete vs incomplete (hexa)
 
 
 
-## Linearity of displacements and stresses
-
-cantilever beam, principal stresses, linearity of von mises
 
 ### ASME stress linearization
 

spectrum-ac.dat

@@ -0,0 +1,9 @@
+0.40  0.10
+1.33  0.40
+2.33  0.90
+3.00  2.10
+8.0   2.10
+13.5  0.80
+24.0  0.80
+35.0  0.50
+100   0.50

spectrum-bd.dat

@@ -0,0 +1,9 @@
+0.40  0.10
+1.33  0.40
+2.33  0.90
+3.00  2.10
+8.0   2.10
+18.5  1.10
+26.0  1.10
+50.0  0.50
+100.0 0.50

spectrum-x.dat

@@ -0,0 +1,9 @@
+0.40  0.10
+1.33  0.25
+7.15  1.00
+10.0  1.00
+13.5  1.35
+22.0  1.35
+35.0  0.60
+50.0  0.40
+100.0 0.40

spectrum-y.dat

@@ -0,0 +1,9 @@
+0.40  0.10
+1.33  0.40
+2.33  0.90
+3.00  2.10
+8.0   2.10
+18.5  1.10
+26.0  1.10
+50.0  0.50
+100.0 0.50

spectrum-z.dat

@@ -0,0 +1,9 @@
+0.40  0.10
+1.33  0.40
+2.33  0.90
+3.00  2.10
+8.0   2.10
+13.5  0.80
+24.0  0.80
+35.0  0.50
+100   0.50

spectrum.ppl

@@ -0,0 +1,24 @@
+set preamble "\usepackage{amsmath}"
+
+set width 14*unit(cm)
+
+set axis x arrow nomirrored
+set axis y arrow nomirrored
+set yrange [0:2.2]
+set key top left
+
+unset ytics
+
+set grid
+
+set terminal pdf
+
+set logscale x
+#set xtics 0.1,10,100
+
+set output "spectrum.pdf"
+set xlabel "Frecuency [Hz]"
+set ylabel "Acceleration"
+plot "spectrum-x.dat" w lp lw 5 lt 2 pt 16 color red   ti "$s_x(f)$",\
+     "spectrum-y.dat" w lp lw 3 lt 3 pt 17 color green ti "$s_y(f)$",\
+     "spectrum-z.dat" w lp lw 1 lt 4 pt 18 color blue  ti "$s_z(f)$"

spectrum.svg

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