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nafems4.md

@@ -994,9 +994,6 @@ $$ K(\vec{x}) \cdot \vec{u}(\vec{x}) = \vec{b}(\vec{x})$$
 
 And this last equation is linear in\ $\vec{u}$. In effect, the discretization step means to integrate over\ $\vec{x}$. As\ $K$, $\vec{u}$ and\ $\vec{b}$ depend only on\ $\vec{x}$, then after integration one gets just numbers with the matrix representation of\ [@eq:kub]. Again, you can either trust me, ask a teacher or go through with the maths.
 
-\medskip
-
-To recapitulate, here are some partial and non-sensitive results of an actual system of a certain nuclear power plant. The main issues to study were the interfaces between a carbon-steel pipe and a stainless-steel orifice plate used to measure the (heavy) water flow through the line.
 
 ::::: {#fig:case-cad}
 ![General view. Carbon steel is gray and stainless steel is green.](case-cad1.png){#fig:case-cad1 width=90%}
@@ -1014,7 +1011,7 @@ Three-dimensional\ CAD model of a section of the piping system between aproppria
 Unstructured grid for the mechanical analysis. Volumetric elements (i.e. tetrahedra) corresponding to carbon steel are magenta and to stainless steel are green. Surface elements (i.e. triangles) corresponding to the internal pressurised face are yellow.
 :::::
 
-![Location of the 15\ SCLs](case-scls.pdf){#fig:case-scls width=50%}
+![Location of the 15\ SCLs](case-scls.pdf){#fig:case-scls width=75%}
 
 ::::: {#fig:case-temp}
 ![Simplified axi-symmetric domain](case-temp-4-0015.png){#fig:case-temp1 width=70%}
@@ -1024,6 +1021,45 @@ Unstructured grid for the mechanical analysis. Volumetric elements (i.e. tetrahe
 Temperature distribution for a certain instant of the transient, computed in the simplified two-dimensional axi-symmetric domain and its generalisation to the three-dimensional mechanical domain as discussed in\ [@sec:thermal].
 :::::
 
+::::: {#fig:case-mode}
+![$i=1$](case-mode1.png){#fig:case-mode-1 width=30%}\ 
+![$i=2$](case-mode2.png){#fig:case-mode-2 width=30%}\ 
+![$i=3$](case-mode3.png){#fig:case-mode-3 width=30%} 
+
+![$i=4$](case-mode4.png){#fig:case-mode-4 width=30%}\ 
+![$i=5$](case-mode5.png){#fig:case-mode-5 width=30%}\ 
+![$i=6$](case-mode6.png){#fig:case-mode-6 width=30%} 
+
+![$i=7$](case-mode7.png){#fig:case-mode-7 width=30%}\ 
+![$i=8$](case-mode8.png){#fig:case-mode-8 width=30%}\ 
+![$i=9$](case-mode9.png){#fig:case-mode-9 width=30%}
+
+First nine natural modes of oscillation of the piping system subject to the boundary conditions the supports provide. 
+:::::
+
+\medskip
+
+To recapitulate, here are some partial non-dimensional results of an actual system of a certain nuclear power plant. The main issues to study were the interfaces between a carbon-steel pipe and a stainless-steel orifice plate used to measure the (heavy) water flow through the line. The steps discussed so far include 
+
+ 1. building a CAD model of the piping section under study (main domain)
+ 2. creating a mesh for the main domain refining locally around the material interfaces
+ 3. defining the number and locations of the SCLs
+ 4. computing a heat conduction transient problem with temperatures as a function of time from the operational transient in a simple domain using temperature-dependent thermal conduction coefficients from ASME\ II (bake)
+ 5. generalising the temperature distribution as a function of time to the general domain
+ 6. performing a modal analysis on the main domain to obtain the main oscillation frequencies and modes (shake)
+ 7. using the floor response spectra and the SRSS method to obtain a distributed force statically-equivalent to the earthquake load (not shown)
+ 8. successively solving the linear elastic problem for different times using the generalised temperature distribution taking into account
+     a. the dependence of the Young Modulus\ $E$ and the thermal expansion coefficient\ $\alpha$ with temperature,
+     b. the thermal expansion effect itself
+     c. the instantaneous pressure exerted in the internal faces of the pipes at the time\ $t$ according to the definition of the operational transient
+     d. the restriction of the degrees of freedom of those faces, lines or points that correspond to mechanical supports located both within and at the ends of the CAD model
+     e. the earthquake load, which according to ASME should be present only during four seconds of the transient: two seconds with one sign and the other two seconds with the opposite sing. This period should be selected to coincide with the instant of highest mechanical stress (conservative computation)
+ 9. computing the linearised stresses (membrane and membrane plus bending) at the SCLs combining them as
+     a. Principal 1
+     b. Principal 2
+     c. Principal 3
+     d. Tresca
+ 10. juxtaposing these linearised stresses for each time of the transient and for each transient so as to obtain a single time-history of stresses including all the operational and/or incidental transients under study    
 
 # Fatigue