fixed item list

nafems4.md

@@ -1047,15 +1047,15 @@ Then we proceed to “shake” the pipes, i.e. to compute the natural frequencie
 
 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](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_II_-_Materials) 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 piping 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 are chosen always to have axially-null displacements.
- 6. The earthquake-equivalent volumetric force\ $\mathbf{f}(x,y,z)$ is 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\ $\mathbf{f}(x,y,z)$ 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 $\mathbf{f}(x,y,z)$ during two seconds and then  $-\mathbf{f}(x,y,z)$ during two further seconds when the main stresses are maximums.
- 8. A number of stress classification lines have to be defined. The locations should be 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 fulfil the requirements ASME\ III\ NB-3126 (see [@sec:in-air] below).  
+  1. The material properties are temperature-dependent (we use data from [ASME\ II](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_II_-_Materials) 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 piping 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 are chosen always to have axially-null displacements.
+  6. The earthquake-equivalent volumetric force\ $\mathbf{f}(x,y,z)$ is 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\ $\mathbf{f}(x,y,z)$ 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 $\mathbf{f}(x,y,z)$ during two seconds and then  $-\mathbf{f}(x,y,z)$ during two further seconds when the main stresses are maximums.
+  8. A number of stress classification lines have to be defined. The locations should be 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 fulfil the requirements ASME\ III\ NB-3126 (see [@sec:in-air] below).
  10. This “break” step is linear.
 
 Is the last bullet right? [Surely you’re joking, Mr.\ Theler!](https://en.wikipedia.org/wiki/Surely_You're_Joking%2C_Mr._Feynman!) Linear problems are simple and almost useless. How can this extremely complex problem be linear? Well, let us see. First, there are two main kinds of non-linearities in FEM: