figuras derivadas

nafems4.md

@@ -810,7 +810,15 @@ In the [displacement-based formulation](http://web.mit.edu/kjb/www/Books/FEP_2nd
 
 Let us focus on the first item and leave the second one for a separate discussion in\ [@sec:two-materials]. The finite-element method computes the principal unknowns at the nodes and then says “interpolate the nodal values inside each element using its shape functions.” It sounds (and it is) great, but a node belongs to more than one element (you can now imagine a 3D mesh composed of tetrahedra but you can also simplify your mind pictures by thinking in just one dimension: a node is shared by two segments). So the slope of the interpolation when we move from the node into one element might (and it never is) the same as when we move from the same node into another element. Neither in linear, nor quadratic nor higher-order elements. So what is the derivative of the displacement\ $v$ in the\ $y$ direction with respect to the $z$ coordinate? One answer would be the average of the derivatives computed in each of the elements that share a common node. But what if one of the elements is very small? Or it has a very bad quality (i.e. it is deformed in one direction) its derivatives cannot be trusted? Should the “average” be weighted? How?
 
-**FIGURA 1D**
+::::: {#fig:derivatives}
+![](slab-1-0.svg){#fig:slab-1-0 width=48%}\  
+![](slab-1-1.svg){#fig:slab-1-1 width=48%}\  
+
+![](slab-2-0.svg){#fig:slab-2-0 width=48%}\  
+![](slab-2-1.svg){#fig:slab-2-1 width=48%}\  
+
+Solution of a problem using FEM using eight linear/quadratic uniform/non-uniform elements. The reference solution is a cosine. Plain averaging works for uniform grids but fails in the non-uniform cases.
+:::::
 
 Detailed mathematics show that the location where the derivatives of the interpolated displacements are closer to the real (i.e. the analytical in problem that have it) solution are the elements’ [Gauss points](https://en.wikipedia.org/wiki/Gaussian_quadrature). Even better, the material properties at these points are continuous (they are usually uniform but they can depend on temperature for example) because, unless we are using weird elements, there are no material interfaces inside elements. But how to manage a set of stresses given at the Gauss points instead of at the nodes? Should we use one mesh for the input and another one for the output? What happens when we need to know the stresses on a surface and not just in the bulk of the solid? There are still no one-size-fits-all answers. There is a very interesting [blog post](http://tor-eng.com/2017/11/practical-tips-dealing-stress-singularities/) by Nick Stevens that addresses the issue of stresses computed at sharp corners. What does your favourite FEM program do with such a case?
 
@@ -1076,18 +1084,51 @@ To recapitulate, here are some partial non-dimensional results of an actual syst
      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, which is what fatigue analysis need (recall\ [@sec:fatigue]).
+ 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, which is what fatigue analysis need (recall\ [@sec:fatigue] and go on to\ [@sec:usage]).
 
 A pretty nice list of steps, which definitely I would not have been able to tackle when I was in college. Would you?
 
 
-# Usage factors {#sec:usage}
+# Cumulative usage factors {#sec:usage}
+
+__Terminamos FEM__
+
+__Hay que aprender!__
 
+1025823
+
+
+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
+
+$$U_j = \frac{n_j}{N_j}$$
+
+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.
 
 ## In air {#sec:in-air}
 
+
+
 ## In water
 
+Environmentally assisted fatigue
+
+fatigue mutiplier
+
+$$F_\text{en} = \frac{N_\text{air}}{N_\text{water}}$$
+
+EPRI:
+
+The general steps for performing an EAF analysis are as follows:
+(1) perform an ASME fatigue analysis using fatigue curves for an air
+environment
+(2) calculate Fen factors for each transient pair in the fatigue analysis
+(3) apply the Fen factors to the incremental usage calculated for each
+transient pair (Ui), to determine the CUFen, using Equation 3-1.
+
+
+
+$$\text{CUF}_\text{en} = U_1 \cdot F_{\text{en},1} + U_2 \cdot F_{\text{en},2} + \dots + U_n \cdot F_{\text{en},n}$$
+
 # Errors and uncertainties
 
 errors and uncertainties: model parameters (is E what we think? is the material linear?), geometry (does the CAD represent the reality?) equations (any effect we did not have take account), discretization (how well does the mesh describe the geometry?)