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| @@ -36,7 +36,7 @@ There are some useful tricks that come handy when trying to solve a mechanical p | |||
| One of the most important ones to use your _imagination_. You will need a lot of imagination to “see” what it is actually going on when analysing an engineering problem. This skill comes from my background in nuclear engineering where I had not choice but to imagine a [positron-electron annihilation](https://en.wikipedia.org/wiki/Electron%E2%80%93positron_annihilation) or an [Spontaneous fission](https://en.wikipedia.org/wiki/Spontaneous_fission). But in mechanical engineering, it is likewise important to be able to imagine how the loads “press” one element with the other, how the material reacts depending on its properties, how the nodal displacements generate stresses (both normal and shear), how results converge, etc. And what these results actually mean besides the pretty-coloured figures.^[A former boss once told me “I need the CFD” when I handed in some results. I replied that I did not do computational fluid-dynamics but computed the neutron flux kinetics within a nuclear reactor core. He joked “I know, what I need are the _Colors For Directors_, those pretty coloured figures along with your actual results.”] | |||
| This journey will definitely need your imagination. We will see equations, numbers, plots, schematics, 3D geometries, interactive 3D views, etc. Still, when the theory says “thermal expansion produces linear stresses” you have to picture in your head three little arrows pulling away from the same point in three directions, or whatever mental picture you have about what you understand thermally-induced stresses are. What comes to your mind when someone says that out of the nine elements of the stress tensors there are only six that are independent? Whatever it is, try to practice that kind of graphical thoughts with every new concept. Nevertheless, there will be particular locations of the text where imagination will be most useful. I will bring the subject up. | |||
| This journey will definitely need your imagination. We will see equations, numbers, plots, schematics, 3D geometries, interactive 3D views, etc. Still, when the theory says “thermal expansion produces linear stresses” you have to picture in your head three little arrows pulling away from the same point in three directions, or whatever mental picture you have about what you understand thermally-induced stresses are. What comes to your mind when someone says that out of the nine elements of the [stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor) there are only six that are independent? Whatever it is, try to practice that kind of graphical thoughts with every new concept. Nevertheless, there will be particular locations of the text where imagination will be most useful. I will bring the subject up. | |||
| Another heads up is that we will dig into some mathematics. Probably they would be be simple and you would deal with them very easily. But probably you do not like equations. No problem! Just ignore them for now. Read the text skipping them, it should work. | |||
| Łukasz Skotny says [you do not need to know maths to perform finite-element analysis](https://enterfea.com/math-behind-fea/). And he is right, in the sense that you do not need to know thermodynamics to drive a car. It is fine to ignore maths for now. | |||
| @@ -97,9 +97,9 @@ We have to accept that there is certain intellectual beauty when complex stuff c | |||
| In any case, what we should understand (and imagine) is that external forces lead to internal stresses. And in any three-dimensional body subject to such external loads, the best way to represent internal stresses is through a $3 \times 3$ _stress tensor_. This is the first point in which we should not fear mathematics. Trust me, it will pay back later on. | |||
| Does the term _tensor_ scare you? It should not. A tensor is just a slightly more complex vector, and I assume you are not afraid of vectors. If you recall, a vector somehow generalises the idea of a scalar in the following sense: a given vector $\mathbf{v}$ can be projected into any direction $\mathbf{n}$ to obtain a scalar $p$. We call this scalar $p$ the “projection” of the vector $\mathbf{v}$ in the direction $\mathbf{n}$. Well, a tensor can be also projected into any direction $\mathbf{n}$. The difference is that instead of a scalar, a vector is now obtained. | |||
| Does the term _tensor_ scare you? It should not. A [tensor](https://en.wikipedia.org/wiki/Tensor) is a general mathematical object and might get complex when dealing with many dimensions (as those encountered in weird stuff like [string theory](https://en.wikipedia.org/wiki/String_theory)), but we will stick here to second-order tensors. They are slightly more complex than a vector, and I assume you are not afraid of vectors. If you recall fresh-year algebra courses, a [vector](https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)) somehow generalises the idea of a [scalar](https://en.wikipedia.org/wiki/Scalar_(mathematics)) in the following sense: a given vector $\mathbf{v}$ can be projected into any direction $\mathbf{n}$ to obtain a scalar $p$. We call this scalar $p$ the “projection” of the vector $\mathbf{v}$ in the direction $\mathbf{n}$. Well, a tensor can be also projected into any direction $\mathbf{n}$. The difference is that instead of a scalar, a vector is now obtained. | |||
| Let me introduce then the three-dimensional stress tensor: | |||
| Let me introduce then the three-dimensional [stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor) (a.k.a [Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy) tensor): | |||
| $$ | |||
| \begin{bmatrix} | |||
| @@ -113,10 +113,12 @@ It looks (and works) like a regular $3 \times 3$ matrix. Some brief comments abo | |||
| * The $\sigma$s are normal stresses, i.e. they try to stretch or tighten the material. | |||
| * The $\tau$s are shear stresses, i.e. they try to twist the material. | |||
| * It is symmetric (so there are only six independent elements) because | |||
| * Due to rotational equilibrium requirements the conjugate shear stresses should be equal: | |||
| - $\tau_{xy} = \tau_{yx}$, | |||
| - $\tau_{yz} = \tau_{zy}$, and | |||
| - $\tau_{zx} = \tau_{xz}$. | |||
| Therefore the stress tensor is symmetricm i.e. there are only six independent elements. | |||
| * The elements of the tensor depend on the orientation of the coordinate system. | |||
| * There exists a particular coordinate system in which the stress tensor is diagonal, i.e. all the shear stresses are zero. In this case, the three diagonal elements are called the [principal stresses](https://en.wikipedia.org/wiki/Principal_stresses), which happen to be the three [eigenvalues](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) of the stress tensor. The basis of the coordinate system that renders the tensor diagonal are the eigenvectors. | |||
| @@ -128,7 +130,7 @@ What does this all have to do with mechanical engineering? Well, once we know wh | |||
| ## An infinitely-long pressurised pipe {#sec:infinite-pipe} | |||
| Let us proceed to our second step, and consider the infinite pipe subject to uniform internal pressure already introduced in\ [@fig:infinite-pipe]. Actually, we are going to solve the mechanical problem on an infinite hollow cylinder, which looks like pipe. This case is usually tackled in college courses, and chances are you already solved it. In fact, the first (and simpler) problem is the “thin cylinder problem.” Then, the “thick cylinder problem” is introduced, which is slightly more complex. Nevertheless, it has an analytical solution which is derived [here](https://www.seamplex.com/fino/doc/pipe-linearized/). For the present case, Let us consider an infinite pipe (i.e. a hollow cylinder) of internal radius $a$ and external radius $b$ with uniform mechanical properties---Young modulus $E$ and Poisson’s ratio $\nu$---subject to an internal uniform pressure $p$. | |||
| Let us proceed to our second step, and consider the infinite pipe subject to uniform internal pressure already introduced in\ [@fig:infinite-pipe]. Actually, we are going to solve the mechanical problem on an infinite hollow cylinder, which looks like pipe. This case is usually tackled in college courses, and chances are you already solved it. In fact, the first (and simpler) problem is the “thin cylinder problem.” Then, the “thick cylinder problem” is introduced, which is slightly more complex. Nevertheless, it has an analytical solution which is derived [here](https://www.seamplex.com/fino/doc/pipe-linearized/). For the present case, let us consider an infinite pipe (i.e. a hollow cylinder) of internal radius $a$ and external radius $b$ with uniform mechanical properties---Young modulus $E$ and Poisson’s ratio $\nu$---subject to an internal uniform pressure $p$. | |||
| ### Displacements | |||
| @@ -294,7 +296,7 @@ Here is an [original example](https://www.toyota-global.com/company/toyota_tradi | |||
| You get the point, even though we know thanks to [Richard Feynmann](https://en.wikipedia.org/wiki/Richard_Feynman) that to [answer a “why” question](https://fs.blog/2012/01/richard-feynman-on-why-questions/) at some point we need to rely on the questioner’s previous experience. We usually assume we have to do what we usually do (i.e. perform finite element analysis). But do we? Do we add a filter or just replace the fuse? | |||
| Getting back to the case study: do we need to do FEM analysis? Well, it does not look like we can obtain the stresses of the transient cycles with just pencil and paper. But how much complexity should we add? We might do as little as axisymmetric linear steady-state conservative studies or as much as full three-dimensional non-linear transient best-estimate plus uncertainties computations. And here is where good engineers should appear: in putting their engineering judgement (call it experience or hunches) into defining what to solve. And it is not (just) because the first option is faster to solve than the latter. Involving many complex methods need more engineering time | |||
| Getting back to the case study: do we need to do FEM analysis? Well, it does not look like we can obtain the stresses of the transient cycles with just pencil and paper. But how much complexity should we add? We might do as little as axisymmetric linear steady-state conservative studies or as much as full three-dimensional non-linear transient best-estimate plus uncertainties computations. And here is where good engineers should appear: in putting their engineering judgment (call it experience or hunches) into defining what to solve. And it is not (just) because the first option is faster to solve than the latter. Involving many complex methods need more engineering time | |||
| 1. to prepare the input data and set up the algorithms, | |||
| 2. to cope with the many more mistakes that will inevitable appear during the computation, and | |||
| @@ -529,7 +531,7 @@ The loads in each cases are applied to the three remaining faces, namely “righ | |||
| ```` | |||
| | | | “right” | | | “back” | | | “top” | | | |||
| | ------ |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:| | |||
| | | $F_x$ | $F_y$ | $F_z$ | $F_x$ | $F_y$ | $F_z$ | $F_x$ | $F_y$ | $F_z$ | | |||
| | | $F_x$ | $F_y$ | $F_z$ | $F_x$ | $F_y$ | $F_z$ | $F_x$ | $F_y$ | $F_z$ | | |||
| | Case A | +10 | 0 | 0 | 0 | +20 | 0 | 0 | 0 | +30 | | |||
| | Case B | 0 | +15 | -15 | +25 | 0 | -5 | -15 | +25 | 0 | | |||
| | Case C | +10 | +15 | -15 | +25 | +20 | -5 | -15 | +25 | +30 | | |||
| @@ -742,7 +744,7 @@ The first thing that has to be said is that, as with any interesting problem, th | |||
| {#fig:cube-struct width=48%}\ | |||
| {#fig:quarter-caeplex width=48%} | |||
| Two of the hundreds of different ways the infinite pressurised pipe can be solved using FEM | |||
| Two of the hundreds of different ways the infinite pressurised pipe can be solved using FEM. The axial displacement at the ends is set to zero, leading to a [plane-strain](https://en.wikipedia.org/wiki/Plane_stress#Plane_strain_(strain_matrix)) condition | |||
| ::::: | |||
| @@ -771,6 +773,8 @@ You can get both the exponential nature of each added bullet and how easily we c | |||
| Error in the computation of the linearised stresses vs. CPU time needed to solve the problem using FEM | |||
| ::::: | |||
| An additional note should be added. The FEM solution, which not only gives the nodal displacements but also a method to interpolate these values inside the elements, does not fully satisfy the original equilibrium equations at every point (i.e. the strong formulation). It is an approximation to the solution of the [weak formulation](https://en.wikipedia.org/wiki/Weak_formulation) that is close (measured in the vector space spanned by the shape functions) to the real solution. Mechanically, this means that the FEM solution leads only to nodal equilibrium but not pointwise equilibrium. | |||
| ## Elements, nodes and CPU {#sec:elements-nodes} | |||
| The last two bullets above lead to an issue that has come many times when discussing the issue of convergence with respect to the mesh size with other colleagues. There apparently exists a common misunderstanding that the number of elements is the main parameter that defines how complex a FEM model is. This is strange, because even in college we are taught that the most important parameter is the size of the stiffness matrix, which is three times (for 3D problems with the [displacement-based formulation](http://web.mit.edu/kjb/www/Books/FEP_2nd_Edition_4th_Printing.pdf) formulation) the number of _nodes_. | |||
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