# Glossary

I remember my frustration from when I was starting to learn about Finite Element Method (FEA). At the beginning I was reading a lot (actually I still do but it is less frustrating ), and it was really hard to go threw all the new vocabulary, especially since few terms sometimes meant the same thing, while some things apparently meant different things depending on context. This is why I believe a good glossary is really needed. Here I will post short definitions to all terms I think needs them (those terms are usually written in italics throughout the blog.

## Abbreviations

I do not intend to use many abbreviations (as this makes things even more complicated), but I will make an exception for analysis types described here.

##### A

**Arc Length** – this is a method of controlling how solver increases loads in nonlinear analysis. I think that the most famous algorithm for this would be *Riks* or *Modified Riks* algorithms but *Crisfield* algorithm is also used. Sometimes software packages simply use *arc-lenght *name for such algorithms.

##### C

**Convergence** – this is a state when outcome of the analysis is enough accurate to fulfil given criteria. I.e. work of forces on received deformations is “equal enough” to work of loads on displacements.

**Critical load** – this is a value of the load that causes *nstability *in the model. This is not a capacity of a model due to instability, but rather equivalent of Euler’s Force for compressed model. This means that *critical load* gives you load-bearing capacity of the model in ideal conditions – in reality capacity is smaller (sometimes even several times smaller). More on this here.

##### D

**DOF** – Degree of Freedom

**Degree of Freedom **– direction of translation or rotation in which *node* can move. Usually in 3D model *nodes* can move in 6 *DOF* – 3 translations (forming X, Y, Z axis) and 3 rotations (around X, Y, Z axis). There are many situations when this is not true: i.e. in 2D model (assuming Z axis is normal to the 2D plane) only 2 translations (X and Y) and 1 rotation (around Z) are possible (since others would move node out of the analysed plane).

##### E

**Eigenvalue** – This is an outcome from LBA analysis. It consists of 2 main parts:

- load factor (sometimes denoted as α) that when applied to load in model results in
*critical load*. Such load causes model to fail due to*instability*. - geometry of
*instability*caused by*critical load*obtained for given load factor

**Equilibrium path** – see *stability path.*

##### F

**Finite element** – this is the smallest “controlled” item in numerical model. Each *finite element* consists from at least 2 *nodes* (theoretically number of nodes in one element is unlimited), and describe how those nodes are “connected” to each other (by means of defining element thickness or material properties).

##### I

**Increment** – This term have 2 meanings:

- First is quite obvious: if I want to implement loading for nonlinear analysis in any model, I will most likely want to have it divided into small pieces. Those pieces will be added to the analysis one by one (to better capture nonlinear behaviour of the model) – you can see an example here. One “piece of load” is
*increment*. So if I divide 100kN loads into 10*increments*each*increment*is 10kN. - Second meaning is harder to explain.
*Increment*is also a “situation” in the model at certain stage (understood for instance as a set of outcomes, when certain load increment was applied). When I write “in third*increment*of the analysis” I actually refer to situation that happen after third load*increment*was implemented into the model. I could say for instance “in third*increment*instability failure occurred” which would mean that after third*increment*of load was applied instability failure occurred. Also various outcomes of numerical analysis are usually marked with “*increment X”*which means that presented outcome (deformation, contour plot etc.) was obtained after X*increment*was added into the analysis. Sometimes this second meaning is also refereed to as step, especially in*Arc-length*analysis.

**Instability** – Situation in which deformations in model rise, even when loads are constant or even get smaller. This is considered to be one of possible failures of model.

**Iteration** – In order to receive outcome for certain *increment *(sometimes called *step)* solver will make few *iterations* in order to receive an outcome that fulfil *convergence* criteria. Last *iteration* leads to *converged solution*, all other *iterations *lead to solutions that did not *converge* and are needed in process of obtaining *converged* solution. There are many strategies concerning how iterations should be made in analysis.

##### L

**Load – displacement plot** – see *stability path.*

**Load multiplier** – is usually denoted as *r* often with analysis type abbreviation in subscript. This is a parameter by which the applied loads in the model are multiplied in the analysis in any given *increment*. Lets imagine we have a model with 10kN load applied. If we would like to apply this load in 10 *increments* (1kN each) then the first *increment* would be 1kN so the load multiplier would be 0.1 (since 10kN * 0.1 = 1kN) and in the second *increment* total load would be 2kN so the load multiplier would be 0.2 and so on. *Load multiplier* can be used on vertical axis of *stability path *instead of loads or constraint forces.

##### N

**Node** – this is the smallest geometry part of numerical model. A single “point” in calculation that is connected to other such points (forming *Finite Elements*)

**Normalized deformation** – usually denote as Δ. This is the ratio of displacement in the selected *DOF* (Degree Of Freedom) in current increment to deformation received in this *DOF* in linear analysis for *load multiplier* *r* = 1.0. *Normalized deformations *can be used instead of deformations on horizontal axis of the *stability path*.

##### S

**Stability path** – this is a chart presenting relationship between loads and displacements in the model. Usually some sort of load indicator is on vertical axis (this could be a value of applied force, total constraint force in certain direction etc or a *load multiplier*). On horizontal axis deformations are shown (at which node/point and on what DOF is up to author, sometimes *normalized deformations* are used). Stability path appear in literature under many namesas: equilibrium path, load-displacement chart, displacement response path etc. For examples see here.