2 Basic terms and algorithms of finite element method
I. In each FEM program the corresponding analysis model is defined at the beginning of the manual. This is usually done in the description of the applied finite element
2.6.2 Interpretation of FEM output data
In order to meet the needs of design practice, sizing, checking of resistance and assessment of serviceability, the outputs of the FEM procedure are further processed either manually or in numerical or graphical postprocessors. The following principle is valid:
FEM programs with elements whose dimension is n D hold in the outputs only such an amount of information of dimension m D, m>n that is embedded into n D finite elements through hypotheses that reduce the dimension from m to n . In particular: FEM programs with 2D elements cannot be used for a detailed 3D stress-state analysis in such places of 2D structures (walls, plates, shells) that require such an analysis (the vicinity of column heads in flat slabs, bridge bearings, concentrated impulses acting on area smaller than h , where 2 h is the thickness of the planar structure, etc.). Even less applicable to the 3D analysis are the outputs from FEM programs that use 1D elements. For example, such programs are not able to tell on which particular part of the cross-section the load or reaction acts (top or bottom flange, etc.), as only the resultant over the cross-section is known. Therefore, it is not possible to perform an accurate analysis of the stress-state in e.g. fixed ends where the specific design of the fixing cannot be taken into account or in connections where the exact welding, bolting or gluing characteristics cannot be considered, etc.
If the real design is modelled through a kind of abstraction, e.g. if a foundation strip is
modelled by a rigid 1D line, the singular outputs from the FEM analysis (that converges to exact singularities) cannot give an accurate idea about the detailed stress-state in the vicinity of the singularities. A typical example: If we define a rigid line support of a bridge deck, either perpendicular or skewed, we get non-designable moments in the corners of the deck, which is a correct trend towards the singularity of the corner. If we want to have realistic outputs, it is sufficient to specify the real stiffness of the supporting beam that can never be infinite, which is included in the definition of a rigid support with w≡0, moreover with a zero area measure σz → ∞. Even the lowest estimate of the stiffness of the line support or the lowest estimate of the spring constant k in older programs that did not allow for such a support type produces usable moments. Hundreds of similar situations can be summed up to express the following principle:
If we want correct FEM results, it is sufficient to exploit the wide capabilities of FEM in order to express correctly the real design and realisation of the structure and its supports and the real distribution of loads.
In this context, mistakes are made by both younger engineers (due to the classical tuition at faculties) and experienced specialists who were – during their long practice – forced to simplify everything (loads, supports, material properties, etc.) as program with the power of present-day FEM systems were missing.
Many engineers learn about the effectiveness of FEM at various courses, but they doubt about the perplexity of input data for complex spatial systems consisting of thousands of 1D and 2D (and possibly also 3D) elements, which usually prevented the full utilisation of FEM in the previous era of development (in the Czech Republic approximately until 1980).
The main idea of the division of a domain into finite elements is however much stronger and it includes also the following capabilities:
g) The analysed structure is first divided into smaller substructures, which makes the division more understandable. The substructures may have technological and production functions, but may also be parts selected only on a formal basis, in order to improve the handling of the model. Sometimes, it is possible to combine (i) parts made of 1D elements and other parts composed of 2D elements, sometimes (ii) individual walls and ceilings, foundation slabs, pile foundations, etc. If we use this division just with the aim to make the handling of inputs more comfortable, it is better to talk about macroelements and sometimes only about segments. If the program creates for these parts separate global stiffness matrices K and vectors of load e parameters f and if the addition theorem is applied to them (no. 13 and 17 of the e overview in art. 2.3.), then we can refer to real substructures in terms of FEM. What is remarkable about this procedure is that the substructure can have much more “internal nodes” than “boundary nodes” through which it is connected to other substructures.
Only the parameters of these joint nodes then appear in the global stiffness matrix K and in the vector of load parameters f of the whole structure. This reduces the number of unknowns N and also the band width BW at the cost of certain preparatory works – resembling pre-elimination – which is just the formation of matrices K and e f . e
h) Creation of what is called superelements by assembling simpler elements, called subelements, is not in fact too different from (a), but it has slightly different technical meaning. One of the oldest superelements is a quadrilateral composed of four triangles
2.6 Main outcome for the users of FEM programs
or a hexahedron (i.e. a brick with the shape of a cube, block, or parallelepiped with either planar or curved faces) composed of five tetrahedrons. The meaning is the following: we apply simple base functions U (no. 1 and 4 in the table in art. 2.3.) in the subelement. These functions usually impose a significant deformation restriction, physical strengthening of the element by means of fictive connections that impose prescribed displacements. This restriction is somewhat relieved for the superelement. Its internal parameters are regularly (in compliance with the energetic principle) eliminated and the result is an element that is “softer in terms of deformation” than it would be after a simple addition of the subelements. Let us present one illustrative example: if we use for a triangle linear functions U, i.e.
monomials 1, ,x y , or for a tetrahedron 1, , ,x y z , then we force it to displace along the plane or hyperplane inserted into 3D space. The created quadrilateral can move along four partial planes (tetrahedron along five partial hyperplanes), which can already approximate more complex, out-of-plane deformation. The reduction of the number of unknowns N and band width BW through the eliminated parameters is also welcome.
In addition to the above-mentioned elimination, also a simple interpolation of parameter values is sometimes used, mainly with the aim to exclude undesired nodes in the middle of the sides. On side 123 we thus set d2 =
(
d1+d3)
2 (similarly for more complex geometry) and linear or quadric interpolation is applied. The principle of compatibility between adjacent elements must be satisfied, i.e. the elimination must generate the same values of deformation parameters in both elements. The users are advised to pay attention to the definition of the element in the manual as it can make clear some characteristics of the element that may show up in the output – interpretation of the results. Let us describe a simple example: let us have d = =w slab deflection without other parameters (angles or derivatives).If d is eliminated through a linear interpolation, then for supporting conditions 2 d1=0,
3 0
d = , side 123 is straight with all possible consequences for the internal forces in the element. If d is eliminated energetically, its value may be different and the boundary of the 2 element can be curved, which produces different internal forces. The situation is different if additional nodal parameters are defined. The requirement of d =0 in nodes does not have to mean w≡0 at the edge of the plate, i.e. exact linear support, but just “skewering” of the plate on the vertex nodes of the element. This has an impact on the output internal forces and must be interpreted appropriately with respect to real support conditions required, see chapter 5.