Finite Element Analysis

What is FEA ?   The FEA method gives an approximate solution for certain problems in engineering and science It is mainly used for problems for which no exact solution is available It is a numerical rather than an analytical method One of the first applications of FEA was to find the stresses and strains in engineering components under load FEA requires large computation resources when applied to any realistic model of an engineering component

What is FEA ?   The development of the FE method has depended on the availability of suitable digital computers for it to run on and graphics technology to display the results The FE method can be applied to a wide range of material properties, linear-elastic (Hookean) behaviour and behaviour involving deviation from Hooke’s law (plasticity or rubber elasticity) Many comprehensive general-purpose FE systems are now available

What is FEA ?   There are more specialised packages for particular applications (e.g. steel framework, piping) Depending on the type and complexity of the analysis, large analyses may benefit from multiple processors FEA can be applied to one-dimensional (line), 2-D (area) or 3-D (volume) problems

What is FEA ?   The FE method is now applied to problems involving a wide range of phenomena  structural and stress analysis dynamics and vibrations heat conduction fluid mechanics electrostatic

What is FEA ?   Any continuous problem can be divided up into a number of smaller areas or volumes which are called finite elements. The process is called discretisation and the assembly of elements is called the mesh

Element types Elements can be various shapes some examples are In two dimensions, quadrilateral or triangular In three-dimensions, brick-shaped, wedge-shaped or tetrahedral

Linear Elastic Analysis The displacement is the quantity that is first found in the analysis at a series of points called nodes Nodes are placed at the corners of the elements and often at the midside – depending on the element formulation Nodes on the boundaries of adjacent elements must be common to the elements that meet there

Linear Elastic Analysis There are a number of incorrect ways of specifying a finite element mesh

Linear Elastic Analysis The analysis calculates the displacement at the nodes for the particular loading applied to the FE model The displacements are found for a finite number of points, namely, the nodes The displacement of each particular node is defined in terms of the displacements of the other nodes of the element

Linear Elastic Analysis The displacement of each node in a 2-D element has two components: one parallel to a reference x axis one parallel to the y axis these are called degrees of freedom Each node has two degrees of freedom associated with it in a 2-D model and three for a 3-D bricked-shaped element The computer time and the cost of the analysis increases as the number of degrees of freedom in the model is increased

Linear Elastic Analysis The solution procedure for the displacement finite element method allows the user to determine a number of unknowns, these are obtained in the following order: Firstly calculate the nodal displacements & reactions the program finds the corresponding strains from the strains the stresses are computed

Pre-processing   Pre-processing is concerned with the creation of the finite element model and the definition of the way in which it is to be constrained and loaded The model can be built up within the pre-processor itself The success of the analysis depends on: The simplifications introduced into the model as compared with the real thing The choice of the mesh and type of elements to be used

Pre-processing The data for the model must contain information about:  Appropriate mechanical or physical properties must be allocated to the elements – e.g. the material of which the structure is made, plate thickness etc. Restrictions to the movement of certain nodes (constraints) The geometry and finite element mesh Control information for the analysis (type of analysis, etc)

Pre-processing   Additional information in the input file specifies the type of analysis, the way it is to be carried out and the output required e.g. static analysis solver to be used Output required - displacements, stresses at nodes, etc  With some systems, an input file can be produced without the use of a pre-processor if the model is particularly simple

Analysis The FE system reads in the data from the input file It carries out certain checks on this information If there are no errors in the input file, the analysis is carried out and output is produced These files can be examined and the relevant information extracted The post-processor can be used to present this information in a user-friendly manner

Post-processing   The post-processor takes in the information from the output files and presents it in a range of different graphical and tabular forms Colour bands may be used to indicate the value of stress on the surface of the component Contour lines of equal stress may be drawn The model may be rotated and examined from different viewpoints Point values may be queried Graphs may be plotted

The FEA Developer’s View The starting point is represented by the differential equations that govern the phenomena to be modelled The FEA is then one method of providing numerical solutions to these equations when analytical solutions are not available Much of the advanced work in this field is not accessible to the non-expert Once the equations for the FE formulation of a problem have been developed, efficient procedures (algorithms) and programmes have to be developed