**Finite Element Analysis**

NB: FURTHER INFORMATION TO BE ADDED

These notes provide some background to finite element analysis (FEA).

Originally most of these notes were writen by Graham Roberts and David Grieve. Later additions are by David Grieve

**i) Introduction to Numerical Techniques**

For stress, thermal and many other types
of analysis, only the simplest of configurations can be solved by exact analytic
methods. For the vast majority of engineering components some alternative method
is needed and the usual approach involves a method of approximation requiring
some numerical technique to solve, which in turn requires the use of a digital
computer.

**ii) Numerical Techniques**

There are 3 main methods of numerical
approximation:

Finite element analysis (FEA)

Boundary element method (BEM)

**FEA**

FEA involves mathematical approximations associated with the
displacement function of a particular element and geometric approximations to
the component shape. An advantage of FEA is that it can be used to analyse
virtually any structural problem in a routine manner, hence non-homogeneous
material can be accommodated as the assembly of elements with different
properties is straightforward.

**FD Method**

The FD method uses a finite difference approximation to
the (partial) differential equations that describe the behaviour of the system.
A significant difficulty in solving the equations is that with fine meshes,
needed for adequate accuracy, there may be instability in determining the
solution. Discontinuous interfaces, such as abrupt changes in material
properties or geometry require special treatment. FD tend to be used more for
Fluids problems than for Mechanics problems.

The two applets on the introductory web page, for computing the load
carrying capacity and pressure distribution of a hydrodynamic tilt pad thrust bearing
and for a journal bearing, are both solved by 2 dimensional FD methods using rectangular
meshes with an iterative solver.

Finite difference method example

**BEM**

The BEM method reduces the dimensionality of the problem by 1,
so the overall size of the computation the smaller than in FEA or FD, but the
matrices produced are assymetric and densely packed and may require just as much
computational effort to solve as when using FEA or FD. The method is
particularly suited to problems with high stress gradients such as fracture
mechanics. Boundaries that are nominally at 'infinity' may readily by handled
whereas such boundaries increase the size of FEA and FD problems. The underlying
mathematics is however more complex than the other two methods.

(N.B. Information about two dimensional stress and strain transformations can be found in the the Mechanics and Engineering Science section).

The notes below relate to FEA.

**References:**

'Finite Elements - A Gentle Introduction', by D Henwood and J Bonet, Macmillan, 1996.

'Basic Principles of the Finite Element Method', by K M Entwistle, Maney Publishing, 2001. ISBN: 978-1902653532.

'Roarks Formulas for Stress and Strain', 6th Ed., Ed by W C Young, McGraw-Hill International Editions, 1989.

'Stress Concentration Factors', by R E Peterson, John Wiley and Sons, 1974.

'Shock and Vibration Handbook', Ed. C M Harris, McGraw-Hill, 1988.

Return to Index of Online Documentation.

David J Grieve, 6th December 2012.