The most common application of FEA
is the solution of stress related design problems. As a result, all commercial
packages have an extensive range of stress analysis capabilities.
What is Stress ??
Stress can be described as a
measurement of intensity of force. As all engineers know, if this intensity
increases beyond a limit known as yield, the component's material will undergo
a permanent change in shape or may even be subjected a to dramatic failure.
From a formal point of view, three conditions have to be met in any stress
analysis, equilibrium of forces (or stresses), compatibility of displacements
and satisfaction of the state of stress at continuum boundaries. These
conditions, which are usually described mathematically in good undergraduate
strength of material texts, are also applicable to non-linear analysis.
How the Result is Achieved
It all starts off with the formulation of the components
'stiffness' matrix. This square matrix is formed from details of the material
properties, the model geometry & any assumptions of the stress-strain field
(plane stress or strain).
Once the stiffness matrix is created, it may be used with
the knowledge of the forces to evaluate the displacements of the structure
(hence the term displacement analysis).
On evaluation of the displacements, they are differentiated
to give six strain distributions, 3 mutually perpendicular direct strains &
3 corresponding shear strains.
Finally six stress distributions are determined via the
stress/strain relationships of the material.
Commercial packages usually go one further & calculate a
range of more usable stress fields from the six stress components such as the
principal stresses & a host of failure prediction stresses as described by
the most common yield criteria (Von Mises/Maxwell/Heckney, Guest/Tresca, etc.). The displacements can be used in conjunction with the
element stiffness’s to determine the reaction forces & the forces internal
to each element (otherwise known as the stress resultants).
A point to note is that at least one of the displacements
must be known before the rest can be determined (before the system of equations
can be solved). These known displacements are referred to as boundary
conditions and are often a zero value. Without these boundary conditions, we
would get the familiar singularity or zero-pivot error message from the solver,
indicating that no unique solution was obtainable.
Non-Linear Analysis
In order to explain non-linearity in stress analyses, let’s
examine the nature of linear solutions. Many assumptions are made in linear
analyses, the two primary ones being the stress/strain relationship & the
deformation behavior. The stress is assumed to be directly proportional to
strain and the structure deformations are proportional to the loads. The second
assumption is oftentimes mistaken to derive from the first; a fishing rod is an
example of a non-linear structure made of linear material. A stress analysis
problem is linear only if all conditions of proportionality hold. If any one of
them is violated, then we have a Non-Linear problem.
Most real life structures, especially plastics, are non-linear, perhaps both in
structure and in material. Most plastic materials have a non-linear stress
strain relationship. The non-linearity arising from the nature of material is
called 'Material Non-linearity'. Furthermore, thin walled plastic structures
exhibit a non-linear load-deflection relationship, which could arise even if
the material were linear (fishing rod). This kind is called geometric
non-linearity.
All non-linearity’s are solved by applying the
load slowly (dividing it into a number of small loads increments). The model is
assumed to behave linearly for each load increment, and the change in model
shape is calculated at each increment. Stresses are updated from increment to
increment, until the full applied load is reached.
In a nonlinear analysis, initial conditions at
the start of each increment are the state of the model at the end of the
previous one. This dependency provides a convenient method for following
complex loading histories, such as a manufacturing process. At each increment,
the solver iterates for equilibrium using a numerical technique such as the Newton-Raphson
method. Due to the iterative nature of the calculations, non-linear FEA is
computationally expensive, but reflects the real life conditions more
accurately than linear analyses. The big challenge is to provide a convergent
solution at minimum cost (the minimum number of increments).
