Engineering | Finite Element Analysis | Stress
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Non-Linear Analysis.
In order to explain non-linearity in stress analyses, lets examine the nature of
linear solutions. Many assumptions are made in linear analyses, the two primary ones
being the stress/strain relationship & the deformation behaviour. 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-linearities 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 is 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.
Linear and Nonlinear Analysis
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