Tuesday, June 21, 2005

Finite Element Method , the principles (1)

Stress & Displacement Analyses



Introduction
    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 analyses.


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 usuable stress fields from the six stress components such as the principal stresses & a host of failure prediction stressess as described by the most common yield criteria (Von Mises/Maxwell/Heckney, Guest/Tresca, Heubner/Thornton, etc.). The displacements can be used in conjunction with the element stiffnesses 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 oftentimes 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.


An alternative solution
    An alternative solution may be obtained via the force matrix method (otherwise known as the flexibility method). In the previous description, the displacements were the unkown, and solution is said to be obtained via the stiffness method. In the force method, the forces are the nodal unknowns, while the displacements are known. The solution is obtained for the unknown forces via the flexibility matrix & the known displacements. The stiffness method is more powerful & applicable than the flexibility approach.


Non-Linear Analyses
    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. The big challenge is to provide a convergent solution at minimum cost (the minimum number of increments).

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