Finite Element Analysis (FEA) is a computer based technique for finding stresses and deflections in a structure using selected load cases. The method divides a structure into small elements with easily defined stress and deflection characteristics based on a series of differential equations. The finite element method solves these equations with global matrices using a computer program. FEA solves mechanical and thermal problems and models complicated features undergoing static and dynamic loading using the following steps:
1) Geometry Development
The first step in Finite Element analysis is the creation of a model that breaks a structure into simple standardized shapes or elements with a common coordi-nate grid system. The coordinate points, called nodes, are locations in the model that provides output data (Figure 30a & b).CAD creates the three-dimensional (3D) representation of the product geometry as previously described. Element selection is a function of product geometry and loading conditions. The element selected affects the results as each element has characteristic properties. A model can use more than one type of element. The list below contains some of the element types and their properties:
- 2D elements = all forces and displacements act in plane
- Axisymmetric elements (2D) = node displacements in radial And axial directions
- 3D elements = forces and displacements in all three dimensions or complex geometry
- specialty elements = shell, plate and beam
2) Material Property Assignment
Modulus of elasticity or rigidity and Poisson's ratio define material properties for each element. A stress-strain curve represents the material properties for a non-linear material condition (elastic-plastic). Section VI-Test Methods and Data defines one such curve.
3) Mesh Generation
Based on the element types selected, automatic mesh generation subdivides the geometry into finite elements. The element density within each segment of the product geometry is either chosen or automatically determined. The nodes and elements defining the geomentry of the structure comprise a mesh. The finite element program calculates nodal stiffness properties for each element and arranges them into matrices. The appropriate matrix transformation generates a global stiffness matrix from the existing element matrix.
4) Boundary Conditions
The appropriate boundary conditions apply constrains to the model (fixed, simply supported, etc.).
5) Load Application
Applies loads to the model (force, pressure, temperature, etc.)
6) Run Analysis
The program processes the equation matrices with applied loads and boundary conditions to calculate displacements, strain, natural frequencies or other data specified by the program. This provides a stress distribution across the entire model. The high stress regions should also have the highest element density. Each individual element should have a small stress gradient across itself. A finer mesh increases the accuracy of the model. However, the computer run time is longer. Various adaptive methods find the critical regions in the model and make the necessary mesh refinement to reduce the error for the next iteration before reaching convergence.
Static analyses such as deflection, stress and strain under a constant set of applied loads are the most common analysis. The material assumption is linear elastics, but analyzing non-linear behavior such as plastic deformation, creep and large deflections is possible.
FEA can predict relative changes in deflection and stress better than absolute deflection and stress. The proportional difference of two structures is more accurate than the absolute results.
Results usually include graphical display of the solution, an output file, and hard copy of the result images. (Figure 31) Typical connector FEA results include :
- Deflected contact shapes during and after mating
- Beam and contact stresses
- Normal forces
- Mating force vs. deflection curve
- Thermal effects
- Behavior under shock and vibration
8) Data Correlation
Experimental data is collected to correlate the FEA model results and to formulate a baseline.
9) Design Optimization
After comparing the baseline results, design modification and remodeling are available. This iteractive process is design optimization. Design optimization combines the engineering requirements, geometric parameter, CAD model and performance goals into a computer simulation to achieve the optimum design.
Caveats for FEA analysis:
Traditional Finite Element Analysis programs analyze linear problems, assuming linear elastic material behavior, small displacements relative to the overall dimensions, and constant boundary conditions. Another requirement when assuming linearity is reversibility of the process modeled. While this condition satisfies most cases in load deformation analysis, the whole premise of contact stamping is to impart permanent and irreversible deformation to the workpiece. The nonlinearities encoum-tered in contact stamping fall into three major categories:
- Material: Elastoplastic material, strain rate sensitivity, and anisotropic behavior due to rolling.
- Boundary Conditions: Nonlinearity due to constantly changing contact between tools and workpieces.
- Geometry: The workpiece undergoes large rotations and deformations.
In order to account for the residual stresses due to the forming history of a connector spring contact, redefine the FEA boundary conditions after forming as follows:
1. Input boundary conditions for forming
2. Use stress-strain curve as material input and then "form" the "part"
3. Elastically remove the forming "tooling" and allow stresses to redistribute
4. Redefine the boundary conditions for the "part" model
5. Deflect the FEA "part" and asses the performance.
One MUST use nonlinear code that can suitably remesh when forming of the node coordinate system. The explicit method takes a large number of steps to represent the severe nonlinearities by many small linearized increments. In addition, use an elastic-plastic material model that includes work hardening for the forming step.
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