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The Finite Element

One of the main challenges in solving partial differential equations is to use equations which are approximate but numerically stable so that error accumulation does not cause the solution to be meaningless. The finite element method is an excellent technique for solving partial differential equations over complex domains. Application of the finite element method in structural mechanics is based on an energy principle, such as the virtual work principle, which provides a general, intuitive and physical basis.
The finite element method originated as a technique used to solve stress analysis problems, but today it can be applied to a multitude of disciplines ranging from fluid mechanics, to heat transfer to electromagnetism.
The buckle of a standard lap belt used in passenger aircrafts has been designed and is ready to undergo testing. In order to be released into the market, the strap system must be able to withstand a 450 kg tensile load. It is assumed that the weakest point of the design is the flat plate of the buckle. Thus, prior to engaging in a costly test scenario, a simple finite element analysis of the buckle is to be made to insure soundness of design, i.e. the material does not exceed its yield strength and no significant distortion occurs. Preparing the problem for analysis first requires definition of assumptions.
Figure 1 is a schematic drawing of the buckle to be analy...
thickness of the part (which is assumed to be constant) is believed to be small enough compared to the width of the part such that shell elements can be used. Displacements are expected to be relatively small such that a linear approximation will be valid.
Figure 1







2.2 Material Properties
The part is manufactured from 2.5 mm stainless steel plate with a Young's modulus of elasticity of 206 GN/m2, Poison's ratio of 0.3, and a yield strength of 580 MN/m2. Homogeneous and isotropic material was assumed with no discontinuities or residual stresses present as a result of manufacturing processes such as forging, rolling and welding. The material is assumed to have linear elastic properties.

2.3 Mesh
A mesh that provides a good representation of the model is critical for an accurate solution; the elements must be well-shaped and close fit. For this analysis, the element type chosen was PLANE82, which is a 2D structural solid element. The element has 8 nodes, which increases calculation time over its 4-node counterpart but also increase accuracy of solution. Eight-noded elements are also known to be more accurate for modelling curved boundaries, which is where the areas of maximum stress were expected in the buckle. The PLANE 82 element type also offered the benefit of accounting for a thickness value in its input properties. Since maximum stress values were expected in the curved sections of the part, two meshing values were utilised, thus providing a denser mesh in critical areas. In the curved areas, a value of 0.25 was used, while 0.51 was used in the rest of the model.

2.4 Boundary Conditions
Determination and application of boundary conditions is critical to the analysis. For this model, it was assumed that displacements would be small enough so as to not ...Show more

Summary

The finite element method (FEM) is a numerical procedure used for finding approximate solutions of partial differential equations (PDE). Problems which use this method are usually too complicated to be solved by classical analytical methods and require significant computational solving capabilities…
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