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By Reid Penner
At Sparta Engineering, we rely heavily in computer simulations, what is technically known as Finite Element Analysis (FEA). These computer simulations work by taking larger, very complex geometry and breaks it up into thousands of tiny, simple parts that interact together (A good analogy of this is a Lego structure, where a complex object is actually comprised of many smaller, simple parts).
One thing first: most metals, steel included, are described as being “ductile” materials. What this means is that as the steel is loaded, it reacts to that load in a linear way, up to a point known as the yield strength. After this point, the steel will behave non-linearly, and will resist very little load before finally reaching the ultimate strength point, and breaking. This non-linear region is known as the plastic zone, and is very hard to describe mathematically and computationally.
Even more specifically, we use a version of computer simulation known as “Linear FEA”. This makes the assumption that the materials we use behave in a linear way, even past the yield strength. While only accurate for loads under the yield strength, we rarely are designing equipment to operate in the plastic zone. As well, a linear FEA requires much less computer resources and time than a non-linear FEA. Using this assumption that the material will behave linearly is accurate for almost all of the designs Sparta produces.
However, the one common case where linear FEA is not accurate is in the case of stress concentrations. At a stress concentration, there is a large stress in the material in a very small area. They are usually caused by sudden changes in the geometry of the equipment (for example, gussets on a beam, or keyways in a driveshaft). Stresses in these areas can be above the yield strength of the material. Because of this sudden increase, linear FEA is not accurate at predicting effects in these stress concentrations. In the past, we have dealt with these areas of apparent high-stress by increasing the material on these areas. This, however, is not ideal, as it adds time and cost to the design process and to the manufacturing process.
In response to this uncertainty, an experiment was undertaken to directly compare a stress concentration to a computer simulation, and determine how much of a stress concentration is acceptable in that computer simulation. While work in this area has been done in the past (most noticeably by Peterson et. al), these works are primarily directed at determining what the stress magnitude is, and not its effects on the structure as a whole. The experiment consisted of a test article of known geometry (in this case a keyway on a shaft), and an arm with a scale that was used to apply a recordable load to the test specimen. Displacements were recorded and compared to the computer simulation.
From this recorded data the yield point of the test specimen was found, and compared to the FEA. With this comparison, we were able to discover that for a linear analysis there is an allowable size of stress concentration that will not affect the overall strength of the component, depending on the loading case. We were also able to determine that the plastic zone in ductile metals is a very good buffer at re-distributing stress in structural components. This is extremely useful in design, as it allows us to scale back materials in these areas, and simplify manufacturing in many areas of the design.
This is not applicable in all cases though, and does not mean that all stress concentrations can now be ignored. These experiments were only run in a static, continuous loading case. If at all a repeated load is occurring, fatigue of the metal becomes a major concern. Stress concentrations have a very prominent effect on metal fatigue, and can lead to cracking and catastrophic failure.
While stress concentrations appear frequently in finite element analysis, the linear assumption used in a linear FEA does not accurately portray what is occurring in these areas. It is not accurate to fail an entire structure or component based on a stress concentration in a linear analysis. From our experiments in this area, there is an acceptable stress allowance before the stress concentration has a significant effect on the overall strength of a structure.