As J.R.R. Tolkien wrote in The Lord of the Rings:
The road goes ever on and on
Down from the road where it began.
Still, around the corner, there may wait
A new road or a secret gate,
And though we may pass them by today,
Tomorrow we may come this way
And take the hidden paths that run
Towards the Moon or to the Sun.
What these few lines basically mean and what my blog is based upon is that discoveries are most often made in the unlikely of circumstances. Or not necessarily discoveries, but merely realizing the existence of something that was always there but we overlooked it, or maybe others knew about it but didn’t bother to tell us simply because they didn’t think it was worth any effort in doing so. But the bright side of it is that sometimes, exploring things on one’s own is a lot more fun and productive! At the time of Leonardo da Vinci, people considered him insane due to his sheer curiosity about everything that he observed in his daily life. Would it make sense for a person diligent in the painting profession to come to you and ask what happens to the tongue of a woodpecker when it’s pecking trees so furiously? Well, da Vinci didn’t mind asking such questions. He explored.
Coming to the topic.
I assume you know what the term “Factor of Safety (FoS)” means in the parlance of engineering. In a simple case where a design element (made of ductile isotropic material) is subjected to static load, we say “yield FoS” is equal to the yield strength of the material divided by the maximum stress that acts on it, because a structural member is no longer considered reliable once it yields (in case of a brittle material, it straightaway fractures). You might have carried out static structural analysis in ANSYS and obtained a FoS there. You get satisfied when it is high enough (not so high that it gets over-engineered). But how is the FoS actually calculated, especially when carrying out hand calculations (and how ANSYS calculates it)? What exactly do we mean by the ‘maximum stress’? I will limit the article to static loading since there is also a ‘fatigue FoS’ when analyzing failure resulting from variable loading and it is calculated in a slightly different way. Also, the material is assumed to be ductile, isotropic and having yield strength that is the same in compression as in tension. Simple enough!
In the first year Mechanics course, we were taught that given a general state of stress at a point referred to the cartesian coordinates, we can calculate the Principal stresses (normal and shear) by using some formulae. The Math used here is the building block of what we subsequently do in the design of mechanical elements. For a simple case of plane stress:
So, we calculated the Principal stresses. What now? As aforementioned, we are dealing with static loads here, and the material of the design element is ductile and isotropic. To prevent the failure of such design elements (failure may mean fracture or permanent deformation due to yielding), certain failure theories were formulated based on what may cause the failure- a critical stress, a critical strain or a critical energy. By far the most widely used theory for ductile behavior is the “Distortion-Energy theory”. What it simply calculates is the well-known “von Mises stress” using the Principal normal stresses. How this theory came into being has an interesting story.
Self-exploration knows no bounds, literally!
You might have heard of “Tension test”, one of the most common mechanical stress-strain tests performed to ascertain several mechanical properties of materials that are important in design. Performing this test at room temperature at atmospheric pressure is alright. But do you know what happens when this test is performed by completely immersing the specimen in a liquid carrying hydrostatic pressure? A spectacular result is observed: the amount of plastic deformation that the test specimen will tolerate without fracture is profoundly increased. It is as if cracks were prevented from forming in the metal by the hydrostatic pressure that pushes the parts together. At hydrostatic pressures of 20,000-30,000 kg/cm^2, tension specimens of ordinary mild steel may show elongations of hundreds of folds without fracture, against elongations at atmospheric pressure of only two to three-fold.
It was then realized that a failure theory needed to be formulated to account for the increased resistance of ductile materials against failure due to hydrostatic stress. Theories available until then were conservative and could not predict ‘no failure’ under such a stress.
Any given state of stress, say referred to the principal normal stresses as the coordinate system, can be resolved into two components: a hydrostatic component (pure normal stresses, pure volume change, no distortion) and a distortional component (pure shear stresses, pure angular distortion, no volume change). Only the latter is responsible for the failure of a ductile design element subject to static loading. The hydrostatic component could only produce if at all, elastic deformation of the body. Hydrostatic stress simple means that the resultant stress on any plane passing through a point is normal to it and equal in magnitude.
In the elementary theory of plastic flow, a simple convention is suggested based on experimental results: a pure hydrostatic pressure is without effect on plastic flow so that it would be legitimate to describe the stress-strain relation by entirely neglecting the hydrostatic pressure of the liquid and treating the specimen as if it were being pulled at atmospheric pressure. Errors do creep in but they are too small to produce any effect. This convention proves to be a very good approximation.
But why is the Distortion-Energy theory related to ‘energy’? Specifically, it is related to the strain energy per unit volume, or the energy stored in an elastic member when external work is done in deforming it. The theory says that it is not the total energy, but rather the energy stored during the distortion of an element which is the criterion for failure. The energy stored during pure volumetric change is simply subtracted from the total energy to give the distortion energy. This theory predicts no failure under hydrostatic stress and agrees well with all data for ductile behavior.
And eventually, according to this theory, yielding occurs when the von Mises stress exceeds the yield strength of the ductile material so that the yield FoS is equal to the yield strength divided by the von Mises stress. And, the FoS displayed in ANSYS for static structural analyses is calculated exactly this way! However, the standard method followed is to first decide a minimum desired FoS and then modify the design element to keep the maximum von Mises stress in ANSYS equal to the yield strength divided by the FoS.
But precaution must be taken when dealing with variable loading, for example when a shaft is under bending as well as torsional stresses. Though the material is ductile, it can fail in a brittle manner (due to fatigue). So yield FoS is not sufficient here, rather it is the fatigue FoS which is lesser of the two and more critical to the design. Also, care should be taken when dealing with materials that are not isotropic, but rather say anisotropic. Calculating FoS has a different theory in such cases. When dealing with the design of gears, in particular, bending and wear stresses and the corresponding FoS are most important which are determined by hand calculations. Performing simulation on ANSYS simple gives you the equivalent von Mises stress and FoS for the overall gear. So hand calculations are equally important to validate the results.
This was all about the Distortion-Energy theory and its relation to the FoS for design purposes.
So the bottom line is that even simple stuff can have an interesting story behind it that laid its foundation. Calculating Factor of Safety might seem like a normal routine while designing mechanical elements, but we most often don’t take a pause and think why it is calculated the way it is. We overlook such simple things, and hence miss the opportunity to learn something new and interesting in the course, something that was already there but we weren’t aware of its existence. As Daniel Kahneman has said: “We can be blind to the obvious, and we are also blind to our blindness.”
An Adventure in Science 