So why are there so many widespread benefits from making the surface harder or stronger? If there is a wear problem, maybe it is pretty obvious that (in many cases) making the surface harder will reduce loss of material due to wear. But even if there is no wear, hardening just the surface will often result in increased durability.
There are many reasons for this, but today we will stick to those most relevant to structural applications, including beams for large civil structures and machinery components, among others. When a piece of round wire is stretched uniformly – if we can ignore the ends where they are somehow attached to something – the load is uniform everywhere. The stress at the surface is the same as the stress in the center with any load below that which causes yielding (permanent shape change).
However, as soon as any bending or twisting happens, or there is any deviation from a completely uniform cross section, that nice simple situation is a thing of the past. Now, in almost all cases (contact stresses behave differently) the surface stress is higher than that below the surface. If you go to Machinery’s Handbook or any other reference giving equations that allow you to calculate the stress in a simple beam with a simple square cross section, you can easily start to get a giant headache.
First of all, the most readily available sources give only the maximum stress to be found in a given situation. For example, take a simple square cross section beam, which is 10 x 10 inches square and 10 feet (240 inches) long. Support it at both ends and press down in the middle. If you use a high enough load, you will be able to measure a deflection. It makes sense that the deflection will be greatest directly under the load (Fig. 1).
It turns out that if you assume that the beam is very long compared to its cross section, then the equations to determine the stress at the position directly under the load – the maximum stress – or any other location for that matter, are not too hard to work with once you find a competent mechanical engineer. (I would like to thank Dr. Julian Raphael of JR Technical Services, Abingdon, Va., for his kind help in this matter.)
I further assumed a load of 200,000 pounds was what was pressing down in the center of the length of the 10 x 10 inch beam. It turns out that what is called “the bending stress,” which is actually what we have called in previous blog entries a NORMAL stress (also called the FIBER stress), can be calculated by the following “simple” equation:
where P is the load, h is the height of the beam (here, 10 inches), y is the distance from the neutral axis (here, the neutral axis is assumed to be at the center of the height of the beam, or at a position of 5 inches from the bottom or top) and x is the distance from the closest end of the beam.
Next time, we will apply this to several examples and see the influence of case hardening.