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In today’s world, products are getting increasingly complex, with downward pressure on overall pricing. This has led to an inevitable increase in the geometric complexity of mechanical components. While this has allowed components to perform multiple functions, it has led to complex load transfers and possible concentrations of stresses.
A stress concentration is defined as a high localized stress, compared to the average stress of the body, and is typically found in a region that has an abrupt geometric change. This article will walk through the basics of a stress concentration, offer real-world examples to illustrate the concept, and outline methods for reducing stress concentrations in your designs.
Stress concentrations are relatively straightforward when it comes to identifying where they will be located and whether they have contributed to failure. They will be located in the small radii and sharp corners that are in a load path. The image below shows a component that will have rollers on the smaller diameter shafts on both ends, with a load applied to the top flat surface.
This setup will almost behave like skateboard trucks, but we’ll be loading to a higher force that is cyclic in nature. The image below is a rendering of the subassembly that this component goes into, including the rollers with bearings.
Now we can see that the smaller diameter shafts meet the center portion of our roller support with a relatively abrupt radius. If we zoom in on this radius, we can see that it comes in at only 0.010”, as shown below.
If you recall from above, a sharp corner or radius will be the location of a stress concentration. In order to visualize this stress concentration, I’ll run a FEA study that will plot out these stresses quite well. The load case has already been discussed, but we can refer to the illustration below for a refresher.
We have a downward force of 500 lbs, and we have fixed the surface of the shaft where the rollers will sit. The only remaining step is to run the simulation and process our pretty pictures! In the image below, we can see the beautiful red spots that indicate a stress concentration. That’s it…it’s as simple as that!
We can see that this is the highest stressed feature, and this would be the first location to fail under cyclic loading (though it would most likely occur on the bottom side loaded in tension, instead of the top compressive side). It would initiate as a small surface crack that eventually propagated through the component to complete failure.
There are also a few basic formulas to define the max stresses in this type of geometry when we have a bending moment applied. The formula in this case is as follows:
K,t – Stress Concentration Factor
σ,ave – average stress in the member
In many simple geometries, the stress concentration factor is defined for a range of geometries. The chart below shows the curves of these factors and the dependency on the ratios of the critical dimensions.
This chart is used by dividing the large diameter by the smaller diameter first. So let’s say we have a 2-inch diameter rod that is stepped down to 1 inch; that gives us a D/d of 2. Next, we look at the ratio of the radius to the smaller diameter.
As another example, let’s say the radius is 0.1 inches, so we have a r/d ratio of 0.1. If we look at the chart above, we need to find the curve that represents a D/d of 2. Then we look at the x-axis to locate the r/d of 0.1 and look across to the Y axis to get a stress concentration factor of about 1.8.
The image below shows all of this in a graphical format.
It’s important to note that this formula changes based on the load case and geometry, so I encourage you to further explore charts for additional geometries. Now that we understand the basics, we can step into some examples of correcting stress concentrations.
In this section, we’ll look at a few real-world examples and how we can reduce the peak stresses at the location of the stress concentration. The first example we’ll look at is our original component with the rollers. If you recall, we had a very small 0.01” radius at the step-up in diameters. Now, we’re going to increase that diameter to 0.08” to see how much we can reduce the stresses.
The first image is the original stresses, and the second is the reduced stresses with the larger radius.
We can see that the stresses have gone from 14,419 psi all the way down to 3,873 psi. While this differential is pretty extreme because of the extremely small original radius, it drives the point home of just how much a stress concentration can influence the stresses in the part.
In the next two examples, we’ll look at some geometries with a tensile load applied, instead of a bending moment. The first part is a support bracket that holds a brass pin. The brass pin typically has an upward load applied, and the base is bolted to a fixed plate, as shown in the setup image below.
As you probably expected, we ran the FEA again to get a baseline stress value before making any changes. The results of this study are shown below.
The stresses at the base with the small radii (0.030”) are definitely a high concentration, ranging all the way above 68,0000 psi. By now, we can guess that a larger radius should help lower the stresses here, even with a much different load case and geometry. In the next simulation, I have increased the radii from 0.030” to 0.080”, and the results are much better, as shown below.
If I wanted to get even lower stresses, I could do so by thickening the flange to which this assembly is attached. You can see the light blue coloring that gives an indication of how this part is deflecting under a load.
In our next example, we have a flat plate with a diamond shaped hole cut out of the center. There is a load applied on one end, while the other end is fixed, as shown in the setup below.
I’m sure by now you have guessed that the highest stresses will be located at the top and bottom corners of the diamond shaped hole. If that is the case, you were absolutely correct. The next image shows the results of this simulation.
We can see the stress concentration is exactly where we expected. We know we can make the radii bigger, but what if we wanted the option of drilling that hole out, instead of requiring a mill or punch? Can we use a larger diameter circle, effectively reducing the total material in the part, and still lower the stresses?
Absolutely! As the image below details, the same simulation was run with a round hole that is greater in diameter than the distance from the top to the bottom of the diamond-shaped hole above.
We can see that while the stress region is larger, the magnitude of peak stress is roughly 1/3 of the original case. In a similar approach but different application, it is common for repair centers to drill a hole at the end of a crack to relieve the high stress concentration associated with the very small radius at the tip of a crack.
I hope by now the idea of locating and reducing stress concentrations is clear, and you are well on your way to improving a design. While I used an FEA program here to determine the magnitude of stresses, there are some general guidelines that can be used to improve a design.
When it comes to common methods of reducing stresses, the following list includes some simple items to get you started quickly:
Make radii in a load path as large as you are comfortable making them.
Limit the ratio of the large feature to the small feature, where possible.
Add stress-reducing holes at the end of slits, sharp angles, or cracks to relieve high stress concentrations.
Refer to stress concentration charts to understand when you are in a region of diminishing returns with respect to radius size.
Some common issues to avoid are:
Do not use sharp corners along a load path.
Do not make a large size transition between loaded features. The stiffness mismatch will drive the stress concentration much higher.
Don’t assume that the same size radius works for all features. Remember that the stress concentration is based on a ratio, not a magnitude.
Don’t place a stress concentration in a high cyclic load if you absolutely must use a sharp corner.
As you may have guessed, most of the “don’ts” are inverse to the “dos”. This list is not fully comprehensive, but it should cover the basic concepts that every designer should know in order to improve their design skills.
Through the examples and analysis above, it should be clear exactly why we need to be concerned with stress concentrations. By incorporating these concepts through your design, you should be able to achieve higher load ratings, reliability, and fatigue life.
I encourage design teams to talk through product requirements and design choices to ensure the proper blend of aesthetics and function. There will always be tradeoffs, but proper analysis can help achieve an optimized solution.