Understanding Stress Concentrations Using Finite Element Analysis

I spent a significant amount of time working with design engineers and occasionally designers. At my company there are probably 5-8 design engineers for every finite element analysis (FEA) engineer. That ratio seems to work out well most of the time. The problem is that design engineers, even structural design engineers, often have trouble visualizing how the stress and strain will “flow” through any given product. This can be a difficult topic because every product, part or assembly is different and is loaded in a different way. If we knew everywhere that something would crack, we would fix it before it cracked.

In my experience it is best to start with simple examples and get more difficult from there. We will start with a simple L shaped corner. In this particular example I started with a 10x10 (mm)  square and deleted a 6x6 section from it. I constrained the five nodes along the lower left vertical edge from all translational motion (but not constrained from any rotation). To each of the five nodes along the horizontal edge on the upper right I applied a 100 (N) force in the X and Y direction (so 141 N to each node at a 45 degree angle). Since I model everything in three dimensional space I applied a thickness of 1 (mm) to the shell (2D) elements. (As a side note Abaqus is unitless so I just put in numbers. The program does not have to have units, it could be meters or inches for distance instead of mm.)

Typical First Design Demonstrating a Square Inside Corner

Running that simulation gives us the pretty pictures below. The main one of the four that we are concerned with is in the lower left corner showing the maximum tensile strain. Tensile strains are often the most damaging and I designed this example to really demonstrate tensile strain.

Clockwise from Upper Left: Displacement; Von Mises Stress; Minimum Principal Strain (Compression); Maximum Principal Strain (Tension)

We quantify this by looking at the strain readings, in this case there is 5000 micro-strain or .005 (mm/mm). That is a huge amount! If this was an actual part, I predict it would break quickly.

One of the main things we try to do is provide a clear path from the load to the constraint. Think of this like standing. It is easier to stand with your legs under you than off to the side. It is easier to hold a weight in your hand at your side than with your arm outstretched. The reason is that the acceleration of gravity exerts a force on the weight that passes directly in line with your shoulder when your arm is down. When your arm is out stretched the force exerted is nowhere close to your shoulder.

A Great Design!

We build a clear path for the stress to flow often by adding a larger radius to an area that is over-stressed. In this example we went from having basically no radius to having a nice 6 (mm) radius. When the same force is applied we get the following images. 

Clockwise from Upper Left: Displacement; Von Mises Stress; Minimum Principal Strain (Compression); Maximum Principal Strain (Tension)

Isn't this great! Looking again at the maximum tensile strain in the lower left we see that there is only about 275 micro-strain! Wow! For an increase of 7.73 sq. (mm) we achieve a strain reduction of 94%! In other words, we increased the total weight of the part from 64 units to 71.73 units a 12% weight increase gave us a 94% strain reduction! 

Now, this is an oversimplification but for every 10% strain reduction the life is approximately double what it was before. That depends enormously on the material and where in the strain life region we are (it's more accurate for long life fatigue estimations in steels). However, using those rough numbers we can estimate that this part will last roughly more than 500 times as long the original design! (Another side note, if I actually used math that simple to build devices I would get fired. We have sophisticated software that helps us make more complex calculations.)

In conclusion, stress "flows" from load to constraint, and more strait the path the lower the stress.