Assuming Linearity: Linear Static vs. Nonlinear

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Background
SolidWorks has several different tools available for analyzing structures. Static analysis (discussed in my previous blog article) is available with the purchase of SolidWorks Simulation (included in SolidWorks Premium). Nonlinear analysis is available with the purchase of SolidWorks Simulation Premium (see Figure 1).


Figure 1 – SolidWorks Simulation study types

When a force is applied to a structure (a pencil, a building, a fixture, etc.) the internal pressure in the structure builds up. We call this internal pressure, stress. As stresses rise, the structure gets longer or shorter. We call this elongation, strain. When we run a Static analysis, we are assuming that the relationship between stress and strain is a simple one; we assume the relationship is linear. By making this assumption, we can potentially make an otherwise difficult or sometimes impossible problem very easy to solve.

However, this assumption isn’t all true. In reality, the relationship between stress and strain is never “perfectly” linear. But this assumption is close enough for many scenarios, and the amount of error it introduces may be negligible.

The Test

So what about if the error isn't negligible? What about those scenarios where we should avoid from making the linear assumption? To understand this, I constructed a very simple example model and ran a linear and a nonlinear analysis on it. My linear analysis results looked like what is shown in Figure 2 below.


Figure 2 – Linear Static stress results

I applied a 100lb load to this small model. This load happened to be quite a lot for this structure. You see the amount of stress before the structure yields is about 90ksi (shown below the chart). This “yield strength” is associated to the material I used (i.e., Alloy Steel). From this result, I noted the maximum stress of about 240ksi. Then I applied the same 100 lb load using nonlinear analysis and saw the results shown in Figure 3.


Figure 3 – Nonlinear stress results

I noted that my maximum stress is now much lower at about 125ksi. This is almost half of that reported for the linear analysis (125 divided by 240). Also, it was interesting for me to take a look at how different the way the stress appeared to be distributed. It was almost as if the stress had to figure out a way to redistribute itself along the cross section so it could somehow bear the load. I then ran the test with smaller loads.

With only 10 lbs applied, I hardly noticed a difference in the results when I compared my nonlinear results to my linear results. I did the same with other loads (15, 25, 35, etc.) as well and still didn’t notice much of a difference. So when did this difference begin to emerge? After running more tests, I summarized my results with the graph seen in Figure 4 below.


Figure 4 – Linear vs. Nonlinear graph

Conclusion
It appears that differences begin to emerge only once the stresses have exceeded the yield strength of the material. Not too far beyond the yield strength, it seems to me that we at least have to consider nonlinear affects if we want accurate stress results. Also, the graph shows that displacements appear to diverge as well, but they appear to take longer to do so in this particular example model.

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Shaun Bentley is a SolidWorks Elite Application Engineer at DASI Solutions, a SolidWorks Value Added Reseller with locations throughout Michigan and Indiana. He is a regular contributor to the DASI Solutions Blog.