Dynamic Analysis Solutions
In this section we will look at three common points of confusion that customers may face when building models for dynamic analysis.
- PROBLEM: Not Enough Member Elements
- PROBLEM: Incorrect Mass
- PROBLEM: Too Few Mode Shapes in Dynamic Response
PROBLEM: Not Enough Member Elements
If you run a static analysis of a typical frame or truss made of member elements, you will get great results without worrying at all about the concept of mesh refinement. When it comes to dynamic analysis however, accurate modeling becomes more difficult.
Consider the relatively simple problem of finding the fundamental frequency of a cantilevered beam. Here is a model that works just fine for a static analysis:
What happens if we run a mode shape analysis on this model?
The mode shape below looks very reasonable and in fact is correct. This is due to VisualAnalysis' use of distributed mass methods. Previous versions of VisualAnalysis simply lumped half the mass at each end of the element giving inaccurate results for non-refined elements. Despite the improved mass distribution, more accurate mode shape frequencies are obtained if the element mesh is refined.
The solution is to split the member elements into pieces. The more pieces you use the more accurate the shape of the vibrating structure will be. This is similar to plate elements during a static analysis where you only get approximate results. To check the accuracy of the model you can run two or more analyses using mesh refinement and comparing the results. When you get close to the theoretical solution, the results will change very little. In the example above the second mode shape frequency was found to be 22.325 Hz for the two element model. Refining the beam by using 5 elements resulted in a frequency of 28.471 Hz. Using 10 elements resulted in 29.784 Hz and using 50 elements resulted in 30.238 Hz. It is clear that not much was gained by refining the beam further than 5 – 10 elements but there was a substantial difference between 2 elements and 5 elements.
PROBLEM: Incorrect Mass
Consider the same cantilevered beam problem. We have now modeled the problem more accurately by splitting each span into multiple member elements.
In addition we have created a dead load case and a live load case with the expected loads on the beam because we remember that the frequency will vary with the mass of the system. Next we run a mode shape analysis and VisualAnalysis calculates the fundamental frequency of vibration.
Unfortunately this is not the correct answer, but VisualAnalysis has no way of letting you know that you have made a mistake. The answer is simply wrong.
The reason the answer is wrong is because the correct mass of the structure was not used. While the mass of the model itself is included, VisualAnalysis does not automatically include loads in the dynamic analysis.
In order to include the dead load and live load mass into the analysis, you need to create a Factored Load Case to combine the two service cases with factors of 1.0 on each. Then when you run the dynamic modal analysis you need to specify this load combination as included mass in the analysis dialog.
PROBLEM: Too Few Mode Shapes in Dynamic Response
When doing a dynamic response analysis of a building frame, the tendency of an inexperienced engineer is to include an arbitrary number of mode shapes and call it good. Part of this reasoning is sound: the response analysis computes dynamic forces by combining the effects of mode shapes. The lower modes however (calculated first) tend to be much more important than the higher modes. The inclusion of too many modes will just increase the analysis time and may not affect the results significantly.
So how do you know if you have enough mode shapes included? Unfortunately there are no warnings or messages in VisualAnalysis. It is up to you to double-check the results.
Building codes typically require that a dynamic response analysis include a certain minimum percentage of participating mass in the response analysis. VisualAnalysis provides a report item called Dynamic Analysis Summary that will show you the mass participation of each mode shape and then total them as a percentage of the mass of the structure. You can then use this to determine if you have enough mode shapes. If not, you will need to increase the number of modes.
Dynamic Analysis Summary
Response Case: Example 10 Mode X-Modal Y-Modal Z-Modal Shape Mass Mass Mass Number Part. (%) Part. (%) Part. (%) 1 0.000 0.000 60.976 2 0.000 0.000 0.000 3 0.000 0.000 32.466 4 80.257 0.000 0.000 5 0.000 0.000 0.000 6 0.000 0.000 6.493 7 15.067 0.000 0.000 8 0.000 0.000 0.000 9 4.606 0.000 0.000
Totals 99.930 0.000 99.934