Initial tests of damping type combinations

Initially with the 2D frames, tests with material damping, global damping and SIM architecture was run. These were done on eight-node plane stress elements (CPS8R).

These tests showed differing results with the SIM architecture and with the traditional

87 architecture (SIM turned off). With SIM architecture turned off, the structural material damping is ignored, and material and global Rayleigh damping yield identical results when added separately.

SIM turned on will include structural damping and Rayleigh damping added as material damping, as well as global Rayleigh damping. The structural damping and the Rayleigh damping seem to work fine as material damping, but the global Rayleigh damping shows unexpected results. The first mode of the frame was considered. This showed a negative damping ratio for a frequency about 10 Hz, and the three following modes showed positive damping ratios.

This was investigated further by adding only the 𝛼_𝑅-part of the Rayleigh damping, only the 𝛽_𝑅-part, and the same 𝛼_𝑅-part with negative sign and the normal 𝛽_𝑅-part. With only 𝛼_𝑅, all the modes yield negative damping, only 𝛽_𝑅 yield positive results as expected, and with negative 𝛼_𝑅 and positive 𝛽_𝑅 all the modes have positive damping. The results for the last case are not the same as global Rayleigh damping without SIM, but close. The same frame was modelled with Timoshenko beam elements and the results with positive and negative damping ratios did not occur for that model when the same tests was run.

Because of these results all the frames were modelled with traditional architecture, one set with material Rayleigh damping and one set with global Rayleigh damping. The results from these analyses are presented in the following. The frame configurations are named “Configuration #Floors#Spans”, i.e. “#Floors” correspond to the first integer in the names, “#Spans” to the second integer.

In the following, results are only included for the frames with beam spans of 6m. Results for the frames with 12m beam spans were similar.

6.2.2 2D frames – 8-node continuum plane stress elements (CPS8R) – material/global Rayleigh damping

Figure 6.6 and Figure 6.7 display the damping ratio, plotted towards the Rayleigh curve, for all calculated modes for all modelled frame configurations with section spans of 6 meter, and modelled with 8-node plane stress elements (CPS8R). The two figures represent damping modelled as material Rayleigh damping or as global Rayleigh damping – the results were identical between the two damping types, giving the same eigenfrequencies and damping ratios. As the figures display, the damping ratios coincide with the Rayleigh curve for all modes for all frame configurations, irrespective of frame size (meaning number of floors and sections). In general, the highest number of

eigenfrequencies are found for higher frames with more floors.

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Figure 6.6: Damping ratio for all modes for 2D frames with section spans 6m modelled with 8-node plane stress elements (CPS8R) and material/global Rayleigh damping. First 10 of 19 frame configurations. Data plotted towards the Rayleigh

curve.

Figure 6.7: Damping ratio for all modes for 2D frames with section spans 6m modelled with 8-node plane stress elements (CPS8R) and material/global Rayleigh damping. Last 9 of 19 frame configurations. Data plotted towards the Rayleigh

curve.

89 6.2.3 2D frames – Timoshenko beam elements (B21) – global vs material Rayleigh damping

Figure 6.8: Damping ratio for all modes for 2D frames with section spans 6m modelled with Timoshenko B21- beam elements and global Rayleigh damping. First 10 of 19 frame configurations. Data plotted towards the Rayleigh curve.

Figure 6.9: Damping ratio for all modes for 2D frames with section spans 6m modelled with Timoshenko B21- beam elements and material Rayleigh damping. First 10 of 19 frame configurations. Data plotted towards the Rayleigh curve.

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Figure 6.10:Damping ratio for all modes for 2D frames modelled with Timoshenko B21-beam elements and global Rayleigh damping. Last 9 of 19 frame configurations. Data plotted towards the Rayleigh curve.

Figure 6.11: Damping ratio for all modes for 2D frames modelled with Timoshenko B21-beam elements and global Rayleigh damping. Last 9 of 19 frame configurations. Data plotted towards the Rayleigh curve.

91 Figure 6.8, Figure 6.9, Figure 6.10 and Figure 6.11 display the damping ratio, plotted towards the Rayleigh curve, for all calculated modes for all frame configurations with section spans of 6 meter, and modelled with Timoshenko 2D beam elements (B21).

Every other figure display damping modelled as respectively global Rayleigh damping and material Rayleigh damping.

Figure 6.8 and Figure 6.10 show that global Rayleigh damping coincide with the mathematical Rayleigh curve, whereas figures Figure 6.9 and Figure 6.11 show that material Rayleigh damping deviates from the Rayleigh curve. These deviations are apparent for all frame configurations for all eigenfrequencies above 200 rad/s (~30 Hz) and are particularly large in the frequency ranges 300-350 rad/s (~50 Hz) and close to 600 rad/s (~100 Hz), approximately. The trend is larger deviations towards the

Rayleigh curve for increasing frequencies. However, at frequencies 370-380 rad/s a

“jump” in damping ratio appears, after which the calculated damping at first lay closer to the Rayleigh curve until the deviations again increase for increasing frequencies, but at a lower rate than before the damping ratio jump.

Although damping ratios are not the same for B21-element models with material damping and equivalent models with global damping, the calculated eigenfrequencies may at first sight seem to be identical by visual inspection of the plots. This is however disproved when investigating the data sets. Although differences are diminishingly small, the frequencies are in fact not numerically identical. The discrepancies between global Rayleigh damped and material Rayleigh damped eigenfrequencies is likely a result of the damping ratios being different. Since the complex eigenfrequencies consider damping and damping has an influence on damped eigenfrequencies.

In general, most eigenfrequencies are found for the highest frames. For the lowest frame configurations (less than three floors) few if any frequencies are found above 400 rad/s (65 Hz).