How to define a constant 5% damping coefficient? 2

[zhddz]

I want to do a Nonlinear Time History Analysis to a building. Because I have non-zero initial conditions, I have to use the Direct Integration method. The problem is that for direct integration method, I do not know how to set a constant damping coefficient of 0.05.
In the following form, what are the numbers I should put under the Mass Proportional Coefficient and the Stiffness Proportional Coefficient boxes to get damping coefficient of 0.05? Or I should use another way to define the damping coefficient?

I tried to put 0.05 in both boxes, it doesn't work. I got overdamped structure.

 

[sebivaq]

Dear zhddz,
SAP 2000 uses "Rayleigh Damping". It considers a mass-proportional and a stiffness-proportional damping in order to achive the real damping (lets say 5%). These two values are the coefficients ao and a1.

C = ao.M + a1.K

Where:
C = damping matrix
M = mass matrix
K = stiffness matrix

You can find the theory in Chopra - Dynamics of structures (Ch. 11), watch the attach, or in another engineering book. Best regards,

Sebastian

 

[zhddz]

Dear Sebastian:
Thank you so much for your help, and your answer is very useful.
I read the Chopra's book you attached, and there is a very nice example of solving a0 and a1. But the problem is that the example structure has point masses, and the stiffness is also easy to calculate; my structure is a real life structure. It is too complex to get the stiffness matrix, mass matrix or the natural frequency by hand.
Without M, K and ?n, I still cannot solve a0 and a1 , so what should I do? Please help me again. Thanks.

Regards,

 

[sebivaq]

Dear zhddz,
with SAP 2000 you can easily find out the first two natural frecuencies (without any damping). Using the solution of Chopra, you can calculate ao and a1, even if you don't known the mass or the stiffness matrix. Regards,

Sebastian

 

[zhddz]

Dear Sebastian:
Thanks again for your response.
You said I can easily find out the first two natural frequencies with SAP2000; did you mean the natural frequency from the first two modes of modal analysis? Following graph is the result of my modal analysis.

If ?0 = 0.5807rad/s, ?1 = 1.1675rad/s as shown above, using Chopra’s method solving matrix, I got a0=0.0388, a1=0.0572. Then, I did the dynamic analysis again, still got an overdamped result (which is impossible for 5% damping ratio).
I am not doing the dynamic analysis for lateral loads or movements. What I am modeling is the vertical vibration of floors caused by sudden failure of a primary column. I don’t know if Chopra’s example works on my case; is this why I got an overdamped result? Or I made mistake somewhere else?

Best Regards

 

[sebivaq]

Dear zhddz,
Yes, I ment the natural frequency from the first two modes of modal analysis. But for this structure, maybe you need to use another natural frecuency (please read page 420 of the previous attach).

In my opinion the natural periods of the structure are very high for a nine-story building, I would check this too.

Best regards,

Sebastian

 

[sebivaq]

I would search which mode includes vertical displacements, using the animation tool. Then you should use the equation with the first and "the vertical mode". The problem is that for vertical modes the damping could be lower, in my opinion.

 

[zhddz]

Thank you very much for all your helps, Sebastian.
I also asked my professor about this issue. My prof. said there is no way to achieve 5% damping for all the modes, which two modes should be chosen is engineer's judgement.