Response Spectra Analysis Question 1

[MarkAJohn]

Can someone tell me what the following state means:
“When you perform the response spectra analysis (RSA), at least 90% of the model's mass must participate in the solution.”
It seems to me that the whole structure is in motion in all modes, so that 100% of the mass is involved in any mode. However this is not the correct way of looking at it.

A good reference (not Penzien) or two would be great too.

Thanks,
MJ

 

[GregLocock]

No 100% of the mass does not participate in each mode, in fact the only mode where that occurs is a rigid body mode. There are at least 3 definitions of modal mass I have seen, so I'd look at the maths they use.

Typically we use the modes up to frequency X, where X is set as a compromise between the analysis time and the fidelity. Your source seems to think that a 90% mass participation is reasonable/accurate/feasible/necessary.



Cheers

Greg Locock


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[Ron]

Mark...refer to Charles Dowding's book on Construction Vibrations. Also, Richart, Hall, and Woods....both are excellent references for this.

For pure vibrational spectra, without regard to source, check Bruehl and Kjaer or Krautkramer.

 

[MarkAJohn]

Thanks for the responses and references.

I should have said my main interest is seismic response in buildings. I think the 90% is required by the building code but I can't seem to put my finger on it at the moment.

I have heard the 90% number forever and have seen it reported in the results from finite analysis programs but don't yet have a intuitive understanding of what is being talked about here.

MJ

 

[Qshake]

As Greg notes above, the only mode that has 100% mass participation is the rigid body mode and that is not of interest for most work in CE structures.

The 90% is actually an accumulative amount for a specified number of modes. Most computer programs SAP2000, GTSTRUDL, SEISAB, STAAD show the mass participation and period and frequency for each mode. At the end of an analysis the mass is summed and that is typically compared to the 90%.

Depending on your structures you may find 90% paritipation within 9 modes, some within 12 modes and others at 30 or more.

I don't know if this rule of thumb is presented in a textbook per se but it is usually in the software guidelines. Some books where I would expect to see this are: Chopra's Dynamics of Structures: Theory and Applications for Earthquake Engineering and Mario Paz's Structural Dynamics texts.

Regards,
Qshake

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[delagina]

hi qshake,

i have a question.
i did a response spectra before for seismic of a certain steel structure a long time ago.
and i was told for x direction, z reaction should be close to zero "0" vice versa, but i'm having a hard time trying to achieve that.

what's the best solution for this.

thanks,

 

[bookowski]

Check out Dynamics of Structures by Chopra.

 

[JoshPlumSE]

Mark -

Look at the different mode shapes for the model and you will notice that various nodes experience higher relative deflection for those mode shapes, right? Then a higher percentage of the mass for those nodes is active for those mode shapes.

The mass particpation factor for mode 'k' is usually represented as:

R_k = Summation from i=1 to n where n is the number of nodes in the model of Phi_i*m_i

From my understanding there really is only one way to define the mass participation (as shown above). You will see different equations sometimes because not all the formulations assume a mass normalized mode shape.

Phi_i is the mass normalized model shape (for mode k of course) displacement for joint i.

m_i is the nodal mass for joint i.

You can see how we are basically just multiplying the mode shape by the mass value to come up with a mass participation.

 

[JoshPlumSE]

Delagina -

When you run a response spectra analysis you doing statistical combination of the individual model results.... usually this is based on a SRSS (square root sum of the squares). This is the source of your Z reactions.

An example to explain further:
The spectral accelartions applied to Mode one produce a Z reaction of 5 kips at N1 and -5 kips at N2. That's a net Z reaction of zero kips.

Then mode 2 gives a -1 kip reaction at N1 and a +1 kip at N2. Still a net zero reaction for this mode.

However, when you combine the two modes together, the negative sign is lost.

SRSS_N1 = sqrt (5^2 + -1^2) = + 5.1 kips
SRSS_N2 = sqrt (-5^2 + 1^2) = + 5.1 kips

Now you have a net Z reaction of 10.2 kips.... The key to understanding this is that the magnitude of any individual reaction is correct. However, you just don't know whether that should be viewed as a positive or negative value.

In the case above (where 83% of the base shear is due to mode #1), you could comfortably use mode shape #1 to better understand which values should be positive and which should be negative.

There are programs (RISA-3D in particular) which overcomes this problem by allowing the user to base the sign convention of the RSA results based on the the "dominant" mode.

 

[JLNJ]

If you are trying to determine the natural frequency of a building you want a mode during which most of the building vibrates. The highest frequencies in a building (the modes first reported in some computer output) might be a single beam or floor or wall plate which is vibrating out of plane and all unto itself.

If just one beam or one floor plate or one portion of the building is vibrating then this is not representative of how the entire building will respond in an earthquake. We are most often interested in the global response, not just the local vibration of a few elements.

Sometimes it can be confusing to sort through the computer's output to find the proper fundamental period. You want to look for the mode where at least 90% of the mass is "participating" or moving during that mode's motion.

 

[delagina]

@josh, thanks. i was using risa 3d at the time. honestly, i still dont understand from your explanation. for now i wont try understand it. i'll research more if i encounter a situation where i need to do this again. hopefully never =)

 

[MarkAJohn]

JLNJ,

I can see that, if one beam on the third floor was going nuts, it wouldn't matter much to the total base shear. You can see that with a program like RISA-3D.

However there has to be a better way to say it than "percent of active mass that is participating". After all, if you model a flag pole (cantelever), the whole thing moves in all modes. So, movement is evidently not the same thing as participation.

It seems more like "the percent that each mode that contribute to the total", or something like that.

MJ

 

[Qshake]

on that last post, consider how you model the cantilever's mass over the length of the beam or flagpole. If you lump it all at the top, pending on how much mass and how much stiffness of the pole you have, the mass will participate in most of the lower modes (associated with the natural frequency and a bit above) but when you get into higher modes there will be a mode in which the mass stays in the same location while the pole displaces to adn fro underneath the mass. In that case, the mass participation is very low or near zero. Just an example to illustrate how the mass can be non-participant.

Regards,
Qshake

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