Mode Shapes and Mass Participation

I should note that I deleted the old post...

The bullet points you listed sound correct to me.

The mass 90% participation is cumulative of all the modes that are examined. For example, if you cut off your modes at 33 Hz, you might miss some mass that is excited at a frequency of 40Hz, another mass that is excited at 50Hz, etc. The bulk of the building (say 60%) might participate in a sway mode at a low frequency; but then smaller masses (like a stiff cantilevered beam that is 0.5% of the building mass) might oscillate by itself at a much higher frequency where the bulk of the building does not oscillate. So cumulatively, with those two modes you have 60.5% mass participation.

- The number of modes a structure has is a function of the number of joints and degrees of freedom of those joints.

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Correct.

- Lower mode shapes tend to capture the structure's natural frequencies.

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All mode shapes show the structure's frequencies. But for a realistic model (of a normal building that includes spring constants for the foundations)....the translational modes will typically be the first few modes.

- The lower mode shapes tend to have longer periods.

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Correct

- The lower mode shapes (longer period) have a larger mass participation than higher mode shapes (higher frequency) because these mode shapes are closer to a structure's natural frequency.

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For most structures we deal with....(loosely speaking, overall) that is correct. But there are individual elements (foundations, beams, non-structural items, etc) that have very high dominant modes. I did (for example) a fan sitting on top of a thick pile cap (with piles put into bedrock) a few months back....and the dominant (vertical) natural frequency was very high.

- As you increase the number of modes you look at, you analyze structural behavior that is further away from the structure's primary natural frequencies.

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Again: it depends on the structure. For most low-rise buildings we deal with, the overall natural frequencies tend to be low. (Usually 1-10 Hz.)

Something I don't understand is, in analyses I've seen so far, you want to increase the number of modes examined to ensure that you capture at least 90% of mass participation. Does this mean that at higher frequencies, more of the mass of the structure is participating?

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Yep.

@WARose
@bones206

Thanks you both!

I'd reiterate most of WARose's comments. Some additional comments of my own.

- The lower mode shapes (longer period) have a larger mass participation than higher mode shapes (higher frequency) because these mode shapes are closer to a structure's natural frequency.

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I'd take some issue with your wording, though it might seem like I'm arguing over semantics. Even higher modes represent natural frequencies of the structure. I prefer to say that more of the mass participates in the response of the lower modes than in the higher order modes. That's still not a theoretically rigorous definition.

Really the idea of mass participation is a convenient way to manipulate the individual "modal response factors" so that you can compare them with each other and so that the total (if all modes were considered) would always add up to 100%. Simply put, a mode with a larger mass participation means that the mode (usually) has a larger effect on the dynamic behavior of the structure.

Something I don't understand is, in analyses I've seen so far, you want to increase the number of modes examined to ensure that you capture at least 90% of mass participation. Does this mean that at higher frequencies, more of the mass of the structure is participating? Or is it cumulative?

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This is the one spot where I disagree with your understanding (and with WARose's response).

Generally the 90% requirement is usually a requirement for a response spectra analysis. The idea is that you should capture enough modes that you can demonstrate that you've caught enough of the dynamic response for your Response Spectra Analysis to be accurate. For vibration design this is not generally important.

Honestly, the newer Seismic codes (ASCE-7 2016) don't put as much emphasis on the 90% number. Instead say you need to capture enough modes to demonstrate that the "missing modes" are in the rigid response. If so, then you can use a simplified method to capture the missing mass or residual rigid response. The problem (especially with some very rigid structures like you see in nuclear work) is that you may find it very easy to get to 75% with just a few modes. But, it might take hundreds of very rigid modes to get from there to 90% mass participation. If those modes are in the rigid response range they offer very little value for a seismic analysis. Therefore, they can be replaced more efficiently with a single "residual mass" mode or correction vector.

MagicFarmer said:

- The lower mode shapes (longer period) have a larger mass participation than higher mode shapes (higher frequency) because these mode shapes are closer to a structure's natural frequency.

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Right or wrong, I've always thought of it more from a work/energy standpoint. By that I mean which mode shape would it take the least amount of effort to deform the structure into.

Think of structure below as a yard stick. Very easy to bend it in its first mode shape, harder to bend in second mode shape, really hard to reverse curvature multiple times and get to third and higher mode shapes.

":bones206 (Civil/Environmental)20 Jul 18 17:01
The bullet points you listed sound correct to me. "

They are not

"
- The lower mode shapes (longer period) have a larger mass participation than higher mode shapes (higher frequency) because these mode shapes are closer to a structure's natural frequency."

No. All the modes you get are at the natural (resonant) frequencies.

"- As you increase the number of modes you look at, you analyze structural behavior that is further away from the structure's primary natural frequencies."

Define 'primary natural frequencies'. I have been involved in dynamic analysis for 35+ years and have never heard that expression.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

@GregLocock

My apologies for not using more precise language. What I meant by 'primary natural frequency' was frequencies which have the highest mass participation. The point I was circling was, as the frequencies increase, less of the mass participates, and you may be exciting specific structural elements, or segments of structural elements (like a joist chord, say) and not the majority of the mass of the structure.

For structural engineers designing buildings, the terms "natural frequency" and "fundamental period" have generally understood meanings. In fact, those terms are used a lot in the ASCE 7 seismic chapters. Without parsing terminology to death, I thought the OP's understanding of these concepts for typical building analysis are pretty much correct.

Nobody said they weren't understood. "Primary natural frequency" was a term that was used I was asking for clarification on.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

Greg -

In our building codes, we structural engineers tend to butcher the terms for structural dynamics. Mostly because buildings behave (in general) much more simply when subjected to earthquakes than mechanical components due when subject to excitation.

Our codes often refer to a single "Fundamental Frequency" for each direction. Hence the reason the term "primary natural frequency" didn't seen unreasonable to a lot of us. It's so close to the relatively poor term in our code that many of us knew exactly what he meant.

For those structural engineers (like me) who really like dynamics, the code terms are imprecise and somewhat inaccurate. Though this is really associated with the Equivalent Lateral Force method. And, a basic assumption with that method is that a single mode dominates the behavior in each direction.

I've taught a many on using RISA's response spectra analysis features over the years. I spend a good amount of time explaining how the code's ELF terms can be misleading. But, then also pointing out the places where the code identifies when other modes may have a significant effect (torsional, mass and stiffness irregularities) and how this then triggers a dynamic analysis instead of ELF.