Seismic Design: Mapped Ground Motions and Resonance

[StrEng007]

Regarding seismic design: Structures with high frequencies subjected to seismic vibrations with high frequencies, will exhibit high amplitude accelerations and resonance. Similarly, structures with long building periods subjected to low frequency earthquakes will be impacted more than adjacent buildings which are very stiff and have shorter periods.

How is this accounted for in the seismic procedures in ASCE 7-10?

From my understanding (using a very brief summary):
Seismic ground motion values, Ss and S1, provided in chapter 22 maps will give us the mapped acceleration parameters for buildings in site class B with a 5% damped system. Let's call this lookup process STEP A.

These values are then adjusted for site conditions, with use of site coefficients Fa & Fv, which will increase or decrease the mapped accelerations. With these coefficients and a 2/3 ratio, we arrive at design values SDs & SD1. (STEP B)

From here, using our building period and design values, a design response spectrum can be created. This will allow us to calculate our seismic response coefficient, Cs, based on the building period. (STEP C)

My questions are:
1. At what point is the ground frequency accounted for? There are two frequencies we're interested in, the vibrations in the ground and the vibrations in the structure. When ground motion values are taken from the tables, they list the acceleration of a structure with 5% damping and a building period of .2 seconds and 1 second.

2.Does this mean that Ss is the acceleration of a building with a period of .2 seconds, when subjected to the maximum available frequency that the bedrock in that location has been known to have? So if the site CAN vibrate at a frequency higher than an equivalent .2 second period building, the map provides the higher acceleration of:

a)the resonance between the frequency of the building when it is in unison with the earth's frequency
b)the acceleration response of the .2 second structure when it feels the maximum available frequency in the earth. (I don't see how this could be higher than a, because it wouldn't resonate)

If that's the case, then would California have mostly the same value for S1... assuming a frequency that exceeds the .2 second structure wouldn't impact it as much? Theoretically the acceleration would cap out.

(Recall, the period is the inverse of frequency T=2(pi)/w).

3. It seems like the magnitude of acceleration is adjusted for with the site coefficients Fa and Fv, due to the fact that different soils vibrate at different frequencies. This leaves me questioning WHAT EXACTLY DOES S1 AND Ss REPRESENT?

I've read through several seismic books, and they don't seem to explain this part. Instead, they just show how to use the formulas and calculate the forces required for design, which is fine.

What I'm trying to do, is draw a correlation between which seismic locations with a certain site class will better support a building with a short, medium or long building period, etc.

For a given design response spectrum diagram, will a building with a long period ever exhibit a higher acceleration than a building with a short period? I know this will happen in areas of low earth frequency, but I don't know how to illustrate it, which brings me back to my original question... Where is the resonance accounted for?

Any clarification would be greatly appreciated.

 

[JStephen]

Several years ago, I took this course, found it informative, and recommend it. The advantage of a live seminar like this is that it makes it easy to ask questions:

http://mylearning.asce.org/diweb/catalog/item/id/1669576/q/t=2115&t=2116&n=3&c=79

You're visualizing the earth shaking at a single specific frequency, and wondering what happens when that frequency also happens to be the natural frequency of your building. The problem is that the earth doesn't shake at a single frequency, it's a random vibration problem, with the actual movement including components of a wide range of frequencies. This is handled with a response spectrum representing those movements and the responses at each different frequency. In some cases, you may derive the natural frequency of the structure, and go to that spectrum for the acceleration. In other cases, you may use the building code approach, which is based on standardized shapes of those spectra. As you work through the design equations, you'll find the design forces may be independent of the frequency, or you may have some equation with the natural frequency or the natural period in it. In those latter cases, they are working off of a portion of the assumed spectrum that is being treated as a straight line (on a logarithmic plot, that is).

The idea of an undamped building being subject to a vibration of exactly the natural frequency doesn't happen, exactly, because the building is always damped to some extent, and the earth vibrations always include a wide range of frequency components, not just that one natural frequency- so you don't get the infinite response that you'd expect.

 

[StrEng007]

Thank you for that clear description.

So the spectrum represents the wide range of fluctuating accelerations that could be experienced by a building in a seismic event at any given location. This means the design spectrum is that of the building frequency response, modified for damping and site conditions.

Since this spectra represents the buildings response, where is the frequency of the earth taken into account? Is it calculated INTO Ss and S1? Or is it settled in the site coefficients Fa and Fv?

 

[MrHershey]

We structurals tend to think of earthquakes as a a very simple two-part 'model' of sorts: the ground and our structure. Probably better to think of it as a three-part model though, as there's two components to the ground that we deal with in determining our input to the structure: source and site. Ss/S1 would be on the source portion of that equation, they reflect the intensity of the ground motion in a general sense. Fa/Fv adjust those values up/down based on site, they reflect how the local soil (site) responds to the ground motion generated at the source (Ss/S1). Then we come along and drop our structure on there and the motion transmitted to the structure is combination of the source (Ss/S1) and site (Fa/Fv) parameters.

All of these factors are of course very simplified to allow us to boil free vibration down into a few simple formulae. My opinion is frequency of the earth (at least right at the building site) mostly lies in Fa and Fv. Those are based on the site class, which is determined usually by a shear wave velocity test that tests how quickly motions/vibrations move through soil. Not a direct measure of frequency, but a useful measuring stick as higher frequency (stiffer) materials tend to transmit vibrations a lot quicker than lower frequency (less stiff) materials. Physical example of this is to imagine like a wave pool at a water park. If you generate a wave at one end, it'll usually take a few seconds for it to get to an observer at the opposite end of the pool. Now replace the water with a solid block of ice that is much stiffer and generate the same 'wave' motion (assume ice is strong enough to not crush). Since the material is much stiffer it takes much less time to respond to the motion and the observer at the other end will get bombarded with ice more or less immediately rather than having to wait a few seconds like they did when it was water.

 

[Shu Jiang PEng SE]

Design response spectrum is a single degree of freedom structure response at given conditions: Soil classification, seismic data and structure frequency. Ground movement, structure damping and structure period are all integrated into the curve. “Dynamics of Structures” by Ray W. Clough is good reference

For equivalent lateral force procedure: The structure is treated as a single degree of freedom. Structure response can be determined per the design spectrum once the structure fundamental frequency is known.

Modal response spectrum analysis: The structure is decomposed into several single degree of freedom structures with different frequencies. Each structure response can be derived per the equivalent lateral force procedure and then combined the responses together according to different methods.