Riddle-like question 3

[BeFEA]

Hello! I have a question for you. I am not sure myself about the answer, but I have some analysis results that make me totally confused. Therefore, the question will look like a quiz-show question or a riddle. Here it goes:

- Let's consider two buildings with the same fundamental period, say 0.5s. One of them has 3 stories, the other has 6 six stories and they both have the same structural system type (say, moment resisting frames). In which of the buildings would we have larger interstorey drift ratios when the two buildings are subjected to the same ground motion accelerogram? Let's assume that the mass is uniformly distributed along the height.

 

[atrizzy]

Sorry for the road block this early but... how would two buildings of such different heights with the same SFRS possibly have the same period?

 

[TehMightyEngineer]

I think he means same type of system but with different stiffness.

Professional Engineer (ME, NH, MA) Structural Engineer (IL)
American Concrete Industries

https://www.facebook.com/AmericanConcrete/

 

[TehMightyEngineer]

I'm going to answer this off the top of my head and say that the drift is the same. Drift doesn't change with mass if I recall correctly and with the same period the drift should be the same assuming the same seismic event.

Professional Engineer (ME, NH, MA) Structural Engineer (IL)
American Concrete Industries

https://www.facebook.com/AmericanConcrete/

 

[atrizzy]

Without working it out, I'll say it's the same drift for both buildings.
Now enlighten us as to how that is!

 

[JGard1985]

Since I'm waiting for IT to repair my e-mail server and website...Here goes

Given natural period/frequency remains the same, the ratio of K/M or M/K must remain constant.

If we double the height, we double mass

To keep the same K/M ratio the 6-story building must be twice as stiff as the 3-storey building.

Since P=Kx and we have same fundamental forcing frequency (and thereby acceleration) our acceleration is the same.

3-Storey Building (Say P=weight of 3 storey building*spectral acceleration, K=lateral stiffness of 3-storey building)

P=K*x

x=P/K

6-Storey Building (Say P=weight of 3 storey building*spectral acceleration, so 2P=weight of 6 storey building*spectral acceleration, )

2*P=2K*x

x=P/K

Storey Drift Will Be the Same.

How'd I do?

Jeff
Pipe Stress Analysis Engineer

http://www.xceed-eng.com

 

[BeFEA]

Thank you for your quick replies. The two buildings can have the same fundamental period, because like TehMightyEngineer said, they can have different stiffness. I added the detail regarding the "structural type" because I would like to say that the mode shapes are similar for the two buildings (this would not be the case if we compared a shear wall building with a frame building). As I said, I don't know the correct answer. I am afraid I might make you biased if I give you the results I have obtained from my trial analyses. If you insist, and if it makes the discussion better, I will give you my preliminary results in another post.

 

[BeFEA]

Thank you for your time JGard1985. I am afraid you are over-simplifying the problem. You are neglecting the distribution of the inertia forces along the height, you are neglecting the contribution of higher modes etc. And don't forget, we are talking about drifts, not the roof displacement. If we consider the distribution along the height, if the base shear is the same for both buildings, the story shear forces must be smaller in the taller building. If this is true, than the drifts are smaller in the taller building. Do you think this is correct?

 

[KootK]

Fun. I feel that we may need to know something about the mass distribution though. Is Jeff's assumption about identical story masses correct? Additionally, what kind analysis are you performing? Response spectrum? Time history? Full building model or SDOF approximation? Linear elastic? Non-linear cracked?

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[JGard1985]

Yes sorry. If roof drift is the same, I agree that intra-story drift will be less in the taller structure (more storeys).

I don't beleive that the base shear is the same for both buildings, since with a fundemntal period that is identical (equal spectral acceleration), the taller structure will have a greater base shear as it is heavier.


Jeff
Pipe Stress Analysis Engineer

http://www.xceed-eng.com

 

[Ron]

Thinking along same line as KootK....if you double the height, you don't necessarily double the mass. In some cases, depending on the frame system, the mass might go more than double....in some cases less. Is it enough to change the drift either way? Who knows....need to know more about the mass. I doubt there would be much drift difference either way.

 

[KootK]

I thought we were talking about two buildings for which only first mode behaviour was important. Once higher modes are considered, the buildings effectively no longer have the same period even if their first mode periods match. In that case, response then depends on the frequency content of the particular seismic record and making predictions gets pretty tough, at least for humans.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[BeFEA]

Sorry for not pointing it out earlier, but yes, let's assume the mass is equal in each storey. I am trying various types of analyses, and I get the same result in all. Linear and nonlinear time history analyses, as well as response spectrum analyses show that the drifts are smaller in the taller building. Do you guys think this is by chance? I have tried this only for two building frames. I can't generalize this conclusion, that's why I am asking here (those that don't get angry by posts like this).

 

[BeFEA]

KootK, the first mode is predominant in the building under consideration. However, I would not like to narrow down the discussion to my specific case. I am trying to figure out if there is ground for generalization of the problem.

 

[drawoh]

Given natural period/frequency remains the same, the ratio of K/M or M/K must remain constant.
...

But the densities of the buildings do not have to be constant. The shorter building must be heavier and/or less stiff. The shorter building may have mass distributed towards the roof. I would say that there is no reason to assume the same drift. I would not go any further than that.

--
JHG

 

[MotorCity]

So here is my take. The fact that the buildings have the same period does not mean they necessarily have the same drift. The drift is determined by stiffness, so while it may take the same amount of time for the building to sway one cycle, each building may sway over a different distance. In other words, one building may be accelerating or "drifting" faster or slower than the other.

 

[BeFEA]

But the densities of the buildings do not have to be constant. The shorter building must be heavier and/or less stiff. The shorter building may have mass distributed towards the roof. I would say that there is no reason to assume the same drift. I would not go any further than that.

In order to be able to solve the problem, let's focus on buildings with uniformly distributed mass along the height. Even simpler, let's focus on buildings with the same mass in each storey.


Are the drifts in the taller building always smaller?

 

[BeFEA]

I need to make another clarification. When I say drift, I mean interstorey drift ratio.

 

[structSU10]

So I took my mathcad sheet for equivalent lateral force procudure, and did a rayleigh period approximation based on the two cases:

3 stories all stories mass 10 kip
6 stories all stories mass 10 kip

both assumed to have a period equal to 1 second

based on the force distribution from the equivalent lateral force procedure, the cumulative drifts are as follow to confirm the 1 second period with the rayleigh method:

3 story building 0.25in, 0.5in, 0.75in
6 story building .125in, 0.25in, 0.375in, 0.5in, 0.625in, 0.75 in

so the interstory drift of the 6 story building is half that of the 3 story building, and they have the same total drift.

This is all based on the very simplified ELF procedure of ASCE, and the quite simplified rayleigh method to estimate fundamental period. To say it could be a general rule from this, I'm not so sure.

 

[BeFEA]

This is all based on the very simplified ELF procedure of ASCE, and the quite simplified rayleigh method to estimate fundamental period. To say it could be a general rule from this, I'm not so sure.

Thank you for your precious time. I ran time history analyses (linear and nonlinear) and they all showed that the drifts in the building with 6 stories are smaller.

Now I invite you all to find a single example/scenario in which the drift ratios would be larger for the taller building, and maybe we can all share a Nobel prize afterwards (just kidding, of course).