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.

 

[BeFEA]

Can someone think of a sound reason why the conclusion reached so far (larger drifts in the three storey building) can't hold for the general case?

 

[KootK]

I think that your proposed theory can be proven fairly rigorously given some unavoidable simplifying assumptions. In the "proof" that follows, I've taken a few things as self evident that may require elaboration. If that's the case, just let me know where you have doubts and I'll expand as required.

1) Subscript 1 = short building; subscript 2 = tall building

2) h2 = 2 x h1 [building heights]

3) Replace discrete floor buildings with vertically cantilevered beams with distributed mass m.

4) As is appropriate for short buildings, only consider shear deflection. No flexural straining.

5) F2 = 2 x F1 [total seismic load on each respective building]

6) K2 = 2 x K1 [building stiffness relationship required to produce equivalent periods (K/M ratio)]

7) (GA)2 = 4 x (GA)1 [ratio of beam shear stiffness per unit length based on #6 for each building]

8) Recognize that for a first mode only shear building, the peak story drift ratio (DR) occurs at the base of the building and is equal the the unit shear strain of the substitute beams (F/GA).

9) DR2/DR1 = ((F2 / (GA)2) / (F1 / (GA)1) = ((2 x F1 / 4 x (GA)1) / (F1 / (GA)1) = 0.5 = structSU10's result-ish

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.

 

[KootK]

Another way to get there would be:

1) Accept Jeff's clever proof that the roof level drifts are the same.

2) Recognize that if the the roof drifts are the same then the whole building drift profile for the taller building is a vertically scaled up copy of that for the shorter building.

3) Recognize that the inverse of drift profile slope = story drift ratio so #2 implies that the taller building has a lower drift ratio on all floors.

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.

 

[KootK]

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

The base shear coefficients would be the same but not the actual base shears.

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]

But wait, why is #5 true? Since both buildings have the same period, the spectral acceleration should be equal for both. You are assuming that the mass of building 2 is twice the mass of buildng 1, right? This doesn't have to be true, am I right?

 

[KootK]

You are assuming that the mass of building 2 is twice the mass of buildng 1, right?

Yup.

But wait, why is #5 true? Since both buildings have the same period, the spectral acceleration should be equal for both.

The spectral acceleration will be the same. But you'll be accelerating twice the mass. Thus twice the base shear.



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.

 

[KootK]

This doesn't have to be true, am I right?

The ratio of the total building masses does not have to be two, or any particular value, so long as the distribution of mass is uniform for both buildings. You could easily repeat my derivation for any ratio of building heights and masses. Or you could repeat it with a variable representing that ration to keep things uber-general.

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]

The two buildings can have the same mass (OK, theoretically) but compensate with stiffness such that both buildings have the same fundamental period... Am I missing something here?

 

[BeFEA]

Sorry, I answered before reading your reply.

 

[sandman21]

A code compliant building with the same lateral system, would have the same interstory drift requirement.

 

[KootK]

Sorry, I answered before reading your reply.

No worries. I'm a little quick on the draw sometimes.

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]

KootK, I am not 100% convinced for your analytical proof yet. Does your proof work if building 1 has distributed mass m1 and buildimg 2 has mass m2 (where m1 is different from m2)? Also, for the 6 storey building I think the highest drifts would be around mid-height.

 

[KootK]

KootK, I am not 100% convinced for your analytical proof yet. Does your proof work if building 1 has distributed mass m1 and buildimg 2 has mass m2 (where m1 is different from m2)?

Sure. The twos and fours just become other numbers accordingly.

Also, for the 6 storey building I think the highest drifts would be around mid-height.

Not possible based on the assumptions that I set forth: first mode, shear deflection only, uniform shear stiffness per unit length (just added that now). To the extent that those assumptions are violated with cantilever action, fixed base plates, etc things may change. That's the thing with rules: they only apply within the limits of when they apply.

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]

I think we reached a conclusion. I think the proof provided by KootK is correct and very nice. I tried to generalize it, and I reached to the conclusion that the drift ratios are related only by the ratio of the building heights, no matter what the mass of each building is! Again, this is valid under the circumstances pointed out by KootK in his proof.

Here it goes:
Let us make the following definitions:
1) α = m₁ / m₂ = k₁ / k₂ (condition for equal fundamental periods)
where m₁ is the uniformly distributed mass of building 1, m₂ is the distributed mass of building 2.

2) β = h₂ / h₁
where h₂ and h₁ are the building heights.

Now, following KootK's formulation, if S is the spectral acceleration (equal for both buildings):
3) F₁ = α m₂ S
F₂ = m₂ S
4) since k₁ = αk₂ (by definition):
GA₁ / h₁ = α GA₂ / (βh₁)
therefore:
GA₂ = βGA₁ / α
5) DR₂/DR₁ = (αm₂S * GA₁ ) / (β GA₁ * α m₂ S) = 1/β = h₁ / h₂.

Besides the simplifications and limitations, I think this is sufficiently correct. Do you have any objection?

 

[KootK]

I bloody love it (no surprise). Thanks for putting in the sweat equity on the generalization of the proof.

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.