mathias

Determination of the fundamental period of vibration (T1) of a building

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Is there any mathematical formula for the approximate calculation of the fundamental period of vibration of a building?

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Ct=0.085 for moment resistant space steel frames,

Ct=0.075 for moment resistant space concrete frames and for eccentrically braced steel frames and

Ct=0.050 for all other structures.

H is the height of the building, in m, from the foundation or from the top of a rigid basement (EN 1998-1 section 4.3.3.2.2).

We can use this approximation for buildings with heights up to 40m provided that they follow the regularity criteria that are described in table 4.1 (EN1998-1 section 4.2.3.1).

Edited by alberto

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An empirical type we use (for concrete structures) is T=n/10 where n=number of stories (typical storey~3m).

So for a 10 storey building T=1 sec.

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There are also a couple of approximation formulae in EC1 (Wind).

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An estimation of the fundamental period can also be made by the following expression: T1=2*(d)^0.5.

d is the lateral displacement of the top of the building in meters due to the gravity loads applied horizontally.

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How does this formula come from?

It seems like a performance based design equation but in that case formula is a little bit different.

(I think T=δ/ks where δ=displacement target for mass center, not displacement at the top.)

T1=2*(d)^0.5

There is a unit problem with that formula except there is a unit decision in advanced.

Edited by rigid_joint

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I really don't know what is the theory behind this expression...

I will search through the internet and my notes... Probably I will find something!

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There is a unit problem with that formula except there is a unit decision in advanced

The unit of the result is in seconds.

This expression works the same way as the formula described by Eurocodes.

I asked it just because I think that I have never seen it before.

No problem at all!

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The unit of the result is in seconds.

And δ in centimeters?

Anyway, It's a strange formula because somebody has to know what δ value is. Only in performance based design seems that it could be used, but in that case formula involves δ and k (stiffness).

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"d is the lateral displacement of the top of the building in meters due to the gravity loads applied horizontally."

Anyway, It's a strange formula because somebody has to know what δ value is.

Yes, I agree!

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The following image describes how this formula works! Unfortunately I couldn't find more information about it...

post-1-1457008957839_thumb.jpg

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I guess it's a simple variation of T=2π(Μ/K)^0.5 for a SDOF system.

F=Kd, F=mg and (g)^0.5=π.

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Marios has right. In general is an equation without any special meaning. It's just for check.

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An estimation of the fundamental period can also be made by the following expression: T1=2*(d)^0.5.

d is the lateral displacement of the top of the building in meters due to the gravity loads applied horizontally.

I just found this formula in EN 1998-1-1 clause 4.3.3.2.2 (5)!

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Equation 4.9. But, it seems that it needs much work to take T when at the same time, you can have it with modal analysis faster and accurate.

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This one shouldn't be used in Europe, because it was formulated on the basis of experimental data of Californian Earthquakes, and California is a high seismic zone. The "formula" that you should use is the Rayleigh formula (you can find the formula on Dynamics of structures books). The masses and displacements there should be, in an approximate way, the gravitational masses in each floor. Don't know if I made my point!

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