Fundamental Periodof a

Tall Building

An application of linear harmonic motion

Dr. Aamer HaqueJune, 2008

Aon Center, Chicago

H=1136 ft

B=194 ft●No published data found for the fundamental period of oscillation of the Aon Center●Fundamental period can be estimated using model/material assumptions●Will not require any information concerning wind loading conditions

Model Assumptions

Linear Harmonic Motion Periodic motion Requires “spring” constant

Linear Elastic Cantilever Beam Building deforms primarily in cantilever mode Models

Continuous beam model Lumped mass truss model

Effective Moment of Inertia Thin-walled tube Requires assumed wall thickness

Linear Harmonic Motion

F=ma

u2 u=0

= km

F=−ku

a=u

T=2

Newton's 2nd Law:

Hooke's Law:

Angular frequency:

2nd Order differential equation

u0=u0

u0=0

Period:

Initial Conditions:

Solution:

ut =u0 cos t

Cantilever Beam Model

Wind Load Tip Deflection Oscillation

Spring Constant

k=3 EIH 3

Continuous Beam Model Truss Model

k=12.36 EIH 3

Effective Moment of Inertia●The building is not solid; entire cross-section does not resist bending●Assume that the structure can be modeled as a thin-walled tube●Must make an assumption for the thickness of the tube (assume 3 inches thick)

B B1 a

B=194 ft

B1=193.5 ft

a=0.25 ft

B

Moment of Inertia:

I=B4−B1

4

12

I≈1.21×106 ft4

Building Mass

=10 lbft3

m=Vg

= B2 Hg

g=32 fts2

m≈13.36×106 slugs

Assumed specific weight for a steel building:

Gravitational acceleration:

Mass of the building:

Truss Model - Fundamental Period

T=2 =2 m

k

T=2 BH 2 g

13 E

1 I

T≈7 s

The fundamental period of oscillation is about 7 seconds

E=29×106 psi

Truss Model - Dimensional Analysis

=H 2 g

13 E

1 I

= ft2 lbft2

s2

ft ft2

lb 1ft4

= sft

T=2B= 2 Bc

2 B=cT

Define:

Dimension:

Period: c= 1Define:

The parameter c has the units of velocity and is the speed at which “information”travels across the cross section and back during one period.

Does this make any physical sense?

Truss Model - More Dimensional Analysis

T=2 BH 2 g

13 E

1 I

c= E/g

I~B4

T~B H 2 1c

1B4

=B H 2 1c

1B2

T~Hc H

B H/c = Time it takes for sound signal to propagate the height of the buildingH/B = Ratio of building height to width

Period:

Sound speed:

Truss Model - Validation

Data on tall buildings indicate the following approximate relationship:

T=N

0.05≤≤0.15

N = Number of floors

N=83

4.15 s≤T≤12.45 s

Thus T=7 s is reasonable!

Conclusions

Models provide “back on the envelope” calculations Continuous beam model would compute a value for

period that is about half that of the truss model Effective moment of inertia highly conjectural More accurate models require large amounts of

numerical computation Many degrees of freedom Nonlinearity Damping

References

Dym, Principles of Mathematical Modeling, 2nd Ed., Elsevier, 2004

Mikhelson, Structural Engineering Formulas, McGraw-Hill, 2004

Newland, Mechanical Vibration, Analysis and Computation, Dover, 2006

Taranath, Structural Analysis and Design of Tall Buildings, McGraw-Hill, 1988