presentacion dos anil k chopra

1

Dynamics of Structures

Lecture 2

Anil K. Chopra

Electricity Generating Authority of Thailand

Bangkok, Thailand

December 7-8, 2010

EGAT Thailand Bangkok, Thailand

EGAT Thailand Bangkok, Thailand 2

Multi-Degree-of-Freedom (MDF) System


Multi-Degree-of-Freedom (MDF) System


MDF System


Equations of Motion

+ + = - gu tmu cu ku m1

= displacement vector ( 1)Nu

= mass, stiffness, damping matricesm, k, c

= influence vector1


Free Vibration

Natural Frequency (Period) of Vibration

EGAT Thailand Bangkok, Thailand 7EGAT Thailand Bangkok, Thailand 7

2secn

n

T

rads/secn

k

m

cps (Hz)

2n

nf


Free Vibration: Mode 1

1 1

1

2Mode Shape : Period:T


Free Vibration: Mode 2

2 2

2

2Mode Shape : Period:T


Eigenvalue Problem

N solutions such that

Eigenvalues and Eigenvectors

Natural period of vibration,

Natural mode of vibration,

2n n nk m

2n n

2n

n

T

n


Example: 5-Story Shear Frame

m = 100 kips/g k = 31.54 kips/in.


Vibration Periods and Modes

1

2.0T 2

0.685T 3

0.435T 4

0.338T 5

0.297 secT


Vibration Periods and Modes


Millikan Library, Caltech


Vibration Generator


Frequency Response Curve


Period & Damping Ratio: N-S Direction

Excitation Roof Acceleration (g) Period, sec Damping, %

Vibration Generator 5x10-3 to 20x10-3 0.51-0.53 1.2-1.8

Lytle Creek Earthquake

0.05 0.52 2.9

San Fernando Earthquake

0.312 0.62 6.4


Recorded N-S Motions, San Fernando Earthquake


Recorded E-W Motions, San Fernando Earthquake


Relative Displacement of Roof


Folsom Dam, Cailfornia


Vibration Generator on Folsom Dam


Frequency Response Curve

Frequency

Acc

ele

ration


Frequencies and Damping Ratios

Resonant Frequency, HZ

Damping, %

4.65 4.0-6.5

5.46 5.6-8.4

6.24 4.0-8.0

7.16 6.3-8.0

8.00 6.0-8.0

8.87 6.0-8.0


Modal Expansion of eff gt u tp m

1 1

N N

n n nn n

m s m

nn n n n n n

n

LL M

M= m m

n th-mode components

eff,n n g n n nt u tp s s m


Example: Modal Expansion of s m


Modal Response History Analysis (RHA)

Modal Response

Total Response

stn n nr t r A t

st

1 1

N N

n n nn n

r t r t r A t


Response History Analysis: Example

Deformations and Pseudo Accelerations nA t nD t


Response History Analysis: Example

Modal and total responses


Modal Response Spectrum Analysis (RSA)

Peak response estimated directly from response or design spectrum

Peak modal responses

Modal combination rule

stno n nr r A

,n n nA A T

1/ 2

2

1

N

o non

r r


0 0.5 1G

1

2

3

4

5

Displacement/height,%

Flo

or

RHA

RSA

1 Mode

2 Modes

3 Modes

0 0.5 1 1.5 2G

1

2

3

4

5

Story drift RSA

or RHA

,%

Sto

ry

RHA

RSA

1 Mode

2 Modes

3 Modes

RSA v’s RHA: One Ground Motion


0 0.5 1G

1

2

3

4

5

Displacement/height,%

Flo

or

RHA

RSA

1 Mode

2 Modes

3 Modes

0 0.5 1 1.5 2G

1

2

3

4

5

Story drift RSA

or RHA

,%

Sto

ry

RHA

RSA

1 Mode

2 Modes

3 Modes

RSA v’s RHA: Median Response to 20 Ground Motions


Finite Element System


2D Finite Element Systems

Degrees of Freedom

node i:

Displacement vector u:

Equations of motion

i iyxu u

i i i and , 1,2yxu u N

+ + = - -g gu t u tmu cu ku m1 m1 yx x y


Analysis of Finite Element Systems

RHA and RSA procedures valid

Extend methods to include dam-water-foundation rock interaction effects

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