presentacion dos anil k chopra
TRANSCRIPT
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Dynamics of Structures
Lecture 2
Anil K. Chopra
Electricity Generating Authority of Thailand
Bangkok, Thailand
December 7-8, 2010
EGAT Thailand Bangkok, Thailand
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Equations of Motion
+ + = - gu tmu cu ku m1
= displacement vector ( 1)Nu
= mass, stiffness, damping matricesm, k, c
= influence vector1
Natural Frequency (Period) of Vibration
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2secn
n
T
rads/secn
k
m
cps (Hz)
2n
nf
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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
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Vibration Periods and Modes
1
2.0T 2
0.685T 3
0.435T 4
0.338T 5
0.297 secT
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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
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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
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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
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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
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Response History Analysis: Example
Deformations and Pseudo Accelerations nA t nD t
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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
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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
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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
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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