1

Chapter20. INTRODUCTION TO EARTHQUAKE RESPONSE OF STRUCTURES §20.1 Introduction References

- “Fundamentals of Earthquake Engineering,” N. M. Newmark and E. Rosenblueth, Prentice-Hall, 1971

- “Earthquake Spectra and Design,” N. M. Newmark and W. J. Hall, Earthquake Engineering Research Institute, 1982

- “Handbook of Earthquake Engineering,” editor, R. L. Wiegel, Prentice-Hall, 1970

- “내진설계기준연구,” 건설교통부, 1996.9

• Characteristics of Earthquakes

focus, center, hypofocus, or hypocenter(진원震源):

- The point in the earth’s crust where calculations indicate that the first seismic waves originated

epifocus, epicenter: - The vertical projection of the focus on the earth’s surface

magnitude(규모):

- A measure of the energy released, denoted by M. -Richter’s magnitude scales are used universally

(Reichter, 1958).

)10millimter( a of thousandth one of amplitude an : edisturbanc

the ofcenter the from 100km of distance aat hseismograp Anderson- Wooda by recorded amplitude maximum the :

log

6-0

010

mA

AAAM =

2

ergs in released energy :

5.18.1log10

WMW +=

intensity(진도):

- A measure of the earthquake’s local destructiveness. - One earthquake will be associated with a single magnitude, while its intensity will vary from station to station.

- Modified Mercali(MM) scale is widely used Intensity Scale - Modified Mercalli Intensity(MMI): modifieded by

Wood and Neumann, 1931 - Modified Mercalli Intensity(MMI): modifieded by C. F. Richter, 1956. Adopted by Federal Emergency Measures Agency(FEMA83)

3

strong-motion earthquakes: can cause structural damage .

4

Types of Earth Waves - Primary(P) waves: longitudinal displacements to

the direction of propagation, body waves - Secondary(S) waves: transverse displacements to

the direction of propagation, body waves - Surface(L) waves: includes Love, Rayleigh, and other types - of waves

Characteristics of Earthquakes Frequency Components and Amplitudes

•Fourier Series

- Real Fourier Series: Periodic Excitation - Complex Fourier Series: ” - Fourier Integral: Non-periodic Excitation - Discrete Fourier Series(DFT): ” - Fast Fourier Series(FFT): ”

•Real Fourier Series: Periodic Excitation

∑∑∞

=

∞

=

Ω+Ω+=1

11

10 )sin()cos()(n

nn

n tnbtnaatp (8.2)

∑∞

=

−Ω+=1

10 )cos(n

nn tnca α

where 1

1

2Tπ

=Ω (8.3)

∫+

= 1 )(:)(1

10

Ttpofvalueaveragedttp

Ta

τ

τ

∫+

≠Ω= 1 0,)cos()(21

1

T

n ndttntpT

aτ

τ (8.4)

∫+

Ω= 1 )sin()(21

1

T

n dttntpT

bτ

τ

5

Figure 8.2. Excitation and response spectra based on Example 8.2.

•Complex Fourier Series: Periodic Excitation

∑∞

−∞=

ΩΩ=n

tnin ePtp )( 1)()( (8.6)

∫+ Ω−= 1 1 )(

1

)(1 T tnin dtetp

TP

τ

τ, L,1,0 ±=n (8.8)

nnn PofconjugatecomplexPP ==−* (8.9)

)()(1 1

10 tpofvalueaveragedttp

TP

T== ∫

+τ

τ (8.10)

6

•Fourier Integral: Nonperiodic Excitation

∫∞

∞−

Ω ΩΩ= dePtp ti)(21)(π

(8.23)

∫∞

∞−

Ω−=Ω dtetpP ti)()( (8.24)

•Discrete Fourier Transforms (DFT): Non-Periodic Excitation

∑−

=

−=Ω=1

0

)/2(

1

1,,1,0,)(1)(N

n

Nmninm NmeP

Ttp Lπ (8.42)

∑−

=

− −=∆=Ω1

0

)/2( 1,,1,0,)()(N

m

Nmnimn NnetptP Lπ (8.40)

•Fast Fourier Transforms (FFT): Non-Periodic Excitation

The fast Fourier transform (FFT) is an efficient numerical algorithm for evaluating the DFT.

7

0 10 20 30 40 50 60 70 80 90 100Time (sec)

-3

-2

-1

0

1

2

3

4A

ccel

erat

ion

(m/s

2 )

0 1 2 3 4 5 6 7 8 9 10Frequency(Hz)

0

1

2

3

4

5

6

7

8

Pow

er S

pect

ral D

ensi

ty

(a) El Centro (1940) Earthquake

0 10 20 30 40 50 60 70 80 90 100Time (sec)

-2

-1

0

1

2

Acc

eler

atio

n (m

/s2 )

0 1 2 3 4 5 6 7 8 9 10Frequency(Hz)

0

1

2

3

4

5

6

Pow

er S

pect

ral D

ensi

ty

(b) Mexico City (1985) Earthquake

0 10 20 30 40 50 60 70 80 90 100Time (sec)

-2

-1

0

1

2

3

Acc

eler

atio

n (m

/s2 )

0 1 2 3 4 5 6 7 8 9 10

Frequency(Hz)

0

1

2

3

4

5

6

7

8

9

Pow

er S

pect

ral D

ensi

ty

(c) Gebze (1990) Earthquake

Time-history and power spectral density of earthquakes

8

9

10

11

12

13

14

Dynamic Structural Analysis Procedures

- Modal Analysis Procedure - Equivalent Lateral Force Procedure

§20.2 Response of a SDOF System to Earthquake Excitation: Response Spectra

Figure 20.2. Ground acceleration, velocity and displacement curves for the El Centro earthquake. (D. E. Hudson, et al., Strong Motion Earthquake Accelerograms-Vol. ΙΙ-Corrected Accelerogrogams and Integrated Ground Velocity and Displacement Curves-Part A, Earthquake Engineering Research Lab., California Institute of Technology, 1971.)

15

Response Spectra

Figure 20.3. SDOF system subjected to base motion.

zuwkwwcum

−==++ 0&&&

kzzckuucum +=++ &&&& or (20.1)

zmkwwcwm &&&&& −=++ more convenient (20.3)

zwww nn &&&&& −=++ 22 ωςω (20.3)

By the Duhamel integral General solution of undamped system

tutu

dtpm

tu

nn

n

t

nn

ωω

ω

ττωτω

sincos

)(sin)( 1)(

00

0

++

−

= ∫

& (6.6)

general solution of damped system

16

( ) teuu

teu

dtepm

tu

dt

nd

dt

t

dt

d

n

n

n

ωςωω

ω

ττωτω

ςω

ςω

τςω

sin1

cos

)(sin)( 1)(

00

0

0

)(

−

−

−−

+

+

+

−

= ∫

&

(6.7)

particular solution of undamped system

∫ −

=

t

nn

dtpm

tu0

)(sin)( 1)( ττωτω

(6.6)

particular solution of damped system

∫ −

= −−t

dt

d

dtepm

tu n

0

)( )(sin)( 1)( ττωτω

τςω (6.7)

ττωτω

τζω dteztw d

t t

d

n )(sin)(1)(0

)( −= ∫ −−&& (20.4)

d

tWω

)(= (20.4)

where ττωτ τζω dteztW n

t tn )(sin)()(0

)( −= ∫ −−&& (20.4)

nnd ωζωω ≈=−= 21

n

md

tWwTSω

ζ )(),( max == spectral displacement (20.5)

where )()( max tWtW m =

dnmv StWwTS ωζ === )(),( max& spectral pseudo-velocity (20.6)

Note that for an undamped or lightly damped system

17

max2

max

maxmax 0wu

kwum

nω−==+

&&

&&

For a lightly damped system, the maximum absolute acceleration

vndna SSuTS ωωζ === 2max),( && spectral pseudoacceleration (20.7)

Plots of avd SSS and , , versus of the natural period ( nT ωπ /2= ) of the

system - pseudovelocity response spectrum. - displacement response spectrum, and - pseudoacceleration response spectrum

aan

ds mSSkkSf =

==

2max)(ω

maximum spring(column) force (20.8)

18

19

Figure 20.4. Pseudovelocity response spectrum for

the N-S component of the El Centro earthquake of May 18, 1940. ( “Strong Ground Motion,” G. W. Houser, Earthquake Engineering, R.L. wiegel, ed., Prentice-Hall, Englewood Cliffs, NJ, 1970.)

Sharp peaks and valleys due to local resonances and antiresonances

20

For design purposes, these irregularities are smoothed out and a number of different response spectra averaged after normalizing them To a standard intensity

Figure 20.5. Average velocity response spectrum,

1940 El Centro Intensity. “Design Spectrum,” (G. W.Housner,

Earthquake Engineering, R. L. Wiegel, ed., Prentice-Hall, Englewood Cliffs, NJ, 1970.)

dnv STS ωζ =),(

vna STS ωζ =),(

dndnv STSTS loglog)2log(loglog),(log +−=+= πωζ

vnvna STSTS loglog)2log(loglog),(log +−=+= πωζ

21

vnvnd STSTS loglog)2log(loglog),(log ++−=+−= πωζ

vnvna STSTS loglog)2log(loglog),(log +−=+= πωζ

Tripartite plot of design spectrum:

Plots of avd SSS and , , versus of the natural period ( nT ωπ /2= )

Figure 20.6. Tripartite plot of design spectrum scaled to 20%g at T=0. (G.W. Housner, “Design Spectrum,” Earthquake Engineering, R.L. Wiegel, ed., Prentice-Hall, Englewood Cliffs, NJ, 1970.) Tripartite(traipa’:rtait)

22

Example 20.1

20.6 Fig. Use1500 5% 0.1 lbmgWsT ==== ζ

a. macw

b. sf

Solution

2sec/.in6617.0sec,/.in5.10.,in7.1 ==== gSSS avd

in.7.1max == dSw

lb255)17.0(1500)( max ==

== aas S

gWmSf

lb255)( max =sf

23

Response of Continuous Systems Discretized as SDOF Systems

Figure 20.7. Earthquake excitation of a SDOF generalized-coordinate cantilever column. Principle of Virtual Work

inertianc WVWW δδδδ +−=′ (20.10)

wdxwEIVL

δδ )(0∫ ′′= (20.11)

wdxzwAWL

δρδ )(0inertia ∫ +−= &&&& (20.12)

)()(),( twxtxw ψ= (20.9)

0)( =++ wzkwwm δµ&&&& (20.13)

µzkwwm &&&& −=+ (20.14)

24

∫

∫′′=

=L

L

dxEIk

dxAm

0

2

0

2

)(ψ

ψρ (20.15)

µzpeff &&−= effective force (20.16)

∫=L

dxA0

ψρµ earthquake participation factor (20.17)

µzkwwcwm &&&&& −=++ (20.18)

zm

www nn &&&&&µωζω −=++ 22

vn

d Sm

Sm

w

=

=

ωµµ

max (20.19)

definition of effective acceleration

),(2 txww ne ω=&& (20.20)

)()(2 twxw ne ψω=&& (20.21)

Figure 20.8. Effective inertia loads on a cantilever column.

25

Base shear

)()()(),0( 2

0

2 twdxxAtwtS n

L

n µωψρω == ∫ (20.22)

max2

max ),0( wtS nµω=

From (20.19)

adn Sm

Sm

tS

=

=

22

max ),0( µµµω (20.23)

26

Example 20.2 A uniform cantilever column

Wweight 5% 0.1 === ζsT 2

)(

=

Lxxψ

a. maxw

gSS ad 17.0sec,/.in5.10Sin.,7.1 v ===

dSm

wtLw

=≡µ

maxmax ),( (1)

∫=L

dxAm0

2ψρ (2)

∫=L

dxA0

ψρµ (3)

== ∫ 5

10

2

gWdx

gLWm

Lψ (4)

== ∫ 3

10 g

WdxgLW L

ψµ (5)

)7.1()5/1()3/1(

max =w (6)

.in8.2max =w (7)

b. ),0(max tS

)17./0()/)(5/1()/()3/1(),0(

222

max ggWgWS

mtS a =

=

µ (8)

WWtS 094.0)17.0(95),0(max =

= WtS %4.9),0(max = (9)

27

§20.3 Response of MDOF Systems

Figure 20.9. Multistory building subjected to earthquake excitation.

0kwwcvm =++ &&& (20.24)

z1+= wv (20.25)

1

==

11111

1

)(pkwwcwm teff=++ &&& (20.26)

)()(p tz1mteff &&−= (20.27)

28

∑=

=Φ=N

rrr t

1)(w ηφη (20.28)

)(12 2 tPM r

rrrrrrr

=++ ηωηωζη &&& (20.29)

)(1m)( tztP Trr &&φ= (20.30)

By analogy with Eqs. 20.16 and 20.17 we can define a modal earthquake

participation factor rµ such that

rTT

rr 11 φφµ mm == (20.31)

)()( tztP rr &&µ= (20.32)

by analogy with (20.2) through (20.4)

)()( tWM

t rrr

rr

=

ωµη (20.33)

where

ττωτ τωζ dteztW rtt

rrr )(sin)()( )(

0−= −−∫ && (20.34)

= ∑

= rr

rrN

rr M

tWω

µφ )(w(t)1

(20.35)

)(2 2 tzM r

rrrrrrr &&&&&

=++

µηωηωζη (20.29)

If )cos()(1

sss

s tZtz θω −= ∑∞

=

&&

∑∞

=

=

1sr

rr M

µη [ ] 2/1222 )2()1( srrsr

s

rr

Z

ζ+− Nr ,..,2,1=

r

ssrr

ωϖ

=

29

Sum of Absolute Maximum Method

),()(wmax rrvrr

rrrt

TSM

t ζω

µφ

= (20.36)

∑∑==

≠N

rrt

N

rr tt

11)(wmax)(wmax (20.37)

Root-Mean-Square(RMS)

2/1

1

2 ]))(max([)(max ∑=

=N

rrtt

tQtQ & (20.38)

2/12

1)],([)(wmax rrv

N

r rr

rr

tTS

Mt ζ

ωµφ

∑=

=& (20.39)

effective modal acceleration

rrer w)w( 2ω=&& (20.40)

contribution of the rth mode to the base shear

)(

)(m1

)(m1

)((t)

2

2

1

2

1

tWM

tWM

t

wmwmS

rr

rr

rr

rrr

T

rrrT

iri

N

rr

N

reirir

=

=

=

== ∑∑==

ωµ

ωµφ

ηωφ

ω&&

(20.41)

),()max2

rrar

rrt

TSM

(tS ζµ

=& (20.42)

2/1

1

22

)],([)max ∑=

=

N

rrra

r

r

tTS

MS(t ζµ

& (20.43)

30

Example 20.3 The four-story building of example 15.1(shown below) Use fig. 20.6

a. ad SSS , , v

b. a root-mean-square estimate of the maximum displacement of the top mass

c. a root-mean-square estimate of the maximum base shear Solution. a.

24.06.103.011.0428.07.207.015.0330.00.414.021.0228.00.860.047.01

)'(sec)/.(.)((sec)r sgSinSinST avdr

b. 2/1

24

1

11 ][)(wmax

= ∑

=vr

r rr

rr

tS

Mt

ωµφ

& (1)

∑=

===N

1ririr

TTrr m11 φφφµ mm (2)

6526.0,3425.15919.1,2565.4

43

21

−=−=−==

µµµµ

2/1

22

22

1

])882.55(642.3

)6.1)(6526.0(1544.0)079.41(367.4

)7.2)(3425.1(9015.0

)660.29(177.2)0.4)(5919.1(1

)294.13(837.2)0.8)(2565.4(1[)max

+

+

+

=&(tw

t

2/11 )00003.00097.07949.0()max +++=&(tw

t

31

.in897.0)max 1 =&(twt

c. 2/1

1

22

][)max

= ∑

=

N

rar

r

r

tS

MS(t µ

& (3)

2/1

2222

2222

]642.3

)386)(24.0()6526.0(367.4

)386)(28.0()3425.1(

177.2)386)(30.0()5919.1(

837.2)386)(28.0()2565.4([)max

+

+

+

=&S(t

t

k3.696)max =&S(tt

32

Example 15.1

tP Ωcos1

3u

4u

2u

1u

23 =m

34 =m

22 =m

./inseck1 21 −=m

.k/in8001 =k

16002 =k

24003 =k

32004 =k

33

−−−−−

−−−

=Φ

=×

=

=

−−−

−−−

=

63688.070797.043761.023506.000000.115859.053989.049655.044817.000000.109963.077910.0

15436.090145.000000.100000.1

882.55079.41660.29294.13

,10

12279.368746.187970.017672.0

3000020000200001

,

73003520

02310011

32 ωω

mk

Solution a.

rrrrTr MKM 2

r ,m ωφφ == (1)

=

23506.049655.077910.00000.1

3000020000200001

23506.049655.077910.00000.1

1

T

M

695.507)87288.2(78.176

87288.2

1

1211

1

===

=

KMK

Mω

34

4.374,1164239.343.736836658.439.191517732.2

695.50787288.2

44

33

22

11

==

====

==

KMKMKMKM

(2)

b.

=

000

P

1P

(3)

PrTrF φ= (4)

14

13

12

11

15436.090145.0

PFPF

PFPF

=

−==

=

(5)

c.

)1(cos)/()( 2r

rrr r

tKFt−

Ω=η (6)

rrr ω

Ω= (7)

d.

∑=

=N

rrr ttu

ˆ

111 )()( ηφ (8)

e.

35

)]79.3122/(1)[4.11374()cos15436.0)(15436.0(

3ˆ

)]46.1687/(1)[43.7368()cos90145.0)(90145.0(

2ˆ

)]70.879/(1)[39.1915()cos)(0.1(

1ˆ)]72.176/(1)[695.507(

)cos)(0.1()(

21

21

21

21

1

Ω−Ω

+

=

Ω−Ω−−

+

=

Ω−Ω

+

=

Ω−

Ω=

tP

N

tP

NtP

NtPtu

(9)

80.2851,402.53,3.1179.44,6486.6,5.0

3

1

=Ω=Ω=Ω=Ω=Ω=Ω

ωω

Constant C in tCPtu Ω= cos)( 11

)10(987.4)10(228.5)10(630.3)10(301.13.1)10(291.3)10(289.3)10(176.3)10(626.25.0)10(604.2)10(602.2)10(492.2)10(970.10

4ˆ3ˆ2ˆ1ˆ

33333

33331

3333

−−−−

−−−−

−−−−

−−−−=Ω

=Ω

=Ω

====

ωω

NNNN

f.

36

§20.4 Further Considerations