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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