REPORT NO. EERC 76-6 MARCH 1976 PB 260 556 EARTHQUAKE ENGINEERING RESEARCH CENTER TIME AND FREQUENCY DOMAIN ANALYSES OF THREE DIMENSIONAL GROUND MOTIONS SAN FERNANDO EARTHQUAKE by TETSUO KUBO JOSEPH PENZIEN Report to the National Science Foundation COLLEGE OF ENGINEERING UNIVERSITY OF CALIFORNIA • Berkeley, California
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REPORT NO.
EERC 76-6
MARCH 1976
PB 260 556
EARTHQUAKE ENGINEERING RESEARCH CENTER
TIME AND FREQUENCY DOMAIN ANALYSES OF THREE DIMENSIONAL GROUND MOTIONS SAN FERNANDO EARTHQUAKE
by
TETSUO KUBO
JOSEPH PENZIEN
Report to the National Science Foundation
COLLEGE OF ENGINEERING
UNIVERSITY OF CALIFORNIA • Berkeley, California
NOTICE
THIS DOCUMENT HAS BEEN REPRODUCED
FROM THE BEST COpy FURNISHED US BY
THE SPONSORING AGENCY. ALTHOUGH IT
IS RECOGNIZED THAT CERTAIN PORTIONS
ARE ILLEGIBLE, IT IS BEING RELEASED
IN THE INTEREST OF MAKING AVAILABLE
AS MUCH INFORMATION AS POSSIBLE.
BIBLIOGRAPHIC DATA 1 1. Report No. SHEET EERC 76-6
3. Recipient's Accession No.
4. Title and Subtitle
"Time and Frequency Domain Analysis of Three Dimensional Ground Motions San Fernando Earthquake"
5. Report Date
March 1976
6.
7. A uthor(s)
Tetsuo Kubo and Joseph Penzien 9. Performing Organization Name and Address
Earthquake Engineering Research Center University of California, Berkeley 1301 S. 46th Street Richmond, California 94804
12. Sponsoring Organization Name and Address
National Science Foundation 1800 G Street Washington, D. C. 20550
15. Supplementary Notes
16. Abstracts
8. Performing Organization Rept. No. --+e-l.Q..".
10. Project/Task/Work Unit No.
11. Contract/Grant No.
GI-36387
13. Type of Report & Period Covered
14.
Principal directions and components are generated for the strong ground motions recorded during the San Fernando earthquake of February 9, 1971. Characteristics of the principal components are investigated using the moving-window technique applied in both the time and frequency domains. A nonstationary random process is defined reflecting these same characteristics in a statistical sense. A computer program for generating principal directions and components of motion, wave-form characteristics, and sample accelerograms from the nonstationary random process is listed.
17. Key Words and Document Analysis. 170. Descriptors
17b. Identifiers/Open-Ended Terms
17c. COSATI Field/Group
18. Availability Statement 19. Security Class (This Report)
UNCLASSIFIED 20. Security Class (This
21. No. of Pages
"-""', Page L-.~~~~~~~~ ________ ~ __________________________ ~~ __ ~U~N~C~L£A~S~S~IF~I~E~D~ __ -L~ __ ~'~-~'~'~'~--J FORM NTtS-35 tREY. 3 72) USCOMM.DC 14952.P72
EARTHQUAKE ENGINEERING RESEARCH CENTER
TIME AND FREQUENCY DOMAIN ANALYSES OF THREE DIMENSIONAL GROUND MOTIONS
SAN FERNANDO EARTHQUAKE
by
Tetsuo Kubo
Joseph Penzien
Report to National Science Foundation
Report No. EERC 76-6
College of Engineering , University of California
Berkeley, California
March 1976
fa.'
ACKNOWLEDGMENT
The study presented herein is an extension of an investigation
previously conducted by the authors in collaboration with Dr. M. Watabe,
Mr. R. Iwasaki, and Mr. K. Ishida at the University of Tokyo. The
authors are indebted to Mrs. Shirley Edwards for her excellence in typing
the original manuscript. The financial support provided by the National
Science Foundation under Grant No. GI-36387 is hereby acknowledged with
sincere thanks and appreciation.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENT . . i
TABLE OF CONTENTS ii
1. INTRODUCTION . . . . .. ~ . 1
A. SCOPE OF INVESTIGATION 1
B. SAN FERNANDO EARTHQUAKE 4
C. STRONG GROUND MOTION ACCELEROGRAMS . 5
II. PRINCIPAL AXES FOR GROUND MOTION 29
A. FIXED PRINCIPAL AXES . . 29
B. TIME DEPENDENT PRINCIPAL AXES 35
C. MAXIMUM VARIANCES AND COVARIANCES 38
III. CHARACTERISTICS OF GROUND MOTION FROM MOVING-WINDOW ANALYSIS 43
A. TIME DOMAIN ANALYSIS 43
1. General 43
2. Results at Station No. 264, Basement of the Millikan Library, CALTECH . . . . • 45
3. Results Through the Time Domain 46
4. Observation of Results 49
B. FREQUENCY Dm-1AIN ANALYSIS 53
1. General 53
2. Results at Station No. 264 54
3. Results Through the Frequency Domain 55
4. Observation of Results .•.. 55
C. MAXIMUM VARIANCES AND COVARIANCES 57
1. General 57
2. Results of Maximum Variances and Covariances 57
iii
3. Observation of Re~3Ults ...••..•.•.
IV. FREQUENCY CONTENT OF GROUND MOTIONS ALONG PRINCIPAL AXES . . . •
58
101
A. GENERAL 101
B. NORMALIZED FOURIER AMPLITUDE SPECTRUM WITH MOVING-WINDOW TECHNIQUE . 103
C. RESULTS AT STATION NO. 264 106
D. RESULTS OF MOVING-WINDOW FOURIER ~WLITUDE SPECTRUM ANALYSIS 107
E. OBSERVATION OF RESULTS 107
V. GENERATION OF THREE COMPONENTS OF GROUND MOTION
A. GENERAL
B. SIMULATION OF A NONgTATIONARY PROCESS
1. Generation of a Rqndom Process Having an Arbitrary Power Spectral Density . . •
2. Generation of a Random Process Having Nonstationary Frequency Content
C. EXAMPLES IN ONE-DIMENSIONAL FOru~ . •
1. Review of Past Simulated Motions
2. Characteristics of Simulated Motions • .
3. Presentation of Examples
4. Observation of Results . .
D. EXAMPLES IN THREE-DIMENSIONAL FORM
1. Characteristics of Simulated Motions •
2. Presentation of Examples
3. Observation of Results ..
ll7
ll7
ll8
ll8
120
123
123
124
127
127
128
128
131
132
VI. CONCLUDING STATEMENT . • • . • • . • . . . . . . . . . .. 148
VII. REFERENCES. • • • • • . . . . . . . . . • • . • . . . .. 149
iv
APPENDIX A Results of (i) time dependent directions of principal axes and square root of principal variances, (ii) frequency dependent directions of principal axes and square root of principal variances, (iii) time dependent principal variances and principal cross correlation coefficients, and (iv) time dependent fre-quency distribution for motions in area F . . . . 0 • • 154
APPENDIX B Results of time dependent directions of principal axes and square root of principal variances at stations not included in area groups A through F. • 164
APPENDIX C Results of frequency dependent directions of principal axes and square root of principal variances for motions in area groups C, D and E ...•.••. 0 185
APPENDIX D Results of time dependent principal variances and principal cross correlation coefficients for motions in area groups C, D and E .......... 0 0 189
APPENDIX E Results of time dependent frequency distribution for motions in area groups C, D and E . 193
APPENDIX F computer program listings . • 197
1
I. INTRODUCTION
A. SCOPE OF INVESTIGATION
Up to the present time, most analytical investigations of the
dynamic response of structural systems subjected to strong earthquake
excitations have considered only one component of ground motion. It is
becoming increasingly evident, however, that responses of some important
structural systems such as three-dimensional piping systems, certain
nuclear power plant components, highway bridge structures and earthfill
dams are significantly affected by more than one component of earthquake
motion. Fortunately, with the recent advances in techniques and
facilities of high speed digital computers, it is becoming possible to
conduct investigations of the dynamic response of such systems con-
sidering the multi-component influence of ground motion. Because of
this awareness, there will obviously be an increasing demand in the
future of dynamic response analyses of selected systems using multi-
components of ground motion excitation.
Ground motion at a point 0 has six components, three trans-
. [* lational and three rotatl0nal 36] . While the three rotational
components, two about horizontal axes (Rayleigh Waves) and one about a
vertical axis (Love Waves), may influence overturing moments and
torsional vibration of structural systems, it is usually sufficien-t to
consider only the three translational components. Presently, the
available earthquake accelerograms are not sufficient to perm:U: a_a
estimate of the rotational components.
A very simple approach to defining the three translational
components of motion would be to assume that certain recorded ground
* Numbers in square brackets refer to corresponding references.
2
motions of a part earthquake are representative of future site ground
motions. This simple approach, however , is subject to question as two
recorded accelerograms even at the same site location often have quite
dissimilar characteristics.
Another approach is to generate accelerograms synthetically
which have proper intensities and appropriate spectral densities.
Recognizing that seismic waves are initiated by irregualr breaks and
slippage along faults followed by numerous random reflections, refractions
and attenuations within the complex ground fOrmations through which they
pass, stochastic modelling of strong ground motions is a realistic form
for practical use. Defining earthquake inputs to a structural system in
this manner has the distinct advantage that analyses yield mean values
and variances of response consistent with the variations to be expected
in ground motion characteristics.
Representive stochastic models for earthquake ground motion
could be established d~rectly by statistical analysis if unlimited data
were available. Unfortunately, strong ground motion data in the form of
accelerograms are quite limited. Therefore, one is forced to hypothesize
model forms and to use the available strong ground motion data primarily
in checking the appropriateness of these forms. A number of stochastic
models, representing both stationary and non stationary random processes,
have been employed. Most of them, however, deal exclusively with only
one translational component of motion [42,32]. One such model, commonly
used in its one-dimensional form [23,37], defines ground accelerations
at a point along three orthogonal axes (x, y and z), usually two horizontal
and one vertical through the relations
3
a (t) I;; x (t) b (t) x x
a (t) = I;;y (t) b (t) (1.1) y y
a (t) = I;;z (t) b (t) z z
where b (t), b (t) and b (t) are stationary random processes and x y z
I;; (t), 1;; (t) and I;; (t) are deterministic intensity functions giving x y z
an appropriate nonstationarity to their respective ground motion processes.
functions,
The use of Eqs. (1.1) requires that the appropriate intensity
1;; (t), I;; (t) x y
and I;; (t) be obtained by statistical analyses z
of real accelerograms and that the realistic power spectral density
functions, or corresponding auto-correlation functions, be established
by similar means for processes b (t), b (t) x y
and b (t). z
When extending
the use of this model to two- or three-dimensional form, the question
immediately arises "Should the components of motions be cross correlated
statistically?". If so, in addition to the power spectral density
functions or corresponding auto-correlation functions, one must establish
appropriate cross-spectral density functions or corresponding cross-
correlation functions for processes b (t), b (t) x y
and b (t).·· z
In this report, applying a procedure similar to the orthogonal
transformation used in stress-state problems, an orthogonal set of
principal axes is defined for three-dimensional earthquake ground motions.
These principal axes are defined along which the components of ground
motion have maximum, minimum and intermediate values of variances and
have zero values of covariances. This property suggests that components
of motions need not be cross correlated statistically provided they are
directed along principal axes, i.e. provided the x, y and z axes in
Eqs. (1.1) are treated as principal axes.
4
In this report, using the concept of an orthogonal set of
principal axes and applying a moving-window technique to the accelerograms
recorded during the San Fernando, California, earthquake of February 9,
1971, analyses of three-dimensional ground motions along principal axes
are carried out. In these analyses, the time-dependent and the frequency
dependent characteristics of the principal values and the corresponding
directions of principal axes of ground motions are determined and time
dependent characteristics of frequency content are examined through a
moving-window fourier amplitude spectrum analysis.
It is concluded that realistic three components of ground
motion can be generated stochastically using statistically uncorrelated
nonstationary random processes along the prinicipal axes provided
appropriate intensity functions and time-dependent frequency characteristics
are used.
B. SAN FERNANDO EARTHQUAKE
The San Fernando, California, earthquake which occurred at
6:00:41.8 a.m. local time on February 9, 1971 has been assigned a
location at 34 0 24' 00" Nand 118 0 23' 42" W, a magnitude of 6.6 on the
Richter scale and a depth of about 13 km. The epicenter of the earth
quake has been located in the San Gabriel Mountains 14 km north of San
Fernando, California. It has been reported that the fault slippage
began at a depth of 13 km and progressed southward and upward at
approximately 45°. A narrow band of surface faulting has been observed
to run east-west in the foothills of the San Gabriel Mountains [3]. It
has been reported that the faults slip zone spread to the south of
epicenter [20] and the "energy center" was located approximately 3 km
southwesterly of the epicenter [14]. The strong motion lasted about
5
12 seconds and a maximum intensity of XI has been assigned to the site of
the Olive View Hospital located north of the Sylmar area [39].
As one can observe in Fig. 1.1, the local geological conditions
around the fault slip zone are quite complex. This complexity is most
likely the cause of weak correlations to exist between certain ground
motion characteristics and the epicenter location as shown subsequently
in this report.
C. STRONG GROUND MOTION ACCELEROGRAMS
The accelerograms used in this investigation were compiled and
issued by the Earthquake Engineering Research Laboratory of the California
Institute of Technology, Pasadena, in a report series entitled "STRONG
MOTION EARTHQUAKE ACCELEROGRAMS: VOLUME II CORRECTED ACCELEROGRAMS AND
INTEGRATED GROUND VELOCITY AND DISPLACEMENT CURVES" parts C through S
[48]. Detailed information, such as directions of accelerometer axes and
fundamental periods and damping ratios of transducers, are also available
in these reports.
Corrections were applied to the recorded accelerograms using a
procedure proposed by Trifunac [46,47,49,50]. In brief, the corrections
were applied to the high and low frequency ranges in such a way that the
resulting accelerograms would correspond to those recorded by accelero
graphs having the characteristics shown in Fig. 1.2. In the first stage
of this procedure, the uncorrected accelerograms were passed through a
Ormsby low-pass filter having a cut-off frequency and roll-off termination
frequency of 25 cps and 27 cps respectively. In the second stage, a base
line correction was performed by passing the accelerograms through a high
pass filter having a cut-off frequency and roll-off termination frequency
of 0.07 cps and 0.05 cps, respectively. In some cases, a cut-off fre
quency of 0.125 cps was used instead of 0.07 cps [17].
6
Hereafter in this report, accelerograph locations are identified
using numbers given in "Annual List of Stations" issued by the Seismological
Field Survey, NOS-NOAA [29]. The locations of accelerograph stations for
which ground motions have been analyzed in this investigation are shown
in Figs. 1.3, 1.4 and 1.5. Figure 1.3 shows the locations of stations in
central and southern California. Figure 1.4 is an enlargement of the
small rectangular area in Fig. 1.3 showing the extended Los Angeles and
San Fernando region. Similarly, Fig. 1.5 is an enlargement of the small
rectangular area in Fig. 1.4 showing the cities of Los Angeles, Hollywood
and Beverly Hills. Table 1.1 summarizes station location, location
coordinates for Figs, 1.3 - 1.5, approximate distances to the epicenter,
directions to the epicenter, building structural types and general site
geology for each station.
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-2
40
.4
-19
1
7-s
tory
RC
2
22
S
ou
th
bu
ild
ing
F
igu
ero
ao
I -----_
.. _------
-----
... _
-----------~-
_l__
___
_ _
__
__
__
_
...
-
SIT
E
GEO
LOG
Y
( )
*1
11
70
0'
± o
f all
uv
ium
(S
)
Mio
cen
e silt-
sto
ne
(I)
70
' o
f all
uv
ium
o
ver
S,O
OO
' o
f se
dim
en
tary
ro
ck
(8)
Gra
nit
ic
(H)
All
uv
ium
(5
)
2S
' o
f all
uv
ium
o
ver
sh
ale
; w
ate
r ta
ble
at
20
' .
(I)
I I i ,
.....
o
TA
BL
E 1
.1
AC
CEL
ERO
GR
APH
S
ITE
IN
FOR
MA
TIO
N
[29
] -
(CO
NT
INU
ED
)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
STR
UC
TU
RA
L
*1
*
11
N
UM
BER
S
ITE
N
AM
E M
AP
KEY
TO
E
PIC
EN
TE
R
TY
PE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
l48
-S0
L
os
An
gele
s,
3 E
-2
40
.4
-18
1
7-s
tory
RC
2
34
S
ou
th
bu
ild
ing
F
igu
ero
a.
lSl-
3
Lo
s A
ng
ele
s,
3 F
-2
41
. 4
-20
IS
-sto
ry S
t.
2S
0 E
ast
bu
ild
ing
F
irst
15
4-6
L
os
An
gele
s,
3 E
-2
41
.1
-19
1
6-s
tory
st.
4
20
S
ou
th
bu
ild
ing
G
ran
d.
15
7-9
L
os
An
gele
s,
3 E
-2
40
.6
-18
3
9-s
tory
St.
4
4S
S
ou
th
bu
ild
ing
F
igu
ero
a.
16
0-2
L
os
An
gele
s,
3 E
-2
40
.6
-18
1
0-s
tory
R
C
S3
3
So
uth
b
uil
din
g.
Fre
mo
nt.
16
3-S
L
os
An
gele
s,
3 E
-3
41
.1
-18
4
3-s
tory
S
t.
61
1
Wes
t b
uil
din
g
Six
th.
I 1
66
-8
Lo
s A
ng
ele
s,
I 3
E-3
4
1.
3 -1
8
8-l
ev
el
RC
, 6
46
S
ou
th
I p
ark
ing
I O
liv
e.
I ~~----
__
L
ram
p.
SIT
E
GEO
LOG
Y
( )
*1
11
2S
' o
f all
uv
ium
o
ver
sh
ale
, w
ate
r ta
ble
at
20
'.
(I)
All
uv
ium
(S
)
Sh
ale
an
d sil
'c-
sto
ne sev
era
l 1
,00
0'
. (
)
Sh
ale
(S
)
All
uv
ium
(S
)
All
uv
ium
(S
)
All
uv
ium
(S
)
I
I I I
I-'
I-'
TA
BL
E 1
.1
AC
CEL
ERO
GR
APH
S
ITE
IN
FOR
MA
TIO
N
[29
] -
(CO
NT
INU
ED
)
STA
TIO
N
APP
RO
XIM
AT
E
DIR
EC
TIO
N
BU
ILD
ING
IDE
NT
IFIC
AT
ION
ST
AT
ION
L
OC
AT
ION
D
IST
AN
CE
TO
ST
RU
CT
UR
AL
*
1
NU
MB
ER
SIT
E
NA
ME
*1
1
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
17
2-4
L
os
An
gele
s,
3 E
-2
40
.4
-19
3
1-s
tory
S
t.
80
0 W
est
bu
ild
ing
. F
irst.
17
5-7
L
os
An
gele
s,
3 E
-3
42
.7
-18
8
-lev
el
RC
80
8
So
uth
p
ark
ing
O
liv
e.
ram
p.
18
1-3
L
os
An
gele
s,
2 D
-4
41
.3
-24
7
-sto
ry
RC
1
64
0
bu
ild
ing
. M
aren
go
.
18
4-6
L
os
An
gele
s,
3 B
-2
37
.9
3 2
7-s
tory
S
t.
19
00
b
Uil
din
g.
Av
enu
e o
f S
tars
.
18
7-9
L
os
An
gele
s,
3 B
-2
38
.5
3 1
9-s
tory
S
t.
19
01
b
uil
din
g.
Av
enu
e o
f S
tars
.
L-
---.----~
--~.---
SIT
E
GEO
LOG
Y
( )
*1
11
Pli
ocen
e
sil
tsto
ne.
(I)
All
uv
ium
(S
)
Ple
isto
cen
e
all
uv
ium
; w
ate
r ta
ble
at
35
'.
(S)
Sil
t an
d
san
d
lay
ers
; w
ate
r le
vel
at
70
'.
(S)
Sil
t an
d
san
d
lay
ers
; w
ate
r ta
ble
at
70
'-8
0'.
(S
)
!
I-'
N
TA
BL
E
1.1
A
CC
EL
ER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
STR
UC
TU
RA
L
*1
*
11
N
UM
BER
S
ITE
N
AM
E M
AP
KEY
TO
E
PIC
EN
TE
R
TY
PE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
19
0-2
L
os
An
gele
s,
2 D
-4
41
. 7
-25
9
-sto
ry
RC
2
01
1
bu
ild
ing
. Z
on
al.
19
6-8
L
os
An
gele
s,
3 D
-2
38
.6
-14
1
2-s
tory
RC
3
34
5
bu
ild
ing
. W
ilsh
ire.
19
9-2
01
L
os
An
gele
s,
3 D
-2
38
.6
-14
7
sto
rie
s;
2 3
40
7
Wes
t st.
an
d
5 S
ixth
. R
C
20
2-4
L
os
An
gele
s,
3 D
-2
38
.5
-13
3
1-s
tory
st.
3
41
1
bu
ild
ing
. W
ilsh
ire.
20
5-7
L
os
An
gele
s,
3 E
-3
43
.2
-14
1
2-s
tory
R
C
34
40
U
ni-
bu
ild
ing
. v
ers
ity
, U
. S.C
.
SIT
E
GEO
LOG
Y
( )
*1
11
Sh
ale
at
east
en
d o
f b
uil
din
g;
8'
of
fill at
west
en
d
(I)
All
uv
ium
(S
)
All
uv
ium
(S
)
Sil
tsto
ne;
wate
r ta
ble
at
base
men
t le
vel.
(I
)
40
0'
of
all
u-
viu
m o
ver
cla
y
an
d sh
ale
; w
ate
r ta
ble
at
27
5'
. (S)
f-'
W
TA
BL
E
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
STA
TIO
N
APP
RO
XIM
AT
E
DIR
EC
TIO
N
BU
ILD
ING
IDE
NT
IFIC
AT
ION
ST
AT
ION
L
OC
AT
ION
D
IST
AN
CE
TO
ST
RU
CT
UR
AL
*
I M
AP *
II
KEY
N
UM
BER
S
ITE
N
AM
E TO
E
PIC
EN
TE
R
TY
PE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
20
8-1
0
Lo
s A
ng
ele
s,
3 D
-2
38
.7
-13
ll
-sto
ry
R
C
34
70
W
ilsh
ire.
bu
ild
ing
.
21
1-3
L
os
An
gele
s,
3 D
-2
38
.6
-13
2
1-s
tory
S
t.
35
50
b
uil
din
g.
Wil
sh
ire.
21
7-9
L
os
An
gele
s,
3 D
-2
38
.5
-12
1
1-s
tory
RC
3
71
0
bu
ild
ing
. W
ilsh
ire.
22
0-2
L
os
An
gele
s,
2 C
-4
29
.4
-6
20
-sto
ry
RC
3
83
8
bu
ild
ing
. L
an
ker-
shim
.
22
3-5
L
os
An
gele
s,
3 D
-2
38
.1
-9
7-s
tory
R
C
46
80
b
uil
din
g.
Wil
sh
ire.
22
6-8
L
os
An
gele
s,
3 D
-l
34
.8
-16
8
-sto
ry
RC
48
67
b
uil
din
g.
Su
nse
t.
-
SIT
E
GEO
LOG
Y
( ) *
III
All
uv
ium
(S
)
All
uv
ium
; w
ate
r ta
ble
at
35
r •
(S)
All
uv
ium
(I
)
Inte
rlay
ere
d
so
ft
san
d-
sto
ne
an
d
sh
ale
. (I
)
All
uv
ium
(I
)
Sh
all
ow
all
uv
ium
ov
er
Mio
cen
e silt-
sto
ne.
(I)
......
>I»
TA
BL
E
1.1
A
CC
EL
ER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
STA
TIO
N
APP
RO
XIM
AT
E
DIR
EC
TIO
N
BU
ILD
ING
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
ST
RU
CT
UR
AL
*
1
*1
1
NU
MB
ER
SIT
E
NA
ME
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
22
9-3
1
Lo
s A
ng
ele
s 2
C-5
5
0.6
-2
7
-sto
ry S
t.
52
60
b
uil
din
g.
Cen
tury
.
23
2-4
L
os
An
gele
s,
3 D
-l
34
.2
-10
1
4-s
tory
St.
6
43
0
bu
ild
ing
. S
un
set.
23
5-7
L
os
An
gele
s,
3 D
-l
34
.2
-10
ll
-sto
ry
S
t.
64
64
b
uil
din
g.
Su
nset.
23
8-4
0
Lo
s A
ng
ele
s,
3 C
-l
33
.5
-8
Il-
sto
ry
R
C
70
80
b
uil
din
g.
Ho
lly
wo
od
.
24
1-3
L
os
An
gele
s,
2 C
-4
21
.1
19
7
-sto
ry
RC
8
24
4
bu
ild
ing
. O
rio
n.
24
4-6
L
os
An
gele
s,
2 C
-5
49
.0
3 1
2-s
tory
R
C
86
39
b
uil
din
g.
Lin
co
ln.
---------
' ~ ~~
~~.
~ -
SIT
E
GEO
LOG
Y
( )
*1
11
All
uv
ium
(S
)
All
uv
ium
; w
ate
r ta
ble
at
55
'.
(S)
All
uv
ium
; w
ate
r ta
ble
at
55
'.
(S)
All
uv
ium
(S
)
All
uv
ium
(S
)
Terr
ace
dep
osit
s--
san
d.
(S)
_ .. -
I I , , I I I
I-'
U1
TA
BL
E
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
(2
9]
-(C
ON
TIN
UE
D)
STA
TIO
N
APP
RO
XIM
AT
E
DIR
EC
TIO
N
BU
ILD
ING
IDE
NT
IFIC
AT
ION
ST
AT
ION
L
OC
AT
ION
D
IST
AN
CE
TO
ST
RU
CT
UR
AL
*
1
*1
1
NU
MB
ER
SIT
E
NA
ME
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
24
7-9
L
os
An
gele
s 2
C-5
5
0.5
-1
1
4-s
tory
RC
9
84
1
bu
ild
ing
. A
irp
ort
B
ou
lev
ard
.
25
3-5
L
os
An
gele
s,
2 C
-4
28
.1
11
1
2-s
tory
R
C
14
72
4
bu
ild
ing
. V
en
tura
.
26
2
Palm
dale
2
D-2
3
2.6
-1
27
S
mall
b
uil
d-
ing
.
26
4-5
P
asa
den
a,
2 D
-4
38
.4
-40
9
-sto
ry
RC
Mil
lik
an
b
uil
din
g.
Lib
rary
, C
.LT
.
26
6
Pasa
den
a,
2 D
-4
34
.7
-36
2
-sto
ry b
uil
d-
Seis
mo
-in
g.
log
ical
Lab
ora
tory
.
26
7-8
P
asa
den
a,
2 D
-4
30
.1
-42
9
-sto
ry S
t.
Jet
Pro
-b
uil
din
g.
pU
lsio
n
Lab
ora
tory
. ~-
SIT
E
GEO
LOG
Y
( )
*1
11
All
uv
ium
(S
)
All
uv
ium
(S
)
All
uv
ium
(S
)
Ap
pro
xim
ate
ly
1,0
00
' o
f all
uv
ium
up
on
g
ran
ite.
(S)
Weath
ere
d
gra
nit
ic.
(H)
San
dy
-gra
vel.
(I
)
I
......
(J)
TA
BL
E
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
ST
RU
CT
UR
AL
*
I *
rI
NU
MB
ER
SIT
E
NA
ME
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
26
9
Pearb
loss
om
2
E-2
4
5.0
-1
06
S
mall
b
uil
d-
ing
.
27
2
Po
rt
1 B
-3
79
.0
69
S
mall
b
uil
d-
Hu
enem
e.
ing
.
27
4
San
B
ern
ar-
1 D
-3
10
7.2
-7
2
6-s
tory
bu
ild
-d
ino
. in
g.
27
8
Pu
dd
ing
sto
ne
2 E
-4
63
.7
-57
S
mall
b
uil
d-
Dam
. in
g.
27
9
Pac
oim
a 2
C-3
7
.2
1 S
mall
b
uil
d-
Dam
. in
g.
SIT
E
GEO
LOG
Y
( ) *
rIr
~---
40
0'
of
all
u-
viu
m o
ver
14
,00
0'
of
sed
imen
tary
ro
ck
. (S
)
All
uv
ium
, >
1
,00
0'.
(S
)
All
uv
ium
, 1
,00
0';
w
ate
r ta
ble
at
30
'.
(S)
Vo
lcan
ic
cla
sti
cs
an
d
intr
usio
ns
wit
h asso
-cia
ted
sh
ale
s.
(H)
Hig
hly
jo
inte
d
dio
rit
e
gn
eis
s.
(H)
f-'
-.J
TA
BL
E
10
1
AC
CEL
ERO
GR
APH
S
ITE
IN
FOR
MA
TIO
N
[29
] -
(CO
NT
INU
ED
)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
ST
AT
ION
L
OC
AT
ION
D
IST
AN
CE
TO
ST
RU
CT
UR
AL
*
I *
II
NU
MB
ER
SIT
E N
AM
E M
AP
KEY
TO
E
PIC
EN
TE
R
TY
PE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
28
0
San
O
no
fre
1 D
-4
13
8.6
-3
4
Sm
all
b
uil
d-
ing
.
28
1
San
ta
Ana
2
E-5
8
7.1
-3
4
3-s
tory
b
uil
d-
ing
.
28
2
San
ta
1 B
-3
13
3.5
9
0
2-s
tory
bu
ild
-B
arb
ara
, in
g.
Un
ivers
ity
o
f C
ali
--fo
rnia
.
28
4-5
S
an
ta F
eli
cia
2
A-3
3
3.3
1
02
E
arth
fil
l d
am;
Dam
. h
eig
ht
20
0',
crest
len
gth
1
,26
0'
•
28
7
San
A
nto
nio
2
F-4
7
1.1
-6
8
Earth
fil
l d
am;
Dam
. h
eig
ht
16
0',
crest
len
gth
3
,85
0' •
28
8
Vern
on
2
D-4
4
8.0
-2
2
6-s
tory
bu
ild
-in
g.
-
SIT
E
GEO
LOG
Y
( ) *
III
Lig
htl
y
cem
en
ted
P
lio
cen
e
san
d-
sto
ne>
3
25
' d
ep
th. (I)
All
uv
ium
(S
)
All
uv
ium
v
en
eer
ov
er
san
dst
on
e.
(I)
San
dst
on
e-
sh
ale
co
m-
ple
x.
(I)
Up
to
15
0'
of
all
uv
ium
ov
er
gra
nit
ics.
(S)
> 1
,00
0'
of
all
uv
ium
; w
ate
r ta
ble
at
>
30
0'.
(S
)
I-' m
TAB
LE
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
ATI
ON
[2
9]
-(C
ON
TIN
UED
)
STA
TIO
N
APP
RO
XIM
ATE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
ST
ATI
ON
LO
CA
TIO
N
DIS
TAN
CE
TO
STR
UC
TUR
AL
*I
NU
MBE
R S
ITE
NA
ME
*II
M
AP
KEY
TO
E
PIC
EN
TE
R
TYPE
EPI
CE
NT
ER
(D
EGR
EE)
(KM
)
289
Wh
itti
er
2 D
-4
52
.7
-37
E
art
hfi
ll d
am;
Nar
row
s h
eig
ht
56
'.
Dam
. cre
st
len
gth
1
4,9
60
' .
290
Wri
ghtw
ood
2 F
-3
70
.1
-86
S
mal
l b
uil
d-
ing
.
29
2-4
H
oo
ver
D
am,
1 G
-l
37
8.1
-1
18
C
on
cre
te d
am.
Nev
.
4ll
P
alo
s V
erd
es
2 C
-5
66
.7
-1
2-s
tory
bu
ild
-E
state
s.
ing
.
41
3-5
L
os
An
gel
es,
3 B
-2
38
.3
0 A
rcu
ate
-ll
77
sh
aped
7
-B
ever
ly
sto
ry R
C D
riv
e.
bu
ild
ing
. I
~ ..
~-.----
---
---
-
SIT
E
GEO
LOG
Y
( ) *
III
Mor
e th
an
1
,00
0'
of
all
uv
ium
. (S
)
All
uv
ium
v
en
eer
on
ig
neo
us
met
amo
rph
ic
com
ple
x.
(I)
Sev
era
l 1
00
' o
f v
olc
an
ic
bre
ccia
o
ver
basalt
. (H
)
Sh
allo
w
Ple
isto
cen
e
san
ds
ov
er
shale
-vo
lcan
ic
com
ple
x.
(I)
All
uv
ium
(S
)
-.
f-J
\D
TA
BL
E
1.1
A
CC
EL
ER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
STR
UC
TU
RA
L
NU
MB
ER *
1
SIT
E
NA
ME
*1
1
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
41
6-8
B
ev
erl
y H
ills
, 3
B-2
3
7.1
-1
1
0-s
tory
R
C
91
00
b
uil
din
g.
Wil
sh
ire.
42
5-7
L
os
An
gele
s,
3 B
-2
37
.5
3 IS
-sto
ry
RC
1
80
0
bu
ild
ing
. C
en
tury
P
ark
E
ast.
42
8-3
0
Lo
s A
ng
ele
s,
3 C
-2
37
.7
-5
3l-
sto
ry S
t.
59
00
b
uil
din
g.
Wil
sh
ire.
43
1-3
L
os
An
gele
s,
3 D
-2
38
.6
-13
1
6-s
tory
R
C
61
6
So
uth
b
uil
din
g.
No
rman
die
.
43
7-9
L
os
An
gele
s 3
E-3
4
1.5
-1
7
10
-sto
ry S
t.
11
50
S
ou
th
bu
ild
ing
. H
ill.
44
0-2
L
os
An
gele
s,
3 B
-2
37
.6
3 1
6-s
tory
S
t.
18
80
b
uil
din
g.
Cen
tury
P
ark
E
ast.
SIT
E
GEO
LOG
Y
( )
*1
11
All
uv
ium
; w
ate
r ta
ble
at
40
'.
(S)
Sil
t an
d
san
d
lay
ers
; w
ate
r ta
ble
at
70
' -8
0'.
(S
)
All
uv
ium
--asp
halt
ic
san
ds.
(I)
All
uv
ium
, sil
tsto
ne at
25
' . (I
)
50
0'
of
gra
vell
y
san
d
ov
er
sh
ale
. (S
)
Sil
t an
d
san
d
lay
ers
; w
ate
r ta
ble
at
I 7
0'-
80
'.
(S)
to
o
TA
BL
E
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9J
-(C
ON
TIN
UE
D)
STA
TIO
N
APP
RO
XIM
AT
E
DIR
EC
TIO
N
BU
ILD
ING
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
STR
UC
TU
RA
L
*1
*
11
N
UM
BER
S
ITE
N
AM
E M
AP
KEY
TO
E
PIC
EN
TE
R
TY
PE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
44
3-5
L
os
An
gele
s,
3 C
-2
37
.6
-5
17
-sto
ry
RC
62
00
b
uil
din
g.
Wil
sh
ire.
44
6-8
L
os
An
gele
s,
3 C
-l
33
.4
-9
22
-sto
ry
RC
1
76
0
No
rth
b
uil
din
g.
Orc
hid
.
44
9-5
1
Lo
s A
ng
ele
s,
3 E
-2
39
.3
-16
1
3-s
tory
RC
2
50
0
bu
ild
ing
. W
ilsh
ire.
45
2-4
B
ev
erl
y H
ills
3
B-2
3
5.8
-1
1
0-s
tory
RC
4
35
N
ort
h
bu
ild
ing
. O
ak
hu
rst.
45
5-7
B
ev
erl
y H
ills
3
B-2
3
6.9
2
10
-sto
ry
RC
4
50
N
ort
h
I b
uil
din
g.
Ro
xb
ury
. " I
I
SIT
E
GEO
LOG
Y
( )
*1
11
Th
in la
yer
of
all
uv
ium
o
ver
asp
halt
ic
san
ds.
(I
)
All
uv
ium
(S
)
All
uv
ium
; sil
tsto
ne at
20
'-3
0'
; an
d w
ate
r ta
ble
at
35
'.
(I)
All
uv
ium
; w
ate
r ta
ble
at
22
'.
(S)
All
uv
ium
(S
)
r-J
f-'
TA
BL
E
1.1
A
CC
ELER
OG
RA
PH
SIT
E
INFO
RM
AT
ION
[2
9]
-(C
ON
TIN
UE
D)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
IDE
NT
IFIC
AT
ION
ST
AT
ION
L
OC
AT
ION
D
IST
AN
CE
TO
ST
RU
CT
UR
AL
*
1
*1
1
NU
MB
ER
SIT
E
NA
ME
MA
P K
EY
TO
EP
ICE
NT
ER
T
YPE
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
45
8-6
0
Lo
s A
ng
ele
s,
2 C
-4
33
.5
-47
C
ircu
lar
7-
15
10
7
sto
ry
RC
V
ano
wen
. b
uil
din
g.
46
1-3
L
os
An
gele
s,
2 C
-4
27
.8
16
1
7-s
tory
S
t.
15
91
0
bu
ild
ing
. V
en
tura
.
46
5
San
Ju
an
2
F-7
1
21
. 3
-33
S
mall
b
uil
d-
Cap
istr
an
o.
ing
.
46
6-8
L
os
An
gele
s,
2 C
-4
28
.1
13
1
2-s
tory
RC
1
52
50
b
uil
din
g.
Ven
tura
.
46
9-7
1
Lo
s A
ng
ele
s,
3 E
-3
40
.5
-16
1
0-s
tory
RC
1
62
5
bu
ild
ing
. O
lym
pic
.
47
2-4
O
ran
ge,
2 E
-5
83
.0
-34
1
9-s
tory
RC
4
00
0
Wes
t b
uil
din
g.
Ch
apm
an.
SIT
E
GEO
LOG
Y
( )
*1
11
All
uv
ium
5
00
';
wate
r ta
ble
at
70
'.
(S)
All
uv
ium
; w
ate
r ta
ble
at
35
'.
(S)
All
uv
ium
(S
)
All
uv
ium
; w
ate
r ta
ble
at
55
'.
(S)
All
uv
ium
(S
)
All
uv
ium
>
3
00
' o
ver
sh
ale
. (S
)
I ! I I
to
to
TA
BL
E 1
.1
AC
CE
LE
RO
GR
APH
S
ITE
IN
FOR
MA
TIO
N
[29
] -
(CO
NT
INU
ED
)
ST
AT
ION
A
PPR
OX
IMA
TE
D
IRE
CT
ION
B
UIL
DIN
G
SIT
E
IDE
NT
IFIC
AT
ION
S
TA
TIO
N
LO
CA
TIO
N
DIS
TA
NC
E
TO
STR
UC
TU
RA
L
GEO
LOG
Y
*1
*
11
(
*1
11
N
UM
BER
S
ITE
N
AM
E M
AP
KEY
TO
E
PIC
EN
TE
R
TY
PE
)
EP
ICE
NT
ER
(D
EG
RE
E)
(KM
)
47
5
Pasa
den
a,
2 D
-4
38
.4
-41
2
-sto
ry b
uil
d-
Ap
pro
xim
ate
ly
C.L
T.
ing
. 1
,00
0'
of
Ath
enae
um
. all
uv
ium
u
po
n g
ran
ite.
(S)
47
6-7
8
Fu
llert
on
, 2
E-5
1
08
.8
-58
la
-sto
ry
RC
All
uv
ium
2
60
0
bu
ild
ing
. (S
) N
utw
oo
d.
48
2-4
A
lham
bra
, 2
D-4
4
1.
7 -3
3
12
-sto
ry S
t.
Few
lO
a'
of
90
0
So
uth
b
uil
din
g.
all
uv
ium
F
rem
on
t.
ov
er
silt-
sto
ne.
(S)
*1
P
erm
an
en
t Id
en
tifi
cati
on
n
um
ber
in
an
nu
al
list o
f sta
tio
ns
issu
ed
b
y th
e
Seis
mo
log
ical
Fie
ld
Serv
ey
, N
OS-
NO
AA
.
*1
1
Map
n
um
ber
s 1
, 2
an
d
3 co
rresp
on
d to
F
igs.
1.3
, 1
.4 an
d 1
.5,
resp
ecti
vely
.
*1
11
)
rep
resen
ts
an
ab
bre
via
ted
sit
e cla
ssif
icati
on
. (S
),
(I)
an
d
(H)
co
rresp
on
d to
so
ft,
in
term
ed
iate
an
d h
ard
sit
e cla
ssif
icati
on
s,
resp
ecti
vely
[5
1].
*IV
R
C
rein
forc
ed
co
ncre
te.
*V
St.
ste
el.
I'J
W
SOn Fernando
Fig. 1.1
I.IJ o ::J f.-J a.. ::a <X
z o fU Z ::J u.. ex: I.IJ
Map 0
24
° 4
~ EPICENTER Kilome1ers N
SAN GABRIEL MOUIv
t
PACOIMA R V.terall' ACCELERQMETE
O~o HO'pltal VETERANS
FAULT 0
." '.""do ~ FAULT ZONE
34°[5' ~-- 1118°15'
f surface fau Ferna It traces, San n do earthquake. [3]
by's filter pass Orms
ff frequency cut-o .
ff terminatlon roll-o frequency
f He 0.07 0.125
by's filter Low pass Orms
ff frequency cut-o . f
LC: ff terminatlon roll-o
f LT : frequency
~ O~-"--Li~~~}5-L-'-LLLLll~--~"-L1LLu1---':3~'~1 'II 1~ILlII .. I I III I 50 100 I I I II Ii -~ - 1;5 10 ~~~~~~D-__ ~_ 100 ... 55 I 0.01
Fig. 1.2
109,0 f, (HZ)
. for functlon Transfe~ f strong correctlons 0
d baseline ment an instru lerograms. motion acce
A
36
°0
0'
35
°00
'
34
°0
0'
33
°0
0'
;() ~0: ~ o
B °0 (C'~.-
.t-
a 2
0
40
lo
w...
lo
w...
S
CA
LE
IN M
ILE
S
c
.BA
KE
RS
FIE
LD
SE
E E
NL
AR
GE
ME
NT
12
0°0
0'
119
00
0'
118°
00
'
D
E
F
G
~ ~
SP
RIN
GS
.10
3
BLY
TH
--..
--.. --.
. ----.
11
7°0
0'
11
6°0
0'
Fig
, 1
.3
Accele
rog
rap
h sit
e lo
cati
on
s
in cen
tral
an
d
so
uth
ern
C
ali
forn
ia.
2 3 4 5
N
Ul
34°45'
34° 30'
34° IS'
34° 00'
26
A B C D E F
·52
.121 LANCASTER
.125 .126
.127
.128 PALMDALE
.269
.284
.290
241.
.104
461 .287
SAN BERNARDINO FWAY .278
POMONA
MALIBU
.108
o 2 4 6 8 10 Iwoo.( Iwoo.( I""",i SCALE IN MILES
118°00'
Fig. 1.4 Accelerograph site locations in extended Los Angeles area.
2
3
4
5
6
7
A
34
" 0
7'
G&
140
B
• 416
142
o :3 CD
c
HO
LLY
WO
OD
SU
NS
ET
B
LV
D
w ;;
w
>
<{
<{ w '" m <
{
44
6 -
23
8 S
AN
TA
M
ON
ICA
-13
3
--' I
B
EV
ER
LY
B
LV
o w
>
<{ z '" w f- ~ I
43
_1
99
...
..-L-
WIL
SH
IRE
I BL
VD
"
20
2 -a.
196 ~ I
44
3
'"'4
28
J -2
28
2
17
· 21
~ -2
08
'
_4
13
PIC
O
BL
VD
WA
SH
ING
TO
N
BL
VD
JEF
FE
RS
ON
B
LV
D
CU
LV
ER
C
ITY
o 0
.5
.... S
CA
LE
IN
M
ILE
S
IIS
O 2
2'
30
" 11
80 2
0'0
010
IIS
O
17
'30
"
Fig
0 L
5 l\
ccele
rog
rap
h sit
e lo
cati
on
s
in cen
tral
Lo
s A
ng
ele
s.
2 3 4
118
01
5'0
0"
tv
-..J
28
AREA B AREA E Q)
3 rd. ai
> J <1 St. c -~
c Q) 0
t; 202 E
~ 431 \.i9 ~ Q)
> Wilshire - !96 Blvd.
2i7 2il-208
L..--_W_it~s.hire Blvd.
.0 o~,u 425'
o ~ '440 o~~ _-184 Blvd.
c" 187
8~ St.
AREA F
Fig. 1.6 Accelerograph site locations along Wilshire Boulevard, Santa Monica Boulevard and Figueroa Street.
29
II. PRINCIPAL AXES FOR GROUND MOTION
A. FIXED PRINCIPAL AXES
a (t), a (t) x y
Suppose three components of ground motion, and
a (t) at a point 0 along an arbitrary set of orthogonal axes x, y and z
z are defined through the following stochastic model
a (t) x
a (t) y
a (t) z
t; (t) b ' (t) x
t; (t) b' (t) y
t; (t) b ' (t) z
where b ' (t), b' (t) and b' (t) are stationary random processes and x y z
(2.1)
t;(t) is a deterministic intensity function. This model represents an
approximation to that defined by Eqs. (1.1). In this model, it is
assumed that the intensity functions in the three directions vary with
time in identically the same manner even though they may differ by a
scalar factor. Considering recorded earthquake motion, these components
normally represent accelerations measured along the instrument axes of
accelerometers. For the purpose of discussion here, however, these
components could equally well represent velocities or displacements.
If a (t), a (t) and a (t) of Eqs. (2.1) are considered to x y z
be zero-mean process, covariance functions defined by
E [a. (t) a. (t+T)] 1 J
t; (t) t; (t+T) E [b~ (t) b~ (t+T)] 1 J
(2.2)
i,j x,y,z
where E denotes ensemble averages, can be used to characterize the com-
plete ground motion process. If this process is Gaussian, these covariance
functions completely characterize the process in a probabilistic sense [5].
30
Since random processes b'(t), b'(t) x y
and b' (t) z
are stationary,
all ensemble averages on the right hand side of Eqs. (2.2) are independent
of time t; therefore, showing dependence only upon the time difference T.
Since in the first approximation, real earthquake accelerograms can be
represented by white noise [19,10,35], these processes demonstrate a very
rapid loss in correlation with increasing values of ITI. Therefore, the
influence of coordinate directions on covariance functions can be
investigated using the approximate relations
E [a. (t) a. (t)] 1 J
2 r;; (t) E [b~ (t) b~ (t)]
1 J (2.3)
i,j x,y,z
Adopting matrix notation, Eqs. (2.3) can be written in the more compact
form
l:! (t)
where
]1 .• (t) E [a. (t) a. (t)] 1J 1 J
S.. E [b~(t) b~(t)] 1J 1 J
i,j = x,y,z
Note that because random processes b'(t), b'(t) x y
and b' (t) z
are
stationary, all nine coefficients in matrix S are time invariant.
(2.4)
(2.5)
(2.6)
If the components of ground motion at point 0 are transformed
from coordinate system x,y and z to a new orthogonal coordinate
system x~y' and z' through the relation
a' (t) ax tt) I x
a' (t) A a (t) (2.7) y
a: tt) J a' (t) z
31
where the transformation matrix A satisfies the condition
I (identity matrix) (2.8)
relations identical to Eqs. (2.4) - (2.6) can be written for the new
coordinate system with
S' (2.9)
(2.10)
This transformation of ground motion is quite identical to the trans-
formation of a three-dimensional state of stress; therefore, it is
apparent that a set of principal axes exist along which the component
variances of motion have maximum, minimum and intermediate values and the
corresponding covariances have zero values. The directions of these
principal axes are found in exactly the same manner as locating the
directions of principal stresses, i.e. by obtaining the eigenvalue
formulation. The resulting three vectors define the principal trans-
formation matrix P; thus, permitting the components of ground motion
along principal axes, 1, 2 and 3 to be given by
al
(t) a (t) x
a2
(t) P a (t) y
(2.11)
a3
(t) a (t) z
where
T I P P (2.12)
32
The corresponding covariance matrix for ground motion becomes
~ (t) -p
=
=
pT ~(t) p - - -
811
r;;2 (t) 0
0
~ll
0
0
0 0
822 0
0 833
0 0
~22 0
0 ~33 (2.13 )
222 The principal values are given by r;; (t) 8
11, r;; (t) 8
22 and r;; (t) 8
33
and the directions of principal axes are given by the corresponding
column vectors of the transformation matrix P, respectively. Since the
covariance matrix 8 in Eqs. (2.13) is time invariant, the coefficients
of the principal transformation matrix P are also time invariant; that
is, the directions of principal axes are fixed during the entire time
history of motion.
Fortunately for most physical phenomena represented by random
processes, the desired properties can often be estimated using a single
member from each process. This is, of course, strictly true only for
ergodic random processes. Therefore, the covariances in Eqs. (2.13) can
be obtained by time averaging over any single member of the process, say
the th
r member. In this case, B .. 1J
defined by Eqs. (2.6) can be
obtained through the relation
8 .. 1J
< b ~ (t) b ~ (t) > 1r Jr
(2.14)
i,j x,y,z
r = 1,2,3, •••.
33
where the triangular brackets denote time average.
To further illustrate the physical meaning of principal axes,
consider two stationary random processes xl (t) and x2
(t). The auto-
and cross-correlation functions of these processes are given by
R .. (T) lJ
E [x. (t) x. (t+T)] l J
i,j 1,2
(2.15 )
where E denotes the ensemble averages. These functions can be decomposed
and expressed in the frequency domain by the equivalent relation
and
S .. (w) lJ
R .. (T) lJ
1 2n
00
f -00
00
J -00
R .. (T) lJ
-iwT e dT
S, . (w) +iwt
dw e lJ
i,j 1,2
(2.16 )
(2.17)
These relations which express R .. ('[) lJ
and s .. (w) lJ
as Fourier transform
pairs are usually called the Wiener-Khintchine relations.
Suppose random processes xl (t) and x2
(t) are considered to be
ergodic processes in which case ensemble averages are equivalent to the
corresponding time averages i.e.
E [xl < x > r
(2.18)
where subscript r denote th
r member of ensemble x. Relations between
ensemble averages and temporal averages which are respectively designated
by R .• (T) lJ
and ill .. (T) (i,j = 1,2) lJ
and those between their corresponding
34
quantities decomposed into the frequency domain s .. (W) 1J
(i,j = 1,2) are obtained by Eqs. (2.15) - (2.18),
R .. (e) E [x. (t) X.(t+e)] 1J 1 J
= < x. (t) x. (t+e) > 1r Jr
= <I> •• (T) 1J
00
S .. (w) 1 I R .. (e) -iwe
de 2n
e 1J 1J
-00
00
1
f <I> •• (e) -iwT
de 2n
e 1J
-00
G .. (w) 1J
i, j = 1,2
and G .. (W) 1J
Principal axes were defined previously along which the components of
(2.19)
(2.20)
motion have maximum and minimum values of variances and have zero values
of covariances for e = O. Intensities of the process along principal
axes can be evaluated by subsituting e = 0 into Eqs. (2.15) - (2.20).
It follows that
<I> •• (0) 1J
T 2
lim 1
IT T T-+<>o
00
I -00
2
G .. (w) dw 1J
x. (t) 1
G (if i j)
o (if i I j)
i,j = 1,2
x. (t) dt J
(2.21)
35
These properties reveal that components of motion along principal axes
have maximum and minimum values of power [2] and zero values of cross
power. Note that cross-correlation function RI2
(T) and R21 (T) are
not generally even functions; therefore, unlike the power spectral
density functions GIl (w) and G22 (w) which are always real and
positive, the cross-power spectral density functions G12
(w) and
G21
(w) are generally complex. Note also that for principal axes the
cross-power r G12
(w) dw and JOO G21
(w) dw are zero, even though -00 -00
the functions G12
(w) and G21
(w) may not be zero.
It can be concluded, however, that for a first approximation
to modelling of ground motions, it is sufficient to accept the concept
of principal axes which do not require one to establish cross powers
between the individual components of motion.
B. TIME DEPENDENT PRINCIPAL AXES
It is easily shown that the intensity functions along an
orthogonal set of axes, x, y and z (usually taken as instrumental
axes), do not have the same identical shape. For example,observe the
shapes of the intensity functions (sigma vs. time) in Figs. 2.2a through
2.2d for the three components of motion along the instrumental axes at
stations Nos. 266, 475, 264 and 267; see Fig. 1.4 for location. These
intensity functions are defined through the relation [22]
1:;. (t , L':.T) 1 0
= It: (0 t
o
i x,y,z
L':.T +
2
D.T 2
2 a. (t) dt
1 (2.22)
with D.t = 5 seconds. The solid, intermediate-dashed and short-dashed
curves represent the components of motion along the north-south,
36
east-west and vertical axes, respectively. The upper set of curves in
these figures (theta and phi vs. time) represent the angles shown in
Fig. 2.1. The intermediate-dashed curve represents 8 (degree) cor-
responding to the left side coordinate of the axis and the short-dashed
curve represents ~ (degree) corresponding to the right side coordinate
of the axis. If the intensity functions s (t), s (t) x y
and s (t) z
are
-1 identically the same in magnitude, angles 8 and ~ are 45° (tan 1) and
-1 1 54.7° (cos --), respectively. Further, if the intensity functions
/3 vary with time in identically the same manner even though they may have
different magnitudes, i.e. satisfy Eqs. (2.1), the values of 8 and ~
are invariant with time. In this case, the curves representing angles
e and ~ become straight horizontal lines. As one can see in Figs.
2.2, the intensity functions for actual strong ground motions do not
change with time in the same manner, i.e. the directions of the principal
axes are not time invariant.
Suppose the three components of ground motion are represented
by the relations
a (t) x
a (t) y
I;; (t) b (t) x x
I;; (t) b (t) y y
a (t) = I;; (t) b (t) z z z
(2.23)
where b (t), b (t) and b (t) are stationary random processes and x y z
I;; (t), I;; (t) and I;; (t) are dissimilar deterministic functions along x y z
a (t), a (t) x y
the x, y and z axes, respectively. Assuming and
a (t) to be zero mean processes and applying the procedures previously z
described, covariance functions are defined by
37
E [a.(t) a.(t+T)] = t;.(t) s.(t+T) E [b.(t) b.(t+T)] 1 J 1 J 1 J
(2.24 )
i, j x,y,z
These covariance functions for a zero value of the time difference T,
can be written in the matrix form
fl (t) I;;(t) 13 I;;(t) (2.25)
where
]1 .. (t) E [a. (t) a. (t) ] 1J 1 J
(2.26)
13 .. E [b. (t) b. (t)] 1J 1 J
(2.27)
s .. (t) 1;;. (t) (i j) 1J 1
0 (i ~ j) (2.28)
i, j = x,y,z
Applying an orthogonal transformation identically similar to the one
previously described, the covariance functions along the principal axes
become
T ]1 (t) fl p p
-p
pT S (t) § f(t) ~
(s (t) p)T 13 (f (t) P)
]1ll 0 0
0 fl22 0 (2.29)
0 0 ]133
Note that because the intensity matrix set) is a diagonal matrix,
T f(t) is equal to set).
38
In Eqs. (2.29), the principal transformation matrix P is
given as a function of time t, i.e. P = P(t). Since the column vectors
of the principal transformation matrix give the direction cosines of the
corresponding principal axes, the directions of the principal axes are
time-dependent when the three components of motion are defined in
accordance with Eqs. (2.23).
C. MAXIMUM VARIANCES AND COVARIANCES
If one assumes each component of motion to be identically the
same, the direction of the instantaneous resultant acceleration vector
will not change in time; thus, the components will be completely cor-
related, i.e. the cross correlation coefficients obtained from the
relation
P .. 1J
E[a. (t) a. (t)] 1 J
I E[a. (t) a. (t)] E[a. (t) a. (t)] 1 1 J J
i ~ j
(2.30)
will be equal to either +1 or -1 depending upon the pair of components
involved. On the other hand, if the components are actual recorded
ground motions, the cross correlation coefficients will be greatly
reduced showing lack of correlation with each other. Note that a cross
correlation coefficient equal to zero, indicates a complete lack of
correlation.
It is interesting to investigate the characteristics of motion
along sets of axes which give maximum correlated components of motion as
well as along the principal axes which give completely uncorrelated
components of motion. A procedure similar to that previously described
can be used to determine the coordinate transformations which yield
39
maximum covariances. When transforming from principal axes 1, 2 and 3,
this procedure leads to the following orthogonal transformation matrices
+ 1 0 0
~l 0 +11 - 2 +)1 - 2
0 -/1 + -2 +11 - 2
+J1 - 2 0 +11 - 2
~2 0 + 1 0
-/1 + -2
0 +/1 - 2
+/1 - 2 +)1 - 2
0
~3 -)1 + -2 +/1 - 2
0
0 0 + 1 -
Substituting Eqs. (2.31) separately into the relation
m
gives principal covariances equal to
and
S -m
1, 2, 3
The corresponding
(2.31)
(2.32 )
variances are ~ll (t), ~22(t) and
1 "2 [~22 (t) + ~33 (t)] ,
~33(t), and the corresponding mean
1 "2 [~ll (t) + ~33 (t) ] and variances are
1 "2 [~ll (t) + ~22(t)], respectively. Using these values, the principal
40
cross correlation coefficients are given by
= [11 11 (t) - 1122 (t) ]
[1111 (t) + 1122 (t) ]
[1122
(t) - 1133 (t) ]
[1122
(t) + 1133 (t) ]
[1111
(t) - 1133 (t)]
P13 [1111
(t) + 1133
(t)]
(2.33)
Note that when two principal variances approach each other in
value, the corresponding principal cross correlation coefficient
approaches zero and the other two cross correlation coefficients approach
the same value. In the limit when these principal variances become equal,
the corresponding principal axes become undefined. This behavior cor-
responds to the three-dimensional stress problem when a deviator stress
along one axis is superposed upon a hydrostatic state of stress resulting
in only one identifiable principal axis which lies along the axis of the
deviator stress. Obviously, it becomes difficult to reliably predict
the directions of the principal axes when principal variances approach
each other in value.
41
UP
OA: INTENSITY RESULTANT
_I Cy(t) e = TAN Cx(t)
COS-I -;=.====:C~Z=( t=)==:~_ ./C~( t) + C~( t) + C~( t)
\ \
\
----------------'*~~~~--_r------~----~NORTH
/
I // I /
/
EAST
/ /
/
/ /
/
Fig. 2.1 Direction angles of intensity function in three-dimensional space.
z o -i
:;c ..., -0
:;
c o o c: " OJ r rn
a S
TA
TIO
N
NO
. Z
66
3
40
8
55
N.I
IB
10
IS
W
CA
lTE
CH
S
EIS
MO
LO
GIC
AL
L
AB
..
PA
SA
DE
NA
. C
AL
.
< 6
0.0
f-4
5.
0
I-3
0.0
8
0.0
CO
MP
SOO
W
« 60
.01
/1--
..
~ 4
0.0
I
/ \
2 a
. 0
CO
MP
S
90
W
CO
MP
DO
WN
7 5
. 0
6 a
. a
:r:
45
.0
5.0
1
0.0
1
5.0
2
0.0
2
5.
0 3
0.
0
TIM
E
(S
EC
.)
c S
TA
TIO
N
NO
. 2
64
3
40
8
12
N.1
18
0
7
30
W
CA
lTE
CH
M
ILL
IKA
N
LIB
RA
RY
. B
AS
EM
EN
T.
PA
SA
DE
NA
. C
AL
.
CO
MP
N
OD
E
CO
MP
N9
0E
C
OM
P D
OW
N
~ :::
: I
\Pl~~~
-c~---
~t--m-
·-·--r
-----·
··--l·
····m·
·-r -
I
:~::I f1~~ I
« 4
a .
0 l:
VI
20
. a
75
. a
60
.0
::I:
45
.0
5.0
1
0.0
1
5.0
2
0.0
2
5.
a 3
0.0
T
IME
(S
EC
.)
b S
TA
TIO
N
NO
. 4
75
3
40
8
ZO
N.1
18
0
7
17
W
CA
LT
fCH
A
TH
EN
AE
UM
. P
AS
AD
EN
A.
CA
L.
CO
MP
NO
OE
C
OM
P N
90
E
CO
MP
DO
WN
'" f-3
0.0
~ :::: I
>+----
--~~t~
~--~~~
J~~~,-
~-~:lc
«r 7
5.0
60
. 0
:c
45
. 0
60
.0
« 4
0 .
a l:
rJ)
20
.0
d « 6
0.0
f- w
45
.0
'" t-J
0 .
0
« l:
(/l
2
0.0
I JJ;L
J .. J J
5
.0
10
.0
15
.0
20
.0
25
. 0
30
. 0
TIM
E
(S
EC
.l
ST
AT
ION
N
O.
Z6
7
34
12
0
1N
.11
8
10
2
SW
JE
T
PR
OP
UL
SIO
N
LA
B .
. B
AS
EM
EN
T.
PA
SA
DE
NA
. C
AL
.
CO
MP
S8
2E
C
OM
P
S0
8W
C
OM
P D
OW
N
I~Fl'······l>! .+
, .
j _
F
....
I i
75
.0
60
.0
:c
"5
.0
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
a .
a T
IME
(S
EC
.)
Fig
. 2
.2
Tim
e d
ep
en
den
cy
o
f in
ten
sit
y fu
ncti
on
s
alo
ng
in
str
um
en
t ax
es.
"'" to
43
III. CHARACTERISTICS OF GROUND MOTION FROM MOVING-WINDOW ANALYSIS
A. TIME DOMAIN ANALYSIS
1. General
In a previous paper [32], variances and covariances of
recorded ground motions were evaluated for successive time intervals
using the relation
< [a. (t) a.] [a. (t) a. ] t2 fl .. >
1J 1 1 J J tl
i,j = X,Y,z
in which the time averages are taken over the interval t < t < t 1 = 2
(3.1)
but where the mean values a. 1
and a. J
are found by averaging a. (t), 1
a. (t) over the entire duration of motion. Locations of principal axes J
and magnitudes of corresponding principal variances were obtained for
earthquake motions recorded at three stations in California and three
stations in Japan. The results show that the directions of principal
axes were not fixed for successive time intervals.
In the present investigation, recognizing that intensity
functions for three components change in a different manner with each
other, va~iances and covariances are obtained as continuous functions
of time t using the so-called "moving-window" technique, i.e. using o
the relation
fl .. (t ,L1T) 1J 0
< [a.(t) - a.][a.(t) - a.] > 1 1 J J
i,j = x,y,z
where the time averages are taken over the interval
t o
t o
L1T
L1T +
2
L1T 2
centered at
time t [9]. Having obtained all nine covariance functions for the o
(3.2)
44
recorded components of motion in accordance with Eqs. (3.2), the
corresponding time dependent directions of principal axes can be
obtained, i.e. giving the principal transformation matrix as a function
of time to and time window length ~T ,i.e. p = p (t , ~T). o - - 0
This time
dependent principal transformation matrix then allows one to obtain the
time dependent directions of principal axes of components of motion
and
and
a (t) and their corresponding principal variances z
One finds increased fluctuations in ~ .. (t , ~T) and the 1J 0
corresponding directions of principal axes as the value of time window
length ~T in Eqs. (3.2) is taken shorter and shorter. In fact, as
~T 7 0, the major principal axis of ground motion coincides with the
instantaneous resultant acceleration vector which changes its direction
rapidly in a random fashion over the entire sphere of space. Therefore,
~T should be taken sufficiently long so that the higher frequency
fluctuations are essentially removed but the slower time dependent
characteristics are retained, i.e. the time average over duration ~T
will be essentially equal to the average taken across the ensemble.
The direction of each principal axis is given by angles ~
and 8 as shown in Fig. 3.1. Angle ~ is the declination of the
principal axis from the vertical axis through point "0"; thus, its value
falls in the range 0° ~ ~ ~ 90°. Angle 8 is measured from the North
axis to the projection of the northerly extension of the principal axis
on a horizontal plane containing point "0". By this definition, 8
lies in the range -90° < 8 < + 90°. The angle 8E
in Fig. 3.1
represents the horizontal direction of an axis passing through the
accelerograph site location (point "a") and the reported epicenter.
Since this angle is measured in a similar manner to that of angle 8,
45
it also lies in the range -90° < 8E ~ + 90°. Length OA in Fig. 3.1
represents the magnitude of the variance Of principal ground motion.
The square root of this quantity (sigma) can be used to represent the
intensity functions of the corresponding non stationary processes [22].
2. Results At Station No. 264 Basement Of The Millikan Library, CALTECH
Direction angles ~ and 8 and the square root of the
principal variance (0) have been obtained as functions of time t for o
the major, minor and intermediate principal axes of the ground motion
at station No. 264, the basement of the Millikan Library at the
California Institute of Technology, Pasadena, California.
'I'he results are shown in Figs. 3. 2a through 3. 2c. These
results of Figs. 3.2a, 3.2b and 3.2c are respectively obtained by using
time window length ~T equal to two, five and ten seconds at discrete
values of one-half second apart. The solid, short-dashed and
intermediate-dashed curves in these figures represent respectively the
major, minor and intermediate principal axes and the horizontal long-
dashed straight line represents the direction 8E
to the reported
epicenter. It should be noted from the definition of 8 that as the
horizontal direction of a principal axis rotates in a continuous manner
through the east-west direction, the value of e changes instantaneously
by 180°, i.e. changes from +90° to -90° or from -90° to +90° depending
upon whether the horizontal projection of the principal axis is rotating
clockwise or counterclockwise. This explains the sudden jumps which
appear in the functions of e which take place over single spacings of
the prescribed discrete values of
spacings.
t , o
namely over one-half second
In the present report, ~T is taken as two seconds for several
short durations of motion, say less than twenty seconds, and taken as
46
five seconds for all other records. Angles ¢ and e and 0 are
evaluated for discrete values of t spaced one-half second apart and o
are interpolated by straight lines.
It should be noticed that if ground motion processes are
represented by the product of stationary random processes and deter-
ministic intensity functions and if the intensity functions for three
components vary with time in the same manner, i.e. satisfy Eqs. (2.1),
any two of the principal variance functions differ from each other by a
fixed constant only in which case the directions of principal axes are
fixed, i.e. they are time invariant over the entire duration.
3. Results Through The Time Domain
The time domain moving-window analysis described above has been
applied to the ground motions recorded at numerous stations during the
San Fernando earthquake of February 9, 1971. Acceleration records at
99 stations have been evaluated. Nearly half of them are located in the
high- or intermediate-rise buildings in the cities of Los Angeles,
Hollywood and Beverly Hills, which are sited about 40 km south of the
epicenter.
Some of these accelerograph locations are in basements and at
the ground level of higher buildings, some are in smaller buildings and
some are on free field. The motions can be considered representative
of the ground motions. The accelerograph at each station has its own
characteristics [48] and was triggered independently of other located
nearby. At station No. 290, two accelerographs were installed, one of
which was a temporary accelerograph.
The accelerograph locations were divided into six area groups,
A through F. certain data associated with these stations, such as
47
station identification number, station location, peak accelerations,
building structural type and local site geology are given in Tables 3.1
through 3.5 and Table A.l,respectively.
Area A (CALTECH)
Area A is located on the campus of California Institute of
Technology which contains four stations, namely No. 267, the Jet
Propulsion Laboratory, No. 266, the Seismological Laboratory, No. 264,
the Millikan Library and No. 475, the Athenaeum. Numerous investigations
on the dynamic behavior and the soil-structure interaction effects of
these buildings have been carried out [27, 12, 21, 15]. The accelero
graphs at these four stations are reported to have been triggered at
the same time, i.e. to have common time bases [24]. The results for
these stations are shown in Figs. 3.3a through 3.3d.
Area B (Wilshire) [12, 17]
Area B having seven stations in the basements of high-rise
buildings is located along Wilshire Boulevard in downtown Los Angeles.
Each instrument in this area was triggered independently upon the arrival
of seismic waves. However, one can estimate a common time base by
reading the arrival time of certain high frequencies in the acceleration
and velocity traces. This procedure is reported to provide an accurate
estimate [17]. The local geology in this area consists of alluvium and
plestocene rock and is classified as an intermediate type or a soft
type. The results for these seven site locations are shown in Figs.
3.4a to 3.4g.
Area C (Lake Hughes)
Area C located in the vicinity of Lake Hughes consists of
four stations array Nos. 1, 4, 9 and 12. The site geology in this area
48
is designated as a hard soil type. Due to the wide separations between
stations in this area, it is impossible to establish a common time base
for the recorded accelerograms. The results for these stations are
shown in Figs. 3.5a through 3.5d.
Area D (Beverly)
Area D contains four stations; two in basements and two at the
first floor level of high-rise buildings. These four stations, Nos. 137,
148, 172 and 145, are located on intermediate geology at the corner of
Beverly Boulevard and Figueroa Street in downtown Los Angeles and are
very close to each other. The distance from the epicenter to the stations
is about 40 km and the direction to the epicenter is about N 20° W.
Figures 3.6a-3.6d show the time dependent characteristics of principal
axes of motion at these stations.
Area E (Santa Monica) [17]
This area contains four stations located in high-rise buildings
having from 15 to 30 stories. These buildings are located on soft geology
about 40 km from the epicenter along Santa Monica Boulevard near Beverly
Hills in the west part of Los Angeles. The results for these stations are
presented in Figs. 3.7a through 3.7d. Although the wave propagation
paths are different, one may be able to investigate the relative in
fluence of intermediate to soft geological conditions on ground motion
characteristics by comparing the results for stations in areas D and E.
Area F (Figueroa)
Area F is located about 40 km S 18° E of the epicenter. The
stations in this area are located in intermediate-and high-rise
buildings along Figueroa Street, south of area D (Beverly) in downtown
49
Los Angeles. It has been reported that the accelerograph at station
No. 154 was installed on the 2nd floor; however, because the floor of
the building is numbered from the adjoining building, this floor actually
corresponds to ground level [29]. The time dependent characteristics of
principal axes of motions for these six stations are included in
Appendix A.
The results obtained for 70 other stations, not included in
area groups A-F, are presented in Appendix B.
4. Observation of Results
Although, the functions of principal transformation shown in
Figs. 3.3 throug-h 3.7 and in Appendices A and B have numerous
unexplainable features, certain correlations should be noted as follows:
(1) Usually during the early periods of low intensity motion,
either the major or the intermediate principal axis is nearly
vertical, i.e. the vertical component represents a large
amount of energy in comparison with the horizontal components.
(2) Later except for several motions recorded in high-rise
building-s and at stations close to the epicenter, the major
and intermediate principal axes shift towards horizontal
positions with the minor principal axis taking the nearly
vertical position; thus, the angle between the horizontal
directions of the major and intermediate principal axes is
about 90°.
(3) Following the shift of the major principal axis towards a
horizontal position, the horizontal directions of the major
and intermediate principal axes are sometimes suddenly inter
changed. 'Ehis interchange which occurs after the period of
50
high intensity motion is due to a corresponding change in the
direction along which the seismic waves have maximum energy.
(4) After the major and intermediate principal axes have moved to
their nearly horizontal positions, the minor and intermediate
principal axes (and in some cases, the minor and major
principal axes) retain approximately the same horizontal
direction angles 8 over large time intervals. This pro
perty suggests geometrically that the plane containing the
minor and intermediate principal axes also contains the
vertical axis. When the minor principal axis takes a nearly
vertical position, slight changes in its direction cause large
fluctuations in the angle 8. Due to this high sensitivity,
the fluctuations in e have little significance in this case.
(5) For many motions measured in high-rise buildings, some of the
more common correlative features related to principal axes
seem to be eliminated due to possible soil-structure inter
action effects. These interaction effects are most apparent
during the strong motion following that time at which the
minor axis shifts to its nearly vertical position.
(6) Usually during the period of high intensity motion, the
horizontal direction of either the major or the intermediate
principal axis is towards the faults slip zone. This
characteristic suggests that the direction along which seismic
waves contain maximum energy either coincides with the
direction to the fault slip zone or is at right angles to it.
(7) The shape of the intensity functions for the minor principal
axis looks fairly flat for those stations located on soft to
intermediate types of geology.
51
For area groups A-F, specific correlations and dissimilar
features can be noted as follows:
(1) In Figs. 3.3, the possible interchange of major and inter
mediate principal axes at stations Nos. 264 and 475 occurs at
a time around 10 second. The directions of principal axes and
corresponding intensities are, however, quite dissimilar with
each other, even though the distance between stations is not
more than 400 m. During the early period of motion, the
direction angle 8 of the major principal axis is approximately
90° for both stations Nos. 267 and 266. Later in the motion,
the direction angle shifts to nearly 45° in each case.
(2) In Figs. 3.4, the general features of principal directions at
various stations agree reasonable well with one another,
especially at stations Nos. 211, 208, 196 and 199 where the
major principal axis is closely directed towards the reported
epicenter during periods of high intensity motion. Also the
intensity functions for stations Nos. 211 and 208 are quite
similar to each other; however for stations Nos. 196 and 199,
they are dissimilar.
(3) In Figs. 3.5, little correlation, if any, can be seen for the
four stations represented. This is to be expected, however,
since these stations are spaced at distances which are large
compared to the significant seismic wave lengths in the
accelerograms and their geological conditions and wave pro
pagation paths could be quite different with each other.
(4) Among the four stations of area group D represented in Figs.
3.6, the intensity functions look quite similar except for
52
station No. 148. The directions of principal axes are quite
similar for stations Nos. 145 and 172 and are also similar for
stations Nos. 137 and 148. They are, however, different
between those two groups.
(5) In Figs. 3.7, the directions of the major principal axes for
stations Nos. 440, 184 and 187 are quite similar with one
principal axis directed towards the reported epicenter in each
case. These same features at station No. 425 are, however,
dissimilar.
Figures 3.8 through 3.11 are maps of areas in southern
California showing the horizontal directions of the major and inter
mediate principal axes at the period when the motions are of highest
intensity. The map in Fig. 3.9 is an enlargement of the small rectan
gular area in Fig. 3.8 showing the extended Los Angeles and San Fernando
regions. Similarly the map in Fig. 3.10 is an enlargement of the small
rectangular area in Fig. 3.9 showing the cities of Los Angeles,
Hollywood and Beverly Hills. The maps in Figs, 3.11a through 3.11c
show area groups B, E and F,respectively.
While the correlation is not strong, there is a tendency of
the directions of the major principal axis or, in some cases, the inter
mediate principal axis to point in the general direction of the fault
slip zone as shown in Fig. 3.9 [20] which is also the general direction
towards the previously reported locations of surface fault traces south
of the epicenter [13]. The concept of intensity defined here is
identical to that defined by Arias [2]. Therefore, one can speculate
that the direction of the major principal axis coincides with the
direction to maximum energy release in the fault slip zone.
53
Obviously, in the case of motions produced by the San Fernando earthquake,
such factors as the complex mechanism of strain energy release, the
dispersion of seismic waves due to variable geological conditions, the
complex geology around the fault slip zone, the closeness of the
recording stations to the fault slip zone and the possible influence of
soil-structure interaction weaken this correlation. It should be
pointed out that the horizontal directions of the major and inter-
mediate principal axes obtained from time averaging over the entire
durations of motion are in most cases quite similar to those shown in
Figs. 3.8 through 3.11.
B. FREQUENCY DOMAIN ANALYSIS
1. General
The moving-window technique, as applied in the time domain
formulation, can be applied in the frequency domain as well. In this
case, however, the variances and covariances are evaluated as con-
tinuous functions of frequency f . o
Using the Fourier integral trans-
formation, variances and covariances are obtained through the relation
].1 •• (f ,b.f) lJ 0
lC~ where
~ {(:+ b.f f + b.f 2 r 2
(2 . f) 2'ITift df + (2 . f) 21Tift A. 1Tl e A. 1Tl e IH
1 b.f 1
2 f -
2 0
b.f f + b.f 2 0 2
(2 . f) 2'ITift A. 'IT 1 e df + J (2 . f) 2'ITift A. 'IT 1 e df} >~ t:.f 2
J
A. (21TH) 1
A. (21Tif) J
J f _ t:.f
o 2
i,j x,y,z
T
J ai
(t)e-2'ITift dt
o T
J aj(t)e-21Tift dt
o
1 dfJ .
(3.3)
(3.4)
54
and where T denotes the total duration of ground motion.
In the frequency domain formulation, the principal trans-
formation matrix P is given as a function of frequency f and o
frequency bandwidth ~f, i.e. P = P (f ,~f). o
This formulation allows
one to investigate directions of principal axes, variances, covariances,
etc. associated with only those frequencies of ground motion in the
range of (f - ~f ) < f < (f + ~f) Hopefully, this approach can be 02- 02'
used to reveal certain characteristic features of the various types of
seismic waves associated with strong ground motions.
2. Results at Station No. 264
Direction angles ~ and e and the square root of the
principal variance (0) have been obtained as functions of frequence
for the major, minor and intermediate principal axes of the ground
motion at station No. 264, located in the basement of the Millikan
f o
Library at the California Institute of Technology, Pasadena, California.
Using the frequency domain formulation, one can obtain
frequency dependent characteristics of principal axes as shown in
Figs. 3.12a through 3.12c. Again the solid, short-dashed and
intermediate-dashed curves represent the results for the major, minor
and intermediate principal axes, respectively. These figures show
properties of principal axes using different values for frequency band-
width (~f) and for spacings between discrete values of frequencies (f ). o
The values of bandwidth and spacings represented in Figs. 3.12a, 3.12b
and 3.12c are, respectively, 0.488 and 0.244, 0.977 and 0.488, and 1.953
and 0.488 Hz. Based on these results, it was judged that values near
0.977 and 0.488 Hz could be used for the motions recorded at other
stations.
55
3. Results Through The Frequency Domain
The moving-window formulation in the frequency domain has been
carried out at 29 site stations in the area groups previously described.
The results for those stations in area groups A through F are shown in
Figs. 3.13 and 3.14 and in Appendices A and C. Information on site
locations can be found in Tables 3.1 through 3.5 and in Table A.l.
4. Observation of Results
While many features of the frequency domain functions shown in
Figs. 3.13 and 3.14 and in Appendices A and C are as unexplainable as
certain features of the time domain functions, there are some
characteristics and correlations which can be identified among area
groups or geological conditions as follows:
(1) Usually in the lower frequency range, the major and inter
mediate principal axes take nearly horizontal positions, while
in the higher frequency range, the minor principal axis shifts
towards a horizontal position with the major principal axis
taking a vertical position.
(2) The major and intermediate principal axes are observed to
interchange their horizontal positions several times at most
stations. At station No. 211 (Fig. 3.14b), they interchange
positions at frequencies around 1.0 and 3.0 Hz. At station
No. 137 (Fig. C.2a), they interchange positions three times
at frequencies around 1.5, 4.5 and 7.0 Hz.
(3) Among several stations, such as Nos. 202, 196, 199, 126, 127,
172, 184 and 187 (Figs. 3.14, C.l, C.2 and C.3), the horizontal
direction of either the major or the intermediate principal
56
axis coincides with the general direction to the reported
epicenter. The relationship between general direction of the
major principal axis and direction to the epicenter is, how
ever, not highly correlated.
(4) Usually, the principal variances along the minor principal axis
are more uniform than the principal variances along the major
or intermediate principal axis. Therefore, it may be con
cluded that the spectral density distribution for the minor
principal axis is approximately uniform, i.e. the spectral
density distribution is similar to white noise.
(5) Based on the results in Figs. 3.13, the motions for sites on
hard geology generally have peak variances in the higher
frequency range and are quite narrow band. However, motions
for sites on soft geology generally have peak variances in the
lower frequency range and are quite wide band. This observa
tion is in general agreement with site dependency effects as
previously reported in the literature [40]. To check further
for evidence of site effects on the dominant frequency, i.e.
the frequency corresponding to maximum intensity, the results
shown in Fig. 3.15 were plotted. This figure shows number of
stations versus dominant frequency for three site conditions
(hard, intermediate and soft) within area groups A through F.
The correlation in this case is quite weak; however, it should
be recognized that the site classifications are not too
reliable.
57
C. MAXIMUM VARIANCES AND COVARIANCES
1. General
Cross correlation coefficients, obtained through the relation
p .. 1J
< [a. (t) - a.l [a.(t) - a.l > 1 1 J J
i,j = x,y,z
cr. cr. 1 J
(3.5)
take values in the range of -1 and +1. If the ground motion processes
are completely dependent, i.e. they are identically the same, the cross
correlation coefficients are either +1 or -1 depending upon the
directions of the processes. On the other hand, if the processes are
completely independent of each other, the cross correlation coefficients
equal zero. As previously shown ground motions along principal axes are
independent in a statistical sense; therefore, the corresponding cross
correlation coefficients equal zero. Axes along which the ground
motions provide maximum cross correlation coefficients can be identified
by the procedure previously described. In the following section of this
report, principal variances and principal cross correlation coefficients
obtained by the moving-window technique are presented.
2. Results of Maximum Variances and Covariances
Principal variances, maximum covariances and principal cross
correlation coefficients are evaluated for motions recorded at stations
in area groups A through F. The results for stations in area groups
A and B are shown in Figs. 3.16 and 3.17, respectively, and those for
other stations are included in Appendicies A and D. Each figure con-
sists of four diagrams showing the following.
58
(1) Principal variances along the major, minor and intermediate
principal axes which are represented by the solid, short-dashed
and intermediate-dashed curves, respectively.
(2) Ratios of the intermediate and minor principal variances to
(3)
the major principal variance which are represented by the
intermediate-dashed and short-dashed curves, respectively.
Mean variances equal to
and 1 2
1 2 []122 (t) + ]133 (t) ] ,
1 2
along axes
1 giving the corresponding maximum covariances 2 []122(t) - ]133(t)],
1 and 2 []111 (t) - ]133(t)] which are shown by the solid,
intermediate-dashed and short-dashed curves, respectively.
(4) Principal cross correlation coefficients P23' P13 and P12
obtained as ratios of the maximum covariance to the cor-
responding mean variance, e.g. P23 (t) = []122(t) - ]133(t)l/
[]122(t) + ]133(t)], which are represented by the solid,
intermediate-dashed and short-dashed curves, respectively.
3. Observation of Results
Based on the time dependent properties of principal variances,
the variances along principal axes vary with time in different manners.
Observed, however, that the ratios of the intermediate and minor
principal variances to the major principal variance are stable during
periods of high intensity motion. Since high energy is represented
during these periods, their statistical properties will be considered of
major importance in characterizing the ground motion process. Therefore,
the most significant statistical properties were evaluated at that time
when the motion was of maximum intensity. Such results obtained for
59
each geological classification are summarized in Table 3.6. It is
apparent that the ratios of the intermediate and minor principal
variances to the major principal variance, though they differ with
geological conditions, are approximately 0.5-0.7 and 0.15-0.25 with the
standard deviations of 0.2 and 0.1, respectively.
Cross correlation coefficients for motions along instrument
for many earthquakes including the San Fernando earthquake were reported
in the. literature [11]. Statistical properties of these cross cor
relation coefficients are shown in Table 3.7. Of the 104 instrument
stations included in this table, several are located at the top of
buildings.
Similar to those of principal variances, statistical pro
perties of cross correlation coefficients evaluated during periods of
high intensity motion are shown in Table 3.8. Using these results,
average ratios of principal variances ~33/~22' ~33/~11 and ~22/~11
were found to be 0.50, 0.22 and 0.43 on hard geology, 0.21, 0.15 and
0.73 on intermediate geology and 0.31, 0.17 and 0.56 on soft geology,
respectively. These ratios correspond with the ratios of principal
variances at the time of highest intensity as shown in Table 3.6.
Although the number of samples is quite limited, it can be concluded
that there is a general correlation between geological classification
and ratios of principal variances. Based on the results in Table 3.8,
the principal cross correlation coefficient between the major and
intermediate principal axes (P12) is smaller for soft geology than
for hard geology. In all cases studied, the minor principal axis at
time of maximum intensity is nearly vertical. When this occurs and the
principal cross correlation coefficient P12
equals zero, the motions
60
in the horizontal plane are statistically independent regardless of the
directions of components.
TA
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TABLE 3.7 STATISTICAL PROPERTIES OF CORRELATION COEFFICIENTS FOR STRONG MOTION ACCELEROGRAMS RECORDED AT 104 SITES [11]
TRUE STANDARD ABSOLUTE ABSOLUTE ABSOLUTE COMPONENTS MEAN DEVIATION MEAN MAXIMUM MINIMUM
HI' H 2
0.0029 0.2116 0.1632 0.6801 0.0014
HI' V 0.0187 0.1774 0.1387 0.4957 0.0004
H2
, V 0.0055 0.1841 0.1321 0.7430 0.0005
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a STATION NO. 217 3403 42N,118 18 24W
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10.0 15,0 20.0 25, 0 30, 0
TIME [SEC.)
Fig. 3.4 (continued)
77
a STATION NO. 125 3440 30N,118 26 24W
LAKE HUGHES, ARRAY STATION 1, CAL.
5. 0 1 O. 0 15. 0 20. 0 25. 0 3 a . 0 a
9 a
~ : ~I f----'-~-f~~~~~~-~ ~:-Ef-------,,-,_,_--+-L-, _-- __ ---l---j'------1. I
« t--
90
45
w a I
t-- _ 4 5
-90
6 a . a
« 4 a . a L Cl
Ul 20. a
« t--
b
I
a
90
60
Q. 30
o
90
45
w 0 I
t-- _ 4 5
-90
6 a . a
« 4 0 . 0 L Cl
Ul 20. 0
0
. 1\/11 i / /' rv Y' ~ ,-- ....... .,.,../ ',,_ ..... _
/----
~5:::-:::::, ,-:::-:::, ,_:::_ ,-==-, _=--, ---l-,-, ,_, ,-"
o 5. a 10.0 15.0 20.0 25. 0 30. a TIME (SEC.)
STATION NO. 126 3438 30N,118 28 48W
LAKE HUGHES, ARRAY STATION 4, CAL.
o 5. 0 10. 0 15.0 20. 0
f----
0 5.0 10.0 15.0 2 a . a TIME (SEC.)
Fig. 3.5 Time dependent directions of principal axes and square root of principal variances in area C.
78
C STATION NO. 127 3 4 36 30 N • 1 1 8 33 42W
LAKE HUGHES, ARRAY STATION 9. CAL.
0 5.0 10. 0 15. 0 20. 0
90
60 I :::r:;r~~~~f= I
I
D- 30
0
90
4 5 I 1\ « "'--~) r- '-" "'----------w 0 . /1
, :r:: - ... ~ ~ ~1 ' . r- 1=---::::-----;. t- +- ~--''--! ---- \----
- 4 5 ..... , .... , I
~ -,
'< -" I -90 ,,~ " /
40.0
30. 0 ~ « -,<\ L t::J 20.0
'-
~ (f)
10. 0 ---'-::::::-= -<:::-:-:::::--:=:: ~ 0
0 5.0 10. 0 15. 0 20. 0
TIME ( SEC. )
d STATION NO. 128 34 34 1 8 N , 1 1 8 33 36W
LAKE HUGHES, ARRAY STATION 12. CAL.
0 5. 0 10. 0 15. 0 20. 0
90
60 I ~--L-=L •
I
D- 30
0
90
« 45
r-w 0 I
r- -45
-90
----J\ ~ /
\ ~ \ /' ,
~
-y f--~.A I- ----"f-:...=....:; ... ,....- ", L-~- ~~.-A- - -,- " ~
" --, I V
\ ,,/ \ /\1 -/ \
100.0
80. 0
« L 60. 0 t::J
4 a . 0 \j~
(f)
2 a . a
a 0 5.0 10.0 15.0 2 a . 0
TIME (SEC.)
Fig. 3.5 ( continued)
79
STATION NO. 137 3403 OON,llB 15 DOW
WATER AND POWER BUILDING. BASEMENT. LOS ANGELES. CAL.
o 5. 0 10. 0 15. 0 20. 0 25. 0 30. 0
90
60 :r: (L 30
0
90
« 4 5
\-w 0 :r: \- - 4 5
-90
./ I '-/ I I I ,....--/ /-./ I ,.-
\ / : I I V '\ I
~ Y'\r ,V-I
h I I ~
I I I
~~ ~_I- J. b~-!-- Nt I " --------, I- ,-...... .,'"- - ~ .... _----; \ , \ I : .. 1 j I 1 ,/ "- 1 - . I ! ,"'<} / /
- \ \ . , . , ~I" __ 1_ r, I ........ \ "
\ ,
'_I '''-'''.,' 'oJ \ I '~ ... - --- '-'
60. 0
« 4 0 . 0 L co
(/) 20. 0
o o 5.0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
b STATION NO. 148 3403 ZON, lIB 15 Z5W
234 FIGUEROA STREET. BASEMENT, LOS ANGELES, CAL.
o 5. 0 10. 0 1 5 . 0 20. 0 25. 0 30. 0
90
« 45
\-w 0 :r: \- - 4 5
-90
80. 0
60. 0 « L co 40. 0
Ul 20. 0
. -', -" .~ -. -:. ::-. ~ ::-.:::-::: ::-::-l:t::--:= .. :=."" .. ~.~ .. ~.=-=*----~ 0 ~-'--'
..... ;.-~--'-
0 5. 0 10.0 15.0 20.0 25.0 30.0
TIME (SEC.)
Fig. 3.6 Time dependent directions of principal axes and square root of principal variances in area D.
c
90
« 45
f-w 0 I
f- - 4 5
-90
60. 0
«40.0 L C)
",20.0
o
d
90
60 I CL 30
0
90
« 4 5
f-w ·0 I
f- - 4 5
-90
60. 0
«40.0 L D
",20.0
o
80
STATION NO. 172 3403 26N,118 15 02W 800 W. FIRST STREET, 1ST FLOOR, LOS ANGELES, CAL.
o 5. 0 1 o. 0 1 5 . 0 20. 0 25. 0 30. 0
o
~ L / V
\
I ,,-'
-'
--I- \;, / -/ \' ' '- \ / "-".,,- - - \- I- T -:- - r - -: -:- - - - -" r ~ /. , ,,~' -_.... ...,,_ ............................ '
~'-"'---------
--... '
5. 0 10.0 15.0 20.0 25. 0 30. 0
TIM E [S E C . J
S TAT ION NO. 145 3 4 0 3 2 5 N, 118 1 5 0 3 W 222 FIGUEROA STREET, 1ST FLOOR, LOS ANGELES, CAL.
o 5. 0 10. 0 I 5 . 0 20. 0 25. 0 30. 0
- _I-!- ~~ v,~ _ f- - r- - 1-. - - - 'r-:- - - ~~ - -\ I \ ..... ;:.i. r I : ~~ \
o 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME [SEC. J
Fig. 3.6 (cant inued)
81
a STATION NO. 425 3403 46N,l18 24 S2W
1800 CENTURY PARK EAST, BASEMENT P-3 LOS ANGELES, CAL.
o 5. 0 10. 0 15. 0 20. 0 Z 5. 0 30.0 i I j
90
60 I
0.- 30
0
90
« 4 5
f--w 0 I
f-- - 4 5
1 ..... -'-/
- 9 0
40. 0
30. 0 « L Cl 20. 0
U)
1 O. 0
--------~
It ,::---- ~_, ~/, I~~-
0 0 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
b STATION NO. 440 34 03 44N,l18 24 SOW
1880 CENTURY PARK EAST. PARKING, 1ST LEVEL, LOS ANGELES, CAL
o 5. 0 1 0 . 0 1 5 . 0 20. 0 25 . 0 30. 0
90
« 4 5 , " //----------.... ~~----- ..... -
f--w 0 :r: f-- - 4 5
\
- 9 0 \ --,
.... , J
60. 0
« 4 0 . 0 L c:J
U) 20. 0 1"---
0 0 5. 0 10.0 15.0 20.0 25 . 0 30. 0
-T I M E (S E C . )
Fig. 3.7 Time dependent directions of principal axes and square root of principal variances in area E.
c
90
60 I [l 30
0
90
« 4 5
I-w 0 I
I- - 4 5
- 9 0
40.0
30. 0 « L Cl 20. 0
Ul 10. 0
0
d
90
60 I [l 30
0
90
« 15 I-w 0 J: I- -15
-90
60. 0
« 1 0 • 0 1: Cl
(J) 20.0
o
82
STATION NO. 184 3 4 0 3 3 5 N, 118 2 4 5 6 W
1900 AVENUE OF THE STARS, BASEMENT, LOS ANGELES, CAL.
o 5. 0 10 . 0 1 5 • 0 20. 0 25. 0 30 . 0
I E] J-----~I~~'2 I 1- _______________ ,--- ---- ------ ___ ------'
,r l.J
"
i f ,,-------, ... ..._-1/-'
-----------------=---~~
0 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME [SEC. 1
STATION NO. IS7 3403 14N.l1S 24 5SW
1901 AVE. OF THE STARS SUBBSMT.. LOS ANGELES, CAL.
o 5. 0 10. 0
· · · " .
, :
15. 0 20. 0 25. 0
-- .......... - ,- .. _# ....... ' • ~-/ --_/-
.",_ _J ........ _~- .... _,). .. ----
_/--- -------......
-- ~ -- - ..... '-------.- '-~ ----,---- -- - -----------~
30.0 35. 0
~I
-- ..... --":::~--:.::.::---= --:"'-"':'::---::-=-- =--.7 __ =-____ ="--:- _______ -s:-::=-_ ---~------ ... -----o 5.0 10.0 15.0 20.0 25.0 J 0 • 0 J 5. 0
TIME (SEC.)
Fig. 3.7 (continued)
10. 0
40. 0
A
36
° 0
0'
35
°00
'
34
°0
0'
33
°0
0'
B
~ 0..
.<"
/0
c D
~INTERMEDIATE
PR
I N
CIP
AL
AX
iS
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JO
R
PR
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AL
A
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RS
FIE
LD
E
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ALE
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ES
_ .. _ .. _
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12
0°0
0'
119
00
0'
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00
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16
°00
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Fig
. 3
.8
Dir
ecti
on
s
of
majo
r an
d in
term
ed
iate
p
rin
cip
al
ax
es
of
sta
tio
ns
in cen
tral
an
d so
uth
ern
C
ali
forn
ia.
2 3 4 5
(J)
w
34°45'
34°30'
84
A B C D E F
f $: INTERMEDIATE
~) PR I NCIPAL AXIS
LANCASTER MAJOR PR I NCIPAL AXIS
::: ~
POMONA
MALIBU
o 2 4 6 8 10 b...f.=.---k I wi SCALE IN MILES
118°30'
Fig. 3.9 Directions of major and intermediate principal axes of stations in extended Los Angeles area.
2
3
4
5
(3
7
34
" 0
7'
34
° 0
5'
34
° 0
3'
34
°0
11
A
B
~INTERMEDIATE
PR
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Fig
. 3
.10
D
irecti
on
s
of
majo
r an
d
inte
rmed
iate
p
rin
cip
al
ax
es
of
sta
tio
ns
in cen
tral
Lo
s A
ng
ele
s.
2 3 4
OJ
lJl
a> > « C La>
AREA B
3rd . S t.
8 th . St.
AREA F
a> > <{
+c o E La> > Blvd.
86
AREA E
Wilshire
Fig. 3.11 Directions of major and intermediate principal axes of stations along Wilshire Boulevard, Santa Monica Boulevard and Figueroa Street.
'" r w
I
a S
TA
TIO
N
NO
. 2
64
3
4
OS
1
2N
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07
3
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LT
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LIK
AN
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NO
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64
3
4
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Fig
. 3
.12
HZ
(l./S
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.
c S
TA
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NO
. 2
64
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8
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qu
en
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ep
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den
t d
irecti
on
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es
an
d
sq
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ro
ot
of
pri
ncip
al
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an
ces at
sta
tio
n
No
. 2
64
.
CD
-.J
«
« I-
a
90
45
w 0 :r: I- _ 4 5
-90
20. 0
15. 0
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Ul 5. 0
0
b
90
60 :r: 0... 30
0
90
« 4 5
I-w 0 :r: I- -45
-90
20.0
15. 0 « L Cl 10. 0
Ul 5. 0
0
88
STATION NO. 267 3 4 1 2 0 1 N , 118 1 0 2 5 W
JET PROPULSION LAB. BASEMENT, PASADENA, CAL.
o 2 . 5 5. 0 7. 5 10. 0
-------::--~--::--::==---::.:::--~::---:::c:
0 2. 5 5. 0 7. 5 10. 0
HZ (1 ISEC.
STATION NO. 266 3408 55N,118 10 15W
CALTECH SEISMOLOGICAL LAB. PASADENA, CAL.
o 2. 5 5. 0 7. 5 10. 0
0 2. 5 5. 0 7. 5 10. 0
HZ (l./SEC.)
Fig. 3.13 Frequency dependent directions of principal axes and square root of principal variances in area A.
c
90
« 45
I-w a :r: I- - 45
-90
2 a . a
15. 0 « L C) 10. 0
(f)
5. 0
0
89
STATION NO. 264 3408 12N.118 07 30W
CALTECH MILLIKAN LIBRARY. BASEMENT. PASADENA. CAL.
a
-~
0
, ,
/ /
2. 5
, , ',-
2. 5
HZ
5. a 7. 5 10. a
---~-~-
i~
5. 0 7 • 5 10. 0
1./SEC.
d STATION NO. 4 75 34 o 8 20 N • 1 1 8 07 17W
CALTECH ATHENAEUM. PASAOENA. CAL.
o 2. 5 5. 0 7. 5 10. 0
90
« 45
I-w 0 :r: I- -45 ----
-90
15. 0
« 1 0 . 0 L (.:)
U) 5. 0 ~ --_ ...
::---~:.=----- -=---------0
0 2. 5 S. 0 7. 5 10. 0
HZ (1./SEC.
Fig. 3.13 (continued)
a
90
« 4 5
f-w 0 I
f- - 4 5
-90
20. 0
1 5 . 0 « L L:l 10. 0
(/)
5.0
0
b
90
« 45
f-w 0 I
f- -45
- 90
15. 0
« 1 0 • 0 L L:l
(/) 5. 0
0
90
STATION NO. 217 3 4 0 3 4 2 N . 118 1 8 2 4 W
3710 WILSHIRE BLVD. BASEMENT. LOS ANGELES. CAL.
o 2. 5 5. 0 7. 5 10. 0
-~~:-~ -- ___ ----- ___ ~:_:-_-_:::_-::: __ c=_--
0 2. 5 5 . 0 7. 5 10. 0 H Z 1 . / SEC.
STATION NO. 211 3 4 0 3 4 2 N. 118 1 8 0 6 W
3550 WILSHIRE BOULEVARD. BASEMENT. LOS ANGELES. CAL.
o
0
2 . 5 5. 0
-/-\ - - -/-+,-~ - -'\ / , \ ", --
---- ~ ...
2. 5 5. 0
HZ (l./SEC.
7. 5 10. 0
,'\ /I '
7 . 5 10. 0
Fig. 3.14 Frequency dependent directions of principal axes and square root of principal variances in area B.
c
90
60 I
Cl- 30
0
90
« 4 5
I-w 0 I
I- - 4 5
-90
20. 0
1 5. 0 « L LJ 1 0 . 0
if)
5. 0
o
d
90
60 I
Cl- 30
0
90
« 4 5
I-w 0 :r: I- - 4 5
- 9 0
20. 0
1 5 . 0 « L LJ 10. 0
(f)
5. 0
o
91
STATION NO. 208 3403 40N,118 17 58W
3470 WILSHIRE BLVD. SUBBASEMENT, LOS ANGELES, CAL.
o 2. 5
o 2. 5
5. 0
/-- \
\,./ " , " ' "
5. 0
, \
" f \ "':/ \
HZ 1 . / SEC.
7.5
f /
7 . 5
1 O. 0
10, 0
STATION NO, 431 3403 45N,118 17 56W
616 S, NORMANDIE AVENUE, BASEMENT, LOS ANGELES, CAL.
o 2. 5 5. 0 7. 5 10. 0
'------- '---------------- ----,---------------~:-:.-::-.::::-~
o 2. 5 5. 0 7 . 5 10. 0
HZ CI./SEC.)
Fig. 3.14 (continued)
e
90
60 :r: 0.- 30
0
90
« 4 5
I-w 0 :r: I- -45
-90
15. 0
« 10. 0 L
'" VI 5. 0
a
f
9.0
60 :r: 0.- 30
a
90
« 4 5
I-w 0 :r: I- - 4 5
- 9 0
15 . 0
« 1 0 . 0 L
'" VI 5. 0
a
92
STATION NO. 202 3403 4SN, I I 8 I 7 5 7 W
3411 WILSHIRE BOULEVARD, 5TH BASEMENT, LOS ANGELES, CAL.
o 2. 5 5. 0 7 • 5 I 0 . 0
--, \
\
" " '----
o 2. 5
STATION NO.
.-.- "--
- -.l.r - - -/'- ~~-=-:-_ \ '-., ,7:/_.\---- I
\
------...... --
\ , I I
5. 0 7. 5 10. a HZ (I./SEC.)
196 3403 4SN,I18 17 43W
3345 WILSHIRE BOULEVARD, BASEMENT, LOS AN~ELES, CAL.
D
\
o
2. 5
.---- ......
\ "-" '-
2 • 5
5. a 7. 5 10. 0
5. 0 7. 5 10. 0
HZ I . / SEC. )
Fig. 3.14 (continued)
9
90
60 ::c Q 30
0
9fJ
..: 45
f-w 0 ::c f- - 45
- 9 0
20. 0
15. 0 ..: L .:> 10. 0
C/ 5. 0
f)
93
STATION NO. 199 3403 45N.118 17 43W
3407 6TH STREET, BASEMENT. LOS ANGELES, CAL.
o 2.5 5.0 7.5 10.0
~ _ --'
0 2. 5
~-'-\ , ' \ , ' , ' ':
\ " /' --- ---- -~ =-----
5. 0 7. 5
1-1 Z 1 • I S [ C . 1
Fig. 3.14 (continued)
10. 0
8
en
6 z o ~
<1
~
en
4
lL. o a::
w
m
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z
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AR
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WIL
SH
IRE
C
LA
KE
H
UG
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S
D
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E
SA
NT
A
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I
FIG
UE
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IN
TE
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Y
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0.4
88
1
.95
3
3.4
18
4
.88
3
DO
MIN
AN
T
FR
EQ
UE
NC
Y
(Hz)
Fig
. 3
.15
C
orr
ela
tio
n
of
do
min
an
t fr
eq
uen
cy
w
ith
so
il cla
ssif
icati
on
.
95
a STATION NO. 267 3 4 1 2 0 1 N • 118 1 0 2 5 W
JET PROPULSION LAB .• BASEMENT. PASADENA. CAL.
o 5. 0 1 0 . 0 15. 0 20. 0 25. 0 30. 0
o 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIM E (S E C . )
b STATION NO. 266 3408 5SN.118 10 15W
CALTECH SEISMOLOGICAL LAB .• PASADENA. CAL.
5. 0 1 0 . 0 1 5 . 0 20. 0 25. 0 30. 0
1 . 0 0
l- . 5 « 0::
0 4 0 0 0
0:: 3 0 0 0 « > 2000
u 1000
0 1 . 0
0
l- . 5 « 0::
0 4000
0:: 3 0 0 0 « > 2 0 0 0
[l.. 1 000
o 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
Fig. 3.16 Time dependent principal variances and principal cross correlation coefficients in area A.
96
c STATION NO. 264 3 4 0 8 1 2 N • 118 0 7 3 0 W
CALTECH MILLIKAN LIBRARY. BASEMENT. PASADENA, CAL.
o 5. 0 10. 0 1 5 . 0 20. 0 25. 0 30. 0
0 5. 0 1 0 . 0 15. 0 20. 0 25. 0 30. 0
TIME ( SEC. )
d STATION NO. 475 3 4 08 20 N , 1 1 8 07 1 7 W
CALTECH ATHENAEUM, PASADENA. CAL.
0 5. 0 10. 0 15. 0 20. 0 25. 0 30. 0
~ .: /----->-~-------->------->;~:-->~--~:-1-------------~--2000 ,-______ ,-____ -, ______ -, ______ -, ______ -. ______ -,
: :::~ I ,£~Et~?ild. I I o 5. 0 10.0 15.0 20.0 25.0 30. 0
TIM E (S E C . )
Fig. 3.16 (continued)
a
b
1 . 0 0
f- . 5 < 0::
0 3000
0::
< 2000 >
10 a 0 u
0 1 . 0
0
f- . 5 < 0::
o
97
STATION NO. 217 3403 42N,118 18 24W
3710 WILSHIRE BLVD. BASEMENT, LOS ANGELES, CAL.
o 5. 0 1 0 . 0 1 5 . 0 20. 0 25. 0 30. 0
o 5. 0 10.0 15.0 20.0 25. 0 30.0 TIME (SEC.)
STATION NO. 211 3403 42N, 118 18 06W
3550 WILSHIRE BOULEVARD, BASEMENT, LOS ANGELES, CAL.
o 5. 0
I: '" I' ' -:::"_'
10. a
--- --_ ..
15. a 2 a . a 25 . 0 30. 0 , i I
.. - ;:_---_ .... _--- -:,,~"' ..... _/
3000
~:::t3f£C:§~.1 I o 5. 0 10.0 15.0 20.0 25 . 0 30. 0
TIME (SEC.)
Fig. 3.17 Time dependent principal variances and principal cross correlation coefficients in area B.
98
c STATION NO. 20B 3403 40N,11B 17 5BW
3470 WILSHIRE BLVO •• SUBBASEMENT. LOS ANGELES. CAL.
o 5. 0 10. 0 15. 0 20. 0 25. 0 30 0
o 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
d STATION NO. 431 3403 45N, IB 17 56W
616 S. NORMANDIE AVENUE, BASEMENT. LOS ANGELES. CAL.
o 5. 0 10. 0 15. 0 20. 0
o 5.0 10.0 15.0 20. 0
TIME (SEC.)
Fig. 3.17 (continued)
99
e STATION NO. 202 3403 45N, 118 17 57W
3411 WILSHIRE BOULEVARD, 5TH BASEMENT, LOS ANGELES. CAL.
o 5. 0 10. 0 15. 0 20. 0 25.0
o 5. a 10. 0 15. a 20. a 25. a TIME [SEC.)
f STATION NO. 196 3403 4SN.118 17 43W
3345 WILSHIRE BOULEVARD, BASEMENT, LOS ANGELES, CAL.
o 5. 0 10. a 1 5 . a 20. 0 25. 0 30.0
o 5. 0 10.0 15.0 20.0 25. a 3 a . a TIME [SEC.)
Fig. 3.17 (continued)
100
9 STATION NO. 199 3 4 0 3 4 ~ N . 118 1 7 4 3 W
3407 6TH STR~ET. BASEM~NT. LOS ANGELES. CAL.
o 5. 0 10 . 0 1 5 • 0 20 . a 25 . 0 30 . 0
~ :::: i > 200 0
u 1000
o
~ :::: I > 2000
Q 1000
o o 5. 0 10.0 15.0 20.0 25. 0 30 . 0
TIME (SEC.)
Fig. 3.17 (continued)
101
IV. FREQUENCY CONTENT OF GROUND MOTIONS ALONG PRINCIPAL AXES
A. GENERAL
The variation of frequency content with time has been examined
previously using the concept of an evolutionary spectrum [33,34,28] and
the concept of a response envelope spectrum [45]. While the use of these
concepts is desirable, they cannot be applied to the moving-window
technique adopted in this study. Applying the moving-window technique,
Fourier amplitude spectra are obtained for the principal components of
motion of duration ~T centered on time t o
The concept of power and power spectrum will be reviewed
briefly [9]. Let a, (t) 1
and A, (W) 1
represent a real time function and
its corresponding Fourier frequency function, respectively. In this
discussion, a, (t) 1
represents acceleration of motion and power will be
regarded as the square of the magnitude of process a, (t) , 1
even though
it does not have units of energy per unit of time. Thus, the square of
a, (t) a t time t, i. e . 1 0
1 a, (t ) 12 1 0
is termed the instantaneous power at time t . o
Correspondingly, the
integral of instantaneous power over the range - 00 < t < + 00, i.e.
co
I la,(t)12
dt 1
-00
is called total energy of process a, (t) 1
(4.2)
when the infinite integral has
a finite value. In practical applications, this integral always has a
finite value since the record length of a, (t) 1
is limited. Theoretically,
this integral converges when the time function is defined only in a
102
finite length or the time function satisfies the Dirichlet's conditions
in a mathematical sense. The average power of time function
obtained through the relation
lim 14«'
1 T J
T 2
T 2
la.(t)12
dt 1.
a.(t) is 1.
(4.3)
Process a. (t) will have a zero value of average power, if it has a 1.
finite value of total energy as given by Eq. (4.2).
Correlation functions are defined through the relation
R .. (T) 1.J
::;
for processes of finite total
R .. (T) l.J
lim T-.+<x>
foo
a. (t) 1.
energy and T 2
a. (t + T) dt J
through the relation
~ f a. (t) a. (t + T) dt
T 2
1. J
(4.4)
(4.5)
for processes of finite average power. For processes of the first type,
the total energy of a. (t) 1.
and its corresponding spectrum can be
related as follows:
00 (X)
f I a. (t) 12 dt ::; J a. (t) a. (t) dt 1. 1. J..
_(X) -(X)
00
[ 2"
00
eiwt
dw ] ::;
J a. (t) J A. (w) dt
J.. J.. _00 _00
00 00
1
J A. (w) dw J a. (t) iwt
dt ::; e 211 1. J..
_(X) _00
(X)
1
f A. (w) A. (-w) dw 211 J.. 1.
_00
When function a. (t) is a real process, 1.
(4.6)
where *
A. (-w) ~
103
'* A. (w) 1
denotes the conjugate property 0
(407)
substituting Eq. (4.7) into
Eq. (4.6), Eq. (406) becomes
00 00
I la.(t)12
dt 1
J * A. (w) A. (w) dw 1 27f 1 1
-00 ~OO
00
1
J IA. (w) 12 dw (4.8) 27f 1
-00
The left hand side of Eq. (4.8) represents the total energy of process
a. (t) 1
and the quantity IA. (w) 12 on the right hand side represents 1
energy density associated with a frequency w which will be called
energy spectral density. This relationship is usually referred to as
Parceval's theorem. For processes of finite average power, the relation
in Eq. (4.8) is slightly changed. Manipulating Eq. (4.5), one finally
obtains the relation T 2
R .. (0) 11
lim 1
I a. (t) T 1
T-W> T 2
[ 2rr lim 2!T LT T-W>
a. (t) dt 1
IA. (w) 12 dw 1
+ J 00
27f T
The quantity IA.(w)12/T in Eq. (4.9) represents average power of t:he
1
process per unit frequency and will be called power spectral density.
B. NORMALIZED FOURIER AMPLITUDE SPECTRUM WITH MOVING-WINDOW TECHNIQUE
Suppose a (t), a (t) x y
and a (t) z
represent respectively the
ground accelerations along the x, y and z axes. The moving-window
104
Fourier amplitude spectrum using a window length ~T centered at time
t is defined by o
If. (w, t , 1. 0
t ~T
+ 2
~T) I
f.(w,t,~T) (4.11) 1. 0
I 0 -iwt
a. (t) e dt ~T
1.
t 0 2
i = x,y,z
Using a similar procedure, a set of moving-window Fourier amplitude
spectra along principal axes, 1, 2 and 3, is given by
A . (w, t , ~T) 1. 0
i
~(w, t , ~T) 0
where
yew, t , liT) = 0
a (t)
If. (w, t , ~T) I 1. 0
1,2,3
6T t +-
2 I 0
~T t
2 0
fl (w,
f2
(w,
f3
(w,
a (t) x
a (t) y
a (t) z
pet ) 0
t , liT) 0
t , liT) 0
t , liT) 0
aCt) -iwt
e dt
and P(t) o
denotes the principal transformation matrix at time
(4.12)
(4.13)
(4.14)
(4.15)
t . o
In
this investigation, a constant window length of five seconds was used.
105
Therefore, ~T can be dropped out of the subsequent equations for
simplicity.
Since the Fourier amplitude spectrum usually oscillates
rapidly, some technique is required to make it smooth. Here, it is made
smooth through convolution weights 1/4, 1/2 and 1/4, i.e. the smoothed
Fourier amplitude spectrum is obtained through the relation
A (w, t ) o
The purpose of the moving-window Fourier amplitude analysis is to
(4.16)
examine the general characteristics of the frequency content with time.
Therefore, the Fourier amplitude spectrum has been made much smoother
by using the filter of convolution weights more than once.
Since intensity of ground motion can be expressed in terms of
variances as previously shown, the moving-window Fourier amplitude
spectrum can be normalized with respect to its maximum value generated
for time t. In this report, the normalized moving-window Fourier o
amplitude spectrum is assigned levels of 0 through 5 by the relation
A. (w, t ) 1 0
5X A. (w, t ) 1 0
A ~ (t ) (4.17)
1 0
i 1,2,3
where A. (w, t ) (i = 1,2,3) represents a so-called normalized moving-1 0
window Fourier amplitude spectrum and
generated at time
A~ (t ) 1 0
t , Le. o
is the maximum amplitude
(4.18)
106
C. RESULTS AT STATION NO. 264
Moving-window Fourier amplitude analysis has been carried out
for the 29 stations previously selected in area groups A through F. The
Fourier transformations have employed the Fast Fourier Transform
technique using 256 samples of each function. Since the time intervals
of digitized data were taken as 0.02 seconds, a time window length of
5 seconds was selected which is identically the same as that used in the
time domain formulation of principal axes.
Spectral amplitudes were evaluated for discrete values of
frequency up to 8 Hz, were smoothed by weighed convolutions and were
normalized using the maximum value obtained over the entire frequency
range. In Figs. 4.1, the time dependencies of frequency content are
shown using three dimensional spectral diagrams in which the x and y
axes denote time and frequency, respectively, and the contour lines
represent levels of the normalized Fourier amplitude spectra. Each line
denotes a level of 1, 2, 3 and 4, which describes the magnitude of
spectral amplitude with respect to the maximum magnitude which is
assigned a value of 5. These figures show the time dependent frequency
content along the principal axes of the accelerogram recorded at station
No. 264. Figures 4.1a, b, c and d present the results obtained by
repeating the smoothing procedure zero, ten, twenty and forty times,
respectively.
The shaded zones in these diagrams are areas where the
normalized Fourier amplitude takes on values greater than four, i.e.
they represent zones in which the Fourier amplitudes are near peak
values. In this investigation, the main purpose is to characterize the
general features of the time dependency of frequency content. For this
107
purpose, the procedure using twenty times filtering was adopted for other
stations.
D. RESULTS OF MOVING-WINDOW FOURIER AMPLITUDE SPECTRUM ANALYSIS
For those stations included in area groups A through F, the
Fourier amplitude spectrum analysis using the moving-window concept has
been carried out. The three dimensional diagrams in Figs. 4.2 and 4.3
and similar diagrams in Appendices A and E represent the time depend-
encies of frequency content for ground motions along principal axes.
The results for stations in area groups A and B are shown in Figs. 4.2
and 4.3, respectively, and the results for stations in area groups C-E
and F are shown in Appendices E and A, respectively. As described
previously, the shaded zones in those figures represent the highest
range of spectral magnitude; thus, indicating the corresponding range
of dominant frequencies at time t . o
E. OBSERVATION OF RESULTS
It is commonly recognized that power spectral density is a
most significant characteristic used in simulating a ground motion
process. Whereas for a nonstationary process most basic statistical
properties cannot be evaluated theoretically, it is often practical to
apply procedures to this process which are similar to those used for a
stationary process and to treat its properties in a statistical manner"
Using Parceval's theorem, one can estimate the power spectral
density of the process at time t and the variation of the spectral o
density with time by taking the square values of amplitude of the time
dependent Fourier spectrum. As these Fourier spectrum diagrams have
been normalized and made smooth by passing a filter weighing 1/4, 1/2
108
and 1/4 convolutions, one can obtain the general time dependency of
dominant frequencies and the general shape of the spectral density
function of the ground motion process. The characteristic features of
these diagrams can be summarized as follows:
(1) In several cases, though the results have been smoothed, the
diagrams show unexplainable complexities which may be caused
by the complex nature of the ground motions as influenced by
energy release mechanism, wave propagation path, local geology
and consequent wave dispersion including possible soil-structure
interaction effect.
(2) The dominant frequency is found to have discreasing values
with time. This property coincides with the results reported
by Saragoni and Hart [38] which were obtained by separating
the accelerogram into several segments and determining the
power spectral density function for each segment.
(3) In some cases, the dominant frequency changes its value
(4)
suddenly at a fixed time which may indicate the arrival of
different types of seismic waves.
Spectral density at time t , o
which can be evaluated by a
cross sectional view of the three dimensional diagram, becomes
higher and more sharply peaked as the frequency parameter
increases towards the dominant frequency. It then gradually
decreases with increasing values of the frequency parameter
beyond the dominant frequency. This tendency can be observed
most clearly for the accelerograms which were recorded on
soft ground.
109
(5) Generally, it is observed that the spectral density functions
derived from the three dimensional diagrams are less sharply
peaked for motions along the minor axis than for motions along
the major and intermediate axes. This tendency agrees with
the results of the moving-window analysis in the frequency
domain which show that principal variances along the minor
axis are more uniform than those along the major and inter
mediate axes.
(6) The frequency content for motions along the major and inter
mediate axes, both of which are nearly horizontal, is similar
to each other. In some cases, however, the frequency content
for motion along the intermediate principal axis is more
uniform than for motion along the major principal axis.
a
B.O
N 6.0 :I:
1.0
w 0:: 2.0
B.O
N 6.0 :I:
1.0
W
a: 2.0
B.O
6.0 :I:
1.0
W
'" 2.0
b
N 6.0 :J:
2,0
N 6.0 :J:
LO
w a:: 2.0
N 6.0
LO
'" w Q; '2. 0
llO
STAT ION NO. 26 4 34 08 1 2 N • 1 1 8 07 30W C STATION NO. 26 4 34 08 1 2 N. 1 18 07
c)' L TECH NJ L II KA N L J BRARY. BASEMENT, P"S .... OEN ..... "'-.- CALTECH MILLIKAN lIe RARY. BA S ENE NT. PASADENA. C .... l.
5.0 ~ 0,, 0 15.0 Z o. ~ TS,' ~ ?O"O 5.0 15.0 20.0 25,0 30.0
MAJOR - ----~ - '.0 - - 'is - • .C= '0
~~~ ~ D4l!2Q =""'" ~ XiV"'iJ2'::s N 6.0
~Q=- ~ --W
a: 2.0
INTERMEDIATE B.O
INTERMEDIATE
N 6,0 :I:
1.0
W cr. 2.0
MIN 0 R B.O
N 6,0 :I:
1.0
W cr. 2.0
5,0 10.0 Ist,o 20.0 30.0 5,0 10.0 15.0 20. 25'.0 30'.0
TIME ( SEC. TIME ( SEC. )
STATION NO. 264 34 08 12 N • 1 18 07 30W d STATION NO. 264 34 08 1 2 N • 1 1 8 07
CA L TECH MI L L J)("" tl L I 51< .... RY. BAS EMENT. PASACEN/I. CAL. CAL TECH M III I KAN LIBRARy. SA S EMENT. PASADENA. CAL.
5.0 10.0 15.0 20.0 25,,0 ?o.o 5.0 10.0 ~ 5 . ~ 20. ~ 25.0 30,,0
MAJOR B.O
~~~~ N 6.0
~~ :J:
1.0
'" W
~~-a:: 2.0
~
INTERMEDIATE B.O
\\~,y~~ N 6.0 :J:
~t~~Ljl~~ 1.0
w a::: 2.0
=-'-MIN 0 R MIN 0 R
B.O
}~~~~tJl ~~ N 6.0
1.0
~~ w a::: z. 0
/ ~ = 5:0 10.0 15,0 20'.0 25.0 30.0 5:0 10'.0 ~ 5'. 0 20'.0 25'.0 30'.0
TIME ( SEC, ) TIME ( SEC. )
Fig. 4.1 Time dependent frequency distribution at station No. 264.
30W
30W
a S
TA
TIO
N
NO
. 2
67
3
41
2
OH
J,1
18
1
0
25
W
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N
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66
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40
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55
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117
v. GENERATION OF THREE COMPONENTS OF GROUND MOTION
A. GENERAL
It is concluded in the previous sections that the three
translational components of ground motion are independent of one another
in a statistical sense, provided they are directed along a set of
principal axes. Therefore, one can simulate three components of ground
motion by generating them, of which intensity and frequency content are
given appropriately, independently along their corresponding principal
axes. One can, if necessary, obtain the three components of ground
motion along a set of principal axes of the structural system for use in
dynamic analysis by transforming them in accordance with horizontal and
vertical rotational angles which may have been determined from the
relationship between the location of the structural system and the
possible location of an earthquake.
In the first part of this chapter, the methodology to generate
a one-dimensional non stationary random process in both intensity and
frequency content is introduced. Four sample processes are produced by
employing this method. In the second part, this methodology is applied
to the simulation of three components of motion along principal axes,
each of which is provided with appropriate intensity and frequency
content. Based upon an assumption of relationship between the location
of the structural system and that of the epicenter, these three com
ponents along a set of principal axes are transformed and three com
ponents along a set of structural principal axes are produced. Finally,
assuming as if these three components were recorded by an accelerograph
of which axes coincide with structural principal axes, directions of
principal axes of ground motion, principal variances of ground motion
118
and frequency content of principal motion are evaluated through the
routine procedures which have been employed in the previous chapters to
analyze the ground motions produced by the San Fernando earthquake.
B. SIMULATION OF A NONSTATIONARY PROCESS
1. Generation of a Random Process Having an Arbitrary Power Spectral Density.
The common procedures to simulate a nonstationary random
process which has an appropriate spectral density are roughly classified
as shown in Table 5.1 [30]. The method employed in this investigation
belongs to the (III)rd group, which has been proposed by Toki [44,16].
The method is to generate a random process having an arbitrary spectral
density by superposing a sufficient number of sinusoidal waves. The
technique is explained briefly in the following. Let a ground motion
process aCt) be represented through the relation
aCt) j[ N - L N
n=l cos (n t + ¢ )
n n (5.1)
in which and ¢n are probabilistic variables which denote the
circular frequency and the phase angle of the sinusoidal wave, respec-
tively. Suppose that the probabilistic distribution of nn is
expressed by probability density function pen) and that the variable
¢n is distributed uniformly over the range 0 < ¢ < 2TI. The auto
correlation function of process aCt) is given by
R (T) = E [aCt) aCt + T)] a
1 2
E [cos n T + cos {n(2t + T) + 2¢}] (5.2)
Since the phase angle ¢ is assumed to be distributed uniformly, the
second part of the expected value in the right hand side of Eq. (5.2)
119
equals zero and Eq. (5.2) becomes
R (T) 1
E[cos n T] a 2
00
1
f • pen) dn 2
cos n T (5.3)
_00
The probability density function pen) is symmetrical with respect to
n 0, therefore, Eq. (5.3) is written in the range n > 0 through
R (T) a
00 I cos n T • pen) dn
o
(5.4)
Let the power spectral density of process aCt) be designated by Sew).
As the power spectral density function is expressed as a Fourier pair
of the corresponding auto-correlation function, the power spectral
density function is given through the relation
00
sew) J R (T) -iwT
dT e a
-00
r 00
pen) dn I -iwT dT cos n T e
0 _00
'IT r p (n) {o(w-n) + o(w+n)} dn
0
'IT p (w) (5.5)
in which 0 denotes a Dirac's 6 function. Equation (5.5) indicates
that the power spectral density of the process is similar to the pro-
bability distribution function of the circular frequency of sinusoidal
waves by 'IT. An average power of one sinusoidal wave is expressed
through
1 T f
o
T 2
cos (l1t+~) dt
120
+ _s_i_n_2_4...:.~....!11-,,-t_+-,-~.:-) ]: (5.6)
Since phase angle ~ is assumed to be uniformly distributed, an average
power is expected to be 1/2. Therefore, if N sinusoidal waves are
superposed, the expected average power will be equal to N/2.
Using Toki's method, one can simulate a ground motion process
which has an arbitrary power spectral density giving an appropriate
probability distribution to the variable circular frequency 11.
2. Generation of a Random Process Having Nonstationary Frequency Content
To simulate a strong ground motion process as a nonstationary
random function in frequency content has been studied by a few, such as
by Bogdonaff et al. [7] or Hart et al. [38,18]. The former method is to
superpose sinusoidal waves having their own deterministic intensity
functions individually. The later one is to devide the process into
several separate segments and to generate a stationary process within
each segment having an appropriate power spectral density independently.
The method employed in this investigation to generate a non-
stationary random process is to divide a process into segments in a
continuous manner and superpose sinusoidal waves of which spectral
density function will coincide with the prescribed time dependent
spectral density function. The procedures are as follows:
(i) Let G denote the time dependent spectral density of motion
which is defined in the range 0 < t < T, where T designates
the duration of motion.
121
(ii) Generate a random number t which is uniformly distributed a
between a and T. Then one can obtain the spectral density
G = G (w,t) at time t. a a
(iii) Applying Toki's method, within a small segment of ~T being
centered at t, generate a sinusoidal wave of which o
circular frequency is n whose probability distribution is
similar to that of spectral density function G (w,t ), a
and
of which phase angle is distributed uniformly over the range
o ~ ¢ < 2n. One cannot always find an appropriate sinusoidal
wave at this step. If not, one should return to step (ii)
and then continue on.
(iv) To satisfy the continuous boundary at the ends of the
segment, the sinusoidal wave is passed through a filter
which has the property
H (t) 0
lim H(t)
~T t -+ t + a 2
In this report, a cosine bell
H(t) 0
t < t a
t > t a
0
function
t < 0
t > ~T
+
H (t) 1 2 (
2nt) 1 - cos l!.T
is adopted for the filtering procedure.
~T
2
~T -2
o < t < 6T
(5.7)
(5.8)
122
(v) Repeat the procedures from (ii) through (iv) and simulate a
nonstationary random function of duration T by superposing
a certain number of sinusoidal waves of short duration ~T.
(vi) Finally mUltiply the process by a deterministic intensity
function which gives an appropriate nonstationarity to the
intensity of motion.
It should be noticed that the variance of the process, which
is given by
E [a(t ) a(t )J o 0 r
-00
G (w,t ) dw o
(5.9)
will be dependent upon time t , o
depending upon the shape of the time
dependent power spectral density function. If the variances of the
process vary with time t , o
one cannot produce random process a(t)
with an appropriate deterministic intensity by multiplying it with
intensity function ~(t).
Identical to the case of recorded strong ground motion, the
integrated velocity and double integrated displacement of the simulated
motion often have unreasonable diverging values with time. The baseline
correction of the simulated accelerograms are performed by fitting a
parabolic curve to the velocity history by a least square method [6].
As previously reported [23J, due to the parabolic curve correction, an
offset of the accelerogram is introduced at the ends of motion. There-
fore, a small linear correction is carried out at the beginning of the
accelerogram which will make it possible for the acceleration to start
from zero. This correction might cause the integrated velocity or
double integrated displacement curves to be altered but it makes
negligible changes in the acceleration curve.
123
C. EXAMPLES IN ONE-DIMENSIONAL FORM
1. Review of Past Simulated Motions
Spectral density function G(w) and deterministic intensity
function ~(t) are commonly recognized to be the most significant
characteristics for simulating a ground motion process. One can find
many types of spectral density and deterministic intensity functions
proposed previously. Most of them have been derived by analyzing actual
records and introducing empirical relations such as
h2
2 1 + 4
w g 2
v G(w)
g B
C 2 2
w + 4 h2 w
2 g 2 v v
g g
(5.10) [43]
G(w) 1 a K 1 a K
= - + 2 2 2 2 2 2
(w-8 ) + a (w+8 ) + a (5.11) [8]
2 (a
2 +8
2)
G(w) 2 w +
B il 4 222 (a. 2 +8 2 ) 2 0
w + 2(a. -8 ) w + (5.12) [4]
_b2w2 2
2 2 -4b w
G(w) Al e + A2 w e (5.13) [22]
2w
2 w
2 p
G(w) w e
3 'IT w
(5.14) [44]
P
to serve as the spectral density function for simulated motions [30].
Similarly, many investigators have established the shapes of intensity
functions which give deterministic nonstationarity to the process. The
functions
S (t)
Set) =
\ :ct ae
I 0 -ct ate
o
e
124
t
t
t
t
I-t/t p
< 0
> 0
< 0
> 0
I (ttlY 0
(5.15) [8]
(5.16) [7]
t < 0
(5.17) [41] t > 0
t < 0
(5.18) [44] t > 0
o < t < t - 1
Set) (5.19) [1] I tl < t < t 0
-c(t-t ) 2
I 2
t2 < t e 0
I (a1 t)
-ct + a2
e 2
leal + a2
t) -ct
e
and those by Jennings, et al. [23] have been employed for this purpose.
2. Characteristics of Simulated Motions
Four samples are produced which have nonstationary properties
in both frequency content and intensity. For a spectral density function,
the type which has been proposed by Kanai and Tajimi [43] is employed,
G G(w,t) 2
1 + 4 h2 w g 2
v = B (5.21)
2 4 h
2 w +
2 g v
g
125
in which w denotes circular frequency and \! g
and h g
represent the
dominant circular frequency and a parameter which indicates sharpness of
the peak, respectively. The dominant circular frequency
to be dependent upon time, having the form
\! \! (t) g g
a c
bt + 1
\! g
is assumed
(5.22)
Coefficients a, band c in Eq. (5.22) are fixed as 5.027 x 101 ,
-2 1.550 x 10 and 2.041, respectively, in which case the dominant circular
frequencies at time t 0, 6 and 20 second become
\! (0 ) 2'Tf • 8 g
\! (6) 2'Tf . 5 g
\! (20) = 2'Tf . 1 g
showing a rapid change in the dominant frequency with time. The parameter
h is fixed at 0.3 over the entire duration of motion. g
For a deterministic intensity function, the shape
-ct e (5.23)
proposed by Iyengar [22] is applied. Coefficients al
, a2
and care
-1 determined as 0, 4.53 x 10 and 1/6, in which case the resulting maximum
intensity of s(6) becomes unity and ratios s(12)/s(6) and s(20)/s(6)
become 0.74 and 0.32, respectively.
As derived from Eq. (5.9), the variances of the process are
given through
E[a(t) aCt )] o 0
=
where V is equal to
(X)
J
(X)
J _(X)
126
G(W,t ) dw o
2 1 + 4 h 2 W
g7 g
+ 4 h2 g
2 B dw
W
2 v
g
V (t ) g 0 2 2 2 2
(l-V) + 4 h v g
v (t ) • C • B g 0
B dV
w/v and C given by the integral can be g
(5.24)
expressed as C = C (h ). g
To give an appropriate deterministic intensity
by the shape function, the variances of the process should be distributed
uniformly, i.e. the integral of power spectral density function should
have identical values at every time t . o
The parameter h g
in this
investigation is assumed to be constant over the entire duration of
motion. Therefore, to yield uniform variances to the process, the
quantity B should be given by
B B (t )
-
o -B
V (t ) g 0
(5.25)
in which quantity B is constant either with circular frequency w or
with time t. Having obtained the process of uniform variances and of o
nonstationarity in frequency content using the spectral density function
G(W,t)
\! g
127
1 + 4
2) 1 <~
\! (t) g
h2 g
+ 4
2 W -
2 \)
h2
2 W
g 2 \)
g
, the process is multiplied by the deterministic
-B--\)
g
(5.26)
intensity function given by Eq. (5.23) and the zero baseline is deter-
mined by a parabolic curve to give reasonable velocity and displacement
time histories. Figures 5.1a and S.lb show the prescribed properties
of the deterministic intensity function and the spectral density func-
tion, respectively. It might be noticed that the deterministic function
is plotted in term of
l;;" (t) 1
l;;(t) (5.27) 12
and the spectral density is expressed in terms of the Fourier amplitude.
3. Presentation of Examples
Based upon the prescribed power spectral density and deter-
ministic intensity, four samples are produced. The time histories and
their spectral diagrams obtained by a Fourier amplitude spectral analysis
with a moving-window technique are presented in Figs. 5.2 through 5.5.
These spectral diagrams are evaluated by passing the filter of con-
volution weights and smoothed. The shaded zones represent the dominant
frequency with time t , o
which were obtained in identically the same
manner described in the previous chapter.
4. Observation of Results
For illustration, the time dependent characteristics of the
dominant frequency is assumed to vary with time in a rapid manner in
128
these examples. The sample random processes seem to be obtained with
much success. The general tendency of the time dependency of frequency
content coincides well with that of the prescribed spectral density and
the intensity agrees well with the prescribed deterministic shape
function.
The method introduced in this investigation cannot always be
applied to the simulation of a random process having an arbitrary non-
stationarity in both frequency content and deterministic intensity. The
difficulty lies in determining a form of time dependent spectral density
function which has an arbitrary form at any time but retains identical
variances, i.e. the same quantity of power over the entire duration of
motion. For example, if in Eq. (5.21) both parameters h and V were g g
assumed to change with time, the evaluation of quantity B to give
uniform variances over the entire duration of motion might become
difficult. It is concluded, however, that if a spectral density function
having a shape which varies with time but having power independent of
time can be found, the random process having nonstationarity in both
frequency content and deterministic intensity can be simulated.
D. EXM1PLES IN THREE-DIMENSIONAL FORM
1. Characteristics of Simulated Motions
Previously general characteristics of motion along a set of
principal axes are evaluated by analyzing recorded motions produced by
the San Fernando earthquake and a method to generate a nonstationary
process both in frequency content and intensity is introduced. As the
principal axes of motion are defined along which the components of
motion are statistically independent of one another, the methodology
used to simulate a one-dimensional nonstationary process can be extended
129
to the simulation of three components of motion. If desired, one can
easily obtain components of motion along structural principal axes for
use in dynamic analysis by multiplying the principal components by a
transformation matrix which defines coordinate rotation between the
principal axes of motion and the principal axes of the structural system.
The shapes of spectral density function along principal axes
are specified to take similar forms given by
4 .h2
2 1 + W
:1. g -.G(w,t) B
(5.28) :1. (1 - 2 2 .\)
W + 4 h W :1. g 2 i g 2
.\) .\) :1. g :1. g
i 1. 2,3
which is identical to that used in the one-dimensional simulation. In
the three-dimensional simulation, the dominant circular frequency which
is assumed to be time dependent will take a form changing moderately
with time, i. e.
. \) . \) (t) :1. g :1. g
2TI (at + b) (5.29)
i 1,2,3
where coefficients a and b are equal to -2/25 and 17/5, respectively,
and the resulting dominant circular frequencies v (5) i g
and \) (30) i g
(i=1,2,3) equal 6TI and 2TI, respectively. The parameter h i g
in the
power spectral density function is assigned values 0.2, 0.3 and 0.6 for
the major, intermediate and minor principal axes, respectively. These
values reflect the finding that the minor component of motion consists
of a wider frequency range than the intermediate component and that the
intermediate component consists of a wider frequency range than the major
130
component. As the number of sinusoidal waves being superposed is
identically the same among the three components, the quantities of
average power of these simulated components are specified to be identical.
Thus, one can obtain the processes of which variances are distributed
uniformly not only over the entire duration of motion but also among the
three components. Therefore, appropriate intensities can be given by
mUltiplying principal motions by their corresponding deterministic
intensity functions. The deterministic intensities of motion along the
major, intermediate and minor principal axes are specified as shown at
the bottom of Fig. 5.6 by the solid, intermediate-dashed and short-dashed
curves, respectively.
After the components'of motion along a set of principal axes
have been generated, assuming the principal axes of the structural system
to be directed to the North-South, East-West and vertical, the components
along a set of structural axes are obtained for use in dynamic analyses
by multiplying the principal motions by the transformation matrix as
follows:
aNS(t) al
(t)
aEW(t) = T(t) a2
(t) (5.30)
aUD(t) a3
(t)
where
sin¢l cosel sin¢2 cose
2 sin¢3 cose3
T sin¢l sinel sin¢2 sin8
2 sin¢3 sin83
(5.31)
cos¢l cos¢2 cos¢3
131
in which ¢. and 8. (i = 1,2,3) represent vertical and horizontal 1 1
directional angles of the corresponding principal axes. The trans-
formation matrix T(t) is mathematically a transpose matrix of the
principal transformation matrix P(t). Though the matrix T contains
six variables, ¢l' 81
, ¢2' 82
, ¢3 and 83
, they are interdependent
through the orthogonal conditions.
(5.32)
If ¢3 equals zero, the minor principal axis is directed vertical in
which case cos¢l and cos¢2 equal zero; therefore, the quantity
cos(81
- 82
) also equals zero, i.e. the major and intermediate principal
axes are directed horizontal and the angle between them is set at 90°.
In the upper diagrams of Fig. 5.6, the time dependent directions of
principal axes are prescribed along with the direction to the possible
epicenter which is indicated by a long-dashed straight line. As
observed in these diagrams, the major principal axis which is shown by
a solid line is prescribed to be directed to the epicenter, N30oE,
during the high intensity motion and the major and intermediate
principal axes are assumed to change their positions with each other
after the period of high intensity motion. It should be pointed out the
horizontal direction angle 8. when the corresponding angle ¢. equals 1 1
zero does not play a significant role in the transformation matrix.
2. Presentation of Examples
Using a digital computer, four sample accelerograms are pro-
duced along a set of principal axes having properties of spectral density
132
and deterministic intensity as shown in Fig. 5.6. A set of structural
principal axes is assumed to be directed to the North-South, East-West
and vertical directions. Assuming the horizontal and vertical direction
angles of principal axes to be time dependent as represented in Fig. 5.6,
the simulated components along the principal axes are transformed to
three components of motion along the assumed axes of the structural
system. The resulting components of motion are shown in Figs. 5.7a
through 5.7d, respectively.
Assuming that these accelerograms were recorded by an
accelerograph installed in the structural system and applying the
identical computer routines which have been used to analyze the ground
motions of the San Fernando earthquake, principal variances, directions
of principal axes and frequency content of motion along the principal
axes are evaluated. The resulting properties of principal variances and
directions of principal axes are presented in Figs. 5.8a through 5.8d
and the corresponding time dependency of frequency content are represented
in Figs. 5.9a through 5.9d, respectively. These diagrams are plotted
in identically the same manner as those previously shown for the
accelerograms recorded during the San Fernando earthquake.
3. Observation of Results
In this investigation, only four samples of the simulated
ground motion process are presented. Their characteristics in terms of
principal variances and directions of principal axes, and frequency
content coincide well with the prescribed properties. Several features
can be noted for these samples as follows:
(1) The direction of the major principal axis oscillates and some
times can deviate up to 30° from the prescribed direction which is
towards the expected epicenter.
133
(2) At the end of sample records when the intensities of motion
are small, the directions of principal axes approach N600W and
N300E which coincides well with the prescribed directions.
(3) The sUdden interchange of directions of the major and inter
mediate principal axes can be explained by the cross-over in
their intensities as measured by principal variances.
(4) The direction of the minor principal axis which is prescribed
to be vertical happens to be nearly horizontal during the later
period of motion when the minor and intermediate principal
variances approach each other in value; thus, allowing the
interchange of position to take place. This same behavior can
be observed for recorded motions, especially, for those obtained
within high-rise buildings near the epicenter.
(5) Since the actual direction of the minor principal axis shifts
slightly from its prescribed vertical position, its horizontal
direction can easily take any position; thus, the horizontal
direction of the minor principal axis is observed to oscillate
significantly. Generally, however, the horizontal direction
of the minor principal axis unexplainably coincides with that
of the intermediate principal axis over large periods of
duration; see sample No.3.
(6) Even though the filtering procedure using convolution weights
was applied twenty times, the evaluated spectral density
shapes oscillate irregularly. The general characteristics of
frequency content as prescribed, however, are suitably
reproduced.
134
As indicated by the above observations, the properties of the
simulated motions reveal complexities similar to the characteristics of
the real motions recorded during the San Fernando earthquake. Therefore,
the simple ground motion model prescribed appears to be adequate.
(I)
(II)
(III)
TA
BL
E
5.1
SC
HE
MA
TIC
D
IAG
RA
M
OF
SIM
UL
AT
ION
O
F N
ON
STA
TIO
NA
RY
RA
ND
OM
PR
OC
ESS
[3
0]
To
mu
ltip
ly b
y
To
pass
th
rou
gh
a
an in
ten
sit
y
fil
ter to
yie
ld a
n
-fu
ncti
on
to
g
ive
ap
pro
pri
ate
p
ow
er -
a d
ete
rmin
isti
c
sp
ectr
al
den
sity
in
ten
sit
y
to th
e p
rocess
To
gen
era
te
a st
ati
on
ary
-
-w
hit
e
no
ise
To
pass
th
rou
gh
a
To
mu
ltip
ly b
y
fil
ter to
yie
ld
an
an in
ten
sit
y
"--
ap
pro
pri
ate
po
wer
fu
ncti
on
to
g
ive -
spectr
al
den
sity
a
dete
rmin
isti
c
to th
e p
rocess
in
ten
sit
y
r---
To
gen
era
te
a T
o m
ult
iply
by
ra
nd
om
pro
cess
an
in
ten
sit
y
hav
ing
an
fu
ncti
on
to
g
ive
ap
pro
pri
ate
po
wer
a
dete
rmin
isti
c
sp
ectr
al
den
sity
in
ten
sit
y
-~--
Sim
ula
ted
M
oti
on
s
I-'
w
U1
,.....,
--l
« t.:> \.oJ
U U
«
,....., N
I I..J
a w a:: lL..
1 • 2
• 6
0
-.6
- 1 • 2
6.0
6.0
1.0
2.0
0
136
DETERMINISTIC INTENSITY FUNCTION IYENGAR S INTENSITY fUNCTION
0
TIME TAJIMI
0 I i
I I 0
5 • 0 10. 0
TIME
DEPENDENT
15.0 20.0
( SEC . )
SPECTRAL S SPECTRAL DENSITY fUNCTION
5. 0 10. 0 15. 0 20. 0 i i i i i i i i i i i i i i i i
SPECTRAL DENSITY
I I I I I I I I I I I I i I I s. 0 1 O. 0 15. 0 20. 0
TIME ( SEC. )
25. 0 :3 O. 0
DENSITY
25. 0 30. 0 i i i i i i
I I I I I 25. 0 30. 0
Fig. 5.1 Prescribed intensity and spectral density function.
1 • '2
,.., ---l . 6 « ~
'-' 0
U
U -.6
<:
137
NON-STATIONARY * SAMPLE 1 * TAJIMI S SPECTRAL DENSITY AND IYENGAR S INTENSITY
fILTERED
-1.2
,.., N
:c ~
a w ~
IJ....
8.0
6.0
1.0
'2.0
0
0 5. 0 1 O. 0
TIME 15.0 20.0
( SEC . )
NON-STATIONARY * SAMPLE TAJIMI S SPECTRAL DENSITY AND IYENGAR
0 5. 0 1 O. 0 15. 0 20.0 I I I I I I I I I I I I I I I I I I I
SPECTRAL DENSITY
I I I I I I I I I I I I I I I I I
0 5. 0 1 O. 0 15. 0 '20. 0
TIME ( SEC. )
'25.0 :3 O. 0
1 * S INTENSITY
25.0 30. 0 I I I I I
I I I I I '25.0 30. 0
Fig. 5.2 Time history and time dependent spectral density for sample 1.
1 • 2
.......
.J • 6 < t.:)
'-J
0
U
U -.6
<
138
NON-STATIONARY * SAMPLE 2 * T~JIMI S SPECTR~L DENSITY ~ND IYENG~R S INTENSITY FILTERED
- 1 • 2
,....., N
:c '-'
a w a:: lL.
8. 0
6. 0
LO
2.0
0
0 5. 0 10. 0
TIME 15.0 20.0
( SEC • J
NON-STATI ONARY * SAMPLE T~JIMI S SPECTRAL DENSITY ~ND IYENGAR
0 5 • 0 10. 0 15. 0 20. 0 I I i I I i I I I i I i i I i i i i I
SPECTRAL DENSITY
I I I I I I I I I I I I I I I I I I I
0 5.0 10. 0 15. 0 20. 0
TIME ( SEC . )
25.0 30.0
2 * S INTENSITY
25. 0 30. 0 i i I i I
I I I I i I I
25. 0 30.0
Fig. 5.3 Time history and time dependent spectral density for sample 2.
1 • 2
r-'I
..J . 6 « t.:)
\..oJ
0
U
U -.6
«
139
NON-STATIONARY * SAMPLE 3 * TAJIMI S SPECTRAL DENSITY AND IYENCAR S INTENSITY fILTERED
- 1 • 2
,-,
N
:c \..J
a w 0::
lL..
0 5.0
NON-STATI
10.0
TIME
ONARY
15.0 20.0 25.0 '30.0
( SEC . )
* SAMPLE 3 * TAJIMI S SPECTRAL DENSITY AND IYENGAR S INTENSITY
0 5 • 0 10. 0 15. 0 '20. 0 '25. 0 i i i , i i i , i i i i i I i i i i i , i i
SPECTRAL DENSITY 8. 0
6. 0
1.0
2. 0
0
I I I I I L , I I I I I I I I I I I I I 0 5.0 10. 0 15. 0 20. 0 25.0
TIME ( SEC . )
Fig. 5.4 Time history and time dependent spectral density for sample 3.
30. 0 i i
I I 30.0
1 • 2
,......
..J • 6 <: t:)
~
0
u U - . 6
<:
140
NON-STATIONARY * SAMPLE 4 * TAJIMI S SPECTRAL DENSITY AND IYENGAR S INTENSITY
fILTERED
- 1 • 2
,...... N
:r: ~
a w Cl::
li.-
B. 0
6 • 0
LO
2 • 0
D
0 5. 0 10. 0
TIME 15.0 20.0
( SEC . )
NON-STATIONARY *" SAMPLE TAJIMI s SPECTRAL DENSITY AND IYENGAR
0 5 • 0 10. 0 1 S • 0 20. 0 I I I I I I I I I I I i i I I I I I
SPECTRAL DENSITY
i I I I I I I I I I I I I I I 0 5.0 1 O. 0 1 S • 0 20. 0
TIME ( SEC . )
25. 0 :3 O. 0
4 *" S INTENSITY
2 S • 0 :3 0 . 0 I I I I I
I I I I I
2 S. 0 :3 o. 0
Fig. 5.5 Time history and time dependent spectral density for sample 4.
Fig. 5.6
141
TIME DEPENDENT CHARACTERISTICS DIRECTION ANO INTENSITY Of PRINCIPAL COMPONENTS
5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
TIME DEPENDENT CHARACTERISTICS SPECTR~L DENSITY fUNCTIONS ALONG PRINCIPAL AXES
B.O
N 6.0 :c
a w
1.0
Ct:: 2.0
8. 0
N 6.0 :c
a w
1. 0
Ct:: 2.0
5. 0 I I I
10. 0 I I ,
15. 0
MAJOR
20. a I , i
INTERMEDIATE
MIN a R
TIME (SEC.)
25.0 , I ,
30. a t I
Prescribed properties of three-dimensional stochastic model.
ON
-ST
AT
ION
AR
Y
* S
AM
PL
E
3-
0 *
NO
N-S
TA
TIO
NA
RY
*
SA
MP
LE
3-~
2 *
TH
RE
E
TR
AN
SL
AT
ION
AL
C
OM
PO
NE
NT
S O
f G
RO
UN
D
MO
TIO
N
TH
RE
E
TR
AN
SL
AT
ION
AL
C
OM
PO
NE
NT
S
Of
GR
OU
ND
M
OT
ION
1.2
C
OM
P N
S 1
. '2
r C
OM
P N
S
-'
.6
-'
.6
<
<
t.:>
t.:>
U
U
U
-.6
U
-
. 6
<
<
-1
.2
-1
.2
5.0
1
0.0
1
5.
0 2
0.0
2
5.0
3
0.
0 5
. 0
10
.0
15
. 0
20
. 0
25
. 0
30
.0
1.
'2 r
CO
MP
EW
" f
CO
MP
EW
I I II
II!
Iii, III ! I
f--'
-'
.6
-'
.6
"'" <
<
[\
J
t.:>
t.:l
U
U
U
-.6
U
-.6
<
<
-1
• 2
-1
• 2
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
0.0
5
. 0
10
.0
15
. 0
20
. 0
25
. 0
30
.0
1.
'2 r
CO
MP
UD
1
.2
r C
OM
P U
O
-'
.6
-'
.6
<
<
t.:>
t.:l
U
U
U
-.6
U
-.6
<
<
-1
. '2
-1
.2
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
0.0
5
.0
10
.0
15
.0
20
.0
25
. a
30
.0
TIM
E
( S
EC
. )
TIM
E
( S
EC
. )
Fig
. 5
.7
Tim
e h
isto
ries
of
sim
ula
ted
accele
rog
ram
.
NO
N-S
TA
TIO
NA
RY
*
SA
MP
L E
3
-0
3 *
NO
N-S
TA
TIO
NA
RY
*
SA
MP
LE
3
-0
4 *
TH
RE
E
TR
AN
SL
.... T
ION
AL
C
OM
PO
NE
NT
S O
f C
RO
UN
O
MO
TIO
N
TH
RE
E
TR
AN
SL
AT
ION
AL
C
OM
PO
NE
NT
S O
f C
RO
UN
O
MO
TIO
N
1.2
C
aMP
N
S 1
. 2 r
CaM
P
NS
-1
.6
-1
.6
« «
t.:>
t.:>
U
U
u -.6
u
-•
6
« «
-1
.2
-1
• 2
5.0
1
0.
0 1
5.
0 2
0.
0 2
5.
0 :1
0.0
S
. 0
10
. 0
15
. 0
20
.0
25
. 0
30
.0
1.
'2 r
CaM
P
EW
1.2
C
aMP
EW
-1
....J
L,
f--'
.6
.6
!II
..,. «
« w
t.:
> t.:
>
U
U
U
-•
6 u
-.6
« «
-1
. '2
-1
. '2
5 •
0 1
0 •
0 1
5 •
0 2
0.0
2
5.
0 3
0.0
S
. 0
10
. 0
15
.0
20
. 0
25
. 0
30
. 0
1.2
C
aMP
uo
1
. 2
CaM
P
uo
-1
• 6
....J
. 6
« «
t.:>
t.:>
U
U
U
-•
6 u
-.
6
« «
-1
• '2
-1
• '2
5 •
0 1
a . 0
1
5.
0 2
0.
0 2
5.
O·
30
.0
S.
0 1
0.
0 1
5.
0 2
0.
0 2
5.0
3
0.0
TIM
E
( S
EC
. )
TIM
E
( S
EC
. )
Fig
. 5
.7
( co
nti
nu
ed
)
90
60 I Q. 30
0
90
<C 45
f-w 0 I f- - '\ 5
-90
• 4
· 3 <C L D · 2
(j)
« fw
.1
0
90
45
o
-90
« L
• 4
· 3
D .2
.1
o
144
NON-STATIONARY * SAMPLE 3-~ * DIRECTION AND INTENSITY OF PRINCIPAL COMPONENTS
o 5. 0 1 O. 0
\ . , J \/~;
15. 0 20. 0
/
25. 0
'-, . ,
30. 0
--:---:.-. , . , --+- --
• r--~/ I
0 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
NON-STATIONARY * SAMPLE 3-D 2 * DIRECTION AND INTENSITY OF PRINCIPAL COMPONENTS
__ ~ __ +- -\ _ I- --\- - - - ~. ",<- c--\ ".- __ = '. " -- " .', / , I \ "'1. / '\ /, \
1 ... / \1, I '/ 'I. V
'1\\ / ~~ ..- .......... ~ ... -~-- ........... _,.- v ,
o 5. 0 10.0 15.0 20.0 25. 0 30. 0
TIME (SEC.)
Fig. 5.8 Time dependent directions of principal axes and square root of principal variances for simulated motions.
9 a 60
:c (L 30
a
9 a
« 4 5
I-w a :c I- - 45
-90
. 3
« .2 L C>
(J) .1
o
90
60 :c Cl. 30
0
90
« 45
I-w 0 :c I- - 45
- 9 a
. 4
.3 « L C> .2
(J)
.1
o
145
NON-STATIONARY * SAMPLE 3-D 3 * DIRECTION AND INTENSITY OF PRINCIPAL COMPONENTS
o
o
, , '" ,
/ ~:
5. a
5. 0
10. a 15. a 20. 0
i·~\/ I~
10.0 15.0 20.0
TIME (SEC.)
NON-STATIONARY * SAMPLE 3-D 4 *
25. a
25. 0
DIRECTION AND INTENSITY OF PRINCIPAL COMPONENTS
o 5. a
o 5. 0
10. 0 15. a 20. a
I"~ I'
~,~
,----------::-:,-
,. , I" I
1 I '" I: '
10.0 15.0 20.0
TIME (SEC.)
Fig. 5.8 (continued)
25. a
25. a
3 a . a
30. 0
3 a . a
30. a
8.
0
N
6.0
:I
:
LO
a w
a: 2
.0
lL.
8.0
N
6.0
:I
:
LO
a w
a: 2
.0
lL.
8.0
N
6.0
:I
:
LO
a w
a: 2
.0
lL.
NO
N-S
TA
TIO
NA
RY
*
SA
MP
LE
3
-D
* N
ON
-ST
AT
ION
AR
Y
* S
AM
PL
E 3-~
2 *
SP
EC
TR
AL
D
EN
SIT
Y
Of
MO
TIO
N
AL
ON
G
PR
INC
IPA
L
AX
ES
S
PE
CT
RA
L
OE
NS
ITY
O
f M
OT
ION
A
LO
NG
P
RIN
CIP
AL
A
XE
S
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
0.0
i
" iii
i I
• • Iii ii,
i s.
0
1 5
.0
20
.0
25
.0
~.o
10
.0
MA
JOR
8
.0
) 0
N
6.0
:I
:
LO
a w
a:
2.0
lL
.
8.
0 v
v
N
6.0
:I
:
1.0
a w
a: 2
.0
lL.
MIN
OR
\J'f~~
8.
0
N
6.0
:I
:
"0
a w
a:
2.0
lL
.
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
0.0
5
.0
10
.0
15
.0
20
.0
25
.0
30
.0
TIM
E
(SE
C.)
T
IME
(S
EC
.)
Fig
. 5
.9
Tim
e d
ep
en
den
t fr
eq
uen
cy
d
istr
ibu
tio
n fo
r sim
ula
ted
mo
tio
ns.
f-'
01::>
(j\
B.
0
N
6.0
::r
:
LO
a w
IX
2.0
u.
.
a. 0
N
6.0
::r
:
'-0
CJ
W
IX
2.0
u.
.
B.
0
N
6.0
::r
:
00
w
1.0
'" 2
.0
u..
NO
N-S
TA
TIO
NA
RY
;I<
S
AM
PL
E
3-D
3
;I<
SP
EC
TR
AL
D
EN
SIT
Y
OF
MO
TIO
N
AL
ON
G
PR
INC
IPA
L
AX
ES
5.
0 1
0.
0 1
5.
0 2
0.
0
INT
ER
ME
DIA
TE
v
V Q~cjD
5.
0 1
0.0
1
5.0
2
0.0
TIM
E
(SE
C.)
25
. 0
25
. 0
\
30
.0
-r-
30
.0
Fig
. 5
.9
8.0
N
6.0
::r
: a w
1.0
'" 2
.0
lL..
8.
0
N
6.0
::r
:
a w
,-0
IX
2.
0 lL
..
B.
0
N
6.0
::r
: '" W
1.0
IX
2.0
lL
..
NO
N-S
TA
TIO
NA
RY
;I>
S
AM
PL
E
3-D
'4
* S
PE
CT
RA
L
DE
NS
ITY
O
f M
OT
ION
A
LO
NG
P
RIN
CIP
AL
A
XE
S
5.
D
10
. 0
15
. a
v ,
20
.0
25
.0
30
.0
,..,
5.0
1
0.0
1
5.0
2
0.0
2
5.0
3
0.0
TIM
E
(S
EC
.)
(co
nti
nu
ed
)
I--' ~
-...J
148
VI. CONCLUDING STATEMENT
This report presents the results of an analysis of the three
dimensional ground motions recorded during the San Fernando earthquake
of February 9, 1971. The methods adopted make use of the concept of
principal axes of ground motion and the moving-window technique in both
the time and frequency domains.
The objective of the analysis is to establish a stochastic
model for three-dimensional ground motions which reflects the significant
statistical properties of the motions recorded during the San Fernando
earthquake. Because of the existance of principal axes, the three
components of motion are statistically independent, provided they are
directed along a set of principal axes. The resulting model is
represented by the product of a deterministic intensity function and a
constant intensity process having a variable frequency content with
time.
It is believed that the general form of the model developed
herein represents an improvement over that most commonly used, i.e.
the product of a deterministic intensity function and a stationary
process. One must be careful, however, in assuming that the quantitative
form of this model would apply to the motions produced by other earth
quakes even when their magnitudes and epicentral distances are similar
to those of the San Fernando earthquake. Only by conducting studies
similar to those reported herein for future earthquake motions can the
quantitative differences and their causes be determined.
149
VI I . REFERENCES
1. Amin, M and A. H.-S. Ang "A Nonstationary Stochastic Model for Strong-Motion Earthquakes," Civil Engineering Studies, Structural Research Series No. 306, University of Illinois, Urbana, Illinois, April 1966.
2. Arias, A. "A Measure of Earthquake Intensity," in "Seismic Design for Nuclear Power Plants," edited by R. J. Hansen, MIT Press, 1970.
3. Barrows, A. G., J. E. Kahle, F. H. Weber, Jr. and R. B. Saul "Map of Surface Breaks Resulting from the San Fernando, California, Earthquake of February 9, 1971," in "San Fernando, California, Earthquake of February 9, 1971," U.S. Department of Commerce, National Oceanic and Atmospheric Administration.
4. Barstein, M. F. "Application of Probability Methods for Design the Effect of Seismic Forces on Engineering Structures," Proceedings of the 2nd World Conference on Earthquake Engineering, Tokyo and Kyoto, July 1960.
5. Bendat, J. S. and A. G. Piersol "Random Data: Analysis and Measurement Procedures," WileyInterscience, 1971.
6. Berg, G. V. and G. W. Housner "Integrated Velocity and Displacement of Strong Earthquake Ground Motion," Bulletin of the Seismological Society of America, Vol. 51, No.2, April 1961.
7. Bogdanoff, J. L., J. E. Goldberg and M. C. Bernard "Response of a Simple Structure to a Random Earthquake-Type Disturbance," Bulletin of the Seismological Society of America, Vol. 51, No.2, April 1961.
8. Bolotin, v. v. "Statistical Theory of the Aseismic Design of Structures," Proceedings of the 2nd World Conference on Earthquake Engineering, Tokyo and Kyoto, July 1960.
9. Bath M. "Spectral Analysis in Geophysics," Elsevier, 1974.
10. Bycroft, G. N. "White Noise Representation of Earthquakes," Journal of the Engineering Mechanics Division, ASCE, EM2, April 1960.
11. Chen, C. "Definition of Statistically Independent Time Histories," Technical Notes, Journal of the Structural Division, ASCE, ST2, February 1972.
150
12. Crouse, C. B. "Engineering Studies of the San Fernando Earthquake," Report of Earthquake Engineering Research Laboratory, EERL 73-04, California Institute of Technology, Pasadena, California, March 1973.
13. Duke, C. M., J. A. Johnson, Y. Kharraz, K. W. Campbell and N. A. Malpiede "Subsurface Site Conditions and Geology in the San Fernando Earthquake Area," School of Engineering and Applied Science, UCLA-ENG-7206, University of California, Los Angeles, California, December 1971.
14. Duke, C. M., K. E. Johnsen, L. E. Larson and D. C. Engman "Effects of Site Classification and Distance on Instrumental Indices in the San Fernando Earthquake," School of Engineering and Applied Science, UCLA-ENG-7247, University of California, Los Angeles, California, June 1972.
15. Foutch, D. A., J. E. Luco, M. D. Trifunac and F. E. Udwadia "Full Scale, Three-Dimensional Tests of Structural Deformations during Forced Excitation of a Nine-Story Reinforced Concrete Building," Proceedings of the U.S. National Conference on Earthquake Engineering, Ann Arbor, Michigan, June 1975.
16. Goto, H. and K. Toki "Structural Response to Nonstationary Random Excitation," Proceedings of the 4th World Conference on Earthquake Engineering, Chile, January 1969.
17. Hanks, T. C. "Strong Ground Motion of the San Fernando, California, Earthquake: Ground Displacements," Bulletin of the Seismological Society of America, Vol. 65, No.1, February 1975.
18. Holman, R. E. and G. C. Hart "Structural Response to Segmented Nonstationary Random Excitation," Journal of American Institute of Aeronautics and Astronautics, Vol. 10, No. 11, November 1972.
19. Housner, G. W. "Characteristics of Strong-Motion Earthquakes," Bulletin of the Seismological Society of America, Vol. 37, No.1, January 1947.
20. Housner, G. W. "General Features of the San Fernando Earthquake," in "Engineering Features of the San Fernando Earthquake, February 9, 1971, edited by P. C. Jennings, Report of Earthquake Engineering Research Laboratory, EERL 71-02, California Institute of Technology, Pasadena, California, June 1971.
21. Iemura, H. and P. C. Jennings "Hysteretic Response of a Nine-Story Reinforced Concrete Building during the San Fernando Earthquake," Report of Earthquake Engineering Research Laboratory, EERL 73-07, California Institute of Technology, Pasadena, California, October 1973.
151
22. Iyengar, R. N. and K. T. S. R. Iyengar "A Nonstationary Random Process Model for Earthquake Accelerograms," Bulletin of the Seismological Society of America, Vol. 59, No.3, June 1969.
23. Jennings, P. C., G. W. Housner and N. C. Tsai "Simulated Earthquake Motions," Report of Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, April 1968.
24. Keightley, W. O. "A Strong-Hotion Accelerograph Array with Telephone Line Interconnections," Report of Earthquake Engineering Research Laboratory, EERL 70-05, California Institute of Technology, Pasadena, California, September 1970.
25. Kubo, T. and J. Penzien "Characteristics of Three-Dimensional Ground Motions, San Fernando Earthquake," Proceedings of the Review Meeting U.S.-Japan Cooperative Research Program in Earthquake Engineering with Emphasis on the Safety of School Buildings, Honolulu, Hawaii, August 1975.
26. Kubo, T. and J. Penzien "Characteristics of Three-Dimensional Ground Motions along Principal Axes, San Fernando Earthquake," submitted to the 6th World Conference on Earthquake Engineering, New Delhi, January 1977.
27. Kuroiwa, J. H. "Vibration Test of a Multistory Building," Report of Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, June 1967.
28 . Li u , S . C . "Evolutionary Power Spectral Density of Strong-Motion Earthquakes," Bulletin of the Seismological Society of America, Vol. 60, No.3, June 1970.
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38. Saragoni, G. R. and G. C. Hart "Nonstationary Analysis and Simulation of Earthquake Ground Motions," School of Engineering and Applied Science, UCLA-ENG-7238, University of California, Los Angeles, California, June 1972.
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154
APPENDIX A
Results of (i) time dependent directions of principal axes and square root of principal variances, (ii) frequency dependent directions of principal axes and square root of principal variances, (iii) time dependent principal variances and principal cross correlation coefficients, and (iv) time dependent frequency distribution for motions in area F.
TA
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A.l
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162
STATION NO. 157 3103 12N.118 15
~1S fICIJERO~ STRE.ET. SUB-BASEMENT, LOS ANGELES. CAL.
5.0 15.0 20.0 25.0 30,0
>.0 , lS'.O 20'.0' , 25',0
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STATION NO. 151 3103 00N.118 15 06W
no S. GRAND AVENUE., 2ND fLOOR, LOS ANGELE.S. CAL.
, O! '\:0' , ~ 5 . 0' k 0'. 0 '2 5 '. 0 '3 0 '. a
TIME [SEC.)
STATION NO. 160 31 03 09N. 118 15 28W d STATION NO. 163 3102 57N.118 15 l6W
5)5 S. FREMONT, AlitNUE. BASEMENT, LOS ANGELES. CAL. 611 WEST S!)ITH STREET. a .... SE.MENT. lOS ..... NGELES. C .... L.
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TIME [SEC.) TIME (SEC.)
Fig. A.4 Time dependent frequency distribution in area F.
163
e STATION NO. 166 3402 SON.IIB IS 14W f STATION NO. 175 3102 07N.IIB 15 03W
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Fig. A.4 (continued)
25.0 30.0
164
APPENDIX B
Results of time dependent directions of principal axes and square root of principal variances at stations not included in area groups A through F.
ST
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185
APPENDIX C
Results of frequency dependent directions of principal axes and square root of principal variances for motions in area groups C, D and E.
a S
TA
TIO
N
NO
. 1
25
3
44
0
30
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. C
.l
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t d
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on
s
of
pri
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squ
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C.
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00
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d
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of
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an
d
sq
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of
pri
ncip
al
vari
an
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are
a
E.
CA
l
.
1-'
ro
ro
189
APPENDIX D
Results of time dependent principal variances and principal cross correlation coefficients for motions in area groups C, D and E.
a S
TA
TIO
N
NO
. 1
25
3
44
0
30
N,1
18
2
6
24
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E.
I-'
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N
193
APPENDIX E
Results of time dependent frequency distribution for motions in area groups C, D and E.
194
a STATION NO. I2S 34 40 30N.118 26 24W b STATION NO. 126 34 38 30N. I 18 28 48W
LAiCE HUGHES, ARR .... Y STATION I, CAL.
INTERMEDIATE
~:: ~I ~~Cffi~~J4 ~::~~~ ..
o W
,"0
0: 2. 0 .. 5 . 0' , ! ~ 0 . 0 ' , t S'. 0 '2: 0'. 0' 2 5'. 0 :) 0 . a
TIME [SEC. I
~
N 6.0 :c
o w
<.0
cr: 2.0 ..
~
N 6.0 :c
o W 0:: 2.0 ..
~
N 6.0 :c
o W
,"0
0: 2. a ..
lAltE HUGHES, ARRAY STATtON i, C .... l.
5.0 10.0 15.0 20.0 :: 5.0 30. a , i, 'i'! iii iii'
0' " 5 . 0' , , I 0 . 0' ~ 5 • 0 2: 0 . 0 ' , 2: 5 • 0 ' , :3 0'. a
TIME [SEC.)
C STATION NO. 127 3436 30N.118 33 42W d STATION NO. 128 3434 18N.I18 33 36W
~
N 6.0 :c
,"0
o W 0:: 2.0
~
N :c ~
0
u.J
'" ..
o w
6.0
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2.0
cr: 2.0
U.
LAteE HUGtlES. ARRAY STATIQIi 9, CAL.
'5:0' , , ;0.0' ;5.0 20',0' 25.0 ' 30',0
TIME [SEC.I
~
N 6.0 :c
o W
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0:: 2.0 ..
~
N 6.0 :c ~
LO
a u.J
'" 2.0 ..
~
N 6.0 I
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U.
L .... KE HUGHES, APR"" STATION 12. C .... l.
i ,sic, i ~o.Q j IS.~ i i ?O"O fS.~ ~.o
a! , , 5.0' 10',0 15.0 ' 20.0 '25.0 30.0
TIME [SEC. I
Fig. E.l Time dependent frequency distribution in area C.
195
a STATION NO. 137 3403 00N,118 15 OOW b STATION NO. 118 3403 ZON, 11815 25W
\.I"'"TER AND POl.'ER BUILDING, BASEMENT. LOS .... NGELES. C .... t.
'--I........I..~.~'--'-_--'-------L.... ! ", '.....l.-J...... :J S.O 10.0 15.0 20.0 2S.::I 30.0
TIME (SEC.]
a w
1.0
0:: 2,0
"-
~
N 6.0 :c
a w
1.0
Q:; 2.0
"-
231 flGlJERO .... STREET. eASEMENT. lOS ANGELES. CAt.
~ ii' ,siD 10.0 15.~ i I ~Oi'O 25.0 30.0
s. a ~ 0 . 0 '1 S', 0 ' , 20', 0 ! , 25', 0 ':3 0'. a TIME (SEC.]
C STATION NO, 172 3403 26N,118 15 02W d STATION NO. 115 3403 Z5N, 11815 03W
aoo w. fIRST STREET, lSI fLOOR, LOS ANGELES, CAt.
5.0 I , lOi'~ , ; 15,'~ ~Oi'~" 25.~ 1,7°,,0
5.0 10.0 J S. 0' 20. 0 ! , h',o 30.0 TIME (SEC.1
~
N 6.0 :c
a w
1.0
cr:: 2.0
"-
~
N 6.0 :c
o W 0: "2. 0
"-
222 FICUEROA STREET, lS1 fLOOR, LOS .... NGlLE.S. Ci\l.
S. Q 10.0 15. a 20. a 25.0 30.0 i • , , i • ii' ( ii' iii iii, Iii. i, i • ( i
, , '5: 0' '10. 0 1 S', 0' 20.0 2S. 0 ':3 0' 0
TIME (SEC.]
Fig. E.2 Time dependent frequency distribution in area D.
196
a STATION NO. 42S 34 03 46N. I 18 24 S2W b STATION NO. 440 34 03 44N. 118 24 SOW
8.0
~
N 6.0 :c
0> W
<.0
a: 2.0
"-
'.0
~
N 6,0 :c
<.0
)800 CE,NltJRY PARK E ... ST. e .... SE.ME.NT P-'). LOS .... NGE.LE.S. C .... l. L8.aa. CENTURY PARI(' E.AST. P.A.RtnNG. 1ST LEVEL, lOS ..... NGE.LES, C .... l
o ,,5 i 0, i • ,10,' 9 i i : 51' 9 ; j 20. 9 ; ; 25,' 9 j lO,'o
o ! , , 's: 0' , , ; 0'. dr, ~ 5 (. 0 ' ( 2 a (. a ' ! 2 s'. 0 I I ') o!. 0 's.o' 'io'.o 15.0' , 20'.0 25',0 30',0
TIME (SEC.) TIME (SEC.)
C STATION NO. 184 34 03 3SN. 11824 S6W d STATION NO. 187 3403 14N.118 24 S8W
o W
8.0
<.0
a: :2.0
"-
8.0
~
N 6.0 :c
o W
<.0
0:: 2. 0 ...
o W
8.0
<.0
a:: 2.0 ...
1900 AVENl!E. Of TNt: ST .... RS. BASEtH.NT. lOS ANGE.LES. CAL.
5.0 10.0 15.0 20.0 25,0 )0.0 iii Ii, I , i • , i •• , , , i , i • I I , • i , i • I ,
0' "5:0!' io.o is',a" 20'.0 ' . h',o ' , 30'.0
TIME (SEC.)
8.0
~
N 6.0 :c
o W
<.0
cr. 2. 0
"-
1901 AVE. Of THE. ST"RS SlJBBSMT •• lOS ANGELES, CAL.
5 • 0, I , : 0t' 9, I 5. 0 i ~ at' 9 , I ? 5 • 0 .? 0,' 0
0' , "s:o 10.0 15,0 20'.0 '25.0 )0.0
TIME (SEC.I
Fig. E.3 Time dependent frequency distribution in area E.
197
APPENDIX F
computer program listings.
PP
OC
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16
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SH
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25
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2
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SH
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SH
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3
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SH
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3
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SH
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3
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SH
PE
3
40
S
HP
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0
SH
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3
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SH
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3
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S
HD
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38
0
SH
DE
3
90
SH
PE
4
00
S
HP
E
41
0
SH
PE
4
2 a
S
HP
E
43
0
SH
PE
4
40
SH
PE
4
50
S
HP
E
46
0
SH
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10
I'J
I-'
-J
EERC 67-1
EERC 68-1
EERC 68-2
EERC 68-3
EERC 68-4
EERC 68-5
EERC 69-1
EERC 69-2
'EERC 69-3
EERC 69-4 •
EERC 69-5
218
EAR'l'HQUAKE ENGINEERING RESEARCH CENTER REPORTS
"Feasibility Study Large-Scale Earthquake Simulator Facility," by J. Penzien, J. G. Bouwkamp, R. W. Clough and D. Rea - 1967 (PB 187 905)
Unassigned
"Inelastic Behavior of Beam-to-Column Subassemblages Under Repeated Loading," by V. V. Bertero - 1968 (PB 184 888)
"A Graphical Method for Solving the Wave ReflectionRefraction Problem," by H. D. McNiven and Y. Mengi 1968 (PB 187 943)
"Dynamic Properties of McKinley School Buildings," by D. Rea, J. G. Bouwkamp and R. W. Clough - 1968 (PB 187 902)
"Characteristics of Rock Motions During Earthquakes," by H. B. Seed, I. M. Idriss and F. W. Kiefer - 1968 (PB 188 338)
"Earthquake Engineering Research at Berkeley," - 1969 (PB 187 906)
"Nonlinear Seismic Response of Earth Structures," by M. Dibaj and J. Penzien - 1969 (PB 187 904)
"Probabilistic Study of the Behavior of Structures During Earthquakes," by P. Ruiz and J. Penzien - 1969 (PB 187 886)
"Numerical Solution of Boundary Value Problems in Structural Mechanics by Reduction to an Initial Value Formulation," by N. Distefano and J. Schujman - 1969 (PB 187 942)
"Dynamic Programming and the Solution of the Biharmonic Equation," by N. Distefano - 1969 (PB 187 941)
Note: Numbers in parenthesis are Accession Numbers assigned by the National Technical Information Service. Copies of these reports may be ordered from the National Technical Information Service, 5285 Port Royal Road, Springfield, Virginia, 22161. Accession Numbers should be quoted on orders for the reports (PB --- ---) and remittance must accompany each order. (Foreign orders, add $2.50 extra for mailing charges.) Those reports without this information listed are not yet available from NTIS. Upon request, EERC will mail inquirers this information when it becomes available to us.
EERC 69-6
EERC 69-7
EERC 69-8
EERC 69-9
EERC 69-10
EERC 69-11
EERC 69-12
EERC 69-13
EERC 69-14
EERC 69-15
EERC 69-16
EERC 70-1
EERC 70-2
219
"stochastic Analysis of Offshore Tower structures," by A. K. Malhotra and J. Penzien - 1969 (PB 187 903)
"Rock Motion Accelerograms for High Magnitude Earthquakes," by H. B. Seed and I. M. Idriss - 1969 (PB 187 940)
"structural Dynamics Testing Facilities at the University of California, Berkeley," by R. M. Stephen, J. G. Bouwkamp, R. W. Clough and J. Penzien - 1969 (PB 189 111)
"Seismic Response of Soil Deposits Underlain by Sloping Rock Boundaries," by H. Dezfulian and H. B. Seed - 1969 (PB 189 114)
"Dynamic stress Analysis of Axisymmetric Structures under Arbitrary Loading," by S. Ghosh and E. L. Wilson - 1969 (PB 189 026)
"Seismic Behavior of Multistory Frames Designed by Different Philosophies," by J. C. Anderson and V. V. Bertero - 1969 (PB 190 662)
"stiffness Degradation of Reinforcing Concrete Structures Subjected to Reversed Actions," by V. V. Bertero, B. Bresler and H. Ming Liao - 1969 (PB 202 942)
"Response of Non-Uniform Soil Deposits to Travel Seismic Waves," by H. Dezfulian and H. B. Seed - 1969 (PB 191 023)
"Damping Capacity of a Model Steel Structure," by D. Rea, R. W. Clough and J. G. Bouwkamp - 1969 (PB 190 663)
"Influence of Local Soil Conditions on Building Damage Potential during Earthquakes," by H. B. Seed and I. M. Idriss - 1969 (PB 191 036)
"The Behavior of Sands under Seismic Loading Conditions," by M. L. Silver and H. B. Seed - 1969 (AD 714 982)
"Earthquake Response of Concrete Gravity Dams," by A. K. Chopra - 1970 (AD 709 640)
"Relationships between Soil Conditions and Building Damage in the Caracas Earthquake of July 29, 1967," by H. B. Seed, I. M. Idriss and H. Dezfulian - 1970 (PB 195 762)
EERC 70-3
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220
"Cyclic Loading of Full Size Steel Connections," by E. P. Popov and R. M. Stephen - 1970 (PB 213 545)
"Seismic Analysis of the Charaima Building, Caraballeda, Venezuela," by Subcommittee of the SEAONC Research Committee: V. V. Bertero, P. F. Fratessa, S. A. Mahin, J. H. Sexton, A. C. Scordelis, E. L. Wilson, L. A. Wyllie, H. B. Seed and J. Penzien, Chairman - 1970 (PB 201 455)
"A Computer Program for Earthqualce Analysis of Dams~" by A. K. Chopra and P. Chakrabarti - 1970 (AD 723 994)
"The Propagation of Love Waves across Non-Horizontally Layered Structures," -by J. Lysmer /!l.Ild L. A. Drake -1970 (FE 197 896)
"Influence of Base Rock Characteristics on Ground Response," by J. Lysmer, H. B. Seed and P. B. Schnabel - 1970 (PB 197 897)
"Applicability of Laboratory Test Procedures for Measuring Soil Liquefaction Characteristics unde~ Cyclic Loading," by H. B. Seed and W. H. Peacock'-1970 (PB 198 016)
"A Simplified Procedure for Evaluating Soil Liquefaction Potential," by H. B. Seed ano. 1. M. Idriss - 1970 (PB 198 009)
"Soil Moduli and Damping Factors for Dynamic Response Analysis," by H. B. Seed and 1. M. Idriss - 1970 (PB 197 869)
"Koyna Earthquake and the Performance of Koyna Dam," by A. K. Chopra and P. Chakrabarti - 1971 (AD 731 496)
"Preliminary In-Situ Measurements of Anelastic Absorption in Soils Using a Prototype Earthquake Simulator," by R. D. Borcherdt and P. W. Rodgers 1971 (PB 201 454)
"Static and Dynamic Analysis of Inelastic Frame structures," by F. L. Porter and G. H. Powell - 1971 (PB 210 135)
"Research Needs in Limit Design of Reinforced Concrete Structures," by V. V. Bertero - 1971 (PB 202 943)
"Dynamic Behavior of a High-Rise Diagonally Braced Steel Building," by D. Rea, A. A. Shah and J. G. Bouwkamp - 1971 (PB 203 584)
EERC 71-6
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221
"Dynamic Stress Analysis of Porous Elastic Solids Saturated with Compressible Fluids," by J. Ghaboussi and E. L. Wilson - 1971 (PB 211 396)
"Inelastic Behavior of Steel Beam-to-Column Subassemblages," by H. Krawinkler, V. V. Bertero and E. P. Popov - 1971 (PB 211 335)
"Modification of Seismograph Records for Effects of Local Soil Conditions," by P. schnabel, H. B. Seed and J. Lysmer - 1971 (PB 214 450)
"static and Earthquake Analysis of Three Dimensional Frame and Shear Wall Buildings," by E. L. Wilson and H. H. Dovey - 1972 (PB 212 904)
"Accelerations in Rock for Earthquakes in the Western United States," by P. B. schnabel and H. B. Seed -1972 (PB 213 100)
"Elastic-Plastic Earthquake Response of Soil-Building Systems," by T. Minami - 1972 (PB 214 868)
"Stochastic Inelastic Response of Offshore Towers to Strong Motion Earthquakes," by M. K. Kaul - 1972 (PB 215 713)
"Cyclic Behavior of Three Reinforced Concrete Flexural Members with High Shear," by E. P. Popov, V. V. Bertero and H. Krawinkler - 1972 (PB 214 555)
"Earthquake Response of Gravity Dams Including Reservoir Interaction Effects," by P. Chakrabarti and A. K. Chopra - 1972 (AD 762 330)
"Dynamic Properties on Pine Flat Dam," by D. Rea, C. Y. Liaw and A. K. Chopra - 1972 (AD 763 928)
"Three Dimensional Analysis of Building systems," by E. L. Wilson and H. H. Dovey - 1972 (PB 222 438)
"Rate of Loading Effects on Uncracked and Repaired Reinforced Concrete Members," by S. Mahin, V. V. Bertero, D. Rea and M. Atalay - 1972 (PB 224 520)
"Computer Program for static and Dynamic Analysis of Linear structural Systems," by E. L. Wilson, K.-J. Bathe, J. E. Peterson and H. H. Dovey - 1972 (PB 220 437)
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222
"Literature Survey - Seismic Effects on Highway Bridges," by T. Iwasaki, J. Penzien and R. W. Clough -1972 (PB 215 613)
"SHAKE-A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites," by P. B. schnabel and J. Lysmer - 1972 (PB 220 207)
"Optimal Seismic Design of Multistory Frames," by V. V. Bertero and H. Kamil - 1973
"Analysis of the Slides in the San Fernando Dams during the Earthquake of February 9, 1971," by H. B. Seed, K. L. Lee, I. M. Idriss and F. Makdisi -1973 (PB 223 402)
"Computer Aided Ultimate Load Design of Unbraced Multistory Steel Frames," by M. B. EI-Hafez and G. H. Powell - 1973
"Experimental Investigation into the Seismic Behavior of Critical Regions of Reinforced Concrete Components as Influenced by Moment and Shear," by M. Celebi and J. Penzien - 1973 (PB 215 884)
"Hysteretic Behavior of Epoxy-Repaired Reinforced Concrete Beams," by M. Celebi and J. Penzien - 1973
"General Purpose Computer Program for Inelastic Dynamic Response of Plane Structures," by A. Kanaan and G. H. Powell - 1973 (PB 221 260)
"A Computer Program for Earthquake Analysis of Gravity Dams Including Reservoir Interaction," by P. Chakrabarti and A. K. Chopra - 1973 (AD 766 271)
"Behavior of Reinforced Concrete Deep Beam-Column Subassemblages under Cyclic Loads," by o. Kustu and J. G. Bouwkamp - 1973
"Earthquake Analysis of Structure-Foundation Systems," by A. K. Vaish and A. K. Chopra - 1973 (AD 766 272)
"Deconvolution of Seismic Response for Linear Systems," by R. B. Reimer - 1973 (PB 227 179)
"SAP IV: A Structural Analysis Program for Static and Dynamic Response of Linear Systems," by K.-J. Bathe, E. L. Wilson and F. E. Peterson - 1973 (PB 221 967)
"Analytical Investigations of the Seismic Response of Long, Multiple Span Highway Bridges," by W. S. Tseng and J. Penzien - 1973 (PB 227 816)
EERC 73-13
EERC 73-14
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223
"Earthquake Analysis of Multi-Story Buildings Including Foundation Interaction," by A. K. Chopra and J. A. Gutierrez - 1973 (PB 222 970)
"ADAP: A Computer Program for Static and Dynamic Analysis of Arch Dams," by R. W. Clough, J. M. Raphael and S. Majtahedi - 1973 (PB 223 763)
"Cyclic Plastic Analysis of Structural Steel Joints," by R. B. Pinkney and R. W. Clough - 1973 (PB 226 843)
"QUAD-4: A Computer Program for Evaluating the Seismic Response of Soil Structures by Variable Damping Finite Element Procedures," by I. M. Idriss, J. Lysmer, R. Hwang and H. B. Seed - 1973 (PB 229 424)
"Dynamic Behavior of a Multi-Story Pyramid Shaped Building," by R. M. Stephen and J. G. Bouwkamp - 1973
"Effect of Different Types of Reinforcing on Seismic Behavior of Short Concrete Columns ," by V. V. Bertero, J. Hollings, O. Kustu, R. M. Stephen and J. G. Bouwkamp - 1973
"Olive View Medical Center Material Studies, Phase I," by B. Bresler and V. V. Bertero - 1973 (PB 235 986)
"Linear and Nonlinear Seismic Analysis Computer Programs for Long Multiple-Span Highway Bridges," by W. S. Tseng and J. Penzien - 1973
"Constitutive Models for Cyclic Plastic Deformation of Engineering Materials," by J. M. Kelly and P. P. Gillis - 1973 (PB 226 024)
"DRAIN - 2D User's Guide," by G. H. Powell - 1973 (PB 227 016)
"Earthquake Engineering at Berkeley - 1973" - 1973 (PB 226 033)
unassigned
"Earthquake Response of Axisymmetric Tower Structures Surrounded by Water," by C. Y. Liaw and A. K. Chopra -1973 (AD 773 052)
"Investigation of the Failures of the Olive View Stairtowers during the San Fernando Earthquake and Their Implications in Seismic Design," by V. V. Bertero and R. G. collins - 1973 (PB 235 106)
EERC 73-27
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224
"Further Studies on Seismic Behavior of Steel BeamColumn Subassemblages," by V. V. Bertero, H. Krawinkler and E. P. Popov - 1973 (PB 234 172)
"seismic Risk Analysis," by C. S. Oliveira - 1974 (PB 235 920)
"Settlement and Liquefaction of Sands under Multi-Directional Shaking," by R. Pyke, C. K. Chan and H. B. Seed - 1974
"Optimum Design of Earthquake Resistant Shear Buildings," by D. Ray, K. S. Pister and A. K. Chopra -1974 (PB 231 172)
"LUSH - A Computer Program for Complex Response Analysis of Soil-Structure Systems," by J. Lysmer, T. Udaka, H. B. Seed and R. Hwang - 1974 (PB 236 796)
"Sensitivity Analysis for Hysteretic Dynamic Systems: Applications to Earthquake Engineering," by D. Ray -1974 (PB 233 213)
"Soil-Structure Interaction Analyses for Evaluating seismic Response," by H. B. Seed, J. Lysmer and R. Hwang - 1974 (PB 236 519)
unassigned
"Shaking Table Tests of a Steel Frame - A Progress Report," by R. W. Clough and D. Tang - 1974
"Hysteretic Behavior of Reinforced Concrete Flexural Members with Special Web Reinforcement," by V. V. Bertero, E. P. Popov and T. Y. Wang - 1974 (PB 236 797)
"Applications of ReliabilitY-Based, Global Cost Optimization to Design of Earthquake Resistant Structures," by E. Vitiello and K. S. Pister - 1974 (FB 237 231)
"Liquefaction of Gravelly Soils under Cyclic Loading Conditions," by R. T. Wong, H. B. Seed and C. K. Chan -1974
"Site-Dependent Spectra for Earthquake-Resistant Design," by H. B. Seed, C. Ugas and J. Lysmer - 1974
EERC 74-13
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EERC 74-15
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225
"Earthquake Simulator Study of a Reinforced Concrete Frame," by P. Hidalgo and R. W. Clough - 1974 (PB 241 944)
"Nonlinear Earthquake Response of Concrete Gravity Dams," by N. Pal - 1974 (AD/A006583)
"Modeling and Identification in Nonlinear Structural Dynamics, I - One Degree of Freedom Models," by N. Distefano and A. Rath - 1974 (PB 241 548)
"Determination of Seismic Design Criteria for the Dumbarton Bridge Replacement Structure, Vol. I: Description, Theory and Analytical Modeling of Bridge and Parameters," by F. Baron and S.-H. Pang - 1975
"Determination of Seismic Design Criteria for the Dumbarton Bridge Replacement Structure, Vol. 2: Numerical Studies and Establishment of Seismic Design Criteria," by F. Baron and S.-H. Pang - 1975
"Seismic Risk Analysis for a Site and a Metropolitan Area," by C. S. Oliveira - 1975
"Analytical Investigations of Seismic Response of Short, Single or Hultiple-Span Highway Bridges," by Ma-chi Chen and J. Penzien - 1975 (PB 241 454)
"An Evaluation of Some Methods for Predicting Seismic Behavior of Reinforced Concrete Buildings," by Stephen A. Mahin and V. V. Bertero - 1975
"Earthquake Simulator Study of a Steel Frame Structure, Vol. I: Experimental Results," by R. W. Clough and David T. Tang - 1975 (PB 243 981)
"Dynamic Properties of San Bernardino Intake Tower," by Dixon Rea, C.-Y. Liaw, and Anil K. Chopra - 1975 (AD/A008406)
"Seismic Studies of the Articulation for the Dumbarton Bridge Replacement Structure, Vol. I: Description, Theory and Analytical Modeling of Bridge Components," by F. Baron and R. E. Hamati - 1975
"Seismic Studies of the Articulation for the Dumbarton Bridge Replacement Structure, Vol. 2: Numerical Studies of Steel and Concrete Girder Alternates," by F. Baron and R. E. Hamati - 1975
EERC 75-10
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226
"Static and Dynamic Analysis of Nonlinear Structures," by Digambar P. Mondkar and Graham H. Powell - 1975 (PB 242 434)
"Hysteretic Behavior of Steel Columns," by E. P. Popov, V. V. Bertero and S. Chandramouli - 1975
"Earthquake Engineering Research Center Library Printed Catalog" - 1975 (PB 243 711)
"Three Dimensional Analysis of Building Systems," Extended Version, by E. L. Wilson, J. P. Hollings and H. H. Dovey - 1975 (PB 243 989)
"Determination of Soil Liquefaction Characteristics by Large-Scale Laboratory Tests," by Pedro De Alba, Clarence K. Chan and H. Bolton Seed - 1975
"A Literature Survey - Compressive, Tensile, Bond and Shear Strength of Masonry," by Ronald L. Maye sand Ray W. Clough - 1975
"Hysteretic Behavior of Ductile Moment Resisting Reinforced Concrete Frame Components," by V. V. Bertero and E. P. Popov - 1975
"Relationships Between Maximum Acceleration, Maximum Velocity, Distance from Source, Local Site Conditions for Hoderately Strong Earthquakes," by H. Bolton Seed, Ramesh Murarka, John Lysmer and I. M. Idriss - 1975
"The Effects of Method of Sample Preparation on the Cyclic Stress-Strain Behavior of Sands," by J. Paul Mulilis, Clarence K. Chan and H. Bolton Seed - 1975
"The Seismic Behavior of Critical Regions of Reinforced Concrete Components as Influenced by Moment, Shear and Axial Force," by B. Atalay and J. Penzien - 1975
"Dynamic Properties of an Eleven Story Masonry Building," by R. M. Stephen, J. P. Hollings, J. G. BoUwkamp and D. Jurukovski - 1975
"State-of-the-Art in Seismic Shear Strength of Masonry -An Evaluation and Review," by Ronald L. .Hayes and Ray W. Clough - 1975
"Frequency Dependencies Stiffness Matrices for Viscoelastic Half-Plane Foundations," by Anil K. Chopra, P. Chakrabarti and Gautam Dasgupt.a - 1975
"Hysteretic Behavior of Reinforced Concrete Framed Walls," by T. Y. Wong, V. V. Bertero and E. P. Popov - 1975
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227
"Testing Facility for Subassemblages of Frame-Wall Structural Systems," by V. V. Bertero, E. P. Popov and T. Endo - 1975
"Influence of Seismic History on the Liquefaction Characteristics of Sands," by H. Bolton Seed, Kenji Mori and Clarence K. Chan - 1975
"The Generation and Dissipation of Pore Water Pressures During Soil Liquefaction," by H. Bolton Seed, Phillippe P. Martin and John Lysmer - 1975
"Identification of Research Needs for Improving a Seismic· Design of Building Structures," by V. V. Bertero - 1975
"Evaluation of Soil Liquefaction Potential during Earthquakes," by H. Bolton Seed, I. Arango and Clarence K. Chan 1975
"Representation of Irregular Stress Time Histories by Equivalent Uniform Stress Series in Liquefaction Analyses," by H. Bolton Seed, I. M. Idriss, F. Makdisi and N. Banerjee 1975
"FLUSH - A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problems," by J. Lysmer, T. Udaka, C.-F. Tsai and H. B. Seed - 1975
"ALUSH - A Computer Program for Seismic Response Analysis of Axisymmetric Soil-Structure Systems," by E. Berger, J. Lysmer and H. B. Seed - 1975
"TRIP and TRAVEL - Computer Programs for Soil-Structure Interaction Analysis with Horizontally Travelling Waves," by T. Udaka, J. Lysmer and H. B. Seed - 1975
"Predicting the Performance of Structures in Regions of High Seismicity," by Joseph Penzien - 1975
"Efficient Finite Element Analysis of Seismic Structure -Soil - Direction," by J. Lysmer, H. Bolton Seed, T. Udaka, R. N. Hwang and C.-F. Tsai - 1975
"The Dynamic Behavior of a First Story Girder of a ThreeStory Steel Frame Subjected to Earthquake Loading," by Ray W. Clough and Lap-Yan Li - 1975
Earthquake Simulator Study of a Steel Frame Structure, Volume II - Analytical Results," by David To. Tang - 1975
"ANSR-I General Purpose Computer Program for Analysis of Non-Linear Structural Response," by Digambar P. Mondkar and Graham H. Powell - 1975
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228
"Nonlinear Response Spectra for Probabilistic Seismic Design and Damage Assessment of Reinforced Concrete Structures," by Masaya Murakami and Joseph Penzien - 1975
"Study of a Method of Feasible Directions for Optimal Elastic Design of Framed Structures Subjected to Earthquake Loading," by N. D. Walker and K. S. Pister - 1975
"An Alternative Representation of the Elastic-Viscoelastic Analogy," by Gautam Dasgupta and Jerome L. Sackman - 1975
"Effect of Multi-Directional Shaking on Liquefaction of Sands," by H. Bolton Seed, Robert Pyke and Geoffrey R. Martin - 1975
"Strength and Ductility Evaluation of Existing Low-Rise Reinforced Concrete Buildings - Screening Method," by Tsuneo Okada and Boris Bresler - 1976
"Experimental and Analytical Studies on the Hysteretic Behavior of Reinforced Concrete Rectangular and T-Beams," by Shao-Yeh Marshall Ma, Egor P. Popov and Vitelmo V. Bertero - 1976
"Dynamic Behavior of a Multistory Triangular-Shaped Building," by J. Petrovski, R. M. Stephen, E. Gartenbaum and J. G. Bouwkamp - 1976
"Earthquake Induced Deformations of Earth Dams," by Norman Serff and H. Bolton Seed - 1976
"Analysis and Design of Tube-Type Tall Building Structures," by H. de Clercq and G. H. Powell - 1976
"Time and Frequency Domain Analysis of Three-Dimensional Ground Motions, San Fernando Earthquake," by Tetsuo Kubo and Joseph Penzien - 1976
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