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UILU-ENG-2000-2013
CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 630
ISSN: 0069-4274
EARTHQUAKE GROUND MOTION SIMULATION AND RELIABILITY IMPLICATIONS
By CHIUN-LiN WU and YI-KWEI WEN
A Report on a Research Project Sponsored by the NATIONAL SCIENCE FOUNDATION (Under Grant NSF EEC-97-01785)
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS JUNE 2000
50272-101
REPORT DOCUMENTATION PAGE
4. Title and Subtitle
1. REPORT NO.
UILU-ENG-2000-2013 2. 3. Recipient's Accession No.
5. Report Date
June 2000 Earthquake Ground Motion Simulation and Reliability Implications
6.
7. Author(s)
C.-L. Wu and Y. K. Wen 9. Performing Organization Name and Address
University of Illinois at Urbana-Champaign Department of Civil and Environmental Engineering 205 N. Mathews Avenue Urbana, Illinois 61801-2352
12. Sponsoring Organization Name and Address
National Science Foundation 4201 Wilson Boulevard Arlington, VA 22230
15. Supplementary Notes
16. Abstract (Limit: 200 words)
8. Performing Organization Report No.
SRS 630 10. ProjectfTask/Work Unit No.
11. Contract(C) or Grant(G) No.
NSF EEC-97 -01785
13. Type of Report & Period Covered
14.
Seismic loads have been a dominant concern in design of buildings and structures in western United States for many years. Mid-America is a region of low to moderate seismicity, infrequent moderate to large events have occurred in the past and can occur again causing large and widespread losses due to relatively slow attenuation. For purpose of performance evaluation of structures, a phenomenological model is developed for generation of earthquake ground motions. The simulation procedure is based on the latest information of seismicity, the most recent ground motion models and simulation methods. A strong emphasis is placed on uncertainty modeling and efficiency in application to performance evaluation and reliability analysis. Site locations of special interest are Memphis TN, Carbondale IL, St. Louis MO and Santa Barbara CA, because they present U.S. cities in Mid-America and western United States of different seismicity. The soil condition of a city is approximately modeled by a generic profile. Ground motions at bedrock, however, are also generated for the Mid-America cities such that if detailed information of local soil variation is available, one can use appropriate soil amplification computer software to obtain the surface ground motions. Suites of ten ground motions are selected to be compatible with the uniform hazard response spectra of a given exceedance probability. They represent the seismic hazard from the causative faults surrounding a given site location. They also allow an accurate and efficient evaluation of structural performance and reliability. The uniform hazard ground motion suites for the three Mid-America cities are also available on the web at http://mae.ce.uiuc.edulResearchIRR-1/GMotions/Index.html.
17. Document Analysis a. Descriptors
Earthquake Simulation, Uniform Hazard Response Spectra, Uniform Hazard Ground Motions, Near-Source Effects, Reliability, Ductility Reduction Factor, Probabilistic Performance Analysis
b. Indentifiers/Open-Ended Terms
c. COSATI Field/Group
18. Availability Statement
Release Unlimited
(See ANSI-Z39.18)
19. Security Class (This Report)
UNCLASSIFIED
21. Security Class (This Page)
UNCLASSIFIED
20. No. of Pages
189
22. Price
OPTIONAL FORM 272 (4-77) Department of Commerce
111
ACKNOWLEDGEMENTS
This report is based on the thesis of Dr. Chiun-lin Wu submitted in partial
fulfillment of the requirements for the Ph.D. degree in Civil Engineering at the University
of Illinois at Urbana-Champaign. The authors would like to thank Gail Atkinston, Igor
Beresnev, David M. Boore, Kevin R. Collins, Douglas A. Foutch, William L. Gamble,
Jamshid Ghaboussi, Youssef Hashash, Robert Herrmann, Howard Hwang, J. P. Singh,
and Paul Somerville for providing computer codes, data, and helpful comments and
suggestions.
This work was supported primarily by the MAE center of Earthquake Engineering
Research Centers Program of NSF under Award Number EEC-970178S.
IV
TABLE OF CONTENTS
CHAPTER 1
INTR 0 D U CTI ON .............................................................................................................. 1
1.1 Overview .................................................................................................................. 1
1.2 Objective And Scope ............................................................................................... 3
1.3 Organization ............................................................................................................. 4
CHAPTER 2
METHOD OF SIMULATION - MID-AMERICA ...................................................... 5
2.1 Overview .................................................................................................................. 5
2.2 Selection of Locations and Soil Profiles .................................................................. 5
2.3 Seismicity and Tectonics ......................................................................................... 6
2.4 Reference Area and Occurrence Model. .................................................................. 6
2.4.1 Number of Earthquakes .................................................................................. 8
2.4.2 Magnitude ....................................................................................................... 8
2.4.3 Epicenter ......................................................................................................... 9
2.4.4 Focal Depth ..................................................................................................... 9
2.5 Modeling of Ground Motions ................................................................................ 10
2.5.1 Point Source Model ....................................................................................... 10
2.5.2 Comparisons of Point Source Model with Broadband Model ...................... 12
2.5.3 Finite Fault Model. ........................................................................................ 13
2.5.4 Uncertainty in Attenuation ............................................................................ 17
2.6 Local Site Effect .................................................................................................... 18
2.7 Baseline Correction ................................................................................................ 21
2.8 Directivity Effects ofMagnitude-8 Events ............................................................ 23
2.9 Final Remarks ........................................................................................................ 23
v
CHAPTER 3
METHOD OF SIMULATION-WESTERN UNITED STATES ............................ .45
3.1 Overview ................................................................................................................ 45
3.2 Selection of Location and Soil Profile .................................................................. .45
3.3 Reference Area, Seismicity and Tectonics ........................................................... .46
3.4 Occurrence and Source Models ............................................................................ .46
3.4.1 Number of Earthquakes ............................................................................... .47
3.4.2 Magnitude ..................................................................................................... 48
3.4.3 Epicenter ....................................................................................................... 49
3.4.4 Focal Depth ................................................................................................... 49
3.5 Ground Motion Modeling ..................................................................................... .49
3.5.1 Duration ........................................................................................................ 50
3.5.2 Fourier Amplitude Spectrum ........................................................................ 50
3.5.3 Spectral Representation of Ground Motions ................................................. 52
3.5.4 Near-Field Effects ......................................................................................... 53
3.6 Simulation Results ................................................................................................. 55
CHAPTER 4
UNIFORM HAZARD RESPONSE SPECTRA AND GROUND MOTIONS ........... 66
4.1 Overview ................................................................................................................ 66
4.2 Uniform Hazard Response Spectra ........................................................................ 67
4.2.1 Linear Elastic Systems .................................................................................. 68
4.2.2 Nonlinear Inelastic Systems .......................................................................... 69
4.2.3 Results ........................................................................................................... 71
4.3 Uniform Hazard Ground Motions .......................................................................... 74
4.3.1 Proposed Selection Procedure ....................................................................... 74
4.3.2 Comparison with Deaggregation Approach .................................................. 74
4.3.3 Suites of Ground Motions for Memphis, TN ................................................ 76
4.3.4 Suites of Ground Motions for Carbondale, IL .............................................. 77
4.3.5 Suites of Ground Motions for S1. Louis, MO ............................................... 78
VI
4.4 Final Remarks ........................................................................................................ 78
CHAPTERS
IMPLICATIONS IN RELIABILITY AND DESIGN ................................................ 109
5.1 Overview .............................................................................................................. 1 09
5.2 Nonlinear Response Estimate by Uniform Hazard Ground Motion Suites ......... 109
5.3 Ductility Reduction Factor ................................................................................... 110
5.4 Peak Ground Acceleration versus Spectral Acceleration in Fragility Analysis .. 113
5.5 Final Remarks ...................................................................................................... 114
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS ....................................................... 141
6.1 Conclusions .......................................................................................................... 141
6.2 Recommendations for Future Studies .................................................................. 143
APPENDIX A
TRUNCATED NORMAL DISTRIBUTION .............................................................. 145
APPENDIXB
MODIFIED TYPE II EXTREME VALUE DISTRIBUTION FUNCTION ............ 147
APPENDIXC
UNIFORM HAZARD GROUND MOTIONS ............................................................ 151
REFEREN CES ............................................................................................................... 17 6
1
CHAPTER 1
INTRODUCTION
1.1 Overview
Seismic loads have been a dominant concern in design of buildings and structures
in western United States for many years. Mid-America is a region of low to moderate
seismicity, infrequent moderate to large events have occurred in the past and can occur
again causing large losses in widespread areas due to relatively slow attenuation. The
performance of the building stocks and other structures under future earthquakes has
received attention of the profession, especially in the Mid-America cities that are in close
proximity to the New Madrid and Wabash Valley seismic zones.
After the recent Northridge and Kobe earthquakes, a main concern of engineers is
the performance-based design. For example, a design fra1l1ework has been proposed in
recent building design documents (e.g. 1997 NEHRP, FEMA 273) to ensure that various
performance objectives according to building categories are met. For this purpose,
uniform hazard ground motions corresponding to various levels of probability of
exceedance are needed for nonlinear time history analyses and structural performance
checks. \Vhen a\'ailable earthquake records are not sufficient to determine the uniform
hazard ground motions, one may: (1) scale existing ground motion records to match
target response spectra, (2) use ground motion records from other seismic zones with
similar tectonic en\'ironments, or (3) generate synthetic ground motions based on
seismotectonic characteristics of the region surrounding the site location.
All three approaches were used in the recent SAC steel proj ect, which provided
ground motion suites for Los Angeles, Seattle and Boston for various probability levels
(Somerville et. aI., 1997). These ground motions are obtained by scalin_g records of past
earthquakes and time histories based on broadband simulations. The median response
2
spectra of the SAC ground motions for stiff and soft soil sites match the target response
spectra in the 1994 NEHRP provisions. The scaling factors used in the SAC ground
motions vary widely from 0.27 to 10.75. Also, due to the large computational efforts
required in the broadband simulation procedure, generation of a large samples (say
thousands) is time consuming and expensive. At most locations in Central and Eastern
United State, the number of ground motion records is generally not enough for the first
two approaches, especially for low levels of probability of exceedance. Simulation of
earthquake motions is therefore necessary. A phenomenological model of simulation is
proposed in this study, which allows an efficient simulation of a large number of ground
motions and probabilistic performance analyses.
The simulation procedure is an extension of that in Collins et al. (1995). It is based
on the latest information of seismicity and the most recent ground motion models and
simulation methods appropriate for engineering applications. In view of the differences
in earthquake records and seismotectonic data available in Mid-America and Western
United States, two slightly different procedures are developed. Since in Western United
States data are more available and seismic zones are better defined, the simulation
procedure for this area is more data based. All earthquakes with distance less than 50 km
to the site are modeled as either a dip-slip or strike-slip fault according to the tectonic
infonnation. In Mid-America, due to the scarcity of records of engineering interest, the
simulation method is largely theory and model based. A strong emphasis is placed on
uncertainty modeling and efficiency in application to performance evaluation and
fragility analysis. The procedure may be improved as more knowledge on earthquakes in
the corresponding regions and more accurate and efficient methods of ground motion
modeling become available, especially in Mid-America.
Site locations of special interest in this study are Santa Barbara CA, Memphis TN,
Carbondale IL, and St. Louis MO. These cities are selected for study because they
present a wide cross section of cities at risk. Since ground motions are strongly
dependent on local soil condition and yet the soil profile variation within a city has not
been mapped in detail, in this study the soil condition of a city will be approximately
3
modeled by a generic profile. Ground motions at bedrock, however, will also be
generated for Mid-America cities such that if detailed information of local soil variation
is available, one can use appropriate computer software to include soil amplification in
the surface ground motions.
1.2 Obj ective And Scope
The objective of this study is to generate uniform hazard response spectra and
ground motions for probabilistic performance evaluation (fragility analysis) and loss
estimation of buildings under future earthquake excitations. To achieve this objective, the
following studies are required:
1. Develop an efficient procedure to generate synthetic ground motions based on
regional seismicity.
2. Develop a procedure to construct uniform hazard response spectra and
generating uniform hazard ground motions for structural performance
evaluation.
3. Verify the efficiency and accuracy of such motions in evaluation of linear and
nonlinear responses of structural systems for probabilistic performance analysis
and loss estimation.
1.3 Organization
In Chapter 2, a simulation procedure based on a 2-comer point source model and a
finite fault model is proposed to generate a large number of ground motions for Mid
America cities. The seismological data from USGS Open File Report 96-532 are used.
Probabilistic distributions of various seismic parameters (magnitude, epicenter, focal
depth, attenuation uncertainty, etc.) are assumed based on field observations and then
used to simulate earthquake sources and path effect. An empirical correlated random
field model is used to simulate asperity in magnitude-8 events in the New Madrid seismic
4
zone. The local site effect is considered by using quarter wavelength method. After
baseline correction following the USGS BAP routine (Converse, 1992), the resulting
ground motions are used for statistical study.
In Chapter 3, a more databased ground motion model is developed for Western
United States. The seismological data from the 1995 WGCEP report (Working Group on
California Earthquake Probabilities) are used. The Fourier spectrum of Trifunac (1994)
was used in simulation and then a near-source factor is incorporated in the simulation to
account for some of the important near-source effects (Sommerville et al. 1997).
In Chapter 4, uniform hazard response spectra results for B/C boundary site are
constructed based on statistical analyses of synthetic ground motions. They are
compared with USGS, FEMA-273 design spectra and similar results in the literature. A
period-independent procedure is then developed to select suites of uniform hazard ground
motions for Mid-America cities at 10% and 2% probabilities of exceedance in 50 years.
Linear structural analyses are performed to verify- that the selected uniform hazard ground
motion suites can be used for unbiased response estimation.
In Chapter 5, nonlinear structural systems under uniform hazard ground motions
are investigated for to ensure that such motions yield unbiased structural response
estimates including the influence of system degradation on ductility reduction factor. An
efficient method of constructing probabilistic performance curve is then proposed. The
accuracy of using spectral acceleration versus that of using peak ground acceleration for
fragility analysis is examined.
In Chapter 6, the significant conclusions of this study and recommendations for
future research are summarized.
5
CHAPTER 2
METHOD OF SIMULATION - MID-AMERICA
2.1 Overview
The procedure proposed by Collins et al. (1995) is extended to simulate future
seismic events and ground motions within a reference area over a given period of time
(e.g. 1 0 years). The ground motion model basically follows the procedure suggested by
Henmann and Akinci (1999), which is based on a point-source simulation method
SMSIM (Boore 1996). To catch some of the important near-source effects due to large
events, however, the finite fault model by Beresnev and Atkinson (1997, 1998) is also
used for magnitude-8 events. The soil amplification is modeled by the quarter
wavelength method by Boore and Joyner (1991, 1997). The tectonic and seismological
data are mainly taken from USGS Open-File Report 96-532 (Frankel et al. 1996).
Possible future earthquake events are generated for a trial period of 10 years, and then
each earthquake source propagates its seismic waves accordingly to site locations of
interest. A total of 9000 simulations of 10-year period are carried out to provide a large
number of ground motions for statistical analysis. The proposed simulation procedure is
shown in Figure 2.1 and details of the simulation method are given in the following.
2.2 Selection of Locations and Soil Profiles
Site locations of special interest In this study are Memphis TN (35.117°N,
90.083°W), Carbondale IL (37.729°N, 89.246°W), and St. Louis MO (38.667°N,
90.1900 W). These cities are selected for study because they present a wide cross-section
of the Mid-America cities at risk. Since ground motions are strongly dependent on local
soil condition and yet the soil profile variation within a city has not been mapped in
6
detail, the soil condition of a city will be approximately modeled by a generic profile.
Ground motions at bedrock, however, will also be generated for these cities so that if
detailed information of local soil variation is available, one can use appropriate soil
amplification computer software to obtain the surface ground motions.
2.3 Seismicity and Tectonics
Based on the seismicity database from USGS Open-File Report 96-532 (Frankel et
al. 1996), the annual occurrence rate of earthquakes with body wave magnitude mb
greater than 5 per 0.1 xO.l degree square for the zone of interest is obtained as shown in
Figure 2.2. Due to the lack of tectonic information, only moment magnitude 8
earthquakes in the New Madrid seismic zone are treated events with finite faults similar
to the 1811-1812 New Madrid earthquakes. However, the faulting mechanism is
simplified as a vertical strike-slip fault with a rupture plane of 140 kIn (along-strike)x 33
km (down-dip) according to Johnston (1996b). A 34.69° azimuth angle is assumed,
which is an approximate estimate based on USGS OFR-96-532. The distance from the
ground surface to the upper edge of the rupture plane is assumed to be 5 Ian.
Attenuation, location of fault rupture, hypocenter, and asperity are varied from event to
event to account for uncertainty in a future Mw-8 event. Figure 2.3 shows seismic zone
of these Mw-8 earthquakes.
2.4 Reference Area and Occurrence Model
All known probable earthquake sources in the Central and Eastern United States
(CEUS) are considered. For a given city, however, only those that fall within the
reference area ( effective zone) of the city and have significant contributions to the
seismic risk are used in the simulation. Different cities may share some of the seismic
sources and hence the same simulated seismic events. Considering the relatively low
attenuation in the CEUS, the effective zone is defined as a circular area with a radius of
7
500 km centered at a specified site location, a value used by most researchers (e.g.
USGS OFR-96-532). The occurrence in time is generated according to a Poisson
process. The magnitude given the occurrence is then generated according to the
magnitude distribution for events of moment magnitude less than 8 following the 1996
USGS Open-File Report.
The magnitude-8 events are generated separately using Poisson process with a
mean recurrence time of 1000 years (Johnston 1996b, Frankel et al. 1996) and with an
epicenter uniformly distributed within the New Madrid fault zone shown in Figure 2.3.
The orientation of the fault is assumed to be along that of the NMSZ. Once rupture
occurs, it propagates toward both ends of the rupture surface.
A total of 9000 simulations of 10-year period are carried out. As a result, 9260
ground motions are generated in Memphis (TN), 9269 in Carbondale (IL) and 8290 in
St. Louis (MO). Figure 2.4 shows the epicenters and magnitudes of earthquakes
corresponding to 600 simulations of 10-year records in the region of interest in the
CEUS. It corresponds, therefore, to about 6000 years of records. Most of the events are
located in the New Madrid, Wabash Valley and East Tennessee seismic zones. The
proximity of the events to each city is shown. According to the simulation results, the
occurrence rates of earthquakes of body wave magnitude 5 and above in the reference
areas for Memphis, Carbondale, and St. Louis are 0.0989, 0.0980 and 0.0882 per year,
respectively. These values are very close to the actual occurrence statistics.
Before the statistics of occurrence and magnitude from the USGS OFR-96-532 can
be used in the simulations, the incremental seismicity rate provided by USGS needs to be
converted into the cumulative seismicity rate using recurrence relation of Herrmann
(1977), and the results are shown in Figure 2.2. In each 0.10 x 0.10 cell during a 10-year
period, the number of earthquakes, magnitudes and hypocenters are determined according
to the procedures described in the following.
8
2.4.1 Number of Earthquakes
To detennine the number of earthquakes in each 0.1 °xO.1 ° cell during a 10-year
period, a random variable Uk with a unifonn distribution between 0 and 1 is generated.
The number of earthquakes is then detennined from:
k = 1,2,.··, Neel' (2.1)
where Vk is the mean annual occurrence rate of earthquakes in cell k with body wave
magnitude greater than 5; nk is the number of earthquakes that occur with body wave
magnitude greater than 5 in cell k during a time interval of t years (t = lOin this study);
and Neel! is the total number of cells in the reference area of a specific site.
2.4.2 Magnitude
In most literature, the body wave magnitude is used for sizing earthquakes; e.g., in
USGS OFR-96-532. The body wave magnitude mb is then converted into moment
magnitude M w using an empirical relation of Johnston (1994):
M w = 3.45 - 0.4 73mb + 0.145mb
2 (2.2)
where mb-5 is equivalent to Mw-4.71. It should be noted that Equation 2.2 differs slightly
from a revised relation of Johnston (1996a). To be consistent with the seismological data
from USGS OFR-96-532, Equation 2.2 is used in this study. The cumulative distribution
function of moment magnitude FMw is then defined as:
(2.3)
9
where mx is the maximum moment magnitude in cell k (Figure 2.5); N(m,mx' vk ) is the
annual cumulative rate of earthquakes greater than moment magnitude m in cell k. Note
that 4.71 ~ m ~ mx •
2.4.3 Epicenter
It is assumed that earthquakes are equally likely to occur anywhere within a cell,
which is numerically done by generating two random numbers of uniformly distributed
and independent of each other along latitudinal and longitudinal directions.
2.4.4 Focal Depth
The focal depth distribution model proposed by Wheeler and Johnston (1992) is
shown in Table 2.1. According to EPRI TR-102293 Report (1993), Wheeler and
Johnston's model is more appropriate in the Eastern North America (ENA) due to their
quality selection criteria. Since the 2-corner point source model (Atkinson and Boore
1995) is widely used in generating earthquake events in the CEUS, the Wheeler-Johnston
distribution (1992) is more appropriate for this purpose. To exclude very shallow
earthquakes within the soil stratum, a cut-off point at 1 km is used. Within each depth
bin, the earthquake focus is then assumed to be equally likely to occur anywhere in the
corresponding depth interval. The focal depth distribution of moment magnitude 8
earthquakes modeled by finite fault is shown in Table 2.2. Since magnitude 8
earthquakes are all assumed to occur within the NMSZ, which is classified as a Stable
Continent Region (Frankel et al 1996), a focal depth distribution model of EPRI TR-
102293 (1993) is used with minor revisions in view of the discretized depths required in
the finite fault model. In the focal depth column of Table 2.2, the figures inside
parentheses stands for the center of the corresponding depth bin and each depth bin
represents a sub-fault segment along the down-dip direction.
10
2.5 Modeling of Ground Motions
2.5.1 Point Source Model
For point sources, the two-comer-frequency model (Atkinson and Boore, 1995) for
generic S-wave type ground motions on hard rock is used since this model has been
shown to give better ground motion prediction in the CEUS (Atkinson and Boore, 1998).
The mathematical form for the Fourier amplitude spectrum of the ground motion using
this two-corner point source model is:
(2.4)
where E(Mo,f) is two-comer-frequency source spectrum; D(Rh,f) is a diminution
factor; P(f) is high-cut filter; I(f) is response indicator; f is cyclic frequency (Hz), Rh
is focal distance (lan), and Mo is moment magnitude (dyne-cm). The various functions
are:
in which,
R{)¢· FS· V C=-----:-3
4ff' Po' flo
R{)¢ = 0.55, average radiation pattern
FS = 2, free-surface amplification
v = 1/ -fi ~ 0.7071 , partition factor
Po = 2.8 gmlcm3, crustal density
fJo = 3.6 km/sec, crustal shear wave velocity
(= weighting parameter
1-( ( ---'--"'7'""2 + 2
1+(J.) l+(iJ (2.5)
JA,jB = corner frequencies (Hz)
log? = 2.52 - 0.637 Mw
10gfA = 2.41- 0.533Mw
10gfB = 1.43 - 0.1 88Mw
M o = 101.5Mw+16.05 (dyne-em)
in which,
11
11 I Rh if Rh ~ 70lan
Dg(Rh) = 1 I 70 if 70 ~ Rh ~ 130lan , geometric attenuation
(1/70)·(130IRh)0.5 if Rh~130lan
_ JrIRh
D m (Rh ,f) = e jJo·Q(f) and Q(f) = 680· fO.36 ,anelastic material attenuation
P(f) = 1 , high-cut filter ~1 + (f / fmax)8
where fmax = 50 Hz is used.
I(f) - 1 , response indicator - (2if)P
where p = 0 for acceleration, 1 for velocity, and 2 for displacement.
(2.6)
(2.7)
(2.8)
Total duration Td (sec) for the strong-motion phase is decomposed into source and
path durations:
(2.9)
where source duration To (sec) is related to corner frequency JA following Boatwright and
Choy (1992):
1 To = -- (sec) 2·fA
(2.10)
12
and the path duration Tp (sec) is of a tri-linear form following Atkinson and Boore
(1995):
0 if Rh ~ 10km
T= 0.16· (Rh -10) if 10 ~ Rh ~ 70km
(2.11 ) p 9.6 - 0.03· (Rh -70) if 70 ~ Rh ~ 130km
7.8 + 0.04· (Rh -130) if 130 ~ Rh ~ 1000km
To incorporate intensity evolution, an exponential window function (Saragoni and
Hart, 1974) is introduced to obtain more realistic accelerograms:
and shape parameters a, band c are defined as:
a_(_e )b c·Tw
b = _ c·RnlJ 1 + c· (Rnc-I) b
c=--coTw
(2.12)
(2.13)
where £ defines the fraction of a specified duration at which the maximum envelope
amplitude will occur; lJ defines the fraction of the maximum envelope amplitude which is
reached at time Tw (Figure 2.6). According to Boore (1983), £= 0.2, and 7J = 0.05 was
found consistent with the envelope function to 22 accelerograms in Saragoni and Hart
(1974) and T" = 'lTd gives a record with a strong phase close to Td. Therefore, they are
used in all applications in this study for point sources.
2.5.2 Comparisons of Point Source Model with Broadband Model
The accuracy and validity of the point-source model can be found in the literature
(Boore 1996, Atkinson and Boore 1998). There have been no records of moderate to
large events near any of the three cities that can be used for comparison with the
simulated ground motions. The closest is the simulation results based on the broadband
13
approach for St. Louis due to events of magnitude 6.5 to 7.5 in the NMSZ by Saikia and
Somerville (1997). The broadband approach considers the details of the geometry of the
fault, rupture surface, and wave propagation for low frequency motion and uses records
for high frequency motion. Large-scale simulations, which require detailed information
of each fault that is generally unknown, are fairly computationally expensive. The results
of these two methods for a rock site at St. Louis are compared. An event based on the
point-source model is chosen such that source and path parameters are comparable to
those of the magnitude-7 event studied by Saikia and Somerville. The response spectra
of hard rock motions based on these two models are compared in Figure 2.7. It is seen
that agreements are fairly good in the period range from 0.2 sec to 3 sec.
2.5.3 Finite Fault Model
This so-called finite fault model emphasizes the effects of a large fault dimension,
including rupture propagation, directivity, and source-receiver geometry, which can have
significant influence on amplitudes, frequency content and duration of ground motion.
The examples of applications in Beresnev and Atkinson (1997, 1998b) showed that the
model produces ground motions that match well field records on rock sites in 4
earthquakes. They are the Mw-8.0 1985/9119 Michoacan (Mexico), the Mw-8.0 1985/3/3
Valparaiso (Chile). the Mw-5.8 1988111/25 Saguenay (Quebec), and the Mw-6.7
1994/1/17 t\orthridge earthquakes. For soil sites (Beresnev and Atkinson, 1998c),
however. their model tends to overestimate the ground motions due to soil nonlinearity,
which is not considered. The application of this recently proposed model is still limited
to simulation of a single strong ground motion without considering surface waves and
spatial \'ariation: therefore, applications in dynamic analyses for bridges on multiple
supports are not possible at present. Despite these disadvantages, this study adopts the
Beresnev-Atkinson model for the simulation task in the CEUS, because
1. The Beresnev-Atkinson model is shown to provide an unbiased fit to 11
earthquake events in the eastern North A_meriea (ENi\) earthquakes, the
magnitudes of which range from 4.0 to 7.3 (Beresnev and Atkinson, 1999).
14
2. Unlike wave propagation based modeling procedures (Saikia and Somerville,
1997), the Beresvnev-Atkinson model requires much less computational efforts,
which allows generation of a large number of ground motions in Monte Carlo
simulation.
3. The Beresnev-Atkinson model requires less faulting parameter inputs, which is
suitable for areas with scarce or no seismotectonic data, such as in CEUS.
4. The accuracy of simulation results by Beresnev-Atkinson model compares
favorably with those by Somerville et al. (1991).
5. The Beresnev-Atkinson model can be applied to simulate two-dimensional
drrma QTrmnrl rrt()tl()nc;:. lf rrt()tl()n~ ~rp. rlp.r:orrtno~p.r1 onto two nnnr:in:::t 1 r1irp.r:ti(m~ U1,..L'-'.L .... O b .... "''-4-.... .lt.~ .......... "" ...... _ ....... a..." ................ _ ...... _ ........... - ... - ---- ........... r- ..... -- -.---- - ~. - r-------r-- --- -.-!-.-!-----~
according to Penzien and Watabe (1976).
It is also possible to simulate the vertical component in the ENA by using the HN
spectral ratio proposed by Atkinson (1993). In addition, spectral ratio in areas of interest
may be also available in the literature, e.g. NMSZ. A 3-component simulation, however,
is not attempted in this study since phase delay in the stochastic model is of random
nature and phase delay due to different arrival times of seismic waves is not accounted
for explicitly as in the wave propagation model.
Figure 2.8 shows the geometry of the finite fault model proposed by Beresnev and
Atkinson (1997, 1998a). A rupture plane of 140 km (along-strike) by 33 km (down-dip)
(J ohnston 1996) with a vertical strike-slip faulting and 34.69° azimuth is assumed, which
is an approximate estimate based on USGS OFR-96-532. The distance from the ground
surface to the upper edge of the rupture plane is assumed to be 5 km. For reference, 0.6
km is used in Saikia and Somerville (1997) and 10 km is used in USGS OFR-96-532.
The fault plane is divided into 64 (16x4) sub-faults. Each sub-fault is then treated as a
one-corner point source (Brune 1970, 1971, Frankel et al. 1996) and may be "triggered" a
few times during an earthquake event. The delay between triggers is of random nature
depending on sub-fault rise time Trise to simulate the complexity in slip process. The
resulting ground motion is therefore a combination of waveforms from different sub
faults accounting for differences in arrival times and path attenuation. The arrival time
15
delay between sub-faults is accounted for by the rupture propagation from the hypocenter
(io)o) to the center of the sub-fault F, and the travel time from the center of the sub-fault
F to the observation point P.
The mathematical form for the Fourier amplitude spectrum of each sub-fault
follows basically Equation (2.4). Source spectrum E(flmo,/) for each sub-fault is
modeled as a single-comer form:
1 (2.14)
flf + flw ( )
3
where flmo = fl()· 2 indicating sub-fault seismic moment (dyne-cm) released
in one trigger; .do-is the Kanamori-Anderson (1975) stress parameter (bars) and is related
to the stress drop of the sub-fault in one trigger; flf and flw are sub-fault length and
width, respectively (Figure 2.8). The comer frequency Ic (Hz) can be expressed as
(2.15)
in which z = 1.68, a calibration constant related to maximum slip rate; /30 is shear wave
velocity of the Earth crust (km/sec); y is the fraction of rupture-propagation velocity to
/30, which ranges from 0.6 to 1.0. In this study, we assume y = 0.8 and anelastic material
attenuation Q(f) = 670*/°·33, following Beresnev and Atkinson (1997, 1999). Geometric
attenuation and high-cut filter take the same form as two-comer point source model. The
total duration Td of each sub-fault consists of sub-fault rise time Trise and path duration Tp
(2.16)
where sub-fault path duration Tp depends on the distance from the center of sub-fault to
the observation point and takes the same form as two-comer model;
16
1 -. (~.e + ~w)
T,.ise = -=2~_V __ -rup
(2.17)
in which rupture velocity V rup = 0.8,00. The delay between two consecutive triggers of a
sub-fault is modeled as (l + u) . T,.ise in which u is a random number uniformly distributed
between 0 and 1. The Saragoni-Hart (1974) exponential window for each sub-fault is
padded with rapid sinusoidal taper zones as shown in Figure 2.9.
The sub-fault spectrum is basically a functional of stress drop and sub-fault size.
To make the resulting ground motion insensitive to stress parameter and sub-fault size,
Beresnev and Atkinson suggested a stress parameter of 50 bars. To determine sub-fault
size, Beresnev and Atkinson (1999) provide a generic formula calibrated according to
earthquake records in the ENA. In this study, 64 sub-faults, each with a stress parameter
of 200 bars, are used in the finite fault model. The model gives results in good
agreements with the USGS OFR-96-532 target spectral accelerations at 0.2-, 0.3- and 1.0-
sec structural periods. A stress parameter of 150 bars is also tested for the finite fault
model. As expected, it causes lower spectral values in the low frequency range in the 2%
in 50 years hazard level.
Before the Beresnev-Atkinson model can be applied in Monte Carlo simulation, the
slip distribution (asperity) needs to be included in the model. Since there are no records
of large events in the NMSZ, observations based on California earthquakes are used. The
slip distribution within the rupture surface is modeled as a correlated random field
according to Saikia and Sommerville (1997), and Sommerville et al (1999), using the
follo\ving wave number spectrum:
(2.18)
17
In which kx and ky are wave numbers in the along-strike and down-dip directions
respectively; n = 2.0 and Cx and Cy are correlation length constants depending on
magnitude:
{IOglO Cx = 1.72 - O.5Mw
loglo Cy = 1.93 - 0.5Mw (2.19)
This random field model is then modulated to ensure larger slip in the middle than at the
edge of the rupture surface, a feature observed in recent earthquakes. Two sample slip
distributions on the rupture surfaces (33 kmx 140 km) based on the correlated random
field model and their corresponding discretizations for the finite fault model are shown in
Figures 2.10-11. It should be noted that the finite fault model uses slip values as weights
to distribute the target seismic moment over the rupture plane. Therefore, only relative
slip values are important in the simulation procedure. In the simulation procedure, the
intensity of seismic motion of an individual sub-fault is determined by its distributed
seismic moment. The distributed moment is also used to determine the number of
triggers in its corresponding sub-fault.
2.5.4 Uncertainty in Attenuation
Path attenuation due to geometric radiation and material damping in the earth crust
is accounted for by the semi-empirical formulae shown above (Eqs 2.6 to 2.8). The
uncertainty in the attenuation is modeled by a truncated lognormal distribution with the
median value given by the above diminution factor as a function of distance and
frequency (Eq. 2.6). A value of 0.75 is used for the natural logarithms of the standard
deviation of PGA following USGS OFR-96-532. To prevent unrealistic large variation,
cut-off limits of mean plus and minus three standard deviations are used. The resulting
peak ground acceleration (PGAsimulation) is then given by,
.en PGAsimulation = .en PGAmedian + 5' ( (2.20)
where (= 0.75 and c (epsilon) is a standard normal random variable, indicating the
deviation from median attenuation. The coefficient of variation- 5= 0.87 from
18
? = ~ t'n(1 + 62). Since peak ground accelerations can be modeled by a lognormal
distribution, In PGA in Equation (2.20) is of normal distribution and &can be modeled by
a truncated normal density function (Appendix A).
2.6 Local Site Effect
Soil amplification due to the local site soil profile is considered. In general,
nonlinear soil properties are needed for an accurate estimate of soil amplification, which
is also earthquake intensity dependent. Detailed information on soil profile is usually
hard to obtain especially for a deep soil column. In view of this, soil is treated
approximately as an elastic medium, which generally gives an overestimate of the
amplification in the high frequency range and an underestimate in the long period range
for a severe event. The quarter wavelength method (QWM) (Joyner et. aI., 1981, Boore
and Joyner, 1991, Boore and Joyner, 1997, Boore and Brown, 1998) is used to model the
soil amplification. Soil profiles are based on boring log data in Memphis, Carbondale
and St. Louis (Hashash 1999, Herrmann 1999). In the QWM, the total amplification is
expressed as:
Amp(f) = A(f) . P(f) (2.21)
where A(f) is the amplification function and P(f) is the attenuation function. The
amplification is approximated by
A(f) - Po . fJo Ps(f)· fJs(f)
(2.22)
in which p and fJ are the density (g/cm3) and shear wave velocity (m/sec) at the source
(subscript 0) and site (subscript s);fis frequency (Hz); over-bar indicates average value.
At the site, the frequency-dependent effective velocity fJs (f) is defined as the average
shear wave velocity from the surface to a depth of quarter wavelength for a given
frequency. If the travel time to the depth ofa quarter wavelength, ttz(/), ~s defined as:
19
(2.23)
Then the depth of a quarter wavelength z can be detennined by solving the following
equations:
(2.24)
where h(i) is the thickness of the ith layer (m); ~(i) is the shear wave velocity of the ith
layer (m/sec); m is the number of layers to satisfy the equality relation. The effective
velocity jJs(f) at a given frequency fis therefore detennined by
(2.25)
A travel-time-weighted average is taken of the density Ps:
(2.26)
(i)
where tt;i)(f) = ;(il ' travel time of shear wave in the ith layer. At high frequency, the s
soil attenuation is accounted for by
P(f) = e -"'KO-/ (2.27)
in which KO is a tenn that accounts for shear velocity and damping over the soil column:
(2.28)
20
where N is the number of soil layers; h is the depth measured from the ground surface
(m); and Q is the quality factor for a frequency-independent measure of damping. For
small material damping, Q is related to the damping ratio ;d as follows:
1 _ 0=- jar Q 1«1 --- 2;d
(2.29)
It should be noted that Q is only a simplified phenomenological description of a complex
process involving an intrinsic and scattering attenuation mechanism. For soil, it is
assumed that Q = 6ho.24 (Hemnann and Akinci, 1999) and Q = 500 for rock.
The "representative" profiles for the three cities used in this study are shown in
Table 2.3 to 2.5. According to QWM, KO = 0.063 sec, 0.043 sec and 0.0076 sec for
Memphis, Carbondale and St. Louis, respectively. The calculated KO value for Memphis
agrees with Akinci and Hemnann (1999) corresponding to a soil deposit of 1000 m. The
resultant soil amplifications as a function of frequency for the three cities are shown in
thick lines in Figures 2.12-14. It is seen that while amplification at St. Louis is restricted
to the high frequency range (> 3 Hz) due to the very shallow soil layer on hard rock, the
much deeper soil layers in Memphis and Carbondale produce much larger amplification
in the longer period range. For comparison, the soil amplification at these three cities
based on the well-known computer software SHAKE is also shown for bedrock ground
motions of 10 % and 2 % probability of exceedance in 50 years. The soil amplification
factor used in the 1997 NEHRP provisions for soil classification of B/C boundary is also
shown since this is the soil condition used for USGS national earthquake hazard maps.
Note that the quarter-wavelength model will give a good estimate of the averaged
amplification and will miss the peaks and valleys. Also, it has a tendency of
overestimating the amplification in the high frequency range for ground motion of very
high intensity when effects of soil nonlinearity may be significant. In spite of these
limitations, the agreements are generally good.
Soil property is a 3-dimensional random field and depends on earthquake intensity.
It varies widely within a given city. For instance, the soil deposit in Memphis City varies
21
from a thickness of 800 m with alluvium surface layer (east of Memphis) to a thickness
of 1000 m with loess surface layer (west/downtown of Memphis) according to compiled
data from Dorman and Smalley (1994), and Hwang et al. (1999). To give a general
understanding of the variation in Memphis, soil amplification for the soil profile at Pier C
of the Hernando Desoto Bridge located at Mud Island (35.1529°N, 90.0579°W) is shown
in Figure 2.15 for comparison with the representative soil amplification. Also shown are
soil amplifications from Herrmann et al. (1999), and Boore and Joyner (1991, 1997)
In passing, it is pointed out that soil amplification considering nonlinear effects is
an extremely complex problem. SHAKE is the most widely used program but is based
on an equivalent linear wave propagation method. It is expected to yield good results
when the excitation intensity is not very high and the soil layer is not very deep. It is
therefore expected to work better for the St. Louis soil profile than those of Memphis and
Carbondale. There are other truly nonlinear programs available for this purpose.
However, they have not been commonly accepted by engineers, therefore are not used for
comparison in this study.
I t is of interest to compare the proposed method of simulation with the broadband
simulation method by Somerville et al. Table 2.7 shows the attributes of the two
simulation methods.
2.7 Baseline Correction
Physics dictates that all earthquake ground motions should start and end with zero
velocity with however a possible pennanent offset of the displacement during a severe
event. Due to instrumental difficulties, however, raw ground motion records usually do
not satisfy this requirement and therefore need baseline correction. In the case of the
stochastic simulation, undesirable nonzero conditions often occur in the simulated ground
motions; therefore, the baseline correction procedure is also needed. To do this, this
study basically follows guidelines from US Geological Survey BAP routine (Converse
1992). The accelerogram is first padded with leading and trailing zeros, then passed
22
through Ormsby filter, and then de-trended by a linear or parabolic function, depending
on which one results in close-to-zero end velocity and displacement. The Onnsby filter
H (f) takes the following mathematical form:
0 0..:;, I..:;, f" Hz . 2( II I - f" ) f" ..:;, I..:;, fz Hz SIn -
2/2 -f" H(f) = 1 fz..:;,I..:;,J;Hz (2.30)
sin2(1l 1- J;) 214 - h
J;..:;,/~ltHz
0 12. It Hz
where jj = 0.1 Hz; h = 0.25 Hz; f3 = 24.5 Hz; 14 = 25.5 Hz. For Mw-8 earthquakes, a
parabolic equation (Brady 1966, Nigam et al. 1968) is adopted:
A* (t) = AU) - Co - C1 t - C2 t2
V*(t)=V(t)-C t-!C t 2 _.lc t 3
° 2 1 3 2
(2.31)
where A (t) is the uncorrected acceleration time history; A * (t) is the corrected acceleration
time history; V(t) is the uncorrected velocity time history; V* (t) is the corrected velocity
time history; Co, C1, C2 are regression coefficients to minimize the mean square of the
corrected velocity V* (t) and will satisfy the following simultaneous equations
(2.32)
For earthquakes smaller than Mw-8, it is observed that a linear function will give
satisfactory results:
Co = A(t)-C1t
C = :LUi - t)(AUi) - Aft)) 1 :L(t
i _ t)2
(2.33)
C2 = 0
where :L = :L:1 ; n is the number of data points. To integrate acceleration over time
into velocity and displacement, the Newmark j3 method with linear variation of
23
acceleration is adopted according to Berg and Housner (1961). Two sample acceleration
time histories after baseline correction are shown in Figures 2.16 and 2.1 7.
2.S Directivity Effects of Magnitude-S Events
The near-source effects USIng the finite fault model are demonstrated by a
comparison of the ground motions at Memphis (Figure 2. 18(a)&(b)) due to two simulated
events, both of magnitude 8. The location, size, and orientation of the faults are the same
with a closest distance of 62 km from the fault surface to the site. The epicenters,
however, are located such that one is far (186 km) from the site with the rupture
propagating toward the site and the other closer (79 km) to the site with the rupture
propagating generally away from the site (Figures 2.10 and 2.11). Sample rupture
surface (33 kmx140 km) slip distributions based on the correlated random field model
and the corresponding discretizations for the finite fault model for the two events are
shown in the figures. The time histories of the ground motion for a soil site at Memphis
are shown in Figure 2.18( a) and (b) for both events. The directivity effects can be clearly
seen that Figure 2.18(a) produces a shorter but more intense ground motion due to the
"Doppler effect" even though the epicenter is much farther away. One accelerogram
simulated for Carbondale, IL due to the same fault rupture in Figure 2.11 is also shown
(Figure 2.18(c)) to further demonstrate the difference in ground motions (between (a) and
(c)) due to the same event but from two observation points.
2.9 Final Remarks
Simulation of strong ground motions in Mid-America is an ongOIng research
problem, particularly in view of the controversy on interpretation of the recent
seismotectonic observations in the NMSZ (Newman et al. 1999, Mueller et al. 1999).
Definitive conclusions may not be reached for some time to come. In this study, for
purpose of performance-based structural analyses, a simulation method is developed
24
following closely the guidelines of USGS OFR-96-532. As more information becomes
available, the procedure can be modified with minor efforts. The ground motion models
used in this study can account for effects of finite fault dimension and have been shown
to produce results which are in good agreement with broadband simulation method in the
frequency range of engineering importance. In the following chapters, these ground
motions will be used for statistical study of structural responses.
25
Table 2.1 Focal depth distribution for point source model (Wheeler and Johnston, 1992).
Depth (km) Weight
1 -- 5 0.250
5--10 0.500
10--15 0.050
15--20 0.050
20--25 0.015
25--30 0.135
Table 2.2 Focal depth distribution for finite fault model (modified from EPRI, 1993).
Depth, km (mean) Weight
5.00 -- 13.25 (9.10) 0.40
13.25 -- 21.50 (17.40) 0.40
21.50 -- 29.75 (25.60) 0.15
29.75 -- 38.00 (33.90) 0.05
26
Table 2.3 Representative soil profile of Memphis, TN (after Hashash, 1999).
Layer Soil column Thickness (m) Vs (m/sec) Density (g/cm3)
1 Alluvium 7.2 360 1.92 2 Alluvium 4.8 360 2.00
3 Alluvium 14.9 360 2.08 4 Loess 9.0 360 2.16
5 Fluvial Deposits 7.9 360 1.98 6 Jackson Formation 47.3 520 2.08
7 Memphis Sand 245.6 667 2.30
8 Wilex Group 83.3 733 2040
9 Midway Group 580 820 2.50
10 Bed Rock Half-Space 3600 2.80
Table 204 Representative soil profile of Carbondale, IL (after Hashash, 1999).
Layer Soil column Thickness (m) Vs (m/sec) Density (g/cm3)
1 Cahokia Alluvium lOA 140 2.0 2 Henry Formation 10.0 250 2.1
3 Henry Formation 25.6 270 2.1
4 Mississippi Embayment 119.0 280 2.3
5 Pennsylvanian Limestone 835.0 2900 2.6
6 Bed Rock Half-Space 3600 2.8
Table 2.5 Representative soil profile of St. Louis, MO (after Hashash, 1999 and Herrmann, 1999).
Layer Soil column Thickness (m) Vs (m/sec) Density (g/ cm3)
1 Modified Loess 5.7 185 1.9 2 Glacio-Fluvial 10.0 310 2.1
3 Mississippian Limestone 984.3 2900 2.6
4 BedRock Half-Space 3600 2.8
27
Table 2.6 Soil profile at Pier C of the Hernando Desoto Bridge located at Mud Island, Memphis, TN (after Hwang et al., 1999).
Layer Soil column Thickness (m) Vs (m/sec) Density (g/cm3)
1 Alluvial Deposits 13.94 170 1.87 2 Alluvial Deposits 15.45 210 1.79 3 Jackson F onnation 21.52 330 1.91 4 Memphis Sand 132.09 425 2.00 5 Memphis Sand 120.00 600 2.12
6 Flour Island F onnation 91.00 708 2.18 7 Fort Pillow Sand 157.00 700 2.18
8 Porters Creek Clay 141.00 789 2.22 9 McNairy Sand 308.00 1050 2.31
10 Knox Dolomite Half-Space 3500 2.70
28
Table 2.7 Comparison of the proposed simulation method with broadband simulation.
Broadband Simulation Stochastic Model Methods (Sommerville et al. 1997, (this study)
Saikia & Sommerville 1997)
Source Model finite fault finite fault (near-field) point source (far-field)
Local Site Effect equivalent linear method quarter-wavelength method
Theoretical hybrid:
spectral representation based on Background
field records (short period) field observations
wave propagation (long period) Computational extensive not excessive Effort Scaling Factor 0.27 ,..., 10.75 0.41",4.48 Median Response 100/0"'" 25% 10% '" 15%
Estimate theoretically rigorous basis, but
Comments not easy for practicing
easy for engineers to use engineers to generate ground motions
29
r-----------~ .. : Initiate the dh simulation I
+ t: II
I Determine no. of earthquakes in the d h simulation I
Generate random numbers for each earthquake sample function e.g. magnitude, focus (epicentral distance Re, focal depth), attenuation uncertainty, and slip distribution (finite-fault), etc.
r------------------------------------------- ------------------------------------------------
far-field 2-corner point source empirical model (Boore, 1996)
soil amplification due to local site effects by quater wave-length method (Joyner et.al. 1981)
near-field ground motion using finite fault model (Beresnev and Atkinson 1997, 1998)
-----------1------------------------1---------------1---------------------------r------------
No
I I
I Baseline correction (USGS BAP or Brady 1966)
,
Sample size is large enough?
Yes
END
Figure 2.1 Ground motion simulation flowchart for Mid-America cities.
I I
30
Seismicity surrounding Memphis TN, Carbondale IL, and St. Louis MO
§. ~
~ .Jl -II" L-V; 0 /,1 ; ~ .7 0 ~
~ cq ~ 0
0
~ ~ ~
~
Figure 2.2 Annual occurrence rate per 0.1 xO.1 degree square of earthquakes of fib 5 or larger.
31
39 ~--------~----------~----------~----------~
38
37
36
35
34 -92
~ St. Louis, MO
~ Carbondale, IL
® Memphis, TN
-91 -90 -89 -88
Longitude
Figure 2.3 Seismic zone of Mw-8 earthquakes.
42
41
40
39
38
36
35
34
33 o o
32 -95
o
o o
o
00
o
o
o
o
-94
o 0 o
0 0
o
o
o o
o
o o
o
o
o o
o o
-93
o
o
o o
o o
00
o
-92
o
o
o o
-91
32
o
o
o
-90
o o
o
o
-89
Longitude
o
o
o
o o
o
o o
o
o 0
o o
o Carbondale, IL o 0
o o
o
o o 0 0
00 CO o 0 o 0
o 0 o 0 000
o o o
o o o 00
o o
o
o
-88 -87 -86 -85
o
o Mw = 4.71 ; fib = 5 Mw = 8 ; fib = 7.47
Figure 2.4 Epicenters and magnitudes of earthquakes of simulated 6000-year record.
CEUS maxilmum-magnitude zones
50 260 270
40 _I
\ \ \,:" ? "--V / I d ___ /7/~j7 \
30 I -\
250 2BO 260 270
Figure 2.5 Mmax zones used in the Central and Eastern United States. Numerical values are moment magnitude (after USGS OFR-96-532).
w W
~ o "'0 ·2
0.9
34
c:= 0.2 11 = 0.05
~ 0.6
~~ 0.0 I---....J ..................... ...................................................... ································,·T··!l··············
--·~I c-Tw I~ I ~ Tw •
-0.3 L--"-_-'-----L_-'-----:..._....I..---l._--I.-_L---l-_.l....---L_....i...---l..----!
0.0 0.2 1.0
Time/Tw (sec)
Figure 2.6 Envelope function used in point source model. The meaning of parameters is shown here. Note that the abscissa has been nonnalized by a quantity proportional to the duration of the motion (Tw =2· Td).
6 4 3 2
,.-... 10-1 OJ)
6 '-"
= 4 0 -- 3 ...... = 2 .. ~ -~ 10-2 CJ CJ
< 6 - 4 = .. 3 ...... C'.J 2 ~ Q.
rJ'J 10-3
6 4 3 2
10-4
10-2
////////
35
~ = 6.9, h = 3.1 km, Re = 264.4 km, E = 0 ( this study) median S. of Saikia & Somerville (1997) range of 16 to 84 percentile (Saikia & Somerville, 1997)
2 3 4 5 67 10-1 2 3 4 5 67 100 2 3 4 567 101
Period (sec)
Figure 2.7. Comparison of response spectra based on point-source model with Saikia and Somerville's broadband model (1997).
36
Surface
Observation r---.L..------t~lR~ ____ __..p~oint
p
d
o
o Origin 81 fault dip CP1 fault strike CP2 azimuth to observation point d depth to fault upper edge F center of subfault ~w subfault width M subfault length i, j subfault number
Hypocenter (io, jo)
NO~X j .. i'.' .. e Surface
P'I .......
I~p rl distance from subfault to observation point
OF = {R. COS(¢2 - ¢l)' R· sin(¢2 - ¢l),-d} OF = {(2i -l)LlI / 2, (2) -l)(Llw /2) sin 5, (2) -l)(Llw / 2) cos 5} ~ --f -f
'1 = OP- OF
'1 = ([R. COS(¢2 - ¢l) - (2i -l)LlI / 2 r + [R. sin(¢2 - ¢l) - (2) -l)(Llw / 2)sin bT +[ d + (2) -l)(Llw /2) cos 5] 2}1/2
Figure 2.8 Beresnev-Atkinson finite fault geometry.
0.9
3 ...... = Q) = 0.3 o Q.. ~ ~
37
7J
s=0.2 11 = 0.2 taper = 0.02
................... l ........................................................................................ ~ ...... I ........................... .
~I o·Td I~ • I taper 'Td ~ Td ~ taper 'Td ~-+--
-0.3 '----'-_..I.------I.._-'------""_-!..-_l..----l-_...l------I-_.......i.-__ --'-_..I.-----.J
0.00 0.02 0.22 1.02 1.04
Time/Td (sec)
Figure 2.9 Envelope function used in subfaults of finite fault model. The meaning of parameters is shown here. The abscissa has been normalized by a quantity proportional to the duration of the sub event (Td in Equation 2.16). Note that the scale of the leading and trailing taper zones is exaggerated.
38
hypocenter
Figure 2.10 Sample of random slip distribution based on correlated random field model and discretization for finite fault model (Mw = 8, h = 25.6km).
39
hypocenter
.J;
-e ...::;,/
~
% ..... ifJ
<6 % ~ ~. ~
~ ~ ~
(.)
Figure 2.11 Sample of simulated slip distribution based on correlated random field model and discretization for finite fault model (Mw = 8, h = 33.9 km).
6
5
-
...... : ... -: ... : .. -: . .; ., ..
... . -: .. :-.:.:.:-: ........ : ..... .
40
1997 NEHRP B/C boundary SHAKE91 + gm(2% in 50 yrs.) SHAKE91 + gm(10% in 50 yrs.) quarter wavelength method
'0 00 2
1 ..... ", . . /
. . . . .... . . . . . . ................. ' .... , -_ ............ .
10-2 2 3 4 5 67 10-1 2 3 4 5 67 100 2 3 4 5 67 101 2 3 4 567 102
Period (sec)
Figure 2.12 Soil amplification factor for representative soil profile, Memphis, TN.
= .: -; u :: c.. E <
9 I I I
8 1997 NEHRP B/C boundary
------- SHAKE91 + gm(2% in 50 yrs.) ....... ~ ~
. --- SHAKE91 + gm(10% in 50 yrs.) :' ........... ·: ... :fH\· .. :. . .......... ..
. -- quarter wavelength method ,- ... - ...... ~ ..... - .. 7 .: .. -: .. : .. :.:. c-:· ...... ·;- .. · '-- -:--: ., .: .. :.;-:. ....... :.,.\ .. : ... : ... :..:. >-:.: ....... ; . . .
4
,:' '~ll:tMj, •••• .', ~" ...... ""'~l ~'-·',I ' fH~': ... " ....... , ....... , ..... : .. :.:.
............ :. :':' : ............. ~ 1./ /1 ; ~( ........ :.:.;.: ........ : .. . . .., l' , . . . ,,, .. "
.. .. t ,~-!-\.,.} r -:lJ:,~\·· f- . ~ . ' ... : .......... : . . . ". . ,\·q(n~1'" (- j-' ...
.... .. ·-:rlfr~V;;'~!\t1Lt ~ \ . .' ... . .. """ ·:t V"" ,' .. \\,}!'~''1:;: .. .j.. .. . .. ..... , ... -.;._./...... 1;"4V~,Ju~:-' .. '" .~ ... - ... - ... -:-,~;;;;:.:;:::..,.....~.~----__ """'
.: ... : .. : :.~!~.:. -.-:,,-:~:~ .... ~ ......... -. .. -..... ,_. -,_ .. --, ~ ..... -............... -,_ .. -,- .
6
5
3
2
.... o l---~-':- --... ~---~
10-2 2 3 4 5 67 10-1 2 3 4 5 67 100 2 3 4 567 101 2 3 4 567 102
Period (sec)
Figure 2.13 Soil amplification factor for representative soil profile, Carbondale, IL.
9
8
7
I: .:: 6 ~ <:oJ :a 5
Q.. S 4 -< ·0 3 rFJ.
2
o
2 3 4 5 67 10-1
41
1997 NEHRP B/C boundary SHAKE91 + gm(2% in 50 yrs.) SHAKE91 + gm(10% in 50 yrs.) quarter wavelength method
..•.. - .....•... - . '-----,.--:---:--,..-,-.,.....,...,,...,...------:--.,.--.,.......-,.--,-:---:-J:!
.. ,' .. ' .. ~ .', " .', ' ........ '. . . . -: - .. ;. .. : . -:. ~ .: . .
.. .: .. : .. ;.: .. :.:.:- ....... : .. .
-, . ~ ...... ' ...
... : .. : .. ;.:..:.:.: ........ :.
2 3 4 5 67 100 2 3 4 567 101
Period (sec)
Figure 2.14 Soil amplification factor for representative soil profile, S1. Louis, MO.
c: . : -eo:::
<:oJ t:
S <: ·0 IJ]
3
. . . . . ~ . . . .'. . . ~ . .' .: .. :.:. ~ .:. . . . .... : .
2
... ./
0 .... -----.::
this study Mud Island (Hwang et a!. 1999) 1997 NEHRP Class D (Boore & Joyner 1997) Mississippi Embayment (Boore & Joyner 1991) Memphis 1000 m soil column (Herrmann et a!., 1999)
2 3 4 5 67 10-1 2 3 4 5 67 100 2 3 4 5 67 101 2 3 4 5 67 102
Period (sec)
Figure 2.15 Comparison of various soil amplifications in Memphis, TN.
42
100 ,-.... N ~=6.7, h = 19.5 km, Re = 137.7 km, c = 0.56 ~
Q) r:I). - 50 e ~ --c
0 .~ ..... e'I: ;... Q)
Q) -50 ~ ~
< -100
0 10 20 30 40
10
,-.... e.J Q) 5 r:I).
e e.J -- 0 ".. ..... . ~ 0
Q) -5 ;;>
-10 0 10 20 30 40
1.5 ,-....
e 1.0 ~ --..... 0.5 ::: Q)
e 0.0 Q) ~ ~ -0.5 c.. r:I).
Q -1.0
-1.5 0 10 20 30 40
Time (sec)
Figure 2.16 Baseline corrected sample 10% in 50 years ground motion for representative soil profile, Memphis, TN.
43
700 -. N ~ = 8, h = 25.6 km, Re= 147.6 km, Rjb= 75.9 km, g = 0.56 ~ ~ t:n - 350 e ~ ---= 0 .s ...... ~
~ Q) -350 ~ ~
< -700
0 10 20 30 40 50 60 70
60
-. 40 ~ ~ t:n - 20 E ~ --- 0 ;,-. ...... . ~
-20 0 ~ :>- -40
-60 0 10 20 30 40 50 60 70
30 -. e 20 u ---...... 10 == Q)
E 0 ~ u ~ -10 Q.. .~ Q -20
-30 0 10 20 30 40 50 60 70
Time (sec)
Figure 2.17 Baseline corrected sample 2% in 50 years ground motion for representative soil profile, Memphis, TN.
44
400 ,-..., N ~ = 8, h = 33.9 km, Re= 186.1 km, Rjb= 61.8 km, e = 0.09 ~
Q) rI:l - 200 5 ~
'-"" :::
0 (a) oS: .... ~ l. Q)
Qj -200 ~ ~
Observation Pt : Memphis (TN) < -400
0 10 20 30 40 50 60 70 80 90 100
,-..., 400 N
~ = 8, h = 2506 km, Re= 79.3 km, Rjb= 61.7 km, e = -0.69 ~ Q) rI:l - 200 5 c:.J
'-"" ::: (b) oS: 0 .... ~ l. Q)
~ -200 c:.J c:.J
Observation Pt : Memphis (TN) < -400
0 10 20 30 40 50 60 70 80 90 100
400 ,-..., N ~ = 8, h = 33.9 km, Re= 117.8 km, Rjb= 114.1 km, e = 0.31 c:.J
Q) rI:l - 200 E c:.J
'-"" :::
0 (C) oS: --~ l. Q)
~ -200 c:.J c:.J
Observation Pt : Carbondale (IL) -< -400
0 10 20 30 40 50 60 70 80 90 100
Time (sec)
Figure 2.18 Near-source "Doppler" effects of simulated acceleration time histories in (a) Memphis, TN, (b) Memphis, TN and (c) Carbondale, IL by the finite fault model.
45
CHAPTER 3
METHOD OF SIMULATION - WESTERN UNITED STATES
3.1 Overview
F or simulation of earthquake ground motions in Western United States, a different
procedure is used in view of much larger number of earthquake records including those
of near-source events and much better defined seismic zones. To simulate future seismic
events and ground motions, the procedure proposed by Collins, Wen and Foutch (1995)
is used again with some modifications. The ground motion model basically follows the
empirical Fourier spectrum proposed by Trifunac (1994), and then a modification factor
is incorporated to account for some of the important near-field effects based on data
(Sommerville et al. 1997). The tectonic and seismological data are taken from the 1995
WGCEP report (Working Group on California Earthquake Probabilities). Possible future
events are generated for a period of 10 years. The site is at Santa Barbara, California. A
total of 1000 simulations of 10-year period are carried out for to construct uniform hazard
response spectra. The simulation procedure is shown in Figure 3.1 and details of the
simulation method are given in the following.
3.2 Selection of Location and Soil Profile
The site location is (34.42°N, 119.700 W) in Santa Barbara, California. The average
shear wave velocity in is shown to be around 406 mlsec according to the average shear
wave velocity map Park and Elrick (1998) generated for the uppermost 30 m soil profile
of Southern California using surface geology. We therefore assume a local site condition
of very dense soil and soft rock corresponding to the 1997 NEHRP Type C site condition
(Vs = 360 -760 mlsec) or Site Class B according to Boore et al. (1993), i.e. rock with Vs
46
= 360 ~ 750 m1sec. The same site condition was used in Collins et al. (1995) for Los
Angeles. This facilitates the comparison of seismic hazard obtained in this study with
Collins et al. (1995) and USGS (1997).
3.3 Reference Area, Seismicity and Tectonics
Considering the relatively faster attenuation in California than in the Mid-America,
the effective zone is defined as a circular area of a radius of 150 km centered in Santa
Barbara. The choice of 150 km is based on Collins et al. (1995), USGS deaggregated
seismic hazard data (1997) and USGS OFR-96-532 (Frankel el al. 1996). In the effective
zone of Santa Barbara (Figure 3 .2(b)), there are 29 seismic zones contributing to the
seismic hazard at the site. The major fault locations are listed in Table 3.1 and plotted in
Figure 3.2 (a). The seismological data are taken from the 1995 Working Group Report
on California Earthquake Probabilities (WGCEP). According to the available geodetic,
geologic and seismic information, the seismic zones are categorized into three types of
seismotectonic zones. Type A zones contain faults for which paleoseismic data suffice to
model characteristic earthquakes as time dependent events; i.e. a renewal model is used.
Type B zones contain faults with insufficient data, so the characteristic earthquakes in
Type B zones are modeled as a Poisson process. Types C zones contain diverse or
hidden faults and therefore there are no characteristic earthquakes. The detailed
information, including source faulting mechanism and seismicity rate, is given in Table
3.2.
3.4 Occurrence and Source Models
To facilitate the proposed simulation procedure, the reference area is further
discretized into 10 kmxl0 km cells as shown in Figure 3.3. In Type A zones,
occurrences of characteristic earthquakes are time dependent and the inter-occurrence
time follows a lognormal distribution. In the B zones, occurrences of earthquakes are
47
time independent (i.e. Poisson distribution) and occurrence time intervals are
exponentially distributed. The main random occurrence parameters are generated as
follows.
3.4.1 Number of Earthquakes
To determine the number of earthquakes in a certain seIsmIC zone, a random
variable, Uk, with a uniform distribution between 0 and 1 is generated and must satisfy:
(3.1)
where nk is the number of earthquakes that occur with a moment magnitude greater than
6 in seismic zone k during a period of t years from present (t = 10 in this study); N sz is
the total number of seismic zones in the reference area of Santa Barbara (Nsz = 29 in this
study); P(X = nk ) is the probability of having nk earthquake occurrences with moment
magnitude greater than 6 in seismic zone k during a period of 10 years. According to this
relationship, for characteristic earthquakes in Type B zones and distributed earthquakes
in all three different types of seismic zones, nk must satisfy:
k = 1,2,.· ·,Nsz (3.2)
where V,I; is the mean annual occurrence rate of earthquakes in seismic zone k with
moment magnitude greater than 6. For characteristic earthquakes in Type A zones,
lognormally distributed random numbers T/s are generated as occurrence time intervals
and must satisfy:
(3.3)
where To is the time elapsed from last event to present; the relationship between Ti and To
is shown schematically in Figure 3.4. The total number of earthquakes in seismic zone k
during a period of 10 years, Nk, can be expressed as
(3.4)
48
where nk c is the number of characteristic earthquakes in zone k; nk d the number of , ,
distributed earthquakes in zone k. In this study, a total number of 1000 simulations of 10-
year periods (starting from the year 2000) are simulated resulting in 1815 ground
motions. The annual probability of no earthquakes greater than moment magnitude 6
based simulation is 0.83, which is close to the target value 0.78 (calculated from the 1995
WGCEP report). 1000 simulations of 10-year period give a satisfactory estimate of
unifonn hazard response spectra, which will be shown in Chapter 4.
3.4.2 Magnitude
For characteristic earthquakes, the moment magnitude Mw is related to the length of
the fault segment L and the characteristic displacement D as follows:
(3.5)
where Jl is the rigidity of the earth crust (assumed to be 3 x 1010 Nm-2); H is the thickness
of the brittle crust (taken to be 11 k..m following here).
F or distributed earthquakes, the magnitude is considered a random variable
following the modified Gutenberg-Richter distribution:
(3.6)
where F M" (M w :::; m) is the cumulative distribution function of moment magnitude Mw
(note that 6:::; m :::; mx and mx indicates the limiting moment magnitude of a given seismic
zone); N (m, mx ' fd) is the cumulative rate of m:::; mx earthquakes and fd is the occurrence
rate of distributed earthquakes with moment magnitude greater than 6. The modified
Gutenberg-Richter distribution is shown in Figure 3.5 for comparison with the original
distribution.
49
3.4.3 Epicenter
It is assumed that earthquakes are equally likely to occur anywhere within a given
seismic zone. To do so, an initial epicenter location is generated from a uniform
distribution function and then a rupture plane within the given seismic zone is drawn
according to this initial epicenter location to have a rupture size comparable with its
magnitude, and appropriate strike and dip angles. However, if this initial epicenter
location fails to satisfy the required rupture condition, another epicenter will be generated
until all requirements are satisfied.
3.4.4 Focal Depth
The focal depth, H (km) is assumed to have a triangular probability density
function with a lower bound of 4 km, an upper bound of 24 km, and a mode of 14 km.
This choice of distribution is based on data compilation from the earthquake catalogue in
the effective area of Santa Barbara (Seekins et al. 1992, NOAA, 1996). To generate
random numbers conforming to a given triangular probability density function, the
inverse transform method is employed. The details can be referred to Ang and Tang
(1990)
3.5 Ground Motion Modeling
For far-field events, a point source is used. For near-field earthquakes, a strike-slip
or dip-slip rupture is assumed according to the tectonic characteristics of the
corresponding seismic zone. The ground motions are modeled as a random process
composed of sinusoidal waves with specified frequencies and random phase delays
(Shinozuka and Jan 1972, Shinozuka and Deodatis 1991), with a Fourier amplitude
spectrum according to the empirical model by Trifunac (1994). The attenuation effect
takes the form of Boore et al. (1993), in which prediction error is also considered. Near
field effects are incorporated according to Somerville et al. (1997a) witp. modification to
50
emphasize long-period motions In the strike-normal direction, which are generally
associated with velocity pulses. The resulting ground motions are then baseline-corrected
using a 2nd-order parabola (Brady 1966, Nigam and Jennings 1968).
3.5.1 Duration
The significant duration of strong motion, td (sec), follows Eliopoulos and Wen (
1991):
10gIO tD = -0.14+ O.2Mw + 0.002Re + Cd (3.7)
where tD is the significant duration of strong motion, i.e., the time interval required to
build up between 5% and 95% of the Arias intensity of the record (Trifunac and Brady,
1975); Mw is moment magnitude; Re is epicentral distance (lan); Cd is the prediction error
that follows normal distribution with a zero mean and a standard deviation of 0.135. The
total duration is therefore determined by the following equation:
(3.8)
where Cj and C2 are random variables intended to model the buildup and decay phases of
the ground motion. The random variables eland C2 are assumed be uniformly distributed
between 0.1 and 0.5, and between 0.5 and 1.0, respectively. The upper bound of tF is set
at 60 seconds.
3.5.2 Fourier Amplitude Spectrum
The median amplitude of the Fourier spectrum, FS(T) , is determined by the
empirical scaling equation by Trifunac (Trifunac and Lee 1985, Trifunac 1994):
where
logici FS( T) = M< + Att(~, M w ' T) + bI (T)· Mo + b2(T)· h + b3(T)· v
+b4 (T)· h v+ bs(T) + b6(T)· Mo2 + b~l)(T)· s2) + b?) . s2)
FS(T) == Fourier amplitude spectrum of acceleration at period T (inches/second)
(3.9)
51
Att( ~,M, T) == frequency-dependent attenuation function ( T= 1 If) of the spectral
amplitudes versus the "representative" source to station distance L1 and magnitude M
(details can be referred to Trifunac, 1994)
h == depth (thickness) of the sedimentary layer beneath the station (lan)
v == indicator variable ( v = ° for horizontal motion, v = 1 for vertical motion)
b i ~ b~2) == scaling coefficient functions of the period T
st) , si2) == indicator variables describing the local soil site condition
sill = {~ if SL = 1 (stiff soil)
otherwise
sf) = {~ if S -; (deep soil) L- .....
otherwise
SL = 0, 1 and 2, representing rock, stiff soil, and deep soil site, respectively.
Re == epicentral distance (km)
H == focal depth (krn)
s == source dimension (details can be referred to Trifunac, 1994)
The above equation is valid only for periods below a certain cut-off period, which
depends on the magnitude under consideration (Trifunac, 1994). For periods above the
cut-off period, the Fourier amplitude spectrum is extrapolated using a straight line with a
slope of 25-M when FS(T) vs T plot is on log-log scale (Collins et al. 1995). The
sensitivity of the Trifunac Fourier amplitude to varying magnitude-distance combinations
is shown in Figure 3.6. It is noted that in this study the Fourier ampJitude is used to
52
describe the general spectral shape due to attenuation and the actual amplitude is scaled
using target peak ground acceleration due to Boore et al. (1993, 1994, 1997) considering
attenuation uncertainty:
In(PGA) = -0.038 + 0.216 ·(Mw - 6) - 0.777 . In(r) + 0.158· GB + 0.254· Gc + cA (3.10)
and
(3.11 )
where Rjb is the closest horizontal distance to the vertical projection of the rupture (km); d
is the fictitious depth for better fit of earthquake records ( d = 5.48 Ian is used); PGA is
the peak ground acceleration (g); Mw is moment magnitude; GB and Gc are parameters
related to site soil condition (GB = 1 and Gc = 0 are used for Santa Barbara); CA is the
uncertainty term modeled as a normally distributed random variable with a zero mean
and a standard deviation of 0.205.
3.5.3 Spectral Representation of Ground Motions
According to Shinozuka and Deodatis (1991), ground motion, A (t), can be
simulated as follows:
and
A(t) = l(t)· {~a/t). Amp(ml1f)· cos[(2mn4()t + ¢ml
+ ~~2(t). Amp(ml1j).cos[(2mn4()t + ¢ml}
Amp(ml1j)=2~4( . FS(m4() tD
(3.12)
(3.13)
where I(t) is intensity modulation function; aI(t) and a2(t) are frequency modulation
functions; FS(mLlj) is the Fourier amplitude spectrum at a certain frequency fm = mLlf; tD
is the significant duration of ground motion (sec); Llf is the cyclic frequency increment
S3
(0.015 Hz is used); ¢is the phase angle modeled as a unifonn distributed random variable
between 0 and 2n; N is the number of frequencies at which values of the Fourier
amplitude spectrum are estimated; p = intergerif,.m/L1j);f,.ms is the central frequency or the
so-called root-mean-square frequency (Hz) defined as:
frms = N (3.14)
LFS(fm) m=1
The intensity modulation function l(t) takes the fonn suggested by Jennings et al. (1968):
( Cl~J2 if t < cltD
l(t) = 1 if cI t D :::; t :::; cI t D + t D (3.1S)
e -K[t-( c1tD+tD)] if t > cltD + tD
where K is defined such that l(tF) = O.OS. To incorporate the time-varying frequency
content, a simple ramping function is introduced for a weak frequency modulation of the
resulting ground motion (Collins et al. 1995):
t a2(t) = 1 if QI(t)=- t < cltD
clD
QI (t) = 1 a/t) = 1 if cl D :::; t :::; cl D + t D (3.16)
QI(t) = 1 a2(t) = 1-t-clD -tD if t > clD +tD
c2tD
3.5.4 Near-Field Effects
F or near field earthquakes, the length and the orientation of the rupture segment
become more important. In view of this, three different models are used to describe the
seismological characteristics of the site. For earthquakes with a closest rupture distance
54
less then 50 km, a faulting mechanism of vertical strike-slip or dip-slip with surface
rupture is used with parameters of the 1995 WGCEP report; otherwise, a point source
model would yield satisfactory results. For a given magnitude, the length and width of
the rupture plane can be expressed by empirical formulae (Wells and Coppersmith,
1994):
{-3.55 + 0.74Mw for strike - slip
10glO (SRL ) = -2.86 + 0.63 M w for dip - slip
{-3.42 + 0.90Mw for strike - slip
10glO(RA) = -3.99 + 0.98Mw for dip - slip
RW=RAISRL for both
(3.17)
(3.18)
(3.19)
where SRL is the surface rupture length (km), RA is the rupture area (km2), RW is the
rupture width (km), and Mw is moment magnitude. After the rupture plane is generated,
the next step is to determine whether the modification due to near-field effects is
necessary. Based on an empirical analysis of near-field data, Somerville et al. (1997a)
proposed a modification due to rupture directivity effect, which indicates larger spectral
acceleration for periods longer than 0.6 second when the rupture propagates toward the
site. The modification can be expressed as:
(3.20)
where y is the natural logarithm of the strike-normal to average horizontal spectral
acceleration ratio; C j , e2, and C3 are period-dependent coefficients (Somerville et aI.,
1997); Rrup is the closest distance to the nlpture plane (km); Mv,I is moment magnitude;
(= B if a strike-slip fault is considered; ('= ¢ if dip-slip fault is considered; B is the
azimuth angle between fault plane and ray path to site (Figure 3.7); ¢ is the zenith angle
hOh~voon f:ault nl-::lnp -::Inri r'.:lU n'.:lth tn C!ltp {PiO'llrp 1 7'\ Prnplrir~l pnll~tl()l1 (1 ?()'\ i.;:: v~liil u\.;lv \.,.<'='~!.!. lL !-'lU.ll\,... UJ.J.U J.UJ P'""''''~~ ""'-' u~ ... '-' \..&. A.ot..t..J.~...,- J- ..L..I.LA. ... .L ......... """' .... .&. ....... '1.. .... _"' .... "'..L .... \_0 __ , .... u .. _ ...... -
only for Mw > 6, Rrup < 50 km and (' < 45°; otherwise, y = O. If spectrum modification is
required, the following iteration procedure (Levy and Wilkinson, 1976 , Shinozuka et. aI.,
1988) is activated:
55
(3.21)
where Ampl[m) is the spectral amplitude after ith iteration at cyclic frequency 1m (= mLJj);
RSr([m) is the target response spectrum at cyclic frequency 1m; RSl[m) is the response
spectrum after ith iteration at cyclic frequency 1m.
3.6 Simulation Results
300 simulations of 10-year records in the reference area of Santa Barbara are
shown in Figure 3.8, which contains 517 earthquakes. By comparison of Figure 3.8 and
Figure 2.4, they both have about the same number of simulated earthquakes, but the
simulation period in Figure 2.4 is doubled and the reference area is 10 times larger,
owing to the lower seismic activities in Mid-America. The very frequent seismic activity
shown to the north of Santa Barbara is in the San Andreas Fault (Zone 3 more precisely);
however, due to the rapid attenuation effect in California, it is the local seismicity that
makes major contribution to the seismic hazard in Santa Barbara (USGS, 1997). A
reference zone of 150 km x 150 km therefore covers all possible hazards of engineering
interest.
A sample acceleration time history generated in Zone 40 is shown in Figure 3.9(b).
Its response spectrum is given in Figure 3.9(a), in which near-field effects are considered.
A permanent displacement offset (Figure 3.9(d)) is observed due to the large magnitude
and very close distance.
The number required of artificial ground motions in a Monte Carlo simulation
depends on: (l) the occurrence rates of events in the seismic zones in the reference area,
(2) required accuracy of the engineering problem under consideration, and (3) the
demand of computational time involved in the engineering problem. More details will be
given in the following chapter.
56
Table 3.1 Locations and zone types of major faults surrounding Santa Barbara.
Major Surface Fault Location Zone Type
Arroyo Parida 46 C
Big Pine 26 B
Garlock 27 B
Kern Front 53 C
Malibu Coast 33 B
N ewport-Inglewood 20 B
Oakridge 46 C
Ozena 65 C
Palos Verdes 34 B
Pine Moutain 40 B
Rinconada 37 B
San Andreas 3,4,5 B,A,A
San Cayetano 46 C
San Gabriel 31,32 B,B
San Juan 50 C
Santa Cruz 35 B
Santa Monica 33 B
Santa Ynez 40 B
Sierra Madre 31 B
White Wolf 25 B
II
Table 3.2 Parameters of seismic source nl0del in the effective area of Santa Barbara. -- ------- -------- - -------
I I Return Period (yr) Last
Limiting fd fc Faulting
Zone Type Rupture
magnitude Mechanism
Range of Strikes Dip Comments Mean Std. Dev. 111 x IO')/yr IO')/yr
3 B 7.05 3.54 13.74 SS 0° ~ 180° 1857/119 Fort Teion m=7.8 RL 4 A 206 86.5 1857 7.51 3.05 SS 0° ~ 180° 5 A 150 71.7 1857 7.56 2.03 P 0° ~ 180° 1894/7/30
20 B 6.79 0.55 0.49 P 90° ~ 180° 25 B 7.29 7.58 0.11 SS 0° ~ 180° 1952/7/21 Kern County R. LL 26 B 6.94 1.48 0.22 DS 30° ~ 150° 45°N 27 B 7.41 2.45 0.38 P 0° ~ 180° 31 B 7.30 1.18 0.15 P 30° ~ 150° 32 B 7.00 2.33 0.40 DS 90° ~ 180° 45°N 197112/9 San Fernando R. LL 33 B 7.10 2.74 0.86 DS 30° ~ 150° 45°N 1855/7111 M=6.0 34 B 7.10 0.75 0.41 P 90° ~ 180° 35 B 6.94 2.19 0.62 DS 30° ~ 150° 45°N 36 C 7.05 1.46 0.00 SS 0° ~ 180° 37 B 6.94 1.38 0.23 P 0° ~ 180° 39 B 7.15 1.93 0.38 P 0° ~ 180° 40 B 7.05 8.16 0.02 DS 30° ~ 150° 45°N 43 C 6.79 1.99 0.00 P 0° ~ 180° 46 C 7.30 16.48 0.00 DS 30° ~ 150° 45°N 49 C 6.43 4.34 0.00 P 90° ~ 150° 50 C 6.94 6.35 0.00 P 0° ~ 180° 51 C 6.79 3.99 0.00 P 0° ~ 180° 1952/11122 M=6.0 52 C 6.79 7.69 0.00 P 0° ~ 180° 1983/5/2 M=6.4 53 C 6.43 9.15 0.00 P 0° ~ 180° 1952/7/29 Kern County R. LL 54 B 6.79 2.91 4.42 DS 30° ~ 150° 45°S 1994/1117 Northridge 55 C 7.05 24.22 0.00 DS 30° ~ 150° 45°S 1925/6/29. 1883/9/5 56, C 7.05 12.60 0.00 P 0° ~J80° 57 C 7.05 3.81 0.00 P 0° ~ 180° 1927111/4 60 C 6.58 1.00 0.00 P 0° ~ 180° 65 C 7.23 16.14 0.00 DS 30° ~ 150° 45°S
P.S. Codes for (1) faulting mechanism: SS == "Strike Slip", DS == "Dip Slip", P == "Point Source", (2) comments: R == "Reverse", LL == "Left Lateral", RL == "Right Lateral ", the subordinate sense of slip is listed after the primary slip type. (Source: WGCEP 1995, Wells and Coppersmith 1994)
Vl -.l
.....-I
+ ~
II
No
58
Initiate the nth simulation
Detennine no. of earthquakes in the nth simulation, i. e. characteristic and distributed earthquakes.
Generate random numbers for each earthquake sample function e.g. magnitude (rupture length, width), distance (Re, Rjb, Rrup),
hypocenter (focal depth), duration, strike/dip, etc.
Detennine peak ground acceleration due to attenuation e.g. Boore et al. 1993
Detennine the general spectral shape due to attenuation (e.g. Trifunac 1994)
r---
Construct intensity and frequency modulations e.g. Jennings et al. 1968, Collins et al. 1995
Build mathematical fonn for stationary ground rnotion e.g. Shinozuka & Deodatis 1991
1 _______________________________________________ J
Modify Response Spectrum due to Somerville et al. 1997
Baseline correction (Brady 1966)
Modify the resulting ground motion to match the response spectrum.
El~D
e.g. iteration procedure due to Levy & Wilkinson 1976, Shinozuka et al. 1988
Figure 3.1 Ground motion simulation flowchart for Santa Barbara, CA.
59
--e ~ 50 --Q)
;:: rIJ
e (a) c ~ Q) C.J = C':I - -50 rIJ
Q
--e ..:.: -Q.I 50 .!:: ..., e Q
c!: (b) G.I '-J
~ .~ Q -50
-150 -150 -50 50 150
Distance from site (km)
o Location of Santa Barbara
Figure 3.2 (a) Major faults (thin black lines) in the effective area of Santa Barbara, (b) Seismic zones contributing to the seismic hazard at Santa Barbara.
60
150
..-,
~ --- 50 Q) ... . -I:I'.l e 0 ~ Q) ~
= ~ ... I:I'.l .- -50 Q
-150 -150 -50 50 150
Distance from site (km)
o Location of Santa Barbara
Figure 3.3 The discretized reference area and seismic zones surrounding Santa Barbara.
last rupture present
., Time (yrs.)
To ~I( t )1 T] )1..: T2 )1-= T3 )1
Figure 3.4 Illustration of time intervals defined in the renewal model.
61
10-2
10-3
,,-"0 ~ --~ (a) ~ U
I .... ~ -- 10-4
._._._._0- Orjginal G-R Modified G-R
10-5
6.0 6.5 7.0 7.5
Magnitude
1.0
= .~
0.8 .... C.J = ::: ~
= .~ 0.6 .... ::: .c (b) .~ ....
rIJ.
Q 0.4 Q,j ;,
;0:: ..:s ::: 0.2 ~ = .-._._._._. Original G-R ::: U Modified G-R
0.0 6.0 6.5 7.0 7.5
Magnitude
Figure 3.5 Comparison of original and modified Gutenberg-Richter distribution functions.
30
~ 25 Q) ~ -= ~ 20
10
5
o
Mw=7.5 Mw=6.5 Mw=5.5
Re=20 km
Focal Depth = 10 km
.... --, , "
I ... ~ ...
.. ..-. ............. .
... " ... ... ....
I
o 5 10 Frequency (Hz)
Focal Depth = 10 km
/
/
" /
.. -- ... ., -.,." '~\
.-\ \
""'-'" \ ". \
Mw=7.5 Mw=6.5 Mw=5.5
'. \
'. \ \\ . \
'. , \ ' . \ ~ , \ .
Frequency (Hz)
62
15
30
25
20
15
" " "
Mw=7
Re=10 km Re=50 km Re=100 km
Focal Depth = 10 km
10 I .... \
5
o o
, , ,
" ... .. ,
... 0. ... ..... .. . ........... :-..... -... .:. '7.~ =."':' ~ - __ _
5 10 Frequency (Hz)
... ..... ...... .....
...........................
Mw=7
Focal Depth = 10 km . , \ \ : \
r-----------,\ \ '. \
Re=10 km Re=50 km Re=100km
Frequency (Hz)
" \
15
Figure 3.6 Sensitivity of Fourier amplitude spectrum by Trifunac (1994).
Vertical
Section
Plan
View
Strike Slip
site
hypocenter
fault
/ /
8 /
I /
I /
I
I I
/ /
site
• /
epicenter
fault
63
Dip Slip
site.
~ epicenter
Figure 3.7 Definition of rupture directivity parameters Bfor strike-slip faults, ¢ for dipslip faults and possible region with hanging wall effect for dip-slip faults (unshaded area).
150 f-
100 r-
50 r-
o f-
-50 -
-100 -
-150 f-
64
Simulated earthquake sources ( 300 10-yrs, 517 earthquakes)
I
o
I
-150
o 0°
I I
o
o o
o
I I
-100 -50
o
I
o 0
o 0
0 0 0
0 0
I
o
I
o o
o
J
50
Distance from Site (km)
I I
N -
t -
-
-
-
-
-
-o
o
o M =6 I M:= 7.47 -
o
, I
100 150
Figure 3.8 Epicenters and magnitudes of earthquakes of simulated 3000-year record in the reference area of Santa Barbara, CA.
65
-. 3 OJ) -- .\
(. Record: s080rOO1 e wi near-field effect .~ - .-._._.- wlo near-field effect ~ 2 ... Cl)
c; (a) C.J
C.J
-< "; 1 ... -C.J Cl)
C. rJJ
""'-. -- . _.-0_.- 0-°-
0 0 1 2 3 4
Period (sec) -.
N 900 C.J Cl) ~ -5 C.J
(b) -- 0 ~ ~ U U
-900 -< 0 5 10 15 20
-. 150 C.J Cl) ~ -5 (c) C.J 0 --~ ~ ;;>
-150 0 5 10 15 20
75
5 C.J -- (d) ~ 0 ~ rJJ -Q
-75 0 5 10 15 20
Time (sec)
Figure 3.9 Artificial near-field ground motion with dip-slip faulting mechanism (PGA = 0.8 g, Mw = 6.15, H = 8 lan, Re = 6.2lan, Rjb = 0 lan, Rrup = 1.2 lan, RL = 10.3 lan, RW = 10.6 lan, strike = 38.5°, dip = 58.1 0).
66
CHAPTER 4
UNIFORM HAZARD RESPONSE SPECTRA AND GROUND MOTIONS
4.1 Overview
In a perfonnance-based design, earthquake ground motions are needed for
evaluation of structural response. The 1997 NEHRP provisions consider two hazard
levels - design earthquake and maximum considered earthquake ground motions. The
fonner is defined as an event of 10% probability of exceedance in 50 years or a mean
return period of 474 years; the latter has a 2% probability of exceedance in 50 years or a
mean return period of 2475 years. FEMA-273 (1997) adopted the same framework and
defined perfonnance levels accordingly to various building categories and rehabilitation
objectives. The Vision 2000 document (SEAOC 1995) proposed a more refined matrix
of perfonnance check as shown in Figure 4.1. At small probability level of 2% in 50
years, recorded motions are scarce, synthetic ground motions are generally necessary for
structural time history analysis (Somerville et al. 1997). It can be done by scaling ground
motion records to match target response spectra using ground motion records from
seismic zones with similar tectonic environments, or simulating ground motions based on
seismotectonic characteristics surrounding the site location.
In scaling ground motions, it should be pointed out that a response spectrum
according to PSHA (Probabilistic Seismic Hazard Analysis) does not, and was never,
intended to represent the response of a single-degree-of-freedom (SDOF) structure to the
ground motion of a single event (Naeim and Lew, 1995). Rather, it is intended to be an
envelope of responses to multiple events that correspond to a specified probability level.
Therefore, the so-called "response spectrum compatible" acceleration time history which
fits a design response spectrum by scaling a given ground motion may contain energy
over a wide range of structural periods that is not seen in actual records. It can lead to
67
gross overestimate of the displacement demand and energy input. It is also noted that the
duration of the scaled ground motion record remains unchanged whereas the duration is
highly dependent on magnitude (rupture size) and distance (wave dispersion) according
to Carballo and Cornell (1998). As a result, scaled ground motions give poor results of
responses that are sensitive to the duration (e.g. hysteretic energy dissipation).
The phenomenological simulation procedure is therefore used to generate suites of
ground motions appropriate for structural performance analyses of general building
stocks. A large number (thousands) of ground motion time histories are simulated at a
given site, from which uniform hazard response spectra (UHRS) are constructed and
corresponding suites of ten ground motions are selected for analysis of nonlinear systems.
They allow efficient probabilistic performance evaluation of structures under future
earthquakes.
4.2 Uniform Hazard Response Spectra
The large number of ground motion time histories generated in Chapters 2 and 3
allow one to obtain the uniform hazard response spectra (UHRS). At a given city, the
response acceleration spectrum for each time history is first calculated. The probability
distributions of spectral acceleration for a given period (50 years) are then obtained from
which one can construct the UHRS. The annual exceedance probability can be
calcualated from probability of exceedance in t years:
( )1/1
PEann = 1- I-PEL (4.1)
where PEann is the annual probability of exceedance; PEt is the probability of exceedance
in t years assuming occurrence independence except for the characteristic earthquakes in
Zone A of California. Conversions from 50-year to annual exceedance probabilities and
return periods are provided in Table 4.1 for quick reference. The UHRS can be used to
evaluate the performance of linear systems using method of modal superposition.
68
4.2.1 Linear Elastic Systems
F or a single degree of freedom oscillator with a linear elastic restoring force, the
governing equation of motion can be written as
(4.2)
where x is the relative displacement of the mass to the ground; Xg is the ground
displacement; COn is the natural frequency of the oscillator; and qd is the damping ratio. A
damping ratio of 0.05 is used. As a measure of severity of structural response we define
a nondimensional spring force coefficient Ce as a function of the maximum relative
displacement Sd and natural frequency 0Jn of the SDOF oscillator
Ce = maximum spring. force = OJ n 2 S d = (2ff)2 S d
mass x gravzty g ~ g (4.3)
where g is the acceleration due to gravity, and Tn is the natural period of the SDOF
oscillator. To determine Ce corresponding to a given probability of exceedance, an
extension of the Type II extreme value distribution is proposed (Appendix B):
(4.4)
where po is the annual probability of no earthquakes greater than mb-5, and v}, lh, VC, k},
k2, kc and ware distribution parameters. Equation (4.4) reduces to the original Type II
extreme value distribution (Ang and Tang, 1990) when w = 0 and ve ~ 00, i.e.
ve » max( Ce ). The design spectral value according to a specified hazard level can be
readily determined by Equation (4.4). The fitting of simulated results by Equation (4.4)
and determination of the required elastic design force coefficient for a given probability
are shown in Figure 4.2.
69
4.2.1 Nonlinear Inelastic Systems
The nonlinear inelastic restoring force of the SDOF oscillator is represented by the
smooth hysteretic model proposed by Wen (1976). The governing equation of motion for
an SDOF oscillator with this type of restoring force can be expressed as
where
i = ~ { Ai - v jj1iiizin-1 z - vy iizin}
17 (4.6)
where a is the post-to-preyield stiffness ratio (or, strain hardening ratio), z is the auxiliary
state variable describing the hysteretic path, and dots indicates time derivative. A, jJ, Y, 17,
v and n are parameters to define the shape of the hysteresis loop, in which A, 17 and v
define the degree of degradation; jJ and y control the level of the yield strength, and n
defines the transition zone between elastic and plastic regions. A system with both
strength and stiffness degradation is shown in Figure 4.3. According to Baber and Wen
(1979), A, 17 and vtake the form
jA = An - 6A E
17= 170 + 6lJE
V= Vo + 6 v E
(4.7)
where Ao, 170 and Vo are initial values (A 0= 170= vo= 1.0 is commonly used); JA, 6'7 and 6v
represent the rate of degradation (JA=6'7=6v=0 for nondegrading systems); E is the
normalized dissipated hysteretic energy (Yeh and Wen, 1989), defined as
where Fy and 4 are the yield strength and yield displacement, respectively. The
integration represents the hysteretic energy dissipated by the system due to inelastic
deformation and ko represents pre-yielding stiffness. J3 and yare related to the yield
displacement as follows:
70
1 jJ=y=--2
2~y (4.8)
n = 5 is used in this study. To better fit the actual hysteretic behavior of buildings, a
minor modification is made on 11 and v:
(4.9)
where A = 1 is used in this study; 811 and 8 y are parameters to define stiffness and
strength degradation rates respectively (c51J
' c5v ;?:: 1.0); E is the normalized dissipated
hysteretic energy; Ec is the threshold hysteretic energy at which 11 reaches 811 (or, v
reaches 8y ). We take Ec as the normalized dissipated hysteretic energy during one
loading cycle. Analogous to the elastic system, a nondimensional yield force coefficient,
Cy , is defined as the ratio of the restoring force at the yield point to the weight of the
SDOF oscillator:
2~ C = OJ --..L y n g
(4.10)
A system with both strength and stiffness degradation is shown in Figure 4.3. Figure
4.4(a) shows the nondegrading hysteresis model used in this study; Figure 4.4(b) shows
the degrading case. We assume 5% stiffness degradation and the strength degradation at
50/0 and 10% in this study. The proposed modified hazard curve can be used to determine
the values of yield force coefficients at various probability levels as follows,
P(p> p/ GIVEN Cy ) =
{[I- Po - w] -[1- exp[ -( ~: r]] + W-[I- exp[ -( ~: r ]]} + -exp[ -( ~: r]} (4.11 )
where Ji is displacement ductility; Jit is target displacement ductility; Po, V], 'V2, VC , k], k2,
kc and ware defined in Equation( 4.4).
71
4.2.2 Results
4.2.2.1 Uniform Hazard Response Spectra (UHRS)
The UHRS is an efficient means of representing seismic hazards for probabilistic
performance (fragility) evaluation of linear and nonlinear structures (e.g., Collins et al.
1995). The UHRS for Memphis, Carbondale, St. Louis and Santa Barbara are obtained
from the simulated ground motions. For a validation of the results, the UHRS in
Memphis obtained in this study are first compared with those of 1997 USGS national
earthquake hazard maps for B/C boundary soil classification (i.e. firm rock with an
average shear velocity of 760 mlsec in the top 30 m). This generic site condition
facilitates the comparison with previous studies by USGS and other researchers. The
UHRS constructed based on the simulated ground motions for three probability levels for
Memphis are shown in Figure 4.5. The spectral accelerations at periods of 0.2 sec, 0.3
sec, and 1.0 sec according to 1997 USGS national earthquake hazard maps for Memphis
are also shown in the figure. The agreements are generally very good. Since the input
seismicity data to these two models are essentially the same, the differences can be
attributed to:
1. The USGS study used a point-source model and the closest distance from the
fault to the site for magnitude-8 events whereas a finite fault model with a
epicenter located randomly within the fault is used in this study.
2. The USGS study used an S-shaped fault trace with a rupture length of more
than 230 km occupy the entire New Madrid seismic zone, whereas this study
uses a straight fault trace with a rupture length of 140 km that may change
from occurrence to occurrence within the New Madrid seismic zone.
3. For point sources, the one-corner-frequency source model was used in the
USGS study whereas the two-corner-frequency source model (Equation 2.5) is
used in this study which has been shown to give better fit to records in the
CEUS (Atkinson and Boore 1998). It is observed that the two-comer model
72
predicts considerably lower ground motions amplitude than one-comer model
in the case of large events (e.g. moment magnitude greater than 7).
4. The Atkinson-Boore attenuation equation generally predicts lower ground
motions than the empirical attenuation model proposed in USGS OFR-96-532
and the 1993 EPRl report.
The UHRS for the three Mid-America cities with the "representative" soil profiles
are shown in Table 2.3 to 2.5 and compared with FEMA 273 recommendations for
design. FEMA 273 spectra are for design check and are based on the 1997 USGS
national earthquake hazard maps. There are five generic soil classifications according to
the upper 30m of soil and the amplification factor is largely based on empirical results of
Lorna Prieta and Northridge earthquakes (e.g., Borcherdt, 1994). The UHRS for the
Memphis representative soil profile are shown in Figure 4.6. It is seen that compared
with those for the B/C boundary, the UHRS are amplified almost by a factor of two for
periods greater than 1.0 sec. and reduced for T < 0.3 sec due to the deep soil layer. The
agreements are generally good for T > 0.7 sec. The differences for T< 0.5 sec. are partly
due to the differences in the source models and attenuation relations as mentioned in the
foregoing and partly due to the differences in soil amplification factors. The UHRS for
the Carbondale representative soil profile are shown in Figure 4.7. The agreements are
generally good. The UHRS for St. Louis representative soil profile are shown in Figure
4.8. There are some major differences. Compared with the FEMA Class C spectra, the
UHRS are much lower for T > 0.2sec and much higher for T < 0.2 sec because of the I
comparatively thin (16m) layer of soil on rock. The current results are more in agreement
with the findings of Saikia and Somerville (1997) indicating a much lower seismic threat
to St. Louis. It is pointed out that the actual soil profile in the St. Louis area may have a
wide variation and could differ significantly from the "representative profile" used in this
study, especially at locations close to the Mississippi River. Deeper soil layers causing
larger soil amplification at longer periods are certainly possible at these locations.
Ground motions at the surface can be generated from the bedrock ground motions using a
proper soil amplification model when detailed information for the soil profile is available.
73
The UHRS obtained for Santa Barbara in this study is also compared with those of
Collins et al. (1995) and USGS results (1997) in Figure 4.9. They are generally in very
good agreement except that in the short period range at the 10% in 50 years hazard level
where this study shows lower spectral amplitude than USGS. According to Stirling and
Wesnousky (1998), the difference can be attributed to th in:
1. The size of maximum magnitude assigned to a given fault.
2. The proportion of predicted earthquakes that are distributed off the major faults.
3. The use of geodetic strain data to predict earthquake rates.
when different seismological databases are used.
4.2.2.2 Nonlinear Inelastic Uniform Hazard Response Spectra
The nonlinear inelastic UHRS of Memphis, TN for nondegrading systems are
shown in Figure 4.10. UHRS for systems with 5% and 10% strength degradations are
shown in Figures 4.11-12. The degrading systems are models for typical dynamic
behavior of steel buildings with a 2% damping and 3% post-to-pre-yield stiffness ratio.
The stiffness degradation rate is held constant at 5% for each cycle. The strength
degradation rate is slightly varied at two different levels, 5% and 10%, considering that
with appropraite detailing design the strength degradation rate should be within 10% for
each loading cycle. The effect due to the difference in degration rates is small and will be
confinned again in Chapter 5. The nonlinear UHRS of nondegrading system is also
shown for Carbondale, IL, St. Louis, MO and Santa Barbara, CA in Figures 4.12-15. For
Santa Barbara, CA, the nonlinear UHRS of degrading systems is given in Figure 4.16 for
comparison with Memphis, TN. The effect of degradation is more obvious in Santa
Barbar, CA.
74
4.3 Uniform Hazard Ground Motions
4.3.1 Proposed Selection Procedure
For nonlinear systems, a time history response analysis is generally required since
response spectra method based on modal superposition principles no longer applies. For
this purpose, suites of ground motions for each probability level are selected. The
selection criterion is that the deviation of the median response spectra of the suite from
the UHRS is minimized. The median value is used because it is less sensitive to sample
fluctuation and facilitates the estimation of the parameters of the underlying lognormal
distribution. To accomplish this, the response spectral accelerations Sa at 10 key
structural periods (0.05 ,...., 1 sec) of the simulated ground motions are compared with
those of the target UHRS for a given probability of exceedance. The ten ground motions
with the smallest mean square natural logarithmic (,en Sa) difference are selected. The
resultant suite will have a median spectral acceleration that best matches the target
UHRS. The matching of the spectral acceleration has been shown by Shome and Cornell
(1999) to be the most effective means of selecting ground motions for probabilistic
nonlinear structural demand analysis. For performance and fragility analysis of
nonlinear structures, one can first calculate the structural responses under the suites of
ground motions of given probability level (e.g. 2% in 50 years). The median value of the
response will then have a probability of exceedance approximately equal to the ground
motion probability. A more detailed performance and fragility analysis can also be
performed using the time history response and an appropriate regression analysis (Wang
and Vi en 1999. Shome and Cornell 1999).
4.3.2 Comparison with Deaggregation Approach
As mentioned earlier, the PSHA-based response spectrum is based on earthquakes
of widely different characteristics. A single earthquake (or, design earthquake) can not
be expected to represent the seismic hazard at a site. The contribution to hazard from
75
individual earthquakes can be better seen by means of "deaggregating" the seismic
hazard into "bins" of three primary parameters, i.e. magnitude M, distance R, and ground
motion deviation c (McGuire, 1995). The deaggregated hazard matrix allows one to
select the representative combination of parameters for a "design" earthquake. Recently,
it has been pointed out in Bazzurro and Cornell (1998) and USGS OFR (1999) that the
deaggregated seismic hazard should be presented on a geographical map such that the
causative faults can be identified It is thus possible to generate suites of ground motions
and conduct nonlinear dynamic time history analysis for structural performance
evaluation with a single or small number of scenario earthquakes (i.e. events of given
magnitude and distance corresponding to the causative fault). Nevertheless, the
seismotectonic characteristics in the western and eastern United States are significantly
different (e.g. source mechanism and path attenuation); as a result, the deaggregated
hazard matrices could be very different. If the hazard matrix "landscape contour" is
relatively smooth with no dominant peaks, the use of scenario earthquakes as defined
above may not be justified. Furtheull0re, the deaggregated seismic hazard is a function
of structural period. For instance, the seismic hazard in S1. Louis at the l.O-sec structural
period is largely from the New Madrid seismic zone (USGS 1999) due to relatively
slower attenuation. The seismic hazard at the O.3-sec period however are due to local
seismic event and large earthquake from the New Madrid seismic zone (USGS 1999).
This implies that different deaggregation-based ground motion suites need to be provided
for structures of different natural periods. The reasons of not using deaggregation
method is this study in selecting the events for ground motion generation are summarized
as follows:
1. Deaggregation works best when there is a dominant event of certain magnitude
(M) and distance (R) and not so well when the M -R "landscape" is flat. The
favorable condition does not necessarily prevail in all sites.
2. Deaggregation is dependent on structural response and probability level, i.e.
events of spectral accelerations of different periods and different probabilities
of exceedance have different M-R "landscape" and hence- could result in
76
different de aggregations and suites of ground motions. It is suitable for purpose
of performance evaluation of a particular structure but not the general building
stock with a wide range of natural frequencies.
3. Magnitude and distance are much less important, compared to spectral
acceleration, in terms of impact on the structural linear and nonlinear responses
(Shome and Cornell 1999).
The locations (epicenter, focal depth) and magnitudes of selected unifonn hazard
ground motions for three Mid-America cities are shown in Figures 4.17-19. For
Memphis, major contribution to the 2% in 50 years hazard comes from the New Madrid
seismic zone (NMSZ), while local moderate events dominate the 10% in 50 years hazard.
For Carbondale, seismic events in NMSZ also make up the 2% in 50 years hazard, while
moderate events in both NMSZ and Wabash Valley Seismic Zone have major
contribution to the 10% in 50 years hazard. For St. Louis, both small events at local areas
and larger events from NMSZ contribute to the 2% in 50 years hazard, while small events
at medium distances and moderate events from NMSZ and Wabash Valley Seismic Zone
have contribute to the 10% in 50 years hazard. The results are quite similar to the USGS
(1999) deaggregation hazard map when more than one spectral accelerations are
considered.
4.3.3 Suites of Ground Motions for Memphis, TN
For each of the two hazard levels, 10% and 2% in 50 years, ten ground motions are
selected from the large number of simulated motions such that the median spectral
accelerations best fit the target UHRS. The selection is done for both ground surface and
the bedrock (or rock outcrop). The suite of ground motion time histories for a 10% in 50
years hazard are shown in Figure C.1 (Appendix C). The source (magnitude, epicentral
distance, and focal depth) and path (attenuation uncertainty) parameters associated with
each ground motion are also shown in the figure. It is seen that at this probability level,
seven ground motions come from magnitude-6 events at some distance, two from
magnitude-5 events and one from a magnitude-8 event. The response spectra of the ten
77
ground motions are shown in Figure 4.20 with the target UHRS. The median constructed
from the 10 sample ground motions and the 16-to-84 percentile band are shown.
According to the theory of regression analysis (Ang and Tang 1990), the uncertainty in
the median response spectra in terms of the 16-to-84 percentile band is about one third
(1/.Jlo) of that shown in the figure. What it entails is that the median value of structural
responses under this suite of ground motions will have very small uncertainty due to
record-to-record variation and will correspond to a probability of exceedance of 10% in
50 years. One can use it in the structural performance evaluation with some confidence.
The suite of ground motion time histories for a 2% in 50 years hazard and the response
spectra comparison are shown in Figure C.2 and Figure 4.21 respectively. It is seen that
at such high intensity and low probability level, all ground motions come from
magnitude-8 events. This observation agrees well with the USGS de aggregation results,
which indicate that 89.2% of the hazard is from magnitude 8 events at I-sec period and
83% contribution for the O.2-sec period. The scatters in the response spectra are larger
but the maximum 16-to-84 percentile uncertainty band for the median value is still
around 10 % or less. The sample ground motion time histories at rock outcrop (or
bedrock) and the response spectra are shown in Figures C.3-4 and Figures 4.22-23.
Without the soil amplification, the frequency content is seen to shift toward shorter
periods and the spectral accelerations are generally much lower. These ground motions
may be used as inputs to soil amplification software to obtain surface ground motions
when detailed information of the site soil profile is available.
4.3.4 Suites of Ground Motions for Carbondale, IL
The time histories and response spectra of ground motions suites for Carbondale,
Illinois soil sites are shown in Figures C.5-6 and Figures 4.24-25. There is a significant
amplification of motion in the long period range because of the deep and soft soil profile.
At the 10 % in 50 years level, all contributions come from events of magnitude 5.8 to 7.1.
At the 2% in 50 years level, all come from magnitude-8 events. The 16-to-84 percentile
bands are reasonably narrow. The suite of ground motions and response spectra for rock
78
site are shown in Figures C. 7 -8 and Figures 4.26-27. The trend is the same that
compositions of the suites are similar to those for the soil site but the intensities are much
lower.
4.3.5 Suites of Ground Motions for St. Louis, MO
The time histories and response spectra of the suites of ground motions at St. Louis
are shown in Figures C.9-10 and Figures 4.28-29. The major feature of the surface
ground motion is the lack of amplification for periods greater than 0.5 sec because of the
thin soil layer. At the 10 % in 50 years level, nine ground motions come from events of
magnitude 6 to 7. At 2% in 50 years level, six come from magnitude-8 events with long
duration. Smaller (magnitude 5 to 7) and closer events with shorter duration make up the
rest. This observation agrees well with the USGS de aggregation results, which indicate
that 68.6% of the hazard is from magnitude 8 events at I-sec period and 36% contribution
for the 0.2-sec period. Since the median spectrum has a better match with the target
value in the long period range, the composition of uniform hazard ground motion suite
agrees better with the USGS contribution percentage at I-sec period. There is
comparatively a much larger scatter at the peak of the response spectra (period from 0.1
to 0.2 sec). The maximum 16-to-84 percentile band for the median, however, is still
around 1 0 ~O. The ground motions and response spectra for rock outcrop (or bedrock) are
shown in Figures C.11-12 and Figures 30-31, respectively. The compositions of the
suites are similar and the ground motion levels are lower than those of the representative
soil site condition.
4.4 Final Remarks
As mentioned in the foregoing, the source and path models and the quarter
wavelength method do not explicitly consider effects of surface waves (e.g., Dorman and
Smalley 1994) and soil nonlinearity. Therefore, the change in frequency content with
time and the nonlinear soil amplification of the ground motions are not modeled in this
79
simulation. However, it is pointed out that comparisons of results by Boore and Joyner
(1991, 1996, 1997) with observations and analytical results generally show the robustness
of their methods. Also there have not been any efficient methods of modeling surface
waves and nonlinear soil effects that can be adapted in a large-scale simulation as
required in this study. In addition, the uniform hazard response spectra based on the
simulated ground motions in Memphis and Carbondale compare favorably with those of
1997 USGS national earthquake hazard maps and the FEMA 273 recommendations. The
response spectra are the most important measure of ground motion potential of causing
severe structural response and damage. The UHRS and suites of simulated ground
motions generated by the proposed method, therefore, represent reasonably well the
seismic hazards to buildings and other structures in these three cities. As efficient
methods for modeling soil nonlinearity and surface waves become available, they can be
incorporated into the simulation method.
80
Table 4.1 Conversion of frequently used annual and 50-year probabilities of exceedance.
rJl ..:l ~
> ~ ..:l
Z 0 E: 0 ~
Q Z ~ 0 ~ e"
50-year PE
50%
10%
5%
2%
Frequent Earthquakes (50% -50 yrs.)
Rare Earthquakes (10% -50 yrs.)
Very Rare Earthquakes (2 % -50 yrs.)
Annual PE Mean Return Period (years)
0.01376730 72.64
0.00210499 475.06
0.00102534 975.29
0.00040397 2475.42
VISION 2000 PERFORMANCE OBJECTIVES
Fully Functional
BUILDING PERFORMANCE LEVELS
Operational Life
Safety Near
Collapse
Figure 4.1 VISION 2000 perfonnance objectives (after Hamburger 1997).
OJ ~ = ~
81
Tn = 1.0 sec, Representative soil profile, Memphis (TN)
o simulation results model
~ 1-PO = 0.0989 ~ 1 0-1 (H--+T-.'~....,... -. _. _. - . _. - . _. - . _. -. _. _. _. _. _. - . _. _. _. _. _. _.
~ ~ ~ o C == 10-2 "Q ~
,.Q o .. ~
'"; 10-3
= = = <
50% in 50 yrs.
10% in 50 yrs.
... ~~o. ~~ .~.Q .Y~~~ .................... .
2 % in 50 yrs.
10-4 ~~~~~~~~~~~~----~~~~~~~
---" 1 0-4 10-2
Elastic Force Coefficient, Ce
Figure 4.2 Comparison of the predicted annual probability of exceeding values of the elastic force coefficient using Equation (4.4) with the simulation results for a 1.0-sec period SDOF system at representative soil profile, Memphis ,TN (vJ = 0.001492, kJ = 1.167, 1h = 0.429, k2 = 1.863, Vc = 1.687, kc = 3.072, and w = 0.0006984).
250
200 Tn = 0.3 see
150
-100
-150
-200
-250
-1.5 -1.0
strength deterioration
stiffness degradation
-0.5
82
pJ---------..--,~ 1-----I----rnT
0.0 0.5
~d=2% a=3% n=5
8p
Cy = 0.2
Dy = 0.45 em
1.0
Displacement (cm)
- oP x100% = Ip'-pi
u u x 100% ~ ~
- ok x 100% = Ik' -k I o 0 x 100%
ko ko
Figure 4.3 Baber-Wen smooth hysteresis model.
1.5
83
(a) Nondegrading System
300
..-200 N
CJ QJ rI:J -e CJ
100 ---rI:J rI:J ~
~ 0 QJ CJ :... 0 ~ OJ:) -100 = 0i: 0 ..... rI:J QJ -200 ~
-300 -5 -4 -3 -2 -1 0 1 2 3 4 5
Displacement (cm)
(b) Degrading System
250
200 ..-
N CJ 150 QJ rI:J -e 100 CJ
---rI:J rI:J 50 ~
~ --- 0 CJ CJ :... Q -50 ~ OJ:)
0: -100 :... Q ..... -150 rI:J QJ
~ -200
-250 -5 -4 -3 -2 -1 -0 1 2 3 4 5
Displacement (cm)
Figure 4.4 Smooth hysteresis model: (a) nondegrading system, (b) degrading system with 5% strength and stiffness degradations.
--Q,() '-'
= .~ ..-~ l-< QJ
~ ~ ~
-< -; l-< ..-~ QJ Q..
ifJ.
2.0
1.5
1.0
0.5
0.0 0.0
~
0.5
84
• 2% in 50 yrs (B/C boundary)
• 50/0 in 50 yrs (B/C boundary) ... 10% in 50 yrs (B/C boundary) ~ 2 % in 50 yrs (USGS) ~ 10% in 50 yrs (USGS)
1.0 Period (sec)
1.5 2.0
Figure 4.5 Uniform hazard response spectra for B/C boundary at Memphis and comparison with USGS national hazard maps results.
2.0
0.0 0.0
, "
0.5
" " "
• •
2% in 50 yrs (representative) 5% in 50 yrs (representative) 100/0 in 50 yrs (representative) 2% in 50 yrs, Class C (FEMA 273) 5% in 50 yrs, Class C (FEMA 273) 10% in 50 yrs, Class C (FEMA 273)
.............................. :: ~:::.~:::.~ ~ ~ ... ~ ....... ~ ....... ~ ....... ~ ....... ~ 1.0
Period (sec) 1.5 2.0
Figure 4.6 Uniform hazard response spectra for representative soil profile at Memphis, TN and comparison with FEMA 273 response spectra.
2.0
0.0 0.0 0.5
• •
85
2% in 50 yrs (representative) 5% in 50 yrs (representative) 10% in 50 yrs (representative) 2% in 50 yrs, Class D (FEMA 273) 50/0 in 50 yrs, Class D (FEMA 273) 10% in 50 yrs, Class D (FEMA 273)
--- ---- ... - . .. - ... -
....................................
1.0 Period (sec)
1.5 2.0
Figure 4.7 Unifonn hazard response spectra for representative soil profile at Carbondale, IL and comparison with FEMA 273 response spectra.
-e.t -C 0 -eo: ... Col
~ U u <: eo: ... -u Col C.
rJ1
1.5
1.0
0.5
0.0 0.0
, , , , .... ....
" .......
0.5
• •
--... .....
2% in 50 yrs (representative) 5% in 50 yrs (representative) 100/0 in 50 yrs (representative) 2% in 50 yrs, Class C (FEMA 273) 5% in 50 yrs, Class C (FEMA 273) 10% in 50 yrs, Class C (FEMA 273)
--- --- ----- ... - .. - .... ---- -----
. ... - ... _ .... - ... - .... -..................... ......... ...................... .
1.0 Period (sec)
1.5 2.0
Figure 4.8 Unifonn hazard response spectra for representative soil profile at St. Louis, MO and comparison with FEMA 273 response spectra. -
2.0 ~
U ..s
1.5 = Q) ...... ~ ~ -Cl) Q
1.0 U Q) ~ ;... Q
~
. ~ 0.5 ....... VJ C': -~
0.0 0
2.0 ~
U ..s c Cl) 1.5 .~
~ -Cl)
0 U 1.0 Cl) ~ ;... 0 ~
.~ 0.5 ....... ~ C': -~
0.0 0
1
86
• P(exceedance in 50 yrs) = 50% • P(exceedance in 50 yrs) = 10% .. P(exceedance in 50 yrs) = 2%
Santa Barbara (this study) L.A. (Collins et a!. 1995)
... ....
2
.... .
......
Period of Oscillator (sec)
--A....- 2% in 50 yrs, Santa Barbara (this study) ~ 10% in 50 yrs, Santa Barbara (this study)
2% in 50 yrs, L.A. (B/C bdry., USGS) ···0·· 10% in 50 yrs, L.A. (B/C bdry., USGS)
1 2
Period of Oscillator (sec)
(a)
3
(b)
3
Figure 4.9 Comparison of the linear elastic uniform hazard response spectra (;d = 5%) of Santa Barbara obtained from this study with those of L.A. for (a) soft rock (Collins et aI., 1995), (b) B/C boundary or firm rock (USGS, 1997).
87
0.25
s: Prob. Level = 10% in 50 yrs • Elastic case
0.20 0 ~=2 >. U • ~=4 I.. 0 ~=6 U" 0.15 lIE ~=8
== (a)
Q,l ·c 0.10 e Q,l 0
U 0.05 Q,l
(;,j I.. 0 r...
0.00 0.0 0.5 1.0 1.5 2.0
0.5
s: Prob. Level = 5% in 50 yrs • Elastic case
0.4 0 ~=2 >. U • ~=4 I.. 0 ~=6 U" 0.3 lIE ~=8
== (b)
Q,l ·c 0.2 e Q,l 0 U
0.1 Q,l (;,j ;;;;:; 1 I..
t 0 r... 0.0
0.0 0.5 1.0 1.5 2.0
1.2
s: Prob~ Level = 2 % in 50 yrs • Elastic case 1.0 0 ~=2
>. U • ~=4 I.. 0.8 ~=6 0
U., lIE ~=8
== 0.6 (c)
Q,l ·c e
Q,l 0.4
0 U
Q,l 0.2 c:.J I.. 0 ~
0.0 0.0 0.5 1.0 1.5 2.0
Period of Oscillator ( sec)
Figure 4.10 Nonlinear unifonn hazard response spectra of nondegrading systems at representative soil condition, Memphis, TN and ~ = 2% and a = 3% are assumed, (a) Exceedance probability = 10% in 50 years, (b) Exceedance probability = 5% in 50 years, (c) Exceedance probability = 2% in 50 years.
88
0.25
S Prob. Level = 10% in 50 yrs • Elastic case
0.20 0 !l=2 >. U • !l=4 .. e !l=6 U" 0.15 )Ii !l=8
== (a)
~ .~ 0.10 IE ~ e
U 0.05 c:.I
c:.I .. e ~
0.00 0.0 0.5 1.0 1.5 2.0
0.5
S Prob. Level = 5% in 50 yrs • Elastic case -- 0.4
(; 0 !l=2 >.
U • !l=4 l-e !l=6 U" 0.3 )Ii !l=8
== (b)
~ .~ 0.2 IE ~ e U
0.1 ~ (,j l-e ~
0.0 0.0 0.5 1.0 1.5 2.0
1.2
S Prob. Level = 2% in 50 yrs • Elastic case
1.0 0 !l=2 '" • !l=4 ~
I- 0.8 !l=6 e
:., )Ii !l=8
c 0.6 (c) C'.< .~
E OA c.. e :., "" 0.2 c.. l-e
"'" 0.0 0.0 0.5 1.0 1.5 2.0
Period of Oscillator ( sec)
Figure 4.11 Nonlinear uniform hazard response spectra of degrading systems (5% stiffness degredation and 5% strength degredation) at representative soil condition, Memphis, TN and C;d = 2% and a = 3% are assumed, (a) Exceedance probability = 10% in 50 years, (b) Exceedance probability = 5% in 50 years, (c) Exceedance probability = 2% in 50 years.
89
0.25
s: Prob. Level = 10% in 50 yrs • Elastic case '-' 0.20 0 ~=2 ..., U • ~=4 I. 0 ~=6
U" 0.15 lIE ~=8 C (a) ~
'r; 0.10 I: .... ~ 0 U
0.05 ~ '-I I. 0 ~
0.00 0.0 0.5 1.0 1.5 2.0
0.5
s: Prob. Level = 5% in 50 yrs • Elastic case '-' 0.4 0 ~=2 ..., U • ~=4 I. 0 ~=6 U" 0.3 lIE ~=8
C (b) ~
'r; 0.2 I: .... ~ 0 U
0.1 ~ '-I I. 0 ~
0.0 0.0 0.5 1.0 1.5 2.0
1.2
s: Prob. Level = 2% in 50 yrs • Elastic case 1.0 0 ~=2 ...,
U • ~=4 I. 0.8 ~=6 0
U" lIE ~=8
C 0.6 (c) Cl,l
'r; I: .... ~
0.4 0 U ~ 0.2 '-I I. 0 ~
0.0 0.0 0.5 1.0 1.5 2.0
Period of Oscillator ( sec)
Figure 4.12 Nonlinear uniform hazard response spectra of degrading systems (10% stiffness degredation and 5% strength degredation) at representative soil condition, Memphis, TN and ;d = 2% and a = 3% are assumed, (a) Exceedance probability = 10% in 50 years, (b) Exceedance probability = 5% in 50 years, (c) Exceedance probability = 2% in 50 years.
90
0.5
~ Prob. Level = 10% in 50 yrs • Elastic case '-' 0.4 0 1l=2 >. U • 1l=4 lo. 0 1l=6
U., 0.3 lE 1l=8
-= (a)
~ .Cj 0.2 I..: ..... ~ 0 U
0.1 ~ C.J lo. 0 ~
0.0 0.0 0.5 1.0 1.5 2.0
1.0
~ Prob. Level = 5% in 50 yrs • Elastic case
0.8 0 1l=2 >. U • 1l=4 I-0 1l=6 U., 0.6 lE 1l=8 c (b) IlJ
.Cj 0.4 I..: ..... ~
~ 0 W 0.2 ~ C"i
; 0.0 [ ~7? J J ~
0.0 0.5 1.0 1.5 2.0
1.50 Prob. Level = 2% in 50 yrs • Elastic case =- 1.25 0 1l=2
:.... • 1l=4 I- 1.00 1l=6 c
:.... lE 1l=8 c 0.75 (c) ~ .~
E 0.50 ~ c
:.... ... 0.25 :: ~
0.00 0.0 0.5 1.0 1.5 2.0
Period of Oscillator ( sec)
Figure 4.13 Nonlinear unifonn hazard response spectra of nondegrading systems at representative soil condition, Carbondale, IL and ~ = 2% and a = 3% are assumed, (a) Exceedance probability = 10% in 50 years, (b) Exceedance probability = 5% in 50 years, (c) Exceedance probability = 2% in 50 years.
91
0.4
-;: Prob. Level = 10% in 50 yrs • Elastic case
-- 0 Jl=2 >. U 0.3 • Jl=4 ;.. 0 Jl=6
Ue.; lIE Jl=8
C 0.2 (a) ~ .~
I:: '-~ 0 0.1 U ~ Cj ;.. 0
r.-0.0
0.0 0.5 1.0 1.5 2.0
0.6
-;: • Elastic case
-- 0.5 0 Jl=2 U>' • Jl=4 ;..
0.4 Jl=6 0
Ue.; lIE Jl=8 ..... 0.3 (b) c ~ .~
I.: 0.2 ....
~ 0 U ~ 0.1 Cj ;.. 0 r.-
0.0 0.0 0.5 1.0 1.5 2.0
1.2
-;: Prob. Level = 2% in 50 yrs • Elastic case 1.0 0 Jl=2
U>' • Jl=4 ;.. 0.8 Jl=6 0
U" lIE Jl=8
C 0.6 (c) ~ .~
I.: 0.4 .... c:.; 0 U CJ 0.2 Cj ;.. 0
1,;1;.
0.0 0.0 0.5 1.0 1.5 2.0
Period of Oscillator ( sec)
Figure 4.14 Nonlinear uniform hazard response spectra of nondegrading systems at representative soil condition, St. Louis, MO and r;d = 2% and a = 3% are assumed, (a) Exceedance probability = 10% in 50 years, (b) Exceedance probability = 5% in 50 years, (c) Exceedance probability = 2% in 50 years.
92
0.5
• Elastic case :::t
>- 0.4 0 1l=2 () • 1l=4 0 1l=6
Cll 0.3 () lIE 1l=8 c" (a) (1)
'u 0.2 :E (1) 0 ()
0.1 (1)
~ 0
LL 0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0
:::t Prob. Level = 10% in 50 yrs • Elastic case >- 0.8 0 1l=2
() • 1l=4 0 Cll
0.6 1l=6
() lE 1l=8 c" (b) (1)
'u 0.4 :E (1) 0 u
0.2 (1) u 0
LL 0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2.0 Prob. Level = 2% in 50 yrs • Elastic case
:::t 0 1l=2
U 1.5 • 1l=4 0 1l=6
Q)
u lE 1l=8 c" 1.0 (c) (1)
'u :E (1) 0 0.5 u (1) u 0
LL 0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Period of Oscillator ( sec)
Figure 4.15 Nonlinear uniform hazard response spectra of nondegrading systems in which ;d = 2%, a = 3% and soft rock site condition are assumed for Santa Barbara, (a) Exceedance probability = 50% in 50 years, (b) Exceedance probability = 10% in 50 years, ( c) Exceedance probability = 2% in 50 years.
93
0.5 ,-... • Elastic case ::t '-" 0.4 0 J..L=2 >-u • /-1=4 0 /-1=6
Q) 0.3 u IE /-1=8 -E (a) (l)
"(:5 0.2 ~ (l) 0 u 0.1 (l) ()
0 ~
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0 ,-... Prob. Level = 10% in 50 yrs • Elastic case ::t
>- 0.8 0 /-1=2 u • /-1=4 0
Q)
0.6 /-1=6
u IE /-1=8 c- (b) (l)
'0 0.4 ~ (l) 0 u
0.2 (l) ()
0 ~
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
2.0 ,.-... Prob. Level = 2% in 50 yrs • Elastic case ::t
>-0 /-1=2
u 1.5 • /-1=4 0 /-1=6
Q)
u IE /-1=8 C 1.0 (c) (l)
'0 ~ (l) 0 0.5 u (l) ()
0 ~
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Period of Oscillator ( sec)
Figure 4.16 Nonlinear uniform hazard response spectra of degrading systems (5% stiffness degredation and 5% strength degradation) in which ;d = 2%, a =
3 % and soft rock site condition are assumed for Santa Barbara, ( a) Exceedance probability = 50% in 50 years, (b) Exceeda~ce probability = 10% in 50 years, (c) Exceedance probability = 2% in 50 years.
94
39 39
St. Louis (MO) +
38 38 + Carbondale (IL)
~ "'0
37 :: = ...... ~
~ 36
35 • : + Memphis (TN)
34 34 -92 -91 -90 -89 -88 -87 0 10 20 30 40
Focal Depth (kIn) 0
E ::::, 10 + Site Location
.... 0 2 % in 50 years 0..
20 • 10% in 50 years <:J
0
0 eo: u 30 Mw =8 0 ~
40 Mw =5.3
-92 -91 -90 -89 -88 -87
Figure 4.17 Locations (epicenter, focal depth) and magnitudes of selected uniform hazard ground motions at two hazard levels for representative soil profile of Memphis, TN (solid circles for 10% and hollow circles for 2% in 50 years).
95
39 r- - 39 St. Louis (MO) + e e •• • • • Q) 38 r- Carbondale (IL)+ - 38
"'0 ::
;:: • --~ • ~ 37 r-
6 - 37
-
36 '- - 36
-Memphis (TN)
35 I , of, L I
35 -92 -91 -90 -89 -88 -87 0 10 20 30 40
Focal Depth (km) 0
-.. E
10 ~ + Site Location ..c: 0 2 % in 50 years Q..
20 • 10% in 50 years Cl.l
~
0 e; • • ~ 30 Mw =8 0 ~
40 • ~=5.8
-92 -91 -90 -89 -88 -87
Figure 4.18 Locations (epicenter, focal depth) and magnitudes of selected uniform hazard ground motions at two hazard levels for representative soil profile of Carbondale, IL (solid circles for 10% and hollow circles for 2% in 50 years).
39
38
37
St. Louis (MO)
+0 o •
96
·A~oe(IL)
39
38
37
33
6
5 ~L~---'------'----~ ~j 33
6
5 ~~ Memphis (TN)
+,
-92 -91 -90 -89 -88 -87 0 10 20 30 40
e :::::, 10 ..c: Q. QJ 20
Q
40 ~~ __ -L--~ __ ~~ __ ~ __ ~~ __ ~~
-92 -91 -90 -89 -88 -87
Focal Depth (kIn)
+ Site Location o 2% in 50 years • 10% in 50 years
o M,.=8
• ~=5.5
Figure 4.19 Locations (epicenter, focal depth) and magnitudes of selected unifonn hazard ground motions at two hazard levels for representative soil profile of St. Louis, MO (solid circles for 10% and hollow circles for 2% in 50 years).
0.4
0.0 0.0
0.3
~ I-U 0.1 ~
c.. cr..
0.0
0.0
0.5
0.5
97
1.0 Period (sec)
simulated ground motions Sa (10% in 50 yrs.)
1.5
target Sa (10% in 50 yrs.) median Sa of simulated motions range of 16 to 84 percentile
1.0 Period (sec)
1.5
2.0
2.0
Figure 4.20 Response spectra of 10% in 50 years ground motion suite for representative soil profile and comparison with target UHRS, Memphis, TN.
1.5
0.0 0.0
1.5
0.5
98
1.0
simulated ground motions Sa (2% in 50 yr)
1.5
Period (sec)
/ /// // Range of 16 to 84 percentile median Sa of simulated motions target Sa (2% in 50 yrs.)
2.0
0.0 1-1 --:-----:----,.---r----r----r----,----,----,----,----,-----r---,-r--r--,....--,---.,.--~
0.0 0.5 1.0
Period (sec)
1.5 2.0
Figure 4.21 Response spectra of 2% in 50 years ground motion suite f-or representative soil profile and comparison with target UHRS, Memphis, TN.
0.4
0.0 0.0
0.4
0.0
0.0
0.5
0.5
99
1.0 Period (sec)
simulated ground motions Sa (10% in 50 yrs.)
1.5
// /// Range of 16 to 84 percentile median Sa of simulated motions target Sa (10% in 50 yrs.)
1.0 Period (sec)
1.5
2.0
2.0
Figure 4.22 Response spectra of 10% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, Memphis, TN. -
1.2
0.0 0.0
1.2
:§ 0.9
.~ -;: ... ~
~ ~ 0.6 -< = ... -(;,J ~
(J; 0.3
0.5
100
1.0
simulated ground motions Sa (2% in 50 yr)
1.5
Period (sec)
/ '/'/ Range of 16 to 84 percentile median Sa of simulated motions target Sa ( 2% in 50 yrs. )
2.0
0.0 ~i ------------~~~--~_r~r_~_r~r_~~~r_~~--r_~
0.0 0.5 1.0
Period (sec)
1.5 2.0
Figure 4.23 Response spectra of 2% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, Memphis, TN.
101
0.7
simulated ground motions 0.6 ----- Sa (10% in 50 yrs.)
---OJ) ---==
0.5 0
"..t: eo: ~ 0.4 C!)
"a) C.J C.J
< 0.3 -; ~ ..... C.J
0.2 C!)
Q.. rF.J.
0.1
0.0 0.0 0.5 1.0 1.5 2.0
Period (sec)
0.7
---_. target Sa (10% in 50 yrs.) 0.6 median Sa of simulated motions
g /// /// range of 16 to 84 percentile c 0.5 .~ ..... eo: ~ 0.4 CJ
~ c.J c.J
<: 0.3 eo: ~ -c.J CJ 0.2 c..
rr.
0.1
0.0 I
0.0 0.5 1.0 1.5 2.0 Period (sec)
Figure 4.24 Response spectra of 10% in 50 years ground motion suite for representative soil profile and comparison with target UHRS, Carbondale, IL.
-; ... .u ~
2.0
0.0 0.0
2.0
9i 0.5
0.0
0.0
0.5
0.5
102
1.0
Period (sec)
simulated ground motions Sa (2% in 50 yr)
1.5 2.0
Range of 16 to 84 percentile median Sa of simulated motions target Sa (2% in 50 yrs.)
1.0
Period (sec)
1.5 2.0
Figure 4.25 Response spectra of 2% in 50 years ground motion suite for representative soil profile and comparison with target UHRS, Carbondale, IL.
0.4
0.0 0.0
0.4
0.0
0.0
0.5
0.5
103
1.0 Period (sec)
simulated ground motions Sa (10% in 50 yrs.)
1.5
/// //
target Sa (10% in 50 yrs.) median Sa of simulated motions range of 16 to 84 percentile
1.0 Period (sec)
1.5
2.0
2.0
Figure 4.26 Response spectra of 10% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, Carbondale, IL. -
104
0.7
simulated ground motions 0.6 ---_. Sa (2% in 50 yr) --~ --:: 0.5 .:: --~
~ 0.4 ~
Q) ~ ~
< 0.3 e; ~ --~ ~ 0.2 c.. ifl
0.1
0.0 0.0 0.5 1.0 1.5 2.0
Period (sec)
0.7
~/// Range of 16 to 84 percentile 0.6 median Sa of simulated motions
§ ----- target Sa (2% in 50 yrs.) :: 0.5 .s ~ ~ 004 ~
C; c.oI c.oI
< 0.3 -= ~ -c.oI s: 0.2 rZ
0.]
0.0
0.0 0.5 1.0 1.5 2.0
Period (sec)
Figure 4.27 Response spectra of 2% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, Carbondale, IL.
105
0.6
simulated ground motions
0.5 ____ a
Sa (10% in 50 yrs.) ,,-., ~ ---C 0 0.4 ~ ~ ;.. ~
'ii 0.3 u u
-< -; ;..
0.2 ~ u ~ c..
rFJ.
0.1
0.0 0.0 0.5 1.0 1.5 2.0
Period (sec)
0.6
// /// Range of 16 to 84 percentile 0.5 median Sa of simulated motions
,,-., ~ ----- target Sa (10% in 50 yrs.) ---c 0 0.4 ~ ~ ;.. ~
'ii 0.3 u u
-< -; ;..
0.2 ~ u ~ C.
rFJ.
0.1
0.0
0.0 0.5 1.0 1.5 2.0 Period (sec)
Figure 4.28 Response spectra of 10% in 50 years ground motion suite for representative soil profile and comparison with target UHRS, St. Louis, MO.
2.5
-.. 2.0 ~ ---= .S --C':S 1.5 ~ QJ
cu c:..J c:..J
< "'; 1.0 ~ --c:..J QJ
C. 7JJ
0.5
0.0 0.0 0.5
1.5
0.0
0.0 0.5
106
simulated ground motions ____ a
Sa (2% in 50 yr)
1.0 1.5
Period (sec)
Range of 16 to 84 percentile median Sa of simulated motions target Sa (2% in 50 yrs.)
1.0
Period (sec)
1.5
2.0
2.0
Figure 4.29 Response spectra of 2% in 50 years ground motion suite for representative soil profile and comparison with target UHRS, St. Louis, MO.
0.20
:§ 0.15 = o :t: ~ ;... Q,)
~ ~ 0.10 < -; ;... ..... u Q,)
rE" 0.05
0.00 0.0
0.20
§0.15 = .S ..... ~ ;... Q,)
~ ~ 0.10 < -; ;... ..... u Q)
~ 0.05
0.00
0.0
0.5
0.5
107
1.0 Period (sec)
simulated ground motions Sa (10% in 50 yrs.)
1.5
/// ///
target Sa (10% in 50 yrs.) median Sa of simulated motions range of 16 to 84 percentile
1.0 Period (sec)
1.5
2.0
2.0
Figure 4.30 Response spectra of 10% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, St. Louis, MO.
-.. en --= 0 ~ ~ ... Q)
Q) C.J C.J
< ~ ... ..-
C.J Q)
C. rJ').
1.0
0.8
0.6
0.4
0.0 0.0
0.5
~ 0.4 --
0.0
0.0
0.5
0.5
108
1.0
simulated ground motions Sa (2% in 50 yr)
1.5
Period (sec)
/~/ /// Range of 16 to 84 percentile median Sa of simulated motions target Sa (2% in 50 yrs.)
1.0
Period (sec)
1.5
2.0
2.0
Figure 4.31 Response spectra of 2% in 50 years ground motion suite for bedrock (hard rock) and comparison with target UHRS, St. Louis, MO.
109
CHAPTERS
IMPLICATIONS IN RELIABILITY AND DESIGN
5.1 Overview
As demonstrated in the foregoing, the median response spectra of linear systems to
the simulated ground motion suites closely match the target elastic spectral accelerations
for a wide range of frequencies. However, whether these ground motion suites can
provide satisfactory estimate of response of inelastic, nonlinear systems has important
implications in performance check of buildings and structures. In addition, it is of
interest to practicing engineers whether the ground motion suites can provide accurate
estimate of structure performance without excessive computational efforts. These
practical implications of the uniform hazard ground motion are examined. The effect of
system degradation on ductility reduction factor which is useful in both analysis and
design of structures in inelastic range is also examined.
5.2 I\onlinear Response Estimate by Uniform Hazard Ground Motion Suites
To \·en fy whether 10 uniform hazard ground motions can reasonably represent the
seismic hazard at a given probability level for nonlinear systems, one can compare the
inelastic response spectra constructed from these 10 simulated ground motions and the
entire population of 9000. In Section 4.2.2.2, uniform hazard response spectra for
nonlinear systems are obtained for ductility ratios from 2 to 8. An iterative procedure is
used to determine the yield capacity of a SDOF system such that ductility under the 10
unifonn hazard ground motions satisfies the target value. For an inelastic SDOF system
of a given fundamental period, an initial yield displacement is assumed and the maximum -
displacement and ductility are calculated. The yield displacement is then modified and
110
the trial and error procedure is repeated until the ductility converges to the target value
within a very small error limit (0.01 is used in this study). The median value of the yield
force coefficients under the 10 uniform hazard ground motions (Equation (4.10)) is then
calculated. In Figures 5.1-4 the median value and the 16th to 84th percentile range are
compared with the nonlinear response spectra for both degrading and non-degrading
systems obtained previously from 9000 ground motions. It is seen that median inelastic
spectral accelerations match the spectral values calculated in Section 4.2.2.2 closely. In
perfonnance evaluation of a given nonlinear system, such iterations are, of course, not
required. For such a system, the median response of 10 time history response analyses
provides an accurate estimate of the system performance. The results imply that, one can
use 10 uniform hazard ground motions instead of 9000 for an accurate estimate of the
nonlinear structural responses corresponding to each probability level. Verification is not
perfonned here for nonlinear MDOF systems due to the large computational effort
required of such systems. However, since the response spectra calculated from the 10
unifonn hazard ground motions match the target values for a wide range of frequency,
these motions are expected to yield response of nonlinear MDOF systems with good
accuracy. They represent realistically the future seismic threat to the site location from
causative faults, which can be clearly seen from the comparison of results with the
deaggregation method in Section 4.3.2.
5.3 Ductility Reduction Factor
F or design purposes, it is impractical to provide in seismic design codes both linear
elastic and nonlinear inelastic uniform hazard response spectra for various values of
structural parameters (e.g., damping and strain-hardening ratio) and for a large number of
sites in the U.S. A common method to consider the inelastic response behavior is the
ductility reduction factor Rp , by which the strength of a nonlinear system can be obtained
that of the elastic system. The ductility reduction factor Rp is defined as:
111
R = Ce
,tJ C y
(5.1)
where Ce is the elastic force coefficient (Equation( 4.3)) and Cy is the yield force
coefficient (Equation(4.10)). Effects of local site soil condition, inelastic behavior of the
structure (e.g. strain-hardening ratio), damping ratio, and fundamental period of the
structure have been investigated by Krawinkler and Nassar (1992), Miranda and Bertero
(1994), Riddell (1995), Krawinkler (1996), Shi (1997) and Han et al. (1999). The
empirical formula proposed by Krawinkler and Nassar (1992) is of the following form:
(5.2)
where
Ta
b c(~,a) = n a +-
1+~ ~ (5.3)
where a damping ratio of 5% is assumed; Tn is structural period; j.t is displacement
ductility; a is the post-to-preyield stiffness ratio. The regression parameters a and b were
obtained for different post-to-preyield stiffness ratios as follows:
a=O%:
a=2%:
a= 10%:
a = 1.00
a = 1.00
a = 1.00
b = 0.42
b = 0.37
b = 0.29
The results from the aforementioned studies indicate that ductility reduction factor
Rj.L generally depends on natural period, soil condition and degree of degradation. The
effect of exceedance probability level on Rji has been investigated by Collins et al. (1996)
and found to be unimportant. The ductility reduction factor based on simulated ground
motions in Mid-America and Santa Barbara will be examined and compared with • •• 1 1 l 1 1 "1'-';: T I r"1 I 1 J /'T'"" J • / ~ 1"'\ '\ 1 / J- "'" '\. "\ preVIOUS ell1pITICal results oasea on Vv eSI Loasl aala tt:,quanons ~J.Lj ana ~J . .:J jj.
In Figure 5.5-10, the ductility factor Rji of the nondegrading and degrading systems
for three exceedance probability levels (solid lines) are shown. It is seen that Rj.L is
insensitive to change in exceedance probability levels, as shown in Collins et al. (1996).
112
There is a significant reduction of Rj1 values due to system degradation at Santa Barbara,
CA ( Figures 5.9-10) but not at Memphis, TN (Figures 5.5-6),. This may be attributed to
the effects of ground motion duration, frequency content and attenuation for each site.
The Santa Barbara results agree with Han et al. (1999), which are largely based on West
Coast data. Also shown for comparison are empirical formulae by Krawinkler and
Nassar (1992) and Miranda and Bertero (1994). Note that Krawinkler and Nassar (1992)
and Miranda and Bertero (1994) did not include degradation in their empirical models.
Since the simulatiuon results fit better the Krawinkler-Nassar formula, their empirical
equation is used.
A regression analyis is performed to obtain the a and b values (Table 5.1) and the
regression results are compared with Krawinkler-Nassar curves in Figures 5.11-16. The
agreements are generally good. Krawinkler and Nassar (1992), however, found that for
MDOF systems an increase in the strength is necessary to meet the drift requirements of
current model building codes due to the higher mode contribution. A more thorough and
systematic investigation of MDOF systems, therefore, is needed before a general
reduction factor for degrading systems can be developed. In addition, it is observed in
Figure 5.14 for St. Louis, that they are much higher than the Krawinkler-Nassar curve in
the period range below 0.25 seconds due to the thin soil layer in St. Louis. It indicates
that the ductility reduction factor is highly site-dependent which needs to be taken into
consideration carefully.
R,tJ factors in current code procedures allow one to obtain nonlinear structural
responses \'ia a linear static analyses. It is a computationally efficient method commonly
used by practicing engineers provided RJ.i is correctly calibrated with respect to building
types. As shown by Wen and Song (1999), RJ.i can also be used to construct probabilistic
performance (fragility) curves for builidngs, e.g. a single story steel moment frame in
Carbondale, Illinois in Figure 5.17. The small solid diamonds are drift ratios calculated
by nonlinear time history analyses at a given probability level and the hollow circles are
the median drift ratios. One can perform first a linear static analysis of the structure and
then use the UHRS and the empirical RJ.i to evaluate the drift ratio. The probabilitisc
113
perf romance curve is then obtained assuming a proper (lognormal) distribution (i.e. dash
lines in Figure 5.17).
5.4 Peak Ground Acceleration versus Spectral Acceleration in Fragility Analysis
Fragility analyses are commonly used for evaluation of structural performance and
loss estimation during future seismic events (e.g., Wen and Song 1999, Abrams and
Shinozuka 1997). Both peak ground acceleration (PGA) and spectral acceleration (Sa)
have been commonly used as the measure of earthquake intensity in the fragility analysis.
They may yield significant different results. The accuracy of these two methods is
investigated by considering the probability of limit state given by:
P(Y> y)= f"'Gy,x(Y> yix)"h(x) dx (5.4)
where P( Y > )') is the probability of structural response Yexceeding a given limit y ; x is
the earthquake intensity, measured by Sa or PGA; Gy1x(Y> y Ix) is the conditional
response probability given the hazard intensity, or commonly referred to as the fragility
curve; h(x) is the seismic hazard probability density at the site (Equation( 4.4)). To
evaluate the conditional probability, the following equation is used:
(5.5)
where Y is the structural response; X is the earthquake intensity; a and jJ are the
regression parameters: and E is the error term with median equal to 1 and standard
deviation equal to ~-. The conditional response probability function Gy1x(Y> y Ix) is
described by lognormal distribution. The fragility obtained from Sa and PGA are
compared with the "'exact" solution based on 9000 simulations.
To compare two approaches, we assume that 100 ground motions with PGA
between 0.15 ,...., 1.15 g are available at the site and are randomly selected from the
population of simulated motions at Memphis. Two nondegrading SDOF systems of a
114
natural period 0.3-sec and l.O-sec are used. The ductility ratio is used for the measure of
structural response Y. The prediction of ductility ratio as a function of spectral
acceleration or PGA is shown in Figures 5.20 to 5.25. The regression parameters a, jJ
and scatter measure 6& are listed in Table 5.2. The hazard functions corresponding to 0.3-
sec and l.O-sec spectral accelerations and PGA are shown in Figure 5.18, Figure 4.2 and
Figure 5.19, respectively. While PGA may be used as a reliable response measure of
short period buildings (e.g., a coefficient of variation (COV) of about 27% at 0.3-sec), its
accuracy drops dramatically for long period buildings (e.g., the COY increases to 108%
at 2.0-sec). On the other hand, the response COY remains almost constant at 27% when
the spectral acceleration is used. When compared the "exact" solutions (Figures 5.26-
28), Sa gives excellent estimates except at the low exceedance probability level due to
the limitation of the power law regression. Other mathematical regression form may be
used to improve the accuracy. On the other hand, PGA gives poor results for
intermediate to long period structures (Figures 5.27-28). In light of the severe
shortcomings, PGA is therefore not recommended for the evaluation of structural
fragility.
5.5 Final Remarks
It is shown that uniform hazard ground motions provide an unbiased estimate of
nonlinear structural dynamic response for SDOF systems. Since they match the target
spectra over a wide range of frequencies, uniform hazard ground motions may be used
for MDOF systems as well. Secondly, system degradation is shown to have significant
influence on the ductility reduction factor for Santa Barbara and is recommended for
further study for nonlinear MDOF systems. An efficient method to estimate structural
probabilistic performance using uniform hazard response spectra, ductility reduction
factor and a lognormal probability fit is demonstrated. Using spectral accelerations to
estimate structural fragility (Shome and Cornell, 1999) is shown to yield satisfactory
results as well.
115
Table 5.1 Regression parameters for ductility reduction factor using the KrawinklerNassar formula (Equations 5.2 and 5.3).
Memphis (TN) Carbondale St. Louis
Santa Barbara (CA) (IL) (MO)
non-degrading
non- non- non-degrading degrading degrading degrading degrading
a 1.50 1.62 0.82 1.1 1.80 2.29
b 0.43 0.46 0.43 0.3 0.40 0.60
Table 5.2 Regression parameters for Sa - Y and PGA - Y relations.
T = 0.3 sec, Cy = 0.37 T = 1.0 sec, Cy = 0.15 T = 2.0, Cy = 0.07
Sa PGA Sa PGA Sa PGA
a 3.88 14.60 7.21 14.61 12.19 14.76
jJ 1.26 1.44 1.00 1.27 0.94 1.59
eSc 0.265 0.261 0.269 0.672 0.266 0.882
(COY) (0.270) (0.266) (0.274) (0.756) (0.271) (1.085)
(2) COY is the coefficient of variation of structural response Y. For quick reference, generally, 6& == COY when 6& < 0.30 .
N 0.16 II
3 u~' 0.12 ..: I::
'0 0.08 ~ !+-c ~ o U ~ CJ ,.. o
0.04
~ 0.00
G' 0.16 II ::1. "-"
U>' 0.12 ;.J'
= ~ '0 0.08 ~ !+-c ~ o U ~ CJ ,.. o
0.04
~ 0.00
S (10% in 50 yrs.) - - • target, a, ,_ I () IJ m.
-- I' S () r '1 IllCC lilli, a I 04'h pcrccntl c ~ 1(" til ('t ~ rallgcof ) ~~~. " s\·slrm .. /, '//, • rlH 109 _ ////.///.' (It) n(Jnd(g
}!f:;1@~,. ( ./~
~ ~
0.0
j; ~
0.5
'// ///
1.0
Period (sec)
1.5
target Sa (I 0% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile
(c) nondegrading systems
2.0
////// ///
/////,
0.0 0.5
//////////////////////////////////////////.
1.0
Period (sec)
1.5 2.0
~ 0.16 II
~ U,O.12
-I:: ~ ·c 0.08 S ~ o U ~ e o
0.04
~ 0.00
00 0.16 1\ ::1.
"-" U>' 0.12 ;.J'
= '0 0.08 ~ !+-c ~ o
U 0.04 ~ e o ~ 0.00
target Sa (10% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile
(b) nondegrading systems
0: , ~
0.0
~//////////.&'////////////////// 0.5 1.0
Period (sec)
1.5
- - - _. target Sa (10% in 50 yrs.) median Sa of 10 g.m.
/ // / range of 16th to 84th percentile (d) nondegrading systems
2.0
~ ///
//////
//////
0.0
///////////////////////////////////// ///",
0.5 1.0
Period (sec)
1.5 2.0
Figure 5.1 Comparison of spectral accelerations calculated from 1 0 ground motions with target values at 1 0% in 50 years probability level for nondegrading SDOF systems, representative soil profile in Memphis, TN.
~
~
0\
Figur 5.2
R 0.8 II :t '-' U~0.6 ,j
target Sa (2% in 50 yrs.) Illcdi<ln Sa of 10 g.m. range of \6 1h to 841h percentile
~ 0.8 II :t '-'
U~0.6
" f~ U ~ cu ~ .,////////, .. ~ ~
r~ (a) nondegrading systems r~ ~ 0.0 I ~
G' 0.8 II :t '-' U~0.6
cu u ~ o ~
0.0 0.5 1.0 1.5 2.0
'// // /
Period (sec)
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84
th percentile
, ~&.«&///&///////////////////////////
( c) nondegrading systems I
0.5 1.0 1.5 2.0
Period (sec)
~ 0.8 II :t '-'
r..J'0.6
cu u ~ o ~
/~/ / '// /
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84
th percentile
.~
~~////////// 0.5
/"' //
1.0
Period (sec)
1.5
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84
th percentile
~ ~//// //~~~~~~~~ / /// //,/'//////////// // // / // / / / / / / / / / / / /
d ' systems (d) nondegra mg
0.5
Period (sec)
Comparison of spectral accelerations calculated from 10 ground motions with target values at 2% in 50 years probability level for nondegrading SDOF systems, representative soil profile in Memphis, TN.
~
~
-.....J
N 0.16 II ::1. ......,
U'ih 0.12
4
= <l)
'0 0.08 ~ I.M <l) o U 0.04 <l)
e o ~ 0.00
co 0.16 II ::1. ......,
U'ih 0.12
4 = '0 0.08 ~ I.M <l) o U <l) u ,.., o
0.04
~ 0.00
,-, 0.16 ~
II S (10% in 50 yrs.) ::1. - - - _. target, of 10 g.m. . ~,., 0.12
medIan Sa th to 84th percentile U ~ /" range of16 i • <al ~egr'ding systems ~ 0.08
~ ~ ~ ~ ~ • ~ u
~&o.oo 0.0
~ ~" ~
0.5 1.0
Period (sec)
1.5
- - - _. target Sa (10% in 50 yrs.) median Sa of 10 g.m.
/ / / // range of 16th to 84th percentile
(c) degrading systems
2.0
//////
////«
////////////////////////////////////////////.
0.0 0.5 1.0
Period (sec)
1.5 2.0
0.04
GO' 0.16 II ::1. ......,
U'ih 0.12
4 = '0 0.08 ~ I.M <l)
o U <l) u ,.., o
0.04
~ 0.00
- - - _. target Sa (10% in 50 yrs.) median Sa of 10 g.m.
/ /// / range of 16th to 841h percentile
(b) degrading systems
~ , ~ ;" ~ij/h'//////////////////////////
0.0
~~ ~,
0.5 1.0
Period (sec)
1.5
- - - - • target Sa (l0% in 50 yrs.) median Sa of 10 g.m .
/ // / / . range of 161h to 841h percenti Ie
(d) degrading systems
2.0
//.:)., /////)-- ,
////,;-
0.0
///////////////////////////////////////////,
0.5 1.0
Period (sec)
1.5 2.0
Figure 5.3 Comparison of spectral accelerations calculated from 10 ground motions with target values at 10% in 50 years probability level for degrading SDOF systems, representative soil profile in Memphis, TN.
~
~
00
R 0.8 II ::::l "-'
UOh 0.6
Q) () ,.", c ~
G' 0.8 II ::::l "-'
UOh 0.6
Q) () ,.", c ~
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile ,. ..
F'/~' ( ~
0.5
/ // /
~
1.0
Period (sec)
1.5
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile
2.0
~~.8?ij/-
~~/§/////////////////////////////// 0.5 1.0 1.5 2.0
Period (sec)
~ 0.8 II ::::l "-'
UOh 0.6
....:-1:1 Q)
'u ~ ~ Q) c U 0.2 Q) () ,.", c ~
QO 0.8 II ::::l "-'
UOhO•6
....:-1:1 .~ ()
~ ~ Q)
c U Q) () ,.", c ~
'/ /// //
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile
~ ~///////////
0.5
'/ // //
~
1.0
Period (sec)
1.5
target Sa (2% in 50 yrs.) median Sa of 10 g.m. range of 16th to 84th percentile
2.0
~ ~///// //u<,,< /////~~/////////////////////////////
(d) degrading systems 1 : 5 2.0
1.0 0.5 ) Period (sec
Figure 5.4 Comparison of spectral accelerations calculated from 10 ground motions with target values at 2% in 50 years probability level for degrading SDOF systems, representative soil profile in Memphis, TN.
I--' I--'
\D
N h 3 ~ ~ o i-' U ~ ~ s:: o
'..t:l u = "C ~
~
£> :s u
= Q
G'
2
1
o 0.0
h 9 ~ ~ .s u ~ ~ s:: o
'..t:l u = "C
~ .£
6
3
(a) w=2
(c) Jl=6
~~
0.5
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
2.0
:s u o ~, ~~------------------------------~ = Q 0.5 0.0 1.0 1.5 2.0
Period (sec)
1( 6 '-...-'
~ I: (:;) ..... (J ~
~~ I: (:;)
'..1= (J :::1 ,:::s cl)
~~ C ~3 (::J
= ~
,..--Q()
4
2
o 0.0
II 12 :::l
~a' :~ 10 10
"'"' I::) I~ 8 ~~
= ~ 6 I::)
~ 4 'Il)
~ ,0 2 ~= ~~ ~ 0 ~ 0.0
(b) ~t=4
0.5
(d) Jl=8
0.5
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
2.0
2.0
Figure 5.5 Comparison of ductility reduction factors of nondegrading systems for uniform hazard response spectra (this study) with empirical formulae by Krawinlker and Nassar (1992) and Miranda and Bertero (1994); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (;d = 2% and a= 3% are assulned for all cases), Memphis TN.
...... N o
-.. -.. N -.:::t \I 3 \I 6 ::l ::l '-" (a) p-2 '-"
(b) 11=4 ~ ~ ~ ~ -0 ---- 0 / ~ --~ .... -----'::--..... <-~ ~---- ....... ..... u 2 /~~ ~~-:: ~ ~ ~ --~ - ~ u 4 t'3 t'3 ~ ~ I: /, .... I: 0 0 :p '..c:: u u ::s ::s 2 "'C "'C Q,) Degrading systems (This study) Q,) If Degrading systems (This study) ~ ~
E Krawinkler & Nassar (1992)
.f' Krawinkler & Nassar (1992)
- Miranda & Bertero, Alluvium (1994) - Miranda & Bertero, Alluvium (1994) :p
0 :p
0 u u ::s 0.0 0.5 1.0 1.5 2.0 ::s
(]I. 0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec) -.. -.. \0 00 \I 9 ~ 12 ::l '-" (c) Jl=6
~ 10 ~ (d) Jl=8 ~
~
~~ 0 ..... u 6 ~ 8 j ~ .. " ~--::::::: t'3 ~ ~ I: 1:1
6 0 0 :p :p u u ::s 3 ::s 4 "'C , "'C Q,) Degrading systems (This study) Q,)
2 ~7 Degrading systems (This study)
~ P:: .f'
Krawinkler & Nassar (1992) .0 Krawinkler & Nassar (1992)
Miranda & Bertero, Alluvium (1994) ... Miranda & Bertero, Alluvium (1994) - S :p 0 0 u u
::s 0.0 0.5 1.0 1.5 2.0 ::s 0.0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec)
Figure 5.6 Comparison of ductility reduction factors of degrading systems for uniform hazard response spectra (this study) with elnpirical formulae by Krawinkler and Nassar (1992) and Miranda and Bertero (1994); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (qd = 2% and a= 3% are assumed for all cases), Memphis TN.
........ N ........
,,-... N
h 3 '-' ~ ;; o ..... ~ 2 ~
= o '.e u ::
"0
~ ,£
1
(a) ~l=2
~---2...Oi--=
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
=E u ::
o '~~~~~~~~~~~~~~~~~~~~ ~
,,-... \0
h 9
0.0
~ (c) ~=6 ;; o ..... ~ 6 ~
= o '.t: u .g 3 Q)
~
£ =E u 0 :: ~ 0.0
0.5
0.5
1.0 1.5
Period (sec)
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
2.0
2.0
,,-... ~
h 6 ~ ;; .s ~ 4 ~
= o '.e u ::
"0
~ .£ =E u :: ~
,,-... QO
2
o 0.0
h 14
(b) ~=4
~ 12 L (d) ~=8 J.. o 't 10 ~
~ 8 = o
'"8 6 :: ~ 4 ~ ,G> .... =E u 0 :: ~ 0.0
0.5
0.5
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
-----
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5
Period (sec)
2.0
2.0
Figure 5,7 Comparison of ductility reduction factors of nondegrading systems for uniform hazard response spectra (this study) with empirical formulae by Krawinkler and Nassar (1992) and Miranda and Bertero (1994); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (~d = 2% and a= 3% are assumed for all cases), Carbondale IL,
....... N N
,-..., N
h 3 '-' ~ ~ o
~ 2 ~
= .~ -C)
= "'0
~ .€
1
(a) fl=2
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
;S C)
o .~~~~~~~~~~~~~~~--~~~ = ~
,-..., \0
h 10
0.0
~ (c) fl=6
~ 8
0.5 1.0 1.5
Period (sec)
.s C) eo:s ~ 6
/?'<;;: ---..r~ ..- -::: ;-::-: ::7_-=-= - - -~~ ---= o .~
~ 4 "'0
<U
~ 2 .€ ;S C) 0 = ~ 0.0 0.5
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
1.0
Period (sec)
1.5
2.0
2.0
,-..., -.::r h 6 ~ ~ .s ~ 4 ~
= o '.t: C)
= "'0
~ E
2
(b) fl=4
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
;S C)
o .~~~~~~~~~~~~~~~~~~~ = ~
,-..., 00
h 14
~ 12 J. o 't 10 eo:s
~ 8 = o ~ C)
= "'0
6
4
0.0
(d) fl=8
0.5 1.0 1.5
Period (sec)
?/..- --:.::::.::-- -~--
Nondegrading systems (this study) Krawinkler & Nassar (1992) Miranda & Bertero, Alluvium (1994)
2.0
~ E 2 ;S o L· ~~~~~~~~~~~~~~~ C)
= ~ 0.0 0.5 1.0
Period (sec)
1.5 2.0
Figure 5.8 Comparison of ductility reduction factors of nondegrading systems for uniform hazard response spectra (this study) with empirical formulae by Krawinkler and Nassar (1992) and Miranda and Bertero (1994); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (C;d = 2% and a= 3% are assumed for all cases), St. Louis MO.
~
N W
,-.., N
~ 3 ~ ~ o .... u ~
~ c .S .... u :3
"0 Q)
~
.0
2
(a) w=2, nondcgrading systcms
6'~=--=-:-:~~-~~-=:---,/----:?~-~
/" "",,,
/"""
This study Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
=E u :3 ~
o ,~------------------------------~
,-.., \0
~ 9 i:a' ~ .s
0.0
u ~ 6 ~ c .S .... u
0.5 1.0 Period (sec)
(c) ~=6, nondegrading systems
1.5 2.0
"..-:: /' --:: :::-~ ::::: -::-:--.: - - - -:
= "0
~ .€' =E
3 I,( ;'
This study Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
u 0 0.0 0.5 1.0 1.5 2.0 = ~
Period (sec)
1 ~ ~ o .... u ~
~ c o .-e u :3
"0 Q)
~
.€' =E u
= ~
,-.., 00
4
2
o 0.0
~ 12 i:a' ~ 10 .s ~ 8 ~ c
6 0 ~ u
= 4 "0 Q)
~ 2 ~; .€' -~ 0 u
= 0.0 ~
(b) ~!=4, nondegrading systems
.. '" '"
---.. -- ---:;--
This study Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
0.5 1.0
Period (sec)
(d) ~=8, nondegrading systems
1.5
/'------
2.0
/' "'''',.------ ---/ .. " ---~ .. ~ '--:';"~~1
0.5
This study Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
1.0 1.5 Period (sec)
2.0
Figure 5.9 Comparison of ductility reduction factors of nondegrading systems for uniform hazard response spectra (this study) with empirical formulae by Nassar and Krawinkler (1992) and Miranda and Bertero (1994); (a) target ductility ratio= 2, (b) target ductility ratio = 4, (c) target ductility ratio = 6, (d) target ductility ratio = 8 (~=5% and a=3% are assumed for all cases), Santa Barbara CA.
........ N ~
-. N ~ 3 ~.~~~--,
~ ~ o ..... u eo::! ~ I: .s .... u ::s
"'0 QJ
~
.f' =E u
= ~
-. \0
2
(a) 11":2
~~7?~~~:;":"-"-"
1· / ,,' / // ,,'
/, -'
Degrading systems (This study) Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
o ~I ~----------------------0.0 0.5 1.0
Period (sec)
1.5
~ 9 -,---.-
~ ~ o ..... CJ eo::! ~ 1:1 o ~ CJ = "0 QJ
~
£> :s CJ
6
3
o I
(c) /1=6
Degrading systems (This study) Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium C1994)
2.0
~ ~ 6 r·~~~----'---~~~-.~~~~--r-~--~~~ '-' ~ (b) W=4
~ 0 .... u eo::! ~ I:
4 "",,--- .--- -:;-.... -----~ ~-7.:
===================~ .s .... u ::s
"'0 QJ
~
.f'
2 ~r7 j
Degrading systems (This study) Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994)
:s u
o L.~~~~~~~~~~~~~~~~~~ = ~
-. 00
~ 12 ~
0.0 0.5 1.0
Period (sec)
1.5
~ 10 .s CJ 8 ",....--------
2.0
: /' --------- --= // -,,/ --~ 6 / .. ,,"" ~ 4 1/ ~==-======-========:==========-===::::=::==========------~::::=---J ., ,/, -, ~ " .0 2 /~ ..... V :s CJ
Degrading systems (This study) Nassar & Krawinkler (1992) Miranda & Bertero, Alluvium (1994) o I.,---'--""'--'-___ .l...-..---'--'---'---L....... _____ --.-...--L-....:.-....~__'
= ~ 2.0 ~
0.-0 ----..---....__'__...L...-............ ____ """'-=-=::-""'--'-.....l..---.._'---'---.I 0.5 1.5 1.0
Period (sec)
0.10 0.5 1.0 1.5 2.0
Period (sec)
Figure 5.10 Comparison of ductility reduction factors of degrading systems for uniform hazard response spectra (this study) with empirical formulae for nondegrading systems by Nassar and Krawinkler (1992) and Miranda and Bertero (1994); (a) target ductility ratio= 2, (b) target ductility ratio = 4, (c) target ductility ratio = 6, (d) target ductility ratio = 8 (~=2% and a=3% are assurned for all cases), Santa Barbara CA.
~
tv Vl
--. --. N ~ II 3 II 6 :1. :1. '-' (a) ~t=2 '-'
r (b) ~=4 ~ ~ J,,:' J,,:' 0 0 ~ ~ u 2 u 4 Co:! Co:! ~ ~ c c 0 0 'C 'C u u = 1 regression curve (nondegrading) = 2r regression curve (nondegrading) "'Q "'Q OJ OJ ~ simulation results (3 prob. levels) ~ simulation results (3 prob. levels) C Krawinkler & Nassar (1992) C Krawinkler & Nassar (1992) .... :a -'C 0 0 u u = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec) --. --. \0 00 II 9 1h. 12 :t '-' (c) ~=6
~ 10 t (d) ~=8 ~ J,,:'
~~~ ........
0 tv ~ 0\ u 6 ~ 8 ~ ~--~ ~ ~ c c
6 0 0 'C 'C u u = 3 regression curve (nondegrading) = :V- regression curve (nondegrading) "'Q "'Q OJ OJ ~ simulation results (3 prob. levels) ~ simulation results (3 prob. levels) C Krawinkler & Nassar (1992) C Krawinkler & Nassar (1992) .... . ... - -'C 0 'C 0 u u = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec)
Figure 5.11 Regression ductility reduction factors of nondegrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (¢d = 2% and a= 3% are assumed for all cases), Memphis TN.
--- ---r-l ~ II 3 II 6 :1. :1. '-" (a) ~l=2 '-" (b) ~=4 ~ ~ ,,; ,,; 0 0 / ~ ------~---... ... u 2 u 4 ~ ~
~ ~
= = 0 0 +::l +::l u u = 1 regression curve (degrading) = 2~ regression curve (degrading) "0 "0 (1) (1)
~ simulation results (3 prob. levels) ~ simulation results (3 prob. levels) C Krawinkler & Nassar (1992) C Krawinkler & Nassar (1992) .....
~ =E 0 0 u u = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 Q Q
Period (sec) Period (sec)
--- ---\0 00 II 9 lh. 12 :1. '-' (c) ~=6 i 10 t (d) ~=8 ~ ,,; :=J 0 ... u 6 ~ 8 ~ ~ ~
= = 6 0 0 +::l +::l u u = 3 regression curve (degrading) =
:~r-regression curve (degrading) "0 "0
(1) (1)
~ simulation results (3 prob. levels) ~ simulation results (3 prob. levels) C Krawinkler & Nassar (1992) C Krawinkler & Nassar (1992) ..... ..... - -+::l 0 +::l 0 u u = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 Q Q
Period (sec) Period (sec)
Figure 5.12 Regression ductility reduction factors of degrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (qd = 2% and a= 3% are assumed for all cases), Memphis TN.
~
tv -.....)
,-.., N
~ 3 ~ ).;' o ..... Co)
1!"3 ~ C o
. .0 Co)
=' "0 ~
~
(a) w2
_-/-------------L-----~
21~ / -=:== ........ ./ ---~ /= /~~CSSiol1-CII~C~110I1dCgrndjl1g)
simulation results (3 prob. levels) Krawinkler & Nassar (1992) .€
=E Co)
o ~,~~~~~~~~~~~~~~~.~~~ = A
-\0
~ 9 ~ ).;'
.s ~ 6 ~
= o • .tl Co)
.g 3
~ ;£ =E u 0
0.0
= A 0.0
0.5
(c) ~l=6
0.5
1.0 Period (sec)
1.5
regression curve (nondegrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
1.0 1.5 Period (sec)
2.0
2.0
1 6 ~ ).;'
.s Co)
CI'3 ~ c .9 .....
Co)
=' "0 ~
~
g
4
2
(b) ~l=4
regression curve (nondegrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
~ u
o ~, ~~~~~~~~~~~--~~~~~~~ = A
--QO
~ 14
~ 12 ,... o 1:) 10 CI'3
~ 8 = o ~ u 6 = ] 4 ~
.€ 2 -~ 0
0.0
= A 0.0
0.5
(d) ~=8
0.5
1.0 1.5 Period (sec)
regression curve (nondegrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
1.0 1.5
Period (sec)
2.0
2.0
Figure 5.13 Regression ductility reduction factors of nondegrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (;d = 2% and a= 3% are assum.ed for all cases), Carbondale IL.
........ tv 00
,-.... ,-N .~
II 3 II 6 ::t ::t -- (a) w:2 ,-
(b) ~1=4 ~ t~ ~ ~ 0 0 .... ..... u u 4 e'!:! e'!:! ~ ~~
= = .~ .~ .... ..... u u :: 1 regression curve (nondegrading)
:: 2r regression curve (nondegrading) '"0 ''0 ~ ~
~ simulation results (3 prob. levels) I~ simulation results (3 prob. levels) .£ Krawinkler & Nassar (1992) .E' Krawinkler & Nassar (1992) =E 0
:E 0 u u
:: 0.0 0.5 1.0 1.5 2.0 :: 0.0 0.5 1.0 1.5 2.0 ~ j~
Period (sec) Period (sec) ,-.... .-, \0 (Xl
~ 10 ~ 14 -- (c) J.l=6
~12~ (d) J.l=8 ~
~ 8 '" ---4 0 t 10 ~ .... u
-----e'!:! e'!:! ~ 6 I~
= = 8 0 0
+= 4 ';:: 6 u u
:: regression curve (nondegrading)
:: 4V- regression curve (nondegrading) '"0 ''0 ~ ~
~ 2 simulation results (3 prob. levels) I~ f -- simulation results (3 prob. levels) £' Krawinkler & Nassar (1992) .0 2 __
Krawinkler & Nassar (1992) =E :E u 0 u 0 :: 0.0 0.5 1.0 1.5 2.0 :: 0.0 0.5 1.0 1.5 2.0 ~ I~
Period (sec) Period (sec)
Figure 5.14 Regression ductility reduction factors of nondegrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (;d = 2% and a= 3% are assumed for all cases), St. Louis MO.
....... N 'D
-. -. N "I't II 3 II 6 :::l :::l
--- (a) ~=2 --- (b) ~=4 ~ ~ ).;' ).;' 0 0 ---.------...... ...... u 2 u 4 ~ ~ ~ ~
== == 0 0 '.c '-C u u = 1 regression curve (nondegrading) =
2 ~I regression curve (nondegrading) "0 "0 Q) Q)
~ simulation results (3 probe levels) ~ simulation results (3 probe levels) .€' Krawinkler & Nassar (1992) £ Krawinkler & Nassar (1992) =E 0 =E 0 u u = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec) -. -. \0 00 II 9 lh. 14 :::l --- (c) ~=6 ~ 12 ~ (d) ~=8 ~ ).;' .------------0
~ 10 ~
...... -------=-=i VJ u 6 0 ~ ~ < ~ ~ 8 == == 0 0
'-C '-C 6 u u = 3 regression curve (nondegrading) = "0 "0 :V regression curve (nondegrading) Q) Q)
~ simulation results (3 probe levels) ~ simulation results (3 probe levels) ;£ Krawinkler & Nassar (1992) ;£ - =E Krawinkler & Nassar (1992) '-C u 0 u 0 = 0.0 0.5 1.0 1.5 2.0 = 0.0 0.5 1.0 1.5 2.0 ~ ~
Period (sec) Period (sec)
Figure 5.15 Regression ductility reduction factors of nondegrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (qd = 2% and a= 3% are assumed for all cases), Santa Barbara CA.
,,-., N
Mt 3 ~ ~ o ... ~ 2 ~
= o '.c u
(a) 11=2
,§ 1 l regression curve (degrading) ~. simulation results (3 prob. levels) .f'. Krawinkler & Nassar (1992) =E u 0
1 6 -~ ~ o ... ~ 4 ~
= o '.c u ::I
"'0 Cl)
~ .0 • ...c
2
(b) 11=4
.-------
regression curve (degrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
=E u o ~, ~~~--~~~~~~~~~~~~~~~
::I Q 0.0 0.5 1.0 1.5 = 2.0 Q 0.0 0.5 1.0 1.5 2.0
,,-., \0
Mt 9 ~ (c) 11=6 ~ o
~ 6 ~
= .S ... u ,§ 3
~ .0
Period (sec)
---------
regression curve (degrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
~ 0 ~.~~~~~~~~-L~~~~~~~~~ = A 0.0 0.5 1.0
Period (sec)
1.5 2.0
,,-., QO
!h. 14 ~ 12 L (d) 11=8 .. o "t 10 ~
~ 8 = o ~ 6 = ~ 4 ~
.f' 2 -~ 0 = Q 0.0 0.5
Period (sec)
--------
regression curve (degrading) simulation results (3 prob. levels) Krawinkler & Nassar (1992)
1.0
Period (sec)
1.5 2.0
Figure 5.16 Regression ductility reduction factors of degrading systems using the Krawinkler-Nassar empirical formula and comparison with the original Krawinlker and Nassar curves (1992); target ductility ratio (a) = 2, (b) = 4, (c) = 6, (d) = 8 (;d = 2% and a= 3% are assumed for all cases), Santa Barbara CA.
~
w ~
132
Cost (million dollars)
10° I I I
• drift ratio under E.Q. excitation 6 rIj
5 0 median drift ratio ~ 4 regression fragility curve ~ \ -----~ 3 \\ ~ , \
0 2 , \ l£I \ \ C \ \ . - @ . ~ 10-1 c:.J
, C \ ,
before retrofit ~ \
, "'''-~ 6 "'0 5 \
~ \ ~ 4 "-
\ "-c:.J "-~ 3 \ " " ~ \ ...............
~ ..........
~ 2 ••• --e~ • 0 ----~ \ -----........ \ ---.- 10-2 \ ----.- \ ~ ~ \ ~ 6 \ after retrofit 0 5 \~ ~
4 C. 3 \:
\ \
2 \ \
10-3
0 1 2 3 4 5 6
Drift Ratio (%)
Figure 5.1 7 Fragility curves for I-story standard buildings before/after retrofit In Carbondale, IL (after Wen and Song, 1999).
133
Tn = 0.3 sec, Representative soil profIle, Memphis (TN)
o simulation results model
I-Po = 0.0989 r---ee:Eer~·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·
50% in 50 yrs.
10% in 50 yrs .
.. . ~.O(o. ~~.~.Q.yX~~........................... . ............... .
2 % in 50 yrs.
1 0 -4 L.-.-'----l--.l.-l....I...l..J..I.I----"--I......I....l...I.J..U..I..--'---I-J....J...,U.J..l..I..--~..J....U...u..u.....___P+_~~
10-4
Elastic Force Coefficient, Ce
Figure 5.18 Comparison of the predicted annual probability of exceeding values of the elastic force coefficient using Equation (4.4) with the simulation results for a 0.3-sec period SDOF system at representative soil profile, Memphis ,TN (Vj
= 0.00485, k j = 1.031, Vc = 1.627, kc = 1.737, and w = 0).
~ u = ~
134
PGA, Representative soil profile, Memphis (TN)
o simulation results model
~ I-Po = 0.0989 ~ 10-1 ltr-++f-¥"l'=~·-· _._._.--._._._.- ._._._._. _._._._._.-._._._.
u ~ ~ ~ o C = 10-2 .",.,.
,.c ~
,.c o .. ~ e; 10-3
= = = -<
50% in 50 yrs.
10% in 50 yrs.
... ~~(o. ~~.~.~.Y~~~ .................... .
2 % in 50 yrs.
1 0-4 L..--~...I.....I....i...:..J.I..--"--"""""""~----l~~i...I..I.-.-..-............... l-ftI--..J..--I.....I..-1....1...1...1
10-4 10-3 10-2 10-1 10°
Peak Ground Acceleration (g)
Figure 5.19 Comparison of the predicted annual probability of exceeding values of peak ground acceleration using Equation (4.4) with the simulation results representative soil profile, Memphis TN (vJ = 0.002, kJ = 1.07, lh = 0.3, k2 =
1.90, Vc = 0.9, kc = 1.20, and w = 0.0002).
135
2.5 '/ ,; 0' /
/ / ,;
0 / ,;
2.0 / .o/IQ ,,-.., 0 /
~ 0 ,; -- C!
C / ,;
.S / ft 0 ...- / " ~ 1.5 /0 l.
" Q) / Q) 0/ U
/ 0/ u /0 ;'
< 9-/ -; 1.0 " l. ...-u T = 0.3 sec, Cy = 0.37 g Q)
c. r:n 0 simulation results 0.5 --- regression curve (median)
._._.-.- 16th_84th percentile bounds
0.0 0 5 10 15 20
Ductility Ratio (DmaJDy)
Figure 5.20 Regression of ductility ratio on median spectral acceleration through 100 simulated ground motions (nondegrading SDOF with T = 0.3 sec and Cy =
0.37 g).
c :: o ...
1.0 o
0.5
o
..... ....... o
T = 0.3 sec, Cy = 0.37 g
simulation results regression curve (median) 16th_84th percentile bounds
0.0 ~----~~~~~~~~~--~~~~~~--~~~~
o 5 10 15 20
Figure 5.21 Regression of ductility ratio on median peak ground acceleration through 100 simulated ground motions (nondegrading SDOF with T = 0.3 sec and Cy
= 0.37 g).
--~ --c: .~ -~ ... ~
~ ~ ~
< -; ... -~ ~ =:..
00.
2.5
2.0
1.5
1.0
0.5
0.0 0
T = 1.0 sec, Cy = 0.15 g
5
D / /
D.""
136
D
"" / /
/
/
D
10
/
'I' ' /
D
D
simulation results regression curve (median) 16th_84th percentile bounds
15
D
20
Figure S .22 Regression of ductility ratio on median spectral acceleration through 100 simulated ground motions (nondegrading SDOF with T = 1.0 sec and Cy = O.lSg).
1.5
1.0
0.5
0.0 o
T = 1.0 sec, Cy = 0.15 g
" /
" " D " " " " D /
" /
/ /
/
" " D
'I' /
D
D
D _._.-.-. ......... ......... ..,. ...... D
[J ......... ...,..
D._ .... ·- -' -.-...,. .....
D
5 10
simulation results regression curve (median) 16th_84th percentile bounds
15 20
Figure S .23 Regression of ductility ratio on median peak ground acceleration through 100 simulated ground motions (nondegrading SDOF with T= 1.0 sec and Cy
=O.lSg).
137
1.5 / /
T = 2.0 sec, Cy = 0.07 g / /
/
-- /
.... ci' OJ) / -- / ."
= /
.8 1.0 o / ...... 0 / ~ / ... <lJ /
"'Z / C) / C) /
~ 0
-; 0 ... 0.5 ~
<lJ 0 0 c.. 00 0
0 simulation results --- regression curve (median) ._._._.- 16th_84th percentile bounds
0.0 0 5 10 15 20
Ductility Ratio {Dma/Dy}
Figure 5.24 Regression of ductility ratio on median spectral acceleration through 100 simulated ground motions (nondegrading SDOF with T = 2.0 sec and Cy = 0.07 g).
c::
c ..
1.0
0.5 o
T = 2.0 sec, Cy = 0.07 g
o
/
/
/
/
/
/
o o
_.-' _.-' .......... _.-'
o
-._._.-.-0-'-'
simulation results regression curve (median) 16th_84th percentile bounds
0.0 ~----~--~~~~--~~~~--~~~~~~~~--~
o 5 10 15 20
Figure 5.25 Regression of ductility ratio on median peak ground acceleration through 100 simulated ground motions (nondegrading SDOF with r = 2.0 sec and Cy
A A"'7 _,
= v.v / g).
-~ = = = <
138
10-2 r-~-----'------r-I-.------r-I-'--~I--~-----~I------r-----~I--~--~ 8 ~ 7 f-:\
6 f-~\ 5 f-:\
4 \\ 3 f- \\
". '.
2 ...... +
". \ .... \ .... "
+
". " ..... +.,. ' ......
' ...... ' .......
"exact" solution from UHRS Sa, Shome & Cornell (1999) PGA, Shome & Cornell (1999)
-
5
4
3
'. ".:.~: 0,.: ~ '.", .~., ~+~.", ". ,,, ... "'.'" :''::::::::::: :::: ~ .::::.:: .. ..
2
T = 0.3 sec, Cy = 0.37 g
10-4 ~~-----_~I~--_~I~--~I--~--~I--~--~I--~--~
o 1 2 3 4 5 6
Ductility Ratio
Figure 5.26 Comparison of using spectral acceleration and peak ground acceleration in fragility analysis (Shome and Cornell, 1999) with "exact" solution (a O.3-sec nondegrading SDOF system is used).
-~ = = = -<
139
10-2 ~
I I I I I -
8 ~: + "exact" solution from UHRS 7 ~: 6 ~ ~ '-'-'-'- Sa' Shome & Cornell (1999) S ~ :, .............. PGA, Shome & Cornell (1999) -
4 ~ ':, -\ ....
3 \ .... \ ....
2 ~"""" '<: ...... . 10-3 ~
"..,L-" 1":'::--'" -..... ", -
8 I-7 I-
6 l-
S
4 I-
3 I-
2
" .~:<"'::.:..::.:. .. ~: :::::.::: ~+'':::::: :~ ... ~."~:.~ .. ,,, .......... ~
T = 1.0 sec, Cy = 0.15 g
10-4 ~~--~I--~--_~I~--~I--~--~I--~--_~I--~~
o 1 2 3 4 5 6
Ductility Ratio
Figure 5.27 Comparison of using spectral acceleration and peak ground acceleration in fragility analysis (Shome and Cornell, 1999) with "exact" solution (a 1.0-sec nondegrading SDOF system is used).
-~ == c c <
140
10-2 ~~--~--~I--~--~I--~--~I--~---~I~--~I--~~
8 3 7 Ii 6 ~ 5 ~ 4 ~~
j
3 ) '.
2 +.
8 7 6 5
4
3
2
"'\. "
"~.:.-:- ..... + ..................
+
".
"exact" solution from UHRS Sa' Shome & Cornell (1999) PGA, Shome & Cornell (1999)
......
-
-
-
-
----
-
-
........... ~:.~.:"'.,,±:~ ... ,,:~........ -~ .... -.........
..............
T = 2.0 sec, Cy = 0.07 g
104 ~~ __ ~J __ ~ __ ~I __ ~ __ ~I __ ~ ___ ~I~ __ ~I __ ~~
o 1 2 3 4 5 6
Ductility Ratio
Figure 5.28 Comparison of using spectral acceleration and peak ground acceleration in fragility analysis (Shome and Cornell, 1999) with "exact" solution (a 2.0-sec nondegrading SDOF system is used).
141
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 Summary and Conclusions
A simulation method is proposed to generate uniform hazard response spectra and
ground motions. It allows efficient evaluation of structural performance and fragility
analysis for loss estimation under future earthquakes. The seismotectonic parameters
used in this study to simulate earthquakes are largely based on USGS OFR-96-532 for
Mid-America and the 1995 WGCEP report for western United States. The method can
be updated as more is known of the seismotectonic features such as in the national
seismic hazard maps updating effort at a 3-year interval by U.S. Geological Survey.
9000 ground motions are generated at three Mid-America cities (Memphis, TN, St. Louis,
MO, Carbondale, IL) and 1815 in Santa Barbara, CA according to the regional
seismicity, from which uniform hazard response spectra are constructed. A least-square
procedure is then proposed to select 10 uniform hazard ground motions for a given
probability level at each site, whose median spectral accelerations match the target
response spectra corresponding to 10% and 2% in 50 years probability of exceedance.
Ground motions for both rock sites and soil sites with a given soil profile are generated.
Based on the results, the following conclusions are drawn:
1. A structural period independent procedure is used to select uniform hazard
ground motions from a large pool of simulated ground motions. Results of
investigation of various nonlinear SDOF systems ( e.g. structural period,
deterioration) indicate that these ground motions can be used for unbiased
estimates of structural responses with small uncertainty (standard error). Since
uniform hazard ground motions match the target response spectra for a wide
142
range of frequencies, they can be used for reliable response estimates of MDOF
systems as well.
2. Since only one component of earthquake motion is generated and the stochastic
source model is based on field observations of shear waves, the uniform hazard
ground motions provided in this study are suitable for structural time history
analyses of standard frame buildings with short to intermediate natural periods
(0.2 - 2 sec) and no significant 3-dimensional and torsional motions.
3. In all three Mid-America cities, small to medium size earthquakes have major
contribution to the hazard at the 10% in 50 years level. At the 2% in 50 years
level seismic hazard in Memphis and Carbondale is dominated by magnitude 8
earthquakes in the NMSZ, while in St. Louis small to medium size earthquakes
from various distances also contribute. These observations are in general
agreement with USGS deaggregation results.
4. Due to the influence of local site condition, spectral shape varies considerably
among three Mid-America cities. In Memphis and Carbondale, there are
significant spectral accelerations in the intermediate to long period range, which
is more damaging to medium- to high-rise buildings. In St. Louis, on the other
hand, large spectral accelerations are primarily within short period range, which
is more damaging to low-rise buildings. In Santa Barbara, the frequency
contents of the uniform hazard spectra indicate that medium- and high-rise
buildings are more vulnerable.
5. The ductility reduction factor RfJ. calculated from the simulated ground motions
for Memphis, Carbondale and Santa Barbara is shown to be in general
agreement with results from recorded earthquake ground motions by
Krawinkler and Nassar (1992) and Miranda and Bertero (1994). One can
construct probabilistic performance (fragility) curves using Rj.J factors within
the framework of current spectra-based code procedures (e.g. 1997 NEHRP)
without having to do a large number of nonlinear time history analyses.
143
6. Spectral acceleration is a more reliable measure of structural responses and
therefore should be used for evaluation of structural fragility. Peak ground
acceleration is a poor measure of long period structure responses and is not
recommended for fragility analysis.
6.2 Recommendations for Future Studies
The simulation methodology proposed herein provides a basic framework for
constructing uniform hazard response spectra and ground motions. However, there are
several issues that require further investigation:
1. The tectonic and seismological information in the CEUS is insufficient for an
accurate estimate of seismic hazard at a low probability level (e.g. 2% in 50
years). Large magnitude (7.5 to 8) earthquakes in the New Madrid seismic
zone are generally dominant at this level but their occurrence rates remain a
controversial topic (Johnston 1996b, Newman et al. 1999, Mueller et al. 1999).
With more geodetic, geologic and seismic information, the proposed simulation
methodology can be refined to generate more realistic ground motions.
2. Soil nonlinearity is important for high intensity earthquakes but has not been
considered in this study. Basically, the quarter wavelength method considers
only elastic soil properties, whereas the program SHAKE accounts for inelastic
soil properties via an equivalent linear model, which can not adequately account
for period shifting and response amplification due to soil nonlinearity, and is
best suited to a shallow soil deposit. Although there are truly nonlinear soil
analysis software available (e.g. D-Mod_D, CyberQuake, etc.), due to very
limited information on soil boring log data, more verification study is still
needed before a reliable nonlinear soil model can be used (Hashash and Park,
1999).
3. The phenomenological stochastic model does not consider surface waves,
which generally have long period motions and cause large responses in long
144
period structures, such as bridges and high-rise buildings. Also, when a
structure deteriorates under earthquake excitation, its fundamental period
lengthens and as a result is more vulnerable to surface waves. To include
surface waves, a wave propagation model such as in Saikia and Somerville
(1997) needs to be used.
4. This study does not consider 2- or 3-component ground motion simulation,
which is important for investigation of 3-dimensional frame building behavior
such as torsion effects, biaxial interaction, etc. (Wang and Wen, 1998).
5. For lifeline engineering (bridges, pipelines, etc) and large-scale structures,
spatial variation of ground motions needs to be considered. F or this purpose,
ground motion coherence function is required in the stochastic approach to
account for differential movement due to phase delay.
6. In reality, a large number of aftershocks often occur within a period of one to
two months after a large magnitude main shock. In such a short period, many
of the damaged buildings are unlikely to be repaired to survive the aftershocks,
e.g. the 1985 Mw-8 Michoacan (Mexico) and the 1999 ML-7.3 Chi-Chi
(Taiwan) earthquakes, etc. For essential and hazardous facilities, this
consideration becomes even more important since they need to be functional
after a damaging earthquake. It is reasonable to consider aftershocks if the
main shock has a magnitude greater than 7 (e.g., at the 2% in 50 years hazard
level).
7. System degradation has important effects on ductility reduction factors and
therefore needs to be considered in the design code. Further investigation on
MDOF systems is recommended to avoid under-design when spectra-based
design procedure is used.
145
APPENDIX A
TRUNCATED NORMAL DISTRIBUTION
Lognonnal distribution generally has a long tail. To avoid physically unrealistic
values in large-scale simulations, cut-off limits are necessary. The lognonnal distribution
is related to the standard nonnal probability density function (PDF) by a logarithmic
transfonnation (Ang and Tang, 1990). To avoid sharp cutoff limits with discontinuity,
the following modified standard nonnal probability density function is used, which gives
smooth truncations (Loh and Jean, 1994):
.JlI(x) = l~ exp( - ~) + A ocos(yx) + B -c:o < x < c:, (A. 1)
o Ixl> Be
where Cc is the cut-off limit; A, Band r are unknown constants and must satisfy the
following boundary conditions:
.J\I(x = -BJ = .J\I(x = Be) = 0 (A.2.1)
[d-:(X)Lc, = [d-:(X)L_c, =0 (A.2.2)
f:,%(X)dx = 1 (A.2.3)
In this study, Be = 3 is assumed, considering the large uncertainty due to lack of
field records in the CEUS. Solving the above equations, one obtains A = 0.01351,
B = 0.00422, and y = 1.3467. The resulting distribution is shown in Figure A.l and
compared with the standard nonnal PDF. To generate the corresponding truncated
lognonnal random numbers, an inverse transfonn method is employed; details can be
referred to Ang and Tang (1990).
A = 0.01351
B = 0.00422
0.4 Y = 1.3467
c .... 0.3 -.... ~ ~ ~ o .. ~ 0.2
0.1
146
Truncated Normal PDF Standard Normal PDF
0.0 __ ""'-"' __ ~--'---"----'----'---....l...--J..--I.--"_J-..~ __ ---""
-4 -3 -2 -1 o 1 2 3 4
Normal Variate, x
Figure A.I Truncated normal probability density function.
147
APPENDIXB
MODIFIED TYPE II EXTREME VALUE DISTRIBUTION FUNCTION
To obtain spectral acceleration according to a prescribed hazard level, one can
construct the hazard curve by running a large number of linear/nonlinear structural
analyses and then calculate the required spectral value through interpolation without
regression analysis. In doing so, a large computational effort is required, especially for
nonlinear analysis. To alleviate the computational burden, this study performs a
moderate number of structural analyses and then constructs hazard curves via a curve
fitting technique.
To determine the tail behavior of simulated results in this study, a generalized
extreme value distribution function proposed by Maes and Breitung (1993) was first
used. It was found that a Type II distribution fits the simulated data the best, which had
been observed by earthquake researchers in the past. However, because of the cut-off
limits in magnitude and attenuation and also a distribution gap between Mw 7.5 and 8, the
resulting distribution is deviate from standard Type II tail behavior. This is especially
true for probability lower than 5% in 50 years. In view of this, the tail of the distribution
is modified as follows:
P~(Cf) = {[l- Po - wl[l-exp[ -( ~~ r ]]+ W-[l-exp[-( ~; r ]]}+-exp[ -( ~; r]} (B.1)
where C;r is design force coefficient (i.e. Cf == Ce in elastic SDOF, Cf == Cy in nonlinear
SDOF); PeA Cd indicates annual exceedance probability; Po is annual probability of
earthquakes with magnitude less than mb 5; w is a weight parameter for the influence of
magnitude 8 earthquakes; VI, 1h and Vc are comer parameters; kI , k2 and kc are slope
parameters. Subscript 1 indicates the influence of earthquakes smaller than magnitude
7.5; subscript 2 indicate the influence of magnitude 8 earthquakes; subscript c indicates
148
the influence of cut-off limits. When w = 0 and Vc -+ 00 (i.e. v c » max( Ce )), Equation
(B. 1) follows the original Type II extreme value distribution (Ang and Tang, 1990).
When w = 0, there is no magnitude gap and therefore no sag segment in the exceedance
probability. When Vc -+ 00, it means no cut-off limits introduced in the simulation and
therefore no secondary slope. The goodness of fit using this generalized exceedance
probability function can be clearly seen in Figure B.1 and parameter values are listed in
Table B.1.
Table B.1 Parameters for the modified extreme value distribution function describing the annual probability of exceeding a target ductility of 4 as a function of yield force coefficient in a case of nondegrading SDOF systems at representative soil site, Memphis, TN.
Period Function Parameters
Vi k i 1h k2 Vc kc w
0.05 0.001836 1.072 0.825 2.828 - - 0.0
0.1 0.001872 1.055 0.687 1.488 - - 0.0
0.2 0.001636 1.035 0.804 3.764 - - 0.0
0.3 0.001039 0.974 0.792 2.438 - - 0.0
0.5 0.001095 1.253 0.183 1.632 0.771 2.057 0.000649
0.7 0.000866 1.394 0.115 1.397 0.654 2.892 0.000899
1.0 0.000374 1.213 0.092 1.426 0.590 3.421 0.000812
1.5 0.000184 1.223 0.060 1.325 0.342 18.017 0.000896
2.0 0.000097 1.130 0.056 1.672 0.278 8.647 0.000800
149
~ 10-~
~ 10-~
~ 5 Tn = 0.1 sec ~ 5 Tn = 0.2 sec = = ~ 3 ~ 3 "0 2 "0 2 ~ ~ ~
10-~ ~
10-~ ~ ~
>< >< ~ 5 ~ 5 r..- 3 r..- 3 0 0
& 2 g 2
10-~ 10-~ :E :E ~ 5 ~ 5
,Q 3 ,Q 3 0 2 0 2 l-c l-c ~
10-~ ~
10-~ '; '; ::s 5 ::s 5 = 3 = 3 = = -< 2 -< 2
10-5 10-5
2 3 456 10-1 2 3 456710
0 2 10-2 2 345610
-1 2 34567100 2
Yield Force Coefficient, Cy Yield Force Coefficient, Cy
10-~ 10-~ ~ ~ ~ 5 Tn = 0.3 sec ~ 6 Tn = 0.5 sec = = ~ 3 ~ 4
"0 2 "0 3 ~ ~ ~
10-~ ~ 2
~ ~
>< >< ~ 5 ~ 10-3 r..- 3 r..- 8 0
2 0 6 g
10-~ g 4
:E :E 3
~ 5 ~ 2 ,Q 3 ,Q 0 2 0 10-4 l-c l-c ~ ~ 8
101 6
'; '; 4 ::s 5 ::s 3 = = = 3 = 2 -< 2 -<
10-5 10-5
10-2 2 3 456 10-1 23456 1 0 0 2 3 4 10-2 2 3 456710
-1 2 3 456710
0
Yield Force Coefficient, Cy Yield Force Coefficient, Cy
Figure B.1 Comparison of the predicted annual probability of exceeding values of the yield force coefficient with the simulation results for nondegrading SDOF systems at representative soil profile, Memphis, TN. (;d = 2%, a= 3%, n = 5, target ductility = 4).
'"; ::: = = <
C': ::: = = <
5 3 2
10-4 5 3 2
10-5
10-3
5 3 2
10-4 5 3 2
10-5
Tn = 0.7 sec
2 3456 10-2 2 3456 10-1 2 3456
Yield Force Coefficient, Cy
Tn = 1.5 sec
10-3 2 :\ 456 10-2 2 3456 10-1 2 3456
Yield Force Coefficient, Cy
150
5 3 2
~ 10-3
~ 5 ,.Q 3 e 2
~ 10-4 ::: 2 -<
5 3 2
10-5
10-3
5 3 2
Tn = 1.0 sec
2 3456 10-2 2 3456 10-1 2 3456 100
Yield Force Coefficient, Cy
Tn = 2.0 sec
10-5
10-4234
Yield Force Coefficient, Cy
Figure B.l (continued).
151
APPENDIXC
UNIFOAAI HAZARD GROUND MOTIONS FOR MID-AMERICA CITIES
100 80 60 40 20
0 -20 -40 -60 -80
-100
80 60 40 20
0 -20 -40 -- -60 N
C.J -80 CJ r:n -100 -5
100 C.J
'-' 80
= 60 0 40
20 ...... 0 ~ -20 ~ -40 CJ -60 CJ -80 C.J -100 C.J
< 100 80 60 40 20
0 -20 -40 -60 -80
-100
100 80 60 40 20
0 -20 -40 -60 -80
-100 0 10
152
~ = 6.7, h = 19.5 km, Re = 137.7 km, E = 0.56
Mw = 6.3, h = 5.2 km, Re = 121 km, E = 1.07
~ = 6, h = 2.1 km, Re = 123 km, E = 1.63
I
~ = 6.8, h = 2.1 km, Re = 92.4 km, E = 0.37
~ = 6.2, h = 3.2 km, Re = 41.2 km, E = 0.34
20
Time (sec)
30 40
Figure C.1 Suite of 10% in 50 years ground motions for representative soil profile, Memphis, TN.
100 80 60 40 20
0 -20 -40 -60 -80
-100
80 60 40 20
0 -20 -40 -. -60 r'I
I:J -80 Q)
-100 t;I) -E 100 I:J -- 80
c 60 0 40
20 ..... 0 co: -20 l. -40 Q)
-60 Q) -80 I:J -100 I:J
< 100 80 60 40 20
0 -20 -40 -60 -80
-100
100 80 60 40 20
0 -20 -40 -60 -80
-100 0
153
Mw = 8, h = 25.6 km, Re = 171 km, Rjb = 113.6 km, E = -1.19
10
Mw = 6.5, h = 11.5 km, Re = 58.8 km, E = 0.24
Mw = 6, h = 8.5 km, Re = 29.6 km, E = 0.23
Mw = 5.3, h = 6.2 km, Re = 32.3 km, E = 1.02
Mw = 5.8, h = 6.7 km, Re = 75.7 km, E = 1.70
20
Time (sec)
Figure C.1 (continued).
30 40
154
700 500 ~ = 8, h = 25.6 km, Re = 147.6 km, Rjb= 75.9 km, c = 0.65
300 100
-100 -300 -500 -700
500 Mw = 8, h = 33.9 km, Re = 186.1 km, Rjb = 61.8 km, c = 0.09
300 100
-100
-. -300 N -500 ~
QJ -700 t"I.l -5 700 ~
--- 500 Mw = 8, h = 25.6 km, Re = 163.2 km, Rjb = 62.7 km, c = -0.1
= 300 0
100 ..... ~ -100 ;...
-300 QJ
QJ -500
~ -700 ~
< 700 500 Mw = 8, h = 9.2 km, Re = 100 km, Rjb = 60.1 km, c = 0.47
300 100
-100 -300 -500 -700
700 500 ~ = 8, h = 9.1 km, Re = 97.6 km, Rjb = 78.3 km, c = 0.48
300 100
-100 -300 -500 -700
0 10 20 30 40 50
Time (sec)
Figure C.2 Suite of 2% in 50 years ground motions for representative soil profile, Memphis, TN.
155
700 500 Mw = 8, h = 17.4 km, Re = 117.6 km, Rjb = 69.3 km, E = 0.57
300 100
-100 -300 -500 -700
500 Mw = 8, h = 9.1 km, Re = 145.7 km, Rjb = 57.8 km, E = -0.02
300 100
-100
-- -300 M -500 ~
ClJ -700 CI:l -E 700 ~ -- 500 ~ = 8, h = 9.1 km, Re = 197 km, Rjb = 92.3 km, E = 0.66
c 300 c 100 ....
~ -100 ;...
-300 ClJ
ClJ -500
~ -700 ~
< 700 500 Mw = 8, h = 9.1 km, Re = 170.5 km, Rjb = 45.9 km, E = -0.16
300 100
-100 -300 -500 -700
700 500 ~ = 8, h = 17.4 km, Re = 187.7 km, Rjb = 83.7 km, E = 0.50
300 100
-100 -300 -500 -700
0 10 20 30 40 50
Time (sec)
Figure C.2 (continued).
156
120
80 Mw = 6.3, h = 5.2 km, Re = 121 km, E = 1.07
40
0
-40
-80
-120
80 ~ = 6.4, h = 6.7 km, Re = 57.5 km, E = 0.64
40
0
-40 -..
N I;.)
-80 Q)
-120 rI.l -E 120 I;.)
'-" 80
Mw = 6.8, h = 18.1 km, Re = 125.1 km, E = O. c 0 40 ..... 0 e-: ~ -40 Q)
-80 Q)
I;.) -120 I;.)
< 120
80 Mw = 6.8, h = 2.1 km, Re = 92.4 km, E = 0.37
40
0
-40
-80
-120
120
80 ~ = 6.2, h = 27 km, Re = 107.1 km, E = 1.62
40
0
-40
-80 1~{\
-~ .. u
4 14 24 34
Time (sec)
Figure C.3 Suite of 10% in 50 years ground motions for bedrock (hard rock), Memphis, TN.
157
120
SO ~ = 6.2, h = 3.2 km, Re = 41.2 km, E = 0.34
40
0
-40
-SO
-120
SO ~ = 6.5, h = 11.5 km, Re = 58.8 km, E = 0.24
40
0
-40
---N -SO CJ QJ
-120 t'I:l -E 120 CJ
'-" ~ = 6.5, h = 23.9 km, Re = 129.1 kIn, E = 0.79 SO
c: 0 40
..... 0 ~ ... -40 QJ
-SO QJ
CJ -120 CJ
< 120 I
SO ~ = 6.3, h = 9.5 km, Re = 166.4 km, E = 1.71
40
0
-40
-80
-120
120
SO ~ = 6.S, h = S.7 km, Re = 35.6 km, E = -0.72
40
0
-40
-SO
-120 4 14 24 34
Time (sec)
Figure C.3 (continued).
158
500 400 300 200 100
0 -100 -200 -300 -400 -500
400 300
Mw = 8, h = 33.9 km, Re = 186.1 km, Rjb = 61.8 km, e = 0.09
200 100
0 -100 -200 .- -300 N
~ -400 <l) (Il -500 -E ~ 500 '-' 400 c 300 Q 200
100
~ = 8, h = 25.6 km, Re = 163.2 km, Rjb = 62.7 km, e = -0.1
...... 0 eo: -100 :.. -200 <l)
-300 <l) -400 ~ -500 ~
< 500 I
400 300
Mw = 8, h = 9.1 km, Re = 169.6 km, Rjb = 79.7 km, e = 0.35
200 100
0 -100 -200 -300 -400 -500
500 400 300
~ = 8, h = 9.1 km, Rb = 97.6 km, Rjb = 78.3 km, e = 0.48
200 100
0 -100 -200 -300 -400 -500
0 10 20 30 40 50
Time (sec)
Figure C.4 Suite of 2% in 50 years ground motions for bedrock (hard rock), Memphis, TN.
159
500 400 300 200 100
0 -100 -200 -300 -400 -500
400 300
~ = 8, h = 17.4 km, Re = 119.2 km, Rjb = 46.8 km, E = -0.
200 100
0 -100 -200 .- -300 N u -400 QJ
r;r., -500 -E 500 u -- 400
c: 300 0 200
100
~ = 8, h = 9.1 km, Re = 145.7 km, Rjb = 57.8 km, E = -0.02
..... 0 ~ -100 ... -200 QJ
-300 QJ -400 u -500 u
-< 500 400 300
~ = 8, h = 9.1 km, Re = 170.5 km, Rjb = 45.9 km, E = -0.16
200 100
0 -100 -200 -300 -400 -500
500 "'00 300
~ = 8, h = 17.4 km, Re = 187.7 km, Rjb = 86.7 km, E = 0.50
200 100
0 -100 -200 -300 -400 -500
0 10 20 30 40 50
Time (sec)
Figure C.4 (continued).
c o
200
100
o
-100
-200
100
o
-100
-200
200
100
-- 0 1---_''\���
~ -100
~ -200
< 200
100
o
-100
-200
200
100
o
-100
-200 o
160
Mw = 7.1, h = 28.5 km, Re = 149.6 km, E = 0.29
Mw = 6.4, h = 2.6 km, Re = 120.8 km, E = 0.93
Mw = 6.1, h = 4 km, Re = 68.5 km, E = 1.23
Mw = 6.4, h = 8.9 km, Re = 103.5 km, E = 1.01
~ = 5.9, h = 7.9 km, Re = 61.7 km, E = 1.51
10 20 30 40 50
Time (sec)
Figure C.5 Suite of 10% in 50 years ground motions for representative soil profile, Carbondale, IL.
200
100
o
-100
-200
100
o I------'MJ
-- -100 M
c:.J QJ ~ -200
5 200
= 100 o
..... 0
... QJ -100
c:.J -200
< 200
100
o
-100
-200
200
100
o
-100
-200 o
161
Mw = 6.2, h = 27.6 km, Re = 119.7 km, E = 1.30
Mw = 5.8, h = 9.1 km, Re = 125.3 km, E = 1.70
Mw = 6.8, h = 10.6 km, Re = 185.6 km, E = 1.06
~ = 6.7, h = 18 km, Re = 185.7 km, E = 1.02
10 20 30 40 50
Time (sec)
Figure C.S (continued).
162
700 500 ~ = 8, h = 17.4 km, Re = 166.4 km, Rjb= 106.2 km, {'; =
300
100 -100
-300
-500
-700
500 ~ = 8, h = 17.4 km, Re = 193 km, Rjb = 71.9 km, {'; = -0.19
300
100
-100
-- -300 N -500 ~ ~
-700 !;I) -5 700 ~
'-" 500 ~ = 8, h = 9.1 km, Re = 166.4 km, Rjb = 95.9 km, {'; = 0.30
c 300 0 100 .....
~ -100 ~
-300 ~
~ -500
~ -700 ~
~ 700
500 ~ = 8, h = 9.1 km, Re = 165.5 km, Rjb = 100.7 km, {'; = 0.20
300
100 -100
-300
-500
-700
700
500 ~ = 8, h = 9.1 km, Re = 122.2 km, Rjb = 110.4 km, {'; = 0.90
300
100
-100
-300 -500
-700 0 20 40 60 80 100
Time (sec)
Figure C.6 Suite of 2% in 50 years ground motions for representative soil profile, Carbondale, IL.
163
700
500 ~ = 8, h = 9.1 km, Re = 137.2 km, Rjb = 124.3 km, I:: = 1.15
300
100
-100
-300 -500
-700
500 ~ = 8, h = 9.1 km, Re = 175.9 km, Rjb = 147.6 km, I:: = 1.39
300
100 -100
-300 N -500 C)
Q)
-700 ~ -E 700 C)
500 ~ = 8, h = 9.1 km, Re = 98.4 km, Rjb = 79.3 km, I:: = 0.69 c
300 0 100
eo: -100 l.
-300 Q)
Q) -500
C) -700 c.J
< 700
500 ~ = 8, h = 9.1 km, Re = 169.3 km, Rjb = 79 km, I:: = 0.41
300
100
-100
-300
-500
-700
700
500 ~ = 8, h = 9.1 km, Re = 183.3 km, Rjb = 128.6 km, I:: = 0.63
300
100
-100
-300 -500
-700 0 20 40 60 80 100
Time (sec)
Figure C.6 (continued)
150
50
-50
-150
50
-50 -.
M ~ Q)
-150 rI.l -= 150 ~
'-'
C Q 50
..... eo: I-c -50 Q)
Q)
~ -150 ~
< 150
50
-50
-150
150
50
-50
-150 0 10
164
Mw = 7.1, h = 28.5 km, Re = 149.6 km, c: = 0.29
~ = 6.4, h = 2.6 km, Re = 120.8 km, c: = 0.93
~ = 6.1, h = 4 km, Re = 68.5 km, c: = 1.23
Mw = 6.4, h = 8.9 km, Re = 103.5 km, c: = 1.01
~ = 5.9, h = 7.9 km, Re = 61.7 km, c: = 1.51
20
Time (sec)
30 40
Figure C.7 Suite of 10% In 50 years ground motions for bedrock (hard rock), Carbondale, IL.
150
50
-50
-150
50
-50 --N CJ Q)
-150 rI.l -E 150 CJ
'-'
c: 0 50
..... ~ ... -50 Q)
Q)
CJ -150 CJ
< 150
50
-50
-150
150
50
-50
-150 0 10
165
Mw = 6.1, h = 1.1 km, Re = 78.7 km, E = 1.68
Mw = 6.2, h = 27.6 km, Re = 119.7 km, E = 1.30
Mw = 5.8, h = 9.1 km, Re = 125.3 km, E = 1.70
Mw = 6.3, h = 4 km, Re = 96 km, c = 1.23
Mw = 6.7, h = 18 km, Re = 185.7 km, E = 1.02
20
Time (sec)
30
Figure C.7 (continued).
40
166
300
200 Mw = 8, h = 17.4 km, Re = 166.4 km, Rjb= 106.2 km, c =
100
0
-100
-200
-300
200 ~ = 8, h = 9.1 km, Re = 166.4 km, Rjb = 95.9 km, c = 0.30
100
0
-100 .-
M c:.J
-200 Col
-300 ~ -E c:.J 300 - 200
~ = 8, h = 25.6 km, Re = 193.2 km, Rjb = 96.5 km, c = 0.23 c Q 100
-- 0 ~ .. -100 Col
-200 Col
c:.J -300 c:.J
-< 300
200 ~ = 8, h = 9.1 km, Re = 187.8 km, Rjb = 82 km, c = 0.23
100
0
-100
-200
-300
300
200 ~ = 8, h = 9.1 km, Re = 165.5 km, Rjb = 100.7 km, c = 0.20
100
0
-100
-200 ,nn -')uu
0 20 40 60 80 100
Time (sec)
Figure C.8 Suite of 2% in 50 years ground motions for bedrock (hard rock), Carbondale, IL.
167
300
200 Mw = 8, h = 9.1 km, Re = 122.2 km, Rjb = 110.4 km, E = 0.90
100
0
-100
-200
-300
200 Mw = 8, h = 9.1 km, Re = 137.2 km, Rjb = 124.3 km, E = 1.15
100
0
-100 --M -200 ~ (l)
-300 rI:J -E 300 ~ -- 200
~ = 8, h = 9.1 km, Re = 175.9 km, Rjb = 147.6 km, E = 1.39 c c 100 .... 0 ~
10.. -100 (l)
-200 (l)
~ -300 ~
< 300
200 Mw = 8, h = 9.1 km, Re = 98.4 km, Rjb = 79.3 km, E = 0.69
100
0
-100
-200
-300
300
200 ~ = 8, h = 9.1 km, Re = 169.3 km, Rjb = 79 km, E = 0.41
100
0
-100
-200
-300 0 20 40 60 80 100
Time (sec)
Figure C.8 (continued).
168
200 Mw = 6, h = 2.7 km, Re = 76.4 km, E = 0.90
100
0
-100
-200
Mw = 6.9, h = 9.3 km, Re = 201.5 km, E = 0.44
100
0
--- -100 M ~ Q)
-200 tI:l -E 200 ~ -- Mw = 7.2, h = 4.4 km, Re = 237.5 km, E = 0.07
c 100 0
.... 0 co: ... Q) -100 Q)
~ -200 ~
< 200 ~ = 6.3, h = 9.8 km, Re = 252.2 km, E = 1.71
100
0
-100
-200
200 ~ = 5.5, h = 2.9 km, Re = 123.1 km, E = 1.81
100
0
-100
-200 0 10 20 30 40
Time (sec)
Figure C.9 Suite of 10% in 50 years ground motions for representative soil profile, St. Louis, MO.
169
200 Mw = 6.2, h = 7.7 km, Re = 207.6 km, E = 1.68
100
0
-100
-200
Mw = 6.9, h = 1.7 km, Re = 193.7 km, E = 0.35
100
0
-- -100 ("l ~ <:lJ
-200 I:I'.l -E 200 ~
--- Mw = 6.2, h = 27.6 km, Re = 174.5 km, E = 1.40 c 100 Q
-- 0 ~ ;..
<:lJ -100
<:lJ
~ -200 ~
~ 200 Mw = 6.2, h = 6.5 km, Re = 221.3 km, E = 1.72
100
0
-100
-200
200 Mw = 6.9, h = 2.7 km, Re = 237.2 km, E = 0.81
100
0
-100
-200 0 10 20 30 40
Time (sec)
Figure C.9 (continued).
700 500 300 100
-100 -300 -500 -700
500 300 100
-100
-- -300 N -500 ~
Cl.l -700 C"'-l -5 700 ~
'-' 500
c: 300 0 100 ......
~ -100 ... -300 Cl.l
Cl.l -500
~ -700 ~
< 700 500 300 100
-100 -300 -500 -700
700 500 300 100
-100 -300 -500 -700
0 10
170
~ = 8, h = 17.4 km, Re = 267 km, Rjb = 223.5 km, c = 1.38
Mw = 8, h = 9.1 km, Re = 229.5 km, Rjb = 197 km, c = 0.64
~ = 7.1, h = 5.5 km, Re = 253.1 km, c = 1.94
~ = 8, h = 17.4 km, Re = 254.3 km, Rjb = 205.8 km, c = 1.1
20 30
Time (sec)
40 50 60
Figure C.1 0 Suite of 2% in 50 years ground motions for representative soil profile, St. Louis, MO.
171
700 500 ~ = 6.8, h = 5.8 km, Re = 224.8 km, l:: = 1.99
300 100
-100
-300 -500
-700
500 Mw = 8, h = 33.9 km, Re = 196.3 km, Rjb = 175.3 km, l:: = 0.7
300 100
-100
..- -300 N -500 u
Q)
-700 ! ! ! I
"-l -E 700 u
500 ~ = 8, h = 9.1 km, Re = 260.7 km, Rjb = 188.4 km, l:: = 0.85 c
300 0
100 ..... ~ -100 l.
-300 Q)
Q) -500
u -700 u
-< 700
500 Mw = 8, h = 9.1 km, Re = 280.5 km, Rjb = 195.6 km, l:: = 0.97
300
100 -100
-300 -500 -700
700
500 ~ = 5.9, h = 4.4 km, Re = 47.7 km, l:: = 1.76
300
100 -100 -300 -500 -700
0 10 20 30 40 50 60
Time (sec)
Figure C.1 0 (continued).
100 80 60 40 20
0 -20 -40 -60 -80
-100
80 60 40 20
0 -20 -40 ,,-.., -60 N
'-i -80 Q) CI:l -100 -5
100 '-i -- 80 c: 60 0 40
20 ..... 0 eo: -20 ... -40 Q)
-60 Q) -80 '-i -100 '-i
< 100 80 60 40 20
0 -20 -40 -60 -80
-100
100 80 60 40 20
0 -20 -40 -60 -80
-100 0
172
~ = 6, h = 2.7 km, Re = 76.4 km, E = 0.90
~ = 6.9, h = 9.3 km, Re = 201.5 km, E = 0.44
~ = 7.2, h = 4.4 km, Re = 237.5 km, E = 0.07
I
Mw = 6.3, h = 9.8 km, Re = 252.2 km, E = 1.71
~ = 6.2, h = 4.7 km, Re = 180.5 km, E = 1.00
10 20
Time (sec)
30 40
Figure C.II Suite of 10% in 50 years ground motions for bedrock (hard rock), S1. Louis, MO.
173
100 80 Mw = 5.5, h = 2.9 km, Re = 123.1 km, g = 1.81 60 40 20
0 -20 -40 -60 -80
-100
80 Mw = 6.2, h = 7.7 km, Re = 207.6 km, g = 1.68 60 40 20
0 -20 -40 ..- -60 M u -80
~ tI:l -100 -E
100 u '-' 80 Mw = 6.9, h = 1.7 km, Re = 193.7 km, g = 0.35 c 60 0 40
20 ...... 0 ~ -20 .... -40 ~ -60 ~ -80 u -100 u
< 100 80 Mw = 6.2, h = 27.6 km, Re = 174.5 km, g = 1.40 60 40 20
0 -20 -40 -60 -80
-100
100 80 60
Mw = 6.2, h = 6.5 km, Re = 221.3 km, g = 1.72
40 20
0 -20 -40 -60 -80
-100 0 10 20 30 40
Time (sec)
Figure C.II (continued).
174
300
200 Mw = 8, h = 9.13 km, Re = 229.5 km, Rjb = 197 km, e = 0.65
100
0
-100
-200
-300
200 Mw = 5.4, h = 2.1 km, Re = 28.7 km, e = 1.63
100
0
-100
---M -200 e.J QJ
-300 rI.l -5 300 e.J -- 200
~ = 7.1, h = 5.5 km, Re = 253.1 km, e = 1.94 c 0 100
..... 0 e'::! I. -100 QJ
-200 c:.l e.J -300 e.J
< 300 I i I
200
iii I I i I I I I I I I iii
MW = 8, h = 25.6 km, Re = 213.9 km, Rjb = 199.9 km, e = 1.1
100
0
-100
-200
-300
300
200 Mw = 6.8, h = 5.8 km, Re = 224.8 km, e = 1.99
100
0
-100 .,nn
-.\lU
-300 ! t
0 10 20 30 40 50
Time (sec)
Figure C.12 Suite of 2% in 50 years ground motions for bedrock (hard_rock), St. Louis, MO.
--f'l c:.J Q) rIJ -E c:.J
c 0
~ ... Q)
Q)
c:.J
c:.J
<:
300
200
100
o -100
-200
-300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
175
~ = 8, h = 33.9 km, Re = 196.3 km, Rjb = 175.3 km, f.: = 0.78
~ = 8, h = 9.1 km, Re = 186.5 km, Rjb = 184 km, f.: = 0.42
~ = 8, h = 9.1 km, Re = 260.7 km, Rjb = 188.4 km, f.: = 0.85
t I I
~ = 8, h = 9.1 km, Re = 280.5 km, Rjb = 195.6 km, f.: = 0.97
~ = 5.9, h = 4.4 km, Re = 47.7 km, f.: = 1.76
0 10 20 30 40 50
Time (sec)
Figure C.12 (continued).
176
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