Improving the Earthquake
Resilience of Buildings:
The worst case approach
Izuru Takewaki
Professor, Kyoto University
Abbas Moustafa
Associate Professor, Minia University,
Egypt
Kohei Fujita
Assistant Professor, Kyoto University
Abstract
Engineers are always interested in the worst-case scenario. One of the most
important and challenging missions of structural engineers may be to narrow
the range of unexpected incidents in building structural design. Redundancy,
robustness and resilience play an important role in such circumstances.
Improving the Earthquake Resilience of Buildings: The worst case approach
discusses the importance of worst-scenario approach for improved
earthquake resilience of buildings. This book consists of two parts. The first
part deals with the characterization and modeling of worst or critical ground
motions on inelastic structures and the related worst-case scenario in the
structural design of ordinary simple building structures. The second part
focuses on investigating the worst-case scenario for passively controlled and
base-isolated buildings. This allows for detailed consideration of a range of
topics including:
• A consideration of damage of building structures in the critical excitation
method for improved building-earthquake resilience,
• A consideration of uncertainties of structural parameters in structural
control and base isolation for improved building-earthquake resilience, and
• New insights in structural design of super high-rise buildings under longperiod ground motions.
This book is a valuable resource for researchers and engineers interested in
learning and applying the worst-case scenario approach in the seismicresistant design for more resilient structures.
Highlights
▶ Considers the elastic-plastic behavior of building
structures in the critical excitation method for
improved building-earthquake resilience
▶ Approaches the uncertainties of structural parameters
in structural control and base-isolation for improved
building-earthquake resilience
▶ Provides new insights in structural design of super
high-rise buildings under long-period ground motions
(case study on tall buildings in mega cities in Japan
during the 2011 off the Pacific coast of Tohoku
earthquake on March 11)
▶ Instructs readers on how to implement the worstscenario approach to provide engineers with effective
design techniques
Chapter 1
1.1 Background and review
1.2 Input ground motion and
worst-case scenario
1.3 Organization of the book
Chapter 2
What is an important measure for
characterizing earthquake ground motion?
acceleration or velocity?
Fig.2.9 Schematic diagram for computing credible bounds
for acceleration and velocity constraints
Chapter 2
Resonance can occur?
Resonance in reality
Fig.2.15(a) Ground acceleration, ground velocity and topstory displacement of a super high-rise building
in Osaka during the 2011 off the Pacific coast
of Tohoku earthquake
Chapter 2
Approach to improving building
earthquake resilience
Improving building resilience
by structural control
Fig.2.26 Plastic-hinge formation in 40-story buildings of
T1=3.6(s) without and with high-hardness
rubber dampers
without dampers
with dampers
Chapter 3
H-E06230
Velocity amplitude spectra (m)
Simulation of Near-Field
Pulse-Like Ground Motion
0.5
0
-0.5
5
(a)
10
Time (s)
15
20
Velocity (m/s)
1
(b)
0.5
0
-0.5
-1
0
5
10
Time (s)
15
20
1.5
Velocity (m/s)
1
0.5
0
-0.5
(c)
-1
0
5
10
Time (s)
15
20
Velocity amplitude spectra (m)
-1
0
Velocity amplitude spectra (m)
Velocity (m/s)
1
SCS052
C02065
TAK000
TCU068W
TCU068N
LGP000
LCN275
TAB-TR
KJM000
RRS228
YPT060
ALS-E
TCU078W
WSM090
JOS090
0.1
0.08
0.06
C02065
0.04
0.02
0
0
ERZ-NS
DZC180
PCD164
(a)
1
2
3
Frequency (Hz)
4
5
H-E06230
0.1
C02065
TAB-TR
RRS228
0.08
0.06
0.04
0.02
0
0
1
2
3
Frequency (Hz)
4
SCS052
TAK000
TCU068N
TCU068W
5
KJM000
YPT060
0.1
0.08
ALS-E
TCU078W
0.06
0.04
0.02
0
0
1
2
3
Frequency (Hz)
4
5
LGP000
Fig. 3.4 Ground velocity and associated
Fourier amplitude spectra
CPM000
LCN275
ERZ-NS
WSM090
DZC180
JOS090
PCD164
(b)
Fig. 3.1 Near-field strong ground motion with distinct velocity pulses:
(a) velocity time history, (b) velocity Fourier amplitude spectra
Critical excitation for
inelastic response
Chapter 4
2
Normalized displacement
Normalized displacement
A cceleration
Acceleration
/ g/ g
0.5
0
-0.5
0
5
10
15
Time s
20
25
1
0
-1
-2
0
30
5
10
15
Time s
20
25
30
4
x 10
3500
3000
1
Energy
Energy NNm
m
Hysteretic
N
Hysteretic force
force N
1.5
0.5
0
-0.5
-1
-1.5
-0.2
Input energy
Hysteretic energy
Damping energy
2500
2000
1500
1000
500
-0.1
0
0.1
Displacement m
0.2
0
0
5
10
15
Time s
20
25
30
Fig. 4.8: Critical earthquake accelerations and
associated critical responses for case 4
(b)
Chapter 5
1
0.2
0.5
0.1
0
0
-0.5
-1
0
Acceleration / g
Characteristics of
Earthquake Ground
Motion of Repeated
Sequences
Acceleration / g
(a)
200
400
NS
600
-0.2
0
1
0.2
0.5
0.1
0
0
-0.5
200
400
600
-0.2
0
1
0.1
0.5
0.05
0
0
-0.5
-1
0
400
Time (s)
200
300
EW
100
200
300
-0.05
UD
200
100
-0.1
EW
-1
0
Acceleration / g
-0.1
NS
UD
600
-0.1
0
100
200
300
Time (s)
Fig 5.1: Ground accelerations with multiple sequences (a) 2004 Niigata earthquake at
Nagaoka-shisho (NIG028) (b) 2004 Niigata earthquake at Koide (NIG020).
Critical excitation for
multiple sequences
Chapter 6
1
Fourier spectra m/s
Acceleration / g
0.5
0
-0.5
0
50
0.8
0.6
0.4
0.2
0
0
100
5
x 10
10
15
4
10
x 10
25
4
Yield energy
Damping energy
2
8
1
Energy (Nm)
Hysteretic force (N)
20
Frequency (Hz)
Time (s)
0
-1
6
4
2
-2
-5
0
Normalized displacement
5
0
0
20
40
60
80
100
Time (s)
Fig. 6.5: Critical ground acceleration of two sequences and
associated response for bilinear inelastic structure
Chapter 7
Mean of input energy (Nm)
15000
10000
5000
0
0
20
40
60
80
100
Variance of input energy (Nm)
Response of Nonlinear SDOF Structures
to Random Acceleration Sequences
0.005
0
-0.005
-0.01
0
20
40
60
Time (s)
80
100
x 10
6
8
6
4
2
0
0
3
Variance of displacement (m2)
Mean of displacement (m)
0.01
10
x 10
20
40
60
80
20
40
60
80
100
-3
2
1
0
0
100
Time (s)
Fig. 7.6: Mean of response quantities for elastic-plastic structure
to ground acceleration of three sequences
Chapter 8
Use of Deterministic and Probabilistic
Measures to Identify Unfavorable
Earthquake Records
(a)
(a)
(b)
(b)
0.12
2 3
Δω = 0.10 rad/s Smax = 10 m /s
Area = 1.0
0.08
0.06
Area = 1.0
0.04
Soft soil
Medium soil
Stiff soil
Rock soil
0.5
S(ω ) m 2/s 3
S(ω ) m 2/s 3
0.1
Narrow band
Kanai-Tajimi
Band-limited
0.4
0.3
Area = 1.0
0.2
Area = 1.0
0.02
0.1
Area = 1.0
0
0
5
10
15
Frequency Hz
20
25
0
0
Area = 1.0
2
Area = 1.0
4
6
Frequency Hz
8
Fig. 8.2: (a) PSD function for ground acceleration models for
medium soil, (b) Kanai-Tajimi PSD function for different soil types
10
Chapter 9
Damage Assessment of Inelastic
Structures Under Worst
Earthquake Loads
(a)
(b)
0.5
Critical input
M4(ω)
Amplitude spectra (m/s)
F o u rie
r a m p litu d e (m /s )
Acceleration /g / g
A cceleration
0.5
0.4
0.3
0
0.2
0.1
-0.5
0
5
10
15
20
Time (s) Hz
Frequency
25(b)
30
0
0
5
10
15
Frequency
Frequency (Hz) Hz
20
Fig. 9.11: Critical acceleration for inelastic structure for case 2
(a) Time history, (b) Fourier amplitude spectrum
25
Chapter 10
Formulation of Evolutionary Waves
Envelope function
1
Body waves
Surface waves
0.8
What is the
critical
combination?
0.6
0.4
0.2
0
0
5
10
15 20 25
Time (s)
30
35
40
Surface waves
4
Acceleration (m/s 2)
Acceleration (m/s 2)
Body waves
2
0
-2
-4
0
5
10
15 20 25
Time (s)
30
35
40
Frequency content:
1.0～10.0Hz
4
2
0
-2
-4
0
5
10
15 20 25
Time (s)
30
35
Frequency content:
0.0～1.0Hz
40
Chapter 11
L2
L1
x2
x1
What is the bi-directional critical input?
Fig.11.13 One sample set of Monte
Carlo simulation of the
2-dimensional ground
motion (2DGM)
Fig.11.14 Comparison between
critically correlated
2DGM and perfectly
correlated ones
Chapter 12
Optimal Placement of Visco-Elastic
Dampers and Supporting Members
Under Variable Critical Excitations
Fig. 12.1: Structural model with viscoelastic
damper including supporting member
Fig.12.9: Variation of objective function in optimal
placement, uniform placement and first-storey
placement; (a) 3-storey model, (b) 10-storey model
Chapter 13
Sustainability
under
uncertainties
Fig.13.1 Concept of sustainable design considering
varied structural performance caused by various
uncertainties of structural parameters
Chapter 14
Earthquake Response Bound
Analysis of Uncertain Base-Isolated
Buildings for Robustness Evaluation
Fig.14.1 N-story baseisolated building model
Fig.14.3 Dependence of critical combination of
uncertain parameters on properties of objective
function, (a) Inclusion monotonic, (b) non-monotonic
Chapter 15
Future Directions
Fig.15.5 Increase of credible bound of input energy for velocity power constraint due
to lengthening of input excitation duration and decrease of damping ratio of structure
Preface
Engineers are always interested in the worst-case scenario. The seismic design of
buildings should ensure structural safety against the worst possible future earthquakes. The features of this monograph are:
(1) Consideration of elastic–plastic behavior of building structures in the critical
excitation method for improved building-earthquake resilience,
(2) Consideration of uncertainties of structural parameters in structural control
and base-isolation for improved building-earthquake resilience, and
(3) New insights into structural design of super high-rise buildings under longperiod ground motions (case study on tall buildings in mega cities in Japan
during the 2011 off the Pacific coast of Tohoku earthquake on March 11).
This book consists of two parts. The first part deals with the characterization
and modeling of worst or critical ground motions on inelastic structures. The
second part of the book focuses on investigating the worst-case scenario for
passively controlled and base-isolated buildings.
Chapter 1 provides an overview of the effects of historic and recent strong
earthquake ground motions on building structures and associated life loss.
Chapter 2 provides comprehensive information about the most recent and
devastating Tohoku earthquake of moment magnitude 9.0 which hit off the pacific
coast of eastern Japan on 11 March 2011. This earthquake and the tsunami following it left severe damage to building structures and caused nearly 20,000 of
losses of life.
As is well known, the robust design of buildings for future earthquake loads
requires reliable understanding of the ground motion characteristics. Accordingly,
Chaps. 3 and 4 report on the characteristics of near-field (near-fault) ground
motions with pulse-like acceleration. Furthermore, these two chapters provide
simple mathematical models for this class of ground motions and associated
structural response. Chapter 3 deals with the simulation of near-field ground
motions with pulse-like acceleration while a critical excitation of multiple
sequences for inelastic responses is discussed in Chap. 4.
v
vi
Preface
Chapters 5–7 deal with the characterization and modeling of earthquake ground
motion of multiple sequences. Recently, this class of ground motions was clearly
observed during the 2011 off the Pacific coast of Tohoku earthquake on March 11.
This research subject is new and has not received adequate attention from
researchers. For instance, most seismic codes specify design ground motions as
single events. However, moderate ground motion with repeated acceleration
sequences could lead to more severe damage to structures than a single sequence
of strong ground motion. The worst-case scenario is studied within the deterministic and probabilistic frameworks. Characteristics of earthquake ground motion
of repeated sequences are made clear in Chap. 5 while critical ground motion
sequences are discussed in Chap. 6. In Chap. 7, responses of elastic–plastic
structures to nonstationary random acceleration sequences are investigated and the
reliability of such structures is evaluated.
A practical problem always arises in the design of buildings against earthquake
loads. It is always difficult to select a suite of suitable earthquake records from a
large set of records as input to the nonlinear time-history analysis of structures.
Chapter 8 provides deterministic and probabilistic measures that can be used to
identify unfavorable accelerograms. This chapter provides simple concepts which
can be utilized to select a suit of appropriate earthquake records for nonlinear timehistory analysis of structures.
Chapters 9 and 10 deal with the worst-scenario of earthquake loads on inelastic
structures with special emphasis on the type of seismic waves of the ground
motion and damage quantification using damage indices.
Chapter 11 deals with the worst-case scenario for bidirectional ground motions.
Most of the current seismic-resistant design codes are based on the simulation of
building response under uni-directional earthquake input. However, bidirectional
input is inevitable for the reliable design of columns.
Chapters 12 and 13 tackle the worst-case scenario for passively controlled
buildings. The structural member stiffness and strength of buildings are uncertain
due to various factors resulting from randomness, material deterioration, temperature dependence, etc. The passive damper systems are also uncertain depending
on various sources. The concept of sustainable building design under such
uncertain structural-parameter environment may be one of the most challenging
issues to be tackled recently. By predicting the response variability accurately, the
elongation of service life of buildings may be possible.
Chapter 14 focuses on the worst-case scenario for base-isolated buildings. The
stiffness and damping of the base-isolation system and the stiffness of the superstructure are selected as uncertain parameters. An efficient methodology is explained to
evaluate the robustness (variability of response) of an uncertain base-isolated building.
The book closes with Chap. 15 on current challenges and future directions on
design of building structures with greater earthquake resilience.
The importance of the worst-scenario approach for improved earthquake resilience of buildings and nuclear reactor facilities has been recognized and demonstrated by the recent great earthquake (March 11, 2011) in Japan. Such
understanding is of extreme significance especially for large or important structures.
Preface
vii
The word ‘unexpected incident’ is often used in Japan after the 2011 great
earthquake. It may be true that the return period of this class of earthquakes at the
same place could be 500–1,000 years and the use of this word may be acceptable to
some extent from the viewpoint of the balance between the construction cost and
the safety level. However, the critical excitation method is expected or has a
potential for enhancing the safety level of building structures against undesirable
incidents drawn from this irrational concept in the future. One of the most important
and challenging missions of structural engineers may be to narrow the range of such
unexpected incidents in building structural design. Redundancy, robustness, and
resilience are expected to play important roles in such circumstances.
Izuru Takewaki
Abbas Moustafa
Kohei Fujita
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Background and Review . . . . . . . . . . . . . . . . .
1.2 Input Ground Motion and Worst-Case Scenario .
1.3 Organization of the Book . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
Earthquake Resilience of High-Rise Buildings: Case Study
of the 2011 Tohoku (Japan) Earthquake . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 General Characteristics of the 2011 Off the Pacific
Coast of Tohoku Earthquake . . . . . . . . . . . . . . . . . . .
2.3 Seismic Response Simulation of Super High-Rise
Buildings in Tokyo . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1
Properties of Ground Motions in Tokyo . . . . .
2.3.2
Measure of Criticality in Long-Period
Ground Motions . . . . . . . . . . . . . . . . . . . . . .
2.3.3
Seismic Response Simulation of Super
High-Rise Buildings in Tokyo . . . . . . . . . . . .
2.4 Seismic Response of High-Rise Buildings
to Simulated Long-Period Ground Motions
(Japanese Government Action) . . . . . . . . . . . . . . . . . .
2.4.1
Characteristics of Simulated Long-Period
Ground Motions . . . . . . . . . . . . . . . . . . . . . .
2.4.2
Response Simulation of Super High-Rise
Buildings Without and With High-Hardness
Rubber Dampers. . . . . . . . . . . . . . . . . . . . . .
2.4.3
AIJ’s Research Result . . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ix
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3
4
5
Contents
Simulation of Near-Field Pulse-Like Ground Motion . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Characterization and Representation of Near-field
Pulse-Like Ground Motions . . . . . . . . . . . . . . . . . . . . . .
3.3 Representation of Near-field Pulse-Like Ground
Motions Using Deterministic Models . . . . . . . . . . . . . . .
3.4 Representation of Near-Field Pulse-Like Ground Motions
Using Probabilistic Models . . . . . . . . . . . . . . . . . . . . . .
3.5 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Critical Characterization and Modeling of Pulse-Like
Near-Field Strong Ground Motion . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Characteristics of Near-Field Pulse-Like Strong Ground
Motion From Another Viewpoint . . . . . . . . . . . . . . . .
4.2.1
Measures Based on Recorded Free-Field
Ground Motion. . . . . . . . . . . . . . . . . . . . . . .
4.2.2
Measures Based on the Structural Response. . .
4.3 Modeling Near-Field Pulse-Like Ground Motion . . . . .
4.3.1
Representation of Pulse-Like Ground Motion
Using Trigonometric Functions . . . . . . . . . . .
4.3.2
Representation of Pulse-Like Ground Motion
Using Trigonometric Functions Modulated
by Envelope Function . . . . . . . . . . . . . . . . . .
4.4 Damage-Based Critical Earthquake Ground Motion
for Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . .
4.4.1
Problem Formulation . . . . . . . . . . . . . . . . . .
4.4.2
Solution Procedures . . . . . . . . . . . . . . . . . . .
4.4.3
Illustrative Example . . . . . . . . . . . . . . . . . . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Characteristics of Earthquake Ground Motion
of Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Characteristics of Earthquake Records
of Repeated Sequences . . . . . . . . . . . . . . . . . . . .
5.3 Characteristics of Free-Field Acceleration Records
of Repeated Sequences . . . . . . . . . . . . . . . . . . . .
5.4 Response Quantities of Inelastic Structures
to Acceleration Sequences . . . . . . . . . . . . . . . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
6
7
8
Modeling Critical Ground-Motion Sequences
for Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Damage Assessment in Inelastic Structures
Using Damage Indices . . . . . . . . . . . . . . . . . . . . . . .
6.3 Modeling Critical Ground-Motion Sequences
for Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . .
6.4 Numerical Illustrations and Discussions . . . . . . . . . . .
6.4.1
Bilinear Inelastic Single-Story Frame Structure
6.4.2
Two-Story Elastic–Plastic Framed Structure . .
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Response of Nonlinear SDOF Structures to Random
Acceleration Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Characteristics of Acceleration Records
of Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Representation of Random Ground Acceleration
with Multiple Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1
Response of Elastic–Plastic Structure to Nonstationary
Random Acceleration Sequences . . . . . . . . . . . . . . .
7.4.2
Reliability of Nonlinear SDOF System
to Random Acceleration Sequences . . . . . . . . . . . . .
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use of Deterministic and Probabilistic Measures to Identify
Unfavorable Earthquake Records . . . . . . . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Use of Entropy as a Measure of Resonance/Criticality
of Probabilistic Earthquake Models. . . . . . . . . . . . . . . . .
8.2.1
Stationary Narrow-Band White Noise Model . . . .
8.2.2
Stationary Band-Limited White Noise Model. . . .
8.2.3
Stationary Kanai-Tajimi Model . . . . . . . . . . . . .
8.2.4
Nonstationary and Evolutionary PSDF Models. . .
8.2.5
Relative Entropy Rate of Two Random Processes
8.3 Dispersion Index and Central Frequency . . . . . . . . . . . . .
8.3.1
Use of Entropy Rate and Dispersion Index
to Measure Resonance of Earthquake Records . . .
8.3.2
Use of Deterministic Measures to Identify
Resonance in Earthquake Records . . . . . . . . . . .
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8.4
Identification of Resonant
of Design Records . . . . .
8.5 Summary . . . . . . . . . . . .
References . . . . . . . . . . . . . . .
9
Accelerations
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Damage Assessment of Inelastic Structures Under Worst
Earthquake Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Damage Assessment for Inelastic Structures
Under Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1
Energy Dissipated by Inelastic Structures . . .
9.2.2
Damage Measures for Inelastic Structures . . .
9.3 Formulation of the Worst Case Scenario . . . . . . . . . .
9.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1
Bilinear Inelastic Frame Structure. . . . . . . . .
9.4.2
Inelastic Two-Story Frame Structure . . . . . . .
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Critical Earthquake Loads for SDOF Inelastic Structures
Considering Evolution of Seismic Waves . . . . . . . . . . . . .
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Dynamic Analysis and Energies Dissipated
by Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . .
10.3 Quantification of Structural Damage
Using Damage Indices . . . . . . . . . . . . . . . . . . . . . .
10.4 Critical Earthquake Loads Considering Evolution
of Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Numerical Results and Discussions. . . . . . . . . . . . . .
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11 Critical Correlation of Bidirectional Horizontal
Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Penzien–Watabe Model and Extended
Penzien–Watabe Model . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Penzien–Watabe Model . . . . . . . . . . . . . . . . .
11.2.2 Extended Penzien–Watabe Model. . . . . . . . . .
11.3 Stochastic Response to 2DGM Described by Extended
Penzien–Watabe Model . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Definition of Nonstationary Ground Motion. . .
11.3.2 Stochastic Response Evaluation
in Frequency Domain . . . . . . . . . . . . . . . . . .
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11.4 Critical Excitation Method for Worst Cross PSD
Function Between 2DGM . . . . . . . . . . . . . . . . . . . . . . .
11.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Response to 2DGM with the Constraint of Sum
of Auto PSD Functions . . . . . . . . . . . . . . . . . . .
11.5.2 Response to 2DGM Described by Extended
Penzien–Watabe Model: Analysis From
the Viewpoint of Critical Incident Angle . . . . . . .
11.5.3 Comparison of Response to Critically Correlated
2DGM with that to Perfectly Correlated 2DGM . .
11.5.4 Analysis of Recorded 2DGM . . . . . . . . . . . . . . .
11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1: Computation of Coherence Function
and Transformation of PSD Matrices . . . . . . . . . . . . . . . . . . . .
Appendix 2: Horizontal Stiffness of Frame . . . . . . . . . . . . . . . .
Appendix 3: Stochastic Response 1 . . . . . . . . . . . . . . . . . . . . .
Appendix 4: Stochastic Response 2 . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Optimal Placement of Visco-Elastic Dampers and Supporting
Members Under Variable Critical Excitations . . . . . . . . . . . .
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Structural Model with Visco-Elastic Dampers
and their Supporting Members . . . . . . . . . . . . . . . . . . . .
12.3 Critical Excitation for Variable Design . . . . . . . . . . . . . .
12.4 Stochastic Response Evaluation in Frequency Domain . . .
12.4.1 3N Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2 N Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Optimal Design Problem . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
12.7 Algorithm for Optimal Damper Placement. . . . . . . . . . . .
12.7.1 Algorithm for Optimal Damper Placement
and Optimal Design of Supporting Members . . . .
12.7.2 Sensitivity with Respect to Damper Area . . . . . .
12.7.3 Sensitivity with Respect to Stiffness
of Supporting Member . . . . . . . . . . . . . . . . . . .
12.7.4 Sensitivities of Axial Force
of Supporting Member . . . . . . . . . . . . . . . . . . .
12.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 1: Equivalent Stiffness and Damping Coefficient
of Damper Unit Including Supporting Member
in N-Model (Eqs. (12.1) and (12.2)). . . . . . . . . . . . . . . . . . . . .
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Contents
Appendix 2: Transformation Matrix from the Nodal
Displacements to the Relative Displacements Between
both Ends of Supporting Members. . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 3: Second-Order Sensitivities of the Equivalent
Stiffness and Damping Coefficient. . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Earthquake Response Bound Analysis of Uncertain Passively
Controlled Buildings for Robustness Evaluation . . . . . . . . . . . . .
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Concept of Sustainable Building Design Under Uncertain
Structural-Parameter Environment . . . . . . . . . . . . . . . . . . . .
13.3 Interval Analysis Methods for Uncertain
Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Interval Analysis Method Based on Approximation
of First-Order Taylor Series Expansion . . . . . . . . . . .
13.3.2 Interval Analysis Method Based on Approximation
of Second-Order Taylor Series Expansion . . . . . . . . .
13.4 Advanced Interval Analysis Method Based on the Information
of the Approximation of Taylor Series Expansion . . . . . . . . .
13.4.1 Reanalysis Approach Based on the Structural
Parameter Set Derived by the Taylor
Series Approximation . . . . . . . . . . . . . . . . . . . . . . .
13.4.2 Varied Evaluation Point Method Considering
the Influence of Initial Value Dependency. . . . . . . . .
13.4.3 Search of the Exact Solution . . . . . . . . . . . . . . . . . .
13.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Earthquake Response Bound Analysis of Uncertain Base-Isolated
Buildings for Robustness Evaluation . . . . . . . . . . . . . . . . . . . . . .
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Modeling of Base-Isolated Buildings and Uncertainty
of Isolators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Past Work on Interval Analysis for Uncertain Input
and Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Interval Analysis Using Taylor Series Expansion . . . . . . . . . .
14.4.1 Interval Analysis Using First-Order
Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4.2 Interval Analysis Using Second-Order
Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4.3 Interval Analysis Considering Non-Monotonic Property
of Objective Function . . . . . . . . . . . . . . . . . . . . . . .
273
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Contents
14.5 Numerical Verification of URP Method . . . .
14.5.1 Property of Base-Isolated Building . .
14.5.2 Input Ground Motions. . . . . . . . . . .
14.5.3 Interval Analysis for Interstory Drift
of Base-Isolation Story . . . . . . . . . .
14.5.4 Interval Analysis for Top-Story
Maximum Acceleration . . . . . . . . . .
14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Earthquake Resilience . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Improving Earthquake Resilience Based on Redundancy
and Robustness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Resonant Response and Random Response . . . . . . . . . .
15.4 Robustness Function for Seismic Performance . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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日本語解説
「極限外乱法」は、米国レンセラー工科大学のDrenick教授およ
びコロンビア大学のShinozuka教授によりそれぞれ独立して1970
年頃に始められた分野であり、「最悪外乱法」とも呼ばれている。
2011.3.11の東北地方太平洋沖地震では、短時間の時間差を伴
う複数断層運動による複数大振幅波形の存在、予想を遥かに
超える津波の発生、原子力施設事故、天井などの２次部材被害、
長周期地震動による超高層建物の長時間大振幅揺れ等、いわ
ゆる想定外の事象が多数発生した。大地震の発生前にすべて
の事象を想定して建物を設計することは極めて困難であり、起こ
り得る事象を可能な限り想定してその中で最悪のケースに対し
て設計することを目指す本手法は信頼性の高い方法であるとい
える。これまで、多数の被害地震を経験してその度に耐震規準
が改訂されているが、今なおこの手順が収束していない状況で
は、このような最悪ケースを想定する方法は被害軽減の上で有
力なアプローチであるとともに、建物レジリエンスの飛躍的な向
上が期待される。