Improving the Earthquake Resilience of Buildings: The worst case … · 2017. 3. 28. · Engineers are always interested in the worst-case scenario. One of the most important and

Improving the Earthquake Resilience of Buildings:

The worst case approachIzuru Takewaki

Professor, Kyoto UniversityAbbas Moustafa

Associate Professor, Minia University, Egypt

Kohei FujitaAssistant Professor, Kyoto University

AbstractEngineers 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 approachdiscusses 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 long-period ground motions.

This book is a valuable resource for researchers and engineers interested in learning and applying the worst-case scenario approach in the seismic-resistant design for more resilient structures.

▶ 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 worst-scenario approach to provide engineers with effective design techniques

Highlights

Chapter 1

1.1 Background and review

1.2 Input ground motion and worst-case scenario

1.3 Organization of the book

Fig.2.9 Schematic diagram for computing credible boundsfor acceleration and velocity constraints

Chapter 2 What is an important measure for characterizing earthquake ground motion?acceleration or velocity?

Fig.2.15(a) Ground acceleration, ground velocity and top-story displacement of a super high-rise buildingin Osaka during the 2011 off the Pacific coastof Tohoku earthquake

Chapter 2

Resonance in reality

Resonance can occur?

without dampers with dampers

Fig.2.26 Plastic-hinge formation in 40-story buildings of T1=3.6(s) without and with high-hardness rubber dampers

Chapter 2

Improving building resilience by structural control

Approach to improving building earthquake resilience

Chapter 3Simulation of Near-Field

Pulse-Like Ground Motion

C02065 TAB-TR RRS228

SCS052 TAK000 KJM000 YPT060

TCU068N TCU068W ALS-E TCU078W

LGP000 LCN275 WSM090 JOS090

H-E06230

C02065 ERZ-NS DZC180 PCD164

(a)

C02065TAB-TR RRS228H-E06230

SCS052 TAK000 KJM000 YPT060

TCU068N TCU068W ALS-E TCU078W

LGP000 LCN275 WSM090 JOS090

CPM000 ERZ-NS DZC180 PCD164

(b)

Fig. 3.1 Near-field strong ground motion with distinct velocity pulses: (a) velocity time history, (b) velocity Fourier amplitude spectra

0 5 10 15 20-1

-0.5

0

0.5

1

Time (s)

Vel

ocity

(m/s

)

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)V

eloc

ity a

mpl

itude

spe

ctra

(m)

0 5 10 15 20-1

-0.5

0

0.5

1

Time (s)

Vel

ocity

(m/s

)

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Vel

ocity

am

plitu

de s

pect

ra (m

)

0 5 10 15 20-1

-0.5

0

0.5

1

1.5

Time (s)

Vel

ocity

(m/s

)

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Vel

ocity

am

plitu

de s

pect

ra (m

)(a)

(b)

(c)

Fig. 3.4 Ground velocity and associated Fourier amplitude spectra

0 5 10 15 20 25 30-2

-1

0

1

2

Time s

Nor

mal

ized

dis

plac

emen

t

0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

Time sEn

ergy

N m

Input energyHysteretic energyDamping energy

0 5 10 15 20 25 30-0.5

0

0.5

Time s

Acc

eler

atio

n / g

-0.2 -0.1 0 0.1 0.2-1.5

-1

-0.5

0

0.5

1

1.5 x 104

Displacement m

Hys

tere

tic fo

rce

NA

ccel

erat

ion

/ g

Nor

mal

ized

dis

plac

emen

t

Hys

tere

tic fo

rce

N

Ener

gy N

m

Fig. 4.8: Critical earthquake accelerations and associated critical responses for case 4

Chapter 4 Critical excitation for inelastic response

Chapter 5Characteristics of

Earthquake Ground Motion of Repeated

Sequences

Time (s)

0 200 400 600-1

-0.5

0

0.5

1

0 200 400 600-1

-0.5

0

0.5

1

EW

UD

0 100 200 300-0.2

-0.1

0

0.1

0.2

0 100 200 300-0.1

-0.05

0

0.05

0.1

Time (s)

0 100 200 300-0.2

-0.1

0

0.1

0.2

NS

EW

UD

0 200 400 600-1

-0.5

0

0.5

1(a) (b)

NSAcc

eler

atio

n / g

Acc

eler

atio

n / g

Acc

eler

atio

n / g

Fig 5.1: Ground accelerations with multiple sequences (a) 2004 Niigata earthquake at Nagaoka-shisho (NIG028) (b) 2004 Niigata earthquake at Koide (NIG020).

Time (s)0 50 100

-0.5

0

0.5

Acc

eler

atio

n / g

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Four

ier s

pect

ra m

/s

-5 0 5

-2

-1

0

1

2

x 104

Normalized displacement

Hys

tere

tic fo

rce

(N)

0 20 40 60 80 1000

2

4

6

8

10 x 104

Yield energyDamping energy

Time (s)

Ene

rgy

(Nm

)

Frequency (Hz)

Fig. 6.5: Critical ground acceleration of two sequences and associated response for bilinear inelastic structure

Chapter 6 Critical excitation for multiple sequences

Chapter 7Response of Nonlinear SDOF Structures

to Random Acceleration Sequences

Time (s) Time (s)

Mea

n of

dis

plac

emen

t (m

)

Var

ianc

eof

inpu

tene

rgy

(Nm

)V

aria

nce

ofdi

spla

cem

ent(

m2 )

0 20 40 60 80 1000

1

2

3 x 10-3

0 20 40 60 80 100-0.01

-0.005

0

0.005

0.01

0 20 40 60 80 1000

5000

10000

15000

Mea

n of

inpu

t ene

rgy

(Nm

)

0 20 40 60 80 1000

2

4

6

8

10 x 106

Fig. 7.6: Mean of response quantities for elastic-plastic structure to ground acceleration of three sequences

Chapter 8Use of Deterministic and Probabilistic

Measures to Identify Unfavorable Earthquake Records

(a) (b)

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

Frequency Hz

S(ω

) m

2 /s3

Narrow bandKanai-TajimiBand-limitedArea = 1.0

Δω = 0.10 rad/s Smax = 10 m2/s3

Area = 1.0

Area = 1.0

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

Frequency Hz

S(ω

) m

2 /s3

Soft soilMedium soilStiff soilRock soil

Area = 1.0

Area = 1.0

Area = 1.0 Area = 1.0

(a) (b)

Fig. 8.2: (a) PSD function for ground acceleration models for medium soil, (b) Kanai-Tajimi PSD function for different soil types

Chapter 9Damage Assessment of Inelastic

Structures Under Worst Earthquake Loads

(b)0 5 10 15 20 25 30-0.5

0

0.5

Frequency Hz

Acc

eler

atio

n / g

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

Frequency HzFo

urie

r am

plitu

de (m

/s)

Critical inputM4(ω)

Time (s)

Acc

eler

atio

n/g

Am

plitu

desp

ectra

(m/s

)Frequency (Hz)

(a) (b)

Fig. 9.11: Critical acceleration for inelastic structure for case 2 (a) Time history, (b) Fourier amplitude spectrum

-4-2024

0 5 10 15 20 25 30 35 40

Acce

lerati

on (

m/s

2 )

Time (s)

-4-2024

0 5 10 15 20 25 30 35 40

Acce

lerati

on (

m/s2

)

Time (s)

Body waves Surface waves

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

Body wavesSurface waves

Enve

lope f

uncti

onTime (s)

Frequency content:1.0～10.0Hz

Frequency content:0.0～1.0Hz

Chapter 10Formulation of Evolutionary Waves

What is the critical

combination?

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

1x

2x

1L2LChapter 11

What is the bi-directional critical input?

Chapter 12Optimal Placement of Visco-Elastic Dampers and Supporting Members Under Variable Critical Excitations

Fig.12.9: Variation of objective function in optimal placement, uniform placement and first-storey

placement; (a) 3-storey model, (b) 10-storey model

Fig. 12.1: Structural model with viscoelastic damper including supporting member

Fig.13.1 Concept of sustainable design considering varied structural performance caused by various uncertainties of structural parameters

Chapter 13

Sustainability under

uncertainties

Chapter 14

Earthquake Response Bound Analysis of Uncertain Base-IsolatedBuildings for Robustness Evaluation

Fig.14.3 Dependence of critical combination of uncertain parameters on properties of objective

function, (a) Inclusion monotonic, (b) non-monotonic

Fig.14.1 N-story base-isolated building model

Chapter 15Future 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 ofbuildings should ensure structural safety against the worst possible future earth-quakes. The features of this monograph are:

(1) Consideration of elastic–plastic behavior of building structures in the criticalexcitation method for improved building-earthquake resilience,

(2) Consideration of uncertainties of structural parameters in structural controland base-isolation for improved building-earthquake resilience, and

(3) New insights into structural design of super high-rise buildings under long-period ground motions (case study on tall buildings in mega cities in Japanduring the 2011 off the Pacific coast of Tohoku earthquake on March 11).

This book consists of two parts. The first part deals with the characterizationand modeling of worst or critical ground motions on inelastic structures. Thesecond part of the book focuses on investigating the worst-case scenario forpassively controlled and base-isolated buildings.

Chapter 1 provides an overview of the effects of historic and recent strongearthquake ground motions on building structures and associated life loss.

Chapter 2 provides comprehensive information about the most recent anddevastating Tohoku earthquake of moment magnitude 9.0 which hit off the pacificcoast of eastern Japan on 11 March 2011. This earthquake and the tsunami fol-lowing it left severe damage to building structures and caused nearly 20,000 oflosses of life.

As is well known, the robust design of buildings for future earthquake loadsrequires reliable understanding of the ground motion characteristics. Accordingly,Chaps. 3 and 4 report on the characteristics of near-field (near-fault) groundmotions with pulse-like acceleration. Furthermore, these two chapters providesimple mathematical models for this class of ground motions and associatedstructural response. Chapter 3 deals with the simulation of near-field groundmotions with pulse-like acceleration while a critical excitation of multiplesequences for inelastic responses is discussed in Chap. 4.

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Chapters 5–7 deal with the characterization and modeling of earthquake groundmotion of multiple sequences. Recently, this class of ground motions was clearlyobserved during the 2011 off the Pacific coast of Tohoku earthquake on March 11.This research subject is new and has not received adequate attention fromresearchers. For instance, most seismic codes specify design ground motions assingle events. However, moderate ground motion with repeated accelerationsequences could lead to more severe damage to structures than a single sequenceof strong ground motion. The worst-case scenario is studied within the determi-nistic and probabilistic frameworks. Characteristics of earthquake ground motionof repeated sequences are made clear in Chap. 5 while critical ground motionsequences are discussed in Chap. 6. In Chap. 7, responses of elastic–plasticstructures to nonstationary random acceleration sequences are investigated and thereliability of such structures is evaluated.

A practical problem always arises in the design of buildings against earthquakeloads. It is always difficult to select a suite of suitable earthquake records from alarge set of records as input to the nonlinear time-history analysis of structures.Chapter 8 provides deterministic and probabilistic measures that can be used toidentify unfavorable accelerograms. This chapter provides simple concepts whichcan be utilized to select a suit of appropriate earthquake records for nonlinear time-history analysis of structures.

Chapters 9 and 10 deal with the worst-scenario of earthquake loads on inelasticstructures with special emphasis on the type of seismic waves of the groundmotion 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 ofbuilding response under uni-directional earthquake input. However, bidirectionalinput is inevitable for the reliable design of columns.

Chapters 12 and 13 tackle the worst-case scenario for passively controlledbuildings. The structural member stiffness and strength of buildings are uncertaindue to various factors resulting from randomness, material deterioration, temper-ature dependence, etc. The passive damper systems are also uncertain dependingon various sources. The concept of sustainable building design under suchuncertain structural-parameter environment may be one of the most challengingissues to be tackled recently. By predicting the response variability accurately, theelongation of service life of buildings may be possible.

Chapter 14 focuses on the worst-case scenario for base-isolated buildings. Thestiffness and damping of the base-isolation system and the stiffness of the super-structure are selected as uncertain parameters. An efficient methodology is explained toevaluate the robustness (variability of response) of an uncertain base-isolated building.

The book closes with Chap. 15 on current challenges and future directions ondesign of building structures with greater earthquake resilience.

The importance of the worst-scenario approach for improved earthquake resil-ience of buildings and nuclear reactor facilities has been recognized and demon-strated by the recent great earthquake (March 11, 2011) in Japan. Suchunderstanding is of extreme significance especially for large or important structures.

vi Preface

http://dx.doi.org/10.1007/978-1-4471-4144-0_5http://dx.doi.org/10.1007/978-1-4471-4144-0_7http://dx.doi.org/10.1007/978-1-4471-4144-0_5http://dx.doi.org/10.1007/978-1-4471-4144-0_7http://dx.doi.org/10.1007/978-1-4471-4144-0_7http://dx.doi.org/10.1007/978-1-4471-4144-0_8http://dx.doi.org/10.1007/978-1-4471-4144-0_9http://dx.doi.org/10.1007/978-1-4471-4144-0_10http://dx.doi.org/10.1007/978-1-4471-4144-0_11http://dx.doi.org/10.1007/978-1-4471-4144-0_12http://dx.doi.org/10.1007/978-1-4471-4144-0_13http://dx.doi.org/10.1007/978-1-4471-4144-0_14http://dx.doi.org/10.1007/978-1-4471-4144-0_15

The word ‘unexpected incident’ is often used in Japan after the 2011 greatearthquake. It may be true that the return period of this class of earthquakes at thesame place could be 500–1,000 years and the use of this word may be acceptable tosome extent from the viewpoint of the balance between the construction cost andthe safety level. However, the critical excitation method is expected or has apotential for enhancing the safety level of building structures against undesirableincidents drawn from this irrational concept in the future. One of the most importantand challenging missions of structural engineers may be to narrow the range of suchunexpected incidents in building structural design. Redundancy, robustness, andresilience are expected to play important roles in such circumstances.

Izuru TakewakiAbbas Moustafa

Kohei Fujita

Preface vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Input Ground Motion and Worst-Case Scenario . . . . . . . . . . . 31.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . 3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Earthquake Resilience of High-Rise Buildings: Case Studyof the 2011 Tohoku (Japan) Earthquake . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 General Characteristics of the 2011 Off the Pacific

Coast of Tohoku Earthquake . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Seismic Response Simulation of Super High-Rise

Buildings in Tokyo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Properties of Ground Motions in Tokyo . . . . . . . . . . 132.3.2 Measure of Criticality in Long-Period

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Seismic Response Simulation of Super

High-Rise Buildings in Tokyo . . . . . . . . . . . . . . . . . 192.4 Seismic Response of High-Rise Buildings

to Simulated Long-Period Ground Motions(Japanese Government Action) . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Characteristics of Simulated Long-Period

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Response Simulation of Super High-Rise

Buildings Without and With High-HardnessRubber Dampers. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 AIJ’s Research Result . . . . . . . . . . . . . . . . . . . . . . . 382.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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3 Simulation of Near-Field Pulse-Like Ground Motion . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Characterization and Representation of Near-field

Pulse-Like Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Representation of Near-field Pulse-Like Ground

Motions Using Deterministic Models . . . . . . . . . . . . . . . . . . 523.4 Representation of Near-Field Pulse-Like Ground Motions

Using Probabilistic Models . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Numerical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Critical Characterization and Modeling of Pulse-LikeNear-Field Strong Ground Motion . . . . . . . . . . . . . . . . . . . . . . . 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Characteristics of Near-Field Pulse-Like Strong Ground

Motion From Another Viewpoint . . . . . . . . . . . . . . . . . . . . . 704.2.1 Measures Based on Recorded Free-Field

Ground Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.2 Measures Based on the Structural Response. . . . . . . . 77

4.3 Modeling Near-Field Pulse-Like Ground Motion . . . . . . . . . . 794.3.1 Representation of Pulse-Like Ground Motion

Using Trigonometric Functions . . . . . . . . . . . . . . . . 794.3.2 Representation of Pulse-Like Ground Motion

Using Trigonometric Functions Modulatedby Envelope Function . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Damage-Based Critical Earthquake Ground Motionfor Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 834.4.2 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . 854.4.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Characteristics of Earthquake Ground Motionof Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Characteristics of Earthquake Records

of Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Characteristics of Free-Field Acceleration Records

of Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Response Quantities of Inelastic Structures

to Acceleration Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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6 Modeling Critical Ground-Motion Sequencesfor Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Damage Assessment in Inelastic Structures

Using Damage Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3 Modeling Critical Ground-Motion Sequences

for Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4 Numerical Illustrations and Discussions . . . . . . . . . . . . . . . . 124

6.4.1 Bilinear Inelastic Single-Story Frame Structure . . . . . 1246.4.2 Two-Story Elastic–Plastic Framed Structure . . . . . . . 129


7 Response of Nonlinear SDOF Structures to RandomAcceleration Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.2 Characteristics of Acceleration Records

of Repeated Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.3 Representation of Random Ground Acceleration

with Multiple Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.4.1 Response of Elastic–Plastic Structure to NonstationaryRandom Acceleration Sequences . . . . . . . . . . . . . . . 143

7.4.2 Reliability of Nonlinear SDOF Systemto Random Acceleration Sequences . . . . . . . . . . . . . 144


8 Use of Deterministic and Probabilistic Measures to IdentifyUnfavorable Earthquake Records . . . . . . . . . . . . . . . . . . . . . . . . 1518.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 Use of Entropy as a Measure of Resonance/Criticality

of Probabilistic Earthquake Models. . . . . . . . . . . . . . . . . . . . 1528.2.1 Stationary Narrow-Band White Noise Model . . . . . . . 1538.2.2 Stationary Band-Limited White Noise Model. . . . . . . 1548.2.3 Stationary Kanai-Tajimi Model . . . . . . . . . . . . . . . . 1548.2.4 Nonstationary and Evolutionary PSDF Models. . . . . . 1548.2.5 Relative Entropy Rate of Two Random Processes . . . 155

8.3 Dispersion Index and Central Frequency . . . . . . . . . . . . . . . . 1588.3.1 Use of Entropy Rate and Dispersion Index

to Measure Resonance of Earthquake Records . . . . . . 1598.3.2 Use of Deterministic Measures to Identify

Resonance in Earthquake Records . . . . . . . . . . . . . . 160

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8.4 Identification of Resonant Accelerations and Selectionof Design Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162


9 Damage Assessment of Inelastic Structures Under WorstEarthquake Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.2 Damage Assessment for Inelastic Structures

Under Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1799.2.1 Energy Dissipated by Inelastic Structures . . . . . . . . . 1809.2.2 Damage Measures for Inelastic Structures . . . . . . . . . 181

9.3 Formulation of the Worst Case Scenario . . . . . . . . . . . . . . . . 1839.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.4.1 Bilinear Inelastic Frame Structure. . . . . . . . . . . . . . . 1879.4.2 Inelastic Two-Story Frame Structure . . . . . . . . . . . . . 194


10 Critical Earthquake Loads for SDOF Inelastic StructuresConsidering Evolution of Seismic Waves . . . . . . . . . . . . . . . . . . . 20310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.2 Dynamic Analysis and Energies Dissipated

by Inelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20510.3 Quantification of Structural Damage

Using Damage Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20610.4 Critical Earthquake Loads Considering Evolution

of Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.5 Numerical Results and Discussions. . . . . . . . . . . . . . . . . . . . 20910.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

11 Critical Correlation of Bidirectional HorizontalGround Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22111.2 Penzien–Watabe Model and Extended

Penzien–Watabe Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22211.2.1 Penzien–Watabe Model . . . . . . . . . . . . . . . . . . . . . . 22211.2.2 Extended Penzien–Watabe Model. . . . . . . . . . . . . . . 223

11.3 Stochastic Response to 2DGM Described by ExtendedPenzien–Watabe Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22511.3.1 Definition of Nonstationary Ground Motion. . . . . . . . 22511.3.2 Stochastic Response Evaluation

in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . 226

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11.4 Critical Excitation Method for Worst Cross PSDFunction Between 2DGM . . . . . . . . . . . . . . . . . . . . . . . . . . 228

11.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22911.5.1 Response to 2DGM with the Constraint of Sum

of Auto PSD Functions . . . . . . . . . . . . . . . . . . . . . . 22911.5.2 Response to 2DGM Described by Extended

Penzien–Watabe Model: Analysis Fromthe Viewpoint of Critical Incident Angle . . . . . . . . . . 234

11.5.3 Comparison of Response to Critically Correlated2DGM with that to Perfectly Correlated 2DGM . . . . . 237

11.5.4 Analysis of Recorded 2DGM . . . . . . . . . . . . . . . . . . 23811.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Appendix 1: Computation of Coherence Functionand Transformation of PSD Matrices . . . . . . . . . . . . . . . . . . . . . . . 242Appendix 2: Horizontal Stiffness of Frame . . . . . . . . . . . . . . . . . . . 243Appendix 3: Stochastic Response 1 . . . . . . . . . . . . . . . . . . . . . . . . 243Appendix 4: Stochastic Response 2 . . . . . . . . . . . . . . . . . . . . . . . . 245References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

12 Optimal Placement of Visco-Elastic Dampers and SupportingMembers Under Variable Critical Excitations . . . . . . . . . . . . . . . 24912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24912.2 Structural Model with Visco-Elastic Dampers

and their Supporting Members . . . . . . . . . . . . . . . . . . . . . . . 25012.3 Critical Excitation for Variable Design . . . . . . . . . . . . . . . . . 25312.4 Stochastic Response Evaluation in Frequency Domain . . . . . . 254

12.4.1 3N Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25412.4.2 N Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

12.5 Optimal Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 25612.6 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25712.7 Algorithm for Optimal Damper Placement. . . . . . . . . . . . . . . 258

12.7.1 Algorithm for Optimal Damper Placementand Optimal Design of Supporting Members . . . . . . . 258

12.7.2 Sensitivity with Respect to Damper Area . . . . . . . . . 26412.7.3 Sensitivity with Respect to Stiffness

of Supporting Member . . . . . . . . . . . . . . . . . . . . . . 26612.7.4 Sensitivities of Axial Force

of Supporting Member . . . . . . . . . . . . . . . . . . . . . . 26612.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26712.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Appendix 1: Equivalent Stiffness and Damping Coefficientof Damper Unit Including Supporting Memberin N-Model (Eqs. (12.1) and (12.2)). . . . . . . . . . . . . . . . . . . . . . . . 270

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Appendix 2: Transformation Matrix from the NodalDisplacements to the Relative Displacements Betweenboth Ends of Supporting Members. . . . . . . . . . . . . . . . . . . . . . . . . 273Appendix 3: Second-Order Sensitivities of the EquivalentStiffness and Damping Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . 273References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

13 Earthquake Response Bound Analysis of Uncertain PassivelyControlled Buildings for Robustness Evaluation . . . . . . . . . . . . . 27713.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27713.2 Concept of Sustainable Building Design Under Uncertain

Structural-Parameter Environment . . . . . . . . . . . . . . . . . . . . 27813.3 Interval Analysis Methods for Uncertain

Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27813.3.1 Interval Analysis Method Based on Approximation

of First-Order Taylor Series Expansion . . . . . . . . . . . 28013.3.2 Interval Analysis Method Based on Approximation

of Second-Order Taylor Series Expansion . . . . . . . . . 28113.4 Advanced Interval Analysis Method Based on the Information

of the Approximation of Taylor Series Expansion . . . . . . . . . 28313.4.1 Reanalysis Approach Based on the Structural

Parameter Set Derived by the TaylorSeries Approximation . . . . . . . . . . . . . . . . . . . . . . . 283

13.4.2 Varied Evaluation Point Method Consideringthe Influence of Initial Value Dependency. . . . . . . . . 284

13.4.3 Search of the Exact Solution . . . . . . . . . . . . . . . . . . 28413.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28513.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

14 Earthquake Response Bound Analysis of Uncertain Base-IsolatedBuildings for Robustness Evaluation . . . . . . . . . . . . . . . . . . . . . . 29514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29514.2 Modeling of Base-Isolated Buildings and Uncertainty

of Isolators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29614.3 Past Work on Interval Analysis for Uncertain Input

and Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 29614.4 Interval Analysis Using Taylor Series Expansion . . . . . . . . . . 298

14.4.1 Interval Analysis Using First-OrderTaylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 299

14.4.2 Interval Analysis Using Second-OrderTaylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 299

14.4.3 Interval Analysis Considering Non-Monotonic Propertyof Objective Function . . . . . . . . . . . . . . . . . . . . . . . 300

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14.5 Numerical Verification of URP Method . . . . . . . . . . . . . . . . 30214.5.1 Property of Base-Isolated Building . . . . . . . . . . . . . . 30314.5.2 Input Ground Motions. . . . . . . . . . . . . . . . . . . . . . . 30414.5.3 Interval Analysis for Interstory Drift

of Base-Isolation Story . . . . . . . . . . . . . . . . . . . . . . 30414.5.4 Interval Analysis for Top-Story

Maximum Acceleration . . . . . . . . . . . . . . . . . . . . . . 30914.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

15 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31315.1 Earthquake Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31315.2 Improving Earthquake Resilience Based on Redundancy

and Robustness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31415.3 Resonant Response and Random Response . . . . . . . . . . . . . . 31415.4 Robustness Function for Seismic Performance . . . . . . . . . . . . 316References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

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「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。

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Improving the Earthquake Resilience of Buildings: The worst case … · 2017. 3. 28. · Engineers are always interested in the worst-case scenario. One of the most important and