Seismic Risk Analysis of Nuclear Power Plants 9781139629010

Table of contents :
Contents......Page 6
Preface......Page 10
Nomenclature......Page 14
1.1 Introduction to Types of Nuclear Power Plants......Page 20
1.2 Important Structures, Systems, and Components in Nuclear Power Plants......Page 24
1.3 Seismic Design Philosophy and Requirements......Page 26
1.4 Seismic Risk Analysis of Nuclear Power Plants......Page 32
2.1 Earthquakes......Page 36
2.2 Case Study − The Great East Japan Earthquake......Page 51
2.3 Strong Ground Motion......Page 56
2.4 Probabilistic Ground-Motion Parameters......Page 62
3.1 Random Processes......Page 76
3.2 Properties of Random Processes......Page 84
3.3 Single Degree-of-Freedom System......Page 96
3.4 Multiple Degrees-of-Freedom Systems......Page 103
3.5 Stationary Response to Random Excitation......Page 105
3.6 Seismic Response Analysis......Page 114
3.7 Nonlinear Systems......Page 124
3.8 Appendix − Method of Residue......Page 127
4.1 Ground Response Spectra......Page 134
4.2 t-Response Spectrum......Page 156
4.3 Appendix......Page 168
5.1 Deterministic Seismic Hazard Analysis (DSHA)......Page 172
5.2 Probabilistic Seismic Hazard Analysis......Page 173
5.3 Seismic Hazard Deaggregation......Page 186
5.4 Treatment of Epistemic Uncertainty......Page 193
5.5 Seismic Design Spectra Based on PSHA......Page 195
5.6 Site Response Analysis......Page 200
6.1 Generating Ground Motions for Seismic Analysis......Page 249
6.2 Spectral Matching Algorithms for Artificial Ground Motions......Page 254
6.3 Spectral Matching Algorithms Based on Recorded Ground Motions......Page 258
6.4 Generating Drift-Free and Consistent Time-Histories Using Eigenfunctions......Page 279
7.1 Introduction......Page 302
7.2 Structural Modelling......Page 303
7.3 Numerical Example......Page 313
8.1 Introduction......Page 347
8.2 Floor Response Spectra......Page 351
8.3 Time-History Method for Generating FRS......Page 354
8.4 Direct Method for Generating FRS......Page 358
8.5 Scaling Method for Generating FRS......Page 384
8.6 Generating FRS Considering SSI......Page 402
9.1 Seismic Fragility......Page 429
9.2 HCLPF Capacity......Page 435
9.3 Methodology of Fragility Analysis......Page 437
9.4 Conservative Deterministic Failure Margin (CDFM) Method......Page 460
9.5 Case Study − Horizontal Heat Exchanger......Page 465
9.6 Masonry Block Wall......Page 502
10.1 Introduction......Page 536
10.2 System Analysis......Page 539
10.3 Seismic Risk Quantification......Page 555
10.4 Seismic Margin Assessment......Page 557
10.5 Seismic Probabilistic Safety Assessment with Screening Tables......Page 565
10.6 Hybrid Method for Seismic Risk Assessment......Page 566
10.7 Estimation of Seismic Risk from HCLPF Capacity......Page 570
10.8 Numerical Examples − ECI System......Page 571
a.1 Normal Distribution......Page 581
a.2 Lognormal Distribution......Page 583
b.1 Sampling......Page 587
b.2 Fourier Series and Fourier Transforms......Page 591
b.3 Digital Signal Processing......Page 599
b.4 Digital Filters......Page 603
b.5 Resampling......Page 605
b.6 Numerical Example − Gaussian White Noise......Page 608
Bibliography......Page 614
Index......Page 628

Citation preview

Seismic Risk Analysis of Nuclear Power Plants

Seismic Risk Analysis of Nuclear Power Plants addresses the needs of graduate students in engineering, practicing engineers in industry, and regulators in government agencies, presenting the entire process of seismic risk analysis in a clear, logical, and concise manner. It offers a systematic and comprehensive introduction to seismic risk analysis of critical engineering structures focusing on nuclear power plants, with a balance between theory and applications, and includes the latest advances in research. It is suitable as a graduate-level textbook, for self-study, or as a reference book. Various aspects of seismic risk analysis, from seismic hazard, demand, and fragility analyses to seismic risk quantification, are discussed, with detailed step-by-step analysis of specific engineering examples. It presents a wide range of topics essential for understanding and performing seismic risk analysis, including engineering seismology, probability theory and random processes, digital signal processing, structural dynamics, random vibration, and engineering risk and reliability.

Seismic Risk Analysis of Nuclear Power Plants WEI-CHAU XIE University of Waterloo

SHUN-HAO NI Candu Energy Inc.

WEI LIU Candu Energy Inc.

WEI JIANG Candu Energy Inc.

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107040465 DOI: 10.1017/9781139629010 © Wei-Chau Xie, Shun-Hao Ni, Wei Liu, and Wei Jiang 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Xie, Wei-Chau, 1964– author. Title: Seismic risk analysis of nuclear power plants / Wei-Chau Xie (University of Waterloo) [and three others]. Description: Cambridge ; New York, NY : Cambridge University Press, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018026579| ISBN 9781107040465 (hardback : alk. paper) | ISBN 9781139629010 (alk. paper) Subjects: LCSH: Nuclear power plants–Earthquake effects. | Nuclear power plants–Risk assessment. | Nuclear power plants–Safety measures. | Earthquake hazard analysis. Classification: LCC TK1078 .S36275 2018 | DDC 621.48/35–dc23 LC record available at https://lccn.loc.gov/2018026579 ISBN 978-1-107-04046-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

11.1

Introduction to Types of Nuclear Power Plants

1

1.2

Important Structures, Systems, and Components in Nuclear Power Plants

5

1.3

Seismic Design Philosophy and Requirements

7

1.4 Seismic Risk Analysis of Nuclear Power Plants

13

22.1

17

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Engineering Seismology . . . . . . . . . . . . . . . . . . . . . . . Earthquakes

1

17

2.2 Case Study − The Great East Japan Earthquake

32

2.3 Strong Ground Motion

37

2.4 Probabilistic Ground-Motion Parameters

43

33.1

57

Basics of Random Processes and Structural Dynamics . . . . . . Random Processes

57

3.2 Properties of Random Processes

65

3.3 Single Degree-of-Freedom System

77

3.4 Multiple Degrees-of-Freedom Systems

84

3.5 Stationary Response to Random Excitation

86

3.6 Seismic Response Analysis

95

3.7 Nonlinear Systems

105

3.8 Appendix − Method of Residue

108

44.1

115

Seismic Response Spectra . . . . . . . . . . . . . . . . . . . . . . Ground Response Spectra

115

4.2 t-Response Spectrum

137

4.3 Appendix

149

55.1

Seismic Hazard Analysis . . . . . . . . . . . . . . . . . . . . . . . Deterministic Seismic Hazard Analysis (DSHA)

153 153

5.2 Probabilistic Seismic Hazard Analysis

154

5.3 Seismic Hazard Deaggregation

167

5.4 Treatment of Epistemic Uncertainty

174

5.5

Seismic Design Spectra Based on PSHA

176

5.6 Site Response Analysis

181

66.1

230

Ground Motions for Seismic Analysis . . . . . . . . . . . . . . . 230 Generating Ground Motions for Seismic Analysis

6.2 Spectral Matching Algorithms for Artificial Ground Motions

235

6.3 Spectral Matching Algorithms Based on Recorded Ground Motions

239

6.4 Generating Drift-Free and Consistent Time-Histories Using Eigenfunctions

260

77.1

Introduction

283

7.2

Structural Modelling

284

7.3

Numerical Example

294

88.1

Modelling of Structures . . . . . . . . . . . . . . . . . . . . . . . 283

Floor Response Spectra . . . . . . . . . . . . . . . . . . . . . . . 328 Introduction

328

8.2 Floor Response Spectra

332

8.3 Time-History Method for Generating FRS

335

8.4 Direct Method for Generating FRS

339

8.5 Scaling Method for Generating FRS

365

8.6 Generating FRS Considering SSI

383

99.1

Seismic Fragility

410

9.2 HCLPF Capacity

416

9.3 Methodology of Fragility Analysis

418

9.4 Conservative Deterministic Failure Margin (CDFM) Method

441

9.5 Case Study − Horizontal Heat Exchanger

446

9.6 Masonry Block Wall

483

vi

Seismic Fragility Analysis . . . . . . . . . . . . . . . . . . . . . . 410

contents

vii

Seismic Probabilistic Safety Assessment 10 10.1

. . . . . . . . . . . . .

Introduction

517 517

10.2 System Analysis

520

10.3 Seismic Risk Quantification

536

10.4 Seismic Margin Assessment

538

10.5 Seismic Probabilistic Safety Assessment with Screening Tables

546

10.6 Hybrid Method for Seismic Risk Assessment

547

10.7 Estimation of Seismic Risk from HCLPF Capacity

551

10.8 Numerical Examples − ECI System

552

Appendix A

Basics of Normal and Lognormal Distributions . . . . . 562

a.1 Normal Distribution

562

a.2 Lognormal Distribution

564

Appendix B

Digital Signal Processing . . . . . . . . . . . . . . . . . . 568

b.1 Sampling

568

b.2 Fourier Series and Fourier Transforms

572

b.3 Digital Signal Processing

580

b.4 Digital Filters

584

b.5 Resampling

586

b.6 Numerical Example − Gaussian White Noise

589

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

595

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Preface Background Earthquakes are among the most destructive natural disasters. The Great East Japan earthquake, measuring 9.0 on the moment magnitude scale, hit Japan on March 11, 2011; the earthquake and the subsequent tsunami caused severe damage to a large number of critical engineering structures. For example, twenty-six Shinkansen bridges were damaged in the earthquake, resulting in major transportation system disruption in Japan for weeks. A total of eleven nuclear reactors shut down automatically following the earthquake. Although seismic forces did not cause any structural failure at the Fukushima Nuclear Power Plant (NPP), the flood caused by the ensuing tsunami led to a series of equipment failures, nuclear meltdowns, and releases of radioactive materials at the Fukushima Daiichi NPP. It was the largest nuclear disaster since the Chernobyl disaster of 1986 and only the second disaster to measure Level 7 on the International Nuclear Event Scale. On the other hand, the Onagawa NPP, which is the closest NPP to the epicentre, rode out the monster earthquake unscathed, demonstrating that the existing seismic design approaches have been tested by a real case of beyond design basis earthquake. In response to the several destructive earthquakes that have occurred in recent decades, seismic risk analysis for critical engineering structures has become one of the most important and popular topics in earthquake engineering. Nuclear energy industries worldwide have launched an unprecedented and extensive re-evaluation of seismic hazards and risk to NPP systems. Furthermore, nuclear energy regulators and utilities are taking a critical look at the existing methods of estimating the seismic risk of NPPs. A number of deficiencies have been recognized in the existing methodologies of seismic risk analysis and design, which need improvements to enhance their reliability and effectiveness. Seismic risk analysis involves a wide range of disciplines and topics, including engineering seismology, probability theory, seismic hazard analysis, seismic design earthquakes, random processes and digital signal processing, structural dynamics and random vibration, seismic fragility analysis, system reliability analysis, and seismic risk assessment. However, there is currently no book that presents a systematic introduction to and discussion on various aspects of seismic risk analysis for engineering structures, in particular NPPs, to graduate students and practicing engineers.

ix

x

preface

Objectives This book addresses the needs of graduate students in engineering, practicing engineers in industry, and regulators in government agencies and aims to achieve the following objectives: ❧ To present the entire process of seismic risk analysis in a clear, logical, and concise manner Seismic risk analysis is an integral and systematic framework, in which all individual components (e.g., seismic hazard analysis, seismic demand analysis, and seismic fragility analysis) not only play their own roles but also interrelate with each other. This book is suitable not only as a textbook for graduate students in civil engineering, mechanical engineering, and other relevant programs but also as a reference book for practicing engineers and government regulators. ❧ To have a balance between theory and applications The book can be used as a reference for engineering graduate students, practicing engineers, and government regulators. As a reference, it has to be reasonably comprehensive and complete. Detailed step-by-step analysis for each topic of seismic risk analysis is presented with engineering examples. ❧ To include the latest research advances and applications Significant progress has been made on most of the topics in seismic risk analysis in the past decades. The latest research advances in improving the existing seismic risk analysis methods, including many contributions from our research team, are presented in the book.

Scope and Organization In Chapter 1, various types of NPPs, important structures, systems, and components (SSCs) in NPPs, general seismic design philosophy, and seismic requirements for NPPs are briefly introduced. In Section 1.4, the procedure of seismic risk analysis of an NPP is outlined, which includes seismic hazard analysis, seismic demand analysis, seismic fragility analysis, system analysis, and seismic risk quantification. In Chapter 2, fundamental principles, definitions, and terminologies in engineering seismology that are essential to the seismic risk analysis of NPPs are presented. In Chapter 3, basic theory of random processes, structural dynamics, and random vibration is presented, which is essential background knowledge to engineering analysts in earthquake engineering. The organization of the remainder of the book follows the general procedure of seismic risk analysis of NPPs as presented in Section 1.4. Chapters 4–6 are on seismic hazard analysis to provide response spectra and spectracompatible ground-motion time-histories for seismic demand. Chapter 4 introduces seismic response spectra, including ground response spectra and t-response spectra,

preface

xi

which are used in the direct method for generating floor response spectra (FRS) in Chapter 8. Chapter 5 presents seismic hazard analysis, including probabilistic seismic hazard analysis (PSHA), seismic hazard deaggregation (SHD), and site response analysis. Chapter 6 introduces various methods for generating spectrum-compatible time-histories, such as Fourier-based, wavelet-based, and Hilbert–Huang transformbased spectral matching algorithms. A new method using eigenfunctions for generating consistent, drift-free, and spectrum-compatible time-histories is also presented. Chapters 7 and 8 are on seismic demand analysis. In Chapter 7, general principles and approaches for modelling a structure into a dynamic 3D finite element model or stick model are presented. Chapter 8 presents methods for generating FRS, which are the seismic input to SSCs in an NPP. The methods presented include time-history method, direct spectra-to-spectra method for fixed-based models and considering soil–structure interaction, and the scaling method. Chapter 9 introduces the general methods for seismic fragility analysis of SSCs, including the method of fragility analysis, high confidence and low probability of failure (HCLPF) values, and conservative deterministic failure margin (CDFM) method for determining HCLPF values. To illustrate the general approach of fragility analysis, two detailed examples on horizontal heat exchanger and masonry block wall are worked using both the fragility method and the CDFM method. In Chapter 10, basic principles and methods of system analysis are introduced first. Two methods of seismic risk quantification, i.e., seismic margin assessment (SMA) and seismic probabilistic safety assessment (seismic PSA), are presented. Appendix A reviews important properties and results of normal distribution and lognormal distribution. In Appendix B, some relevant topics in digital signal processing are presented, including sampling, Fourier transforms, digital filter, and resampling a signal at a different rate, which are important in processing real earthquake records and generating spectra-compatible artificial ground-motion time-histories.

Acknowledgements First and foremost, our sincere appreciation goes to Candu Energy Inc. (formerly Atomic Energy of Canada Limited), in particular Han Ming, who has always supported the training and growth of students and graduates from the University of Waterloo. We are very grateful to Dr. Binh-Le Ly, who has offered many insights and directions on seismic analysis and design. We appreciate the support and collaborations of our colleagues at Candu Energy Inc. We are grateful to the members of CSA N289 Technical Committee on Seismic Design for their encouragement and feedback on our research progress. The Collaborative R&D grants from the University Network of Excellence in Nuclear Engineering (UNENE) and Natural Sciences and Engineering Research Council

xii

preface

(NSERC) in Seismic Risk Analysis of Nuclear Power Plants are greatly appreciated. These grants helped support collaborative research between the University of Waterloo and Candu Energy Inc. and training of PhD students at the University of Waterloo. Dr. Zhen Cai has carefully read the book and made many helpful and critical suggestions. A number of graduate students at the University of Waterloo have reviewed and commented on portions of various drafts of this book. Our sincere appreciation goes to Peter Gordon, former Senior Editor, and Steven Elliot, Senior Editor, Engineering, Cambridge University Press, for their trust, encouragement, and hard work to publish this book.

Wei-Chau Xie: I am grateful to my former graduate students, collaborators, coauthors, and friends, Wei Liu, Shun-Hao Ni, and Wei Jiang for their hard work during this long process and for contributing their expertise in various areas of seismic risk analysis to make this book possible. This book is dedicated in the loving memory of my beloved mother, who passed away on Good Friday of 2016. She had always unconditionally loved and supported me. I thank my wife Cong-Rong for her love, encouragement, and support. I am very grateful to my lovely daughters, Victoria and Tiffany, for their love and encouragement. I am thrilled that we have a positive influence on their value system; they have developed great work ethics and, through hard work, have achieved great success in their academic and professional careers.

Shun-Hao Ni: I would especially like to express my sincere appreciation to Dr. WeiChau Xie, my professor and the leading author of this book, who has led me into the world of seismic-related research of nuclear power plants. His encouragement, guidance, and support not only enabled me to develop an understanding of the subject, but also brought us together to initiate a plan for writing this book. Many thanks to all who have inspired, supported, and helped me during the course of writing this book. I would like to extend my gratitude to many people who have supported and helped me in various ways in my professional career, including my graduate co-supervisor, Professor Mahesh D. Pandey of the University of Waterloo. I would like to acknowledge with gratitude, the unflagging love, support, and encouragement from my family, especially my mother, my wife, Qi Sun, and my lovely son, Kai.

Wei Liu: First and foremost, I would like to thank my mentor, Professor Xie, for his inspiration and motivation in my life and career development. Special thanks to my family, especially my two daughters, Catherine and Helen, for always cheering me up and keeping me going. I hope that one day they can read this book and understand the seismic issues that I have been working on. Wei Jiang: I would like to express my deepest gratitude to Professor Wei-Chau Xie, who is the supervisor for my PhD study, for enabling me to be a part of this book. I

preface

xiii

would like to thank him for inspiring me and for allowing me to grow in all aspects of life. His guidance on both research and my life have been invaluable. I am also truly grateful to my friends and colleagues at University of Waterloo and Candu Energy Inc. for their continuous help and valuable suggestions on my career. I want to thank my wife, Bingqian Zhou, who understood, supported, and encouraged me despite all the time it took me away from her. It was a long and difficult journey for her. I thank my parents, my parents-in-law, and my family. This chapter of my life would be less fulfilling without their unflagging love and unconditional support throughout these years. We appreciate hearing your comments via email ([email protected]).

Nomenclature ACI

American Concrete Institute

AEP

annual exceedance probability

AFE

annual frequency of exceedance

ARS

acceleration response spectrum/spectra

ASCE

American Society of Civil Engineers

BH

borehole

BNSP

balance of nuclear steam plant

BOP

balance of plant

BWR

boiling water reactor

CD

core damage

CDF

core damage frequency cumulative distribution function

CDFM

conservative deterministic failure margin

CENA

Central and Eastern North America

CI

conventional island

CLCS

consequence limiting control system

CMS

conditional mean spectrum/spectra

CoV

coefficient of variation, equals mean value divided by standard deviation

Cov(X, Y) covariance of random variables X and Y

CQC

complete quadratic combination

CRDM

control rod drive mechanism

CSA

Canadian Standard Association

CSIS

containment spray injection system

DBE

design basis earthquake

DFT

discrete Fourier transform

DMF

dynamic magnification factor

DOF

degrees-of-freedom

DRS

design response spectrum/spectra

DS

damage state

xiv

nomenclature

DSHA

deterministic seismic hazard analysis

DTFT

discrete-time Fourier transform

ECC

emergency core cooling

ECI

emergency coolant injection

EMD

empirical mode decomposition

ENA

Eastern North America

EPRI

Electric Power Research Institute

ESD

energy spectral density

EWS

emergency water supply

FA

fragility analysis

FAS

Fourier amplitude spectrum/spectra

FE

finite element

FEM

finite element method

FIR

finite impulse response

FIRS

foundation input response spectrum/spectra

FLIRS

foundation level input response spectrum/spectra

FRS

floor response spectrum/spectra

FT

Fourier transform

GMP

ground-motion parameter

GMPE

ground-motion prediction equation

GMRS

ground-motion response spectrum/spectra

GRS

ground response spectrum/spectra

GWN

Gaussian white noise

HAS

Hilbert amplitude spectrum

HCLPF

high confidence and low probability of failure

HCSCP

hazard-consistent, strain-compatible properties

HES

Hilbert energy spectrum

HSA

Hilbert spectral analysis

HTS

heat transport system

IDFT

inverse discrete Fourier transform

IDTFT

inverse discrete-time Fourier transform

IFT

inverse Fourier transform

IMF

intrinsic mode functions

IRVT

inverse random vibration theory

xv

xvi

nomenclature

LERF

large early release frequency

LLOCA

large loss of coolant accident

LOCA

loss of coolant accident

LOOP

loss of offsite power

MCR

main control room

MDOF

multiple degrees-of-freedom

MMI

modified Mercalli intensity

NBCC

National Building Code of Canada

NEP

nonexceedance probability

NGA

next generation attenuation

NI

nuclear island

NPPs

nuclear power plants

NPS

nuclear power stations

NRCAN

Natural Resources Canada

NSP

nuclear steam plant

NUREG

Nuclear Regulatory (U.S. Nuclear Regulatory Commission)

PDF

probability density function

PEER

Pacific Earthquake Engineering Research

PGA

peak ground acceleration

PGD

peak ground displacement

PGV

peak ground velocity

PHWR

pressurized heavy water reactor

PMF

probability mass function

PSA

probabilistic safety assessment

PSD

power spectral density

PSHA

probabilistic seismic hazard analysis

PWR

pressurized water reactor

RB

reactor building

RBD

reliability block diagram

RE

reference earthquake

RLE

review level earthquake

RS

response spectrum/spectra

RVT

random vibration theory

RWST

refueling water storage tank

nomenclature

SA

spectral acceleration

SAM

seismic anchor movements

SB

service building

SCA

secondary control area

SD

standard deviation

SDE

site design earthquake

SDOF

single degree-of-freedom

SHD

seismic hazard deaggregation

SIS

safety injection system

SMA

seismic margin assessment

SME

seismic margin earthquake

SPRA

seismic probabilistic risk assessment

SPT

standard penetration test

SRHA

seismic response history analysis

SRSA

seismic response spectrum analysis

SRSS

square root of sum of squares

SSCs

structures, systems, and components

SSE

safe shutdown earthquake

SSEL

safe shutdown equipment list

SSI

soil–structure interaction

TP

test pit

tRS

t-response spectrum/spectra

TSCR

truncated soil column response

UHS

uniform hazard spectrum/spectra

USNRC

U.S. Nuclear Regulatory Commission

VPSHA

vector-valued probabilistic seismic hazard analysis

WNA

Western North America

ZPA

zero period acceleration

xvii

C

H

1 A

P

T

E

R

Introduction

1.1 Introduction to Types of Nuclear Power Plants Nuclear Fission One of Einstein’s greatest discoveries is that the law of conservation of energy must be generalized to include mass as a form of energy

e = mc 2 ,

(1.1.1)

where e is energy, m is mass, and c is the speed of light. Any change in mass in a reaction is accompanied by release or intake of energy. Nuclear energy comes from changes in the nuclei of atoms, which produces energy by mass conversion. The purpose of a nuclear power plant (NPP) is to generate electricity safely, reliably, and economically. In nuclear reactors, nuclear fission releases heat energy by splitting atoms; it takes place when a large, somewhat unstable isotope, is bombarded by highspeed neutrons, causing it to undergo fission (break into smaller particles). In fission reaction, energy is released, and the process has potential of being self-perpetuating because neutrons that emerge from fission can induce more fissions. A reactor is a device to maintain and control nuclear fission chain reactions and convert the nuclear energy released by fission to heat energy.

Reactor Types There are more than 400 nuclear power reactors, representing about 16 % of the total electricity production of the world. The types of reactors could be categorized according to the purpose, coolant type, moderator type, and fuel. The main design of a reactor for NPP is the pressurized water reactor (PWR), which has water at over 300◦ C under pressure in its primary cooling/heat transfer circuit and generates steam in a secondary circuit. The less numerous boiling water reactor (BWR) 1

2

makes steam in the primary circuit above the reactor core, at similar temperatures and pressure. Both PWR and BWR use enriched uranium as fuel and water as both coolant and moderator to slow down neutrons. Because water normally boils at 100◦ C, they have robust steel pressure vessels or tubes to enable the higher operating temperature. In Canada, CANDU (CANada Deuterium Uranium) reactors are employed, which is a pressurized heavy water reactor (PHWR) type. Heavy water is used as both moderator and coolant in CANDU reactors.

PWR Figure 1.1 shows a schematic diagram depicting a typical working process of PWR NPP. Water carries heat from the fission heat generated from the reactor vessel and becomes high-temperature and high-pressure water. It then flows into U tubes of the steam generators (primary coolant loop) and exchanges heat with the feeder water outside the U tube (secondary loop), which becomes saturated steam. The main steam lines direct the steam, which powers the turbine generator. The cooled coolant is then pumped back to the reactor to be reheated; this circulation is iterated and forms a closed heat absorption and release loop called first loop, also known as a nuclear steam supply system.

BWR In a BWR reactor, a steam–water mixture is produced when reactor coolant (pure water) moves upward through the core, absorbing heat. The steam–water mixture leaves the top of the core and enters the two stages of moisture separation, where water droplets are removed before the steam is allowed to enter the steam line, which directs the steam to power turbine generator. The major difference between a PWR and a BWR is that a PWR has water at over 300◦ C under pressure in its primary cooling/heat transfer circuit and generates steam in a secondary circuit, while a BWR makes steam in the primary circuit above the reactor core.

PHWR (CANDU) Figure 1.2 shows a schematic diagram illustrating the working process of a CANDU 6 reactor. A CANDU 6 nuclear steam supply system’s power production process starts like that of any other PWR nuclear steam supply system, with controlled fission in the reactor core. However, unlike other reactors, the CANDU 6 is fuelled with natural uranium fuel that is distributed among 380 fuel channels. Each six-meter-long fuel channel contains 12 fuel bundles. The fuel channels are housed in a horizontal cylindrical tank (called a calandria) that contains cool heavy water (D2 O) moderator at low pressure. Fuelling machines connect to each fuel channel as necessary to provide on-power refuelling; this eliminates the need for refuelling outages.

3

Figure 1.1

Pressurized water reactor (PWR).

1.1 introduction to types of nuclear power plants

4

Figure 1.2

CANDU reactor.

1.2 important structures, systems, and components in nuclear power plants

5

1.2 Important Structures, Systems, and Components in Nuclear Power Plants In this section, only the structures, systems, and components (SSCs) of PWR and PHWR are described.

General Description According to functions of the building structures, PWR NPP could be grouped into ❧ Nuclear island (NI) or nuclear steam plant (NSP), ❧ Conventional island (CI) or turbine island, ❧ Balance of plant (BOP). PHWR NPP consists of ❧ Nuclear steam plant (NSP), including the balance of nuclear steam plant (BNSP), ❧ Balance of plant (BOP). Per definitions of NI, CI, and BOP for PWR and NSP, BNSP, and BOP for PHWR ❧ NI of PWR = NSP (including BNSP) of PHWR, ❧ (CI + BOP) of PWR = BOP of PHWR. The NI/NSP part of the NPP is defined as all equipment required for the production of steam, including the nuclear reactor, relevant safety systems, and their auxiliaries. The main functionality of NI is to utilize the energy from nuclear fission to generate steam. ❧ In PWR NPP, NI includes five main building structures: • • • • •

reactor building, fuel building, electrical building, connecting building, nuclear auxiliary building,

as well as other building structures, such as an emergency diesel generator building. ❧ In PHWR CANDU 6 NPP, NSP (including BNSP) includes • • • • • • •

reactor building, service building, secondary control area, emergency water supply pump house, emergency power supply building, high-pressure emergency core cooling building, D2 O upgrading tower.

6

1DA

9L

1L 1W

1RE

9NA

9NB

9NC

9ND

9NE

9NF

2W

1R

2R

1K

2RE

2K

1DB

2DB

1 Unit 1 Main building structures R Reactor Building L N

2DA

2L

9

Shared by two units

Electrical Building Nuclear Auxiliary Building

Other building structures D Emergency Diesel Generator Building Figure 1.3

2

Unit 2

K

Fuel Building

W

Connecting Building

RE

Auxiliary Feed Water Storage Building

PWR NPP layout.

14

8 7 6

3

2 12

2

Unit 1 1

8 7 6

3

Unit 2 1

12

5

5

4 11 10 1 3 5

4 11

9

13

Reactor Building Turbine Building Crane Hall, Decontamination

10 2 4 6

9

Service Building Irradiated Fuel Bay Water Treatment Plant

12

8 Auxiliary Boiler Diesels Emergency Power Supply Building 10 Secondary Control Area Emergency Core Cooling Accumulator Building D2O Upgrading Tower

13

Administration Building

7 9 11

Figure 1.4

14

Switchyard

Candu PHWR NPP layout.

1.3 seismic design philosophy and requirements

7

Conventional island (CI), also called turbine island, is the general term for the turbinegenerator set and its supporting facilities in an NPP. The main function of the conventional island is to convert the thermal energy of the steam produced at the nuclear island into mechanical energy of the steam turbine and then into electrical energy through the generator. For all NPPs, CI are similar. Schematic layouts of a PWR plant and a CANDU 6 PHWR plant are shown in Figures 1.3 and 1.4, respectively. In this book, the main focus is on the NI/NSP, where safety-related SSCs are located.

Reactor Building The reactor building (RB) houses the primary coolant loop together with its associated auxiliary and safety systems. For PHWR, unique moderator systems and fuel-handling systems are also housed inside the reactor building. To minimize the seismic response, the heavy equipment is located close to low elevations of the reactor building, and the structural and equipment layouts are designed to be as symmetrical as possible. To optimize the seismic response of the reactor building, the containment structure of the reactor building is designed as ellipsoid. The reactor building is divided into three major structural components: ❧ prestressed concrete containment structure, ❧ internal reinforced concrete structure, ❧ reinforced concrete reactor vault (PHWR only). The containment structure is the main component of the containment system. This system is provided to ensure that public exposure to radiation is prevented beyond the station’s exclusion area in the event of the accidents postulated for the reactor. The containment structure is typically a prestressed concrete building comprising three structural components: ❧ a base slab, ❧ a cylindrical perimeter, ❧ a spherical segmental dome. The internal structure supports reactor process systems and is therefore a major nuclear support structure.

1.3 Seismic Design Philosophy and Requirements In this section, seismic design philosophy and high-level seismic requirements for NPPs are briefly introduced. The philosophy is general and applicable to all types of nuclear reactors, but it is presented here in the terminology for CANDU NPPs.

8

Seismic Levels For CANDU plants, two levels of earthquake are defined as design envelopes for achieving the safety objectives (CSA N 289.3, CSA, 2010a): ❧ Design Basis Earthquake (DBE)−an engineering representation of potentially severe effects at the site due to earthquake ground motions having a selected probability of exceedance of 1×10–4 per year. ❧ Site Design Earthquake (SDE)−an engineering representation of the effects at the site of a set of possible earthquakes with an occurrence rate, based on historical records, not greater than 1×10–2 per year.

☞

DBE or SDE ground motion is usually referred to as an “earthquake” and can take the form of a response spectrum or time-history.

Seismic Categories Two categories, “A” and “B”, are used in design to establish the extent to which components must remain operational during and/or after an earthquake: ❧ Category “A” Components−Those that must retain their pressure boundary integrity or structural integrity or passive function (e.g., cables) and are not required to change state during and/or following an earthquake. ❧ Category “B” Components−Those that must retain their pressure boundary integrity and remain operable during and/or following an earthquake.

Site Considerations When selecting an NPP site, the following seismic-related aspects must be considered: ❧ Seismicity is a major item for site selection. ❧ Seismic requirements generally refer to the ability of the facility to withstand movement in three orthogonal directions (two horizontal and one vertical). ❧ The seismic requirement for a generic NPP design is typically 0.2g peak ground acceleration (PGA) for GEN II and 0.3g PGA for GEN III reactors. ❧ For seismic considerations, an NPP is selected to be founded in a geologically stable zone and should not be near fault lines. ❧ The space requirements between the structures will be determined by seismic movements of the structures.

1.3.1 Safety Functions Nuclear safety requires that the radioactive products from the nuclear fission process of the reactor be contained, both within the plant systems for the protection of the plant workforce and outside the plant structure for the protection of the public. The safety-

1.3 seismic design philosophy and requirements

9

related SSCs, which are necessary to ensure the four basic safety functions that have to be performed both during and after a severe earthquake, are seismically qualified to perform these safety functions. These four safety functions are essentially the same for all NPPs, but for different types of reactors the descriptions are somewhat different because the systems and components are different. For a CANDU NPP, the four major safety functions are as follows: 1. Shut down the reactor and maintain it in a safe shutdown condition ❧ Shutdown system #2 is qualified to shut down the reactor but is not qualified to be recocked. Shutdown system #1 is qualified to drop the rods into the core if electrical power to the clutch assemblies is lost. ❧ Even though the shutdown systems can shut down the reactor under any circumstances, the reactor regulating system is qualified to ensure that a seismically induced failure in this system will not cause an increase in positive reactivity exceeding the capability of the shutdown systems and heat removal systems. 2. Remove decay heat ❧ The primary heat transport system (HTS), including the fuel channels, headers, pumps, steam generators, and connected subsystems, are qualified to ensure that a loss of coolant accident (LOCA) does not occur as a result of the earthquake. ❧ The pumps may not remain operable, in which case the fuel is cooled by natural circulation− “thermosyphoning”. The pumps remain freewheeling immediately after the earthquake to enable pump rundown to assist thermosyphoning. ❧ Because a reduction in coolant inventory may be caused by shrinkage due to cooling, minor leaks, or transfer of coolant to the bleed condenser, the emergency water supply (EWS) system is qualified to provide a light water makeup after a period of about thirty minutes. ❧ If the normal unqualified feedwater system fails, an alternative qualified source of emergency water is available, either from the dousing tank, or from the EWS. ❧ A portion of the main steam piping and the main steam safety valves is qualified to ensure that the residual heat is discharged to the atmosphere. ❧ The recovery portion of the emergency core cooling (ECC) system is qualified to cater for the occurrence of an SDE following a LOCA. Cooling water to the ECC heat exchanger is supplied by the EWS. ❧ Electrical power is supplied by the emergency power supply system.

10

❧ The high- and intermediate-pressure portions of the ECC system need not be qualified because the earthquake is not postulated to occur immediately after a LOCA. 3. Maintain a barrier to limit the release of radioactive material ❧ Structures or components outside the reactor building whose failures could result in the release limits being exceeded are seismically qualified. This includes the spent fuel storage bay. The reactor building containment system is qualified to remain available after an earthquake. ❧ Releases of radiation within containment may be caused by minor leaks in the HTS (possibly existing prior to the earthquake) or by interruption of the cooling to spent fuel being transferred in the fuelling machine. The containment system must also remain functional for the occurrence of an SDE following a LOCA. ❧ The containment needs not be qualified to withstand peak building pressure coincident with an earthquake. However, the containment structure is qualified to withstand a “reduced accident pressure” (due to the failure of piping or components that are not qualified and that may contain high energy) combined with the DBE. 4. Perform essential safety-related control and monitoring functions The secondary control area (SCA) and the control and monitoring systems associated with it are qualified. ❧ Electrical power is supplied from the qualified emergency power system. ❧ A qualified source of instrument air is supplied, in the form of qualified local air tanks, where required for essential control functions. ❧ A sufficient number of control and monitoring functions are provided in the SCA, or in areas accessible after a seismic event, to shut down the reactor and maintain it in a safe shutdown condition. ❧ Structures and components that may pose a hazard to seismically qualified systems are also qualified. In addition to the systems and components noted earlier, structures that house and support them also have a safety function to maintain their structural integrity during and following the earthquake. These structures include • the reactor building, • the service building, • the secondary control area building, • the structure housing the main steam safety valves.

1.3 seismic design philosophy and requirements

11

Other structures, such as the turbine building, are qualified if their failure during an earthquake could indirectly affect the qualification of another building (e.g., service building) or qualified component (e.g., main steam lines).

1.3.2 Safety Objectives The safety objective of the seismic design of the plant is to have sufficient capability to perform the essential safety functions to ensure that: 1. During and/or following a DBE: a. The reactor can be shut down and maintained in a safe shutdown state. b. The HTS integrity can be maintained for fuel cooling, (i.e., no LOCA as a result of an earthquake). c. Fuel in the reactor can be cooled by thermosyphoning to the steam generators. d. The containment boundary can be maintained. e. The plant can be controlled and monitored from the seismically qualified SCA. f. The main control room (MCR) remains available to the extent necessary to protect the operator, and a qualified route is provided for safe access to the SCA. g. Critical structures and systems outside containment are maintained so that radioactivity releases beyond allowable accident limits are not caused. 2. During and/or following an SDE occurring 24 hours or more after a LOCA: a. The reactor fuel can continue to be cooled. b. Essential variables can continue to be monitored from the SCA. The preceding objectives reflect the following concepts: 1. Based on the low frequency of each independent event, a DBE is not considered to occur simultaneously with, or following, a LOCA. 2. Based on the low probability of an SDE occurring within the first 24 hours after a LOCA, the SDE is considered to occur not less than 24 hours after a LOCA.

1.3.3 Systems Required to Satisfy the Safety Objective The reactor is required to be shut down only if an earthquake causes failure requiring shutdown. Equipment not seismically qualified cannot be credited for postseismic functions. To meet safety objectives, a detailed listing of the SSCs requiring seismic qualification is developed. The following requirements, based on the safety functions to be maintained, shall also be satisfied:

12

❧ Reactor Shutdown 1. Both shutdown systems shall be seismically qualified to DBE: a. Shutdown system #2 shall be fully qualified. b. Shutdown system #1 shall be qualified to permit the shut-off rods to drop into the core on loss of electrical power to their clutch assemblies. 2. Provisions shall be made to enable manual initiation of the shutdown systems from the SCA following an earthquake. ❧ Residual Heat Removal 1. The HTS, including the pressure tubes, fuel, headers, feeders, end fittings, pumps, steam generators, and connected subsystems, shall be qualified to ensure that a LOCA would not occur as a result of the earthquake. The pumps may not remain operable, in which case the fuel shall be cooled by natural circulation. 2. The HTS pressure boundary shall ensure retention of the inventory necessary for natural circulation following a DBE. The designer shall identify the seismically qualified boundary and shall ensure that open valves on the boundary will be closed automatically or can be closed from a seismically qualified area. 3. The heat transport pumps shall be designed to free-wheel to establish thermosyphoning following a pump trip after an earthquake. 4. Emergency water to the steam generators and cooling water to the ECC heat exchangers shall be provided by the EWS. 5. Makeup capability to inject water into the HTS shall be provided to compensate for small leaks that will continue following an earthquake. 6. To maintain HTS cool-down capability: a. The steam generator water inventory shall be sufficient to maintain adequate fuel cooling until initiation of the EWS system. b. Main steam safety valves shall be seismically qualified and shall be provided with means to open them and keep them open. 7. The ECC system shall be seismically qualified to cater for the possibility of an SDE, 24 hours or more after a LOCA. ❧ Barrier to Radioactive Release To contain radioactive releases, the plant design shall provide a succession of seismically qualified barriers, namely, the fuel sheath, the HTS boundary, and the containment boundary.

1.4 seismic risk analysis of nuclear power plants

13

❧ Control and Monitoring 1. A sufficient number of qualified control and monitoring functions shall be provided in the SCA, or in the areas accessible after a DBE or SDE, to shut down the reactor and maintain it in a safe shutdown condition. 2. The SCA shall be qualified to be habitable for operators following an earthquake. Design measures, such as ventilation isolation and radiation shielding, shall be provided, if required for the operation of qualified components. 3. Electrical power shall be supplied from a qualified power supply. 4. A qualified source of instrument air shall be supplied, where required for essential control functions. ❧ Plant Layout 1. Structures and components that may pose a hazard to seismically qualified systems shall also be qualified to the same earthquake level as the qualified system, or the seismically qualified systems shall be suitably protected. 2. There shall be a qualified route from the MCR to the SCA.

1.4 Seismic Risk Analysis of Nuclear Power Plants Section 1.3 presents the seismic design philosophy and requirements of NPPs subjected to the design basis conditions. The seismic design basis of NPPs could be called into question as new information on seismology may show a higher seismicity than the level of design basis during the lifetime of the NPPs. To demonstrate the design concepts of redundancy and defense-in-depth and to quantify the actual margin to failure probabilistically, seismic risk analysis of NPPs subjected to beyond-designbasis earthquake is performed. Although many aspects in terms of methodologies and procedures as well as of seismic design and seismic risk analysis overlap, this book mainly focuses on the seismic risk analysis. Seismic risk analysis is a broad and complex process, involving a number of science and engineering disciplines. This complex procedure is illustrated schematically in Figure 1.5, which is divided into five areas: seismic hazard analysis, seismic demand analysis, seismic fragility analysis, system analysis, and seismic risk quantification.

1. Seismic Hazard Analysis The objective of seismic hazard analysis is to determine the seismic hazard curves and seismic input at the site of interest in terms of ground response spectra (GRS) and spectra-compatible time-histories that can be used for seismic analysis and design. To achieve this objective, it is essential to understand the fundamentals of engineering

14

Figure 1.5

General procedure of seismic risk analysis of NPP.

1.4 seismic risk analysis of nuclear power plants

15

seismology: seismic sources, seismic source mechanism (the theory of plate tectonics and elastic rebound), propagation of seismic waves, strong ground motions, and ground-motion parameters. Ground response spectra have been widely used in earthquake engineering practice for analysis and design of structures. Seismic hazard analysis gives a quantitative estimation of ground-shaking hazards at a particular site. Probabilistic seismic hazard analysis (PSHA) provides a framework in which the uncertainties in locations of earthquake, magnitudes of earthquake, rates of occurrence of earthquakes, and variations of ground motion characteristics with magnitude and location can be identified, quantified, and combined in a mathematically rigorous manner to provide a complete picture of the seismic hazard. The result of PSHA gives the seismic hazard curves, which are the annual probabilities of exceedance of a ground-motion parameter. Because structures are usually founded on soil (the term soil herein represents a broad range of foundation medium, on which soil–structure interaction [SSI] effects cannot be ignored), the effect of soil layers on seismic wave propagation is important. Seismic hazard curves and response spectra at a desired elevation (such as foundation level and ground level) are determined as seismic input to the structures, by site response analysis based on seismological, geological, and geotechnical conditions of the site.

2. Seismic Demand Analysis The objective of seismic demand analysis is to determine the seismic demands or inputs to important systems, structures, and components (SSCs) in the NPP in terms of floor response spectra (FRS). Dynamic finite element models of the structures in the NPP are established first, and responses of the structures at desired locations are determined through structural dynamic analysis. There are two approaches: the time-history analysis method and the direct spectra-to-spectra method. To apply the time-history analysis method, spectra-compatible time-histories need to be generated. If the NPP is founded on soil, soil–structure interaction must be considered when performing structural dynamic analysis.

3. Seismic Fragility Analysis The objective of seismic fragility analysis is to determine the seismic fragility curves or the HCLPF values of important individual SSCs using the seismic demands obtained. Seismic fragility is the conditional probability that the damage of an SSC exceeds a specified limit state for a given level of seismic hazard. HCLPF is the high confidence low probability of failure of an SSC, which is used in seismic margin analysis (SMA).

16

4. System Analysis The objective of system analysis is to determine the seismic fragility curves or the HCLPF value of the plant damage state, such as core damage. The plant system and accident sequences are studied through logic trees (including fault trees and event trees).

5. Seismic Risk Quantification The objective of seismic risk analysis of a nuclear power facility is to determine the probability distribution or the frequency of occurrence of adverse consequences, such as core damage frequency (CDF) or large early release frequency (LERF), due to the potential effects of earthquakes. In SMA of an NPP, it is required to demonstrate that the plant can withstand the review level earthquake (RLE) with high confidence and to identify seismic vulnerabilities and any potential weak links. Seismic capacity of SSCs on the critical path is estimated in terms of HCLPF values based on FRS. SSCs with HCLPF values lower than the RLE are improved to meet their safety targets. This process ensures that the NPP has sufficient seismic margin to achieve an acceptably low seismic risk. Seismic risk analysis of NPPs is performed under the general framework of seismic probabilistic safety assessment (seismic PSA) or seismic probabilistic risk assessment (SPRA). Seismic PSA is the formal process in which the randomness and uncertainty in seismic hazard, structural responses, and material capacity variables are propagated through engineering modelling to determine a probability distribution or frequency of occurrence of failure or other adverse consequences due to earthquakes. Seismic PSA is an invaluable tool to identify weak links in a system or facility, which can guide the efficient allocation of funds to strengthen or modify an existing NPP.

C

H

2 A

P

T

E

R

Engineering Seismology

This chapter presents a brief introduction to the basics of seismology and engineering seismology. The fundamental principles, definitions, and terminologies are crucial to the seismic risk analysis of nuclear power plants (NPPs) and are used throughout this book. More details on seismology and engineering seismology can be found in standard textbooks, such as Kramer (1996).

2.1 Earthquakes 2.1.1 Internal Structure of the Earth The earth is nearly spherical, with an average radius of approximately 6,400 km. The earth consists of three different structural layers from its surface inward: crust, mantle, and core (Figure 2.1). The crust is the outermost layer of the earth. Its thickness ranges from about 5 km (beneath the oceans) to about 80 km (under some young mountain ranges). The average thickness of the crust ranges from 30 km to 40 km under the continents. The crust is only a very small fraction of the earth’s diameter but most earthquakes occur within the crust. The layer beneath the crust is the mantle. The boundary between the crust and the underlying mantle is known as the Mohorovicic Discontinuity, or the Moho, named after the seismologist who discovered it in 1909. Within the range of 40 km to 70 km beneath the Moho is a layer of solid rock. This layer of solid rock together with the crust forms the so-called lithosphere with a thickness of about 150 km. Underneath the lithosphere, there is a layer of partially molten rock with a thickness of about 450 km, named asthenosphere. The layer of the solid rock together with the asthenosphere is called the upper mantle. Beneath the upper mantle is the lower mantle, which appears to be structurally and chemically homogeneous. No earthquakes have been recorded in the lower mantle. 17

18

Figure 2.1

Internal structure of the earth.

The core consists of the outer core (or liquid core) and the inner core (or solid core). The outer core is liquid, which cannot transmit shear waves, while the inner core is a very dense solid, made up of nickel–iron material. With the depth of the earth increases from its surface to its inside, in general, the densities, temperatures, and pressures of the earth layers increase significantly.

2.1.2 Plate Tectonics and Boundaries Theory of Plate Tectonics The theory of continental drift was proposed based on the observations of similarity between the coastlines and geology of eastern South America and western Africa, and the southern part of India and northern part of Australia, as shown in Figure 2.2. Wegener, a German geophysicist, believed that a supercontinent (Pangea) existed about 250 million years ago and that Pangea broke apart into the continents about 200 million years ago; the broken pieces of the Pangea then slowly drifted into the present configuration of the continents. Based on the theory of continental drift, the modern theory of plate tectonics started to evolve, which became one of the foundations of modern geology. The basic hypothesis of plate tectonics is that the surface of the earth consists of a number of large and intact blocks, called plates, and that these plates move with respect to each other. Two major pieces of evidence support the theory of plate tectonics: paleomagnetism and the map of earthquake epicentres (more details in Kramer, 1996).

2.1 earthquakes

19

Figure 2.2

Pangea and the evolution of the continents.

20

Figure 2.3

Plate boundaries.

While the theory of plate tectonics assumes movements of large continental plates, the sources of such movements have been of great interest to the researchers in this area. The most widely recognized explanation of the sources of the plate movements is based on the thermomechanical equilibrium of the materials of the earth. The upper portion of the mantle is in contact with the cool crust and the lower portion of the mantle is in contact with the hot outer core, which creates significant temperature gradient within the mantle. As the denser magma rises, it becomes cooler. Due to the unstable situation that the denser (cooler) material rests on top of the less dense (warmer) material, the cooler and denser material begins to sink under the action of gravity. The sinking material gradually becomes warmer and less dense; eventually, it moves laterally and begins to rise. Such convection currents in the semimolten rock of the mantle impose shear stresses on the bottom of the plates, “dragging” them in various directions across the surface of the earth, as shown in Figure 2.3. Plate Boundaries Depending on the direction in which the plates move, three types of plate boundaries have been identified: divergent boundary, convergent boundary, and transform fault boundary. The characteristics of the plate boundaries lead to different natures of the earthquakes that occur along them. When the plates tend to separate, the divergent boundary, also called spreading ridge boundary, forms. As shown in Figure 2.3, the molten rock from the underlying mantle rises to the surface, cools in the gap formed by spreading plates, and becomes part of the spreading plates. When the plates tend to come together, the convergent boundary, or subduction zone boundary, is formed. As shown in Figure 2.3, at the contact points of the plates, one plate subducts beneath the other plate. Because the size of the earth remains constant, the consumption of the plate material at the subduction zone boundary is considered to balance the creation of new plate material at the spreading ridge boundary.

2.1 earthquakes

21

Figure 2.4

San Andreas Fault.

Figure 2.5 An example of a fault exposed to the ground surface.

Figure 2.6 Accumulation and release of elastic strain energy in the vicinity of boundary.

When the plates slide horizontally past each other, the transform fault boundary is formed. As illustrated in Figure 2.3, the transform faults are usually found in offsetting spreading ridges. Fault is defined as a break in earth’s crust along which displacement of rock occurs. The well-known San Andreas Fault, shown in Figure 2.4, is a continental transform fault that extends roughly 1,300 km through California and forms the tectonic boundary between the Pacific Plate and the North American Plate. An example of a fault exposed to the ground surface is shown in Figure 2.5.

22

2.1.3 Seismic Source Mechanism Elastic Rebound Theory The occurrence of earthquakes can be explained by the plate tectonic movements and the elastic rebound theory. Relative movement of the plates causes stresses to build up on their boundaries. As illustrated in Figure 2.6, when relative movement occurs, elastic strain energy accumulates in the vicinity of the boundaries. When the shear stress reaches the shear strength of the rock along the fault, the rock fails, and the accumulated strain energy is released. The effects of the failure depend on the nature of the rock along the fault. If the rock is weak and ductile, strain energy stored will be released relatively slowly, and the movement will occur aseismically. If the rock is strong and brittle, the failure will be rapid. Rupture of the rock will release the stored energy explosively, producing earthquakes. The theory of elastic rebound describes this process of the successive buildup and release of strain energy in the rock adjacent to faults. The size of an earthquake depends on the amount of energy released. Faults At a particular location, a fault is assumed to be planar with an orientation described by strike and dip. As shown in Figure 2.7, the strike is the compass direction relative to north, and the dip is the angle of inclination or slope relative to horizontal plane. The focus is the point on the fault plane where an earthquake originates, and the epicentre is the point on the earth’s surface located directly above the focus of an earthquake. Figure 2.7 also shows the hanging wall, which is the block positioned over the fault plane, and the foot wall, which is the block positioned under the fault plane. Various fault movements are illustrated in Figure 2.8. Fault movement occurring primarily in the direction of the dip (or perpendicular to the strike) is referred to as dip-slip movement. Normal faults occur when the ground is stretched and when the hanging wall moves downward relative to the foot wall. Reverse faults occur when the ground is compressed and when the hanging wall moves upward relative to the foot wall. Fault movement occurring parallel to the strike is referred to as strike-slip movement. Right-lateral strike-slip fault occurs when an observer would observe the material on the other side of the fault moving to the right. The San Andreas Fault shown in Figure 2.4 is a right-lateral strike-slip fault. Left-lateral strike-slip fault occurs when an observer would observe the material on the other side of the fault moving to the left. In most cases, fault slip is a mixture of strike-slip and dip-slip and is called oblique fault. Interplate Earthquakes The seismic source mechanism of the interplate earthquakes, which occur at the plate boundaries, has been well explained by the theories of plate tectonics and elastic

2.1 earthquakes

23

Figure 2.7

Definition of fault orientation.

Figure 2.8

Fault movements.

24

rebound. Because the deformation occurs predominantly at the boundaries between the plates, the locations of earthquakes have been concentrated near plate boundaries. Earthquakes occurring at convergent plate boundaries contribute more than 90 % of the world’s release of seismic energy. Most of the largest earthquakes have originated in the subduction regions a result of the thrusting of one plate under another. About 10 % of the world’s earthquakes occur along the divergent ocean-ridge system, and contribute only about 5 % of the world’s total seismic energy. The (Pacific) Ring of Fire is an area where large numbers of earthquakes and volcanic eruptions occur in the basin of the Pacific Ocean (Figure 2.9). In a 40,000-km horseshoe shape, it is associated with a nearly continuous series of oceanic trenches, volcanic arcs, volcanic belts, and plate movements. It contains 452 volcanoes (over 75 % of the world’s active and dormant volcanoes). About 90 % of the world’s earthquakes and 80 % of the world’s largest earthquakes occur along the Ring of Fire. Intraplate Earthquakes Some earthquakes do not necessarily occur at the plate boundaries; instead, they occur in the interior of the tectonic plates and are called intraplate earthquakes. The recurrence frequency of the intraplate earthquakes is much lower than the interplate earthquakes. The seismic damage of the intraplate earthquakes could be tremendous because the human population is more concentrated within the plates. Moreover, the interior of the tectonic plates generally has thick layers and high strength, which may produce large and shallow-focus earthquakes. Intraplate earthquakes arise from more localized systems of forces in the crust, perhaps associated with ancient geological structural complexity, or with anomalies in temperature and strength of the lithosphere.

2.1.4 Seismic Waves When an earthquake occurs, different types of seismic waves are produced: body waves and surface waves (Figure 2.10). ❧ Body Waves (P-waves and S-waves) can travel through the interior of the earth along paths influenced by the material properties (density and stiffness), resembling the reflection and refraction of light waves (Figure 2.11). • P-waves (primary, compressional, or longitudinal waves) involve successive compression and dilation of the materials through which they pass. They are analogous to sound waves; the motion of an individual particle is parallel to the direction of the P-wave. Like sound waves, P-waves can travel through any type of material. The speed of body waves varies with the stiffness of the materials they travel through. Because geologic materials are stiffest in compression, P-waves travel faster than other seismic waves (at nearly twice the speed of S-waves) and are therefore the first to arrive at a particular site.

Figure 2.9

Ring of Fire.

2.1 earthquakes 25

26

P-Wave

Compression Compression Dilation Dilation Dilation

Undisturbed material

Direction of wave propagation

S-Wave

Undisturbed material

Direction of wave propagation

Rayleigh Wave

Undisturbed material

Direction of wave propagation

Love Wave Undisturbed material

Direction of wave propagation

Figure 2.10

Seismic waves.

2.1 earthquakes

27

Figure 2.11

Figure 2.12

Light beams.

Seismic waves travelling through the earth.

• S-waves (secondary, shear, or transverse waves) cause shearing deformations as they travel through a material. The motion of an individual particle is perpendicular to the direction of the S-wave. The direction of particle movement divides S-waves into two components: SV (vertical plane movement) and SH (horizontal plane movement). S-waves are slower than P-waves, with speeds typically around 60 % of that of P-waves in any given material. An S-wave shakes the ground surface vertically

28

and horizontally, which is particularly damaging to structures. Fluids, which have no shearing stiffness, cannot sustain S-waves. ❧ Surface waves (Rayleigh waves and Love waves) result from the interaction between body waves at the surface and surficial layers of the earth. They travel along the earth’s surface with amplitudes that decrease roughly exponentially with depth. Due to the nature of the interactions required to produce them, surface waves are more prominent at distances farther from the source of the earthquake. At distances greater than about twice the thickness of the earth’s crust, surface waves, rather than body waves, will produce peak ground motions. The most important surface waves, for engineering purposes, are Rayleigh waves and Love waves. • Rayleigh waves are produced by the interaction of P-waves and SV-waves with the earth’s surface. They involve both vertical and horizontal particle motions. Rayleigh waves are also called “ground rolls”, similar to water waves produced by a rock thrown into a pond. They are slower than body waves, roughly 90 % of the velocity of S-waves for typical homogeneous elastic media. • Love waves result from the interaction of SH-waves within a soft surficial layer. They have no vertical component of particle motion. They usually travel slightly faster than Rayleigh waves, about 90 % of the S-wave velocity, and have the largest amplitude. The horizontal shaking of Love waves is particularly damaging to the foundations of structures. Love waves do not propagate through liquids. As the different types of seismic waves travel through the earth, they are refracted and reflected at boundaries between different layers, as light beams (Figure 2.11). Whenever a wave is reflected or refracted, some of the energy is converted to waves of the other types (as shown in Figure 2.12). Seismic waves reach different points on the earth’s surface by different paths. In strong ground shaking on land, after the first few shakes, a combination of the two kinds of waves, body waves and surface waves, is usually felt. It is noted that studies of the refractions and reflections of the seismic waves actually brought to light the layered structure of the earth and provided insight into the characteristics of each layer.

2.1.5 Size of Earthquakes The size of an earthquake is a crucial parameter and has been described in different ways. The oldest measure of earthquake size is the earthquake intensity. The intensity is a qualitative description of the effects of the earthquake at a particular location, as evidenced by observed damage and human reactions. The most famous earthquake intensity is the Modified Mercalli intensity (MMI) scale originally developed by the Italian seismologist Mercalli and modified later by Richter (1958). The MMI scale is listed in Table 2.1.

2.1 earthquakes

29 Table 2.1

Modified Mercalli intensity (MMI) scale.

I

Not felt

II

Felt only by persons at rest

III–IV

Felt by persons indoors only

V–VI

Felt by all; some damage to plaster, chimneys

VII

People run outdoors, damage to poorly built structures

VIII

Well-built structures slightly damaged; poorly built structures suffer major damage

IX

Buildings shifted off foundations

X

Some well-built structures destroyed

XI

Few masonry structures remain standing; bridges destroyed

XII

Damage total; waves seen on ground; objects thrown into air

Seismic instruments have allowed an objective, quantitative measurement of earthquake size in terms of earthquake magnitude. Most magnitude scales are based on measured ground-motion characteristics. The concept of earthquake magnitude was first introduced by Richter (1935), named Richter local magnitude ML , i.e., ML = log10 A − log10 A0 (δ),

(2.1.1)

where the empirical function A0 (δ) depends only on the epicentral distance of the station δ. The base-10 logarithm of the maximum amplitude of waves (as shown in Figure 2.13) recorded by seismographs, is adjusted to compensate for the variation in the distance between the various seismographs and the epicentre of the earthquake. There are other instrumental magnitude scales: surface wave magnitude Ms , which is based on the amplitude of Rayleigh waves; body wave magnitude mB , which is based on the amplitude of P-waves that is not affected by the focal depth of the source. For strong earthquakes, the measured ground-shaking characteristics become less sensitive to the size of the earthquake than for smaller earthquakes. This phenomenon is known as the magnitude saturation. Body wave magnitude mB and Richter local magnitudes ML saturate at magnitudes of 6 to 7, and surface wave magnitude Ms saturates at Ms = 8, as shown in Figure 2.14. Because these amplitudes tend to reach limiting values, they may not accurately reflect the size of very large earthquakes. The moment magnitude (denoted as Mw or M ), introduced by Hanks and Kanamori (1979) to describe the size of any earthquake, is defined as Mw = M = 23 log10 M0 − 10.7, (2.1.2) which is not obtained from ground-motion characteristics. M0 is the seismic moment (in dyne-cm) of an earthquake given by ¯ M0 = μA D,

(2.1.3)

30

Maximum Amplitude A t

Maximum amplitude of ground motion.

Figure 2.13 9

Mw (M )

Ms

8 ML Magnitude

7

mb

6 5 4 3 3

4

5 6 7 8 9 Moment Magnitude Mw (M )

Figure 2.14

10

Earthquake magnitudes.

Table 2.2 Energy released in an earthquake.

ML

e (Joules)

6 7 8 9

6.310× 1013 1.995× 1015 6.310× 1016 1.995× 1018

Number of Nuclear Bombs 0.8 25 794 25,099

where μ is the rupture strength of the material along the fault, A is the rupture area, and D¯ is the average amount of slip. Because the seismic moment is a measure of the work done by the earthquake, the moment magnitude is the only magnitude scale not subject to saturation. Figure 2.14 shows a comparison between different magnitude scales. Because earthquake magnitude scales are base-10 logarithmic, a unit change in magnitude corresponds to a 10-fold change in the magnitude parameter (groundmotion characteristic or seismic moment). The total seismic energy (in ergs = 10−7 joules) released during an earthquake is often estimated as (2.1.4) log10 e = 11.8 + 1.5 ML . A unit change in magnitude corresponds to a 101.5 = 32-fold change in energy released by an earthquake. The energy released in an atomic bomb of the size used at Hiroshima is 19,000-ton TNT equivalent (1 ton of TNT = 4.184×109 joules). Table 2.2 lists the relationship between energy released and the Richter magnitude ML in an earthquake.

2.1 earthquakes

31 Before Earthquake

Lock ed

Tsunami

Tsunami Water pushed up During Earthquake

Subside Spring upward Slippe d

Figure 2.15 Wave Length 213 km

Generation of tsunami. 10.6 km 23 km

50 m 4,000 m

10 m

Depth (m) Velocity (km/hr) Wave Length (km) 7,000 943 282 4,000 713 213 2,000 504 151 200 159 48 50 79 23 10 36 10.6

Figure 2.16

Propagation of tsunami.

2.1.6 Generation of Tsunami As relative movement of the tectonic plates occurs, the plates are locked together at the boundary, and tremendous elastic strain energy is built up in the materials near the boundary, as illustrated in Figure 2.15. When the shear stress reaches the shear strength of the rock along the fault, the rock fails, and the accumulated strain energy is released. If the rock is strong and brittle, rupture of the rock will release the lock at the boundary, and the top plate springs upward explosively. The sudden displacement in the seabed is sufficient to cause the sudden raising or lowering of a large body of water. Following the initial disturbance to the sea surface, water waves spread in all directions. Figure 2.16 illustrates the propagation of tsunami waves. Their speed of travel in  deep water is given by gH, where H is the sea depth. In the open sea, tsunami waves are characterized by very large wavelength, reaching tens to hundreds of kilometers. Their velocities, depending on the water depth, can

32

reach as high as 800 to 900 km/hour. They can propagate for thousands of kilometers without dissipating much of their energy and can travel long distances. Tsunami waves are usually unnoticed because of their heights commonly not exceeding 1 m. When tsunamic waves approach the coast, their velocities decrease directly proportionally to the water depth; their heights increase and can reach tens of meters.

2.2 Case Study − The Great East Japan Earthquake The Great East Japan Earthquake (東日本大震災, Higashi Nihon Daishinsai), also known as the 2011 T¯ohoku Earthquake, occurred at 14:46 JST (05:46 UTC) on Friday, March 11, 2011. It was an undersea mega thrust earthquake with a moment magnitude Mw = 9.0. The epicentre was at 38.322◦ N, 142.369◦ E, near the east coast of the island of Honshu, with the hypocenter at an underwater depth of approximately 32 km. The nearest major city to the epicentre is Sendai, on the main island of Honshu, 130 km away. The duration of the quake was six minutes. It was the most powerful known earthquake to have hit Japan; the earthquake and the accompanying tsunami provoked the largest crisis that Japan has encountered in the 65 years since the end of World War II. It is the fourth most powerful earthquake in the world, since modern record keeping began in 1900. The largest PGA (three components vector sum) among K-NET and KiK-net sites was recorded at MYG004 K-NET station (latitude: 38.7292◦ , longitude: 141.0217◦ , distance to epicentre: 173.3 km) reaching 2.99g. Japan lies at the crossing of four tectonic plates: Eurasian, North American, Philippine, and Pacific plates, as shown in Figure 2.17. Details of the Kanton Triple Junction are shown in Figure 2.18. It is clearly seen that the Philippine plate subducts under the Eurasian plate at the Nankai Trench, and the Pacific plate subducts under both the Philippine and Eurasian plates at the Japan Trench and Izu-Bonin Trench. As a result, Japan endures 20 % of the world’s powerful earthquakes. The northern part of Japan belongs to the North American plate but the relative displacement between the Eurasian and the North American plates in this region is relatively low. The earthquake was an extremely massive event; the fault moved upwards of 30−40 m and slipped over an area approximately 300 km long (along-strike, N-S) by 150 km wide (in the down-dip direction, E-W) caused multiple ruptures of seismic sources. The rupture zone is roughly centred on the earthquake epicentre along strike, while peak slips were up-dip of the hypocentre, towards the Japan Trench axis. As shown in Figures 2.19 and 2.20, the main earthquake was preceded by a number of large foreshocks and followed by hundreds of aftershocks. The first major foreshock was a 7.2 Mw event on March 9, approximately 40 km from the location of the March 11 quake, with another three on the same day in excess of 6.0 Mw .

2.2 case study − the great east japan earthquake

33

Figure 2.17 Tectonic plates around Japan.

c Pa

e lat nP

te

ras ia

Pla

ch Tren

ific

n Japa

Japan

in on c Ar

ch

n re iT

N

a nk

a

Ph

Figure 2.18

e pin

te

Pla

ilip

Kanton triple junction.

ch en Tr

B u-

in

n Bo

Eu

u-

Iz

Iz

Sagami Trench

34

Magnitude (Mw)

9.0

8.0

7.0

6.0

5.0 03/09

03/11

03/13

03/15

Date (MM/DD) 03/17

Figure 2.19 Time distribution of earthquake magnitudes.

Figure 2.20

Location distribution of earthquake magnitudes.

An upthrust of 6 to 8 m along the rupture fault resulted in a major tsunami that brought destruction along the Pacific coastline of Japan’s northern islands. Figure 2.21 shows the distribution of height of tsunamic waves along the Pacific coastline of Japan.

Fukushima Nuclear Power Stations Table 2.3 lists the PGA values at Fukushima Nuclear Power Stations (NPS) and the corresponding design basis earthquake (DBE) accelerations. It is seen that PGA in E-W direction at Daiichi Units 2, 3, and 5 exceeded the design tolerances. Ground acceleration thresholds for automatic reactor shutdown are 1.35−1.50 m/sec2 in the horizontal directions or 1.00 m/sec2 in the vertical direction. When the earthquake occurred, the reactors of Units 1, 2, and 3 were operating. They were successfully

2.2 case study − the great east japan earthquake

Figure 2.21

35

Height of tsunami waves.

Table 2.3 Ground acceleration at Fukushima Nuclear Power Stations

Fukushima

Daiichi

Daini

Unit 1 2 3 4 5 6 1 2 3 4

Peak Ground Accel. (m/sec2 ) Horizontal Vertical N-S E-W 4.60 4.47 2.58 3.48 5.50 3.02 3.22 5.07 2.31 2.81 3.19 2.00 3.11 5.48 2.56 2.98 4.44 2.44

Design Basis Accel. (m/sec2 ) Horizontal Vertical N-S E-W 4.87 4.89 4.12 4.41 4.39 4.20 4.49 4.41 4.29 4.47 4.45 4.22 4.52 4.52 4.27 4.45 4.48 4.15

2.54 2.43 2.77 2.10

4.34 4.28 4.28 4.15

2.30 1.96 2.16 2.05

3.05 2.32 2.08 2.88

4.34 4.29 4.30 4.15

5.12 5.04 5.04 5.04

shut down by the automatic systems installed as part of the design of the NPS to detect earthquakes (called SCRAM).Although all off-site power was lost when the earthquake occurred, all available emergency diesel generator power systems were in operation, as designed. Units 4, 5, and 6 had already been shut down for periodic inspection. To protect Fukushima Daiichi NPS again tsunami (Figure 2.22), the design basis was 5.7 m (height of seawall). However, the first of a series of large tsunami waves reached the site about 46 min after the earthquake. The maximum height of tsunami was 14 to 15 m. The ground level is 10 m at Units 1−4, and the units were inundated by about 4

36 O.P. — Onahama Port Construction Base Level Inundation Height Turbine Building Approximately O.P. +14–15 m Seawall Site Level O.P. +4 m Height O.P. +5.7 m Site Level Base Level O.P. +10 m O.P. 0 m (Units 1–4) Water Intake Site Level of Units 5 & 6 O.P. +13 m Seawall

Reactor Building

Figure 2.22

Elevations of Fukushima Daiichi NPS.

to 5 m of water. The ground level is 13 m at Units 5 and 6, and the units were inundated by up to 1 m of water. For Fukushima Daini NPS, the design basis was 5.2 m. The height of tsunami was about 6.5 to 7 m. Although the ground level at the site is 12 m above sea level, the run-up height of the wave in the main building area reached about 14 to 15 m on the south side of the Unit 1 buildings, with flooding of about 2 to 3 m. The area on the ocean side, where seawater pumps were located, was inundated. Although the plants withstood the effects of the earthquake, the tsunami caused the loss of all power sources, except for one emergency diesel generator providing emergency power shared by Units 5 and 6. The station blackout rendered the loss of all instrumentation and control systems at Units 1−4: safety systems for cooling of the reactor cores and control of containment pressures. The tsunami and associated large debris caused widespread destruction of many buildings, doors, roads, tanks, and other site infrastructure, including loss of heat sinks. The operators had to work in darkness, with almost no instrumentation and control systems, to secure the safety of reactors and associated fuel pools, a common fuel pool, and dry cask storage facilities. In summary, at Units 2, 3, and 5 of the Fukushima Daiichi NPS, the acceleration response spectra of seismic ground motion observed on the basemat exceeded the design basis in a part of the period-band. Although damage to the external power supply was caused by the earthquake, no damage caused by the earthquake to systems, equipment, or devices that are important for nuclear reactor safety, at nuclear reactors, has been confirmed. On the other hand, the tsunamis that hit the Fukushima Daiichi NPS were 14−15 m high, substantially exceeding the height assumed under the design of construction permit or the subsequent evaluation. The tsunamis severely damaged seawater pumps and other equipment, causing the failure to secure the emergency diesel power supply and reactor cooling function. The procedural manual did not assume flooding from a tsunami, but rather only stipulated measures against a backrush. The design against tsunamis has been based on tsunami folklore and indelible traces of tsunami, not on adequate consideration of the recurrence of large-scale earthquakes in relation to a safety goal to be attained. The assumption on the frequency and height of tsunamis was insufficient; therefore, measures against large-scale tsunamis were not prepared adequately.

2.3 strong ground motion

37

2.3 Strong Ground Motion 2.3.1 Strong-Motion Measurement Ground motion, also known as ground shaking, is a vibration near or at the ground surface of the earth induced by the seismic waves released from the seismic sources. Although the world’s first seismic instrument (seismoscope) was invented in 132 B.C., the first measurements of ground motions, which are important from an engineering perspective, were made only several decades ago. The current widely used instruments for ground-motion measurements are generally divided into two categories: seismographs and accelerographs. The seismographs are used to measure relatively weak ground motions, and the records they produce are called seismograms. Seismologists usually use seismograms to investigate and better understand geophysics or seismology-related areas, such as seismic source mechanisms, earth mediums along the seismic wave paths, and seismic wave propagations. The accelerographs are used to measure strong ground motions and to produce the records in the form of accelerograms in most cases. The strong-motion records are of great interest to engineers who work on engineering seismology, earthquake engineering, or structural engineering. They may be used as a direct seismic input to analyze engineering structures or used as inputs for seismic hazard analysis and for determining the design earthquakes (as discussed in Chapters 4 and 5). The main purpose of an accelerograph is to measure the entire strong groundmotion process near or at the ground surface of the earth during an earthquake. Ground-motion acceleration is usually selected as the physical quantity to describe the ground-motion process with time based on several reasons: 1. Acceleration is closely relevant to the seismic inertia forces, which are the main parameter considered in earthquake and structural engineering. 2. Recording apparatus for acceleration-based accelerographs can be readily manufactured. 3. Ground-motion velocity and displacement processes can be easily obtained mathematically from the recorded acceleration process. As a result, most accelerographs record strong ground motions in terms of acceleration. Early strong-motion recording instruments are analog accelerographs, which transform the ground motions to the motion of a physical mechanism using a stylus on paper. Although the strong-motion records produced by these analog accelerographs have played an extremely important role in the development of earthquake engineering, several deficiencies of analog accelerographs have been widely recognized: 1. Because the analog accelerographs are triggered to start recording by the exceedance of a small threshold acceleration at the beginning of the earthquake motion, they are unable to record any vibrations that have preceded the triggering.

38

Figure 2.23 An example of ground motion recorded in Manitoba, Canada.

2. The accuracy and capacity of the analog accelerographs are not adequate to record strong motions with long-period (low-frequency) components and high amplitudes of acceleration. 3. The physical recording mechanism of the analog accelerographs results in considerable errors in the digitization of the records (including corrections for instrument response and baseline), which may require significant effort on processing the strong-motion records. Compared to analog accelerographs, modern digital accelerographs are able to overcome these deficiencies. Digital accelerographs are capable of recording strongmotions with much wider ranges of frequency and amplitude and have significantly improved the accuracy. Digital accelerographs are able to record a process with several seconds prior to the arrival of the seismic waves, such that a complete strong-motion record (starting with P-wave, containing the maximum amplitude of the P-wave, and having complete processes of S-wave and surface waves) can be obtained. An example of a complete ground motion recorded in Manitoba, Canada, is shown in Figure 2.23.

2.3.2 Ground-Motion Parameters Earthquake engineers are interested in strong ground motion, i.e., motion of sufficient strength to affect people and their environment. At a given point, ground motions produced by earthquakes can be completely described by three components of translation (two orthogonal horizontal directions and one vertical direction, e.g., East–West, North–South, Vertical) and three components of rotation, which are usually neglected in engineering practice due to their complexity and minor effect. For engineering purposes, three characteristics of earthquake motion are of primary significance: amplitude, frequency content, and duration of the motion. It is important

2.3 strong ground motion

39

to identify a number of ground-motion parameters that reflect these characteristics in a compact and quantitative form by analyzing the strong-motion records. Figure 2.24 shows the East–West component of ground acceleration, velocity, and displacement of the 1940 El Centro Earthquake. The acceleration time-history shows a significant proportion of relatively high frequencies. Because velocity is the integration of acceleration and integration produces a smoothing or filtering effect, the velocity time-history shows substantially less high-frequency motion than the acceleration time-history. Displacement time-history is dominated by relatively low-frequency motion. Peak ground acceleration (PGA), as illustrated in Figure 2.24, is the most commonly used parameter for describing ground motion, because of the relationship between acceleration and inertial force. Ground motion has two orthogonal horizontal and one vertical components. In seismic design and analysis practice, it is usually assumed that the peak ground accelerations in the two horizontal directions are equal. The maximum vertical acceleration is assumed to be a factor (e.g., two-thirds) of the peak horizontal component. The largest dynamic forces induced in certain types of structures (very stiff structures) are closely related to the PGA. Because the velocity is less sensitive to the higher-frequency components of the ground motion, peak ground velocity (PGV) can characterize ground-motion amplitude more accurately at intermediate frequencies. For structural systems with dominant intermediate frequencies, such as tall or flexible buildings, PGV provides a more accurate indication of the potential for damage than PGA. Peak ground displacement (PGD) can characterize ground-motion amplitude more accurately at low frequencies. For flexible structural systems with dominant low frequencies, PGD provides a more accurate indication of the potential for damage. Because the PGA is considered as a random number of a strong-motion process, the sum of all the squared acceleration values from a strong-motion record can be used as a more reliable parameter to represent the statistical characteristics of the amplitude of a strong-motion record, i.e., the Arias intensity defined as  T   π A(t) 2 dt, (2.3.1) IA = 2g

0

where T is the duration of the earthquake. A plot of the buildup of Arias intensity with time, i.e,  t   π IA (t) = A(s) 2 ds, 0  t  T, 2g 0

(2.3.2)

or the cumulative normalized Arias intensity IA (t)/IA is known as a Husid plot, and it serves to identify the interval over which the majority of the energy is imparted. The duration of a strong-motion is related to the time required for release of accumulated strain energy by rupture along the fault. As the length, or area, of fault rupture

0.2 0.1 0 −0.1 −0.2 −0.3

Time Interval Δt = 0.01s T = 53.45s Ground Acceleration

PGA= 0.2107g

40 20 0

V (cm/s)

A (g)

40

D (cm)

−20 −40 30 20 10 0 −10

Ground Velocity

PGV =31.31 cm/s PGD = 24.15 cm

Ground Displacement

0

10 20 30 40 Figure 2.24 East–West component of the 1940 El Centro Earthquake.

Time (s) 50 53.45

Cumulative Normalized Arias Intensity IA(t) / IA (%)

A (g)

Strong-Motion Duration t5-95 0.2 0.1 0 −0.1 −0.2 −0.3

PGA 0

10 20 Strong-Motion Duration t5-75

2.14

Ground Acceleration

0.2107g 26.29

30

40

T= 53.45 s Time (s) 50

100 95

IA = IA(53.45 s) = 1.1687 m/s

80 75

60

40

20 5

Time (s) 19.87 26.29 30 10 20 40 50 53.45 0 2.14 Figure 2.25 Arias intensity and strong-motion duration of the El Centro Earthquake. 0

increases, the time required for rupture increases. The duration is proportional to the cubic root of the seismic moment. For engineering purposes, only the strong-motion portion of the accelerogram is of interest. A popular definition of a strong-motion duration tm used in nuclear industry is t5−75 (the time span between 5 % and 75 % of the Arias intensity) or t5−95 (the time span between 5 % and 95 % of the Arias intensity).

Fourier Amplitude Spectrum (m/s)

Fourier Amplitude Spectrum (m/s)

Fourier Amplitude Spectrum (m/s)

2.3 strong ground motion

41

3.5 3

(a) Frequency Resolution = Δf = 0.0122 Hz

2.5 2 1.5 1 0.5 0

0

5

10

Frequency (Hz)

15

20

25

20

25

20

25

2.5 2

(b) Frequency Resolution = 4 Δf = 0.0488 Hz

1.5 1 0.5 0

0

5

10

Frequency (Hz)

15

2 (c) Frequency Resolution = 8 Δf = 0.0976 Hz

1.5 1 0.5 0

0

5 Figure 2.26

10

Frequency (Hz)

15

FAS of the El Centro Earthquake.

Figure 2.25 presents the cumulative normalized Arias intensity IA (t)/IA of the East–West component of the El Centro Earthquake and the strong-motion duration     t5−75 = 2.14, 19.87 ⇒ 17.73 s or t5−95 = 2.14, 26.29 ⇒ 24.15 s. The amplitudes of the ground motions, such as PGA, PGV, PGD, or Arias intensity, are individual parameters to represent the strength of the ground motion. The frequency contents, which have been widely recognized as important parameters to engineering structures, are usually described by ground response spectra (GRS, see Chapter 4) or Fourier amplitude spectra (FAS, see Section 3.1.4 and Appendix B, in particular, Sections B.2 and B.6).

42

The duration of the E-W component of the El Centro Earthquake is T = 53.45 s with time interval s = 0.01 s, resulting in 53.45/0.01 = 5345 data points. To perform fast Fourier transform (FFT) to obtain FAS, the number of data points must be 2N ; hence, 0s are padded at the end of the earthquake to extend the duration to T = 81.92 s so that there are 213 = 8192 data points. FAS of the E-W component of the El Centro Earthquake is shown in Figure 2.26(a) with frequency resolution F = 1/T =1/81.92 = 0.0122 Hz. Because F = 1/T, the longer the duration T of the earthquake time series, the higher the frequency resolution (the smaller the value of F ). It is seen in Figure 2.26(a) that, at such a high-frequency resolution, there are clusters of sharp spikes in the FAS, because it is able to differentiate harmonics at 0.0122 Hz. As discussed in Section B.6, for engineering applications, it is necessary to increase the value of frequency interval F to obtain a smoother FAS; Figure 2.26(b) and (c) shows FAS with frequency resolutions 4F and 8F . As discussed in Section 3.1.3, power spectral density (PSD) function SXX (ω) is very important in characterizing a stationary random process X(t). For a transient nonstationary random process, FAS is often used (Section 3.1.4). Nevertheless, PSD is frequently employed in earthquake engineering to characterize the frequency contents of strong-motion portions of earthquake ground motions because the strong-motion portion of an earthquake ground motion can be reasonably modelled as a stationary Gaussian process (Section 3.5.2). For an acceleration time-history A(t), the one-sided PSD is defined by equation (2-1) in ASCE/SEI 4-16 (ASCE/SEI, 2017) or equation (1) in Appendix B of SRP 3.7.1 (USNRC, 2012b): 2  2A(ω) SAA (ω) = , (2.3.3) 2πtm     A(t) is the Fourier transform of A(t) over the duration tm , A(ω) where A(ω) = is the one-sided FAS, and tm = t5−75 is the strong-motion duration required for the Arias intensity to rise from 5 % to 75 %. For the E-W component of the El Centro Earthquake, the one-sided PSD and the average one-sided PSD are shown in Figure 2.27(a) and (b). To obtain average PSD, at any frequency F, the average PSD is computed over a frequency bandwidth of ±20 % centred on the frequency F, i.e.,   0.8 F, 1.2 F .

F

GRS are used extensively in earthquake engineering practice. A plot of the peak value of a response quantity, such as acceleration, velocity, or displacement, as a function of the natural vibration period T or natural frequency F of a linear single degree-offreedom (SDOF) system is called the earthquake response spectrum for that quantity. Each such plot is for SDOF systems having a range of natural frequencies, such as 0  F  100 Hz, and a fixed damping ratio ζ ; several such plots for different values of ζ are included to cover the range of damping values encountered in engineering structures. A detailed study of GRS is presented in Chapter 4.

2.4 probabilistic ground-motion parameters

43

Power Spectral Density (m2/s3)

0.05

(a) One-Sided PSD

0.04

0.03

0.02

0.01

0 0

5

10

Frequency (Hz)

15

20

25

Power Spectral Density (m2/s3)

0.016 0.014

(b) Average One-Sided PSD 0.012 0.010 0.008 0.006 0.004 0.002 0 0.3

0.4

0.5

0.6 0.7 0.8 0.9 1

Figure 2.27

Figure 2.28

2

3

Frequency (Hz)

4

5

6

7

8 9 10

20

25

PSD of the El Centro Earthquake.

Commonly used source–site distance measures.

2.4 Probabilistic Ground-Motion Parameters 2.4.1 Magnitude and Distance Effects Much of the energy released in an earthquake takes the form of stress waves. Because the amount of energy released is strongly related to the earthquake magnitude, the characteristics of the stress waves are strongly related to magnitude. The characteristics

44

of the stress waves are also strongly related to source–site distance. As stress waves travel away from the source of an earthquake, they spread out and are partially absorbed by the materials they travel through. Energy per unit volume decreases with increasing distance from the source. A number of commonly used source–site distance measures are illustrated in Figure 2.28. Observations of the relationships between the stress waves and the earthquake magnitude and source–site distance have provided a strong link between the ground-motion parameters and the earthquake magnitude and source–site distance. This link is the basis for deriving ground-motion prediction equations (GMPE) for seismic hazard analysis.

2.4.2 Ground-Motion Prediction Equations GMPEs for ground-motion intensity measures (i.e., ground-motion parameters), such as PGA, and spectral acceleration at individual vibration period, as functions of earthquake magnitude, source–site distance, and some other variables, are important tools in seismic hazard analysis. In this section, for illustration purposes, spectral accelerations for horizontal components are used as the intensity measures. GMPEs are typically developed empirically by regression analyses of recorded strong-motion amplitude data versus magnitude, distance, and possibly other predictor variables. Some GMPEs were also determined based on simulated ground motions (Atkinson and Boore, 2006). GMPEs provide a connection between the intensity measures of earthquake-induced ground motions (e.g., spectral accelerations), which are directly correlated to seismic analysis and design, and the parameters of earthquakes (e.g., magnitude and distance). A typical GMPE for spectral acceleration SA(Tj ) at vibration period Tj can be expressed as (2.4.1) ln SA(Tj ) = F(m, r, Tj , θ ) + σ ε(Tj ), where F(m, r, Tj , θ) is the expected value of the prediction equation, in which, besides the essential parameters (earthquake magnitude m and source–site distance r), various predictor variables θ, such as fault type, hanging wall effect, seismic wave propagation path, and local site condition, may be considered. σ is the total standard deviation of the prediction equation model, which could be a combination of intra-event (ground motions within the same earthquake event) and inter-event (ground motions between different earthquake events) aleatory uncertainties and a function of m, Tj , and other predictor variables. ε(Tj ) is the number of standard deviations σ by which the logarithmic spectral acceleration ln SA(Tj ) deviates from the expected value F(m, r, Tj , θ) (Abrahamson and Silva, 1997; Atkinson and Boore, 2006; Boore and Atkinson, 2008). Given a set of predictor variables m, r, and θ, ε(Tj ) has been verified to follow standard normal distribution; consequently, ln SA(Tj ) follows normal distribution

2.4 probabilistic ground-motion parameters

45

conditional on m, r, and θ (Baker and Jayaram, 2008). Detailed discussion on the probability distribution of spectral accelerations is presented in Section 2.4.4. The functional form of the prediction equation F(m, r, Tj , θ) in equation (2.4.1) is usually selected to reflect the mechanisms of the ground-motion process as closely as possible. However, ground motions are complicated; they are influenced by, and consequently reflect, characteristics of the seismic source, the rupture process, the source–site travel path, and local site conditions (Kramer, 1996). Various functional forms of the prediction equation have been adopted to account for different groundmotion mechanisms, based on the availability of regional recorded ground motions and engineering practice (Douglas, 2011). To introduce the basic concept of GMPE, three examples of typical GMPEs are presented.

☞

As discussed in Section 2.1.5, there are many definitions to quantify earthquake magnitude, such as Richter local magnitude ML and moment magnitude M . A specific set of GMPEs are usually developed in terms of a particular definition of earthquake magnitude; moment magnitude is the most popular. When a set of GMPEs are developed specifically in terms of moment magnitude, the notation M is used for moment magnitude.

☞

In general, earthquake magnitude is a random variable in probabilistic seismic hazard analysis (PSHA). The formulations in PSHA are general and are applicable to any earthquake magnitude definition as long as it is consistent with that for which GMPEs were developed. Following the conventions in probability, the uppercase M is used to denote earthquake magnitude as a random variable, and the lowercase m is used to denote a specific magnitude value that M takes.

GMPEs by Abrahamson and Silva Abrahamson and Silva (1997) developed GMPEs for spectral accelerations. The GMPEs are derived for the geometric average of two horizontal components and vertical component for shallow earthquakes in active tectonic regions. A database of 655 sets of tridirectional recorded ground motions from 58 earthquakes are used in the regression analysis. The functional form F(m, r, Tj , θ ) in equation (2.4.1) of the GMPE is given by ln SA = F 1 (M , rrup ) + F · F 3 (M ) + HW · F 4 (M , rrup ) + S · F 5 (PGArock ),

(2.4.2)

where SA is spectral acceleration at a specific vibration period, M is moment magnitude, rrup is the closest distance to the rupture plane, F is a coefficient for the fault type (1 for reverse, 0.5 for reverse/oblique, and 0 otherwise), HW is a coefficient for hanging wall sites (1 for sites over the hanging wall and 0 otherwise), and S is a coefficient for local site conditions (0 for rock or shallow soil and 1 for deep soil).

46

Function F 1 (m, rrup ) in equation (2.4.2), which is the basic functional form of the GMPE for strike-slip events recorded at rock sites, is given by    F 1 (M , rrup ) = a1 + a(M −6.4) + a12 (8.5− M )n + a3 + a13 (M −6.4) ln r2rup + c24 , (2.4.3) where a = a2 for M  6.4 and a = a4 for M > 6.4. Function F 3 (M ) in equation (2.4.2), which allows for magnitude and period dependence of the type of fault, is given by ⎧ for M  5.8, a , ⎪ ⎪ ⎨ 5 a6 −a5 (2.4.4) F 3 (M ) = a5 + , for 5.8 < M < 6.4, ⎪ 6.4−5.8 ⎪ ⎩ for M  6.4. a6 , Function F 4 (M , rrup ) in equation (2.4.2) that allows for magnitude and distance dependence of the effect of hanging wall is given by F 4 (M , rrup ) = F HW (M ) · F HW (rrup ), where

(2.4.5)

⎧ ⎨ 0,

for M  5.5, F HW (M ) = M −5.5, for 5.5 < M < 6.5, ⎩ 1, for M  6.5, ⎧ ⎪ 0, for rrup  4 km, ⎪ ⎪ ⎪ ⎪ 1 ⎪ for 4 km < rrup  8 km, ⎪ ⎨ 4 a9 (rrup −4), for 8 km < rrup  18 km, F HW (rrup ) = a9 ,   ⎪ ⎪ ⎪ ⎪ a9 1− 1 (rrup −18) , for 18 km < rrup  24 km, ⎪ 7 ⎪ ⎪ ⎩ 0, for r > 24 km. rup

The nonlinear soil response term F 5 (PGArock ) in equation (2.4.2) is modelled by

F 5 (PGArock ) = a10 + a11 ln PGArock +c5 , (2.4.6) where PGArock is the expected value of peak ground acceleration (PGA) on rock, as predicted by the expected prediction equation (2.4.2) with S = 0. The total standard deviation σ in equation (2.4.1) of the GMPE, including intraevent and inter-event aleatory uncertainties, is magnitude-dependent modelled by ⎧ for M  5.0, ⎪ ⎨ B5 , (2.4.7) σtotal (M ) = B5 − B6 (M −5), for 5.0 < M < 7.0, ⎪ ⎩ B5 − 2 B6 , for M  7.0. In equation (2.4.7), the total standard deviation σtotal (M ) for the horizontal ground motions is determined by calculating the geometric average of two horizontal components. The total standard deviation can also be obtained for arbitrary horizontal

2.4 probabilistic ground-motion parameters

47

components of ground motions by inflating σtotal (M ) using a functional fit developed by Baker and Cornell (2006c). Consequently, the standard deviation for arbitrary horizontal components, in terms of magnitude M and vibration period T, is given by  2 σtotal, arb (M , T) = σtotal (M ) . (2.4.8) 1.78 − 0.039 lnT

In equations (2.4.2) to (2.4.7), a1 , . . . , a6 , a9 , . . . , a13 , c4 , c5 , n, B5 , and B6 are perioddependent parameters of regression analysis.

GMPEs by Atkinson and Boore Atkinson and Boore (2006) developed GMPEs for horizontal ground motions at hard rock and soil sites in eastern North America (ENA). The GMPEs are obtained by regression analysis of a database of simulated ground motions using a stochastic finitefault model. The GMPE model incorporates information obtained from new ENA seismographic data, including tridirectional broadband ground motions that provide new information on ENA source and path effects. The functional form of the GMPE is given by log10 SA = c1 + c2 M + c3 M 2 + (c4 +c5 M ) · F 1 + (c6 +c7 M ) · F 2 + (c8 +c9 M ) · F 0 + c10 rcd + S, (2.4.9)    

R 10 F0 = max log10 r , 0 , F1 = min log10 rcd , log10 70 , F2 = max log10 cd , 0 , 140

cd

where SA is spectral acceleration at a specific vibration period, M is the moment magnitude, and rcd is the closest distance to the fault rupture. To account for soil amplification in both linear and nonlinear ranges, a piecewise soil amplification factor S in equation (2.4.9), as a function of shear wave velocity in the upper 30 m (V30 ), is modelled as ⎧ 0, for hard-rock sites, ⎪ ⎪ ⎪    ⎪ ⎪ 2 V 60 ⎨ , for pgaBC  60 cm/s , log10 exp Blin · ln 30 + Bnl · ln 760 100 . S= (2.4.10) ⎪ ⎪ ⎪    ⎪ ⎪ ⎩ log exp B · ln V30 + B · ln pgaBC , for pgaBC > 60 cm/s2 , lin nl 10 760

100

where pgaBC is the predicted value of PGA for V30 = 760 m/s, Blin is the linear factor of soil amplification, and the nonlinear soil amplification factor Bnl is given by ⎧ for V30  180 m/s, B , ⎪ ⎪ ⎪ 1 ⎪ ⎪ lnV30 − ln300 ⎪ ⎪ ⎪ (B1 −B2 ) + B2 , for 180 m/s < V30  300 m/s, ⎨ ln180 − ln300 Bnl = (2.4.11) ⎪ lnV − ln760 ⎪ 30 ⎪B , for 300 m/s < V30  760 m/s, ⎪ 2 ⎪ ⎪ ln300 − ln760 ⎪ ⎪ ⎩ 0, for V30 > 760 m/s.

48

In equation (2.4.9), the spectral acceleration is predicted for an event with fault stress of 140 bars. For stress values other than 140 bars (within the tested range from 35 to 560 bars), the spectral acceleration log10 SA, adj is obtained by adjusting log10 SA in equation (2.4.9) as log10 SA, adj = log10 SA + SF1 · log10 SF2 , SF1 =

log10 Stress − log10 140 

log10 2

(2.4.12a)

,

(2.4.12b) 

 · max (M −M1 ), 0 log10 SF2 = min +0.05, 0.05+ Mh −M1



 .

(2.4.12c)

The standard deviations of log10 SA for all vibration periods are 0.3. In equations (2.4.9) to (2.4.12), c1 , . . . , c10 , Blin , B1 , B2 , , M1 , and Mh are period dependent parameters of regression analysis.

GMPEs by Boore and Atkinson Boore and Atkinson (2008) developed GMPEs for the geometric average values of two horizontal components of ground motions as a function of earthquake magnitude, source–site distance, local average shear wave velocity, and fault type. The geometric average is determined from the 50th percentile value of the geometric mean values computed for all non-redundant rotation angles, which is approximately identical to the geometric mean in most cases (Boore et al., 2006). The GMPEs are derived by regression analysis of an extensive strong-motion database compiled by the Next Generation Attenuation (NGA) project of the Pacific Earthquake Engineering Research (PEER) Center. For periods less than 1 s, the analysis uses 1574 recorded ground motions from 58 main-shocks in the distance range from 0 km to 400 km (the number of available data decreases with increasing vibration period). The functional form F(m, r, Tj , θ) in equation (2.4.1) of the GMPE is given by ln SA = FM (M ) + FD (rJB , M ) + FS (V30 , rJB , M ),

(2.4.13)

where FM , FD , and FS represent the magnitude scaling, distance, and site amplification functions, respectively. M is the moment magnitude, rJB is the Joyner-Boore distance (defined as the closest distance to the surface projection of the fault, which is approximately equal to the epicentral distance for events of M < 6), and V30 is the average shear wave velocity from the surface to a depth of 30 m. The magnitude scaling function FM in equation (2.4.13) is given by  e1 · U+e2 · SS+e3 · NS+e4 · RS+e5 (M −MH )+e6 (M −MH )2, for M  MH , FM (M ) = for M > MH , e1 · U + e2 · SS+e3 · NS+e4 · RS+e7 (M −MH ), (2.4.14)

2.4 probabilistic ground-motion parameters

49

where U, SS, NS, and RS are coefficients (1 for unspecified, strike-slip, normalslip, and reverse-slip fault type, respectively, and 0 otherwise) and MH , the “hinge magnitude” for the shape of the magnitude scaling, is a coefficient to be determined during the analysis. The distance function FD in equation (2.4.13) is given by      2 + H2 + c 2 + H2 − 1 . FD (rJB , M ) = c1 + c2 (M −4.5) ln rJB r (2.4.15) 3 JB The site amplification function FS in equation (2.4.13) is given by FS (V30 , rJB , M ) = FLIN + FNL ,

(2.4.16)

where the linear term FLIN is FLIN = Blin ( lnV30 − ln760), and the nonlinear term FNL is   ⎧ 0.06 ⎪ ln , for pga4nl  0.03g, B ⎪ nl ⎪ 0.1 ⎪ ⎪ ⎪   2    pga4nl 3  ⎪ ⎪ ⎨ B ln 0.06 + c ln pga4nl + D ln , nl 0.1 0.03 0.03 FNL = ⎪ ⎪ ⎪ for 0.03g < pga4nl  0.09g, ⎪ ⎪ ⎪  pga4nl  ⎪ ⎪ ⎩ B ln , for pga4nl > 0.09g, nl

c=

(2.4.17)

(2.4.18)

0.1

  1 3Bnl ( ln0.09− ln0.06) − Bnl ( ln0.09− ln0.03) , 2 ( ln0.09− ln0.03)

  −1 2B ( ln0.09− ln0.06) − B ( ln0.09− ln0.03) , nl nl ( ln0.09− ln0.03)3 ⎧ for V30  180 m/s, B , ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ (B1 −B2 )( lnV30 − ln300) ⎪ ⎪ ⎪ + B2 , for 180 m/s < V30  300 m/s, ⎨ ln180− ln300 Bnl = ⎪ B ( lnV30 − ln760) ⎪ ⎪ 2 ⎪ for 300 m/s < V30 < 760 m/s, ⎪ ⎪ ln300− ln760 , ⎪ ⎪ ⎪ ⎩ 0, for V30  760 m/s.

D=

The total standard deviation σ for ln SA in equation (2.4.13), including intra-event and inter-event aleatory uncertainties, has been tabulated for fault type specified and unspecified (Boore and Atkinson, 2008). In equation (2.4.18), pga4nl is the predicted PGA for V30 = 760 m/s using equation (2.4.13) with the site amplification function FS = 0. In equations (2.4.14) to (2.4.18), e1 , . . . , e7 , c1 , . . . , c3 , B1 , B2 , MH , H, and Blin are period-dependent parameters of regression analysis.

50

As can be seen in these three examples of GMPE, various functional forms have been used to characterize the ground-motion mechanisms based on the availability of recorded ground motions and the corresponding geographical locations, over which the recorded ground motions were collected. Historically, the functional forms of GMPE used in regression analysis have been revised significantly with the increase of recorded ground motions since 1960s (Douglas, 2011). Most GMPEs are updated in the literature every 3 to 5 years or shortly after the occurrences of major earthquakes in well-instrumented regions (Kramer, 1996). In this situation, it is not feasible to have the “correct” functional form of a GMPE with 100 % confidence. In other words, each GMPE contains epistemic uncertainty to a varying extent. It is therefore important to include multiple GMPEs in seismic hazard analysis or risk analysis with weights (McGuire, 2004). One useful procedure is 1. Determine the magnitude-distance range most critical to seismic hazard analysis. 2. Collect a set of recorded ground motions relevant to the site of interest. 3. Compare the selected GMPEs with the ground-motion data in the magnitudedistance range. GMPEs that fit the ground-motion data better are given high weights, and GMPEs that fit less well are given low weights. The weight can be calculated proportionally to the inverse of the residual of the data around each GMPE (McGuire, 2004). On the other hand, when applying any GMPE, it is very important to ensure that parameters, such as spectral acceleration, magnitude, and distance, are defined and used consistently with the application. For example, the horizontal spectral acceleration of a GMPE can be predicted for the geometric average of two horizontal components, arbitrary horizontal component, or the 50th percentile values of the geometric means of two horizontal components computed for all non-redundant rotation angles. Typically, GMPE for arbitrary horizontal component should be used for horizontal ground motion when structural responses are predicted by engineers (Baker and Cornell, 2006c).

2.4.3 Correlation of Ground-Motion Parameters GMPEs describe the probability distributions of spectral accelerations at individual vibration periods, given a set of predictor variables, such as magnitude and distance. However, the statistical correlations between spectral accelerations at multiple periods are not addressed by GMPEs. Because the correlations of spectral accelerations at multiple periods are required in determining the joint distribution functions of spectral accelerations in vector-valued PSHA, the Pearson product-moment correlation coefficients of spectral accelerations have been obtained empirically based on a large number of recorded ground motions (Baker and Cornell, 2006a; Baker and Jayaram, 2008). In equation (2.4.1), only ε(Tj ) and ln SA(Tj ) are random variables, which follow standard normal distribution and normal distribution conditional on predictor vari-

2.4 probabilistic ground-motion parameters

51

ables (m, r, and θ), respectively. Hence, the determination of correlation coefficients of spectral accelerations is equivalent to that of correlation coefficients of ε. By rearranging equation (2.4.1), ε(Tj ) can be expressed as 



ε(Tj ) = σ ln SA(Tj ) − F(m, r, Tj , θ ) . 1

(2.4.19)

For a given sample (a recorded ground motion with known SA(Tj ), m, r, and θ), ε(Tj ) in equation (2.4.19) is a known number. Equation (2.4.19) then eliminates the impact of predictor variables (m, r, and θ ) on the variability of ε(Tj ), i.e., the variability of ε(Tj ) is independent of the predictor variables. In practice, it is thus convenient to study ε(Tj ), instead of SA(Tj ), in statistical and regression analysis (Baker and Cornell, 2006a). It is noted that, because the normalized residual ε(Tj ) is used in the analysis, any outcomes are in principle dependent on the GMPEs selected. The Pearson product-moment correlation coefficient between ε(Tu ) and ε(Tv ), at vibration periods Tu and Tv , is given by N  

ρε(T

u ), ε(Tv )

=

I=1 N  

I=1

ε(I) (Tu )−ε(Tu )

ε(I) (Tu )−ε(Tu )



2



ε(I) (Tv )−ε(Tv )

N   I=1



, ε(I) (Tv )−ε(Tv )



(2.4.20)

2

where ε (I) (Tu ) is the number of standard deviations departing from F(m, r, Tu , θ) of the Ith recorded ground motion at period Tu using equation (2.4.19), ε(Tu ) is the mean value of ε of all N recorded ground motions at Tu , and N is the total number of recorded ground motions used in the statistical and regression analysis. Using equation (2.4.20), a symmetric positive semi-definite matrix of the correlation coefficients can be determined from all the combinations of ε at different periods. Given a set of predictor variables (m, r, and θ ) in equation (2.4.19), by substituting equation (2.4.19) into equation (2.4.20), the correlation coefficient between spectral accelerations ln SA(Tu ) and ln SA(Tv ) at any two periods Tu and Tv is obtained as N  

ρ ln S

A(Tu ),

ln SA(Tv )

=

I=1 N   I=1

(I)

ln SA (Tu )− ln SA(Tu ) (I) A (Tu )−

ln S

ln SA(Tu )



(I)

ln SA (Tv )− ln SA(Tv )

N  2  I=1

(I) A (Tv )−

ln S



ln SA(Tv )

2

,

(2.4.21) where ln S is the natural logarithmic spectral acceleration of the Ith recorded ground motion at period Tu , and ln SA(Tu ) is the mean value of logarithmic spectral acceleration of all N recorded ground motions at Tu . Hence, the correlation coefficients of ε and those of spectral accelerations conditional on a given scenario earthquake (known predictor variables m, r, and θ) are identical. (I) A (Tu )

52

Using equations (2.4.19) and (2.4.20), a number of models of correlation coefficients of spectral accelerations (abbreviated as spectral correlation models) have been developed based on different databases of recorded ground motions (Inoue and Cornell, 1990; Baker and Cornell, 2006a; Baker and Jayaram, 2008). Two recent correlation models are described below.

Spectral Correlation Model by Baker and Cornell Baker and Cornell (2006a) developed a spectral correlation model empirically based on 267 sets of recorded ground motions. An approximate analytical equation for the correlation coefficients between horizontal spectral accelerations at any two vibration periods is determined using nonlinear regression. The spectral correlation model is equally valid for both arbitrary component and geometric mean of two horizontal components of spectral accelerations. The valid range of vibration period of this correlation model is between 0.05 s and 5 s. It is observed that the resulting spectral correlations do not vary significantly when different GMPEs are used, which suggests that the correlation model are applicable regardless of the GMPEs chosen. The correlation coefficient between the horizontal spectral accelerations ln SA(Tu ) and at any two periods Tu and Tv (Tu  Tv ) is estimated as      π  Tu T · ln v , − 0.359 + 0.163 i (Tu < 0.189) · ln ρ ln S (T ), ln S (T ) = 1 − cos 0.189 Tu A u A v 2 (2.4.22) where i (Tu < 0.189) is an indicator function equal to 1 if Tu < 0.189 s and equal to 0 otherwise. The matrix of correlation coefficients of spectral accelerations of this model is shown as contours in Figure 2.29. As can be seen in Figure 2.29, the spectral correlations decrease with increasing separation of vibration periods when the periods are larger than around 0.2 s, which cover the range of period of general engineering interest.

Spectral Correlation Model by Baker and Jayaram Baker and Jayaram (2008) developed a spectral correlation model empirically using the NGA ground-motion library and new NGA GMPEs (Abrahamson and Silva, 2008; Boore and Atkinson, 2008; Campbell and Bozorgnia, 2008; Chiou and Youngs, 2008). The spectral correlation model is valid for a variety of definitions of horizontal spectral acceleration, including spectral acceleration of arbitrary component, the geometric average of spectral accelerations from two orthogonal horizontal components, and the 50th percentile value of the geometric means computed for all non-redundant rotation angles. The valid range of vibration period of this model is between 0.01 s and 10 s. The correlation model of spectral correlations has the following properties: 1. It is not sensitive to the choice of accompanying GMPE models.

2.4 probabilistic ground-motion parameters

53 0.0

5

0.1

Period (sec)

0.2 0.3

1

0.4 0.5 0.6 0.7 0.8

0.1 0.05 0.05

Figure 2.29

0.9 0.1

1

5

1.0

Period (sec) Correlation coefficients of spectral accelerations by Baker and Cornell (2006). 10

0.0 0.1

Period (sec)

0.2 0.3

1

0.4 0.5 0.6 0.1

0.7 0.8 0.9

0.01 0.01

Figure 2.30

0.1

1

10

1.0

Period (sec) Correlation coefficients of spectral accelerations by Baker and Jayaram (2008).

2. Intra-event (ground motions in the same earthquake event) error, inter-event (ground motions between different earthquake events) error, and total error in the regression analysis of the GMPEs all exhibit similar correlation structure. The correlation coefficient between the horizontal spectral accelerations ln SA(Tu ) and ln SA(Tv ) at any two periods Tu and Tv (Tu  Tv ) is estimated as ⎧ if Tv < 0.109, C2 , ⎪ ⎪ ⎪ ⎪ ⎨C , else if Tu > 0.109, 1 ρ ln S (T ), ln S (T ) = (2.4.23) A u A v ⎪ min(C2 , C4 ), else if Tv < 0.2, ⎪ ⎪ ⎪ ⎩ else, C4 , where



  Tv π

, − 0.366 ln C1 = 1 − cos 2 max Tu , 0.109

54

C2 =

 ⎧ ⎨ 1 − 0.105 1 − ⎩ 

C3 =

1 1+ exp(100Tv −5)



0,

 Tv −Tu , if Tv < 0.2, Tv −0.0099 otherwise,

C2 , if Tv < 0.109, C1 , otherwise,

C4 = C1 − 0.5



  π T 

u . C3 −C3 1 + cos 0.109

The matrix of correlation coefficients of spectral accelerations of this model is shown as contours in Figure 2.30.

2.4.4 Probability Distribution of Ground-Motion Parameters In Sections 2.4.2 and 2.4.3, the GMPEs and correlation coefficients of spectral accelerations are described. In this section, they are used to determine probability distributions of spectral accelerations, which are the core components in probabilistic seismic hazard analysis (PSHA). For a given set of GMPEs for spectral accelerations at different vibration periods as in equation (2.4.1), based on a large number of recorded ground motions, the same number of samples is obtained for ε(Tj ) at each period Tj . Using normal Q-Q plots, in which “Q” stands for quartile, univariate normality of the samples of ε(Tj ) (including inter-events and intra-events) at individual period Tj are tested. The normal Q-Q plots show strong linearity, indicating that the residuals ε(Tj ) are well represented by a standard normal distribution marginally. Thus, spectral acceleration SA(Tj ) at individual period Tj follows lognormal distribution marginally, for a given scenario earthquake in terms of m, r, θ (Jayaram and Baker, 2008),

P



  SA(Tj ) > sj  m, r = 1 −

 lns − μ j ln S

A(Tj )

σ ln S

A(Tj )

   m,r

 m

,

(2.4.24)

where (·) is the standard normal distribution function, sj is the threshold value, and the mean and standard deviation of ln SA(Tj ) are μ ln S

A(Tj )

  m,r

= F(m, r, Tj , θ ),

σ ln S

A(Tj )

 m

= σ,

(2.4.25)

where F(m, r, Tj , θ) and σ are obtained from the GMPE in equation (2.4.1). It is noted that the marginal lognormal distribution in equation (2.4.24) is conditional on a scenario earthquake, which may be represented by magnitude m, source–site distance r, and other predictor variables θ. Only m and r are expressed explicitly in equation (2.4.24) to represent a scenario earthquake, because m and r are treated as random variables in the PSHA in most cases. In this context, m and r are used

2.4 probabilistic ground-motion parameters

55

to represent a scenario earthquake explicitly, while other predictor variables θ are considered implicitly. Based on the samples of ε(Tj ) (including inter-events and intra-events) at different periods Tj (j = 1, 2, . . . , K), using Henze-Zirkler test, Mardia’s tests of skewness and kurtosis, it is shown that the residuals ε(Tj ) at different periods Tj follow a multivariate standard normal distribution. Consequently, spectral accelerations SA(Tj ) at different periods Tj conditional on a scenario earthquake follow multivariate lognormal distribution (Jayaram and Baker, 2008) 



P SA(T1 ) > s1 , . . . , SA(TK ) > sK  m, r 

=

∞

sK



...

∞ s1

F SA(T



S

1 ), ..., A(TK )



 s1 , . . . , sK  m, r ds1 · · · dsK , (2.4.26)

where s1 , . . . , sK are threshold values of spectral accelerations at different periods and

 F SA(T ), ..., SA(T ) s1 , . . . , sK  m, r is the PDF of multivariate lognormal distribution of K 1 K spectral accelerations conditional on m and r. In the conditional multivariate lognormal distribution of spectral accelerations in equation (2.4.26), the mean and standard deviation values of ln SA(Tj ) in terms of m and r are obtained from the GMPE for each vibration period using equation (2.4.25). The correlation coefficient between the natural logarithmic spectral accelerations at any two periods given a scenario earthquake (m and r), as in equation (2.4.21), has been empirically obtained as discussed in Section 2.4.3. The marginal and joint probability distributions of spectral accelerations governed by equations (2.4.24) and (2.4.26), respectively, will be used in the PSHA. ❧

❧

In this chapter, the fundamentals of engineering seismology are presented. ❧ Tectonic plates drift due to the convective motion of magma underneath the earth’s crust. When two tectonic plates move towards each other, one plate subducts under the other. The plates grind against each other, causing stresses to build up at the boundary. When the shear stresses accumulated reach the shear strength of the rock at the interface, the rock fails. If the rock is strong and brittle, rupture of the rock releases the stored energy explosively, resulting in earthquake. The theory of tectonic plates and the theory of elastic rebound theory explain most of the major earthquakes in the world. ❧ When earthquake occurs, body waves (P-waves and S-waves) and surface waves (Rayleigh waves and Love waves) are produced. When seismic waves propagate, they refract and reflect at boundaries between different layers of the earth and convert to waves of the other types. S-waves shake the ground surface vertically and horizontally and are particularly damaging to structures. Love waves shake the

56

ground surface horizontally and are particularly damaging to the foundations of structures. ❧ The size of an earthquake is measured by its magnitude. Most definitions of earthquake magnitude are based on amplitudes of seismic waves. The moment magnitude Mw or M is independent of seismic wave (ground-motion) characteristics and is not subjected to magnitude saturation; it has been widely used in establishing ground-motion prediction equations (GMPEs). ❧ When an earthquake occurs due to an explosive release of lock at the boundary of tectonic plates, the upward spring motion of the top plate displaces a large body of water, producing tsunami waves. The speed of water waves is proportional to square-root of the sea depth. When tsunami waves approach the coast, their speeds decrease and their heights increase dramatically, capable of reaching tens of meters. ❧ Strong ground motions are measured in terms of PGA, PGV, and PGD. Arias intensity, which is the sum of squared acceleration values, is a more reliable statistical measure of the amplitude of a strong motion record. The time span between 5 % and 75 % (or 95 %) of the Arias intensity is usually defined as a strong-motion duration. ❧ Fourier amplitude spectra (FAS) and power spectral density (PSD) functions of earthquake time-histories have been used to characterize the frequency contents of earthquake ground motions. Ground response spectra (GRS) are used extensively in seismic analysis and design and will be studied in detail in Chapter 4. ❧ GMPEs provide empirical relations between ground motion intensity measure (e.g., PGA or spectral acceleration at individual vibration frequency) and earthquake magnitude, source–site distance, and possibly other predictive variables, based on statistical analysis of real earthquake ground-motion records and sometimes simulated ground-motion time-histories. Statistical correlation between spectral accelerations at two different frequencies are also obtained from statistical analysis. Spectral accelerations at different frequencies conditional on a scenario earthquake (given earthquake magnitude and source–site distance) follow multivariate lognormal distribution. These results are used in probabilistic seismic hazard analysis (PSHA) in Chapter 5.

C

H

3 A

P

T

E

R

Basics of Random Processes and Structural Dynamics 3.1 Random Processes A random process is a continuous physical process influenced by nondeterministic factors. In the Kth experiment, a random process X(t) generates a record X (K)(t), which is the Kth realization or sample function of the process. The random nature of the process is reflected in the fact that no two records are identical in every aspect, as shown in Figure 3.1. The collection of all possible realizations of the random process is called the ensemble of realizations. While each realization is a definite function of t in the sense that X (K)(t) is an ordinary function for fixed K, the ensemble of all realizations can be specified only statistically. For any particular value of t, X(t) is a random variable. A random process may be a function of a single variable such as time (e.g., the vertical displacement of the support of equipment as a function of time during an earthquake) or may depend on several independent variables (e.g., the vertical acceleration of soil surface as a function of both time and location during an earthquake).

3.1.1 Probability Distribution and Density Functions A random process is described by its various probability distribution functions. With reference to Figure 3.1, at a fixed time instance t, the first-order probability distribution function of X(t) is defined as   (3.1.1) F1 (x, t) = P X(t) < x . If the ensemble consists of N sample functions, out of which there are n realizations with X(t) < x, then the first-order probability distribution function defined in equation 57

58

Ensemble averaging X (3) t

X (2) t

X (1) t Time averaging t1 Figure 3.1

t2

Sample functions

X (K)(t)

of a random process.

(3.1.1) is approximately, for N large, F1 (x, t) ≈

n . N

Now consider two time instances, t1 and t2 . The probability that X(t1 ) < x1 and X(t2 ) < x2 is known as the second-order probability distribution function, i.e.,   F2 (x1 , t1 ; x2 , t2 ) = P X(t1 ) < x1 , X(t2 ) < x2 .

(3.1.2)

If n12 is the number of sample functions with X(t1 ) < x1 and X(t2 ) < x2 , for large N F2 (x1 , t1 ; x2 , t2 ) ≈

n12 . N

Similarly, the probability distribution function of order n is defined as   Fn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn ) = P X(t1 ) < x1 , X(t2 ) < x2 , . . . , X(tn ) < xn . (3.1.3) A random process is said to be completely specified if its distribution functions of all orders, i.e., n = 1, 2, . . . , are known. The corresponding probability density function of order n, n = 1, 2, . . . , is defined by pn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn ) =

∂ n Fn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn ) , ∂x1 ∂x2 · · · ∂xn

i.e., pn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn )dx1 dx2 · · · dxn

(3.1.4)

3.1 random processes

59

  = P x1 < X(t1 ) < x1 +dx1 ; x2 < X(t2 ) < x2 +dx2 ; . . . ; xn < X(tn ) < xn +dxn . It is usually either unnecessary or impossible to specify the probability distribution functions of all orders. For many practical purposes, the knowledge of only the firstorder and second-order probability distribution functions is sufficient. In particular, for a Gaussian random process, the probability distribution functions of the first two orders describe the process completely, as will be seen in Section 3.1.6. If the probability distribution functions Fn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn ) are invariant under a change of time origin, i.e., if Fn (x1 , t1 ; x2 , t2 ; . . . ; xn , tn ) = Fn (x1 , t1 +τ ; x2 , t2 +τ ; . . . ; xn , tn +τ ),

(3.1.5)

for all orders n and any value of τ , the random process is said to be stationary. This implies that the first-order probability distribution is independent of time, and the second-order probability distribution depends only on the time difference, i.e. F1 (x, t) = F1 (x); F2 (x1 , t1 ; x2 , t2 ) = F2 (x1 , x2 ; t2 −t1 ) = F2 (x1 , x2 ; τ ), τ = t2 −t1 .

(3.1.6)

A random process can be expected to be stationary when the physical factors influencing it do not change with time. For example, the wind pressure on a building will be stationary when the wind flow is steady, whereas ground accelerations of an earthquake are nonstationary random processes.

3.1.2 Averages and Moments For many physical applications, it is extremely difficult to determine all the probability distribution functions from the available data. Furthermore, such detailed information is often unnecessary. In these circumstances, one may have to be content with knowing only certain average properties of the random process. Two types of average can be defined for a random process X(t): ❧ The ensemble average for a fixed value t of the time parameter, denoted by E[ X(t) ] or  X(t), is obtained by evaluating the average of the random variable X(t) over the ensemble of possible realizations: N 1  (K) E[ X(t) ] = lim X (t). N→∞ N

(3.1.7)

K=1

¯ ❧ The time average, denoted by X(t), is determined by selecting a particular realiza(K) tion X (t) and computing the average of X (K)(t) over a large time period:  1 T (K) ¯ = lim X (t)dt. (3.1.8) X(t) T→∞ T 0

60

Both of these averages can also be defined for any functions of a random process. The simplest of such functions are the polynomials X K1(t1 )X K2(t2 ) · · · X Kn(tn ), whose averages are called the moments. The most important of the moments obtained from the first-order probability distribution are the mean (or expected value) and the mean-square value defined by  +∞ x p1 (x, t)dx, μX (t) = E[ X(t) ] = −∞

2 (t) Xrms

= E[ X (t) ] = 2



+∞

−∞

(3.1.9) 2

x p1 (x, t)dx,

where Xrms is the root-mean-square value of X(t). The variance of X(t) is defined by  2  = E[ X 2 (t) ] − μ2X (t), (3.1.10) σX2 (t) = E X(t) − μ(t) which is the expectation of the square of the deviation from the mean. The positive square root of the variance σX (t) is called the standard deviation. For a random process with zero mean value, Xrms (t) = σX (t). For a stationary random process, because p1 (x, t) is independent of t, all of these averages are also independent of t.

3.1.3 Correlation Functions and Power Spectral Density Functions The most important average obtained from the second-order probability distribution is the autocorrelation function  +∞ x1 x2 p2 (x1 , t1 ; x2 , t2 )dx1 dx2 . (3.1.11) RXX (t1 , t2 ) = E[ X(t1 )X(t2 ) ] = −∞

The prefix auto- indicates that the two random variables considered, X(t1 ) and X(t2 ), belong to the same random process. A related quantity is the covariance function defined by     KXX (t1 , t2 ) = E[ X(t1 )−μ1 · X(t2 )−μ2 ] = RXX (t1 , t2 ) − μ1 μ2 , (3.1.12) where μI = E[ X(tI ) ], I = 1, 2. The counterpart of the autocorrelation function is the cross-correlation function, defined as  +∞ x y pXY (x, t1 ; y, t2 )dxdy, (3.1.13) RXY (t1 , t2 ) = E[ X(t1 )Y(t2 ) ] = −∞

where X(t1 ) and Y(t2 ) belong to two different random processes X(t) and Y(t). Two random processes X(t1 ) and Y(t2 ) are independent if pXY (x, t1 ; y, t2 ) = pX (x, t1 ) pY ( y, t2 ),

3.1 random processes

61

and equation (3.1.13) becomes  +∞  E[ X(t1 )Y(t2 ) ] = x pX (x, t1 )dx −∞

+∞

y pY ( y, t2 )d y = E[ X(t1 ) ] E[ Y(t2 ) ].

−∞

For a stationary random process, the autocorrelation function depends on the time difference t2 −t1 only and is usually denoted by RXX (τ ), where τ = t2 −t1 . Without loss of generality, suppose that X(t) has zero mean value. The autocorrelation function RXX (τ ) possesses the following properties: 2 (t) = σ 2 , the mean-square value of X(t). 1. RXX (0) = E[ X 2 (t) ] = Xrms X

2. RXX (τ ) is symmetric about τ = 0, i.e., RXX (τ ) = RXX (−τ ).   3. It can be shown that RXX (0)  RXX (τ ). Hence, R XX (0), if it exists, must be zero and R XX (0), if it exists, is negative. 4. If there is no periodic component, then lim RXX (τ ) = 0. τ →±∞

5. RXX (τ ) may be expressed in the following form RXX (τ ) =

F

−1





1 SXX (ω) = 2π



+∞

−∞

SXX (ω) ei ωτ dω,

(3.1.14a)

i.e., RXX (τ ) is the inverse Fourier transform of SXX (ω), where SXX (ω) is a real positive even function of frequency ω. Hence, SXX (ω) is the Fourier transform of RXX (τ ) given by

SXX (ω) = F





RXX (τ ) =



+∞

−∞

RXX (τ ) e−i ωτ dτ.

(3.1.14b)

Function SXX (ω) is known as the power spectral density (PSD) function of the random process X(t). Equations (3.1.14) connect the autocorrelation function and the power spectral density function of a stationary random process. The meaning of the term “power spectral density” becomes clear when looking at the relation   +∞ 1 +∞ SXX (ω)dω = SXX ( F )dF, ω = 2πF. (3.1.15) . E[ X 2 (t) ] = RXX (0) = 2π −∞ −∞ If X(t) is considered as a current in an electrical circuit, then E[ X 2 (t) ] is the average power dissipated in a unit resistor, and SXX ( F ) is the contribution to this power at the frequency F from a band of width dF. Some common examples of autocorrelation functions and their corresponding power spectral density functions are: 1. Periodic functions: RXX (τ ) = A2 cosω0 τ ,

62

SXX(ω)

RXX(τ)

τ

0

(a)

–ω0

ω0

0 SXX(ω) α1

RXX(τ) α1 0, and H(x) = 0, if x < 0.   ˙ ˙ = δ X(t)−u X(t), where δ(t) is Differentiating Z(t) with respect to t yields Z(t) ˙ is a positive delta function when there is an upcrossing of the Dirac delta function. Z(t) level X = u, and a negative delta function when there is a downcrossing of level X = u.   ˙ ˙ by H X(t) eliminates downcrossing of level X = u. Integrating Multiplying Z(t)   ˙ ˙ H X(t) results in the counting process NX+ (u, t), which is the total number of Z(t) upcrossings in time [0, t]:  t   ˙ H X(s) ˙ NX+ (u, t) = Z(s) ds. 0

Hence, the expected rate of upcrossing level X = u is the expected value of the derivative of NX+ (u, t), i.e., νX+ (u, t)





      dNX+ (u, t) ˙ H X(t) ˙ ˙ H X(t) ˙ =E = E Z(t) = E δ X(t)−u X(t) dt  +∞  +∞ = δ(x−u) x˙ H(x) ˙ pXX˙ (x, x) ˙ dx dx˙

[

x= ˙ − ∞ x= − ∞

 =

+∞

x= ˙ −∞

x˙ H(x) ˙ pXX˙ (u, x) ˙ dx˙ =

]



+∞ 0

[

x˙ pXX˙ (u, x) ˙ dx. ˙

]

3.2 properties of random processes

67



˙ = pX (x) pX˙ x˙  X(t) = x pXX˙ (x, x)

Using

yields the expected rate of upcrossing the level X = u:  +∞  +∞ 

+ νX (u, t) = x˙ pXX˙ (u, x) ˙ dx˙ = pX (u) x˙ pX˙ x˙  X(t) = u dx. ˙ 0

(3.2.6)

0

Equation (3.2.6) is very general, applicable to any stationary or nonstationary process with any probability distribution. ˙ < 0 is downcrossing. Reversing the sign of Recall that the event X(t) = u with X(t) ˙ X(t) in equation (3.2.6) gives the expected rate of downcrossing the level X = u:  0  0 

x˙ pXX˙ (u, x) ˙ dx˙ = − pX (u) x˙ pX˙ x˙  X(t) = u dx. ˙ (3.2.7) νX− (u, t) = − −∞

−∞

Stationary Gaussian Process with Zero Mean For a stationary Gaussian process with zero mean, its joint probability density is    1 1 x2 x˙2 . ˙ = exp − + 2 pXX˙ (x, x) 2π σX σX˙ 2 σ2 σX˙ X Substituting into equation (3.2.6) gives   +∞      1 σX˙ 1 u2 x˙2 u2 d x ˙ = , exp − x ˙ exp − exp − νX+ (u) = 2π σX σX˙ 2π σX 2σX2 2σX˙2 2σX2 0 i.e.,

  u2 ± ± , νX (u) = νX (0) exp − 2σX2

1 σX˙ 1 νX± (0) = = 2π σX 2π



λ2 . λ0

(3.2.8)

Rate of Occurrence of Peaks ˙ = 0 and X(t) ¨ < 0, the rate of occurrence Because a peak of X(t) occurs whenever X(t) ˙ of peaks of X(t) is the rate of downcrossing of the level X˙ = 0 by X(t). Hence, from equation (3.2.7), the rate of occurrence of peaks of X(t) is  0  0 

˙ = 0 dx. ¨ (3.2.9) x¨ pX˙ X¨ (0, x)d ¨ x¨ = − pX˙ (0) x¨ pX¨ x¨  X(t) νp (t) = νX−˙ (0, t) = − −∞

−∞

Between any two upcrossing of the same level u, at least one peak must occur. Hence, νp  νX+ (u, t) for any u and for any process X(t) with continuous time derivatives. For a narrow-band process, the rate of occurrence of peaks is expected to be only slightly larger than the rate of upcrossings of the mean μX = E[ X(t) ]. This property is commonly used to provide a measure of bandwidth through the irregularity factor:

I=

ν+ X (μX , t) , νp (t)

For a narrow-band process, I tends to 1.

0 < I  1.

(3.2.10)

68

Stationary Gaussian Process with Zero Mean For a stationary Gaussian process with zero mean, referring to equation (3.2.8),  νp = νX−˙ (0) = and the irregular factor is

I=

1 σX¨ 1 = 2π σX˙ 2π

σ ˙2 ν+ X (0) = X νp σX σX¨

=⇒

λ4 , λ2

I= 

λ2

λ0 λ4

(3.2.11)

.

(3.2.12)

3.2.3 Probability Distribution of Peaks 



Define νp t, X(t)  u as the expected rate of occurrence of peaks not exceeding the level u. During an infinitesimal time interval dt, there is either one peak or none. Hence, the expected number of occurrences in the interval dt is the same as the probability of one occurrence in dt, i.e.,       νp t, X(t)  u = P Peak  u during [t, t+dt] =⇒ νp (t)dt = P Peak during [t, t+dt] ,





where νp (t) = lim νp t, X(t)  u is the total expected rate of peak occurrences. Using u→∞

conditional probability, one has   P Peak  u during [t, t+dt]      = P Peak during [t, t+dt] · P Peak  u  Peak during [t, t+dt] .    Because P Peak  u  Peak during [t, t+dt] = Fp(t) (u), where Fp(t) is the cumulative distribution function for a peak at time t, hence   P Peak  u during [t, t+dt] ν (u, t)   = p Fp(t) (u) = . (3.2.13) P Peak during [t, t+dt] νp (t) 



˙ To determine νp (u, t), define a process Z(t) = H − X(t) as shown in Figure 3.4.     ˙ ¨ ˙ = δ − X(t) · − X(t) , which is a Differentiating Z(t) with respect to t yields Z(t) positive delta function when there is a peak and a negative delta function when there is   ˙ by H − X(t) ¨ a valley. Multiplying Z(t) eliminates all negative Dirac delta functions   or valley. Multiplying the result by H u−X(t) further eliminates all peaks above the level u. Integrating the result leads to the counting process Np (u, t), which is the total number of peaks not exceeding the level u in time [0, t]:  t       ¨ · δ − X(s) ˙ ¨ − X(s) · H − X(s) · H u−X(s) ds. Np (u, t) = 0

Hence, the rate of occurrence of peaks not exceeding the level u is the expected value of the derivative of Np (u, t), i.e.,





      dNp (u, t) ¨ · δ − X(t) ˙ ¨ = E − X(t) · H − X(t) · H u−X(t) . (3.2.14) νp (u, t) = E dt

[

]

3.2 properties of random processes

69

X(t) u 0

t

H[ X(t)] 1 t

0 H[−X(t)] 1

t

0 δ[−X(t)]·[−X(t)]

t

0

−X(t) · δ[−X(t)]· H[−X(t)] t

0 −X(t) · δ[−X(t)]· H[−X(t)]· H[u −X(t)]

t

0 3 2 1 0

Np(u,t)

t Determination of Np (u, t).

Figure 3.4

Substituting into equation (3.2.13) yields Fp(t) (u) =

[











¨ · δ − X(t) ˙ ¨ E − X(t) · H − X(t) · H u−X(t)

[









]



]

¨ · δ − X(t) ˙ ¨ E − X(t) · H − X(t)  +∞  +∞  +∞       − x(t) ¨ δ − x(t) ˙ H − x(t) ¨ H u−x(t) pXX˙ X¨ (x, x, ˙ x)dx ¨ dx˙ dx¨ x= ¨ − ∞ x= ˙ − ∞ x= − ∞ =  +∞  +∞     − x(t) ¨ δ − x(t) ˙ H − x(t) ¨ pX˙ X¨ (x, ˙ x) ¨ dx˙ dx¨  =

0



x= ¨ − ∞ x= ˙ −∞

u

     

x= ¨ − ∞ x= − ∞  0 

x¨ pXX˙ X¨ (x, 0, x) ¨ dx dx¨

    

x= ¨ −∞

x¨ pX˙ X¨ (0, x) ¨ dx¨

.

(3.2.15)

70

Differentiating with respect to u gives the probability density function for the peak  0     x ¨  pXX˙ X¨ (u, 0, x) ¨ dx¨ −∞ . (3.2.16) pp(t) (u) =  0     x ¨  pX˙ X¨ (0, x) ¨ dx¨ x= ¨ −∞

Equations (3.2.15) and (3.2.16) give the probability distribution of peak that occurs within the vicinity of time t. Other quantities can be easily obtained, such as the mean, the mean-square value  +∞  +∞ 2 u pp(t) (u)du, E[ p (t) ] = u2 pp(t) (u)du, (3.2.17) μp = E[ p(t) ] = −∞

−∞

and the variance σp2 = E[ p2 (t) ] −μ2p .

☞

Equations (3.2.15) to (3.2.17) are the conditional probability distribution and conditional moments given the existence of peaks.

☞

To find the probability distribution of the peak p(t), one needs the joint proba˙ , and X(t) ¨ because the occurrence of a peak p(t) bility distribution of X(t), X(t) ˙ = 0, and X(t) ¨ < 0. at level u requires the intersection of the events X(t) = u, X(t)

Stationary Gaussian Process with Zero Mean





˙ is independent of X(t), X(t) ¨ For a stationary Gaussian process X(t), X(t) . pX˙ (0) can be factored out of both the numerator and denominator of equation (3.2.16)  0     √ x ¨  pXX¨ (u, x)d ¨ x¨    2π 0 x¨ = − ∞   x = ¨  pXX¨ (u, x)d ¨ x¨ pp (u) =  0 σ   x ¨ = − ∞ ¨   X x ¨  pX¨ (x)d ¨ x¨ √

x¨ = − ∞

2π = p (u) σX¨ X



0



x¨ pX¨ x¨  X = u dx. ¨

     

x¨ = − ∞

Using conditional probability

The conditional probability density function of a Gaussian process is also Gaussian, i.e.,   

1 (x¨ − μ) ˆ 2  pX¨ x¨ X = u = √ exp − , 2 σˆ 2 2π σˆ ¨ are where the conditional mean and standard deviation of X(t) μˆ = ρXX¨ ·

σX¨

σX

· u,

σˆ = σX¨

Hence, 1 p (u) pp (u) = − σX¨ σˆ X





1−ρX2X¨ ,

ρXX¨ = −

  (x¨ − μ) ˆ 2 dx¨ x¨ exp − 2 σˆ 2 x¨ = − ∞ 0

σX˙2

σX σX¨

= − I.

(3.2.18)

3.2 properties of random processes

71

  μˆ 2  √ μˆ  μˆ  σˆ exp − 2 − 2π

− = pX (u) σX¨ 2 σˆ σX¨ σˆ 

= pX (u)



 1− I 2 ξ 2 exp −





√ I 2 u2 Iu Iu + 2π

√ 2 2 σX 2(1− I )σX 1− I 2 σX

 .

Substituting the Gaussian form for pX (u) gives the probability density function for the peaks of a mean zero stationary Gaussian process X(t) √

      u2 Iu Iu u2 1− I 2 + 2 exp − 2 √ . (3.2.19) pp. (u) = √ exp − 2(1− I 2 )σX2 σX 2σX 2π σX 1− I 2 σX

The corresponding cumulative distribution function is given by       u u2 Iu − I exp − 2 √ . Fp (u) = √ 2σX 1− I 2 σX 1− I 2 σX

(3.2.20)

These results are commonly referred to as the Rice distribution. For the limiting case of narrow-band process with I →1− , noting that (− ∞) = 0 and (∞) = 1, one has the Rayleigh distribution:     u2 u2 u Fp (u) = 1 − exp − 2 . (3.2.21) pp (u) = 2 exp − 2 , σX 2σX 2σX

The Highest Peak in a Time Interval Considering a time interval [0, t], the total number of peaks is Np = νp t, where νp is the rate of occurrence of peaks given by equation (3.2.11). Among these Np peaks, let Xmax be the highest peak, which is also the maximum value of X(t) in time interval [0, t]. Assuming that the peaks are independent, the distribution of Xmax is 

FXmax (x) = Fp (x)



Np

,

(3.2.22)

where Fp is the probability distribution function given by equation (3.2.20). The probability distribution of Xmax is given by pX

max

(x) =

 d d  FXmax (x) = dx dx



Fp (x)



Np

!

.

(3.2.23)

The expected value of the maximum peak is given by  +∞  +∞   E[ Xmax ] = x · pX (x)dx = x · d Fp (x) Np max

−∞



=−

0 −∞



Fp (x)



 Np

−∞

+∞

dx +

in which the first integral is negligible.

0



1 − Fp (x)



Np

! dx,

(3.2.24)

72

Note that the asymptotic expansion of (x) is, for x large,  x   1 1 1 1 2 2

(x) = √ e−x /2 dx = 1 + √ e−x /2 − + 3 − · · · . x x 2π 2π −∞

(3.2.25)

Hence, the probability distribution function Fp given by equation (3.2.20) can be approximated as   x2 (3.2.26) Fp (x) ≈ 1 − I exp − 2 . 2σX Equation (3.2.24) becomes E[ Xmax ] =



+∞

0



 Np  x2 1 − 1 − I exp − 2 dx. 2σX 

(3.2.27)

The peak factor Pf, defined as the ratio of the peak value E[ Xmax ] and root-meansquare value σX of random process X(t), is given by

Pf =

E[ Xmax ]

σX

1 =√ 2



+∞ 0



1 − 1 − I e−θ



Np

!

θ −1/2 dθ ,

θ=

x2 , (3.2.28) 2σX2

or √ E X Pf = [ max ] = 2 σX

☞

 0

+∞



2 N p

1 − 1 − I e−z

! dz,

z= √

x 2 σX

.

(3.2.29)

Equation (3.2.28) is the same as equation (6.8) in Cartwright and LonguetHiggins (1956); whereas equation (3.2.29) is different from equation (29) in √ Boore (2003) by a factor of 2.

3.2.4 Extreme Value Distribution Define a new stochastic process Y(t) that is the extreme value of X(t) during the period [0, t], i.e., (3.2.30) Y(t) = max X(s). 0st

The extreme value distribution is then the distribution of Y(t). Note that even for a stationary process X(t), Y(t) is generally nonstationary because larger and larger values of X(t) will generally occur when the period [0, t] is extended. The cumulative distribution function of Y(t) is     (3.2.31) FY(t) (u) = P Y(t)  u = P X(s)  u for 0  s  t . The probability density function of the extreme value is given by pY(t) (u) =

 ∂  FY(t) (u) . ∂u

(3.2.32)

3.2 properties of random processes

73

Let TX (u) > 0 denote the time at which X(t) has the first upcrossing of the level     u, i.e., X TX (u) = u, X˙ TX (u) > 0, and there has been no crossing in the interval 0  t < TX (u). For any given u value, TX (u) is a random variable.





Note that the extreme value problem X(s)  u for 0  s  t is equivalent to the   first passage time problem X(0)  u, TX (u)  t . Taking the probabilities gives      FY(t) (u) = P TX (u)  t  X(0)  u P X(0)  u    = P TX (u)  t  X(0)  u FY(0) (u). (3.2.33)    In many problems, the condition in P TX (u)  t  X(0)  u can be neglected:   ❧ In some problems, P X(0)  u = 1, such as when the system is known to start at X(0) = 0, and conditioning by a sure event can always be neglected. ❧ In other situations, although the distribution of TX (u) depends on X(0), the effect of X(0) may be significant only for a small period of time. Equation (3.2.33) can be written as 





P TX (u)  t  X(0)  u = 1 −

FY(t) (u) FY(0) (u)

.

Differentiating with respect to t gives

  pT t  X(0)  u = − X

 ∂  FY(t) (u) . FY(0) (u) ∂t

1

(3.2.34)

It is often convenient to define FY(t) (u) as   t  FY(t) (u) = FY(0) (u) exp − ηX (u, s) ds .

(3.2.35)

0

Differentiating equation (3.2.35) with respect to t yields   t     ∂  FY(t) (u) = FY(0) (u) exp − ηX (u, s) ds · −ηX (u, t) = −FY(t) (u) ηX (u, t), ∂t 0 which leads to  FY(t) (u) − FY(t + t) (u) ∂  1 FY(t) (u) = lim FY(t) (u) ∂t t t→0 FY(t) (u)    1 P t  TX (u)  t+t  X(0)  u    = lim . (3.2.36) P TX (t)  t  X(0)  u t→0 t

ηX (u, t) = −



1



The event t  TX (u)  t+t means that the first upcrossing of level u is in the time   interval t, t+t . This event is the intersection of the event that there is no upcrossing   prior to t and the event that there is an upcrossing in the time interval t, t+t , i.e.,







 







t  TX (u)  t+t = TX (u)  t ∩ Upcrossing in t, t+t .

74

Equation (3.2.36), in which the ratio is the conditional probability, can be written as ηX (u, t) = lim

    P Upcrossing in t, t+t  X(0)  u ∩ No upcrossing prior to t t

t→0

 E[ Upcrossing in t, t+t  X(0)  u ∩ No upcrossing prior to t ] 

= lim



t

t→0

. (3.2.37)

Hence, ηX (u, t) is the conditional rate of upcrossing of level u, given that the initial condition is below u and that there is no prior upcrossing. Using the conditional probability density function, ηX (u, t) can be written as  +∞    x˙ pXX˙ u, x˙  X(0)  u ∩ No upcrossing in t, t+t dx. ηX (u, t) = ˙ x=0 ˙

However, the conditional probability density function is generally unknown, and some approximations must be made. Note that most physical processes have only a finite memory in the sense that X(t) and X(t−τ ) can generally be considered independent if τ > T for some large T value. Hence, for t > T,

  pXX˙ u, x˙  X(0)  u ∩ No upcrossing in [0, t] ≈ pXX˙ u, x˙  No upcrossing in [t−T, t] . If X(t) is a stationary process, the approximate conditional probability density is stationary because it is independent of the choice of the origin of the time axis, except for the restriction that t > T. This means that ηX (u, t) approaches asymptotically to

 a stationary value of ηX (u) as pXX˙ u, x˙  X(0)  u ∩ No upcrossing in [0, t] tends to

 pXX˙ u, x˙  No upcrossing in [0, t] . This asymptotic behaviour of ηX (u, t) implies that equation (3.2.35) can be approximated as FY(t) (u) ≈ F0 e

−η (u) t X

,

for large t.

(3.2.38)

This limiting behaviour for large t is also applicable if X(t) is a nonstationary process that has finite memory and that becomes stationary with the passage of time.

Poisson Approximation The most widely used approximation of the extreme distribution problem is to neglect   the conditioning event X(0)  u ∩ No upcrossing prior to t . Hence ηX (u, t) ≈ Rate of upcrossings of the level u = νX+ (u, t),   t  + FY(t) (u) ≈ FY(0) (u) exp − νX (u, s) ds .

(3.2.39) (3.2.40)

0

If X(t) is a stationary process, equation (3.2.40) reduces to +

FY(t) (u) ≈ FY(0) (u) e−νX (u) t .

(3.2.41)

3.2 properties of random processes

75

If the crossing rate is independent of the past history of the process, the time intervals between upcrossings are independent, which makes the integer-valued counting process NX+ (u, t) a Poisson process. Hence, the approximate of equations (3.2.39) to (3.2.41) is commonly called the Poisson approximation of the extreme value or the first-passage problem. Substituting equation (3.2.41) into (3.2.34) yields, for stationary process X(t), pT (t) = − X

1

∂FY(t) (u)

FY(0) (u)

∂t

+

≈ νX+ (u) e−νX (u) t .

(3.2.42)

Hence, the Poisson approximation gives an exponential distribution for the firstpassage time of a stationary process X(t). The mean first-passage time can be easily obtained from the exponential distribution and is given by E[ TX+ (u) ] ≈

1 νX+ (u)

,

for stationary process X(t).

(3.2.43)

☞

For a narrow-band process X(t), an upcrossing of level u at time t is very likely to be associated with another upcrossing approximately one period later, due to the slowly varying amplitude of X(t). Such a relationship between the upcrossing times is inconsistent with the Poisson approximation that the times between upcrossings are independent. Hence, the Poisson approximation is not appropriate when the process X(t) is narrow-band.

☞ ☞

When u is very large, the assumption of independent crossing becomes better. The Poisson approximation is best when the process X(t) is broadband and the level u is large.

For a stationary Gaussian process zero-mean with zero mean, νX+ (u) and the zeroupcrossing rate ν0 = νX+ (0) is given by equation (3.2.8). The probability distribution function FY(t) (u) is, with FY(0) (u) = 1, +



FY(t) (u) = FY(0) (u) e−νX (u) t = exp −ν0 t e− u

2 /(2 σ 2 ) X



.

(3.2.44)

   ∂   ∂  2  2 P Y(t)  u = exp −ν0 t e− u /(2 σX ) . ∂u ∂u

(3.2.45)

The probability density function is pY(t) (u) = Defining ξ = ν0 t e− u

2 /(2 σ 2 ) X

,

(3.2.46)

equation (3.2.46) becomes pY(t) (u) =

dξ ∂e−ξ dξ · = e−ξ . ∂ξ du du

(3.2.47)

76 Peak Factor Pf

4.0 3.5 3.0 2.5 2.0 1.5

ν0 t 0

100

200

300

400

500

Figure 3.5

600

700

800

900

1000

Peak factor.

When u→ + ∞, ξ →0, and when u→0, ξ →ν0 t. The expected value of Y(t) is  +∞  νt 0 u pY(t) (u) du = u e−ξ dξ. (3.2.48) E[ Y(t) ] = E[ Xmax ] = 0

0

Solving for u from equation (3.2.48) gives u = σX = σX

 √

2( lnν0 t− lnξ ) = σX  2 lnν0 t

1 − 12

√

 2 lnν0 t ·

1−

lnξ lnν0 t

  lnξ   lnξ 2 1 − 8 − ··· . lnν0 t lnν0 t

Because ξ  ν0 t, equation (3.2.49) can be approximated as  √ lnξ u ≈ σX , 2 lnν0 t − √ 2 lnν0 t equation (3.2.48) gives the peak factor   ν t √ 0 E[ Xmax ] lnξ e−ξ dξ Pf = ≈ 2 lnν0 t − √ σX 0 2 lnν0 t   +∞  √ lnξ e−ξ dξ , 2 lnν0 t − √ ≈ 0 2 lnν0 t

(3.2.49)

(3.2.50)

(3.2.51)

which yields √ E X Pf = [ max ] ≈ 2 lnν0 t + √ σX

γ 2 lnν0 t

,

γ = 0.5772,

(3.2.52)

where γ is the Euler number. This result was first obtained in Davenport (1964) and is plotted in Figure 3.5. Der Kiureghian (1980) determined an empirical reduced zero-upcrossing rate νe , representing an equivalent rate of statistically independent crossings, given by 

1.63 q0.45 − 0.38 ν0 , q < 0.69, (3.2.53) νe = q  0.69, ν0 ,

3.3 single degree-of-freedom system

77

where q is the spectral parameter defined in equation (3.2.5). In the peak factor given by equation (3.2.52), ν0 is replaced by the reduced rate νe . Equations (3.2.53) and (3.2.52) with ν0 replaced by νe are applicable for 0.1  q  1 and 5  ν0 t  1, 000, which are of interest in earthquake engineering. Resulting error in the estimated peak factor is generally within 3 %.

Double-Barrier Problem In many engineering applications, it is required to determine large excursions of X(t) in either the positive or negative direction. For example, for earthquake ground motion excitation u¨g(t), the positive or negative sign of u¨g(t) has no real significance, and it is   important to determine peak ground acceleration u¨g(t)max . The event of X(t) remaining between −u and +u is exactly the same as the event   of X(t) remaining below the level u. Following equation (3.2.35), one can write   t  FY(t) (u) = FY(0) (u) exp − η|X| (u, s) ds , 0

  Y(t) = max X(s). 0st

(3.2.54)

The terms double-barrier problem and single-barrier problem are often used to distin  guish between the upcrossings by X(t) and X(t), respectively. The Poisson approximation of the symmetric double-barrier problem of equation (3.2.54) is simply to replace η|X| (u, s) with + (u, s) = νX+ (u, s) + νX− (−u, s). ν|X|

˙ is symmetric, this gives ν + (u, s) = 2νX+ (u, s). If the distribution of X(t) and X(t) |X|

3.3 Single Degree-of-Freedom System 3.3.1

Equations of Motion

Consider a single-storey shear building consisting of a rigid girder of mass m, supported by weightless columns with combined stiffness K. The columns can take shear forces but not bending moments. In the horizontal direction, the columns act as a spring of stiffness K. As a result, the girder can move only in the horizontal direction; its motion can be described by horizontal displacement u(t), and the system is single degree-offreedom (SDOF). The building is subjected to a dynamic load P(t), and the base of the building is subjected to a dynamic displacement ug(t), as shown in Figure 3.6. The elastic and damping forces applied on the girder are, respectively, K(u−ug ) and c(u− ˙ u˙g ). Newton’s Second Law requires that → ma =











F =⇒ m u(t) ¨ = P(t) − K u(t)−ug(t) − c u(t)− ˙ u˙g (t) .

78

u(t) Rigid girder

Reference position u(t)

m k c

m

P(t)

P(t) k (u ug)

c (uug)

Weightless columns

k c

ug(t) Ground displacement

ug(t)

x(t)

Figure 3.6 A single-storey shear building.

u(t)

ug(t)

m

u(t)

k

k, c

c

m

ω, ζ

x(t)

ug(t)

u, u, u

u(t) m

ug(t)

k, c

c

Figure 3.7

k(u−ug)

k m

c (u −ug)

m

SDOF oscillator under ground excitation.

Let x(t) = u(t)−ug(t) be the relative displacement between the girder and the base. In terms of the relative displacement x(t), the equation of motion becomes m( x¨ + u¨g ) = P(t) − Kx − c x˙

=⇒

m x(t) ¨ + c x(t) ˙ + Kx(t) = P(t) − m u¨g(t). (3.3.1)

The equivalent loading on the girder created from ground excitation is −m u¨g(t), which is proportional to the mass of the structural system m and the ground acceleration u¨g(t). The equation of motion (3.3.1) can be written in the standard form as 1 ˙ + ω02 x(t) = m F(t), x(t) ¨ + 2ζ0 ω0 x(t)

K c ω02 = m , 2ζ0 ω0 = m ,

(3.3.2)

where ω0 is the natural circular frequency, ζ0 is the damping ratio, and the forcing is F(t) = P(t)−m u¨g(t). In earthquake engineering, it is convenient to use an SDOF oscillator as illustrated in Figure 3.7 to model an SDOF system under base excitation ug(t) or u¨g (t), with P(t) = 0.

3.3.2

Free Vibration

For free vibration, equation of motion (3.3.2) is reduced to ˙ + ω02 x(t) = 0. x(t) ¨ + 2ζ0 ω0 x(t)

(3.3.3)

3.3 single degree-of-freedom system

a

79

xC(t) a e−ζω0 t

ϕ ωd t ae−ζω0 t cos(ωd t − ϕ) −ae−ζω0 t (Envelope)

−a Figure 3.8

Response of underdamped free vibration.

Impulse I = F(τ)τ

F(t)

F(t) I

t–τ τ

t

τ xP(t,τ)

t

τ

t xP

τ Figure 3.9

t

t –τ

t t

Response of underdamped SDOF system due to an impulse.



The characteristic equation is λ2 +2ζ0 ω0 λ+ω02 = 0, giving λ = ω0 −ζ0 ± ζ02 −1 . Most engineering structures are underdamped with 0 < ζ0 < 1. The roots of the characteristic equation become  

λ = ω0 −ζ0 ± i 1−ζ02 = −ζ0 ω0 ± iωd , ωd = ω0 1−ζ02 , where ωd is the damped natural circular frequency. The response of free vibration is xC (t) = e − ζ0 ω0 t (A cosωd t + B sinωd t),

(3.3.4)

where constants A and B are determined from the initial conditions x(0) = x0 and x(0) ˙ = v0 , resulting in   v +ζ ω x xC (t) = e − ζ0 ω0 t x0 cosωd t + 0 0 0 0 sinωd t , 0  ζ0 < 1, (3.3.5) ωd = ae − ζ0 ω0 t cos(ωd t − ϕ), 

where a=

x20 +

 v +ζ ω x 2 0 0 0 0 , ωd

(3.3.6) −1

ϕ = tan

 v +ζ ω x  0 0 0 0 . ωd x0

The response of free vibration of an underdamped system with 0 < ζ0 < 1 is shown in Figure 3.8, which decays exponentially and approaches zero as t→∞ and is called transient response. Because its value becomes negligible after some time, its effect is small and is not important in practice.

80

3.3.3 Forced Vibration − Duhamel Integral The response of forced vibration due to force F(t) is the particular solution xP (t) of the differential equation (3.3.2). Consider the response of the SDOF system (3.3.2) under a general dynamic force F(t), as shown in Figure 3.9. Let xP (t, τ ) be the displacement at time t due to an impulse I = F(τ )τ applied at time τ . The initial velocity v0 imparted by the impulse I is v0 =

F(τ )τ I = , m m

(3.3.7)

and the initial displacement x0 = 0. At time t, the displacement xP (t, τ ), due to an impulse I at time τ , is the response of free vibration with initial displacement x0 = 0 and initial velocity v0 . Using equation (3.3.5), one has   v0 +ζ0 ω0 x0 sinωd (t−τ ) xP (t, τ ) = e − ζ0 ω0 (t − τ ) x0 cosωd (t−τ ) + ωd = e − ζ0 ω0 (t − τ )

F(τ )τ sinωd (t−τ ). m ωd

(3.3.8)

Summing up the effect of all such impulses from τ = 0 to t due to force F(t) yields 

sinωd (t−τ ) F(τ )dτ m ωd 0  t  t H(t−τ ) F(τ )dτ = H(t) ∗ F(t) = H(τ ) F(t−τ )dτ , = t

xP (t) =

e − ζ0 ω0 (t − τ )

0

(3.3.9)

0

where H(t) is the unit impulse response function of the SDOF system given by H(t) = e−ζ0 ω0 t

sinωd t , m ωd

 ωd = ω0 1−ζ02 .

(3.3.10)

Integral of the form (3.3.9) is called a convolution integral or Duhamel integral. For  lightly damped system, ζ0 1, ωd = ω0 1−ζ02 ≈ ω0 .

Response of SDOF System under Base Excitation For the case when the SDOF oscillator under base excitation u¨g (t) only, the forcing function F(t) = −mu¨g (t). The relative displacement x(t) of the oscillator, given by equation (3.3.9), becomes  t sinωd (t−τ ) u¨g (τ )dτ. (3.3.11) x= − e − ζ0 ω0 (t − τ ) ωd 0 In earthquake engineering, the negative sign has no real significance with regard to earthquake excitation and can be dropped; hence, equation (3.3.11) can be written as  t sinωd (t−τ ) u¨g (τ )dτ = h(t) ∗ u¨g (t), (3.3.12) e − ζ0 ω0 (t − τ ) x(t) = ωd 0

3.3 single degree-of-freedom system

81

where h(t) is referred to as the impulsive response function with respect to base excitation in this book and is given by h(t) = e − ζ0 ω0 t

☞

sinωd t . ωd

(3.3.13)

The difference between functions H(t) and h(t) is the mass term m. h(t) is used in structures under base excitation (earthquake) because the mass term m in H(t) is cancelled with the mass in the equivalent earthquake load −m u¨g (t).

Equation (3.3.12) can be rewritten as  t 1 e−ζ0 ω0 (t−τ ) sinωd (t−τ ) u¨g(τ ) dτ. x(t) = ωd 0 Taking time derivative of equation (3.3.14) gives the relative velocity  t ζ ω x(t) ˙ =− 0 0 e−ζ0 ω0 (t−τ ) sinωd (t−τ ) u¨g(τ ) dτ ωd 0  t + e−ζ0 ω0 (t−τ ) cosωd (t−τ ) u¨g(τ ) dτ.

(3.3.14)

(3.3.15) (3.3.16)

0

Substituting equations (3.3.14) and (3.3.16) into (3.3.2) yields the absolute acceleration u(t) ¨ = x(t) ¨ + u¨g(t) = −ω02 x(t) − 2ζ0 ω0 x(t) ˙   2ζ 2 ω2 = −ω0 + 0 0 I s (t) − 2ζ0 ω0 I c (t), ωd where



I

s

I

c

(t) =

t

0

 (t) =

(3.3.17)

e−ζ0 ω0 (t−τ ) sinωd (t−τ ) u¨g(τ ) dτ , (3.3.18)

t

e

−ζ0 ω0 (t−τ )

0

cosωd (t−τ ) u¨g(τ ) dτ.

For small damping ζ0 1, ωd ≈ ω0 , and h(t) ≈ e − ζ0 ω0 t

1 x(t) ≈ ω0



t

0

 x(t) ˙ ≈

t

0

 u(t) ¨ ≈ −ω0

0

t

sinω0 t , ω0

(3.3.13 )

e−ζ0 ω0 (t−τ ) sinω0 (t−τ ) u¨g(τ ) dτ = h ∗ u¨g ,

(3.3.14 )

e−ζ0 ω0 (t−τ ) cosω0 (t−τ ) u¨g(τ ) dτ ≈ ω0 h ∗ u¨g ,

(3.3.16 )

e−ζ0 ω0 (t−τ ) sinω0 (t−τ ) u¨g(τ ) dτ = −ω02 h ∗ u¨g .

(3.3.17 )

82

3.3.4 Forced Vibration − Harmonic Excitation Externally Applied Load P(t) = P0 e i ωt Consider the case when the SDOF system (3.3.2) is subjected to an externally applied harmonic load P(t) = P0 e i ωt . Substituting xP (t) = xˆP e i ωt into equation (3.3.2) yields xˆP = H(ω) P0

=⇒

xP (t) = H(ω) P0 e i ωt ,

where H(ω) is the complex frequency response function given by  ∞ 1  . H(ω) = H(t) e−i ωt dt =  2 m (ω0 −ω2 ) + i2ζ0 ω0 ω −∞

(3.3.19)

(3.3.20)

If dynamic effect is not considered, i.e., if only static terms are considered in equation (3.3.2), one obtains xstatic = P0 /K, which is the static displacement of the structure under static force P0 . The dynamic magnification factor (DMF) is defined by   x (t) 1 ω P max DMF = = , r= ω , (3.3.21) xstatic 0 (1−r 2 )2 +(2ζ0 r)2 where r is the frequency ratio. D-MF is plotted in Figure 3.10 for various values of the damping ratio ζ0 ; it is one of the most important quantities describing the dynamic behavior of an underdamped SDOF system under harmonic excitation. ❧ When r→0 (ω ω0 ), D-MF →1. The dynamic excitation is effectively a static force and the amplitude of dynamic response approaches the static displacement.

DMF →0 or the dynamic response approaches zero. When r ≈ 1 (ω ≈ ω0 ), DMF tends to large values for small damping. DMFmax occurs when d(DMF)/dr = 0, giving r ≈ 1−ζ02 , for ζ0 1, and

❧ When r→∞ (ω  ω0 ), ❧ ❧

-

-

-

-

DMFmax -

  1 1    = . ≈ DMF r=1 =   2 2 2 2ζ (1−r ) +(2ζ0 r) r=1 0 -

(3.3.22)

Hence, the smaller the damping ratio, the larger the amplitude of dynamic response. ❧ When ζ0 = 0 and ω = ω0 , the system is in resonance, and the amplitude of the response grows linearly with time.

Ground Excitation ug(t) = u0 e i ωt When the SDOF system is subjected to only the ground excitation ug(t) = u0 e i ωt, the

equivalent earthquake load is F(t) = −m u¨g (t) = mω2 u0 e i ωt . Referring to equation (3.3.19), the response of the SDOF oscillator under ground excitation is given by

(3.3.23) x(t) = H(ω) mω2 u0 e i ωt = H(ω) ω2 u0 e i ωt ,

3.3 single degree-of-freedom system

Dynamic Magnification Factor (DMF)

7

83 ζ=0

6 5

ζ=0.1

4 3

ζ=0.2

2 1

r ≈1 – ζ2

r = ωω

ζ=0.3

0

0

1 2 0.5 1.5 2.5 Figure 3.10 DMF of SDOF system under externally applied force.

Dynamic Magnification Factor (DMF)

7

ζ=0

6 5

ζ=0.1

4 3 ζ=0.2

2 1

r ≈1 + ζ2

ζ =0.3

r = ωω

0

0

1 2 0.5 1.5 2.5 Figure 3.11 DMF of SDOF system under ground excitation.

where

H(ω) is the complex frequency response function with respect to base excitation H(ω) =



∞

−∞

h(t) e−i ωt dt =

1 (ω02 −ω2 ) +

i2ζ0 ω0 ω

.

(3.3.24)

The D-MF, characterizing the magnification of the dynamic displacement response am  plitude x(t)max of the SDOF oscillator in terms of the ground displacement amplitude u0 , is defined as   x(t) r2 ω max = (3.3.25) DMF = , r= ω . u0 0 (1−r 2 )2 +(2ζ0 r)2

84

The D-MF is plotted in Figure 3.11 for various values of the damping ratio ζ0 .

❧ When r→0 (ω ω0 ) or when the SDOF oscillator is very stiff, D-MF →0. The SDOF oscillator moves with the ground as a rigid body, and the relative displacement between the mass and the ground approaches 0. ❧ When r→∞ (ω  ω0 ) or when the SDOF oscillator is very flexible, D-MF →1. The mass m does not move, and the relative displacement between the mass and the ground approaches the ground displacement.

DMF tends to large values for small damping. DMFmax occurs when d(DMF)/dr = 0, giving r ≈ 1+ζ02 , for ζ0 1, and

❧ When r ≈ 1 (ω ≈ ω0 ), ❧

-

-

-

DMFmax -

  1 r2    = . ≈ DMF r=1 =   2 2 2 2ζ (1−r ) +(2ζ0 r) r=1 0 -

(3.3.26)

❧ When ζ0 = 0 and ω = ω0 , the system is in resonance.

3.4 Multiple Degrees-of-Freedom Systems The equation of motion of a multiple degrees-of-freedom (MDOF) structure with N degrees-of-freed (DOF) under the excitation of ground motion can be written as M x¨ (t) + C x˙ (t) + Kx(t) = −M I u¨g(t),

(3.4.1)

 T  T where x = x1 , x2 , . . . , xN is the relative displacement vector, I = 1, 1, . . . , 1 is the N-dimensional influence vector, M, C, K are the mass, damping, and stiffness matrices of dimension N×N, respectively. Matrices M and K are symmetric, i.e., MT = M, KT = K, and positive definite.

3.4.1 Free Vibration Consider the undamped free vibration governed by M x¨ (t) + Kx(t) = 0.

(3.4.2)

Seeking a solution of the form x(t) = ϕ e i ωt and substituting into equation (3.4.2) yield an eigenvalue problem (3.4.3) (K−ω2 M)ϕ = 0. To have nonzero solutions for ϕ, the determinant of the coefficient matrix must be zero det(K−ω2 M) = 0.

(3.4.4)

The Ith root (eigenvalue) ωI (ω1 < ω2 < · · · < ωN ) is the natural frequency of the Ith mode of the system or the Ith modal frequency.

3.4 multiple degrees-of-freedom systems

85

Corresponding to the Ith eigenvalue ωI , a nonzero solution ϕ I of system (3.4.4), (K−ωI2 M)ϕ I = 0,

I = 1, 2, . . . , N,

(3.4.5)

is the Ith eigenvector or the Ith mode shape. Construct the modal matrix  as ⎤ ⎡ ϕ11 ϕ12 · · · ϕ1N ⎢ϕ   ϕ22 · · · ϕ2N ⎥ 21  = ϕ1 ϕ2 · · · ϕN = ⎢ (3.4.6) . .. .. .. ⎥ ⎦, ⎣ . . . . . ϕN1 ϕN2 · · · ϕNN where the first subscript I of element ϕI j refers to the node number and the second subscript j corresponds to the mode number.  has the following orthogonal relations





T M = diag m¯ 1 , m¯ 2 , . . . , m¯ N = m, ¯





 2

T K = m ¯ 2 = diag m¯ 1 ω12 , m¯ 2 ω22 , . . . , m¯ N ωN ,

(3.4.7) (3.4.8)



where  = diag ω1 , ω2 , . . . , ωN , and m¯ 1 , m¯ 2 , . . . , m¯ N are the modal masses. It is usually assumed that





T C = diag c¯1 , c¯2 , . . . , c¯N ,

c¯n = 2ζn ωn m¯ n ,

(3.4.9)

i.e., the structure has classical damping and the modal matrix  can diagonalize the damping matrix.

3.4.2 Forced Vibration Substituting x(t) = q(t) into the equation of motion (3.4.1) and multiplying T from the left yield (T M) q¨ + (T C) q˙ + (T K)q = −T M I u¨g . Applying the orthogonal relations (3.4.7) to (3.4.9) gives m¯ n q¨ n + m¯ n 2ζn ωn q˙ n + m¯ n ωn2 qn = − Ln u¨g ,

n = 1, 2, . . . , N,

or q¨ n + 2ζn ωn q˙ n + ωn2 qn = −n u¨g ,

n =

Ln m¯ n

,

(3.4.10)

where Ln , called the earthquake excitation factors, are given by

Ln = ϕ Tn M I. Using Duhamel integral, the solution of equation (3.4.10) is n qn (t) = − ω Vn (t), n

Vn (t) =

 0

t

e−ζn ωn (t−τ ) sin ωn (t−τ ) u¨g(τ )dτ.

(3.4.11)

86

The relative displacement vector x(t) becomes x(t) = q(t) = −

N  n=1

n ϕ n ω Vn (t).

(3.4.12)

n

The elastic forces associated with the relative displacements are given by f e (t) = Kx(t) = Kq(t).

(3.4.13)

Noting that K = M2 , one has

 T f e (t) = M2 q(t) = Fe,1 (t), Fe, 2 (t), . . . , Fe, N (t) ,

(3.4.14)

where Fe,n (t) is the elastic force at the nth floor given by Fe,n (t) = −Mϕ n n ωn Vn (t).

(3.4.15)

Having obtained the elastic forces Fe,n (t) at any time t during the earthquake, any desired force results, such as base shear and overturning moment, can be determined.

3.5 Stationary Response to Random Excitation When a structural or mechanical system is subjected to random excitation, the response will be a random process. The general response problem is to determine the statistical properties of the response process in terms of the given statistical properties of the excitation and the system parameters. The complete solution would require the determination of the distribution functions of all orders of the response. Such exhaustive information is quite often either difficult to obtain or unnecessary, and in many practical problems a knowledge of the first few moments of the response is adequate. In the important case of a linear system subjected to Gaussian excitation, it can be shown that the response is also Gaussian, which is then completely defined by the mean and the auto-correlation function only. In general, the response depends on the state of the system when the excitation is applied, i.e., on the initial conditions. These conditions may be specified uniquely, as in deterministic problems, or only statistically. However, in an important class of problems in which only the asymptotic behaviour for t→∞, i.e., the stationary response, is of interest, a knowledge of the initial conditions is unnecessary. In a stable system, this is the motion that will persist after the initial transient has died away.

3.5.1 Single DOF Systems under Random Excitations Consider a single DOF oscillatory system governed by equation (3.3.2), i.e., 1 F(t), X¨ + 2ζ ω0 X˙ + ω02 X = m

ω02 =

K , m

2ζ ω0 =

c , m

(3.5.1)

3.5 stationary response to random excitation

87

where F(t) is assumed to be a stationary random process. The response of system (3.5.1) is given by the Duhamel integral (3.3.9). It can be shown that, for stationary response, the Duhamel integral can be written as  ∞  ∞ F(τ ) H(t−τ ) dτ = F(t−τ ) H(τ ) dτ. (3.5.2) X(t) = −∞

−∞

Taking the expectation of both sides of equation (3.5.2), the mean response is  +∞ H(τ ) E[ F(t−τ ) ] dτ. E[ X(t) ] = −∞

Because F(t) is stationary, E[ F(t) ] = mF ; thus, mX = E[ X(t) ] = mF



+∞

−∞

H(τ ) dτ.

(3.5.3)

If F(t) has zero mean, i.e., mF = 0, then the mean response is zero, i.e., mX = 0. The auto-correlation function of X(t) is  +∞  RXX (τ ) = E[ X(t) X(t+τ ) ] = E H(τ1 )F(t−τ1 )dτ1  =

+∞ +∞

−∞

−∞

−∞



+∞ −∞

H(τ2 )F(t+τ −τ2 )dτ2

H(τ1 ) H(τ2 ) RFF (τ +τ1 −τ2 ) dτ1 dτ2 ,

(3.5.4)

where RFF (τ ) = E[ F(t) F(t+τ ) ] is the auto-correlation function of F(t). Direct evaluation of RXX (τ ) using (3.5.4) is seldom performed. Instead, one takes the Fourier transform of both sides to obtain the power spectral density of the response

SXX (ω) =



−∞

 =

+∞

RXX (τ ) e−i ωτ dτ

+∞ +∞ +∞

−∞

−∞

−∞

H(τ1 ) H(τ2 ) RFF (τ +τ1 −τ2 ) e−i ωτ dτ dτ1 dτ2 .

Writing τ3 = τ +τ1 −τ2 and rearranging, one obtains  +∞  +∞  i ωτ1 −i ωτ2 SXX (ω) = H(τ1 ) e dτ1 H(τ2 ) e dτ2 −∞

−∞

+∞

−∞

RFF (τ3 ) e−i ωτ3 dτ3

= H ∗ (ω) H(ω) SFF (ω), where H(ω) is the frequency response function given by equation (3.3.20), H∗ (ω) is the complex conjugate of H(ω), and SFF (ω) is the power spectral density (PSD) function of the excitation. Thus, 



SXX (ω) = H(ω)2 SFF (ω).

(3.5.5)

88

The response auto-correlation function can be calculated from (3.5.5) by inverse Fourier transform  +∞  +∞   1 1 H(ω)2 S (ω) e i ωτ dω. (3.5.6) SXX (ω) e i ωτ dω = RXX (τ ) = FF 2π −∞ 2π −∞ The mean-square of the response is given by  +∞   1 H(ω)2 S (ω) dω. E[ X 2 (t) ] = RXX (0) = FF 2π −∞

(3.5.7)

If the excitation F(t) is a stationary Gaussian process, it can be shown that the response X(t) is also Gaussian; the mean and auto-correlation function are then sufficient to describe the response random process completely. For the SDOF system described by equation (3.5.1), for which H(ω) is given by (3.3.20), the response power spectral density is, by (3.5.5),

SXX (ω) =

m2

1



(ω02 −ω2 )2

+ (2ζ ω0 ω)2



SFF (ω).

The auto-correlation function, given by equation (3.5.6), is  +∞ 1 1 i ωτ   S (ω) e dω, RXX (τ ) = 2 2 2 2 2π −∞ m (ω0 −ω ) + (2ζ ω0 ω)2 FF and the mean-square response is, using equation (3.5.7),  +∞ 1 1   E[ X 2 (t) ] = RXX (0) = 2π −∞ m2 (ω02 −ω2 )2 + (2ζ ω0 ω)2

SFF (ω) dω.

(3.5.8)

(3.5.9)

(3.5.10)

Suppose that F(t) is a white noise process with SFF (ω) = S0 and zero mean. Then the mean response given by equation (3.5.3) is mX = 0. The auto-correlation function and the mean-square response given by equations (3.5.9) and (3.5.10) are S0  +∞ e i ωτ RXX (τ ) = dω, (3.5.11) 2πm2 −∞ (ω02 −ω2 )2 + (2ζ ω0 ω)2 S0  +∞ 1 2 2 σX = E[ X (t) ] = dω. (3.5.12) 2 2 2 2 2πm −∞ (ω0 −ω ) + (2ζ ω0 ω)2 The integrals in equations (3.5.11) and (3.5.12) may be evaluated by the method of residues (see Section 3.8, in particular equation (3.8.11)) to give σX2 = and

S0

π 2ζ ω03

S0

S0

.

(3.5.13)

  ζ ω0 RXX (τ ) = σX2 e−ζ ω0 τ cosωd τ + sinωd τ . ωd

(3.5.14)

2πm2

·

=⇒

σX2 =

4 m2 ζ ω03

=

2cK

3.5 stationary response to random excitation

89

For a general (coloured noise) case, the integration in (3.5.9) may have to be per 2 formed numerically. However, if the damping is light with ζ 1, the function H(ω) is sharply peaked near the frequency ω = ω0 (see Figure 3.12). If further SFF (ω) does not vary too rapidly in the neighbourhood of ω = ω0 as in Figure 3.13, the excitation may be approximated by a white noise with spectral density equal to SFF (ω0 ). One may then write 2 S (ω )  +∞  S (ω ) S (ω ) H(ω) dω = FF2 0 3 = FF 0 . (3.5.15) σX2 ≈ FF 0 2π 2cK 4 m ζ ω0 −∞

3.5.2 MDOF Systems under Random Excitations As derived in Section 3.4.2, the response of an N-DOF system in the nth normal mode to a single component of earthquake input u¨g (t) is governed by equation (3.4.10), where u¨g (t) is a random process with zero mean describing the ground acceleration.

Assumptions on the Ground Motion and Structural Response To develop the response spectrum method, the following assumptions are made on the ground motion process u¨g (t) and the response process. 1. The ground motion u¨g (t) is a stationary, Gaussian process with a wide-band power spectral density. ❧ Whereas earthquake-induced ground motions are inherently nonstationary, the strong phase of such motions is usually nearly stationary, as shown in Figure 3.14. Because the peak response generally occurs during this phase, it is reasonable, at least for the purpose of developing a response spectrum method, to assume it to be a stationary process. This assumption would clearly become less accurate for short-duration, impulsive earthquakes. ❧ The assumption of Gaussian excitation is acceptable on the basis of the central limit theorem because the earthquake ground motion is the accumulation of a large number of randomly arriving pulses. ❧ The wide-band assumption for the earthquake motion has been verified based on recorded motions and is generally accepted. 2. The response of the linear structure is a stationary process. ❧ It is well known that the response of a not-too-lightly damped oscillator to a wide-band input reaches stationarity in just a few cycles. Thus, this assumption is acceptable for structures whose fundamental periods are several times shorter than the strong-phase duration of the ground motion. It is clear from the preceding discussion that the response spectrum method for earthquake loading will be most accurate for earthquakes with long stationary phases of

90

H(ω) 2

1 (2kζ)2

2ζω0 1 2(2kζ)2

1 k2

Figure 3.12

ω

ω0

−ω0

Frequency response function.

H(ω) 2 White noise approximation

PSD of excitation

SFF(ω)

H(ω) 2 White noise approximation

SFF(ω)

H(ω) 2 Not suitable for white noise approximation

SFF(ω)

Figure 3.13 Approximation of a random process by a white noise process.

strong shaking and for not-too-lightly damped, not-too-flexible structures (whose fundamental periods are several times shorter than the duration of earthquake).

Response of the Structure The solution of equation (3.4.10) is given by the Duhamel integral (3.5.2), i.e.,  ∞ −n u¨g (t−τ ) hn (τ ) dτ , (3.5.16) qn (t) = −∞

3.5 stationary response to random excitation

91

u(t)

t (s) Stationary Figure 3.14 A sample of earthquake ground motion.

where hn (t) = e − ζn ωn t

sinωn, d t , ωn, d

 ωn, d = ωn 1−ζn2 ,

(3.5.17)

is the impulse response function with respect to base excitation of the nth mode, and its Fourier transform is the complex frequency response function with respect to base excitation given by  ∞ 1 . (3.5.18) Hn (ω) = hn (τ ) e−i ωτ dτ = 2 2 (ωn −ω ) + i2ζn ωn ω −∞ Mean Response Taking the expectation of both sides of equation (3.5.16), the mean response is  ∞ E[ qn (t) ] = −n hn (τ ) E[ u¨g (t−τ ) ] dτ = 0, (3.5.19) −∞

because u¨g (t) is stationary with mean zero, i.e., E[ u¨g (t) ] = 0. Covariance of Response The covariance of responses produced by modes m and n is given by E[ qm (t)qn (t+τ ) ]  ∞  =E −m u¨g (t−τ1 ) hm (τ1 )dτ1 −∞



= m n

where Ru¨

g u¨g



∞

−∞ −∞

 = m n

∞

∞



∞

−∞ −∞

∞ −∞

 −n u¨g (t+τ −τ2 ) hn (τ2 )dτ2

hm (τ1 ) hn (τ2 ) E[ u¨g (t−τ1 ) u¨g (t+τ −τ2 ) ] dτ1 dτ2 hm (τ1 ) hn (τ2 )Ru¨g u¨g (τ +τ1 −τ2 ) dτ1 dτ2 ,

(3.5.20)

(τ ) = E[ u¨g (t) u¨g (t+τ ) ] is the auto-correlation function of u¨g (t). Taking

Fourier transform of both sides yields  ∞ Sqm qn (ω) = E[ qm (t)qn (t+τ ) ] e−i ωτ dτ −∞

= m n



∞



∞



∞

−∞ −∞ −∞

hm (τ1 ) hn (τ2 ) Ru¨g u¨g (τ +τ1 −τ2 ) e−i ωτ dτ1 dτ2 dτ.

92

Writing τ3 = τ +τ1 −τ2 =⇒ τ = τ3 +τ2 −τ1 , one has  ∞  ∞  i ωτ1 −i ωτ2 Sqm qn (ω) = m n hm (τ1 )e dτ1 hn (τ2 )e dτ2 −∞

−∞

∞ −∞

Ru¨g u¨g (τ3 )e−i ωτ3 dτ3

= m n Hm∗ (ω) Hn (ω) Su¨g u¨g(ω), where

(3.5.21)

Su¨g u¨g(ω) is the power spectral density of earthquake excitation u¨g (t), which is

the Fourier transform of Ru¨

g u¨g

E[ qm (t)qn (t+τ ) ] =

(τ ). Hence, taking the inverse Fourier transform gives

1 2π



∞

−∞

  = m n 2π



Sqm qn (ω) e i ωτ dω ∞

−∞

Hm∗ (ω) Hn (ω) Su¨g u¨g(ω) e i ωτ dω,

(3.5.22)

Hm∗ (ω) Hn (ω) Su¨g u¨g(ω) dω.

(3.5.23)

and, by setting τ = 0, E[ qm (t)qn (t) ] =

m n 2π



∞

−∞

As a special case, when m = n, one obtains the auto-correlation function    2 ∞  E[ qn (t)qn (t+τ ) ] = n Hn (ω)2 Su¨g u¨g(ω) e i ωτ dω, 2π −∞

(3.5.24)

and the mean-square response E[ q2n (t) ] =

n2 2π



∞

−∞

   H (ω)2 S (ω) dω. n u¨g u¨g

(3.5.25)

Approximation of Covariance of Response To evaluate equation (3.5.22), consider the following cases.

  ❧ When frequencies ωm and ωn are well separated, the narrow peaks of  Hm (ω)   and  Hn (ω) do not overlap for lightly damped systems. In this case, the numerical value of the integral is relatively small, and the covariance E[ qm (t)qn (t) ] is very small compared to the mean-square values E[ q2m (t) ] and E[ q2n (t) ].   ❧ When frequencies ωm and ωn are very close together, the narrow peaks of  Hm (ω)   and  H (ω) overlap sufficiently so that the covariance E[ q (t)q (t) ] becomes m

n

n

of similar order of magnitude to the mean-square values E[ q2m (t) ] and E[ q2n (t) ]. Because the frequencies ωm and ωn must become very close to each other for this to happen, the value of Su¨g u¨g(ω) will not vary greatly in the neighbourhood of these closely spaced frequencies, i.e., E[ qm (t)qn (t) ] =

Su¨g u¨g(ωm ) ≈ Su¨g u¨g(ωn ) ≈ Smn . Hence,

m n 2π

Smn



∞

−∞

Hm∗ (ω) Hn (ω)dω.

(3.5.26)

3.5 stationary response to random excitation

93

Using equation (3.8.8), equation (3.5.26) becomes E[ qm (t)qn (t) ] =

m n 2π

π 1 ρ =  3 ω3 mn 2 ζm ζn ωm n

Smn ·

Smn 4

  ρ ,  m n 3 ω3 mn ζm ζn ωm n (3.5.27)

where ρmn is given by equation (3.8.9). Noting that m =

Lm m¯ m

=

Km Lm 2 Lm · = ωm , m¯ m Km Km

equation (3.5.27) can be written as E[ qm (t)qn (t) ] =

Smn 4

L

L

m 2 ωm · ωn2 n Km Kn ρmn =  3 ζm ζn ωm ωn3

Smn

Lm Ln

4

Km K n



ωm ωn ρ . (3.5.28) ζm ζn mn

For the special case when m = n, ωm = ωn , ζm = ζn , r = 1, ρnn = 1, equation (3.5.27) yields the mean-square response

Snn

E[ q2n (t) ] =

where

4

n2 , ζn ωn3

(3.5.29)

Snn = Su¨g u¨g(ωn ), or the root-mean-square value of qn (t) 

σqn =

Snn 2

     n . ζn ωn3

(3.5.30)

Using equation (3.5.30) and noting that Smm ≈ Snn ≈ Smn , equation (3.5.27) can be written as   S E[ qm (t)qn (t) ] = mn  m n ρ 3 ω3 mn 4 ζm ζn ωm n       Smm m  Snn n    = · · ρmn ·  m n     3 3 2 2  m n  ζm ωm ζn ωn   αmn =  m n  = sgn(m n ).    m n

= αmn ρmn σqm σqn ,

(3.5.31)

3.5.3 CQC and SRSS Combination of Modal Responses Consider a response z(t), which has contributions from all N normal modes, given by z(t) =

N  n=1

An qn (t),

(3.5.32)

where coefficients An are known for the structural system under consideration. Squaring both sides of equation (3.5.32) yields z2 (t) =

N  N  m=1 n=1

Am An qm (t) qn (t).

94

Taking expected value of both sides gives the mean-square response E[ z2 (t) ] = σz2 =

N  N  m=1 n=1

Am An E[ qm (t) qn (t) ].

Using equation (3.5.31) results in  σz =

N  N 

m=1 n=1

αmn Am An ρmn σqm σqn .

(3.5.33)

The maximum value of the response z(t) is, according to Section 3.2,   z(t)

max

= Pfz · σz ,

Pfz = Peak factor.

(3.5.34)

Similarly,   q (t) = Pfqn · σqn , n max

Pfqn = Peak factor,

n = 1, 2, . . . , N.

Substituting equations (3.5.34) and (3.5.35) into (3.5.33) results in (     ) q (t) q (t) N  N )   m n max max * z(t) = Pfz αmn Am An ρmn · · . max

Pfqm

m=1 n=1

Pfqn

(3.5.35)

(3.5.36)

For responses in earthquake engineering, the values of the peak factors Pfqm , Pfqn , and

Pfz do not differ significantly, i.e., Pfz2 / Pfqm · Pfqn ≈ 1; hence, equation (3.5.36) can be approximated as   z(t) ≈ max



N  N 

m=1 n=1

    αmn Am An ρmn · qm (t)max · qn (t)max .

(3.5.37)

❧ CQC Method. The combination method of (3.5.37) for evaluating maximum total response from the individual maxima of modal responses is known as the complete quadratic combination (CQC) method. When the major contributing modes have frequencies close together, the corresponding cross terms in equation (3.5.37) can be very significant and should be retained. ❧ SRSS Method. If the frequencies of the contributing modes are well separated, the cross terms in equation (3.5.37) are negligible, i.e., ρmn 1, m  = n. In this case, equation (3.5.37) reduces to   z(t)

max

 =

N 

n=1

 2 A2n qn (t)max .

which is known as the square root of sum of squares (SRSS) method.

(3.5.38)

3.6 seismic response analysis

95

It is very important to note that using the CQC or the SRSS combination method must always be the last step in evaluating the maximum value of any response quantity. In other words, one cannot use the maximum value of one response quantity, obtained using the CQC and the SRSS method, to evaluate the maximum value of another response quantity.

3.6 Seismic Response Analysis N

un,6

Ground response spectrum

un,3 un,2 n

2 1

un,5

xGi (t)= uGi (t)− ugi (t) (Relative)

un,1 un,4 Multiple DOF primary structure

ω0, ζ0 SDOF oscillator

ugi (t) ug3(t)

ug2(t) ug1(t)

Figure 3.15

SAi (ω0, ζ0)=max |uGi (t) |

uGi (t) (Absolute)

Tridirectional ground excitations

Multiple DOF structure under tridirectional seismic excitations.

Consider a three-dimensional model of a structure with N nodes. A typical node n has six DOF: three translational DOF un,1 , un,2 , and un,3 , and three rotational DOF un,4 , un,5 , and un,6 . The structure is subjected to tridirectional seismic excitations (Figure 3.15). The relative displacement vector x of dimension 6N is governed by M x¨ (t) + C x˙ (t) + Kx(t) = −M where

⎧ ⎫ x ⎪ ⎪ ⎪ ⎬ ⎨ x1 ⎪ x = .2 , . ⎪ ⎪ ⎪ ⎩ . ⎪ ⎭ xN

⎧ ⎫ x ⎪ ⎪ ⎪ ⎪ n,1 ⎪ ⎪ ⎨ xn,2 ⎬ xn = . , ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩x . ⎪ ⎭ n,6

3  I=1

⎧ I⎫ 1⎪ ⎪ ⎪ ⎬ ⎨1I ⎪ I . I = . , ⎪ .⎪ ⎪ ⎭ ⎩ I⎪ 1

I I u¨gI (t), ⎧ ⎫ δ ⎪ ⎪ ⎪ ⎬ ⎨δI1 ⎪ I2 I 1 = . , .⎪ ⎪ ⎪ ⎭ ⎩.⎪ δI6

(3.6.1)

(3.6.2)

M, C, and K are, respectively, the mass, damping, and stiffness matrices of dimension 6N×6N, xn is the relative displacement vector of node n, I I is the influence vector of seismic excitation in direction I, and δIj denotes the Kronecker delta function, i.e., δIj = 0, if I = j, and δIj = 1, if I = j. Let x = x1 + x2 + x3 , where x I is the relative displacement vector due to earthquake excitation ugI (t) in direction I. Hence, xn,I j = un,I j − ugI δIj , where xn,I j and un,I j

96

are, respectively, the relative and absolute displacements of node n in direction j due to earthquake excitation in direction I. Because the system is linear, from equation (3.6.1), x I is governed by M x¨ I (t) + C x˙ I (t) + Kx I (t) = −M I I u¨gI (t),

I = 1, 2, 3.

(3.6.3)

In response time-history methods, equation (3.6.3) may be solved by modal superposition method or by direct time integration method; whereas in response spectrum method, equation (3.6.3) are solved by modal superposition method.

3.6.1

Modal Superposition Method

Free Vibration Consider first the undamped free vibration with M x¨ I (t) + Kx I (t) = 0.

(3.6.4) 



Let ω1 , ω2 , . . . , ω6N be the 6N natural frequencies and  = ϕ 1 , ϕ 2 , · · · , ϕ 6N be the   modal matrix, where ϕ K = ϕ T1,K , ϕ T2,K , . . . , ϕ TN,K T is the mode shape of the Kth mode,   with ϕ n,K = ϕn,1; K , ϕn,2; K , . . . , ϕn,6; K T . In element ϕn, j; K , the first subscript n refers to the node number, the second subscript j indicates the direction of response, and the third subscript K is the mode number. The modal matrix  has the following orthogonal relations   T M = diag m¯ 1 , m¯ 2 , . . . , m¯ 6N T = m, ¯   2 T ¯ 2 = diag m¯ 1 ω12 , m¯ 2 ω22 , . . . , m¯ 6N ω6N , T K = m    = diag ω1 , ω2 , . . . , ω6N T ,

(3.6.5)

where m¯ 1 , m¯ 2 , . . . , m¯ 6N are the modal masses. Assume that the structure has classical damping so that the modal matrix  can also diagonalize the damping matrix   c¯K = m¯ K · 2ζK ωK . (3.6.6) T C = diag c¯1 , c¯2 , . . . , c¯6N T ,

Forced Vibration Apply the transformation x I (t) = Q I (t), where

I = 1, 2, 3,

(3.6.7)

⎧ I ⎫ ⎧ I⎫ ⎧ I I ⎫ ⎧ I I ⎫ ⎧ I⎫ xn,1 ⎪ ⎪ Q1 ⎪ ⎪ 1 q1 ⎪ x1 ⎪ n,1 qn,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Q I ⎬ ⎨ I q I ⎬ ⎨x I ⎬ ⎨ I q I ⎬ ⎨x I ⎪ ⎬ n,2 n,2 n,2 , x I = 2 2 2 , xI = Q I = .2 = , (3.6.8) , QnI = . . . n .. .. ⎪ .. ⎪ .. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ I⎭ ⎩ I I ⎭ ⎩ I⎭ ⎩ I I ⎭ ⎪ ⎭ ⎩xI ⎪ n,6 qn,6 QN 6N q6N xN n,6

3.6 seismic response analysis

xn,I j =

N  6  ν=1 δ=1

97

ϕn, j; 6(ν−1)+δ ν,I δ qν,I δ =

I T I Ln, j = ϕ 6(n−1)+j M I , I  n, j =

I Ln, j

m ¯ 6(n−1)+j

,

or

or KI =

6N  K=1

ϕn, j; K KI qKI ,

LKI = ϕ TK M I I , LKI m ¯K

=

ϕ TK M I I . ϕ TK Mϕ K

(3.6.9) (3.6.10) (3.6.11)

For ease of presentation, the two-subscript-notation n, j (node, direction) and the one-subscript-notation K = 6(n−1)+j are used interchangeably; the former is advantageous in describing the meaning of the quantity in terms of node and direction, and I the latter gives the position of the quantity in the corresponding vector. Ln, j is the earthquake excitation factor, quantifying the contribution of earthquake excitation in I .  I is the modal participation factors; if  I the Ith direction to the modal response qn, j K K is small, then the contribution of mode ϕ K to the structural response due to excitation in the Ith direction is small. Substituting equation (3.6.7) into (3.6.3) and multiplying T from the left yield ¨ I (t) + (T C) Q ˙ I (t) + (T K)QI (t) = −T M I I u¨gI (t). (T M) Q Using equations (3.6.5), (3.6.6), (3.6.10), and (3.6.11) gives m¯ K · KI q¨KI + m¯ K 2ζK ωK · KI q˙KI + m¯ K ωK2 · KI qKI = − LKI u¨gI (t), or q¨KI (t) + 2ζK ωK q˙KI (t) + ωK2 qKI (t) = − u¨gI (t), K = 1, 2, . . . , 6N, I = 1, 2, 3. (3.6.12) The solution of equation (3.6.12) is given by Duhamel integral  t 1 I I I e−ζK ωK (t−τ ) sin ωK (t−τ ) u¨gI (τ )dτ. qK (t) = − ω VK (t), VK (t) = K

(3.6.13)

0

The modal response Q IK (t) is I

Q IK (t) = KI qKI = − ωK VKI (t). K

3.6.2

(3.6.14)

Seismic Response History Analysis

The set of 6N coupled differential equations in (3.6.3) in nodal displacements xKI (t) is transformed to the set of 6N uncoupled differential equations given by (3.6.12) in modal coordinates qKI (t). Equation (3.6.12) governs the equation of motion of the Kth mode (an SDOF system) of the structural system with damping ratio ζK and natural frequency ωK subjected to ground acceleration u¨gI (t) in the Ith direction. The response (relative displacement with respect to the ground) of the linear 6N DOF system as shown in Figure 3.15 subjected to earthquake ground motion u¨gI (t) (I = 1, 2, 3) can then be obtained using equations (3.6.7) and (3.6.14).

98

The seismic response history analysis (SRHA) procedure is concerned with the calculation of structural response as a function of time when the system is subjected to a set of tridirectional ground acceleration u¨gI (t) (I = 1, 2, 3). For illustration purpose, the nodal inertia forces of the 6N-DOF system are computed using SRHA based on the modal superposition method. From equations (3.6.7) and (3.6.14), the acceleration vector of modal masses (DOF) of the system relative to the ground is given by x¨ I (t) =

2N 

ϕ K KI q¨KI (t),

K=1

I = 1, 2, 3,

(3.6.15)

where q¨KI (t) is the acceleration of the Kth mode of the system relative to the ground. The vector of the earthquake-induced ground motion in the Ith direction can be written as u¨ gI (t) = I I u¨gI (t) =

2N 

K=1

ϕ K KI u¨gI (t),

II =

2N 

K=1

ϕ K KI .

(3.6.16)

The nodal inertia forces of the 6N-DOF system subjected to ground acceleration in the Ith direction is given by   (3.6.17) F I (t) = −M u¨ gI (t) + x¨ I (t) , 



where u¨ gI (t)+ x¨ I (t) is the vector of the nodal absolute accelerations of the system subjected to ground acceleration in the Ith direction. Substituting equations (3.6.15) and (3.6.16) into equation (3.6.17) gives FI (t) = −M

2N 

K=1





ϕ K KI u¨gI (t)+ q¨KI (t) ,

(3.6.18)

which is a function of time. The vector of time-histories of the nodal inertial forces of the system due to the tridirectional ground acceleration can be obtained by algebraic summation of the response time-histories at each time step due to the ground acceleration in individual direction, i.e., F(t) =

3.6.3

3 

I=1

F I (t).

Direct Time Integration Method

In Section 3.6.2, coupled equations of motion (3.6.1) or (3.6.3) of a linear structural system with classical damping are decoupled using modal analysis, such that the solutions to a set of coupled differential equations are transformed to the solutions to a set of uncoupled differential equations of equivalent SDOF system. In this case, numerical methods are only involved in solving the equations of motion (3.6.12) of SDOF system when subjected to earthquake ground motion. However, uncoupling of equations of motion is not possible if the structural system has nonclassical damping or it responds into the nonlinear range. In this section, the basic concepts of direct time integration method for solving the uncoupled equations of motion of such structural system are introduced.

3.6 seismic response analysis

99

The objective is to numerically solve the system of differential equations of the multiple DOF system given by equation (3.6.1), M x¨ (t) + C x˙ (t) + Kx(t) = p(t),

p(t) = −M

3  I=1

I I u¨gI (t)

(3.6.19)

with equivalent earthquake loading p(t) and initial conditions x = x(0), x˙ = x(0) ˙ at t = 0. The solution will provide the displacement vector x(t) as a function of time. By direct time integration, the equation of motion (3.6.19) is solved using numerical integration schemes. For linear response analysis of multiple DOF systems, central difference method and Newmark’s method are two popular direct time integration methods, which are detailed in Chopra (2012). For nonlinear systems, the solution algorithm involves an iteration process to converge at each time step. Direct time integration is usually done using a commercial finite element analysis package, such as STARDYNE and ANsys.

3.6.4

Seismic Response Spectrum Analysis

The seismic response spectrum analysis (SRSA) is concerned with procedures for computing the peak response of a structure during an earthquake directly from the earthquake response (or design) spectra without the need for time-history analysis of the structure. This procedure does not give the exact peak response, but it provides an estimate that is sufficiently accurate for structural analysis and design applications. Based on equation (3.6.14), the maximum absolute value of Q IK (t) is, using response spectra studied in details in Chapter 4, 







 I   I   I   I I K K I I Q (t)   S (ζ , ω ), = S (ζ , ω ) = S (ζ , ω ) = A K D K K K K K K K V max ωK ω2

(3.6.20)

K

where

S

I V (ζK , ωK )

  = V I (t) K

max

 t    −ζK ωK (t−τ ) I  = e sin ωK (t−τ ) u¨g (τ )dτ  max

(3.6.21)

0

is the velocity response spectrum in direction I, and

SAI (ζK , ωK ) =

S VI (ζK , ωK ) ωK

,

SDI (ζK , ωK ) = ωK S VI (ζK , ωK )

(3.6.22)

are the acceleration and displacement response spectra, respectively, in direction I. The relatively displacement xn,I j of node n in direction j due to seismic excitation in direction I is given by (3.6.9). The elastic force vector f sI associated with the relative displacement x I (t) due to seismic excitation in direction I is f sI (t) = Kx I (t) = KQ I (t).

(3.6.23)

100

In undamped free vibration, the elastic forces can be expressed in terms of the equivalent inertial forces (3.6.24) K x I = M x¨ I =⇒ K = M2 . Hence, f sI (t) = KQ I (t) = M2 Q I (t).

(3.6.25)

In general, for a response quantity z I (t) due to seismic excitation in direction I z I (t) =

6N  K=1

AIK qKI (t),

(3.6.26)

its maximum absolute value can be obtained using CQC as  6N  6N  I   I     I AI AI ρ z (t) q (t) · q I (t) , = α · K K K K K K K K max max max K=1 K =1

(3.6.27)

where αK K is given by equation (3.5.31) and ρK K is given by equation (3.8.9) I I αK K =  KI KI  = sgn(KI KI ),    K K  8 ζK ζK (ζK +rζK )r3/2 ρK K = , (1−r 2 )2 + 4 ζK ζK r (1+r 2 ) + 4 (ζK2 +ζK2 )r 2 or using SRSS as  I  z (t)

max

 =

6N 

K=1

(3.6.28) ω

r = ωK , K

 2 (AIK )2 qKI (t)max .

(3.6.29)

(3.6.30)

  The directional maximum absolute value z I (t)max , I = 1, 2, 3, are combined using   SRSS to obtain z(t)max as   z(t)

max

=

  z1 (t)2

max

 2  2 + z 2 (t)max + z 3 (t)max .

(3.6.31)

Example Consider a frame ABC with a rigid right-angle at B and clamped to ground at support A as shown in Figure 3.16(a). It supports a lumped mass m at end A with three DOF u1 , u2 , and u3 . The weight of the frame is negligible. The flexural rigidity of both members AB (in both directions 1 and 2) and BC (in both directions 2 and 3) is EI. Both members AB and BC are rigid in the axial direction and in torsion. The frame is subject to tridirectional ground excitations u¨g1 (t), u¨g2 (t), and u¨g3 (t). The ground response spectra (GRS) follow USNRC R.G. 1.60 (USNRC, 2014), with GRS in both horizontal directions being the same and anchored at PGA = 0.3g and vertical GRS anchored at PGA = 0.2g. The modal damping is 5 % for all modes .

3.6 seismic response analysis u3

(a)

B

101 L,EI

(b)

u2

l, EI

δB

m u1

C

3

L,EI L, EI ug2(t)

ug3(t)

A

θB

P

M L,EI

ug1(t)

2

δB = PL 3EI

θB = PL 2EI

2 δB = ML 2EI

θB = ML EI

Figure 3.16 A frame under tridirectional seismic excitations. (1)

f11

f11

−f31

2 θ= L 2EI F=1

L3 3EI

f11 F=1

θ

lL2 2EI

l

f11

L,EI

A F=1

(2)

f22

F=1

F=1

L3 3EI

l, EI

3

l 3EI

L,EI

A F=1

(3)

−f13

−f13

θ= lL EI f33

lL2 2EI

M =1. l L,EI

l2L EI

l θ F =1 l,EI

A

Figure 3.17

l3 EI

Flexibility.

Use the parameters L = 2L, L = 4 m, m = 5 kg, and EI = 106 N · m2 . Using the response spectrum method, determine the maximum absolute values of the relative displacements x1 = u1 −u1g , x2 = u2 −ug2 , and x3 = u3 −ug3 and the maximum overturning moments M1 and M2 in directions 1 and 2. Because there is only one lumped mass at end A, the mass matrix is   M = diag m, m, m . It is easy to determine the flexibility matrix using the definition of flexibility: F I j is the displacement along DOF I due to a unit force applied along DOF j, while forces along

102

all other DOF being zero. Some useful results for bending of cantilever are shown in Figure 3.16(b). Referring to Figure 3.17, apply a unit force F = 1 along each of DOF in turn, the elements of flexibility matrix F are F11 =

L3 , 3EI

F21 = 0,

F12 = 0, F13 = −

L3 +L 3 , 3EI

F22 = LL2 , 3EI

F31 = − F32 = 0,

F23 = 0,

F33 =

Using L = 2L, the stiffness matrix is given by ⎡ K = F−1 =

21 ⎢ 20

0

9 10

0

EI ⎢ ⎢0 L3 ⎣

LL2 , 3EI

L3 L2L + . EI 3EI

⎤

9 10 ⎥

⎥ 0 ⎥. ⎦

1 3

6 5

Due to the tridirectional ground excitations, the inertial forces applied on the mass m are FI = −m(¨xI + u¨gI ), I = 1, 2, 3. Using the flexibility matrix, the relative displacements xI = uI −ugI , I = 1, 2, 3, due to the inertial forces are 











xI = FI1 −m(¨x1 + u¨g1 ) + FI2 −m(¨x2 + u¨g2 ) + FI3 −m(¨x3 + u¨g3 ) , or, in the matrix form, ⎧ ⎫ ⎡ x F11 F12 ⎪ ⎨ 1⎪ ⎬ ⎢ x2 = − ⎣ F21 F22 ⎪ ⎩ ⎪ ⎭ x3 F31 F32

F13

⎤⎡

m

⎥⎢ F23 ⎦ ⎣ 0 F33

0

⎤ ⎧x¨ + u¨ 1 ⎫ ⎪ g⎪ ⎪ ⎪ ⎨ 1 ⎬ ⎥ 2 m 0 ⎦ x¨ 2 + u¨g . ⎪ ⎪ ⎪ ⎪ 0 m ⎩x¨ 3 + u¨g3 ⎭ 0

0

Multiplying the equation by K = F−1 yields ⎤⎧ ⎫ ⎡ ⎡ ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫ m 0 0 ⎨1⎬ m 0 0 ⎨0⎬ m 0 0 ⎨0⎬ Kx = −M x¨ − ⎣ 0 m 0 ⎦ 0 u¨g1 − ⎣ 0 m 0 ⎦ 1 u¨g2 − ⎣ 0 m 0 ⎦ 0 u¨g3 , 0 0 m ⎩0⎭ 0 0 m ⎩0⎭ 0 0 m ⎩1⎭ or M x¨ + Kx = −M I 1 u¨g1 − M I 2 u¨g2 − M I 3 u¨g3 , where

⎧ ⎫ ⎨1⎬ I1 = 0 , ⎩0⎭

⎧ ⎫ ⎨0⎬ I2 = 1 , ⎩0⎭

The eigenequation is given by

  21 −λ  20    2 K − ω M = 0 =⇒  0   9  10

0 1 −λ 3

0

⎧ ⎫ ⎨0⎬ I3 = 0 . ⎩1⎭ 

9  10 

 0  = 0,   6 −λ 5

λ = ω2 ·

mL 3 . EI

3.6 seismic response analysis

103

The eigenvalues and eigenvectors are λ1 = 0.2219, λ2 = 0.3333, λ3 = 2.0281, ⎫ ⎫ ⎧ ⎧ ⎫ ⎧ ⎨ −1.0868⎬ ⎨0⎬ ⎨0.9201⎬ 0 0 ϕ1 = , ϕ2 = 1 , ϕ3 = ⎭ ⎩ ⎩0⎭ ⎩ 1 ⎭ 1 The modal frequencies are determined as  EI λI ωI = , mL 3 which give ω1 = 26.3320 rad/s, F1 = 4.1909 Hz,



ωI , 2π

FI =

ω2 = 32.2749 rad/s, F2 = 5.1367 Hz,



=⇒  = ϕ 1 ϕ 2 ϕ 3 .

ω3 = 79.6108 rad/s, F3 = 12.6704 Hz.

It is easy to determine that     T M = diag m¯ 1 , m¯ 2 , m¯ 3 = diag 10.9057, 5, 9.2332   T K = diag 7561.7302, 5208.3333, 58518.9990 . Using equation (3.6.11), the modal participation factors are 11 = −0.4983,

21 = 0,

31 = 0.49823,

12 = 0,

22 = 1,

32 = 0,

13 = 0.4585,

23 = 0,

33 = 0.5415.

The critical points of USNRC R.G. 1.60 GRS are listed in Table 3.1; each spectrum is obtained by connecting the critical points linearly in the log-log scale. Using linear interpolation, the spectral values at the modal frequencies can be determined, with ζ1 = ζ2 = ζ3 = 5 %:

SA1 (ζ1 , F 1 ) = 0.8957g, SA1 (ζ2 , F 2 ) = 0.8723g, SA1 (ζ3 , F 3 ) = 0.6761g, SA2 (ζ1 , F 1 ) = 0.8957g, SA2 (ζ2 , F 2 ) = 0.8723g, SA2 (ζ3 , F 3 ) = 0.6761g, SA3 (ζ1 , F 1 ) = 0.5862g, SA3 (ζ2 , F 2 ) = 0.5729g, SA3 (ζ3 , F 3 ) = 0.4508g. The GRS are shown in Figure 3.18. Using equation (3.6.20), the maximum modal responses can be determined    1   1  Q (t) Q (t) = 0.006307 m, Q 12 (t)max = 0, = 0.000521 m, 1 3 max max   2   2   Q (t) Q (t) = 0, = 0.008206 m, Q 23 (t)max = 0, 1 2 max max  3     3  Q (t) Q (t) = 0.003798 m, Q 32 (t)max = 0, = 0.000377 m. 1 3 max max

104 Table 3.1

USNRC R.G. 1.60 GRS with 5 % damping

Horizontal F (Hz) 0.1 0.25 2.5 9 33 100

SA ( g) 0.02264 0.14149 0.93900 0.78300 0.30000 0.30000

1

Acceleration (g)

Vertical

SA ( g) 0.01009 0.06304 0.59600 0.52200 0.20000 0.20000

F (Hz) 0.1 0.25 3.5 9 33 100

0.8957 0.8723 0.5862 0.5729

0.6761

0.4508

Horizontal 0.3 Vertical

0.2

0.1

0.01

f1 =4.1909

0.1

Figure 3.18

f2 =5.1367 f3 =12.6704

10 1 Frequency (Hz) 100 USNRC R.G. 1.60 GRS with 5 % damping.

The relative displacement vector is given by equation (3.6.7), ⎫ ⎡ ⎤⎧ I ⎫ ⎧ I + 0.9201 Q I Q −1.0868 Q ⎪ ⎪ −1.0868 0 0.9201 ⎪ 1 3⎪ ⎪ ⎪ ⎨ 1⎪ ⎬ ⎪ ⎬ ⎨ ⎢ ⎥ I I I I 0 1 0 ⎦ Q2 = x (t) = Q (t) = ⎣ . Q2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ I⎭ ⎩ ⎭ 1 0 1 Q3 Q I1 + Q I3 From SRSS, the maximum relative displacements are       I 2  I 2 2  I 2 x  1 max = 1.0868 Q 1 max + 0.9201 Q 3 max ,  I x 

2 max

  = Q I2 max ,

which give  1 x  1 max = 0.006872 m,  1 x  2 max = 0,  1 x  3 max = 0.004143 m,

     I Q I 2 + Q I 2 , x  = 3 max 1 max 3 max

 2 x  1 max = 0,  2 x  2 max = 0.008206 m,  2 x  3 max = 0,

 3 x  1 max = 0.006329 m,  3 x  2 max = 0,  3 x  3 max = 0.003817 m.

3.7 nonlinear systems

105

Applying SRSS, the maximum relative displacements are         x 1 2 + x 2 2 + x 3 2 , K = 1, 2, 3, x  = K max K max K max K max       x  x  x  1 max = 0.008024 m, 2 max = 0.008206 m, 3 max = 0.007391 m. The elastic force vector is give by equation (3.6.25) f sI (t) = M2 Q I (t) ⎡

m

⎢ = ⎣0

0 m

0

⎤⎡

⎥⎢ 0 ⎦⎣

−1.0868 0 0.9201 0

1

0

⎤⎧ I ⎫ ⎤⎡ 0 0 26.33202 ⎪ ⎪ ⎪Q 1 ⎪ ⎥⎨ I ⎬ ⎥⎢ 2 32.2749 0 ⎥ ⎦⎢ ⎣ 0 ⎦⎪Q 2 ⎪ ⎪ ⎭ ⎩ I⎪ 0 0 79.61082 Q3

0 0 m 1 0 1 ⎫ ⎧ I I ⎪ ⎪ ⎪−3767.8051 Q 1 + 29158.4301 Q 3 ⎪ ⎬ ⎨ I . = 5208.3333 Q 2 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 3466.8813 Q I1 + 31689.3687 Q I3

The overturning moments at support A are  I     M1 −41666.6667 Q I2 − F s2I · L = = . M2I −44009.9660 Q I1 + 106509.9662 Q I3 F s1I · L − F s3I · L Applying SRSS, the maximum overturning moments are   ⎫  I   ⎧ 41666.6667 Q I2 max ⎬ ⎨ M1 max  . =  I     ⎩ M  2 Q I 2 2 Q I 2 ⎭ 44009.9660 + 106509.9662 2 max 1 max 3 max The maximum overturning moments due to directional earthquake excitations are  3  2  1   M  M  1 max = 341.9358 N · m, M1 max = 0, 1 max = 0,  3  2  1 M  M    2 max = 171.9290 N · m. 2 max = 283.0883 N · m, M2 max = 0, Applying SRSS, the maximum overturning moments are         M  M 1 2 + M 2 2 + M 3 2 , K = 1, 2, K max = K max K max K max     M  M  1 max = 341.9358 N · m, 2 max = 331.2078 N · m.

3.7 Nonlinear Systems The method of equivalent linearization, which replaces a nonlinear system by an equivalent linear system, is useful in determining the mean-square response to stationary excitation. Consider the following SDOF system ˙ + g(X) = F(t). X¨ + F(X)

(3.7.1)

106

Property of Gaussian Random Variables 1. If X1 (t), X2 (t), …, Xn (t) are Gaussian random variables with mean zero, then E[X1 X2 X3 ] = 0, E[X1 X2 X3 X4 ] = E[X1 X2 ] E[X3 X4 ] + E[X1 X3 ] E[X2 X4 ] + E[X1 X4 ] E[X2 X3 ].

In general, for m = 2, 3, . . . , E[X1 X2 · · · X2m−1 ] = 0,

E[X1 X2 · · · X2m ] =



E[XI Xj ] E[XK XL ],

(3.7.5)

where the summation involves (2m)!/(2m m!) terms and is to be taken over all different ways by which 2m elements can be grouped into m distinct pairs. 2. If X(t) is a stationary process, then RX X (τ ) = E[ X(t) X(t+τ ) ], ˙ ˙ ], ) ] = E[ X(t−τ ) X(t) RX X (τ ) = RX X˙ (τ ) = E[ X(t) X(t+τ

(3.7.6)

˙ X(t+τ ˙ ˙ ˙ ]. ) ] = − E[ X(t−τ ) X(t) RX X (τ ) = −RX˙ X˙ (τ ) = − E[ X(t)

Replacing equation (3.7.1) by the equivalent linear system X¨ + β X + ω02 X = F(t),

(3.7.2)

the error of linearization, which is also a random process, is 







˙ . E = ω02 X − g(X) + β X˙ − F(X)

(3.7.3)

If the error term is zero, the response of the linear system (3.7.2) is given by (3.5.10)  +∞ 1 SFF (ω) 2 E[ X (t) ] = dω. (3.7.4) 2 2π −∞ (ω0 −ω2 )2 + (βω)2 The parameters β and ω02 are chosen in such a way that some average function of the error E is minimized. ˙ = β X− ˙ F(X), ˙ α = ω02 . The usual choice is to miniLet E1 (X) = α X− g(X), E2 (X) mize the mean-square errors E[ E12 ] = α 2 E[ X 2 ] − 2α E[ X g(X) ] + E[ g 2 (X) ],

i.e., ∂ E[ E12 ] =0 ∂α

E[ E22 ] = β 2 E[ X˙

=⇒

α=

2

˙ ], ˙ ] + E[ F 2 (X) ] − 2β E[ X˙ F(X)

E[ X g(X) ] E[ X 2 ]

,

∂ E[ E22 ] =0 ∂β

=⇒

β=

˙ ] E[ X˙ F(X) E[ X˙

2

]

. (3.7.7)

3.7 nonlinear systems

107

Equations (3.7.4) and (3.7.7) are solved to determine the mean-square response E[ X 2 ], and parameters β and ω02 = α. As an example, consider the following SDOF system with nonlinear damping 3 X¨ + β (X˙ + ε X˙ ) + ω02 X = F(t),

(3.7.8)

where F(t) is a white noise process with SFF (ω) = S0 . Replace the equation of motion by the linear equation (3.7.2). Because F(t) is a white noise process, then RFF (τ ) = S0 δ(τ ), and 



SXX (ω) = H(ω)2 SFF (ω) =

S

0 (ω02 −ω2 )2 + (βω)2

Therefore, from equation (3.5.14),  +∞ 1 RXX (τ ) = SXX (ω) e i ωτ dω 2π −∞   1 β = σX2 e− 2 βτ cosωD τ + sinωD τ , 2 ωD

.

(3.7.9)

1/2 ωD = ω02 − 41 β 2 ,

(3.7.10)

and, from equation (3.5.13), the mean-square response of system (3.7.2) is given by  +∞   1 2 2 H(ω)2 S dω = S0 . E[ X ] = σX = RXX (0) = (3.7.11) 0 2π −∞ 2βω02 Differentiating equation (3.7.10) and using equation (3.7.6) result in RXX (τ ) = RX X˙ (τ ) = −

ω02 2 − 1 βτ σ e 2 sinωD τ , ωD X

(3.7.12)

(0) = −RX˙ X˙ (0) = −ω02 σX2 . RXX

(3.7.13)

From equation (3.7.13) 2 E[ X˙ ] = RX˙ X˙ (0) = −R XX (0) = ω02 σX2 =

S0 2β

.

(3.7.14)

The error of approximation is ˙ E2 (X) = β X˙ − F(X),

˙ = β (X˙ + ε X˙ ). F(X) 3

(3.7.15)

Minimizing the mean-square error E[ E22 (X) ], one obtains, from equation (3.7.7), ˙ ] = β E[ X˙ E[ X˙ F(X)

2

]=

S0 2

.

(3.7.16)

Because ˙ ] = E[ X˙ · β (X˙ + ε X˙ 3 ) ] = β E[ X˙ 2 ] + ε · E[ X˙ 4 ] E[ X˙ F(X)

!

108

= β E[ X˙

2

] + ε · 3 E[ X˙ 2 ] E[ X˙ 2 ]

!

,

(3.7.17)

in which equation (3.7.5) is used. Substituting equation (3.7.14) into (3.7.17) gives 



β ω02 σX2 + ε · 3 · ω02 σX2 · ω02 σX2 =

S0 2

,

(3.7.18)

which yields the mean-square response σX2 =

☞

3 S0  1 S0 S0  2 2 . (3.7.19) 1−ε ≈ (1−ε · 3ω σ ) = 0 X 2β 1+ε · 3ω02 σX2 2βω02 2βω02

S0 2βω02

Although this result has been obtained for a Gaussian white noise excitation, it can be expected to yield a reasonable approximate solution when the spectral density of the excitation is slowly varying in the neighbourhood of ω02 and when the damping in the system is light.

3.8 Appendix − Method of Residue Theorem – Method of Residue Suppose P(x) and Q(x) are polynomials that are real-valued on the real axis and for which the degree of Q(x) exceeds the degree of P(x) by 2 or more. If Q(x)  = 0 for all real x, then  ∞  P(x)   P(x) (3.8.1) Res dx = 2π i ;z , Q(x) j −∞ Q(x) U

P(x) where the sum is taken over all poles of that lie in the upper half-plane Q(x)   U = z : Im (z) > 0 . The complex frequency response function is given by equation (3.5.18). Hence,  ∞  ∞ dω ∗    Hm (ω) Hn (ω)dω = Imn = 2 2 2 2 −∞ −∞ (ωm −ω )− i2ζm ωm ω (ωn −ω )+ i2ζn ωn ω  =



∞

2 −ω2 ) + i2ζ ω ω (ωm m m



−∞





2 −ω2 ) − i2ζ ω ω (ω2 −ω2 ) + i2ζ ω ω (ωm m m m m m 

×



(ωn2 −ω2 ) − i2ζn ωn ω 





(ωn2 −ω2 ) + i2ζn ωn ω (ωn2 −ω2 ) − i2ζn ωn ω

R I = Imn + iImn ,



dω (3.8.2)

3.8 appendix − method of residue

109

where  R Imn

= =

2 −ω2 )(ω2 −ω2 ) + 4ζ ζ ω ω ω2 (ωm m n m n n  dω,  2 −ω2 )2 + (2ζ ω ω)2 (ω2 −ω2 )2 + (2ζ ω ω)2 (ωm n n n m m



−∞

 I Imn

∞



∞

2 −ω2 ) 2ω ζm ωm (ωn2 −ω2 ) − ζn ωn (ωm



−∞

(3.8.3)





2 −ω2 )2 + (2ζ ω ω)2 (ω2 −ω2 )2 + (2ζ ω ω)2 (ωm n n n m m



dω.

(3.8.4)

Because 2 2 4 −ω2 )2 + (2ζm ωm ω)2 = ω4 − 2ωm (1−2ζm2 )ω2 + ωm , F(ω) = (ωm

the roots of F(ω) = 0, solved by treating the equation as a quadratic equation in ω2, are  2 (1−2ζ 2 )± 4ω4 (1−2ζ 2 )2 − 4 · 1 · ω4 

2ωm m m m m 2 2 = ωm ω = 1−2ζm2 ± i 2ζm 1−ζm2 2  2 cosθ = 1−2ζm2 , sinθ = 2ζm 1−ζm2 . = ωm ( cosθ ± i sinθ), It is easy to evaluate that cos 2

1 + (1−2ζm2 ) 1+ cosθ θ = = 1−ζm2 , = 2 2 2

sin 2

1 − (1−2ζm2 ) 1− cosθ θ = = ζm2 . = 2 2 2

The roots of F(ω) = 0 are then given by  θ +2Kπ θ +2Kπ  ω = ωm cos ± i sin , K = 0, 1, 2 2   

θ θ ω = ωm cos ± i sin = ωm 1−ζm2 ± iζm , 2 2        θ θ θ θ ω = ωm cos +π ± i sin +π = ωm − cos ∓ i sin 2 2 2 2

 = ωm − 1−ζm2 ∓ iζm .

K = 0: K = 1:

R and I I , there are four of total eight poles that lie in the upper Hence, for both Imn mn half-plane given by





ω = ωm ± 1−ζm2 + iζm , ωn ± 1−ζn2 + iζn .

(3.8.5)

Using equation (3.8.1) and a symbolic computation software package, such as Maple, R and I I can be easily evaluated to yield the integrals Imn mn I = 0, Imn R Imn =

4π(ωm ζm + ωn ζn ) 4 2 2 4 2 +ω2 ) + 4ω2 ω2 (ζ 2 +ζ 2 ) (ωm −2ωm ωn +ωn ) + 4ωm ωn ζm ζn (ωm n m n m n

(3.8.6)

110

ωn  1  ζ + ζ m 3 ωm ωm n =  ω2 ω4  ω ω2  ω2 1−2 2n + 4n + 4 n ζm ζn 1+ 2n + 4 2n (ζm2 +ζn2 ) ωm ωm ωm ωm ωm 4π ·

=

ζm + rζn 4π · , 3 2 2 ωm (1−r ) + 4ζm ζn r(1+r 2 ) + 4(ζm2 +ζn2 )r 2

r=

ωn . ωm

(3.8.7)

Hence, equation (3.8.2) becomes  ∞ R I Hm∗ (ω) Hn (ω)dω = Imn = Imn + iImn −∞

ζm + rζn 4π · 3 2 2 ωm (1−r ) + 4ζm ζn r(1+r 2 ) + 4(ζm2 +ζn2 )r 2  8 ζm ζn (ζm +rζn )r3/2 1 4π = 3 ·  · , ωm 8 ζm ζn r3/2 (1−r 2 )2 + 4ζm ζn r(1+r 2 ) + 4(ζm2 +ζn2 )r 2

=

 ∴

∞

−∞

Hm∗ (ω) Hn (ω)dω = Imn =

1 π · ρ , 3 ω3 mn 2 ζm ζn ωm n

(3.8.8)

where ρmn

 8 ζm ζn (ζm + rζn )r3/2 = , (1−r 2 )2 + 4ζm ζn r(1+r 2 ) + 4(ζm2 +ζn2 )r 2

r=

ωn . ωm

For the special case when m = n, ωm = ωn , ζm = ζn , r = 1, one has    8 ζm ζn (ζm +rζn )r3/2  = 1, ρmm =  (1−r 2 )2 + 4ζm ζn r(1+r 2 ) + 4(ζm2 +ζn2 )r 2 m=n

(3.8.9)

(3.8.10)

r=1

and equation (3.8.8) reduces to 

   H (ω)2 dω = I = π · 1 . m mm 3 2 ζm ωm −∞ ∞

(3.8.11)

Maple Program > restart: I and I R , expressed in Q is the denominator of the integrands of integrals Imn mn

terms of poles. > Q:=(w-omega[m1])*(w-omega[m2])*(w-omega[m3])*(w-omega[m4]) *(w-omega[n1])*(w-omega[n2])*(w-omega[n3])*(w-omega[n4]); Q := (w−ωm1 )(w−ωm2 )(w−ωm3 )(w−ωm4 )(w−ωn1 )(w−ωn2 )(w−ωn3 )(w−ωn4 ) I of the integrand of integral I I . IP is the numerator Pmn mn

3.8 appendix − method of residue

111

> IP:=2*w*(zeta[m]*omega[m]*(omega[n]^2-w^2)-zeta[n]*omega[n] *(omega[n]^2-w^2)); IP := 2w(ζm ωm (ωn2 −w2 ) − ζn ωn (ωn2 −w2 )) > FI:=IP/Q; FI :=

I . FI is the integrand of integral Imn

2w(ζm ωm (ωn2 −w2 ) − ζn ωn (ωn2 −w2 )) (w−ωm1 )(w−ωm2 )(w−ωm3 )(w−ωm4 )(w−ωn1 )(w−ωn2 )(w−ωn3 )(w−ωn4 )

R of the integrand of integral I R . RP is the numerator Pmn mn

> RP:=(omega[m]^2-w^2)*(omega[n]^2-w^2) +4*zeta[m]*zeta[n]*omega[m] *omega[n]*w^2; 2 −w2 )(ω2 −w2 ) + 4ζ ζ ω ω w2 RP := (ωm m n m n n

> FR:=RP/Q; FR :=

R . FR is the integrand of integral Imn

2 −w2 )(ω2 −w2 ) + 4ζ ζ ω ω w2 (ωm n m n m n (w−ωm1 )(w−ωm2 )(w−ωm3 )(w−ωm4 )(w−ωn1 )(w−ωn2 )(w−ωn3 )(w−ωn4 )

I and I R . Q is the denominator of the integrands of both integrals Imn mn

> Q:=((omega[m]^2-w^2)^2 +(2*zeta[m]*omega[m]*w)^2)*((omega[n]^2-w^2)^2 +(2*zeta[n]*omega[n]*w)^2); 2 −w2 )2 + (2ζ ω w)2 )((ω2 −w2 )2 + (2ζ ω w)2 ) Q := ((ωm m m n n n I , IR . omega[m1],..., omega[n4] are eight poles of integrands of integrals Imn mn

> omega[m1]:=omega[m]*(sqrt(1-zeta[m]^2)+I*zeta[m]);  ωm1 := ωm ( 1−ζm2 + iζm ); > omega[m2]:=omega[m]*(sqrt(1-zeta[m]^2)-I*zeta[m]);  ωm2 := ωm ( 1−ζm2 − iζm ); > omega[m3]:=omega[m]*(-sqrt(1-zeta[m]^2)+I*zeta[m]);  ωm3 := ωm (− 1−ζm2 + iζm ); > omega[m4]:=omega[m]*(-sqrt(1-zeta[m]^2)-I*zeta[m]);  ωm4 := ωm (− 1−ζm2 − iζm ); > omega[n1]:=omega[n]*(sqrt(1-zeta[n]^2)+I*zeta[n]);  ωn1 := ωn ( 1−ζn2 + iζn ); > omega[n2]:=omega[n]*(sqrt(1-zeta[n]^2)-I*zeta[n]);  ωn2 := ωn ( 1−ζn2 − iζn ); > omega[n3]:=omega[n]*(-sqrt(1-zeta[n]^2)+I*zeta[n]);  ωn3 := ωn (− 1−ζn2 + iζn );

112

> omega[n4]:=omega[n]*(-sqrt(1-zeta[n]^2)-I*zeta[n]);  ωn4 := ωn (− 1−ζn2 − iζn ); > RIm1:=residue(FI,w=omega[m1]):

Evaluate the residue of FI at ωm1 .

> RIm3:=residue(FI,w=omega[m3]): > RIn1:=residue(FI,w=omega[n1]):

Evaluate the residue of FI at ωm3 . Evaluate the residue of FI at ωn1 .

> RIn3:=residue(FI,w=omega[n3]):

Evaluate the residue of FI at ωn3 .

I . Use Theorem A.1 to determine the integral Imn

> INTI:=simplify(2*Pi*I*(RIm1+RIm3+RIn1+RIn3)); INTI :=0 > RRm1:=residue(FR,w=omega[m1]):

Evaluate the residue of FR at ωm1 .

> RRm3:=residue(FR,w=omega[m3]):

Evaluate the residue of FI at ωm3 . Evaluate the residue of FI at ωn1 . Evaluate the residue of FI at ωn3 .

> RRn1:=residue(FR,w=omega[n1]): > RRn3:=residue(FR,w=omega[n3]):

R . Use Theorem A.1 to determine the integral Imn

> INTR:=simplify(2*Pi*I*(RRm1+RRm3+RRn1+RRn3)):

☞

When a function is in the form of a complex expression divided by another complex expression, it is difficult to simplify the function on its own. The numerator and denominator of the function must be simplified separately. R , expand, and factorize the result, called NINT. Extract the numerator of Imn

> NINT:=factor(expand(numer(INTR))): R , expand, and factorize the result, called DINT. Extract the denominator of Imn

> DINT:=factor(expand(denom(INTR))): > INTR:=NINT/DINT; INTR :=

R is obtained. A simple expression of Imn

4π(ωm ζm + ωn ζn ) 4 2 2 4 2 +ω2 ) + 4ω2 ω2 (ζ 2 +ζ 2 ) (ωm −2ωm ωn +ωn ) + 4ωm ωn ζm ζn (ωm n m n m n

❧

❧

In this chapter, fundamentals of random processes and structural dynamics that are the theoretical foundations for a number of topics in this book were presented. ❧ A random process is described by its various probability distribution functions. If the probability distribution functions are invariant under a change of time origin, the random process is stationary. A random process can be practically modelled as stationary when the physical factors influencing it do not change with time.

3.8 appendix − method of residue

113

❧ For many practical applications, a random process can be satisfactorily described by its moments, in particular the first two moments (mean and mean-square value or variance). A random process is ergodic if its time average is equal to its ensemble average; for a random process to be ergodic, stationarity is a necessary condition. ❧ Autocorrelation function is one of the most important averages in random vibration. Power spectral density (PSD) is the Fourier transform of the autocorrelation correlation function; PSD is very important in characterizing the frequency content of a stationary random process. A random process can be satisfactorily modelled as a white noise process if it has a nearly constant PSD over the frequency range of interest. For a transient nonstationary random process, Fourier amplitude spectrum (FAS) is more appropriate for characterizing its frequency content. ❧ There is a significant gap between the theory of random vibration and the practice in earthquake engineering. In random vibration, responses of a system are determined in terms of averages, such as mean responses and mean-square responses. On the other hand, in earthquake engineering, responses of a structure are determined in terms of peak values, such as peak accelerations. The peak value of a random process is related to its mean-square value through the peak factor. However, there is no exact analytical expression available for the peak factor. Various assumptions, approximations, and statistical approaches have been applied to obtain approximate or empirical expressions. ❧ Modal analysis is important in studying the responses of multiple DOF systems. The strong-motion portion of an earthquake ground motion time-history can be reasonably modelled as a stationary, Gaussian process with a wide-band PSD; hence, the theory of random vibration can be applied to study the responses of a multiple DOF system under earthquake ground motion excitation. CQC and SRSS combination rules are developed based on results from random vibration to formulate the method of seismic response spectral analysis. ❧ The maximum responses of a multiple DOF structure can be determined using the time-history method or the response spectrum method. For a given ground motion time-history, the time-history method gives the numerically exact response. However, depending on how the spectrum-compatible time-histories are generated, there may be large uncertainties in the generated time-histories (see Chapter 6); as a result, there may be large variabilities in responses obtained from the time-history method. The response spectrum method does not give the exact peak responses; however, it gives sufficiently accurate results for practical engineering applications.

114

❧ There are many engineering applications in which nonlinearity has to be considered. The method of equivalent linearization is a popular approach to replace a nonlinear system by an equivalent linear system (equivalent in the sense that the error of approximation is minimized).

C

4

H

A

P

T

E

R

Seismic Response Spectra 4.1 Ground Response Spectra 4.1.1 Definitions The calculations of response spectra, for assessing the impact of ground motion on structures, from available earthquake records were started by George Housner in 1941 at Caltech. A response spectrum gives the level of seismic force or displacement as a function of natural period (or frequency) of vibration of the structure and its damping. For simplicity of presentation of the formulations, consider a lightly damped (i.e., ζ0 1) single degree-of-freedom (SDOF) oscillator with natural circular frequency ω0 or period T0 under ground excitation u¨g(t), as shown in Figure 3.7. From Section 3.3.3, the relative displacement x(t), relative velocity x(t), ˙ and absolute acceleration u(t) ¨ are given by x(t) =

1 ω0



t

0

 x(t) ˙ =

t

0

 u(t) ¨ = −ω0

0

t

u¨g(τ ) sinω0 (t−τ ) e−ζ0 ω0 (t−τ ) dτ = h ∗ u¨g ,

(3.3.14 )

u¨g(τ ) cosω0 (t−τ ) e−ζ0 ω0 (t−τ ) dτ ≈ ω0 h ∗ u¨g ,

(3.3.16 )

u¨g(τ ) sinω0 (t−τ ) e−ζ0 ω0 (t−τ ) dτ = −ω02 h ∗ u¨g .

(3.3.17 )

The spectral relative displacement, spectral relative velocity, and spectral absolute acceleration are defined as Spectral relative displacement





SD (ζ0 , T0 ) = x(t)max ,

(4.1.1a) 115

116

Spectral relative velocity





S V (ζ0 , T0 ) = x(t) ˙ max ,

(4.1.1b)

  Spectral absolute acceleration SA(ζ0 , T0 ) = u(t) ¨ max ,

(4.1.1c)

where T0 = 2π/ω0 is the natural period of the SDOF system. It is usually necessary to evaluate only the pseudo-velocity response spectrum defined by   t   −ζ0 ω0 (t−τ )  dτ  SpV (ζ0 , T0 ) =  u¨g(τ ) sinω0 (t−τ ) e 0

,

(4.1.2)

max

which is the maximum value of the shaded term in equation (3.3.16), except the cosine function is changed to the sine function. It can be shown that

SpV (ζ0 , T0 ) differ very little numerically for undamped system,

S V (ζ0 , T0 )

and

except for oscillators

with very long period (very small ω0 ). For damped systems, the difference between

S V (ζ0 , T0 ) and SpV (ζ0 , T0 ) is considerably larger and can differ by as much as 20 % for ζ0 = 0.2.

From equation (3.3.14 ), it is seen that

SD (ζ0 , T0 ) =

1 ω0

SpV (ζ0 , T0 ).

(4.1.3)

For damping values over the range 0 < ζ0 < 0.2, one has, from equation (3.3.17 ),

SA(ζ0 , T0 ) ≈ ω0 SpV (ζ0 , T0 ).

(4.1.4)

The right-hand side of equation (4.1.4) is called the pseudo-acceleration response spectrum, i.e.,

SpA (ζ0 , T0 ) = ω0 SpV (ζ0 , T0 ). The pseudo-acceleration response spectrum

(4.1.5)

SpA (ζ0 , T0 ) is particularly significant be-

cause it is a measure of the maximum spring force developed in the SDOF system, i.e., Fs,max = K SD (ζ0 , T0 ) = m ω02 SD (ζ0 , T0 ) = m SpA (ζ0 , T0 ).

(4.1.6)

The procedure of determination of response spectra is shown schematically in Figure 4.1. An earthquake excitation u¨g(t) is input to a series of SDOF oscillators with a given damping ratio and various natural period (or frequency); a plot of the maximum response versus the natural period (or frequency) is called a ground response spectrum (GRS). It is applicable to displacement response spectrum velocity response spectrum

SpA (ζ0 , T0 ).

SpV (ζ0 , T0 ),

SD (ζ0 , T0 ),

pseudo-

and pseudo-acceleration response spectrum

4.1 ground response spectra

For the displacement response spectrum

117

SD (ζ0 , T0 ),

the pseudo-velocity response

spectrum SpV (ζ0 , T0 ), or the pseudo-acceleration response spectrum SpA (ζ0 , T0 ), only one needs to be generated for any prescribed single component of ground motion; the other two spectra can be easily obtained from equations (4.1.3) and (4.1.5).

☞

Without causing confusion, the subscript “0” for the oscillator parameters, ω0 , ζ0 , T0 , or F 0 may be dropped in various response spectra.

4.1.2 El Centro Earthquake As an example, consider the earthquake of El Centro, California, in 1940. The three components, i.e., the north–south component, the east–west component, and the up–down component, are shown in Figure 4.2. The north–south component has the largest peak ground acceleration (PGA) of 0.313g and will be considered in the remaining of this section. Figure 4.3 shows the time-histories of the ground velocity and displacement of the north–south component. To determine the response spectra, scale the peak ground acceleration (PGA) to a prescribed value to give u¨g(t); in this case, the PGA of the north–south component of the El Centro Earthquake is scaled to 0.35g, as shown in Figure 4.4. For a given damping ζ , Duhamel integral of equation (3.3.14) is evaluated for various values of the natural period T or natural frequency F = ω/(2π ) of the SDOF oscillator to yield the displacement response time-histories. For example, Figure 4.4 shows the displacement response time-histories for damping ratio ζ = 0.05 and natural periods T = 0.05, 0.1, 0.2, 0.4, 1 s. For each displacement time-history with damping ratio ζ and natural period T, the   maximum displacement gives the displacement spectrum S (ζ , T) = maxx(t). For a given value of damping ratio ζ, a plot of

SD (ζ , T)

D

versus natural period T is the

displacement response spectrum, as shown in Figure 4.5(a).

SD (ζ , T), the pseudo-velocity spectrum SpV (ζ , T) and pseudo-acceleration spectrum SpA (ζ , T) can be determined Having obtained the displacement response spectrum

using equations (4.1.3) and (4.1.5), i.e., 2π S (ζ , T), T D g SpA (ζ , T) = ω SpV (ζ , T) = 2 π SpV (ζ , T) = 9.81× SpA (ζ , T), T

SpV (ζ , T) = ω SD (ζ , T) =

where

g SpA (ζ , T) is the pseudo-acceleration in g = 9.81 m/s2 .

(4.1.7a) (4.1.7b)

118 t-response spectrum

Max response

ζ

Ground response spectrum

Max response

f ζ t

f f1

t

f2

t

t

f3

f2

f1 t

t

f3

Response time histories

f1 f1 ζ

f2 ζ

f3 ζ SDOF oscillators

ζ

f1 ζ

f2 ζ

f3 ζ SDOF oscillators

f2 ζ

f3 ζ

Perfect-tuning, uncoupled SDOF structures

t ug(t)

Earthquake input

Figure 4.1 Acceleration ( g)

0.2

Response spectra.

Up–Down

0.1 0 −0.1

Acceleration (g)

−0.2 0.3 0.2 0.1 0 −0.1 −0.2 −0.3

East–West

Acceleration (g)

0.4

North–South 0.2 0 −0.2 −0.4

PGA=0.281g 0

10

Time (sec) 20

30

40

50

60

Figure 4.2 Three components of the El Centro Earthquake. 30

Velocity (cm/sec)

20 10 0 −10 −20 −30

PGV=30.9 cm/sec

−40 6

Displacement (cm)

4 2 0 −2 −4 −6 −8 −10 0

PGD=8.7 cm 10

Time (sec) 20

30

40

50

Figure 4.3 The north–south component of the El Centro Earthquake.

60

4.1 ground response spectra

119

0.4

Acceleration ( g)

El Centro Ground Motion, North–South, scaled to PGA =0.35g 0.2

0

−0.2

Time (sec)

0.35 g

−0.4

0

10

0.05

20

30

40

50

ζ =0.05, T =0.05 sec

0.022 cm

0

60

−0.05 0.2

ζ =0.05, T =0.1 sec

Displacement (cm)

0 −0.184 cm

−0.2 1

ζ =0.05, T =0.2 sec

0 −0.774 cm

−1 5

ζ=0.05, T =0.4 sec

0 −3.034 cm

−5 20

Time (sec) 0

Figure 4.4

2

4

6

8

10

12

14

16

18

Displacement histories of single DOF systems under the El Centro Earthquake. Pseudo-acceleration (g) Pseudo-velocity (cm/sec) Displacement (cm)

−20

ζ=0.05, T =1 sec

14.546 cm

0

40

(a)

30 20 10 0 120 100 80 60 40

(b)

20 0 1.5

(c) 1 0.5 0

0

0. 5

1

1. 5

2

2. 5

3

3. 5

Period (sec) Figure 4.5 Response spectra of El Centro Earthquake.

4

20

120

4.1.3 Tripartite Taking logarithm of base 10 of equations (4.1.7) yields 



log10 SpV (ζ , T) = − log10 T + log10 2π SD (ζ , T) ,   9.81 g log10 SpV (ζ , T) = + log10 T + log10 SpA (ζ , T) . 2π

(4.1.8a) (4.1.8b)

❧ When log10 SpV (ζ , T) is plotted versus log10 T, equation (4.1.8a) is a straight

line in the direction of −45◦ for a given value of log10 SD (ζ , T), as shown in

Figure 4.6. Some critical points can be easily found from equation (4.1.7a): for

T, SD (ζ , T) = (0.01, 0.001), (0.1, 0.01), (1, 0.1), (10, 1), (100, 10),

SpV (ζ , T) =

2π T

SD (ζ , T) = 2π ×0.1 = 0.628 m/s.

❧ When log10 SpV (ζ , T) is plotted versus log10 T, equation (4.1.8b) is a straight g

line in the direction of +45◦ for a given value of log10 SpA (ζ , T), as shown in

Figure 4.6(b). Some critical points can be easily found from equation (4.1.7b): for

g T, SpA (ζ , T) = (0.01, 10), (0.1, 1), (1, 0.1), (10, 0.01), (100, 0.001),

SpV (ζ , T) = 9.81×

T 2π

g SpA (ζ , T) =

9.81×0.1 = 0.156 m/s. 2π

Figure 4.6 is a four-way logarithmic plot, called tripartite, so that the displacement, pseudo-velocity, and pseudo-acceleration spectra can be plotted in the same figure as a function of the natural period of vibration. Figure 4.7 shows the response spectra of the El Centro Earthquake, for various values of damping ratio ζ = 0, 0.05, 0.1, and 0.2, along with the peak ground displacement     (PGD) maxug , peak ground velocity (PGV) maxu˙g , and peak ground acceleration   (PGA) maxu¨g , obtained in Figure 4.5, plotted in the tripartite. A typical point shown in Figure 4.5 is, for ζ = 0.05 and T = 2 s, 2π 2π ×0.25 = 0.79 m/s, SD (ζ , T) = T 2 2π 2π g ×0.79 = 0.25 g. SpA (ζ , T) = SpV (ζ , T) = 9.81 T 9.81×2

SD (ζ , T) = 0.25 m, SpV (ζ , T) =

Some properties of the response spectra can be observed: 1. For an SDOF oscillator with very short period, say T < 0.03 s, or very high natural frequency F > 33 Hz, it is very stiff or essentially rigid; the system moves rigidly with the ground, and there is little relative displacement between the mass and the

4.1 ground response spectra

121

e Ps

D isp lac em en t( m )

10

ud 0

ce

10

ac

10

oler 10

1 1 0.

1

)

0.628

(g

0.156

0. 01

01 0.

0. 1

0.1

0.

00

00

00

0.

01

01

0.

00

00

0.

0.01

1

1 00 0.

Pseudo-velocity (m/sec)

n io at

1

1

0.001 0.01

0.1

10

1 Period (sec)

100

Figure 4.6 Tripartite. e Ps

D isp lac em en t( m )

10 ud

10

ce ler

10

ac

0

o-

ζ=0

n io at 10

1

0.1

25 m

0.2

0.

0.05 1

0.

Peak ground velocity 0.38 m/sec

1 0.

g 35

0.

01

ler ce Pe a

0.

00

00

00

0.

01

01

0.

00

00

0.

1

k

gr ou nd

1

ac

00

0.01

1m .1 t0 en em lac sp di

at io n

nd ou gr

0.

ak Pe

0.

25

g

1

01 0.

0.1

0.

Pseudo-velocity (m/sec)

) (g

1 0.79 m/sec

1

0.001 0.01

Figure 4.7

2 sec

0.1

1 Period (sec)

10

Response spectrum of the El Centro Earthquake.

100

122

ground. As a result, the pseudo-acceleration spectrum SpA (ζ , T) approaches the   PGA maxu¨ , and the displacement spectrum S (ζ , T) is extremely small. g

D

2. For an SDOF oscillator with very long period, say T > 33 s, or very low frequency F < 0.03 Hz, it is extremely flexible, and the mass would remain essentially stationary; the relative displacement between the mass and the ground is the displacement of the ground. As a result, the pseudo-acceleration spectrum small, and the displacement spectrum

SpA (ζ , T) is extremely  

SD (ζ , T) approaches the PGD maxug .

3. For an SDOF oscillator with intermediate period, say 0.5 < T < 3 s, or frequency 0.33 < F < 2 Hz, the pseudo-velocity spectrum SpV (ζ , T) is larger than the PGV   maxu˙g . In this region, the pseudo-velocity may be approximated by a straight horizontal line amplified by a factor αV , which depends on the value of the damping   ratio ζ, from the PGV maxu˙ . g

4.1.4

Newmark Elastic Design Spectra

The response spectra of several earthquakes in California (scaled to PGA = 0.4 g): 1940 El Centro Earthquake, 1989 Loma Prieta earthquake, 1994 Northridge earthquake, and 1971 San Fernando earthquake are shown in Figure 4.8; the mean response spectrum of these four response spectra is also plotted. It is clearly seen that there is great variability in response spectra for different earthquakes. Newmark and Hall (1982) constructed an “idealized” seismic response spectrum based on response spectra generated from a number of real earthquake records. This idealized seismic response spectrum is further developed into a design response spectrum (DRS) for structural design. The original design response spectrum and its modifications have been widely used in seismic structural design all over the world. For a given level of nonexceedance probability (NEP), a DRS can be derived from statistical studies of actual earthquakes, expressed conveniently as a set of amplification factors applied to the peak parameters of ground motions, as shown in Figure 4.9. The procedure of Newmark elastic DRS is as follows:   1. Determine the PGA maxu¨g  based on probabilistic seismic hazard analysis (PSHA). The PGV and PGD can be determined from the empirical equations   2   maxu˙g  maxu¨g        , , maxug  = c2 maxu˙g  = c1 g maxu¨g   48 in/s, for competent soil sites; c2 = 6. c1 = 36 in/s, for rock sites;       Draw the lines corresponding to maxug , maxu˙g , and maxu¨g .

(4.1.9)

4.1 ground response spectra

123

e Ps

D isp lac em en t( m )

10 ud

El Centro (1940)

ce

10

ac

0

oler

10

San Fernando (1971) Loma Prieta (1989)

n io at

Northridge (1994)

(g

Parkfield (1966)

)

1

10

1

1

0. 1

5% Damping Scaled to 0.4 g (PGA)

0. ce

ler

at io n

01

0.

4g

1

0.

0.1

01 0. 0. 00 1

0.01

0.

00

00

00

0.

01

1

0 00 0.

Pe a

k

gr ou n

1

d

00

ac

0.

Pseudo-velocity (m/sec)

Average

1

0.001 0.01

0.1

10

1 Period (sec)

100

Response spectra of several earthquakes in California.

Figure 4.8

Velocity sensitive c

αV · max | ug |

| ug

|

Acceleration sensitive

d

4

A

α

| | ug (lo gs

ca le)

m ax n io at

t

ler

en em

ce

en

em 10 sec 1/10 Hz

33 sec 1/33 Hz

10 Te

Tf

ax

m

Pe

|

ak

|u g

le)

D

ca

isp 1/8 sec 8 Hz

t

lac

5

gs (lo

1/33 sec 33 Hz

0.01

f

lac

n

ac

sp di

io at

nd

nd ou

gr ler

ou

6

k ce

gr

| a Pe

ac

o-

5

e

1

6

a

|u g

Peak ground velocity max | ug |

b

d eu Ps

Pseudo-velocity (log scale)

ax

·m

ax

3

·m αD

2

Displacement sensitive

Ta

0.1Tb

Tc Td 1 Period (sec, log scale)

Figure 4.9

Newmark design spectra.

100

124

  2. From TB = 1/8 s, draw line αA · maxu¨g  parallel to the line of PGA.   3. Draw line αV · maxu˙g  parallel to the line of PGV.   4. Up to Te = 10 s, draw line αD · maxug  parallel to the line of PGD. These three lines form a polyline between TB = 1/8 s and Te = 10 s.   5. Up to Ta = 1/33 s, draw a line coinciding with maxu¨g .   From T = 33 s, draw a line coinciding with maxu . F

g

6. Draw connecting line from Ta = 1/33 s and TB = 1/8 s. Draw connecting line from Te = 10 s and TF = 33 s. ❧ At point c,

SA = ω S V =

2π Tc

SV

=⇒

❧ At point D,

S V = ω SD =

2π Td

SD

=⇒

SV = 2π · SA S Td = 2π · D = 2π · SV Tc = 2π ·

αV · PGV . αA · PGA αD · PGD . αV · PGV

Procedure for Determining Spectrum Amplification Factors 1. Select Ground Motions To determine the spectrum amplification factors for a given site, ground motions observed at the site of interest are required. However, to determine the spectrum amplification factors for a given type of site conditions, ground motions observed at sites with similar geological conditions are required and can be selected from earthquake events around the world using the following selection criteria: ❧ Ground motions with PGA less than 0.05g are not selected because they do not usually cause structural damage to buildings (Mohraz, 1976). Furthermore, weak ground motions usually have larger spectrum amplification factors (NUREG-0003, USNRC, 1976), which will affect the statistical results of spectrum amplification factors for ground motions of engineering interest. ❧ Only ground motions with complete information, including three components records and site classifications are considered. ❧ In order to study characteristics of response spectra at frequencies higher than 33 Hz, only ground motions with usable frequency greater than 33 Hz are selected. ❧ Pulse-like ground motions are not selected. 2. Calculate Ground Response Spectra Response spectra SA of selected ground motions are calculated for periods or frequencies uniformly spaced over the logarithmic scale of a given period range (e.g., 0.01 s to 10 s) or frequency range (e.g., 0.1 Hz to 100 Hz).

4.1 ground response spectra

125

Plot response spectra of the selected ground motions on tripartite. To determine statistically the spectrum amplification factors, relative response values rather than absolute response values are required. Because different ground motions have different PGA, PGV, and PGD values, normalization of ground motions is required to eliminate the effects of ground motion parameters. Ground motions are normalized by PGA in high-frequency (short-period) band, by PGV in intermediate-frequency (period) band, and by PGD in low-frequency (long-period) band, as shown in Figure 4.10. ❧ If ground motions are normalized by only one ground motion parameter, there is significant variation over the entire frequency range of interest. ❧ If ground motions are normalized to PGA, the variation of spectrum amplification factors is small in the short-period (high-frequency) band, but quite large in the intermediate- and long-period bands. ❧ If ground motions are normalized to PGD, the variation of spectrum amplification factors is small in the long-period (low-frequency) band, but quite large in the intermediate and short-period bands. ❧ If ground motions are normalized to PGV, the variation of spectrum amplification factors is nearly constant over the whole frequency range. However, the variations of spectrum amplification factors in the short- and long-period bands are larger than those of ground motions normalized to PGA and PGD, respectively. 3. Spectrum Amplification Factors Suppose that M ground motions are used. At a given frequency, the spectrum amplification factors are assumed to follow normal distributions. The displacement, velocity, and acceleration sensitivity frequency bands used for calculating spectrum amplification factors are chosen as follows: ❧ Displacement sensitivity frequency band: 0.2 to 0.4 Hz (2.5 to 5 s) ❧ Velocity sensitivity frequency band: 0.4 to 2 Hz (0.5 to 2.5 s) ❧ Acceleration sensitivity frequency band: 2 to 6 Hz (0.166 to 0.5 s) The procedures for determining spectrum amplification factors αA , αV , and αD are similar. In the following, the steps and formulations for determining αA are presented. (1) For a damping value ζK and at frequency F I , the acceleration amplification factor for the jth ground motion, normalized by PGA, is αA, j (ζK , F I ) =

SA, j (ζK , F I ) PGA

,

I = 1, 2, . . . , NA ,

K = 1, 2, . . . , K.

(4.1.10)

126

(2) The mean value and standard deviation (SD) of αA (ζK , F I ) at frequency F I can be determined from the M sample values αA, j (ζK , F I ), j = 1, 2, . . . , M: μα (ζK , F I ) =

M 1  αA, j (ζK , F I ), M

σα2 (ζK , F I ) =

M 2 1  αA, j (ζK , F I )−μα (ζK , F I ) . A M−1

A

A

j=1

(4.1.11)

j=1

(3) The median value αA50% (ζK , F I ) and the mean-plus-one-SD (84.1 % NEP) value αA84.1% (ζK , F I ) can be determined from the normal distribution: αA50% (ζK , F I ) = μα (ζK , F I ), A

αA84.1% (ζK , F I ) = μα (ζK , F I ) + σα (ζK , F I ). A

A

(4.1.12)

(4) The median αA50% (ζK ) and the mean-plus-one-SD (84.1 % NEP) value αA84.1% (ζK ) are obtained by averaging the corresponding values in the acceleration sensitive frequency band: αA50% (ζK )

NA 1  αA50% (ζK , F I ), = NA

αA84.1% (ζK )

I=1

NA 1  = αA84.1% (ζK , F I ), (4.1.13) NA I=1

where NA is the number of discrete frequency values used in this band. (5) For K damping values, regression analysis is applied to αA50% (ζK ) and αA84.1% (ζK ), respectively, to obtain statistical relationships αA50% (ζ ) and αA84.1% (ζ ). (6) Knowing αA50% (ζ ) and αA84.1% (ζ ), spectrum acceleration amplification factor p

αA (ζ ) for any desired level p of NEP is obtained from normal distribution:

μα (ζ ) = αA50% (ζ ), A

p αA (ζ )

σα (ζ ) = αA84.1% (ζ ) − αA50% (ζ ), A

= μα (ζ ) + σα (ζ ) · −1 ( p) A

A

−1

( p) · αA84.1% (ζ ) +

=





1− −1 ( p) · αA50% (ζ ).

(4.1.14)

Newmark’s Spectrum Amplification Factors In WASH-1255 (USNRC, 1973; see also Newmark et al., 1973), 28 horizontal ground motions were used to determine the probabilistic distributions of horizontal spectrum amplification factors, and 14 vertical ground motions were used to determine the probability distribution of vertical spectrum amplification factors. Information of the 28 ground motions used is listed in Table 4.1. It is found, in WASH-1255 (USNRC, 1973), that the velocity and acceleration amplifications factors obtained using ground motions with PGA > 0.1g are smaller than those

4.1 ground response spectra 10

127

ud o-

ler

10

ce n io at

10 0

ac

D isp lac em en t( m )

e Ps

20 ground motions normalized to PGA of 0.287g 5% damping

) (g 1

10

1 0.

A PG

D PG

01 0.

0.1

0. 1

PGV

1

0.

00

01

0.

0.

84.1%

1

0.

00

00

50%

01

01

0.

00

00

1

0.

0.01

00

Pseudo-velocity (m/sec)

1

1

Period (sec)

0.001 0.01 10

0.1

10

1

ud

20 ground motions normalized to PGV of 0.256 m/sec 5% damping

o-

D isp lac em en t( m )

e Ps

ler

10

ce

0

ac n io at

10

100

) (g 1

10

1

0.

A PG

D

01

1

0.

00

1

0.

00

00

84.1%

0.001 0.01

01

0.

50%

00

01

0.

00

00

1

0.

0.01

PG

01

0.

0.1

0. 1

PGV

0.

Pseudo-velocity (m/sec)

1

1

Period (sec) 0.1

1

10

100

e Ps ud o-

20 ground motions normalized to PGD of 0.094 m 5% damping ler

10

ce n io at

10 0

ac

D isp lac em en t( m )

10

128

) (g 1

10

1 0.

A PG

01 0.

01

1

0.

00

00

84.1%

01

0.

50%

00

01

0.

00

00

1

0.

0.01

D PG

01 0.

0.1

0. 1

PGV

0 0.

Pseudo-velocity (m/sec)

1

1

Period (sec)

0.001 0.01

0.1

Idealized response spectra based on 20 ground motions 5% damping n io at

0

ler

10

ce

ac

o-

ud

e Ps 10

100

D isp lac em en t( m )

10

10

1

Newmark response spectra.

Figure 4.10

(g ) 1

10

1

2. 1 71 ×P G 2× PG A A

1

A

0. 1

2. 1

01

D

PG

01

PG

0.

0.1

1.

0.

PGV

D G × P GD 01 P 2. 39×

1.65×PGV

0.

00

0.

Pseudo-velocity (m/sec)

2.30×PGV

1

1 0.

00

00

84.1%

01

0.

50%

00

01

0.

00

00

0.

0.01

1

0.001 0.01

1/33 sec 33 Hz

1/8 sec 8 Hz 0.1

Figure 4.11

Period (sec)

10 sec 1/10 Hz

1

10

Idealized response spectra.

33 sec 1/33 Hz 100

4.1 ground response spectra Table 4.1 Earthquake

Eureka Eureka Northern Calif Northern Calif El Alamo  El Alamo  Hollister 

28 ground motions used by Newmark in 1973.

Date

Station Name

22/03/1957 22/03/1957 09/02/1971 09/02/1971 09/02/1971 09/02/1971 19/05/1940 19/05/1940 08/10/1951 08/10/1951 21/07/1952 21/07/1952 21/07/1952 21/07/1952 21/12/1954 21/12/1954 21/12/1954 21/12/1954 09/02/1956 09/02/1956 09/04/1961 09/04/1961 09/04/1968 09/04/1968 09/02/1971 09/02/1971 09/02/1971 09/02/1971

1117 Golden Gate Park 1117 Golden Gate Park Old Ridge Route Old Ridge Route 126 Lake Hughes #4 126 Lake Hughes #4 117 El Centro Array #9 117 El Centro Array #9 1023 Ferndale City Hall 1023 Ferndale City Hall Hollywood Stor FF Hollywood Stor FF Hollywood Stor Lot Hollywood Stor Lot CA-Federal Building CA-Federal Building 1023 Ferndale City Hall 1023 Ferndale City Hall 117 El Centro Array #9 117 El Centro Array #9 1028 Hollister City Hall 1028 Hollister City Hall 117 El Centro Array #9 117 El Centro Array #9 279 Pacoima Dam 279 Pacoima Dam 15250 Ventura Blvd 15250 Ventura Blvd

Event San Francisco San Francisco San Fernando San Fernando San Fernando San Fernando Imperial Valley Imperial Valley Northwest Calif Northwest Calif Kern County  Kern County  Kern County  Kern County 

129

Component Magn.

Rrup (km)

Site Cond.

PGA ;(g)

GGP010 GGP100 ORR021 ORR291 L04111 L04201 I-ELC180 I-ELC270 B-FRN224 B-FRN314 HOL090 HOL180 PEL090 PEL180 N11W N79E H-FRN044 H-FRN314 ELC180 ELC270 B-HCH181 B-HCH271 A-ELC180 A-ELC270 PCD164 PCD254 N11E N79W

8.0 8.0 24.9 24.9 24.2 24.2 8.3 8.3 56 56 120.5 120.5 120.5 120.5 23.5 23.5 31.5 31.5 130 130 12.6 12.6 46 46 2.8 2.8 23.4 23.4

USGS(A) USGS(A) USGS(B) USGS(B) USGS(B) USGS(B) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C) USGS(C)

0.095 0.112 0.324 0.268 0.192 0.153 0.313 0.215 0.105 0.110 0.044 0.057 0.042 0.058 0.153 0.258 0.159 0.189 0.033 0.052 0.074 0.196 0.130 0.057 1.226 1.160 0.225 0.149

5.3 5.3 6.6 6.6 6.6 6.6 7.0 7.0 5.8 5.8 7.4 7.4 7.4 7.4 6.6 6.6 6.5 6.5 6.8 6.8 5.6 5.6 6.8 6.8 6.6 6.6 6.6 6.6

Hollister Borrego Mtn Borrego Mtn  San Fernando San Fernando San Fernando San Fernando  Ground motions not considered in calculating spectrum amplification factors because PGA < 0.1 g.

Table 4.2

Equations for spectrum amplification factors of horizontal elastic design spectra.

αA αV αD Table 4.3

Damping ζ ( %) 0 0.5 1 2 3 5 7 10 20

Median (50 %)

Mean-plus-one-sigma (84.1 %)

3.21 − 0.68 lnζ 2.31 − 0.41 lnζ 1.82 − 0.27 lnζ

4.38 − 1.04 lnζ 3.38 − 0.67 lnζ 2.73 − 0.45 lnζ

Spectrum amplification factors of horizontal elastic design spectra.

αA 4.30 3.68 3.21 2.74 2.46 2.12 1.89 1.64 1.17

Median (50 %) αV 2.85 2.59 2.31 2.03 1.86 1.65 1.51 1.37 1.08

αD 2.15 2.01 1.82 1.63 1.52 1.39 1.29 1.20 1.01

Mean-plus-one-sigma (84.1 %) αA αV αD 6.30 4.40 3.30 5.10 3.84 3.04 4.38 3.38 2.73 3.66 2.92 2.42 3.24 2.64 2.24 2.71 2.30 2.01 2.36 2.08 1.85 1.99 1.84 1.69 1.26 1.37 1.38

130

obtained using all 28 ground motions. It was concluded that “the strong motion data clearly indicate a decrease in amplification, especially for the velocity and acceleration regions, as compared to the case where low intensity excitation is included.” Hence, 8 ground motions listed in Table 4.1 with PGA < 0.1g were not used. From the 20 ground motions with PGA > 0.1g, the average PGA, PGV, and PGD are calculated as 0.287g, 0.256 m/s, and 0.094 m, respectively. Prior to calculating the spectrum amplification factors, these 20 ground motions are normalized to PGA = 0.287g in the short-period (high-frequency) band, to PGV = 0.256 m/s in the intermediateperiod (frequency) band, and to PGD = 0.094 m in the long-period (low-frequency) band, as shown in Figure 4.10. The amplification factors at the median level (50 % NEP) and the mean-plus-one-SD level (84.1 % NEP) given in Newmark and Hall (1982, pages 35 to 36) are presented in Table 4.2 with numerical values for various damping ratios given in Table 4.3. Following the procedure for determining the spectrum amplification factors presented on pages 124 to 126, the spectrum amplification factors obtained based on 20 ground motions are given in Table 4.4. The idealized response spectra constructed using the procedure presented on pages 122 to 124 are shown in Figure 4.11, which are also plotted in Figure 4.10 to illustrate the validity of the idealization.

☞

Because the averaged PGA = 0.287g , PGV = 0.256 m/s, and PGD = 0.094 m can be determined from the 20 ground motions, they are used in constructing the idealized response spectra instead of using equations (4.1.9). However, for a specific site with unknown PGV and PGD, empirical equations (4.1.9) are usually used to determined PGV and PGD for a specified PGA.

☞

The small discrepancy between the results presented in Table 4.4 and those given in Newmark and Hall (1982, pages 35 to 36; see Table 4.2) could be due to the difference in ground motions used: Newmark performed baseline correction and digital filtering to the 28 ground motions because these ground motions obtained from the Department of Commerce or the California Institute of Technology were raw data (USNRC, 1973). Because the original ground motions used by Newmark cannot be obtained, the ground motions used for constructing Table 4.4 are obtained from the Pacific Earthquake Engineering Research Center (PEER) strong motion database and the Center for Engineering Strong Motion Data. Baseline correction and digital filtering on the ground motions obtained from PEER ground motion database have been performed by the supplying agency (PEER, 2010); whereas ground motions obtained from the Center for Engineering Strong Motion Data are raw

4.1 ground response spectra

131

data and are processed prior to being used. The difference in data processing methods could have an effect on the ground motions.

☞

As discussed in WASH-1255 (USNRC, 1973), the numbers of discrete frequencies NA , NV , and ND used in the acceleration sensitive, velocity sensitive, and displacement sensitive regions, respectively, and thevalues of the frequencies also have an effect on the results obtained.

☞

Using equation (4.1.14), spectrum amplification factors of any level of NEP can be determined from αA50% (ζ ) and αA84.1% (ζ ). From Table 4.4, 



αA90% (ζ ) = −1 (0.9) · αA84.1% (ζ ) + 1− −1 (0.9) · αA50% (ζ ) = 1.2816×(4.41−1.01 lnζ ) − 0.2816×(3.21−0.67 lnζ ) = 4.747−1.106 lnζ ,

(4.1.15a) 



αV90% (ζ ) = −1 (0.9) · αV84.1% (ζ ) + 1− −1 (0.9) · αV50% (ζ ) = 1.2816×(3.31−0.62 lnζ ) − 0.2816×(2.28−0.37 lnζ ) = 3.600−0.690 lnζ ,

(4.1.15b) 



90% 84.1% 50% αD (ζ ) = −1 (0.9) · αD (ζ ) + 1− −1 (0.9) · αD (ζ )

= 1.2816×(2.64−0.42 lnζ ) − 0.2816×(1.82−0.24 lnζ ) = 2.871−0.471 lnζ , (4.1.15c) which compare extremely well with the results obtained from statistical analysis

as listed in Table 4.4. This comparison verifies that the spectrum amplification factors follow the normal distribution. Table 4.4 Spectrum amplification factors obtained from 20 ground motions.

αA αV αD

Median (50 %)

Mean-plus-one-sigma (84.1 %)

90 %

3.21 − 0.67 lnζ 2.28 − 0.37 lnζ 1.82 − 0.24 lnζ

4.41 − 1.01 lnζ 3.31 − 0.62 lnζ 2.64 − 0.42 lnζ

4.74 − 1.11 lnζ 3.59 − 0.70 lnζ 2.87 − 0.48 lnζ

Based on the results in WASH-1255 (USNRC, 1973) and NUREG-0003 (USNRC, 1976) on spectrum amplification factors for ground motions in horizontal and vertical directions, NUREG/CR-0098 (USNRC, 1978, page 11) recommends that “the design motions in the vertical direction be taken as two-thirds of the value in the horizontal direction across the entire frequency range.”

Plot of Ground Response Spectra Ground response spectra are piecewise straight lines in the log-log scale as shown in Figure 4.12, especially for those extracted from tripartite. However, in nuclear industry

132

1

NUREG-0098 84.1%

R.G. 1.60 84.1% CSA

CENA UHS

Acceleration (g)

NUREG-0098 50%

0.1

0.01

0.2

1

Frequency (Hz)

10

GRS (5 % damping) plotted in the log-log scale.

Figure 4.12

1.0 Horizontal

NUREG R.G. 1.60 84.1% Vertical CSA

NUREG-0098 84%

0.8

100

CENA UHS

Acceleration (g)

Soil site NUREG-0098 50%

Rock site

0.6

Rock site Soil site 0.4

0.2

0.0

0.2

1 Figure 4.13

Frequency (Hz)

10

100

GRS (5 % damping) plotted in the log-linear scale.

practice, they are often plotted in the log-linear scale for better visualization in the range of engineering interest. In the log-linear scale, the spectra become piecewise curves as shown by dashed lines in Figure 4.13. However, for ease of application, the critical (corner) points are commonly connected by straight lines as shown by the solid lines in Figure 4.13 for analysis and design. It can be seen that the GRS become smoother and more conservative after the adjustment.

4.1 ground response spectra

133

1.0 0.9 0.8

84.1% NEP

Spectral acceleration (g)

0.7

50% NEP

0.6 0.5 0.4 0.3 0.2

NUREG/CR-0098

Rock site Soil site

0.1

USNRC RG 1.60

Horizontal Vertical

0

0.1

1

Figure 4.14 Table 4.5

Frequency (Hz)

10

Design response spectra with 5 % damping.

NUREG/CR-0098 DRS with 5 % damping anchored at 0.3 g PGA.

Rock Site 50 % NEP F (Hz) 0.1 0.34 2.19 8 33 100

100

SA ( g) 0.0086 0.0983 0.6347 0.6347 0.3 0.3

Soil Site 84.1 % NEP

F (Hz) 0.1 0.33 2.01 8 33 100

SA ( g) 0.0124 0.1321 0.8119 0.8119 0.3 0.3

50 % NEP F (Hz) 0.1 0.25 1.64 8 33 100

SA ( g) 0.0152 0.0983 0.6347 0.6347 0.3 0.3

84.1 % NEP F (Hz) 0.1 0.24 1.51 8 33 100

SA ( g) 0.0220 0.1321 0.8119 0.8119 0.3 0.3

USNRC NUREG/CR-0098 Spectra Newmark design response spectrum (DRS) and the spectrum amplification factors determined by Newmark and coauthors in 1973 (Newmark et al., 1973; USNRC, 1973) have been adopted by many nuclear and nonnuclear standards, such as NUREG/CR0098 (USNRC, 1978). The spectrum amplification factors are the same for both rock and soil sites, as given in Tables 4.2 and 4.3. For a given PGA value, the corresponding PGV and PGD can be determined using equation (4.1.9) for rock or soil site. Following the procedure presented on pages 122 to 124, the spectral shapes can be determined. For 5 % damping and PGA = 0.3g, the results are listed in Table 4.5 and shown in Figure 4.14.

134 Table 4.6

USNRC R.G. 1.60 horizontal DRS with 5 % damping anchored at 0.3 g PGA.

USNRC R.G. 1.60

Newmark et al. (1973)

84.1 % NEP F (Hz) 0.1 0.25 2.5 9 33 100 Table 4.7

84.1 % NEP

SA ( g) 0.023 2 2.05 0.141 1 3.13 0.939 2.61 0.783 1.0 0.3 0.3 8 αA

αD

F (Hz) 0.1 0.25 2.9 9 32 →33 7 100

αA

3.1 2.6 1.0

50 % NEP

SA ( g) 0.023 5 0.141 3 0.93 0.78 0.3 0.3 8

F (Hz) 0.1 0.25 2.9 8.33 →9 6 28 →33 7 100

αA

2.3 2.0 1.0

SA ( g) 0.017 5 0.105 4 0.69 0.6 0.3 0.3 8

USNRC R.G. 1.60 vertical DRS with 5 % damping anchored at 0.3 g PGA.

USNRC R.G. 1.60 Horizontal 50 % NEP 4

Vertical 84.1 % NEP F (Hz) αA αD 0.1 0.25 1.37 1 3.5 2.98 9 2.61 33 1.0 100

SA ( g) 0.015 3 0.095 2 0.894 0.783 0.3 0.3

F (Hz) 0.1 0.25 2.9 9 33 100

SA ( g) 0.017 0.105 0.69 0.6 0.3 0.3

αA

αD 1.52 5

2.3 2.0 1.0 1.0

Vertical 50 % NEP F (Hz) 0.1 0.25 3.5 6 9 33 100

αA

αD 1.01 9

2.25 7 2.0 8 1.0 8 1.0 8

SA ( g) 0.011 11 0.070 10 0.675 0.6 0.3 0.3

USNRC R.G. 1.60 Spectra In 1973, USNRC (formerly U.S. Atomic Energy Commission) issued Regulatory Guide 1.60 (USNRC, 2014) entitled Design Response Spectra for Seismic Design of Nuclear Power Plants. Horizontal Design Response Spectra For 5 % damping and PGA = 0.3g, the results are listed in Table 4.6 and shown in Figure 4.14. Because USNRC R.G. 1.60 gives only the spectral shape with 84.1 % NEP, the accompanying paper by Newmark et al. (1973) is used to determine the spectral shape with 50 % NEP. However, because the information provided in USNRC R.G. 1.60 and Newmark et al. (1973) is not complete, some approximations are made to obtain the complete spectral shapes as explained in the following:

☞ The entries in boldface are given by the respective documents. ☞ 1USNRC R.G. 1.60 specifies that a PGA of 1g corresponds to a PGD of 36 in or

0.914 m. For 0.3g PGA, SD (0.25 Hz) = αD · PGD = 2.05×(0.3×0.914) = 0.562 m.

Using SA(ω) = ω02 SD (ω),

SA(0.25 Hz) = (2π×0.25)2 ×0.562/9.81 = 0.141g.

4.1 ground response spectra

135

☞ 2This entry is determined by assuming SD (0.1 Hz) = SD (0.25 Hz) = 0.562 m. This 0.1 and 0.25 Hz. Hence,

S

D is constant between 2 A(0.1 Hz) = (2π×0.1) ×0.562/9.81 = 0.023g.

assumption follows Figure 1 in USNRC R.G. 1.60, in which

S

☞ 3Assume that S V (0.25 Hz) = S V (2.9 Hz). Because both 0.25 Hz and 2.9 Hz are outside the velocity sensitive region, this is obviously an approximation. Based on this assumption, SA (0.25 Hz) with 84.1 % NEP is determined to be 0.080g. However,

the corresponding value in USNRC R.G. 1.60 is SA(0.25 Hz) = 0.141 with a ratio of 0.141/0.080 = 1.763.

Because USNRC R.G. 1.60 is the final document, SA (0.25 Hz) obtained by assuming

S V (0.25 Hz) = S V (2.9 Hz) is multiplied by 1.763 to give SA(0.25 Hz) = 0.141. ☞ 4Following Note 3, for 50 % NEP, SA (0.25 Hz) found by assuming S V (0.25 Hz) = S V (2.9 Hz) is 0.059, which is multiplied by 1.763 to give SA(0.25 Hz) = 0.105. This result seems quite reasonable: SA(0.25 Hz) of USNRC R.G. 1.60 are slightly higher than the corresponding values of soil site given by NUREG/CR-0098 for both 84.1 % and 50 % NEP.

☞ 5Following Note 2, SA (0.1 Hz) is obtained by assuming SD (0.1 Hz) = SD (0.25 Hz) = 0.105×9.81/(2π×0.25)2 = 0.417 m, which gives

SA(0.1 Hz) = 0.017g.

☞ 6The control points given in Newmark et al. (1973) are in terms of period. For this entry, the corresponding frequency is 8.33 Hz. To be consistent with USNRC R.G. 1.60 and also slightly more conservative, the control frequency is set to 9 Hz.

☞ 7In USNRC R.G. 1.60, SA is taken as PGA when frequency F  33 Hz, which is also the common practice in the nuclear industry. To be consistent, the control frequency is changed to 33 Hz, which is also slightly more conservative.

☞ 8Because SA( F ) = PGA when F  33 Hz, the spectra are extended to 100 Hz. Vertical Design Response Spectra For 5 % damping and PGA = 0.3g, the vertical DRS are listed in Table 4.7 and shown in Figure 4.14. USNRC R.G. 1.60 gives only the vertical spectrum amplification factors with 84.1 % NEP at the same control frequencies. The complete spectral shapes with 84.1 % and 50 % NEP are constructed following the principles set forth in USNRC R.G. 1.60, as explained in the following:

☞ The entries in boldface are given by USNRC R.G. 1.60. ☞ USNRC R.G. 1.60 specifies that “Acceleration amplification factors for the vertical design response spectra are equal to those for horizontal design response spectra at a given frequency, whereas displacement amplification factors are 2/3 those for horizontal design response spectra.”

136

☞ 1The control points in the horizontal DRS are αA (2.5 Hz) = 3.13, αA (9 Hz) = 2.61. To find αA (3.5 Hz), linear interpolation in the semilogarithmic scale is used: log10 3.5− log10 9 αA (3.5 Hz)−2.61 = 3.13−2.61 log10 2.5− log10 9

=⇒

αA (3.5 Hz) = 2.99 ≈ 2.98.

☞ 2As in Note 1 for horizontal DRS, SD (0.25 Hz) = αD · PGD = 1.37×(0.3×0.914) = 0.376 m, and

SA(0.25 Hz) = (2π×0.25)2 ×0.376/9.81 = 0.095g.

☞ 3As in Note 2 for horizontal DRS, this entry is determined by assuming SD (0.1 Hz) = SD (0.25 Hz) = 0.376 m, which gives

SA(0.1 Hz) = 0.015g.

☞ 4Horizontal DRS with 50 % NEP is obtained from Table 4.6, which is listed here to obtain the acceleration and displacement amplification factors.

☞ 5For SA(0.25 Hz) = 0.105g, SD (0.25 Hz) = 0.105×9.81/(2π×0.25)2 = 0.417 m, and αD = SD (0.25 Hz)/PGD = 0.417/(0.3×0.914) = 1.52.

☞ 6Note that, for horizontal DRS, the control frequency with 50 % NEP is higher than the control frequency with 84.1 % NEP (2.9 Hz versus 2.5 Hz). There is no information available for this control frequency for vertical DRS with 50 % NEP; it is taken as 3.5 Hz following the vertical DRS with 84.1 % NEP, which is slightly more conservative than a control frequency higher than 3.5 Hz.

☞ 7As in Note 1, the control points in the horizontal DRS with 50 % NEP are αA (2.9 Hz) = 2.3, αA (9 Hz) = 2.0. Linear interpolation in the semilogarithmic scale is used to find αA (3.5 Hz): log10 3.5− log10 9 αA (3.5 Hz)−2.0 = 2.3−2.0 log10 2.9− log10 9

=⇒

αA (3.5 Hz) = 2.25.

☞ 8αA for vertical DRS and horizontal DRS are the same at a given frequency. 2 ☞ 9αD for vertical DRS is equal to 3 of αD for horizontal DRS at a given frequency; αD = 23 ×1.52 = 1.01.

☞ 10 As in Note 1 for horizontal DRS, SD (0.25 Hz) = αD · PGD = 1.01×(0.3×0.914) = 0.277 m, which gives

SA(0.25 Hz) = 0.070g.

☞ 11 As in Note 2 for horizontal DRS, this entry is determined by assuming SD (0.1 Hz) = SD (0.25 Hz) = 0.277 m, which gives

SA(0.1 Hz) = 0.011g.

CSA N289.3 Spectra CSA N289.3 spectra were issued in 1981 (CSA, 1981; see also CSA, 2010a) for seismic design and qualification of nuclear power plants (NPPs), as shown in Figure 4.15. The amplification factors for the standard GRS were developed using California earthquakes

4.2 t-response spectrum

137

that were predominantly of magnitude 6 to 7, recorded on soil and soft rock sites at distances from 10 to 50 km, available up to and including the San Fernando earthquake of February 9, 1971, based on the study of Mohraz, Hall, and Newmark published as WASH-1255 (USNRC, 1973). Spectral shapes with 90 % NEP are adopted, as opposed to spectral shapes with 84.1 % NEP in USNRC R.G. 1.60 (USNRC, 2014) and NUREG/CR0098 (USNRC, 1978). GRS with 5 % damping is also plotted in Figures 4.12 and 4.13 for comparison.

4.2 t-Response Spectrum 4.2.1 Definition In seismic design, qualification, and evaluation of critical structures, systems, and components (SSCs) performing operational and safety-related functions in NPPs, it is crucial to determine floor response spectra (FRS) at various elevations or floors of the supporting structures where these SSCs are mounted. When the direct spectrato-spectra method based on Duhamel integral, developed in Chapter 8, is applied to generate FRS, the following quantity is required when the equipment and supporting structure are in resonance (tuning)   SAt (ω, ζ ) = 12 −ω2 t e−ζ ωt cosωt ∗ u¨g (t) + ω e−ζ ωt sinωt ∗ u¨g (t)

max

.

(4.2.1)

Due to the presence of a time variable t in the first convolution term, it is difficult to obtain an analytic expression for equation (4.2.1) in terms of GRS. Analogous to GRS defined in equations (4.1.2) to (4.1.5), equation (4.2.1) is defined as t-response spectrum (tRS), in which “t” indicates “tuning” or the extra “t ” term in equation (4.2.1) as compared to GRS. Comparing the concepts of GRS and tRS (illustrated in Figure 4.1), one has, under an earthquake excitation u¨g (t), ❧ GRS

SA( F, ζ ) is the maximum acceleration response of an SDOF oscillator (with

frequency F and damping ratio ζ ) mounted directly on ground. ❧ tRS

SAt ( F, ζ ) is the maximum acceleration response of

an SDOF oscillator (with

frequency F and damping ratio ζ ) mounted on top of an SDOF structure (with the same F and ζ ) that is mounted on ground. The identical SDOF oscillator and SDOF structure are uncoupled and are in resonance or tuning. To apply the direct spectra-to-spectra method to generate FRS, tRS corresponding to the given GRS are required. To establish statistical relationships between tRS and GRS, a large number of ground motions recorded with different site conditions are

138

60 50

D isp 00

40

en

la ce m 00

30

20

10

0

8

60

6

20

4

10

8

35

0.3

6

0.

06

0.4

0. 04

2

4

0. 10 0. 08

2

1.0

0 1.

4 8

0. 0.

0.

6 0.

6

0.

8

4 0.

0

1.

0.

2

0.3

6

06

4

8

0.2

10

5

6

20

7

60

0.1 .08

40

10

80

0

.06 .05

10

20

0

io n

20

at

ce le r

1 0 % ga Pe ccel a k erat gr i ou on n d ac

8

.04

40

33

30

0

.03 0.02

60

0

40

80

0

10

00

50 60

A, V, D constant Spectral Acceleration (mm/s2), Spectral Velocity (mm/s), Spectral Displacement (mm)

0.

08

4

0.

3

1 0.

A 2 πV

2

0.6 0.5 0.4

β = 2%

β = 0%

0.8

Period (s)

0.

0 .8 1 .0

FU =

Frequency (Hz)

0.5 0.6

2

Peak ground velocity

71 mm/s velocity

β = 5%

en

0.

02

m )

m ce la sp

0 .2

V 2 πD

2

β = 10%

t( m

00 0

di

0 .1

FL=

CSA N289.3 standard-shape ground response spectra.

00

0.01

80 100

20

g)

(%

n

tio

500 40 20 10 0 0

10 80 40

t en

0

60

d

n ou gr

80 40 20

t

em Figure 4.15

ra

le ce Ac

400 300

200

100 90 80 70 60

ak

lac di sp

m

50

Pe

0.03 0.04 .05 .06

0.08

m .5

40 30

20

10 9 8 7 6 5 4 3

2

1 0.01

0.02

30

100 80 1000 900 800 700 600

Velocity (mm/s)

4.2 t-response spectrum

139

selected (Li et al., 2015): 49 horizontal and 49 vertical ground motions recorded at B sites, 154 horizontal and 154 vertical ground motions recorded at C sites, and 220 horizontal and 220 vertical ground motions recorded at D sites are selected from the PEER Strong Motion Database and the European Strong-Motion Database (Ambraseys et al., 2002). The site categories B, C, and D follow the National Earthquake Hazard Reduction Program (NEHRP) site classification criteria (ASCE/SEI, 2010; IBC, 2012).

4.2.2 Statistical Relationships between tRS and GRS To construct the statistical relationship between tRS and GRS, tRS and GRS of the selected ground motions are calculated first. Suppose the regression model is of the form ln SAt ( F, ζ ) = c1 (ζ , F ) + c2 (ζ , F ) · ln SA( F, ζ ) + ε · σ ln S t , A

(4.2.2)

where c1 (ζ , F ) and c2 (ζ , F ) are coefficients of regression, ε is the number of standard

deviations of ln SAt ( F, ζ ) deviating from the mean value of ln SAt ( F, ζ ), and σ ln S t is A the standard deviation. In practice, for a given GRS SA(ζ , F ), with or without a prescribed NEP, tRS at each

SAt, p(ζ , F ) with any NEP

frequency is modelled using lognormal distribution. tRS

p

corresponding to the given GRS can be estimated as: t, p

ln SA (ζ , F ) = c1(ζ , F ) + c2(ζ , F ) · ln SA(ζ , F ) + σ ln S t (ζ , F ) · −1( p). A

(4.2.3)

The results of c1 (ζ , F ), c2 (ζ , F ), and σ ln S t (ζ , F ) are summarized in Tables 4.8 to 4.10. A

For frequencies not listed, the coefficients and standard deviations can be obtained using linear interpolation in the logarithmic scale of frequency.

Procedure to Establish Statistical Relationships between tRS and GRS The procedure to establish the statistical relationship is as follows: 1. All selected ground motions in a suite are scaled to a constant PGA = 0.3g. 2. For a fixed damping ratio ζ, calculate GRS SA( F, ζ ) and tRS SAt ( F, ζ ) for frequencies F uniformly spaced over the logarithmic scale of a required frequency range, e.g., from 0.1 to 100 Hz. 3. Calculate amplification ratios

AR( F, ζ ) = -

SAt ( F, ζ ) , SA( F, ζ )

(4.2.4)

R for all ground motions in the suite, and determine AR - 50 % and A - 84.1 % with 50 % and 84.1 % NEP, respectively.

140

AR = tRS GRS

10

1

Horizontal ground motions 5% damping ratio 84.1% ratio obtained from statistical calculation Median ratio obtained from statistical calculation

0.1

0.1

Figure 4.16

0.5

1 Frequency (Hz)

5

8

10

16

25 33

50

100

Ratio of tRS to GRS for the 49 horizontal ground motions at B sites.

AR= tRS GRS

10

1

Vertical ground motions 5% damping ratio 84.1% ratio obtained from statistical calculation Median ratio obtained from statistical calculation

0.1

0.1

Figure 4.17

0.5

1 Frequency (Hz)

5

8

10

15

25 33

50

100

Ratio of tRS to GRS for the 49 vertical ground motions at B sites.

4. Analyze the trend of the median amplification ratios AR - 50 % . Two examples are

shown in Figures 4.16 and 4.17 to illustrate the trend of the amplification ratio AR -

for 5 % damping ratio. It is seen that relationships between tRS and GRS are different for horizontal and vertical motions. ❧ 0.5 Hz < F < 5 Hz for horizontal and 0.5 Hz < F < 8 Hz for vertical motions:

AR is almost constant; all random tRS and GRS are respectively grouped to-

gether, and one regression analysis is performed. ❧ 0.1 Hz < F < 0.5 Hz: AR is set as the value at 0.5 Hz. ❧ 5 Hz < F < 50 Hz for horizontal and 8 Hz < F < 50 Hz for vertical motions: perform frequency-by-frequency regression analysis. ❧ F > 50 Hz: AR - = 1 and tRS is considered to be equal to GRS without variation. ❧ 25 Hz < F < 50 Hz: because the coefficient of variation of AR reduces from large values to zero, it is difficult to quantify these variations by statistical relationships. For real ground motions, tRS or GRS usually either reduce rapidly

4.2 t-response spectrum

141

from large values to PGA or remain close to PGA over 25 to 50 Hz. Considering the special variations of AR - , tRS is obtained by linear interpolation in the logarithmic-linear scale between 25 and 50 Hz to avoid possible nonconservatism.

4.2.3

Relationship between Horizontal tRS and GRS

For each suite of ground motions, there are a total of 68 regression equations describing the horizontal statistical relationship between tRS and GRS over the frequency range from 0.1 to 100 Hz. For easy of engineering applications, a simplified yet robust horizontal statistical relationship is developed by considering the following factors: ❧ Frequencies lower than 0.5 Hz are not very important for structures and equipment in nuclear power plants, and there are extremely large variations in the amplification ratio AR as shown in Figure 4.16. Hence, for frequencies lower than 0.5 Hz, the horizontal relationship is conservatively taken as that at frequency 0.5 Hz. ❧ In the frequency range from 5 to 50 Hz, horizontal statistical relationship at critical frequencies 8, 10, 16, 25, 33, and 50 Hz are used to characterize the horizontal statistical relationship over this frequency range. ❧ For frequencies greater than 50 Hz, because the amplification ratio AF - approaches 1, tRS are considered to be equal to GRS. It is shown in Li et al. (2015) that the simplified horizontal statistical relationships are suitable to replace the complete relationships. It is also concluded that the influence of site conditions on the horizontal statistical relationships between tRS and GRS is small for damping ratios less than 20 %.

General Horizontal Statistical Relationship Because site conditions have negligible effect on the horizontal statistical relationship between tRS and GRS, the 28 horizontal ground motions for NUREG/CR-0098, the 49, 154, and 220 horizontal ground motions recorded at B, C, and D sites are combined into one suite of horizontal ground motions to obtain a more reliable statistical result. Simplified relationships between tRS and GRS for 20 damping ratios (1 %, 2 %, . . . , 20 %) are established; based on the regression coefficients for the 20 damping ratios, equations for regression coefficients against damping ratio are obtained by curve fitting using the least-square method and are listed in Table 4.8. Coefficients and standard deviations for other frequencies not listed in the table can be obtained using linear interpolation in the logarithmic-linear scale.

142

16.0

10.0

8.0

0.1∼5.0

Frequency (Hz)

0.21( ln ζ )3 − 0.22( ln ζ )2 − 3.16 ln ζ + 7.23

0.39( ln ζ )3 − 1.74( ln ζ )2 + 0.16 ln ζ + 6.33

−0.08( ln ζ )2 − 0.45 ln ζ + 3.32

0.06( ln ζ )2 − 0.80 ln ζ + 2.99

0.10( ln ζ )2 − 0.93 ln ζ + 3.01

0.06( ln ζ )2 − 0.92 ln ζ + 3.03

Coefficient c1

0.20( ln ζ )3 − 0.38( ln ζ )2 − 2.15 ln ζ + 6.58

0.35( ln ζ )3 − 1.66( ln ζ )2 + 0.77 ln ζ + 5.58

−0.22( ln ζ )2 + 0.58 ln ζ + 2.24

−0.06( ln ζ )3 + 0.21( ln ζ )2 + 1.45

−0.01( ln ζ )3 + 0.07( ln ζ )2 + 0.03 ln ζ + 1.35

0.02( ln ζ )3 − 0.04( ln ζ )2 − 0.02 ln ζ + 1.12

Coefficient c2

0.04( ln ζ )2 − 0.31 ln ζ + 0.49

0.02( ln ζ )3 − 0.04( ln ζ )2 − 0.21 ln ζ + 0.60

0.02( ln ζ )3 − 0.12( ln ζ )2 + 0.07 ln ζ + 0.43

−0.01( ln ζ )3 + 0.01( ln ζ )2 + 0.28

−0.01( ln ζ )3 + 0.02( ln ζ )2 − 0.02 ln ζ + 0.27

−0.01( ln ζ )2 − 0.05 ln ζ + 0.30

Standard deviation σ ln S t

Table 4.8 Equations for coefficients and standard deviations of horizontal statistical relationship.

25.0

1

Coefficient c1

−0.04( ln ζ )3 + 0.19( ln ζ )2 − 0.13 ln ζ + 1.24

0.01( ln ζ )4 − 0.06( ln ζ )3 + 0.12( ln ζ )2 − 0.12 ln ζ + 1.15

Coefficient c2

−0.01( ln ζ )3 + 0.05( ln ζ )2 − 0.05 ln ζ + 0.2

0.01( ln ζ )2 − 0.06 ln ζ + 0.28

Standard deviation σ ln S t

A

33.0 0

0

50.0∼100.0

0.04( ln ζ )2 − 0.89 ln ζ + 3.09

Table 4.9 Equations for coefficients and standard deviations of vertical statistical relationships for hard sites. Frequency (Hz) 0.07( ln ζ )2 − 0.90 ln ζ + 3.08

25.0

15.0

0.17( ln ζ )3 − 0.98( ln ζ )2 + 0.51 ln ζ + 3.83

−0.03( ln ζ )2 − 0.52 ln ζ + 3.25

0.10( ln ζ )2 − 0.90 ln ζ + 3.06

0.17( ln ζ )3 − 1.10( ln ζ )2 + 1.28 ln ζ + 3.26

−0.03( ln ζ )3 − 0.02( ln ζ )2 + 0.29 ln ζ + 2.28

−0.03( ln ζ )3 + 0.13( ln ζ )2 + 0.08 ln ζ + 1.35

0.01( ln ζ )2 − 0.26 ln ζ + 0.65

0.01( ln ζ )3 − 0.05( ln ζ )2 − 0.12 ln ζ + 0.60

−0.01( ln ζ )3 + 0.01( ln ζ )2 + 0.25

A

10.0

0.5∼8.0

33.0

0.01( ln ζ )3 − 0.03( ln ζ )2 − 0.04 ln ζ + 1.17

Coefficient c2

−0.01( ln ζ )3 + 0.06( ln ζ )2 − 0.09 ln ζ + 0.22

−0.04 ln ζ + 0.32

Standard deviation σ ln S t

0

0

1

50.0∼100

Coefficient c1

−0.02( ln ζ )4 + 0.09( ln ζ )3 − 0.14( ln ζ )2 + 0.14 ln ζ + 1.24

Equations for coefficients and standard deviations of vertical statistical relationships for soft sites.

0.04( ln ζ )2 − 0.90 ln ζ + 3.13

Table 4.10 Frequency (Hz)

0.05( ln ζ )2 − 0.90 ln ζ + 3.08

33.0

25.0

15.0

0

−0.19( ln ζ )2 + 3.21

−0.08( ln ζ )2 − 0.27 ln ζ + 3.15

0.09( ln ζ )2 − 0.95 ln ζ + 3.05

1

0.07( ln ζ )4 − 0.34( ln ζ )3 + 0.17( ln ζ )2 + 0.65 ln ζ + 2.62

0.04( ln ζ )4 − 0.25( ln ζ )3 + 0.37( ln ζ )2 + 0.06 ln ζ + 2.63

0.01( ln ζ )4 − 0.07( ln ζ )3 + 0.15( ln ζ )2 − 0.03 ln ζ + 1.40

0

0.02( ln ζ )3 − 0.07( ln ζ )2 − 0.13 ln ζ + 0.58

0.01( ln ζ )3 − 0.02( ln ζ )2 − 0.24 ln ζ + 0.74

−0.01( ln ζ )3 − 0.01( ln ζ )2 + 0.04 ln ζ + 0.24

A

10.0

0.5∼8.0

50.0∼100.0

4.2 t-response spectrum

143

In developing a regression model, it is necessary to restrict the coverage of the regression model to some interval or region of values of the predictor variables (Neter et al., 1996). The horizontal statistical relationship developed in this study is valid for GRS falling between the minimal and maximal values of predictor variable SA used for regression analysis. An example of valid coverage of GRS for the horizontal statistical relationship for 5 % damping ratio is shown in Figure 4.18. It is clearly seen that the horizontal design spectra in NUREG/CR-0098 and USNRC R.G. 1.60 fall within the valid coverage of horizontal statistical relationship. The spectral shapes of NUREG/CR-0098 and USNRC R.G. 1.60 are similar to those of uniform hazard spectra (UHS) in Western North America (WNA), but are quite different from those of UHS in Central and Eastern North America (CENA; Green et al., 2007; Silva et al., 1999; USNRC, 2001). Therefore, the horizontal statistical relationships are suitable to estimate tRS corresponding to horizontal design spectra in NUREG/CR-0098 and USNRC R.G. 1.60, horizontal UHS in WNA, and any horizontal GRS falling inside the valid coverage of horizontal statistical relationship.

Amplification Ratio for UHS with Large High-Frequency Components For horizontal UHS with significant high-frequency spectral accelerations, such as the standard UHS in CENA shown in Figure 4.18 (Atkinson and Elgohary, 2007), they may fall outside the valid coverage of horizontal statistical relationship. An approach using amplification ratio AR - = tRS/GRS is applied to estimate tRS: ❧ For F  50 Hz, a constant amplification ratio is determined by

AR p( F h0 , ζ ) = -

SAt, p( F h0 , ζ ) , SAmean ( F h0 , ζ )

(4.2.5)

p ( F h0 , ζ ) is the amplification ratio with NEP p, SA ( F h0 , ζ ) is where F h0 = 5 Hz, AR t, p

the tRS with NEP p calculated using equation (4.2.3) with and

SA( F, ζ ) = SAmean ( F h0 , ζ ),

SAmean ( F h0 , ζ ) is the mean value given by the regression relation SAmean ( F h0 , ζ ) = 0.02



2

ln(100ζ ) − 0.28 ln(100ζ ) + 1.14.

(4.2.6)

❧ At very high frequencies, the amplification ratio AR should be equal to 1; hence, -

AR = 1 is assumed at 100 Hz. -

❧ For frequencies between 50 and 100 Hz, tRS is obtained by linear interpolation in the logarithmic-logarithmic scale using known tRS at 50 and 100 Hz.

144 2

Max of predictor variable

Spectral acceleration (g)

NUREG/CR-0098, soil USNRC R.G. 1.60 5% damping 1

84.1%

CENA UHS

50% Mean Min of predictor variable 0 0.1

100 10 Frequency (Hz) Figure 4.18 Valid region of GRS for the horizontal statistical relationship.

Spectral acceleration (g)

2

1

NUREG/CR-0098 Max of predictor variable Hard

USNRC R.G. 1.60 Mean of predictor variable 5% damping 1

84.1%

Soft CENA UHS

50%

Min of predictor variable 0 0.1

1

100 10 Frequency (Hz) Figure 4.19 Valid region of GRS for the vertical statistical relationships.

Hence, for horizontal UHS with significant high-frequency spectral accelerations, the corresponding tRS is determined as

SAt, p( F, ζ ) = SAUHS ( F, ζ ) × AR p ( F h0 , ζ ), -

t, p log10 A ( F, ζ )

S

F  50 Hz,

(4.2.7a)

t, p

=

log10 PGA− log10 SA (50, ζ ) log10 2

· log10 F

t, p

+

log10 (PGA)( log10 2−2)+2 log10 SA (50, ζ ) log10 2

,

50 Hz < F  100 Hz, (4.2.7b)

where SAUHS ( F, ζ ) represents spectral acceleration of horizontal UHS anchored to PGA. tRS corresponding to horizontal UHS estimated using equation (4.2.7) should be conservative over the entire frequency range, especially for high frequencies from 50

4.2 t-response spectrum

145

to 100 Hz. However, when using the direct method to generate FRS, the effect of the conservatism in tRS at high frequencies is not significant for structures in NPP.

4.2.4 Relationship between Vertical tRS and GRS For a suite of vertical ground motions, a total of 62 regression equations can be obtained from regression analysis over the frequency range from 0.1 to 100 Hz, defining the complete vertical statistical relationship. As for horizontal motions, simplified vertical statistical relationships are established by considering the following factors: ❧ Because frequencies lower than 0.5 Hz are not important for structures and equipment in nuclear power plants, the vertical relationship for frequencies lower than 0.5 Hz is conservatively taken the same as that at 0.5 Hz. ❧ In the frequency range between 8 and 50 Hz, the vertical statistical relationship at critical frequencies 10, 15, 25, 33, and 50 Hz are used to characterize the relationship over this frequency range. ❧ For frequencies greater than 50 Hz, as the amplification ratio AR is close to 1, tRS is considered to be equal to GRS. It is shown in Li et al. (2015) that the simplified vertical statistical relationships are suitable to replace the complete relationships. There is 10 % difference between the estimated tRS for soft sites (D sites) and for hard sites (mainly B and C sites).

General Vertical Statistical Relationship To obtain more reliable vertical statistical relationships between tRS and GRS, the 14 vertical ground motions used in WASH-1255 (USNRC, 1973) for NUREG/CR-0098, the 49 and 154 vertical ground motions recorded at B and C sites are combined into one suite of ground motions for hard sites, and the 220 vertical ground motions recorded at D sites are used for soft sites. The procedure is similar to that for horizontal components, and the results are listed in Tables 4.9 and 4.10, respectively, for hard and soft sites. Coefficients and standard deviations for frequencies not listed in the tables can be obtained using linear interpolation in the logarithmic scale for frequency. The vertical statistical relationship should be valid for GRS falling within the minimal and maximal values of vertical predictor variable SA. An example of valid coverage of GRS for the vertical statistical relationship with 5 % damping ratio is presented in Figure 4.19. Similar to the horizontal case, the vertical statistical relationships are suitable for determining tRS corresponding to vertical design spectra in NUREG/CR-0098 and in

146

USNRC R.G. 1.60, vertical UHS in WNA, and any GRS falling inside the valid coverage of vertical statistical relationship.

Amplification Ratio for UHS with Large High-Frequency Components For vertical UHS with significant high-frequency spectral accelerations (e.g., UHS in CENA) such that they fall out the valid coverage of vertical statistical relationship, the amplification ratio approach is applied to estimate tRS. For frequency F  50 Hz, a constant amplification ratio is determined as

AR -

p

( F v0 , ζ ) =

SAt, p( F v0 , ζ ) , SAmean ( F v0 , ζ )

(4.2.8)

p where F v0 = 8 Hz, AR ( F v0 , ζ ) is the amplification ratio with NEP p, -

the tRS with NEP p calculated by equation (4.2.3) using

SAmean ( F v0 , ζ ) is the mean value of

SAt, p( F v0 , ζ ) is

SA( F, ζ ) = SAmean ( F v0 , ζ ), and

vertical predictor variable at 8 Hz given by the

regression relation ⎧   ⎨0.03 ln(100ζ ) 2 −0.36 ln(100ζ )+1.19, for hard sites; mean SA ( F v0 , ζ ) = ⎩0.04  ln(100ζ )2 −0.38 ln(100ζ )+1.24, for soft sites.

(4.2.9)

Hence, tRS in the vertical direction is estimated as

SAt, p( F, ζ ) = SAUHS ( F, ζ ) × AR p ( F v0 , ζ ), -

where

F  50 Hz,

(4.2.10)

SAUHS ( F, ζ ) is the vertical UHS anchored to PGA. For 50 < F  100 Hz, SAt, p( F, ζ )

is obtained by linear interpolation in the log-log scale, given by equation (4.2.7b). tRS corresponding to vertical UHS estimated by the amplification ratio should be conservative over almost the entire frequency range, especially for high frequencies from 50 to 100 Hz. However, when using the direct method to generate FRS, the effect of the conservatism of tRS at high frequencies is negligible.

4.2.5 Examples of Estimating tRS Example 1 – USNRC R.G. 1.60 Design Spectra GRS are taken from the 5 % horizontal and vertical design spectra in USNRC R.G. 1.60 (USNRC, 2014). For each GRS, a set of 30 spectrum-compatible time-histories are generated following Approach 2 in USNRC-0800 SRP Section 3.7.1 (USNRC, 2012b) using the Hilbert–Huang transform (HHT) method (Section 6.3.2). tRS are simulated using the generated time-histories; tRS with 50 % and 84.1 % NEP are statistically calculated and used as benchmarks, as shown in Figure 4.20. Considering the special variation

4.2 t-response spectrum

147

over the frequency range between 25 and 50 Hz, tRS calculated using the statistical relationship are replaced by straight lines connecting tRS at 25 Hz to tRS at 50 Hz to avoid possible nonconservatism. Horizontal GRS: 1. tRS with 50 % NEP estimated by the horizontal statistical relationship closely matches the benchmark tRS except for the frequency range from 25 to 50 Hz. 2. tRS with 84.1 % NEP estimated by the horizontal statistical relationship closely matches the 84.1 % NEP benchmark tRS over the frequency range from 0.3 to 8 Hz, displays some degree of conservatism from 8 to 20 Hz, and is slightly below the benchmark tRS from 20 to 25 Hz. Overall, the estimated tRS with 84.1 % NEP is acceptable for frequencies lower than 25 Hz comparing to the benchmark tRS. 3. tRS with 50 % and 84.1 % NEP estimated by the horizontal statistical relationship are somewhat below their benchmark tRS for frequencies from 25 to 50 Hz. As seen over this frequency range. in Figure 4.16, this is due to the special variation of AR The adjusted tRS by linearly connecting tRS at 25 Hz to tRS at 50 Hz are more conservative than their corresponding benchmark tRS. Vertical GRS: 1. tRS estimated by the vertical statistical relationship closely match the benchmark tRS over the frequency ranges from 0.3 to 4 Hz and from 8 to 25 Hz for 50 % NEP, from 0.3 to 3 Hz and from 10 to 25 Hz for 84.1 % NEP. The estimated tRS are conservative from 4 to 8 Hz for 50 % NEP and from 3 to 10 Hz for 84.1 % NEP. 2. tRS with 50 % and 84.1 % NEP estimated by the vertical statistical relationship are somewhat below their benchmark tRS from 25 to 50 Hz, due to the special variation over this frequency range as seen in Figure 4.17. The adjusted tRS by linearly of AR connecting tRS at 25 and 50 Hz are more conservative than their corresponding benchmark tRS. It is concluded that the horizontal and vertical statistical relationships between tRS and GRS are acceptable in practice to estimate tRS and generate FRS for GRS falling within the valid coverage of the statistical relationship.

Example 2 – Standard UHS for CENA The 5 % standard UHS proposed by Atkinson and Elgohary (2007) for CENA sites is taken as the horizontal GRS. Thirty tRS are simulated using time-histories compatible with CENA UHS, generated following the requirements of CSA N289.3 (CSA, 2010a)

148

t-Spectral acceleration (g)

8 Horizontal 84.1%

6

50%

4 2

From statistical relationship

0 0.2

1

From 30 tRS of TH analysis 10 Frequency (Hz)

25

50

100

25

50

100

t-Spectral acceleration (g)

8 Vertical 6

84.1% 50%

4 2

From statistical relationship From 30 tRS of TH analysis

0 0.2

1 Figure 4.20

t-Spectral acceleration (g)

6

Frequency (Hz)

10

tRS for USNRC R.G. 1.60 GRS.

Amplification ratio method

5

30 tRS of TH analysis

84.1%

4 50%

3 2 1 0 0.2

1 Figure 4.21

Frequency (Hz)

10

100

Comparison between horizontal tRS and GRS.

10

84.1%

AR = tRS GRS

50%

Amplification factor method 30 UHS-compatible THs

1 0.2

1

Frequency (Hz)

10

Figure 4.22 Amplification ratios of horizontal tRS corresponding to UHS.

100

4.3 appendix

149

by the HHT method (Section 6.3.2). tRS with 50 % and 84.1 % NEP are statistically calculated and used as benchmarks, as shown in Figure 4.21. From Figure 4.18, it is seen that CENA UHS does not fall within the valid coverage of the horizontal statistical relationship in the high-frequency range. The amplification ratio method presented in Section 4.2.3 is applied to estimate tRS. From Figure 4.21, the following observations can be made: 1. The estimated tRS with 50 % and 84.1 % NEP match their corresponding benchmark tRS within a 5 % relative error over the frequency range from 1 to 25 Hz, which is important for SSCs of NPPs, and very well from 50 to 100 Hz. 2. The estimated tRS are larger than the corresponding benchmark tRS from 33 to 50 Hz for 50 % NEP tRS, and from 0.2 to 50 Hz for 84.1 % NEP tRS. However, the effect of this conservatism on FRS is negligible for SSCs in NPPs, as shown in Chapter 8. The amplification ratios obtained from the amplification ratio method and calculated from the ratios of 30 generated tRS to the UHS are presented in Figure 4.22. The small discrepancies between the amplification ratios from the two methods cause the discrepancies between the estimated and the benchmark tRS in Figure 4.21. The amplification ratio method is acceptable to estimate tRS and to generate FRS under UHS with significant high-frequency spectral accelerations.

4.3 Appendix 4.3.1 Numerical Evaluation of Response Spectra Consider an accelerogram A(t), which could be an earthquake ground acceleration or acceleration response of a floor. The acceleration response spectrum SA(ζ0 , ω0 ) of A(t) is the maximum absolute value of the absolute acceleration response u(t) ¨ of an SDOF oscillator (with natural circular frequency ω0 and damping ratio ζ0 ) under the base excitation of A(t). Referring to equation (3.3.17),   2ζ 2 ω2 u(t) ¨ = −ω0 + 0 0 I s (t) − 2ζ0 ω0 ωd

I c (t),

(4.3.1)

where 

I

s

I

c

(t) =

t

0

 (t) =

e−ζ0 ω0 (t−τ ) sinωd (t−τ ) A(τ ) dτ , (4.3.2)

t

e 0

−ζ0 ω0 (t−τ )

cosωd (t−τ ) A(τ ) dτ.

150

Suppose that A(t) has duration T and is discretized into N equal interval: t0 = 0, t1 = , . . . , tI = I, . . . , tN = N = T, where  = T/N. From equation (4.3.2), one has

I s (tI ) =



tI

0



e−ζ0 ω0 (tI −τ ) sinωd (tI −τ ) A(τ ) dτ

tI−1

= 0

e−ζ0 ω0 (tI−1 −τ + ) sinωd (tI−1 −τ +) A(τ ) dτ + I s (tI )   tI−1

= e−ζ0 ω0  cosωd  ·

 + sinωd  ·  = e−ζ0 ω0  cosωd  · where I s (tI ) =



0 tI−1 0

e−ζ0 ω0 (tI−1 −τ ) sinωd (tI−1 −τ ) A(τ ) dτ 

e−ζ0 ω0 (tI−1 −τ ) cosωd (tI−1 −τ ) A(τ ) dτ + I s (tI )

I s (tI−1 ) + sinωd  · I c (tI−1 ) tI tI−1



+ I s (tI ),

e−ζ0 ω0 (tI −τ ) sinωd (tI −τ ) A(τ ) dτ.

(4.3.3)

(4.3.4)

Similarly,

I c (tI ) = e−ζ0 ω0  I c (tI ) =



tI tI−1



cosωd  ·

I c (tI−1 ) − sinωd  · I s (tI−1 )



+ I c (tI ), (4.3.5)

e−ζ0 ω0 (tI −τ ) cosωd (tI −τ ) A(τ ) dτ.

(4.3.6)

Between tI−1 and tI , A(t) can be determined using linear interpolation to yield A(t) = AI−1 + KI (t−tI−1 ), KI =

AI −AI−1 , AI−1 = A(tI−1 ), AI = A(tI ). (4.3.7) 

Equations (4.3.4) and (4.3.6) become I s (tI ) =



tI

tI−1





e−ζ0 ω0 (tI −τ ) sinωd (tI −τ ) A(tI−1 ) + KI (τ −tI−1 ) dτ

   1 = − 3 e−ζ0 ω0  ω0 ζ0 ω0 AI−1 +(1−2ζ02 ) KI sinωd  ω0  !

+ ωd −2ζ0 KI +ω0 AI−1 cosωd  + ωd (2ζ0 KI − ω0 AI ) , I c (tI ) =



tI

tI−1





e−ζ0 ω0 (tI −τ ) cosωd (tI −τ ) A(tI−1 ) + KI (τ −tI−1 ) dτ

   1 = 3 e−ζ0 ω0  −ω0 ζ0 ω0 AI−1 +(1−2ζ02 ) KI cosωd  ω0

(4.3.8a)

4.3 appendix

151

 ! 

+ ωd −2ζ0 KI +ω0 AI−1 sinωd  + ω0 (1−2ζ02 ) KI + ζ0 ω0 AI ) . (4.3.8b) From equation (4.3.1),   2ζ 2 ω2 u(t ¨ I ) = −ω0 + 0 0 ωd

I s (tI ) − 2ζ0 ω0 I c (tI ),

I = 1, 2, . . . , N,

(4.3.9)

and the response spectrum is

SA(ζ0 , ω0 ) = max

☞

    !  u(t ¨ 1 ), u(t ¨ 2 ), . . . , u(t ¨ N ) .

(4.3.10)

Based on the assumption that acceleration changes linearly between two adjacent sampling times (equation (4.3.7)), the formulation presented in this section is exact; there are no other assumptions and approximations.

Procedure RS — Generating Response Spectrum

❧ Input: • Accelerogram A(t) discretized at times with equal time interval  . Time: t0 = 0, t1 = , . . . , tI = I, . . . , tN = N = T . Acceleration: A0 = 0, A1 , . . . , AI , . . . , AN = 0 • Frequencies of the required response spectrum: ωm , m = 1, 2, . . . , M • Damping ratio ζ0 ❧ Output: • Response spectrum: 1. Evaluate: ζ1 = 1−2ζ02 ,

SA(ζ0 , ω1 ), SA(ζ0 , ω2 ), . . . , SA(ζ0 , ωM )

ζ2 = 2ζ0 ;

KI =

AI − AI−1 , I = 1, 2, . . . , N. 

2. LOOP 1 For frequency ωm , m = 1, 2, . . . , M, evaluate ω0 = ωm , c = ζ0 ω0 ,

 ωd = ω0 1−ζ02 , c2 = 2c,

S = sin(ωd ),

3 = ω0−3 ,

E = e−c ,

C = cos(ωd ), K = −ω0 +

2c 2 . ωd

Initialize: ( I s )0 = 0, ( I c )0 = 0. 2.1 LOOP 2 . For time tI , I = 1, 2, . . . , N (I s )I = −3

E







 ω0 c AI−1 +ζ1 KI S + ωd −ζ2 KI +ω0 AI−1 C

! + ωd ζ2 KI −ω0 AI ,

152

E

(I c )I = 3 ( I s )I =

E

( I c )I =

E







 −ω0 c AI−1 +ζ1 KI C + ωd −ζ2 KI +ω0 AI−1 S

! + ω0 ζ1 KI + c AI , 

C · ( I s )I−1 + S · ( I c )I−1 + (I s )I ,



u¨I = K · 2.2 Spectrum:





C · ( I c )I−1 − S · ( I s )I−1 + (I c )I ,

I s (tI ) − c2 · I c (tI ).

SA(ζ0 , ωm ) = max

     ! u¨ , u¨ , . . . , u¨  . 1 2 N

❧

❧

In this chapter, ground response spectra (GRS) and t-response spectra (tRS) are introduced. ❧ For a given damping ratio, the maximum acceleration SA( F, ζ ), maximum velocity

S V ( F, ζ ), maximum displacement SD ( F, ζ ) of an SDOF oscillator (with frequency

F and damping ratio ζ ) mounted directly on ground under the specified ground motion are determined for a series of values of frequency F over the entire frequency range of interest, e.g., 0.1 Hz  F  100 Hz.

SA( F, ζ ), S V ( F, ζ ), SD ( F, ζ ),

and F are

plotted in a four-way logarithmic plot, called tripartite. ❧ “Idealized” Newmark elastic design spectra are constructed based on tripartite plots. There are a number of standard design spectra, such as those in R.G. 1.60 and CSA N289.3. In nuclear engineering practice, design spectra are usually plotted for maximum acceleration SA( F, ζ ) versus frequency F. ❧ tRS SAt ( F, ζ ) is the maximum acceleration response of an SDOF oscillator mounted on top of an SDOF structure that is mounted on the ground under the specified ground motion. The identical SDOF oscillator and SDOF structure (with the same F and ζ ) are uncoupled and are in resonance or tuning. Through statistical analysis, empirical expressions are obtained for

SAt ( F, ζ ),

which are essential in the direct

spectra-to-spectra method for generating FRS in Chapter 8.

C

H

5 A

P

T

E

R

Seismic Hazard Analysis Seismic hazard analyses involve the quantitative estimation of ground-shaking hazards at a particular site. Seismic hazards may be analyzed deterministically using deterministic seismic hazard analysis (DSHA), or probabilistically using probabilistic seismic hazard analysis (PSHA). In this chapter, both DSHA and PSHA are presented.

5.1 Deterministic Seismic Hazard Analysis (DSHA) DSHA involves the determination of a particular scenario earthquake, called controlling earthquake, on which a ground-motion hazard evaluation is based, through a ground-motion prediction equation (GMPE) as discussed in Section 2.4.2. The scenario earthquake consists of the postulated occurrence of an earthquake having a specified size and occurring at a specified location. A typical DSHA can be performed following four steps (Reiter, 1990; Kramer, 1996): 1. Identify and characterize all seismic sources, capable of producing significant ground motions, surrounding the site of interest, including determining the geometry (the seismic source zone) of each seismic source and earthquake potential (earthquake magnitude m) of each source. 2. Select a source–site distance parameter r for each source, usually the shortest distance between the seismic source zone and the site of interest. The distance may be expressed as the closest distance to the rupture plane, the closest distance to the fault rupture, the Joyner-Boore distance, or others, depending on the measure of distance of the GMPE used in Steps 3 and 4.

153

154

3. Select the controlling earthquake in terms of magnitude and source–site distance. For each seismic source I, determine the ground-motion intensity measure, such as spectral acceleration

SA, I (Tj ) at a specified vibration period Tj

at the site of

interest, produced by an earthquake with magnitude mI (identified in Step 1) and occurring at source–site distance rI (identified in Step 2) using a selected GMPE. The earthquake occurring at the seismic source producing the largest groundmotion spectral acceleration SA, I (Tj ) is the controlling earthquake. 4. Determine the seismic hazard at the site of interest in terms of ground-motion intensity measures, such as PGA or design response spectrum, by substituting the controlling earthquake (identified in Step 3) into the GMPE. The procedure of DSHA is shown schematically in Figure 5.1. When applied to a nuclear power plant (NPP) for which failure could have catastrophic consequences, DSHA provides a straightforward framework for evaluating the worst-case design earthquakes. However, it provides no information on the likelihood of occurrence of the controlling earthquake, the likelihood of occurrence at the assumed location, the level of shaking that might be expected during a finite period of time, and the effects of aleatory and epistemic uncertainties in the various steps in the determination of the resulting ground-motion characteristics (Kramer, 1996).

5.2 Probabilistic Seismic Hazard Analysis To analyze seismic hazard at a given site under a probabilistic framework rather than deterministically as in DSHA in Section 5.1, probabilistic seismic hazard analysis (PSHA) can be performed. PSHA is a primary component of seismic risk analysis of NPPs. It mainly produces inputs for seismic analyses, such as uniform hazard spectra (UHS) and seismic hazard curves to be convolved with seismic fragility curves to obtain seismic risk. The primary advantage of PSHA is that it provides a framework, in which aleatory and epistemic uncertainties in the locations, the sizes, and the rates of occurrence of earthquakes, and the variation of ground-motion characteristics with earthquake size and location can be identified, quantified, and combined in a mathematically rigorous manner to describe the seismic hazard at a given site. The framework of PSHA was established by Cornell (1968). The current well-known PSHA method is recognized as the Cornell-McGuire PSHA (McGuire, 2004). Generally, PSHA consists of four steps (Reiter, 1990; Kramer, 1996; McGuire, 2004), as illustrated in Figure 5.2, each of which is similar to the step of DSHA procedure:

5.2 probabilistic seismic hazard analysis

Step 1

Source 1 m1

155

So

ur ce m4 4

r2

So

ur

ce

4

Site

ur

ce

ce m2

So

ur

r4

2

2

Site

So

Source 1 r1

Step 2

m3

Source 3

Step 3

SA,i (Tj) SA,3(Tj) SA,1(Tj) SA,2(Tj)

Source 3

Controlling earthquake

m3 m2 m4 m1

Combination of m3 and r3 produces the largest value of SA,3(Tj) Ground-motion prediction equations

SA,4(Tj) r1

r3

r2

r4

r

Step 4

SA,i (Tj) m 3 SA,3(Tj)

r3 Ground-motion prediction equation Figure 5.1

SA(T)

Controlling earthquake

r

T Seismic design spectrum at the site

Procedure of deterministic seismic hazard analysis.

1. Identify and characterize potential seismic sources surrounding the site of interest. This is identical to the first step of DSHA, except that the probability distribution of potential rupture locations within the source must be characterized. In most cases, uniform probability distribution is assigned to each seismic source zone, implying that earthquakes are equally likely to occur at any point within the source zone. The distribution of the rupture locations is then combined with the source geometry to obtain the corresponding probability distribution of source–site distance. DSHA, on the other hand, implicitly assumes that the probability of occurrence is 1 at

156

Step 1

Source 1

fR(r) r

So

ur

fR(r)

ce

4

r

ce

2

Site

So

ur

fR(r)

fR(r)

r

r

log10(# earthquakes exceeding m)

Source 3

Step 2

fM (m)

2

1

4

3

m0

Magnitude m

mmax Magnitude m

Spectral acceleration SA

Step 3 P {SA > s | m, r }

Step 4 fSA( s)

s M =m

P {SA > s} s Spectral acceleration SA

r Distance r

Figure 5.2

Procedure of probabilistic seismic hazard analysis.

the point in each source zone closest to the site, and zero elsewhere. This step is elaborated in Section 5.2.1. 2. Characterize seismicity of each seismic source (i.e., temporal distribution of earthquake recurrence, from which mean rate of earthquake occurrence and probability distribution of earthquake magnitude are obtained). The temporal distribution of earthquake recurrence is called recurrence relationship, specifying the average rate at which an earthquake of some magnitude will be exceeded. DSHA, on the other hand, implicitly assumes that the maximum magnitude earthquake will occur with probability 1. This step is discussed in Section 5.2.3 3. Determine the ground motions (in terms of ground-motion intensity measures, such as spectral accelerations) produced at the site by earthquakes of any possible

5.2 probabilistic seismic hazard analysis

157

Site

Source Source Source

Site

Site (a) Short fault modelled as a point source

Figure 5.3

(b) Shallow fault modelled as a line source

(c) Three-dimensional source zone

Geometries of seismic source zones modelled in PSHA.

magnitude, occurring at any possible point in the source zone, using GMPEs. The randomness inherent in the GMPEs is also considered. This step has been discussed in Section 2.4. 4. Obtain the probability that ground-motion intensity measures will be exceeded marginally (scalar PSHA) or jointly (vector-valued PSHA) during a time period by combining the aleatory and epistemic uncertainties in the location of earthquakes, the size of earthquakes, the rate of occurrence of earthquakes, and the variation of ground-motion intensity measures with earthquake size and location. Details of this step are presented in Sections 5.2.4 and 5.2.5.

5.2.1 Probability Distribution of Source-Site Distance The geometries of seismic sources depend on the tectonic processes. For example, zones near volcanoes, in which earthquakes associated with volcanic activities originate, are small enough to be characterized as point sources. Well-defined fault planes, on which earthquakes can occur at many different locations, can be considered as two-dimensional seismic area sources. Areas where earthquake mechanisms are poorly defined, where faulting is so extensive as to preclude distinction between individual faults, or where diffuse seismicity that are not amenable to modelling by specific faults exists, can be treated as three-dimensional seismic volume sources (Kramer, 1996). For seismic hazard analysis, a seismic source zone may be an approximation of the actual source, depending on the relative geometry of the source and site of interest and the quality of information available for the source. For example, as shown in Figure 5.3(a), the relatively short fault can be modelled as a point source because the distance between any point along the fault and the site of interest is almost constant. Similarly, as shown in Figure 5.3(b), the depth of the fault plane is sufficiently small so that it does not significantly affect the source–site distance along the direction of the depth; as a result, the planar source can be approximated as a line source. In Figure 5.3(c),

158

C

l 2

C Epicenter

l 2

Site

B

F

D

d

h B

A

rce ou s t l Fau

(a) Perspective of the line source

A l 2

A

r0

F Hypocenter

Hypocenter F

l 2

X

C

B R

d

r0 D

Site

(b) ABD plane of the perspective of the line source Figure 5.4

Geometry of line seismic source zone.

the available data are not sufficient to determine accurately the actual geometry of the source so that it is represented as a volume source (Kramer, 1996). The occurrences of earthquakes are usually assumed to be uniformly distributed within a seismic source zone, i.e., earthquakes are considered equally likely to occur at any location within the source. It is also assumed that all the energy of the earthquake is released at the hypocenter of the earthquake. Based on these two assumptions, the probability distribution of the source–site distance can be determined using the geometrical relations between the source and the site of interest. For illustration purpose, a simple line seismic source (linear fault) is used to obtain the probability distribution of source–site distance (Cornell, 1968). As shown in Figure 5.4(a), the site of interest is assumed to be located at a perpendicular distance  from line A C on the ground surface, along which the epicenter F of the scenario earthquake is expected to lie. Line A C is vertically above the fault AC of length L, located at the focal depth H. The site D is located symmetrically with respect to the fault AC. Figure 5.4(b) shows the ABD plane of the fault. The geometrical relations between the fault and the site of interest are  D = H2 + 2 ,

R=



D2 + X2,

(5.2.1)

5.2 probabilistic seismic hazard analysis

159

where D is the perpendicular slant distance from site D to fault AC, and R is the hypocentral distance, which is the distance from site D to any scenario hypocenter F located at a distance X from point B. By setting point B as the origin, it is observed that − L/2  X  L/2 in Figure 5.4(b). Thus, the source–site distance R is restricted to D  R  r0 , in which r0 =

 D 2 + (L/2)2 .

(5.2.2)

Based on the assumption that an earthquake is equally likely to occur anywhere along the fault, the location variable X is uniformly distributed on the interval [−L/2, L/2],   and X is then uniformly distributed on the interval [0, L/2]. Thus, the Cumulative   Distribution Function (CDF) of random variable X can be expressed as  2|x|   L , 0  |x|  . P X  |x| = L

(5.2.3)

2

By solving for X in the second equation of (5.2.1) and substituting X into equation (5.2.3), the CDF of source–site distance R is determined as ! 2 √r 2 −D 2     2 2 2 2 FR (r) = P R  r = P , R −D  r −D = L

D  r  r0 , (5.2.4)

where r is threshold value of R. By differentiating equation (5.2.4), the probability density function (PDF) of R is given by F R (r) =

2r d , FR (r) = √ dr L r 2 −D 2

D  r  r0 .

(5.2.5)

For seismic source zones with more complex geometries, which are always the cases in industry practices, the PDF F R (r) can be evaluated numerically. By dividing the irregular source zone into a large number of discrete elements with equal length (for line source), area (for area source), or volume (for volume source), distance R from the site of interest to the center of each element is obtained. The percentages that the source–site distances R fall into predefined bins of the source–site distance are then determined. A histogram, in which the percentage of distance in each bin is plotted against the distance R, is then constructed to approximate F R (r) (Kramer, 1996).

5.2.2

Probability Distribution of Earthquake Occurrence

For a seismic source zone, the temporal uncertainty of distribution of earthquakes with time is usually modelled as a Poisson distribution, i.e., the occurrence of earthquake in time follows a Poisson distribution (Kramer, 1996). The Poisson distribution is based on the assumptions that an earthquake can occur at random and at any time, and the

160

occurrence of an earthquake in a given time interval is independent of that in any other non-overlapping intervals. If λ is the mean rate of occurrence of earthquake, which is the number of occurrences of earthquake in an unit time interval, then λt is the average (mean) number of occurrences of earthquake in time interval t. Letting the random variable Ne be the number of occurrences of earthquake in time interval t, then Poisson distribution gives 



P Ne = K = 5.2.3

(λt)K e−λt , K!

K = 0, 1, 2, . . . .

(5.2.6)

Probability Distribution of Earthquake Magnitude

When a seismic source zone is identified, the earthquake magnitudes that the source zone is expected to produce need to be evaluated. In general, the source zone will produce earthquakes of different magnitudes up to the maximum possible value, with smaller earthquakes occurring more frequently than larger ones. In practice, to characterize the temporal distribution of earthquake recurrence in terms of earthquake magnitude, occurrence relationships are established based on the database of historical seismicity. Gutenberg and Richter (1944) organized the data of historical seismicity in southern California according to the number of earthquakes that exceeded different earthquake magnitudes over a period of many years. By dividing the number of exceedance of each magnitude by the length of the time period, a mean annual rate λm of occurrence of an earthquake exceeding earthquake magnitude m is determined. When the logarithm of the mean annual rate of exceedance λm for southern California earthquakes is plotted against earthquake magnitude m, a linear relationship is observed. Consequently, the Gutenberg-Richter recurrence relationship is expressed as log10 λm = a − B m.

(5.2.7)

For the special case of m = 0, equation (5.2.7) gives λ0 = 10 a , which is the mean annual number of earthquakes of magnitude larger than zero, or the mean annual rate of occurrence of earthquake (of any magnitude). B describes the relative likelihood of large and small earthquakes. Generally, parameters a and B are obtained by regression analysis on a database of historical seismicity for the source zone of interest. The Gutenberg-Richter recurrence relationship governed by equation (5.2.7) can also be expressed as λm = 10 a−Bm = e α−β m ,

(5.2.8)

5.2 probabilistic seismic hazard analysis

161

Mean annual rate of exceedance λm

1 10–1 10–2 10–3 ν =1.0, β =2.0

10–4 10–5

mmin = 4

10–6 3

4

Figure 5.5

mmax = 7 5

8

6 7 8 Earthquake magnitude m

9 9

10

Bounded Gutenberg-Richter recurrence relationship.

where α = a ln10 and β = B ln10. The Gutenberg-Richter recurrence relationship (5.2.8) implies that earthquake magnitudes are exponentially distributed and magnitude m is valid for the semi-infinite range of [0, +∞). For engineering purpose, it is common to disregard very small earthquakes because they seldom cause significant damage. On the other hand, a seismic source zone has a maximum earthquake magnitude mmax that it is capable to produce due to its geological conditions. Thus, the bounded Gutenberg-Richter recurrence relationship was established to eliminate earthquake magnitudes lower than threshold mmin and larger than threshold mmax for the source zone of interest (McGuire, 2004) λm = ν ·

e−β(m−mmin ) − e−β(mmax −mmin ) , 1 − e−β(mmax −mmin )

mmin  m  mmax ,

(5.2.9)

where ν = λmin = exp(α −β mmin ) is the mean annual rate of occurrence of earthquakes above the minimum earthquake magnitude mmin for the source zone. The bounded Gutenberg-Richter recurrence relationship (5.2.9) is illustrated in Figure 5.5 for ν = 1 and β = 2. Based on equation (5.2.9), the CDF of earthquake magnitude M between mmin and mmax is given by λmmin −λm    1−e−β(m−mmin ) = . FM (m) = P M  m  mmin  m  mmax = . λmmin 1−e−β(mmax −mmin ) (5.2.10) By differentiating equation (5.2.10), the PDF of M is given by FM (m) =

d β e−β(m−mmin ) . FM (m) = 1−e−β(mmax −mmin ) dm

(5.2.11)

162

PSHA is usually conducted by considering seismic hazard environment surrounding the specific site. The earthquake-generating characteristics of individual faults are then important. Individual faults repeatedly generate earthquakes of similar size (within about one-half magnitude unit), known as characteristic earthquakes, at or near their maximum earthquake magnitude. Geological evidence indicates that the characteristic earthquakes occur more frequently than implied by extrapolating the Gutenberg-Richter recurrence relationship from high rates of exceedance (low magnitude) to low rates of exceedance (high magnitude). This results in a more complex recurrence law governed by seismicity data at low magnitudes and geologic data at high magnitudes (Kramer, 1996).

5.2.4

Scalar Probabilistic Seismic Hazard Analysis

Having characterized the randomness in the locations of earthquakes in Section 5.2.1, the magnitudes and rates of occurrence of earthquakes in Section 5.2.3, and the variation of individual ground-motion intensity measure in Section 2.4.1, scalar PSHA can be performed in a mathematically rigorous manner. Take spectral acceleration

SA(Tj ) at vibration period Tj

as the ground-motion in-

tensity measure, and assume that earthquake magnitude M and source–site distance R are statistically independent. For a given occurrence of an earthquake at a seismic source zone I, the probability that

SA(Tj ) exceeds a threshold sj

at the site of interest

can be calculated using the total probability theorem 





P SA(Tj ) > sj  an earthquake occurring at source I  

= r m

where P



P



   SA(Tj ) > sj  m, r FM (m) F R (r) dm dr , (5.2.12) I

  SA(Tj ) > sj  m, r is the CDF of spectral acceleration SA(Tj ) conditional on a

scenario earthquake in terms of m and r, given by equation (2.4.21), and FM (m) and F R (r) are the PDF of magnitude M and source–site distance R, determined in Sections 5.2.3 and 5.2.1, respectively. Multiplying equation (5.2.12) by νI (the mean annual rate of occurrence of earthquakes above the minimum earthquake magnitude mmin for the source zone I), the mean annual rate that

SA(Tj ), produced by source I, exceeds a threshold sj

of interest is given by   (I) λsj = νI

r m

P



    SA(Tj ) > sj m, r FM (m) F R (r) dm dr . I

at the site

(5.2.13)

5.2 probabilistic seismic hazard analysis

163

For the seismic hazard evaluated at a site having Ns potential seismic source zones, the mean annual rate of exceedance of spectral acceleration SA(Tj ) is λsj =

Ns  I=1

λ(I) sj

=

Ns  I=1

  νI

r m

P



   SA(Tj ) > sj  m, r FM (m) F R (r) dm dr .

(5.2.14)

I

In equation (5.2.13), the mean annual rate of spectral acceleration

SA(Tj ) exceeding

a threshold value sj is calculated by summing up the probabilities for all possible earthquake magnitudes and distances, multiplied by the earthquake occurrence rate, for source zone I. The seismic hazards for all the sources are then combined in equation (5.2.14) to obtain the aggregate hazard at the site. It is noted that for regions where the causative structures of seismicity are largely unknown, smoothed historical seismicity method can be used to obtain a combination of magnitude and distance distributions, and the mean rate of occurrence, without drawing seismic source zones (Frankel, 1995). Having obtained the mean annual rate of exceedance of spectral acceleration SA(Tj ) in equation (5.2.14), the temporal randomness of the occurrence of such earthquakes are often modelled using Poisson process (Kramer, 1996). The probability of n events

SA(Tj ) exceeding sj ) occurring in a time

(i.e., earthquakes with spectral acceleration

period of t is given by, from equation (5.2.6), 



P Ne (t) = n =

(λsj t)n n!

e

−λsj t

,

n = 0, 1, 2, . . . .

(5.2.15)

From equation (5.2.15), the probability of at least one event occurring (i.e., the event of spectral acceleration SA(Tj ) exceeding sj occurring at least once) during time period t is given by









P SA(Tj ) > sj = 1 − P Ne (t) = 0 = 1 − e−λsj t .

(5.2.16)

In most earthquake engineering practices, the time period t is taken as one year or 50 years. For simplicity of notation, in this book, the time period t is taken as one year. When the value of the mean annual rate of exceedance λsj in equation (5.2.14) is   small, which is almost always the case in reality, λsj and the AEP (t = 1) P SA(Tj ) > sj in equation (5.2.16) are numerically identical. Hence, for λsj in equation (5.2.14), the commonly used terminology “annual exceedance probability” (AEP) and “mean annual rate of exceedance” are used interchangeably. From equation (5.2.14), the annual probability (or mean annual rate) of exceedance plotted against spectral acceleration

SA(Tj )

is known as a seismic hazard curve, as

shown in Figure 5.7. A seismic hazard curve is used to convolve with the seismic fragility curve (as presented in Chapter 9) in the probabilistic seismic risk analysis (as presented in Chapter 10). For a given annual probability of spectral acceleration

164

SA(Tj ) exceeding sj , a plot of the threshold sj for a number of vibration periods Tj of engineering interest at a site gives a uniform hazard spectrum (UHS). The properties and limitations of UHS are discussed in Section 5.5.1. Because the simultaneous exceedance of spectral accelerations at multiple periods in vector-valued PSHA can also be treated as an event of interest in the time interval t, the Poisson process in equation (5.2.16) and the terminology of “annual probability of exceedance” are also employed in vector-valued PSHA presented in Section 5.2.4. Scalar PSHA is demonstrated by a numerical example, which is based on a hypothetical configuration of seismic source zones as shown in Figure 5.6. The probability distribution of source–site distance in equation (5.2.5) is used for each seismic source zone. Equation (5.2.11) is taken as the probability distribution of earthquake magnitude, in which mmin , mmax , and β are assumed to be 5, 6, and 2.07, respectively, for Source 1, and 5, 8, and 2.07, respectively, for Source 2. The mean annual rates ν for Sources 1 and 2 are taken as 0.01 and 0.09, respectively. It is noted that the site of interest can be located inside an area source, which is the most common situation in eastern North America (ENA), for example. The GMPE, proposed by Abrahamson and Silva (1997) as presented in Section 2.4.2, is used for obtaining the mean and standard deviation values in the conditional probability distribution of spectral acceleration at individual vibration period in equation (2.4.21). In the selected GMPE, parameters are set for rock site condition, reverse fault, any geological condition except for hanging wall, and sigma for arbitrary component. By performing the scalar PSHA for the hypothetical configuration of seismic source zones, the seismic hazard curves for spectral accelerations at 0.1 and 1 s are plotted in Figure 5.7. For a given AEP, e.g., 4×10−4 , the corresponding threshold values of spectral accelerations at 0.1 and 1 s can be determined as 0.464g and 0.155g, respectively, in Figure 5.7. When plotted on a log-log scale, seismic hazard curves are close to linear, as shown in Figure 5.8. Over at least any tenfold difference in exceedance frequencies, a seismic hazard curve may be approximated by a straight line log10 H(a) = −Kh log10 a + Ki ,

(5.2.17)

H(a) = Ki a−Kh .

(5.2.18)

which can be written as − Kh is the slope of the straight line given by Kh =

1 1 = , log10 a2 − log10 a1 log10 AR

(5.2.19)

5.2 probabilistic seismic hazard analysis

165

500

Source 2

Distance (km)

100

mmin = 5, mmax = 8, ν = 0.09, Depth = 5 km Source 1

mmin = 5, mmax = 6, ν = 0.01, Depth = 5 km 10 Site of interest

0

0

Figure 5.6

10

20

30 40 Distance (km)

50

60

Hypothetical configuration of seismic source zones.

Annual exceedance probability

10−1 Period of 0.1s 10−2 Period of 1 s 10−3 4×10−4

10−4 0.01 Figure 5.7

H(a)

10–1 –1

0.1 0.155 Spectral acceleration (g)

0.464

1

Seismic hazard curve for spectral acceleration.

log10 H(a)

10–2 –2

10–3 –3

10–4 –4

log10 a

10–5 –5 –1

0

10–1

100

Figure 5.8

log10 a1

Seismic hazard curve.

log10 a2

1 101

a

166

in which AR = a2 /a1 is the ratio of ground motions corresponding to a tenfold reduction in exceedance frequency. A large value of AR represents a shallow-sloped hazard curve, whereas a small value of AR represents a steep hazard curve. From equation (5.2.18), Ki is given by Ki = H(a1 ) a1Kh = H(a2 ) a2Kh .

(5.2.20)

5.2.5 Vector-Valued Probabilistic Seismic Hazard Analysis The scalar PSHA for a specific site, as discussed in Section 5.2.4, provides the marginal annual probability of exceeding a threshold value of spectral acceleration at an individual period. To improve the accuracy in predicting structural response induced by earthquakes, vector-valued PSHA, from which the joint annual probability of simultaneously exceeding threshold values of spectral accelerations at multiple vibration periods can be determined, was first proposed by Bazzurro and Cornell (2002). In the K-dimensional case, replace the marginal probability distribution of spectral acceleration conditional on a scenario earthquake in equation (5.2.13) using the joint conditional probability distribution governed by equation (2.4.23). The joint annual probability (joint mean annual rate in actuality) of simultaneously exceeding spectral accelerations

SA(T1 ), . . . , SA(TK ) at vibration periods T1 , . . . , TK for the source zone I

is given by .

λ(I) s1 ...sK

  = νI

r m

P



    SA(T1 ) > s1 , . . . , SA(TK ) > sK m, r FM (m) F R (r)dm dr . (5.2.21) I

For the seismic hazard evaluated at a site having Ns potential seismic source zones, the joint probability of simultaneously exceeding spectral accelerations is λs1 ...sK

  Ns  = νI I=1

r m

P



   SA(T1 ) > s1 , . . . , SA(TK ) > sK  m, r FM (m) F R (r)dm dr . I

(5.2.22) To illustrate vector-valued PSHA, the hazard analysis for the hypothetical configuration of seismic source zones, shown in Figure 5.6, is performed for spectral accelerations at vibration periods of 0.1 and 1 s. In the joint conditional probability distribution governed by equation (2.4.23), the matrix of spectral correlation developed by Baker and Jayaram (2008) is adopted. All other information required in the hazard evaluation is the same as used in Section 5.2.4. Figure 5.9 shows the contours of the vector-valued PSHA for the hypothetical seismic hazard environment. The contours denote the joint annual probability of exceeding spectral accelerations at 0.1 and 1 s simultaneously. For the marginal AEP of 4×10−4 ,

5.3 seismic hazard deaggregation

167

1×10 –

4

3.24×10–5

0.464

1×

10 –

5

10 –

–4

10 4×

1×

–3

10 2×

3

–3

10 5×

0.1

–2

10 1×

Spectral acceleration (g) at period of 0.1 s

1

0.01 0.01 Figure 5.9

0.1 0.155 Spectral acceleration (g) at period of 1 s

1

Contours (denoting the joint AEP) of vector-valued PSHA.

spectral accelerations at 0.1 and 1 s are 0.464g and 0.155g, respectively. As can be seen in Figure 5.9, the joint probability that the events of spectral acceleration at 0.1 s exceeding 0.464g and spectral acceleration at 1 s exceeding 0.155g occur at the same time is 3.24×10−5 , which is much smaller than the marginal probability for spectral accelerations at individual vibration periods.

5.3 Seismic Hazard Deaggregation PSHA combines all possible and relevant earthquake scenarios and probability levels through integration, which is an “aggregation” procedure. To study the contributions of parameters (such as earthquake magnitude, source–site distance, epsilon, and the rate of occurrence) of earthquake scenarios to the seismic hazard at a given site, a “deaggregation” procedure for equation (5.2.14) or (5.2.22), known as seismic hazard deaggregation (SHD), is often performed (McGuire, 1995; Bazzurro and Cornell, 1999). In engineering practice, the results of a SHD in terms of earthquake magnitude, source–site distance, epsilon, or the rate of occurrence are used to assist in the selection of recorded ground motions for seismic analysis (as presented in Chapter 6) and to generate some of the PSHA-based seismic design spectra (as presented in Section 5.5). SHD can be classified as scalar SHD and vector-valued SHD, corresponding to scalar PSHA and vector-valued PSHA, respectively. Scalar and vector-valued SHD are discussed in Sections 5.3.1 and 5.3.2, respectively.

168

5.3.1 Scalar Seismic Hazard Deaggregation To investigate not only the relative seismic hazard contributions from earthquake magnitude m and source–site distance r but also the number of standard deviations ε in equation (2.4.1), the standard formulation of scalar PSHA given by equation (5.2.14) can be extended to    Ns  νI λsj = I=1

ε r m









P SA(Tj ) > sj  m, r, ε FM (m) F R (r) F  (ε)dm dr dε . (5.3.1) I

In equation (5.3.1), F  (ε) is the PDF of standard normal distribution as discussed in Section 2.4.4, and the first term in the integrand is the Heaviside step function,        P SA(Tj ) > sj  m, r, ε = H ln SA(Tj )  m, r, ε − lnsj , (5.3.2)    which is equal to 1 if ln SA(Tj )  m, r, ε , as computed from equation (2.4.1), is greater than lnsj and 0 otherwise. From equations (5.3.1) and (5.3.2), the relative seismic hazard contributions can be calculated towards the probability of

SA(Tj )

exceeding

(not equal to) sj . In some cases, the relative seismic hazard contributions towards the probability of

SA(Tj ) equal to (not exceeding) sj

may be desired (McGuire, 1995; Baker and Cornell,

2006b). Equation (5.3.2) can be replaced by        P SA(Tj ) = sj  m, r, ε = i ln SA(Tj )  m, r, ε , lnsj ,

(5.3.3)

where i (x, y) denotes an indicator function, which is equal to 1 when x = y and 0 otherwise. Because the probability of a continuous random variable equal to a threshold value is always zero, equation (5.3.3) exits only when numerical discretization is applied so that equation (5.3.3) becomes a probability mass function (PMF). Using equation (5.3.1) along with equation (5.3.2) or (5.3.3), the annual probability of spectral acceleration exceeding (or equal to) a given value sj (determined from a specified AEP using a seismic hazard curve) for the intervals mx−1  m  mx , ry−1  r  ry , and εz−1  ε  εz (1  x  xN , 1  y  yN , and 1  z  zN , in which xN , yN , and zN are the numbers of intervals for m, r, and ε, respectively) is given by  Ns  εz  ry  mx      λsj , x, y, z = νI P SA(Tj ) > sj m, r, ε FM (m) F R (r) F  (ε)dm dr dε . I=1

εz−1 ry−1 mx−1

I

(5.3.4) Dividing these annual probabilities of exceedance (or equal) λsj , x, y, z for different cubic intervals of m-r-ε by the total AEP (or equal) λsj in equation (5.3.1), a four-dimensional histogram of relative contributions to the seismic hazard against the coordinates of m, r, and ε can be plotted.

5.3 seismic hazard deaggregation

169

Representative Earthquake From the relative seismic hazard contributions to a given spectral acceleration sj , the representative earthquake, in terms of modal or mean values of earthquake magnitude m, source–site distance r, and the number of standard deviations ε, can be obtained. For the cubic interval having the largest relative hazard contribution, the corresponding set of m ˆ j -ˆrj -ˆεj is known as the modal (most-likely) earthquake towards sj . The mean earthquake towards sj , represented by the weighted average values of magnitude m¯ j , distance r¯j , and the number of standard deviations ε¯ j , is expressed as

m¯ j = r¯j = ε¯ j =

xN  yN  zN  mx−1 +mx λsj , x, y, z x=1 y=1 z=1

2

λsj

xN  yN  zN r  y−1 +ry λsj , x, y, z x=1 y=1 z=1

2

λsj

yN  zN xN   εz−1 +εz λsj , x, y, z x=1 y=1 z=1

2

λsj

,

(5.3.5a)

,

(5.3.5b)

,

(5.3.5c)

where λsj , x, y, z and λsj are given by equations (5.3.4) and (5.3.1), respectively. In engineering practice, to achieve certain engineering objective, modifications or manipulations may be applied to the procedure of the scalar SHD. Beta Earthquake McGuire (1995) proposed a procedure of scalar SHD to obtain the most-likely combination of mβ , rβ , and εβ , known as the beta earthquake, to accurately replicate uniform hazard spectrum (UHS) in some manner. The procedure for generating the beta earthquake is summarized as follows. ❧ Calculate seismic hazard curves of spectral accelerations at 0.1 s (representing shortperiod portion of UHS) and 1 s (representing long-period portion of UHS) for each seismic source zone separately using equation (5.2.13) and for all the seismic zones using equation (5.2.14). ❧ Specify an AEP (e.g., 4.0×10−4 ) and determine the spectral acceleration threshold corresponding to this specified AEP on each calculated seismic hazard curve. ❧ One Dominant Seismic Source: If one seismic source is the dominant hazard contributor at both 0.1 and 1 s, i.e., the largest threshold values of spectral accelerations at both 0.1 and 1 s are from the same seismic source (dominant seismic source), one beta earthquake is used to represent the seismic hazard. In this case,

170

1. Draw two 4-dimensional histograms of hazard contributions by m, r, and ε, using equations (5.3.3) and (5.3.4), for vibration periods of 0.1 and 1 s at the specified annual probability of exceedance for the dominant seismic source. 2. Draw a composite four-dimensional histogram from the two histograms obtained in Step 1 by • considering only intervals in which there is a nonzero contribution in both histograms, • adding the contributions in the corresponding intervals for the two histograms, and • assigning zero to the remaining intervals, in which one or both histograms have a zero contribution. 3. Determine the modal (or most-likely) earthquake in terms of mβ , rβ , and εβ from the composite 4-dimensional histogram obtained in Step 2. 4. Substitute mβ , rβ , and εβ , obtained in Step 3, into the GMPEs in equation (2.4.1) at both 0.1 and 1 s. Adjust the value of εβ so that the resulting spectral accelerations at 0.1 and 1 s are greater than or equal to the threshold values of spectral accelerations (corresponding to the specified AEP) calculated for all the seismic zones at 0.1 and 1 s, respectively. 5. Obtain the beta earthquake in terms of mβ , rβ , and the adjusted εβ . ❧ Two Dominant Seismic Sources: If different seismic sources are the dominant hazard contributors at 0.1 and 1 s, i.e., the largest threshold values of spectral accelerations at 0.1 and 1 s, respectively, are from two different seismic sources (dominant seismic sources), two beta earthquakes are used to represent the seismic hazard at the portions of short period and long period, respectively. In this case, 1. Draw a four-dimensional histogram of hazard contributions by m, r, and ε, using equations (5.3.3) and (5.3.4), for vibration period of 0.1 s at the specified AEP for the corresponding dominant seismic source. 2. Determine the modal (or most-likely) earthquake in terms of mβ , rβ , and εβ from the four-dimensional histogram obtained in Step 1. 3. Substitute mβ , rβ , and εβ , obtained in Step 2, into the GMPE in equation (2.4.1) at 0.1 s and adjust the εβ so that the resulting spectral acceleration at 0.1 s is equal to the threshold value of spectral acceleration (corresponding to the specified AEP) calculated for all the seismic zones at 0.1 s.

5.3 seismic hazard deaggregation

171

4. Obtain the beta earthquake in terms of mβ , rβ , and the adjusted εβ for spectral acceleration at 0.1 s. 5. Repeat Steps 1 to 4 to determine the beta earthquake in terms of mβ , rβ , and the adjusted εβ for spectral acceleration at 1 s. There are initially two distributions (for spectral accelerations at 0.1 and 1 s) contributing to seismic hazard by m, r, and ε. If the two distributions are close, similar earthquakes drive the hazard at both 0.1 and 1 s, as treated in the case of one dominant source. If the distributions are different, different earthquakes drive the hazards for 0.1 and 1 s, as treated in the case of two dominant sources. The beta earthquakes determined are used for replicating the UHS, as discussed in Section 5.5.2. Representative Earthquake for CMS-ε Baker and Cornell (2005) proposed the concept of Conditional Mean Spectrum considering ε (CMS-ε). CMS-ε is obtained based on GMPEs, correlation coefficients of spectral accelerations, and the representative earthquake from the scalar SHD. The representative earthquake in terms of mean values m¯ j , r¯j , and ε¯ j is obtained for spectral acceleration at the fundamental period of the structure of interest using equations (5.3.3)–(5.3.5). The value of ε¯ j is then adjusted so that the resulting spectral acceleration at the structural fundamental period using GMPE matches the UHS at the fundamental period. CMS-ε is discussed in Section 5.5.3. Controlling Earthquake As presented previously, to replicate

SA(Tj ) on a UHS,

representative earthquake in

terms of m, r, and ε are substituted into a GMPE. However, ε is often reassigned to the value that gives a prediction of the target SA(Tj ) because the resulting SA(Tj ) using

GMPE dose not necessarily match the target SA(Tj ) on the UHS (McGuire, 1995; Baker

and Cornell, 2005). In this situation, it may be unnecessary to disaggregate the seismic hazard for ε. On the other hand, as shown in equation (2.4.1) and discussed in Section 2.4.4, the random variable ε(Tj ) in a GMPE characterizes only the probability level of

SA(Tj )

conditional on the occurrence of an earthquake in terms of m and r. The randomness of the occurrence of the earthquake itself, i.e., the occurrence rate of the earthquake,

SA(Tj ) on the UHS using GMPE. From equation (5.2.14), the probability level of SA(Tj ) on a UHS is a summation of

is not considered in the prediction of

the products of the annual probabilities of exceedance conditional on the earthquake occurrence and the earthquake occurrence rates. It is therefore appropriate to predict

172

SA(Tj )

on a UHS by using not only the GMPE and ε, but also the occurrence rate

of such an earthquake; in other words, it is reasonable to explicitly characterize the probability level of

SA(Tj ) by both the occurrence rate and ε(Tj ).

Ni et al. (2012) proposed the concept of controlling earthquake in terms of earthquake magnitude mc , source–site distance rc , and the rate of occurrence νc . The values of mc and rc are determined for the exceedance of a threshold of spectral acceleration using equations (5.3.5a), (5.3.5b), and (5.3.2). Similar to equations (5.3.5), the weighted average value of the rate of occurrence νc is given by νc =

Ns  I=1

(I)

νI

λsj

λsj

,

(5.3.6)

(I)

where λsj is the seismic hazard contribution from seismic source I in equation (5.2.13), and λsj is given by equation (5.2.14). It is noted that the concept of “controlling earthquake” proposed by Ni et al. (2012) is different from the one mentioned in DSHA in Section 5.1. For the hypothetical configuration of seismic source zones illustrated in Figure 5.6, the three-dimensional histograms of relative seismic hazard contributions from m and r for spectral accelerations at vibration periods of 0.1 and 1 s, respectively, are shown in Figures 5.10 and 5.11. The AEP is selected as 4.5×10−3 , and all other parameters are the same as used in Section 5.2.4. In Figure 5.10, Source 1 (producing small near-field earthquake) is the dominant seismic hazard contributor to the spectral acceleration at 0.1 s (short-period portion) so that the resulting controlling earthquake has relatively small magnitude and short distance. In contrast, as shown in Figure 5.11, Source 2 (producing large far-field earthquake) is the dominant seismic hazard contributor to the spectral acceleration at 1 s (long-period portion) so that the resulting controlling earthquake possesses relatively large magnitude and long distance. This phenomenon has been widely observed in seismic hazard analysis (McGuire, 1995; Frankel, 1995; Halchuk and Adams, 2004; Atkinson and Elgohary, 2007).

5.3.2 Vector-Valued Seismic Hazard Deaggregation In Section 5.3.1, the scalar SHD does not consider the simultaneous exceedance of spectral accelerations at multiple vibration periods, as shown in equation (5.3.4), i.e., the representative earthquakes are extracted for spectral accelerations at individual periods separately. In this section, the scalar SHD procedure is extended to vector-

5.3 seismic hazard deaggregation

Source 1

80

Mean: mC = 6.00, rC = 43.00 km, νC = 0.029

60 Source 2

40

5 0

Figure 5.10

20

40 60 Source-site distance (km)

80

100

Scalar SHD for spectral acceleration at vibration period of 0.1 s.

Annual probability of exceedance of 4.5 ×10–3 Period = 1s

100 80

Mean: mC = 6.18, rC = 80.44 km, νC = 0.063

60 Source 1

Source 2

40

0

itu d

7

e

9 20

M ag n

5 0

Figure 5.11

20

40 60 Source-site distance (km)

80

100

Scalar SHD for spectral acceleration at vibration period of 1 s.

Annual probability of exceedance of 4 ×10–4 Period = 0.01, 0.02, 0.05, 0.1, 0.3, 0.5, 1, 5 s Mean: mC = 6.16, rC = 54.00 km, νC = 0.052

100 80 60

Source 2

Source 1 40

M ag n

0

7

e

9 20

itu d

Hazard contribution (%)

itu d

7

e

9 20 0

Hazard contribution (%)

Annual probability of exceedance of 4.5 ×10–3 Period = 0.1 s

M ag n

Hazard contribution (%)

100

173

5 0

20

40 60 Source-site distance (km)

80

100

Figure 5.12 Vector-valued SHD for spectral accelerations at multiple vibration periods.

valued SHD for determining the controlling earthquake that contributes seismic hazard to spectral accelerations at multiple periods simultaneously (Ni et al., 2012). In comparison with the scalar SHD in equation (5.3.4), replacing the conditional marginal distribution of spectral acceleration at individual period in equation (2.4.21) by the conditional joint distribution of spectral accelerations at multiple periods in

174

equation (2.4.23), the formulation of the vector-valued SHD is expressed as λs1 , ..., sK , x, y, z =

Ns  I=1

 νI

ry



mx

ry−1 mx−1

P



    SA(T1 )>s1 , . . . , SA(TK )>sK m, r FM (m) F R (r)dm dr . I

(5.3.7) It is noted that, in equation (5.3.7), the variable ε is not disaggregated when compared to equation (5.3.4). The quotient of AEP λs1 , ..., sK , x, y, z for various pairs of m and r divided by the total seismic hazard λs1 , ..., sK obtained from equation (5.2.22) gives a three-dimensional histogram of the relative seismic hazard contributed by different m-r pairs. Because the multivariate distribution of spectral accelerations is used in equation (5.3.7), the simultaneous exceedance of spectral accelerations at multiple periods is then considered in the vector-valued SHD, which corresponds to the vector-valued PSHA in equation (5.2.22). For example, each bar in the histogram in Figure 5.12 (for the same hazard configuration and parameters used in Section 5.3.1) represents the relative seismic hazard contribution to the spectral accelerations s1 , s2 , . . . , sK simultaneously, provided by the corresponding m-r pair. Having obtained the distribution of the relative seismic hazard contributions, the weighted average values of magnitude and distance, which contribute the dominant seismic hazard to the spectral accelerations s1 , s2 , . . . , sK simultaneously, is given by . mc =

yN xN   (mx−1 +mx ) λs1 , ..., sK , x, y, z x=1 y=1

2

λs1 , ..., sK

, rc =

yN (r xN   y−1 +ry ) λs1 , ..., sK , x, y, z x=1 y=1

2

λs1 , ..., sK

. (5.3.8)

Similarly, the weighted average value of the rate of occurrence for the resulting earthquake (mc and rc ) can be obtained νc =

Ns  I=1

(I)

νI

λs1 , ..., sK λs1 , ..., sK

.

(5.3.9)

5.4 Treatment of Epistemic Uncertainty In previous discussions, all the aleatory uncertainties in the location of earthquakes, the size of earthquakes, the rate of occurrence of earthquakes, and the variation of ground-motion characteristics with earthquake size and location are characterized by the corresponding probability distribution models. However, the determination of these distribution models may vary with the knowledge of professionals; this uncertainty due to variation in professional knowledge is known as epistemic uncertainty. On the other hand, the intrinsic randomness described by a certain distribution model is known as the aleatory randomness.

5.4 treatment of epistemic uncertainty

GMPE model

175

Magnitude distribution

mmax 7.0 (0.2)

Gutenberg-Richter (0.7) Abrahamson & Silva (0.5)

Characteristic earthquake (0.3)

7.5 (0.6) 8.0 (0.2) 7.0 (0.2) 7.5 (0.6) 8.0 (0.2) 7.0 (0.2)

Gutenberg-Richter (0.7) Atkinson & Boore (0.5)

Characteristic earthquake (0.3)

7.5 (0.6) 8.0 (0.2) 7.0 (0.2) 7.5 (0.6) 8.0 (0.2)

Weight 0.07 0.21 0.07 0.03 0.09 0.03 0.07 0.21 0.07 0.03 0.09 0.03 1.00

Figure 5.13

Logic tree for characterizing epistemic uncertainty of seismic hazard models.

To properly characterize the epistemic uncertainty, the logic tree approach is often performed in seismic hazard analysis (Kramer, 1996; McGuire, 2004). It allows the use of alternative models, each of which is assigned a weight factor that is interpreted as the relative likelihood of the model being correct. A logic tree consists of a series of nodes, representing points at which models are specified, and branches, representing the different models specified at each node. The sum of the probabilities of all branches connected to a given node (i.e., all the branches on the right-hand side of the given node) is 1. A simple logic tree, considering the epistemic uncertainty in selection of models for GMPE, probability distribution of earthquake magnitude, and maximum magnitude, is illustrated in Figure 5.13. In this logic tree, two GMPEs, according to Abrahamson and Silva (1997) and Atkinson and Boore (2006), are considered equally likely to be correct; each GMPE is therefore assigned a relative likelihood of 0.5. Proceeding to the next level of nodes, the Gutenberg-Richter model is considered to be more likely to be correct than the characteristic earthquake model. At the final level of nodes, different values of relative likelihood are assigned to the maximum magnitude. As can be seen in Figure 5.13, the logic tree terminates with a total of 2×2×3 = 12 (number of GMPE models × number of magnitude distribution models × number of maximum magnitudes) branches. The relative likelihood of the combination of models and parameters implied by each terminal branch is given by the product of the relative

176

likelihood of the terminal branch and all prior branches leading to it. For example, the relative likelihood of the combination of Abrahamson and Silva (1997) model, Gutenberg-Richter model, and maximum magnitude of 7.0 is 0.5×0.7×0.2 = 0.07. The sum of the relative likelihoods of the terminal branches is equal to 1. To use the logic tree, a PSHA or SHD is performed for the combination of models and/or parameters associated with each terminal branch. The resulting quantity (e.g., AEP for a given spectral acceleration, spectral acceleration from PSHA, and representative earthquake in terms of m, r, ε, and ν from SHD) is weighted by the relative likelihood of this terminal branch. The resulting quantities associated with their corresponding weight factors (relative likelihoods of the terminal branches) give the PMF of the quantity. The mean, median, or any percentile value of the quantity can then be obtained using the PMF. For the procedure of determining the beta earthquake discussed in Section 5.3.1, when different models (e.g., GMPEs, magnitude distributions, or maximum magnitudes) are considered to account for the epistemic uncertainties, the mean values of the beta earthquakes from all the combinations of the models, in terms of mβ , rβ , and εβ , are obtained first using the logic tree approach presented earlier. The resulting beta earthquake is substituted into each GMPE. The value of εβ is then adjusted so that the weighted average values of spectral accelerations using the GMPEs satisfy the target matching criteria as presented in Section 5.3.1 (McGuire, 1995).

5.5 Seismic Design Spectra Based on PSHA PSHA has been widely used for the establishment of seismic design spectra, including uniform hazard spectrum (UHS) (ASCE/SEI, 2005; CSA, 2010a; CSA, 2010b; CSA, 2014; NBCC, 2015), predicted spectrum based on GMPEs (McGuire, 1995), and conditional mean spectrum considering ε (CMS-ε) (Baker and Cornell, 2006b; EPRI, 2011). In earthquake engineering, a seismic design spectrum is generally a postprocessed and smoothed spectrum when comparing with the earthquake response spectrum, which is derived directly from a recorded ground motion and has peaks and valleys in its shape, as discussed in Section 4.1. The term “seismic design spectrum” is usually used to distinguish with the earthquake response spectrum, although the seismic design spectrum can actually be applied as the seismic input in both design and analysis. In this section, the properties and limitations of these PSHA-based seismic design spectra are presented and discussed.

5.5 seismic design spectra based on psha

177

SA(T1) Annual probability of exceedance 0.0004

Annual probability of exceedance 0.002

s1

Spectral acceleration

SA(T2)

s2

s1

s2

PDF of

SA(T1)

PDF of

SA(T2) T2 Vibration period

T1 Figure 5.14

Spectral acceleration (g)

0.5

Concept of uniform hazard spectrum.

Dominant period 0.1 s

Prediected spectrum based on GMPEs

0.4 0.3

CMS-ε

0.2

UHS 0.1 Annual probability of exceedance 4 ×10–4 0.0

0.01

0.1 Figure 5.15

Period (sec)

1

10

Seismic design spectra.

5.5.1 Uniform Hazard Spectrum As discussed in Section 5.2.4, based on the scalar PSHA in equation (5.2.14), for a given probability pm (the subscript “m” standing for Marginal) of spectral acceleration

SA(Tj )

exceeding sj , a plot of the threshold sj for a number of vibration periods

Tj (j = 1, 2, . . ., K) of engineering interest at a site gives a uniform hazard spectrum (UHS). The concept of UHS is schematically illustrated in Figure 5.14. It is called “uniform hazard” spectrum because the AEP pm of spectral acceleration at each period is consistent. Figure 5.15 shows a UHS based on the hypothetical seismic source zones in Figure 5.6 and the hazard information presented in Section 5.2.4.

178

The AEP pm for an individual spectral acceleration, however, is not the AEP for the entire design spectrum with spectral accelerations at multiple periods, i.e., a UHS does not provide probabilistic knowledge about the simultaneous exceedance of spectral accelerations at multiple periods. Furthermore, for the same probability level pm , the controlling earthquakes in terms of magnitude and source–site distance extracted from a UHS at different vibration periods through scalar SHD are often different if the dominant hazard contributors to high and low vibration period ranges are different seismic sources (Halchuk and Adams, 2004). As a result, a UHS generally represents an envelope over different earthquakes that contributes to the seismic hazard at the site of interest and possesses a broadband spectral shape. The seismic response spectral analysis of structures is based on seismic design spectra and modal superposition methods. When a modal superposition method is applied to predict the peak response of a structure, the peak responses of the vibration modes of the structure are assumed to occur at the same time subjected to the same ground excitation. Because the spectral accelerations at different vibration periods on a UHS are generally induced by different earthquakes, it may not be appropriate to obtain the peak first mode response from one earthquake, the peak second mode response from a different earthquake, and the peak third mode response from yet another earthquake, on the UHS (Ni et al., 2012). The controversy on the UHS has existed for decades in the aspects of the basic concept of multiple dominant earthquakes, broadband spectral shape, the calibration of probability level, and the high amplitudes at high frequencies (CSA, 2014). In the nuclear power industry, however, it is still considered one of the most suitable seismic inputs for seismic analysis and design. The UHS is a product of a comprehensive seismic hazard investigation (i.e., PSHA) and a site-specific seismic input. It provides sufficient margin in the seismic design and analysis of an NPP due to its low probability level and broadband spectral shape. To connect the probability level of a UHS with the seismic risk of an NPP, a performance-based approach has been proposed and included in a seismic design standard (ASCE/SEI, 2005) to calibrate the design level of the UHS. A number of studies have also been performed to understand and mitigate the effects of the high-frequency contents in UHS to Structures, Systems, and Components (SSCs) of an NPP (EPRI, 1993a; EPRI, 2005; EPRI, 2007a; EPRI, 2007b; EPRI, 2014; EPRI, 2015). In seismic risk analysis of an NPP, one of the major tasks is to identify the design redundancies in SSCs. Attempts have been made to derive the relatively narrow-

5.5 seismic design spectra based on psha

179

band seismic design spectra from the UHS, each of which is considered to represent an individual earthquake, and thus more physically meaningful and realistic. Two representative such attempts are presented in Sections 5.5.2 and 5.5.3, respectively.

5.5.2 Predicted Spectrum Based on GMPEs In equation (5.2.14), the AEP λsj is obtained by integrating over all possible occurrences of earthquakes surrounding a given site. To obtain an individual design earthquake based on scalar PSHA, λsj can be disaggregated at predefined probability level pm , and the most-likely combination (modal values) of magnitude mβ , source–site distance rβ , and εβ , called “beta earthquake”, can then be obtained, as discussed in Section 5.3.1. If one seismic source is the dominant contributor for spectral accelerations at both 0.1 and 1 s, one beta earthquake is used to represent the seismic hazard over the entire period range. If different seismic sources are the dominant contributors at 0.1 and 1 s, two beta earthquakes are used to represent the hazards over the short-period range and the long-period range, respectively. In general, one small near-field earthquake and one large far-field earthquake are regarded as sufficient to represent the short-period range and the long-period range of a UHS, respectively (McGuire, 1995; Atkinson and Beresnev, 1998). Having obtained one beta earthquake (in terms of mβ , rβ , and εβ ) for the case of one dominant source, by substituting the beta earthquake into the GMPEs in equation (2.4.1) for spectral accelerations at different periods individually, the resulting design earthquake (in terms of seismic design spectrum), which closely matches the UHS over the entire period range, can be constructed. For the case of different dominant sources, by substituting two resulting beta earthquakes into equations (2.4.1), respectively, two seismic design spectra, which closely match the short-period range and the long-period range of the UHS, respectively, can be generated. In this case, the resulting design spectrum is relatively narrow-band centred at the specified period in comparison with the UHS. This can be observed in Figure 5.15, in which a predicted design spectrum based on GMPEs is anchored to the UHS at the vibration period of 0.1 s. It is noted that if more than one GMPE is used, predefined weight should be assigned to each GMPE to obtain the weighted average design spectra. A predicted design spectrum based on GMPEs can be interpreted as a single design earthquake in terms of the mβ -rβ pair. However, the primary advantage of the PSHA of integrating all possible earthquake occurrences surrounding the site of interest is not completely reflected in the predicted spectra because the seismic hazard of the

180

site is simply represented by the beta earthquakes. Because each point on a predicted spectrum is obtained independently through the beta earthquake and the prediction equation, the probabilistic knowledge about the simultaneous occurrence of these points on a spectrum is not provided.

5.5.3 Conditional Mean Spectrum Considering Epsilon Similar to the predicted design spectra, to account for the relationship between spectral acceleration

SA(T1 )

at fundamental period T1 and spectral accelerations

SA(Tj )

at

other periods Tj (j = 2, 3, . . ., K), Baker and Cornell (2006b) proposed the concept of conditional mean spectrum considering ε (CMS-ε) based on the assumption that spectral accelerations at two different periods are jointly lognormally distributed for a given scenario earthquake, which was verified later by Jayaram and Baker (2008). Assume that the variance in conditional spectral accelerations ln SA(Tj ) is primarily due to ε rather than variations in magnitude and distance (Baker and Cornell, 2005). For a specified AEP pM , the logarithmic mean and standard deviation of the CMS-ε

SA(T1 ) = s1 can be approximated by

conditional on the occurrence of .μ ln S

  SA(T ) = s , m, 1 1 ¯ r¯

= μ ln S

  ¯ A(Tj ) SA(T1 ) = s1 , m

2 = σ ln S

A(Tj )

2 σ ln S

A(Tj )

  m, ¯ r¯

  ¯ A(T ) m j



+ ρ ln S

A(T1 ),

1 − ρ 2ln S

ln SA(Tj )

A(T1 ),

ε¯S (T ) σ ln S (T )  m¯ , (5.5.1a) A 1 A j

ln SA(Tj )



,

(5.5.1b)

where s1 is the spectral acceleration on a UHS at the structural fundamental period ¯ r¯, and ε¯S (T ) are the mean values of the T1 obtained using equation (5.2.14). m, A 1 results of the scalar SHD towards the occurrence of SA(T1 ) = s1 (not > s1 ), as presented in Section 5.3.1. The mean μ ln S (T )  m, and standard deviation σ ln S (T )  m¯ of ¯ r¯ A j A j ln SA(Tj ) can be determined using GMPEs in equation (2.4.1), and ρ ln S (T ), ln S (T ) A

1

A

j

is the correlation coefficient between ln SA(T1 ) and ln SA(Tj ), as discussed in Section 2.4.3. Because the scenario earthquake is obtained from the spectral acceleration at the fundamental period, the CMS-ε is centred at this period with narrow-band spectral shape when compared to the corresponding UHS. This can be observed in Figure 5.15, in which a CMS-ε is anchored to the UHS at the vibration period of 0.1 s. Based on the assumption of joint lognormal distribution, the probabilistic meaning of equation (5.5.1a) is that, given the occurrence of the causal earthquake in terms of m, ¯ r¯, and ε¯S (T ) , and the spectral acceleration SA(T1 ) = s1 at the fundamental period A 1 T1 , the annual probability of logarithmic spectral acceleration ln SA(Tj ) at vibration T  is 50 % . The vector s , s , . . ., s period Tj exceeding lnsj = μ ln S (T )  S (T ) = s , m, 1 2 K A j A 1 1 ¯ r¯ gives the CMS-ε for the specified marginal AEP pm at fundamental period T1 . For the

5.6 site response analysis

181

conditional AEP other than 50 %, the randomness of SA(Tj ) conditional on SA(T1 ) = s1 can be characterized by the logarithmic standard deviation in equation (5.5.1b). When

SA(Tj )

is the spectral acceleration at the fundamental period Tj (j = 1),

the correlation coefficient ρ ln S (T ), ln S (T ) is one and equation (5.5.1a) then reA 1 A j duces to the GMPE in equation (2.4.1). For spectral acceleration SA(Tj ) at period Tj (j = 1), the difference between equations (5.5.1a) and (2.4.1) is characterized by ρ ln S

A(T1 ),

ln SA(Tj )

· ε¯S

ρ ln S

A(T1 ),

ln SA(Tj )

· ε¯S

A(T1 ) A(T1 )

in equation (5.5.1a), instead of ε(Tj ) in equation (2.4.1). denotes the mean value of ε(Tj ) conditional on ε¯S

A(T1 )

. With

the decrease of the linear correlation between ln SA(T1 ) and ln SA(Tj ) in terms of the

correlation coefficient, the deviation (second) term on the right-hand side of equation (5.5.1a) reduces. As a result, the hazard level of spectral acceleration SA(Tj ) on a CMSε, controlled by the deviation term in equation (5.5.1a), depends on the statistical correlation between spectral accelerations at the fundamental period T1 and period Tj . For a CMS-ε, the marginal AEP pm of spectral acceleration

SA(T1 ) at fundamen-

tal period T1 is predefined, and the threshold value s1 of this spectral acceleration and the causal earthquake in terms of m, ¯ r¯, and ε¯S (T ) can then be determined. Given A 1 SA(T1 ) = s1 and the causal earthquake, the conditional AEP of any other spectral accel-

eration SA(Tj ) (j  = 1) on a CMS-ε is 50 %. However, the probability of exceeding all the spectral accelerations on a CMS-ε simultaneously, based on the complete information available from PSHA, still remains unknown.

5.6 Site Response Analysis 5.6.1 Introduction Site response analysis aims to assess the effects of soil conditions on the ground shaking at a specific elevation level, such as free surface or foundation level, due to the propagation of shear waves in the soil deposit. It is generally performed based on onedimensional elastic wave propagation in the soil column, incorporating the nonlinear effects of the soil deposit. Figure 5.16 illustrates the procedure of site response analysis: 1. Developing input ground motions at reference hard rock ❧ Apply an appropriate stochastic ground motion model to determine the Fourier amplitude spectra (FAS) at reference hard rock. ❧ Apply the random vibration theory (RVT) to calculate the acceleration response spectra (ARS) corresponding to the FAS.

182

Figure 5.16 A general site response analysis procedure.

5.6 site response analysis

☞

183

Because the effect of soil amplification depends on loading levels, a minimum number of eleven response spectra with peak acceleration values spanning from 0.01g to 1.5g are required to cover the loading range, and uncertainties in input motions are considered.

❧ Develop the input ground motions • Frequency-domain dynamic response analysis: An inverse random vibration theory (IRVT) is used to convert the ARS to FAS, which are defined as the input motions at reference hard rock. • Time-domain dynamic response analysis: Time-histories spectrum-compatible with the ARS are generated as the input motions at reference hard rock. 2. Establishing geotechnical model for the site Geotechnical model and uncertainties of the site are established, including • modelling layer thickness HI , • modelling shear-wave velocity Vs, I , • modelling unit weight ρI , • modelling shear modulus reduction (G/Gmax )I , • modelling damping ratio, including initial damping DI and hysteretic damping. 3. Calculating amplification functions and strain-compatible soil properties ❧ Frequency-domain dynamic response analysis • Using FAS at the reference rock as input, employ an RVT-based program, such as RVT-SHAKE, to calculate FAS at the elevation levels of interest. • Apply RVT to determine ARS corresponding to the output FAS. ❧ Time-domain dynamic response analysis • Using ARS-compatible time-histories at the reference rock as the input motions, employ a time-history-analysis-based program, such as SHAKE 91, to calculate the output time-histories at the elevation levels of interest. • Generate ARS from output time-histories. Calculate amplification functions by comparing the input and output ARS. Dynamic response analysis will also generate strain-compatible soil properties, which are used in SSI analysis.

184

4. Determining seismic hazard curves at foundation level ❧ Determine seismic hazard curves at reference hard rock from PSHA reports. ❧ Convolve the seismic hazard curves at reference hard rock with the amplification functions at free surface (foundation level) to obtain seismic hazard curves at free surface (foundation level). 5. Determining the Ground Motion Response Spectrum (GMRS) and Foundation Input Response Spectra (FIRS) ❧ For a given level of AEP, e.g., 1×10−4, interpolate the seismic hazard curves at free surface (foundation level) to obtain GMRS (FIRS) in the horizontal direction. The vertical GMRS (FIRS) is obtained by multiplying the horizontal GMRS (FIRS) by the site-specific V/H ratios. NUREG/CR-6728 (USNRC, 2001) provides a comprehensive guideline on developing hazard- and risk-consistent ground motion spectra on soil. Appendix B of EPRI1025287 (EPRI, 2013) offers a guidance on how to develop site response on soil. Rathje and Kottke (2008) gives a procedure for RVT-based site response analyses. Based on these works, a general procedure of site response analysis in Eastern North America (ENA) is presented in this section.

5.6.2

Developing Input Ground Motions

Ground motions in ENA are different from empirical earthquake records in California. Direct empirical observations in ENA at high ground motion levels are too sparse to justify a data-based approach in that region. Therefore, it is necessary to use a theoretical-empirical modelling method in ENA (EPRI TR-102293-V1, EPRI, 1993b). This approach uses a stochastic ground motion model, which has been validated using data largely from California, where instrumental records are available over a wide range of magnitudes and distances (McGuire et al., 1984), to estimate ground motion amplitudes in the frequency band of interest to engineering analysis and design.

5.6.2.1 Stochastic Ground Motion Model For sites in ENA, a point-source stochastic model is used to determine the Fourier amplitude spectrum (FAS) FD (M0 , R, F ) of ground motion displacement due to earthquake sources

FD (M0 , R, F ) = E(M0 , F ) · P(R, F ) · G( F ),

(5.6.1)

5.6 site response analysis

185

where E(M0 , F ) is the Brune point-source spectrum, P(R, F ) represents the propagation path effects, and G( F ) is the modification due to site effects. M0 is seismic moment M0 = 101.5 (M +10.7) , where

(5.6.2)

M is the moment magnitude. R is hypocenter distance determined by R=



R2epi + D 2 ,

(5.6.3)

where Repi is the epicenter distance, and D is the source depth (the perpendicular distance between the source and generic hard rock surface). ❧ The Brune point-source spectrum E(M0 , F ) can be expressed as (Boore, 2003) E(M0 , F ) = C M0 S( F ),

C=

Rθφ VF , 4 πρs βs3 R0

(5.6.4)

where Rθ φ = 0.55 is the shear-wave radiation pattern average over the focal sphere, √ V = 1/ 2 is the partition of total shear-wave energy into two horizontal components, F = 2 is the effect of the free surface, ρs and βs are the density and shear-wave velocity in the vicinity of the earthquake source, and R0 = 1 km is the reference distance. For sites in ENA, Mid-Continent Crustal Model with ρs = 2.71 g/cm3 and βs = 3.52 km/s in Table B-5 of EPRI-1025287 is used. S( F ) is the source spectrum, which can be obtained from a single-corner frequency source model (1c) or an empirical doublecorner frequency source model (2c): • Single-Corner Frequency Source Model or ω-Square Model S( F ) =

1 , 1 + ( F/ F c )2



F c = 4.9×106 βs σ/M0 1/3,

(5.6.5)

where F is frequency in Hz, F c is the corner frequency, and σ is the stress drop, which is taken as 110 bars in Table B-4 of EPRI-1025287. Figure 5.17 shows E(M0 , F ) using single-corner frequency source model. • Double-Corner Frequency Source Model (Boore, 2003) S( F ) = Sa ( F )×S b ( F ),

(5.6.6)

in which Sa ( F ) and S b ( F ) are given in Table 5.1, the parameters F a , F b , and  are listed in Table 5.2.

☞

Source distance R starts at R = R0 ; when R0 = 1 km, R  1 km. The Brune point source model is also the Fourier displacement spectrum without considering the path effect, i.e., R = R0 = 1.

186

Table 5.1

Model † BC92 AB95, AS00

Shape of source spectra S( F ) = Sa ( F ) × Sb ( F ).

Sa ( F ) 1, F < F a F a / F, F  F a 1−  + 1 + ( F/ F a )2 1 + ( F/ F b )2 

NSHM96 # 

H96



N97 Table 5.2

1 + ( F/ F a )8 1 + ( F/ F a )2

BC92 AB95 NSHM96 # H96 N97

1 + ( F/ F b )2

−1/2

1

−1

1

−1/8



−3/4



1 + ( F/ F b )8 1 + ( F/ F b )2

−1/8 −1/4

Corner frequencies and moment ratios.

log10 F a

Model

AS00

1 + ( F/ F a )2

Sb ( F ) 

log10 F b

3.409− 0.681 M , M  5.3‡ 2.452− 0.5 M , M < 5.3 2.41−0.533 M , M  4.0∗ 2.678− 0.5 M , M < 4.0 2.623− 0.5 M 2.3 −0.5 M 2.312− 0.5 M 2.181− 0.496 M , M  2.4‡ 1.431− 0.5(M − 2.4), M < 2.4

log10 

1.495− 0.319 M − 2.452− 0.5 M − 1.43−0.188 M 2.52−0.637 M 2.678− 0.5 M 0.0 − − 3.4 −0.5 M − 3.609− 0.5 M − 2.41−0.408 M 0.605− 0.255 M 1.431− 0.5(M − 2.4) 0.0

†

References of models: BC92: Boatwright and Choy (1992); AB95: Atkinson and Boore (1995); NSHM96: Frankel et al. (1996); H96: Haddon (1996); N97: NUREG CR-6372, USNRC (1997); AS00: Atkinson and Silva (2000). ‡ The specified magnitude corresponds to the point at which F = F . a b ∗ The specified magnitude corresponds to the point at which  = 1.0. # This is the single-corner frequency source model or ω-square model, a special case of double-corner   frequency source mode, log10 F a =1.341 + log10 βs (σ )1/3 − 0.5 M , βs = 3.6 km/s, σ =150 bars. Table 5.3

Frequency F (Hz)

Site amplification factors from the ENA stochastic model.

0.1

0.2

0.3

0.5

0.9 1.25 1.8

3.0

5.0

8.0

14.0 100

Amplification A( F ) 1.02 1.03 1.05 1.07 1.09 1.11 1.12 1.13 1.14 1.15 1.15 1.15 Table 5.4

Parameters in the ENA stochastic ground motion model.

Parameter Density, ρs (g/cm3 ) Shear-wave velocity, βs (km/s)

ENA 2.71 3.52

Stress drop † , σ (bar) Diminution parameter, κ0 (s)

110 0.006 1/R, R  70 km 1/70, 70  R  130 km (130/R)0.5 /70, R > 130 km 670 F 0.33

Geometric spreading function, Z(R) Anelastic attenuation, Q( F ) †

Stress drop σ corresponds to the single-corner frequency source model (1c). This parameter is not included in the double-corner frequency source model (2c).

5.6 site response analysis

187

Brune point source spectra (cm)

103 R = R0 =1.0 102 101

M =5.5

M =6.5

100 10−1

M =4.5

10−2 10−3 0.01

0.1

1 Frequency (Hz)

Fourier acceleration spectrum (cm/s2)

Figure 5.17

104

100

10

100

Brune point source spectra.

M =6.5

R0 =1.0 103

M =5.5

102

M =4.5 101

100 0.01

0.1 Figure 5.18

Fourier acceleration spectrum (cm/s2)

10

102

1 Frequency (Hz)

FAS at R0 = 1 km for different magnitudes.

R =10

M = 6.5

R = 30

101

R =100 10

0

R =300

10−1 10−2 10−3 0.01

0.1 Figure 5.19

1 Frequency (Hz)

10

FAS for different hypocenter distances.

100

188

❧ Propagation path effects P(R, F ) is given by   π FR P(R, F ) = Z(R) · exp − , Q( F )βs

(5.6.7)

where Z(R) is geometric spreading function, given in Boore (2003), ⎧R 0 ⎪ ⎪ R  R1 , ⎨R,  pI Z(R) = RI ⎪ ⎪ ⎩Z(RI ) , RI  R  RI + 1 , I = 1, 2, . . . , n. R

(5.6.8)

Three-segment geometric spreading operator is usually used in GMPEs in ENA. For example, in Atkinson and Boore (1995), R0 =1 km, R1 = 70 km, p1 = 0, R2 = 130 km, and p2 = 0.5 are used. Seismic quality factor Q( F ) is given by Q( F ) = Q0 · F η .

(5.6.9)

For sites in ENA, Q0 = 670 and η = 1/3 in Tables B-4 and B-7 of EPRI-1025287 are usually used. ❧ Site effects G( F ) is given by G( F ) = A( F ) · D( F ),

(5.6.10)

where A( F ) is amplification factor relative to source depth velocity conditions; in practice, amplification factors given in Table 5.3 are usually used (Table 4, Campbell, 2003). D( F ) accounts for the path-independent loss of high-frequency energy in ground motions and can be obtained by (EPRI-1025287) D( F ) = e−πκ0 F ,

(5.6.11)

where the diminution parameter κ0 = 0.006 is used for sites in ENA. An alternative F max filter (Boore, 2003), 

D( F ) = 1 + ( F/ F max )8

−1/2

,

(5.6.12)

can be combined with (5.6.11), and F max = 50 Hz may be used for sites in ENA.

☞

FAS in equation (5.6.1) is Fourier displacement spectrum. When Fouriervelocity or acceleration spectrum is required, equation (5.6.1) is multiplied by I( F ) = (2 iπ F )n ,

(5.6.13)

where n =1, 2 are for Fourier velocity and acceleration spectra, respectively, i.e., . FV (M0 , R, F ) = 2π F FD (M0 , R, F ), FA (M0 , R, F ) = (2πF )2 FD (M0 , R, F ). (5.6.14)

5.6 site response analysis

189

The seismological parameters for the stochastic ground motion model are summarized in Table 5.4 (EPRI-1025287). Given a pair of earthquake magnitude M and hypocenter distance R, and seismological parameters, FAS at reference hard rock can be obtained from equation (5.6.1). Figure 5.18 shows Fourier acceleration spectra at reference distance R0 = 1 km for different earthquake magnitudes, comparing with Figure 5.17 that shows the Fourier displacement spectra. Figure 5.19 shows FAS for several hypocenter distances, given earthquake magnitude

M = 6.5.

5.6.2.2 Developing Acceleration Response Spectra Determining Earthquake Magnitude

M

It is recognized that soil response is governed primarily by the earthquake magnitude

M

and the level of rock motions (NUREG/CR-6728, USNRC, 2001). Bazzurro and

Cornell (2004) demonstrate that soil amplification af ( F ) is virtually independent of

M

earthquake magnitude

except when frequencies F are less than initial resonant

frequency Fsc of soil column. It is required that a sufficient depth be taken so that Fsc  0.5 Hz to ensure that site response has no influence on frequencies greater than 0.5 Hz (EPRI-1025287). Furthermore, sensitivity analysis also shows that the difference in the derived amplification functions for different earthquake magnitudes is minor. Hence, in practice, a representative earthquake magnitude is obtained from seismic hazard deaggregation (SHD) with a mean AEP of 1×10−4 . If the representative earthquake magnitude is very close to

M 6.5, then M 6.5 is used; otherwise, the actual

representative earthquake magnitude is used. Developing Acceleration Response Spectra from RVT Having obtained FAS, random vibration theory (RVT) is often employed to obtain acceleration response spectra (ARS). Consider a single degree-of-freedom (SDOF) oscillator (with circular frequency ω0 and damping ratio ζ0 ) that is mounted on the reference hard rock and under the

excitation of reference hard rock motion in terms of FAS FA (M0 , R, ω ) =

F



u¨hr (t)



of acceleration u¨hr (t). The equation of motion is given by equation (3.3.2), in which the equivalent earthquake load is F(t) = −m u¨hr (t). Taking Fourier transform of both sides of (3.3.2) yields

F





x(t) =



F



F(t)



m (ω02 −ω2 ) + i2ζ0 ω0 ω



=



(−m)

F



u¨hr (t)



m (ω02 −ω2 ) + i2ζ0 ω0 ω



;

190

hence

     X(ω) =  H(ω) FA (M , R, ω ), 0

  in which  X(ω) is FAS of relative displacement x(t), and

(5.6.15)

H(ω) is complex frequency

response function with respect to base excitation of the SDOF oscillator given by equation (3.3.24). Using equations (3.3.14) and (3.3.17), FAS of the absolute acceleration u(t) ¨ = x(t)+ ¨ u¨hr (t) is      U¨ (ω) = ω2  H(ω) FA (M , R, ω ). 0 0

(5.6.16)

Applying equation (3.1.20), the mean-square response of the absolute acceleration is   Trms 2 1 1 ∞ 4  1 2 2 u¨ (t)dt = · ω0 H(ω ) FA2(M0 , R, ω)dω u¨rms = Trms 0 Trms 2π 0  ∞   1 2 = F 4  H( F ) FA2(M0 , R, F )dF, (5.6.17) Trms 0 0 in which

H( F ) =

1 ( F 02 − F 2 ) +

i2ζ0 F 0 F

.

By considering the responses of the oscillator and using results from time-domain numerical simulations, Boore and Joyner (1984) proposed to determine the rootmean-square duration Trms as  Trms = Tgm + To

 κn , κn + α

κ=

Tgm , To

(5.6.18)

where To = 1/(ω0 ζ0 ) is the duration of the SDOF oscillator, and Tgm = Ts +Tp is the duration of ground motion. Trms approaches Tgm and Tgm +To , respectively, for small and large earthquakes. Ts = 1/ F c is the source duration with F c being the corner frequency obtained from equation (5.6.5). Tp is path duration given by, for sites in ENA (Atkinson and Boore, 2006), ⎧ ⎪ 0, ⎪ ⎪ ⎨0.16 (R−10), Tp = ⎪ 9.6 − 0.03 (R−70), ⎪ ⎪ ⎩ 7.8 + 0.04 (R−130),

R  10, 10 < R  70, 70 < R  130, R > 130.

(5.6.19)

In equation (5.6.18), the constants n = 3 and α = 1/3 are used by Boore (2003). Liu and Pezeshk (1999) suggest that n = 2 and α is taken as 



λ2 α = 2π 1 − 1 λ0 λ2

1/2 ,

(5.6.20)

5.6 site response analysis

where

 λK =

∞

0

191

 2 (2π F )K F 04  H( F ) FA2(M0 , R, F )dF,

K = 0, 1, . . . .

(5.6.21)

Having obtained the root-mean-square response u¨rms from equation (5.6.17), peak

acceleration response | u¨ |max or ARS can be determined through the peak factor Pf

SA( F 0 ) = | u¨ |max ≈ Pf · u¨rms .

(5.6.22)

Because it is a double-barrier problem, from equation (3.2.29) the peak factor Pf can be obtained using

| u¨ |max u¨rms

√  ≈ Pf = 2

+∞

0



1 − 1 − I e−z

in which Ne is given by Ne = (2νP ) · Tgm

Tgm = π



2 N e

! dz,

λ4 , λ2

(5.6.23)

(5.6.24)

νP is the rate of occurrence of positive peaks given by equation (3.2.11). Figure 5.20 shows ARS for different hypocenter distances, given earthquake magnitude

M = 6.5.

To cover the range of loading levels, a minimum of eleven expected (median) peak acceleration values at reference hard rock (usually taken at F 0 = 100 Hz) are needed to span from 0.01g to 1.50g (i.e., 0.01g, 0.05g, 0.10g, 0.20g, 0.30g, 0.40g, 0.50g, 0.75g, 1.00g, 1.25g, 1.50g). Given the earthquake magnitude

M

and seismological

parameters, changing the hypocenter distance R can result in the ARS with these eleven peak acceleration values.

5.6.2.3 Developing Input Ground Motions Having obtained ARS in Section 5.6.2.2, input ground motions at reference hard rock can be determined, which can be expressed in frequency-domain or time-domain based on the type of subsequent dynamic response analysis. Frequency-Domain Dynamic Response Analysis An IRVT method is applied to convert ARS to FAS, which are used as the input ground motions at reference hard rock. The IRVT technique proposed by Gasparini and Vanmarcke (1976) and further developed by Rathje and Kottke (2008) gives the square of FAS at F 0 of an SDOF oscillator (with frequency F 0 and damping ratio ζ0 ) as    F 2 0 Trms SA2 ( F 0 ) 1 2   · FA ( F 0 ) ≈  ∞  − FA ( F ) d F , (5.6.25) 2 2 2 P f 0 4  F 0 H( F ) dF − F 0 0

192 Acceleration response spectra (g)

102

M =6.5

R=10

101

R=30 100

R=100

10−1

R=300

10−2 10−3

Frequency (Hz) 0.1

1

10

100

Fourier acceleration spectrum (cm/s2)

Figure 5.20 ARS for several hypocenter distances. 101

M =6.5 R =50

100

Initial FAS FAS after 1 iteration FAS after 5 iterations Final FAS

10−1

Frequency (Hz)

10−2 0.01

0.1

Figure 5.21

Spectral acceleration (g)

100

1

10

100

FAS determined using inverse RVT

UHS (1% in 100 year) ARS (1c model) ARS (2c model)

10−1

UHS anchor to 0.1g ARS (1c) anchor to 0.1g ARS (2c) anchor to 0.1g

10−2

10−3

Frequency (Hz) 0.1

1

10

100

Spectral shape comparison of ARS and UHS

Figure 5.22

in which SA( F 0 ) is the target ARS obtained from equation (5.6.22). It can be shown that  ∞  2 π F0 π = . (5.6.26) F 04  H( F ) dF = F 04 · 3 4ζ0 4ζ0 F 0 0 Hence, equation (5.6.25) can be simplified as    F 0 1 Trms SA2 ( F 0 ) 2 2  · FA ( F 0 ) ≈  π − FA ( F )dF . 2 Pf 2 0 F 0 4 ζ −1 0

(5.6.27)

5.6 site response analysis

193

To solve for FAS, equation (5.6.27) is applied first to determine F( F 0 ) at a sufficient low frequency, e.g., F 0 = 0.01 Hz. At this low frequency, the integral in equation

(5.6.27) is assumed to be zero. The peak factor Pf is assumed to be a preset value, e.g.,

Pf = 2.5. Having obtained F( F 0 ) at 0.01 Hz, FAS values at frequency

(I)

F 0 , I  1, can

be determined by FA2 F 0(I)

 1 Trms  π  ≈ · (I) 2 F0 4 ζ − 1 0

SA2 ( F 0(I) )

Pf 2

−

I 



 FA2 F 0(K−1) · F 0(K) − F 0(K−1)

 .

K=1

(5.6.28) The accuracy of the estimated FAS FA ( F 0 ) is improved iteratively by comparing the

SArvt ( F 0 ) determined from the estimated FAS FA ( F 0 ) using the IRVT technique and the target ARS SA( F 0 ):

ARS

(I) 1. Initial FAS FA F 0 , I  0, is determined by equations (5.6.27) and (5.6.28). 2. Calculate the ARS

SArvt ( F 0 ) associated with the initial FAS using the RVT presented

in Section 5.6.2.2, i.e., equations (5.6.17) and (5.6.22). 3. Determine the correction factor by

C ( F 0) =

SA( F 0 ) . SArvt ( F 0 )

(5.6.29)

Multiplying the FAS by C ( F 0 ) results in a new FAS. 4. Based on the new FAS, new peak factor Pf can be obtained from equations (5.6.21) and (5.6.23); the new ARS

SArvt ( F 0 ) is then determined using equations (5.6.17)

and (5.6.22). 5. Steps 3 and 4 are repeated until one of following conditions is met: • a preset maximum number of iterations (e.g., 30) is reached; • the absolute error of root-mean-square response between | u¨ |max corresponding rvt

to

SArvt ( F 0 ), determined using equations (5.6.17), and | u¨ |max corresponding to

SA( F 0 ) is less than a prespecified tolerance (e.g., 0.005);

• change in the error of root-mean-square response is less than a prespecified value (e.g., 0.001). Figure 5.21 shows FAS determined using the IRVT technique compared with the final FAS, given a pair of

M = 6.5 and R = 50 km.

Time-Domain Dynamic Response Analysis Time-histories spectrum-compatible with the ARS (corresponding to eleven acceleration values at 100 Hz) are generated as the input ground motions at reference hard

194

rock using a suitable generating method, such as the Hilbert–Huang transform method presented in Section 6.3.2.

5.6.2.4 Application In this subsection, input ground motions for a hypothetical site are developed in accordance with the procedure in Sections 5.6.2.1 to 5.6.2.3. Determining ARS PSHA results at the hypothetical site show that the representative earthquake magnitudes obtained from SHD with a mean AEP of 1×10−4 ranges from

M 6.4 to M 6.6 for low and high frequencies, respectively. Because they are very close to M 6.5, M 6.5

is used in the determination of ARS at reference hard rock. Having obtained the earthquake magnitude

M,

two source models, i.e., single-

corner frequency source model (1c) and double-corner frequency source model (2c), may be employed to generate ARS. From SHD results, for intermediate frequency (5 ∼10 Hz), the magnitude

M and source-to-site distance R of the controlling earth-

quake are 6.4 and 15 km, respectively. Inputting

M = 6.4,

R = 15 km, and seismological parameters in Table 5.4, ARS

using two source models are developed, in which the Atkinson and Boore (1995) model is used to calculate corner frequencies for 2c source model. Figure 5.22 shows the sitespecific UHS and ARS using these two source models. It can be seen that the spectral shape of ARS using the 1c source model is much more similar to that of the UHS. Therefore, in this application, the 1c source model is used to generate ARS with peak acceleration values spanning from 0.01g to 1.50g. Given earthquake magnitude

M 6.5, adjusting hypocenter distances R can result in

eleven ARS. A suite of source epicentral distances and depths with respect to eleven ARS are listed in Table 5.5. ARS and FAS determined using the IRVT technique for the eleven acceleration values at 100 Hz are shown in Figures 5.23 and 5.24. Generating Spectrum-Compatible Time-Histories Hilbert–Huang transform method (Section 6.3.2) is applied to generate time-histories compatible with ARS shown in Figure 5.23. As an example, the ARS with an acceleration value of 0.1g at 100 Hz in Figure 5.23 is selected. Compatible time-histories are shown in Figure 5.25. The time increment and duration of the time-histories are 0.005 s and 23 s, respectively. The Arias intensity of the time-history is shown in Figure 5.26. The significant ground motion duration D5 -75 (5 % to 75 % Arias intensity) is 11.2 s, which satisfies the compatibility requirement. Furthermore, the PSD of the time-history is

5.6 site response analysis

195

0.40 0.50 0.75 1.00 1.25 1.50 Acceleration response spectra (g)

0

10

0.30 0.20 0.10

10−1

0.05

0.01

10−2

10−3

Frequency (Hz) 0.1

1

10

100

Figure 5.23 ARS for eleven loading levels.

M =6.5

Fourier acceleration spectrum (cm/s2)

0.40 0.50 0.75 1.00 1.25 1.50 102

0.30 0.20

101

0.10 0.05

10

0

0.01 −1

10

10−2 0.01

0.1

Figure 5.24

1

Frequency (Hz) 10 100

FAS for eleven loading levels.

Table 5.5 Suite of rock peak accelerations, source epicentral distances, and depths

( M 6.5, Single-Corner Source Model) Expected Peak Acceleration ( g) Distance (km) Depth (km) 0.01 210.00 8.00 0.05 58.00 8.00 0.10 38.00 8.00 0.20 23.50 8.00 0.30 16.50 8.00 0.40 12.50 8.00 0.50 9.60 8.00 0.75 4.00 8.00 1.00 0.0 7.00 1.25 0.0 5.75 1.50 0.0 4.85

compared to the minimum PSD requirement (USNRC-0800 SRP Section 3.7.1, USNRC, 2012b), as shown in Figure 5.27. It is seen that the PSD of the time-history satisfies the PSD requirement.

Displace. (cm) Velocity (cm/s) Acceleration (g)

196 0.10 0.05 0 −0.05 −0.10 4 2 0 −2 −4 2 1 0 −1 −2

Time (s) 5

0

10

15

20

Figure 5.25 Time-history spectrum-compatible with target response spectrum.

Arias Intensity (%)

100 75 50 25

Power Spectral Density (m2/s3 )

5 0 0

Time (s) 5 10 15 20 Figure 5.26 Significant duration of Arias intensity.

25

100 10−1 10−2

Time History PSD

80% CEUS-Rock PSD

−3

10

10−4

Frequency (Hz) 10−5 0.1 1 10 100 Figure 5.27 Comparison between time-history PSD and CEUS-rock minimum PSD.

5.6.3 Developing Geotechnical Model In this subsection, the geotechnical model for the site response analysis is determined. The model consists of a sufficient number of horizontal layers characterized by shearwave velocity Vs , dynamic material properties (normalized shear modulus G/Gmax and damping ratio), and density.

5.6 site response analysis

197

EPRI-1025287 (EPRI, 2013) provides a guidance on establishing the geotechnical model, such as how to choose the data and how to capture the epistemic uncertainty and aleatory randomness. The basic data are provided by geologic and geotechnical reports. Generally, the procedure includes five steps: 1. determining layer thickness and boundaries, 2. modelling shear-wave velocity profiles, 3. modelling nonlinear dynamic material properties, 4. modelling layer density profiles, 5. capturing epistemic uncertainty and aleatory randomness in the model.

5.6.3.1

Determining Layer Thickness and Boundaries

Layer boundaries are usually determined based on the site investigation, such as borehole logs; layer thickness and depth to bedrock are determined by measured data. If significant uncertainty exists in the thickness of soil, it should be treated as an epistemic uncertainty. The total thickness of the layers depends on Vs and soil column frequency Fsc . A sufficient depth is required such that Vs reaches 2830 m/s (9200 ft/s) or Fsc reaches 0.5 Hz. Fsc of soil can be estimated by Fsc = 1/Tsc , where Tsc is the natural period of the soil deposit given by (Villaverde, 2009) Tsc = 4

H , Vs

H=

n  I=1

HI ,

(5.6.30)

where H is the total thickness of the soil deposit, HI is the thickness of the Ith layer, n is the number of soil layers, and Vs is averaged shear-wave velocity in the soil deposit which can be determined by Vs =

H , t

t=

n  I=1

tI =

n H  I I=1

Vs, I

,

(5.6.31)

where t is the total time duration that shear-wave propagates in the soil deposit, and Vs, I is the shear-wave velocity in the Ith layer. Combing equations (5.6.30) and (5.6.31) gives the natural period Tsc Tsc = 4t = 4

n H  I I=1

Vs, I

.

(5.6.32)

198

5.6.3.2 Modelling Shear-Wave Velocity Developing Base-Case Shear-Wave Velocity Profile Shear-wave velocity Vs profiles are generally based on measured data from site investigation. For firm rock sites (typically 914  Vs  2438 m/s or 3000  Vs  8000 ft/s) with sparse or very limited data, a constant shear-wave velocity gradient of 0.5 m/s/m (0.5 ft/s/ft) is used to estimate the base-case shear-wave velocity profile. If Vs in soil is unavailable, two approaches to estimate Vs are provided by EPRI1025287 (EPRI, 2013) and PEER (2012). The EPRI approach is based on Vs30 , whereas the PEER approach is based on standard penetration test (SPT) and effective stress σv . EPRI-1025287 recommends that Vs in soil from shallow depths to hard rock can be developed by a suite of profile velocity templates parameterized with Vs30 , shown in Figure 5.28 to a depth of 305 m (1000 ft): For both soil and soft rock sites, the profile with the closest velocities over the appropriate depth range should be adopted from the suite of profile templates and adjusted by increasing or decreasing the template velocities or, in some cases, stripping off material to match the velocity estimates provided. Figure 5.29 illustrates how to use available information to develop a mean base-case shear-wave velocity profile. The available data gives shallow Vs over the upper 30 m (100 ft) with Vs30 = 450 m/s (1475 ft/s); firm rock is present at a depth of 45 m (150 ft) with Vs ≈ 1525 m/s (5000 ft/s). The closest template profile to Vs30 = 450 m/s is the 400 m/s profile. Velocities in the 400 m/s template are scaled by a factor of 450/400 =1.125 to adjust to the desired Vs30 value. The 400 m/s template and the estimate profile are listed in Table 5.6. Because firm rock is at 45 m (150 ft) depth, a discontinuous Vs of 1525 m/s (5000 ft/s) is inserted. Below this depth, Vs increases by 0.5 m/s/m. The equations to estimate Vs recommended by PEER (2012, Table 4.11) are listed in Table 5.7; N60 is SPT blow count corrected for hammer efficiency and σv is in kPa. Epistemic Uncertainty in Shear-Wave Velocity Profile Because limited information is available, there is considerable uncertainty in the shearwave velocity profile. Vs is assumed to be lognormally distributed. The estimate for epistemic uncertainty in Vs is taken as σln , which is applied throughout the profile: ❧ The value of σln is usually taken as 0.35. ❧ For sites where Vs measurements are sparse, e.g., based on inference from geotechnical or geological information rather than geophysical measurements, σln = 0.5.

5.6 site response analysis

0

0

199

Shear-wave velocity (m/s) 1000 1500 2000

500

2500 2830 3000 0

Overly hard rock 100

Residual soils 50

400

200

190

300

Vs =900 m/s

100

270

150

500

Depth (ft)

Depth (m)

400

600 200 700

800

Vs =760 m/s

250

900

Vs =560 m/s 300

0

0

2000

4000 6000 8000 Shear-wave velocity (ft/s) Figure 5.28 Template shear-wave velocity profiles

0

500

Shear-wave velocity (m/s) 1000 1500 2000

Vs30 Measured

1525 m/s 5000 ft/s

9200

2500

3000

2830 m/s 9200 ft/s

400 m/s −40 45m template Estimated firm rock −60 depth

Base-Case Profile

Depth (m)

Depth (m)

−20

Adjusted template

Firm rock gradient

−80 Figure 5.29

Illustration of developing a mean base-case profile.

200 Table 5.6 Base-case shear-wave velocity profile developed by template profiles.

Depth (m) 0.0 1.0 2.1 3.5 4.5 5.6 7.1 8.1 9.2 10.5 11.7 12.8 14.2 15.3 16.9 18.8 21.0 22.6 24.5 26.7 28.4 30.2 32.7 35.6 38.1 40.6 43.1 45.6

Template Shear-Wave Velocity (m/s) 240.0 242.4 259.3 271.6 291.5 303.8 316.1 334.6 346.9 366.8 379.1 396.0 409.9 428.3 440.6 449.9 466.8 485.3 503.7 516.1 534.5 554.5 565.3 577.6 597.6 616.1 628.5 640.8

Estimated Shear-Wave Velocity (m/s) 270.0 272.7 291.7 305.5 328.0 341.8 355.7 376.4 390.2 412.7 426.5 445.6 461.1 481.9 495.7 506.1 525.2 545.9 566.7 580.6 601.3 623.8 636.0 649.8 672.4 693.1 707.0 720.9

Table 5.7 Equations to estimate shear-wave velocity.

All soils

Shear-Wave Velocity Holocene Scaling Pleistocene Scaling (m/s) Factors Factors 0.215 0.275 Vs = 30 N60 σv 0.87 1.13

Clays and silts

0.17 0.32 Vs = 26 N60 σv

Soil Type

Sands

Vs =

Gravels − Holocene

Vs =

Gravels − Pleistocene

Vs =

0.23 0.23 30 N60 σv 0.19 0.18 53 N60 σv 0.17 0.12 115 N60 σv

0.88

1.12

0.90

1.17

−

−

−

−

❧ For sites with an intermediate level of information available, such as a single shearwave velocity profile of high quality or shear-wave velocities inferred from measured compressional-wave velocities, σln = 12 ×0.35 = 0.175.

5.6 site response analysis

201

To represent epistemic uncertainty in shear-wave velocity profiles with a minimum number of cases, three profiles, i.e., 50th-percentile best-estimate (BE), 90th-percentile upper-bound (UB), and 10th-percentile lower-bound (LB), are used, with the weights of 0.4 for BE and 0.3 for LB and UB. Because Vs is assumed lognormally distributed, the 10th- and 90th-percentiles can be obtained by multiplying the median BE profile by a factor a = e σln

−1 (Q)

, for

Q = 10 % and Q = 90 %, respectively. Because Q(0.1) = −1.2816 and Q(0.9) = 1.2816, one has σln = 0.175 :

Q = 10 %, a = 0.80;

Q = 90 %, a = 1.25;

σln = 0.35 :

Q = 10 %, a = 0.64;

Q = 90 %, a = 1.57;

σln = 0.5 :

Q = 10 %, a = 0.53;

Q = 90 %, a = 1.90.

5.6.3.3 Modelling Nonlinear Dynamic Material Properties According to EPRI-1025287, two sets of G/Gmax and hysteretic damping are developed to characterize epistemic uncertainty in material properties for soil (cohesionless soils comprised of sands, gravels, silts, and low plasticity clays) and rock (Cenozoic or Paleozoic sedimentary rocks including shale, sandstones, and siltstones). For soil conditions, EPRI soil curves in EPRI TR-102293-V2 (EPRI, 1993c), which accommodate with more nonlinear soils (Figure 5.30), and Peninsular Range curves (Silva et al., 1996; Walling et al., 2008), which accommodate with more linear soils, are used. The two sets of soil curves are given equal weights. The Peninsular Range curves reflect a subset of the EPRI soil curves, with the 51 to 120 ft (15 to 37 m) EPRI curve applied to the 0 to 50 ft (0 to 15 m) depth range and the EPRI 501 to 1,000 ft (153 to 305 m) curve applied to the 51 to 500 ft (15 to 152 m) depth range. For rock conditions, EPRI rock curves in EPRI-1025287 (Figure 5.31) and linear response are used. In the model of linear response, damping remains constant with cyclic shear strain at input loading levels up to and beyond 1.5g. For all sites where soil and firm rock extended to depth exceeding 150 m (500 ft), linear response can be assumed in the deep portions of profiles. The two sets of material properties are summarized in Table 5.8.

5.6.3.4 Modelling Layer Densities Densities are always measured from site investigation. If measured data are unavailable, a model proposed in EPRI-1025287 based on Vs of the mean base-case profile can be used (Table 5.9). Because densities play a minor role in the site-specific amplification, the density profile is held constant for BE, UB, and LB profiles.

202 1

G/Gmax

0.8 0.6 0.4 0.2 Shear strain (%)

0 30

0.0001

0.001

0.0001

0.001

0.01

0.1

0.01

0.1

1

Damping (%)

25 20 15 10 5 0

Shear strain (%)

Figure 5.30

1

Shear modulus and hysteretic damping curves for cohesionless soil.

1

G/Gmax

0.8 0.6 0.4 0.2 Shear strain (%)

0 0.0001

0.001

0.01

0.1

0.0001

0.001

0.01

0.1

1

Damping (%)

40 30 20 10 0

Shear strain (%)

Figure 5.31

1

Shear modulus and hysteretic damping curves for cohesionless rock.

5.6.3.5 Capturing Aleatory Variability in the Model Aleatory variability in the geotechnical model can be accounted by randomizing (at least 30 times) various parameters of soil properties. Randomizing Layer Thickness and Depth to Bedrock Assumptions of the probability distributions (e.g., normal, uniform, or lognormal) of layer thickness and depth to bedrock are based on measured information, such as data

5.6 site response analysis

203 Table 5.8

Dynamic property values

Layers Soil Rock in top 150 m (500 ft) depth Deeper rock Table 5.9

Curve Set 1 EPRI soil EPRI rock Linear response

Model to estimate density from shear-wave velocity

Shear-Wave Velocity (m/s) < 500 500 to 700 700 to 1500 1500 to 2500 > 2500 Table 5.10

Parameter ρ0  ρ200 D0 b

Curve Set 2 Peninsular range Linear response Linear response

Density (g/cm3 ) 1.84 1.92 2.10 2.20 2.52

Parameters for shear-wave velocity correlation coefficient.

Geomatrix A and B 0.96 13.10 0.96 0.00 0.095

C and D 0.99 8.00 1.00 0.00 0.160

Table 5.11

Vs30 (m/s) 600 m/s) or a very thin veneer (less than 5 m) of soil overlying rock material. B. Shallow (Stiff) Soil Instrument is founded in/on a soil profile up to 20 m thick overlying rock material, typically a narrow canyon, near a valley edge, or on a hillside. C. Deep Narrow Soil Instrument is found in/on a soil profile at least 20 m thick overlying rock material in a narrow canyon or valley no more than several kilometers wide. D. Deep Broad Soil Instrument is found in/on a soil profile at least 20 m thick overlaying rock material in a broad canyon or valley. E. Soft Deep Soil Instrument is found in/on a deep soil profile that exhibits low average shear-wave velocity (Vs < 150 m/s).

in borehole logs. For example, when both properties follow normal distributions, the random values for the Ith layer thickness and depth to bedrock can be determined by

204

HI = H¯ I + εI,1 · σH,I ,

BI = B¯ I + εI,2 · σB,I ,

where H¯ I and σH,I are the mean and standard deviation of the Ith layer thickness, B¯ I and σB,I are the mean and standard deviation of depth of the Ith layer to bedrock, and εI,1 , εI,2 are random values following the standard normal distribution. The random values should be checked by comparing to measured data. The variation in the depth to bedrock is accommodated by adjusting the thickness of the deepest soil layer. Randomizing Shear-Wave Velocity According to EPRI-1025287, random field models are used to generate Vs profiles. The model assumes that Vs at mid-depth of the layer follows lognormal distribution and correlates between adjacent layers. The empirical standard deviation σln of the natural logarithm of Vs is about 0.25 and decreases to 0.15 below 15 m (50 ft). A bound of 2 σln should be imposed throughout the profile, and Vs should be limited to 2830 m/s (9200 ft/s). The random values of Vs, I are given by Vs, I = Vs e ZI σln ,

(5.6.33)

where Vs, I is the Ith random value of Vs , and ZI is a number limited between −2 and 2 and generated from Z1 = ε1 ;

 ZI = ZI−1 · ρ + εI 1−ρ 2 ,

I > 1,

(5.6.34)

in which εI is a random value following the standard normal distribution. ρ is the correlation coefficient and is a function of depth D and thickness H of the layer: 



ρ(D, H) = 1 − ρD (D) ρH (H) + ρD (D),

(5.6.35)

where ρH and ρD are the thickness-dependent and depth-dependent correlations: ρH (H) = ρ0 e

H −

,

⎧  D + D B ⎨ρ 0 , D < 200 m, 200 200 + D ρD (D) = 0 ⎩ ρ200 , D  200 m.

(5.6.36)

B and D0 are parameters given in Geomatrix and Vs30 classifications. The parameters are listed in Table 5.10 and Geomatrix classification is given in Table 5.11 (Toro, 1995). Randomizing Dynamic Material Properties According to EPRI-1025287, aleatory variability in dynamic material properties is accommodated by randomizing G/Gmax and hysteretic damping curves. Lognormal distributions are assumed with σln of 0.15 and 0.3 for G/Gmax and hysteretic damping, respectively, at a cyclic shear strain of 0.03 %. A bound of 2 σln is applied. The random

5.6 site response analysis Table 5.12

205 BE shear-wave velocity profiles.

Shear-Wave BE Profile (P1) LB Profile (P2) UB Profile (P3) Velocity Top of Depth to Top of Depth to Top of Depth to (ft/s) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) 760 45 0 45 0 45 0 2880 19 26 32 13 8 37 7940 9 36 22 23 −2 47 8040 −191 236 −178 223 −202 247 8140 −391 436 −378 423 −402 447 8240 −591 636 −578 623 −602 647 8340 −791 836 −778 823 −802 847 8440 −991 1036 −978 1023 −1002 1047 8540 −1191 1236 −1178 1223 −1202 1247 8640 −1391 1436 −1378 1423 −1402 1447 8740 −1591 1636 −1578 1623 −1602 1647 8840 −1791 1836 −1778 1823 −1802 1847 8940 −1991 2036 −1978 2023 −2002 2047 9040 −2191 2236 −2178 2223 −2202 2247 9140 −2391 2436 −2378 2423 −2402 2447 9200 −2511 2556 −2498 2543 −2522 2567 Table 5.13

UB shear-wave velocity profiles.

Shear-Wave BE Profile (P1) LB Profile (P2) UB Profile (P3) Velocity Top of Depth to Top of Depth to Top of Depth to (ft/s) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) 940 45 0 45 0 45 0 3600 19 26 32 13 8 37 9130 9 36 22 23 −2 47 9200 −131 176 −118 163 −142 187

curves are generated by computing the change of G/Gmax and hysteretic damping at 0.03 % cyclic shear strain and applying this factor at all strains. The factor should be reduced near the end of the range to preserve the general shape of the base-case curves. Hysteretic damping at 0.03 % shear strain should not exceed 15 % in applications. G/Gmax and hysteretic damping curves can be developed by G/Gmax, I = G/Gmax e0.15 εI,1 ,

DI = D e0.3 εI,2 ,

(5.6.37)

where G/Gmax, I and DI are the Ith random modulus reduction and hysteretic damping, respectively, and εI,1 and εI,2 are independent standard normal random numbers.

5.6.3.6 Application Determination of Layer Thickness and Boundaries Three soil models are assumed for the hypothetical site: 1. Profile P-1: A best-estimate (BE) base-case, in which the thickness of the soil layer is 26 ft, representing the average value.

206 Table 5.14 LB shear-wave velocity profiles.

Shear-Wave BE Profile (P1) LB Profile (P2) UB Profile (P3) Velocity Top of Depth to Top of Depth to Top of Depth to (ft/s) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) Layer EL (ft) Top (ft) 590 45 0 45 0 45 0 2300 19 26 32 13 8 37 6900 9 36 22 23 −2 47 7000 −191 236 −178 223 −202 247 7100 −391 436 −378 423 −402 447 7200 −591 636 −578 623 −602 647 7300 −791 836 −778 823 −802 857 7400 −991 1036 −978 1023 −1002 1047 7500 −1191 1236 −1178 1223 −1202 1247 7600 −1391 1436 −1378 1423 −1402 1447 7700 −1591 1636 −1578 1623 −1602 1647 7800 −1791 1836 −1778 1823 −1802 1847 7900 −1991 2036 −1978 2023 −2002 2047 8000 −2191 2236 −2178 2223 −2202 2247 8100 −2391 2436 −2378 2423 −2402 2447 8200 −2591 2636 −2578 2623 −2602 2647 8300 −2791 2836 −2778 2823 −2802 2847 8400 −2991 3036 −2978 3023 −3002 3047 8500 −3191 3236 −3178 3223 −3202 3247 − − −3391 3436 −3378 3423 8600 − − − − 8700 −3578 3623 9200 −3496 3541 −3671 3716 −3352 3397

2. Profile P-2: An upper-bound (UB) base-case, in which the soil layer is 13 ft, representing the average value between 0 ft and 20 ft. 3. Profile P-3: A lower-bound (LB) base-case, in which the soil layer is 37 ft, representing the average value between 20 ft and 60 ft. The weights are 0.4 for BE profile and 0.3 for both UB and LB profiles. In general, with the increase of depth of bedrock, the fluctuations in Vs decrease, indicating that the thickness of sublayer in the bedrock can take a relative large value. In this example, 200 ft is taken as the thickness of sublayer in bedrock. The total thickness can be determined after establishing the Vs profiles. Modelling of Shear-Wave Velocity In this example, it is assumed that site investigation provides detailed information about Vs ; hence σln is taken as 0.175. The 10th- and 90th-percentiles can be obtained by multiplying the mean base-case profile by 0.80 and 1.25, respectively. The profiles are presented in Tables 5.12 to 5.14. As discussed in Section 5.6.3.1, the depth of soil deposit requires that either shearwave velocity Vs of the soil layer reaches 9200 ft/s or resonant frequency Fsc of the

5.6 site response analysis

207

soil column approaches 0.5 Hz. It is seen in Tables 5.12 and 5.13 that, for BE and UB shear-wave velocity profiles, Vs reaches 9200 ft/s at relative shallow depths. However, for LB shear-wave velocity profile, when Vs = 8700 ft/s, the natural period T1 of the soil column of P-1 profile is given by   10 200 200 105 26 + + = 2.045s, +···+ + T1 = 4 6900+7000 8500+8600 8600+8700 590 2300 2

2

2

giving that Fsc of the soil column approaches 0.5 Hz before Vs reaches 9200 ft/s. Therefore, for P-1 profile in LB shear-wave profile, the depth of soil deposit is taken as 3436 ft. Similarly, for both P-2 and P-3 profiles in LB shear-wave velocity profile, Fsc approaches 0.5 Hz before Vs reaches 9200 ft/s. Modelling Nonlinear Dynamic Material Properties EPRI soil/rock curves, Peninsular Range curves, and Stokoe curves (Stokoe et al., 2003) are used to represent the uncertainty in G/Gmax and damping of soil and rock in the top 500 ft depth. Specifically, for the soil above the weathered rock, EPRI soil curves and Peninsular Range curves are used with equal weights; for the weathered rock, EPRI rock curves and Stokoe curves are used with equal weights; for the competent rock in the top 500 ft, Stokoe curves and 1 % linear response are used with equal weights. Stokoe curves are presented in Tables 5.15 and 5.16. Dynamic properties for the hypothetical site are listed in Table 5.17. Note that 1 % is the minimum damping ratio in Stokoe curves. Modelling Layer Densities The unit weight of soil and rock is assumed to be 135 pound per cubic foot (pcf) and 165 pcf, respectively. Poisson’s ratios are 0.4 for soil and 0.3 for rock. Capturing Aleatory Variability The soil layer thickness, the depth to bedrock, shear-wave velocity Vs , and dynamic material properties are randomized for 60 times to capture aleatory variability. The soil thickness is assumed to follow a uniform distribution. The randomized soil thickness values of P-1 profile are shown in Figure 5.32. To capture aleatory variability in Vs , Vs30 is required. For BE P-1 profile, there are 26 ft soil on the top, 10 ft weathered rock below the soil, and 98.4−26−10 = 62.4 ft competent rock beneath the weathered rock. Hence, the total time period that shear-wave propagates in the top 30 m (98.4 ft) and Vs30 are determined as t=

26 10 + + 750 2880

1 2

62.4  = 0.046 s, × 7940+(7940+0.5×62.4) 

(5.6.38)

208 Table 5.15

Stokoe G/Gmax − log10 γ curve and D − log10 γ curve for weathered shale.

Shearing Strain γ ( %) 1.0 ×10−6 5.0 ×10−6 1.0 ×10−5 5.0 ×10−5 1.0 ×10−4 1.7 ×10−4 3.4 ×10−4 5.0 ×10−4 1.0 ×10−3 2.0 ×10−3 3.0 ×10−3 5.0 ×10−3 7.0 ×10−3 1.0 ×10−2 Table 5.16

Normalized Shear Modulus G/Gmax 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.95 0.91 0.83 0.77 0.67 0.59 0.50

Damping Ratio D ( %) 3.04 3.04 3.05 3.10 3.16 3.25 3.46 3.65 4.23 5.29 6.25 7.91 9.32 11.05

Stokoe G/Gmax − log10 γ curve and D − log10 γ curve for unweathered shale.

Shearing Strain γ ( %) 1.0 ×10−6 1.0 ×10−5 1.0 ×10−4 1.0 ×10−3 3.0 ×10−3 5.0 ×10−3 1.0 ×10−2 2.0 ×10−2 3.0 ×10−2 4.0 ×10−2 5.0 ×10−2 6.0 ×10−2 7.0 ×10−2 8.0 ×10−2 8.5 ×10−2 1.0 ×10−1 1.5 ×10−1 2.0 ×10−1 Table 5.17

Layers

Normalized Shear Modulus G/Gmax 1.00 1.00 1.00 1.00 0.99 0.98 0.95 0.91 0.87 0.83 0.80 0.77 0.74 0.71 0.70 0.67 0.57 0.50

Damping Ratio D ( %) 1.00 1.00 1.01 1.06 1.19 1.31 1.61 2.19 2.73 3.25 3.74 4.21 4.66 5.08 5.29 5.88 7.59 9.02

Dynamic property values for the hypothetical site.

Curve Set 1 Curve Set 2 Soil EPRI soil Peninsular range Weathered rock EPRI rock Stokoe weathered rock Competent bedrock in top 500 ft Stokoe unweathered rock Linear 1 % Deeper competent rock Linear 1 % Linear 1 %

5.6 site response analysis

0

209

0

1

2

0

1

2

Shear-wave velocity (×1000 ft/sec) 3 4 5 6 7

8

9

10

8

9

10

Depth (ft)

20 40 60 80 0

3

4

5

6

7

500

Depth (ft)

1000

1500

2000

2500

3000 Figure 5.32

Realizations of shear-wave velocity for BE P-1 Profile.

1

G/Gmax

0.8 0.6 0.4 0.2 0 30

0.0001

0.001

0.01

0.1

1

0.1

1

Damping (%)

25 20 Damping ratio shall not exceed 15%

15 10 5 0 0.0001

0.001

0.01

0.03

Shear strain (%)

Figure 5.33

Realizations of shear modulus and hysteretic damping curves for 0–20 ft soil.

210

Vs30 =

98.4 = 2139 ft/s = 652 m/s. 0.046

(5.6.39)

Knowing Vs30 , the parameters from Table 5.10 are used to randomize the Vs values in accordance with the procedure in Section 5.6.3.5. Note that the random value of ZI is between −2 and 2 for soil and weathered rock, whereas it ranges from −2 to 0.056 for competent rock to ensure that the Vs  9200 ft/s. Vs in all BE profiles reaches 9200 ft/s at depth of 2556 ft. Figure 5.32 shows 60 realizations of Vs for BE P-1 profile. When randomizing G/Gmax and hysteretic damping curves, the random factor is adjusted at both beginning and end of the curves to preserve the general shape of the base-case curve. The adjustment of the random factor for G/Gmax curves is illustrated in the following; hysteretic damping curves are adjusted similarly. ❧ For the random factor larger than 1, G/Gmax is taken as 1 when it exceeds 1. The random factor is taken as 1 at the end of the curves, and the potion over 1 is reduced by 50 % (e.g., the random factor is reduced to 1.1 from the original value of 1.2) for the neighbouring two points to the end to keep the curves from sudden change. ❧ For the random factor smaller than 1, G/Gmax is taken as the minimum value on the curve when it is smaller than the minimum value. The random factor is taken as 1 at the beginning, and the portion below 1 is reduced by 50 % (e.g., the random factor is increased to 0.9 from the original value of 0.8) for the neighbouring two points to the beginning. The random curves of EPRI soil from 0 ft to 20 ft are shown in Figure 5.33. In this case, the bound of ε for damping is set at 1.684 so that the damping ratio at shear strain 0.03 % does not exceed 15 %.

5.6.4 Developing Foundation Input Response Spectra In this subsection, the procedure to obtain the amplification functions af is presented first. Seismic hazard curves at reference hard rock and af are convolved to calculate the seismic hazard curves at free surface or foundation level. Finally, foundation input response spectra (FIRS) in the horizontal and vertical directions are determined.

5.6.4.1 Amplification Function

 Amplification function af is defined as the ratio of spectral acceleration SAs F  a on  soil to SAhr F  a at reference hard rock, given the amplitude of rock motion a, i.e.,  af F  a =



SAs F  a  . SAhr F  a

(5.6.40)

5.6 site response analysis Earthquake Magnitude

Point Source Model

Shear Wave Velocity Profile

211 Depth to Bedrock

Shear Modulus and Damping

Random Profiles

Amplification Function

Lower range (0.3) Single corner (0.5)

m =6.5

Best estimate (0.4)

Lower range (0.3)

Upper range (0.3)

Best estimate (0.4)

Double corner (0.5)

Upper range (0.3)

Curve set 1 (0.5)

Generate 30 random profiles

Analyze 30 profiles

Curve set 2 (0.5)

Figure 5.34 An example logic tree for site response analysis.

Figure 5.34 gives an example logic tree for determining site-specific amplification functions af , in which there are 36 epistemic branches accounting for epistemic uncertainty and aleatory randomness in seismic source model (Section 5.6.2) and sitespecific geotechnical model (Section 5.6.3). For each branch, there are a minimum of 30 random realizations generated to capture aleatory randomness in the model. Recall that eleven ground motion levels are considered to account for the influence of ground motion intensities on af . Therefore, there are a total of 36×30×11 =11880 analyses required to calculate af . The outputs of the analyses are response spectra or timehistories and af at a specific elevation level (such as free surface or foundation level), and strain-compatible soil properties. Site response analysis can be conducted by time-history analysis method with the input ground motions being acceleration time-histories, or RVT method with the input ground motions being FAS. Based on the shear modulus reduction and hysteretic damping curves, strain-compatible soil properties can be obtained after several iterations starting at initial values. These soil properties are then treated as small strain shear modulus Gmax and 5 % damping ratio. In the recent past, several computer programs, e.g., SHAKE91, SHAKE2000, RVT-SHAKE, DEEPSOIL, and Strata, have been developed to conduct site response analysis. For embedded structures analyzed as surface structures,

SAs

 F  a is equal to Trun-

cated Soil Column Response (TSCR) (DC/COL-ISG-017, USNRC, 2010). In order to determine TSCR, the soil column with no truncation is carried out first to develop strain-compatible soil properties. The layers corresponding to the embedment depth are then removed and site response analysis is conducted for a second round based on the strain-compatible soil properties without iteration. The response spectra of the truncated surface is TSCR. Furthermore, a lower bound of 0.5 is implemented for  af F  a (EPRI-1025287).  af F  a is usually assumed to be lognormally distributed given the amplitude of rock motion a (EPRI-1025287). For the Ith epistemic branch, the logarithmic mean

212 Table 5.18 BE, UB, and LB soil properties.

Soil Properties G

BE exp(μln, g )

UB exp(μln, g + σln, g )

LB exp(μln, g − σln, g )

D

exp(μln, d )

exp(μln, d + σln, d )

exp(μln, d − σln, d )

 μI and logarithmic standard deviation σI of af F  a can be obtained from the set of at least 30 random profiles. Then the total μ ln af  a and σ ln af  a for each frequency given the amplitude of rock motion a are calculated by    

2 2  = (5.6.41) wI · μI , σ ln wI · μI − μ ln af  a + σI2 , μ ln af  a = af  a I

I

where wI is the weight for the Ith epistemic branch. Hazard-Consistent Strain-Compatible Material Properties (HCSCP) Hazard-consistent strain-compatible material properties (HCSCP) are developed sim  ilarly to af F  a . They are also assumed lognormally distributed and use the same  weight for each epistemic branch as for af F  a . For each branch, hazard consistent μln, I and σln, I are corresponding to the rock motion a, which is determined by interpolating site-specific seismic hazard curves with respect to a specific AEP level, such as 1×10−4 . The final median μln and logarithmic standard deviation σln of HCSCP are computed using equation similar to (5.6.41). The coefficient of variation (CoV) should be no less than 0.5 for well investigated  sites, giving σln = ln(1+0.5 2 ) = 0.472, and 1 for not well-investigated sites, giving  σln = ln(1+1 2 ) = 0.833. BE, UB, and LB soil properties are obtained from μln and σln as in Table 5.18. Meanwhile, the shear modulus for LB should meet the design demand for foundation settlement under static loads, and the UB shear modulus should be no less than the values at low strain in BE. The damping ratio is bounded to 15 %. To examine consistency in strain-compatible properties across structural frequency, HCSCP are evaluated at PGA (typically 100 Hz) and at low frequency (typically 1 Hz). If the differences in properties at high frequency and low frequency are less than 10 %, the high-frequency properties are used because this frequency range typically has the greatest impact on soil nonlinearity. If the difference exceeds 10 % , the HCSCP developed at PGA and at 1 Hz may be combined with equal weights. HCSCP are developed as a function of bedrock SAhr ( F ) and are used as input for SSI analysis.

5.6.4.2

Horizontal Foundation Input Response Spectrum

Horizontal Foundation Input Response Spectrum (FIRS) is obtained from seismic hazard curves at foundation level. NUREG/CR-6728 (USNRC, 2001) provides four

5.6 site response analysis

213

Annual frequency of exceedance

10−1 10−2 −3

10

H1 H2 Hi

Seismic hazard curve

Hi–1 Hi+1

10−4 Hn–1 −5

10

Hn ai–1

10−6 0.001

0.01 Figure 5.35

ai+1

1 an–1 aU

aL a2 ai 0.1 Spectral acceleration (g)

10

Discretization of seismic hazard curve.

approaches to obtain seismic hazard on soil, among which Approach 3 is widely used in practice. It calculates seismic hazard on soil by convolving amplification functions and seismic hazard curves at reference hard rock, i.e.,  ∞  !  z  HZs (z) = P af > a  m, r, a FM, R  A (m, r  a) hAhr (a)dm dr da, 0

(5.6.42)

r m

where af is the amplification function depending on m, r, and a. a is the amplitude of  rock motion, and FM, R  A (m, r  a) is the joint probability density function of M and R given the amplitude a of rock motion. hAhr (a) is the absolute value of the derivative of the seismic hazard at reference hard rock. Bazzurro and Cornell (2004) demonstrate that af is virtually independent of earthquake magnitude M and source-to-site distance R when frequencies F are greater than initial resonant frequency Fsc of soil column, i.e., F

 Fsc .

In practice, Fsc  0.5 Hz is

usually satisfied (see Section 5.6.3.1). Hence, equation (5.6.42) can be simplified as  HZs. (z)

∞

= 0

P

   ∞ z  ! d HAhr (a)  z  ! hr f da = af > a  a  P a > a  a hA (a) da. (5.6.43) da  0

In analysis, GMP a is usually truncated at a lower-bound value aL and an upperbound value aU with respect to preset AEP levels. For example, as shown in Figure 5.35, aL = 0.026g is truncated at AEP of 1×10−2 , and aU = 2.206g is truncated at

AEP of 1×10−5 . Hence, equation (5.6.43) is expressed as  a U z  ! s HZ (z) = P af > a  a hAhr (a) da. aL

(5.6.44)

214

Discretizing GMP a into aL = a1 < · · · < aI < · · · < an = aU , equation (5.6.44) can be approximated by . HZs (z) ≈

 all aI

P af >

 z  ! hr z  ! hr f h (a ) a = P a > a  I I I A aI aI  aI pA (aI ),

(5.6.45)

all aI

where pAhr (aI ) is the probability that bedrock motion has amplitude aI and is given by, referring to Figure 5.35,

⎧ 1 ⎪ (H − H2 ), ⎪ ⎪ ⎨2 1 pAhr (aI ) = hAhr (aI ) aI = 12 (HI−1 − HI+1 ), ⎪ ⎪ ⎪ ⎩ 1 (H − H ), 2

n−1

n

I = 1, 2  I  n−1,

(5.6.46)

I = n.

Because the absolute value of the slope of seismic hazard curve increases with the increase of GMP a, to make the values pAhr (aI ) in different intervals comparable, the lengths of intervals can be set non-uniformly, i.e., the lengths of intervals decrease with z  ! the increase of GMP a. P af > a  aI can be determined by assuming that af is I lognormally distributed given the amplitude aI of rock motion ⎫ ⎧

⎨ ln z/aI − μ ln af  a ⎬ z  ! I , (5.6.47) P af > a  aI = 1 −

 ⎭ ⎩ σ I  ln af aI

where μ ln af  a and σ ln af  a are median value and standard deviation of ln af given I I the amplitude aI of rock motion, which can be obtained from equation (5.6.41). EPRI-1025287 adopts equations (5.6.45) and (5.6.47) to calculate seismic hazard curves on soil. In practice, amplification functions can be obtained from site response analysis (Section 5.6.4.1); seismic hazard curves at reference hard rock are given in PSHA reports for the site of interest. Note that PSHA reports usually provide seismic hazard curves for only several representative frequencies. These frequency points are too sparse to construct the entire horizontal FIRS spanning from 0.1 Hz to 100 Hz. Therefore, UHS at reference hard rock from PSHA reports are needed to define seismic hazard curves using several discrete points for different AEP levels. Once amplification functions and seismic hazard curves at reference hard rock for a number of frequencies (e.g., 30) have been obtained, seismic hazard curves at foundation level can be calculated using equation (5.6.45). Horizontal FIRS for a specific AEP, e.g., 1×10−4 , can then be determined by interpolating seismic hazard curves at foundation level for a number of frequencies.

5.6 site response analysis

215

5.6.4.3 Vertical Foundation Input Response Spectrum Vertical FIRS is usually developed by multiplying the V/H ratios to the horizontal FIRS. NUREG/CR-6728 provides recommended V/H ratios for sites in the ENA (Table 4-5, USNRC, 2001), in which V/H ratios are given for frequencies from 0.1 Hz to 100 Hz as functions of PGA ranges (< 0.2g, 0.2g to 0.5g, and > 0.5g), as shown in Figure 5.36. Reference site conditions for the ENA are defined in terms of average shear-wave velocity (Vs30 ) over the upper 30 m of the earth and the diminution parameter κ0 corresponding to shallow crustal damping over the upper 1 km to 2 km. Diminution parameter κ0 for reference hard rock in the ENA is 0.006 s. Site-specific κ0 can be obtained from site investigation. ❧ If the site-specific κ0 is close to 0.006 s, recommended V/H ratios in Figure 5.36 can be used with respect to the PGA range. ❧ Otherwise, empirical V/H ratios from GMPEs in the western United States (WUS) should also be applied to calculate V/H ratios for the site of interest. The weights for GMPEs are based on experience and judgement. ❧ A transfer function should be used to adjust the difference in ground motion between two region types (equation (4-9), page 4-11, NUREG/CR-6728, USNRC, 2001). The V/H ratios are shifted to higher frequencies so that the peak corresponds to about 60 Hz. Meanwhile, a factor of 1.5 should be used to adjust the ratios for low frequencies less than 3 Hz. Finally, a smoothed function incorporating recommended V/H ratios in Figure 5.36 with weighted average V/H ratios from GMPEs is obtained to represent the site-specific V/H ratios.

5.6.4.4 Application Amplification Function As shown in Figure 5.37, six epistemic branches are developed to account for epistemic uncertainty in soil profiles and soil property curve sets. For each branch, there are a minimum of 60 random realizations generated to capture aleatory randomness in the geotechnical model. Because eleven ground motion levels are considered to account for the influence of ground motion levels on af , there are a total of 6×60×11 = 3960 analyses required. In this example, computer software SHAKE91 is used to perform the site response analysis. Response spectra at the ground surface (EL+45 ft) and strain-compatible soil properties for all realizations are obtained when the analyses are complete. Because the building structures at the site are assumed to be treated as surface structures in the

216

Recommended V/H ratios

1.6

> 0.5g

1.4 1.2

0.2 ~ 0.5g

1.0

< 0.2g

0.8 0.6

Frequency (Hz) 0.1

1

Figure 5.36 Sesimic Source Model

10

100

Recommended V/H ratios for CEUS rock site condition. Soil Profile

Shear Modulus and Damping

Random Profiles

Curve set 1 (0.5)

Generate 60 random profiles

Amplification Function

Lower range (0.3) Single corner (m =6.5)

Best estimate (0.4) Upper range (0.3)

Figure 5.37

Analyze 60 profiles

Curve set 2 (0.5) Logic tree for site response analysis at hypothetical site.

subsequent SSI analyses, the horizontal FIRS at the foundation level (assumed to be EL+25 ft) should be taken as the TSCR. To determine the TSCR, site response analyses need to be conducted for a second round (strain-compatible soil properties are taken from the values of the first round) based on truncated soil column (without the soil above foundation level, EL+25 ft). The response spectra from the second round is taken to calculate the af at foundation level. μI and σI of af are calculated for each case given the ground motion and frequency. The cumulative distribution functions of empirical distribution and lognormal distribution for af given the ground motion with PGA of 1g and frequency of 10 Hz is shown in Figure 5.38, which shows that the lognormal distribution is well adapted to describe af . μ ln af  a and σ ln af  a are then determined using equation (5.6.41). Following the procedure in Section 5.6.4.1, af at three percentiles (16th, 50th, and 84th percentiles) for two elevation levels (EL+45 ft and +25 ft) are determined and are shown in Figures 5.39 and 5.40. μ ln af  a and σ ln af  a of af given eleven loading levels at EL+45 ft and EL+25 ft are presented in Figures 5.41 to 5.44. Horizontal GMRS and FIRS Having obtained af and seismic hazard curves at reference hard rock, GMRS and FIRS for a specific AEP level can be determined.

Cumulative probability

5.6 site response analysis

217

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0

0.2

Case 1 1

1.5

2

2.5

3

3.5

0

4

1 1

Cumulative probability

1 0.8

0.8

0.6

0.6

0.4

0.4

0.2

Cumulative probability

1

1.5

2

2.5

1

0.8

0.8

0.6

0.6

0.4

0.4

0 0.5

1.5

2

2.5

4

3

3.5

0

5

Case 4 1

1.5

0.2

Case 5 1

3

0 0.5

3

1

0.2

2

0.2

Case 3

0 0.5

Case 2

2

Case 6 1

2

3

Amplification factor

4

Amplification factor

Lognormal distribution Empirical distribution 16th percentile 50th percentile 84th percentile

Figure 5.38 Table 5.19

Cumulative distribution functions of amplification function.

Calculation of

SAs ( F ) at 100 Hz with respect to AEP of 1 × 10−4

Loading

aI ( g)

pAhr (aI )

μ ln af  a

1 2 3 4 5 6 7 8 9 10 11

0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.75 1.00 1.25 1.50

7.67×10−3 1.33×10−3 8.09×10−4 2.92×10−4 1.30×10−4 6.17×10−5 7.28×10−5 2.64×10−5 1.14×10−5 6.05×10−6 1.19×10−5

0.452 0.533 0.471 0.418 0.291 0.220 0.190 0.0978 0.129 0.025 0.079

I

σ ln af  a 0.248 0.184 0.148 0.135 0.136 0.162 0.159 0.145 0.226 0.235 0.213

I

5

5.5

Amplification function

218 3.5

3.5

3.5

3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1.5

1

1

1

Amplification function

0 0.1

0.01 g 1

10

100

0 0.1

0.05 g 1

10

100

0 0.1

3.5

3.5

3

3

3

2.5

2.5

2.5

2

2

1.5

1.5

1

1

0.1g 1

10

100

2 1.5

0.5 1

0 100 0.1

10

1 0.5

0.5

0.2 g

0 0.1

Amplification function

0.5

0.5

0.5

0.3g 1

10

0.4 g 0 100 0.1

3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

1

0.75g 0 100 0.1

10

10

100

0.5

0.5 g 0 0.1

1

1

0 100 0.1

10

1g 1

10

Amplification function

Frequency (Hz) 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

84th percentile 50th percentile 16th percentile

0.5

1.25 g 0 0.1

1

10

Frequency (Hz)

0 100 0.1

1.5g 1

10

100

Frequency (Hz)

Figure 5.39 Amplification functions for eleven loading levels at surface EL + 45 ft.

100

Amplification function

5.6 site response analysis

219

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0.01g

Amplification function

0.1

1

100

10

0.1g 0

0

0

0.1

1

100

10

0.1

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0.2g 0 1

10

0.4g

1

10

100

0.1

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0.5g 0.1

1

10

100

100

10

0 0.1

100

0

1

0.3g 0

0.1

Amplification function

0.05g

1

10

1g

0.75 g 0 0.1

100

0 1

10

100

0.1

1

10

100

Amplification function

Frequency (Hz) 1.5

1.5

1

1

84th percentile 50th percentile 16th percentile

0.5

0.5

1.25g 0 0.1

1

10

Frequency (Hz)

100

1.5g 0 0.1

1

10

100

Frequency (Hz)

Figure 5.40 Amplification functions for eleven loading levels at foundation EL + 25 ft.

220

2 Amplification factor

10 Hz 1.8 100 Hz

1.6 1.4

5 Hz

2.5 Hz

0.5 Hz

1.2

1 Hz

1 0.01

Logarithmic standard deviation

25 Hz

0.1 Loading level a (g) Figure 5.41 Median amplification factors at EL + 45 ft.

0.6

1

2

10Hz

0.5 0.4

25Hz

0.3 5Hz 0.2 0.1 0

0.01

2.5Hz

100Hz

0.5 Hz

1Hz 1

0.1 Loading level a ( g) Figure 5.42 Logarithmic standard deviations at EL + 45 ft.

2

Amplification factor

1.4

5Hz

1 100Hz

0.8 0.6 0.4

Logarithmic standard deviation

0.5Hz

1.2

1Hz

2.5Hz

10Hz

25Hz 0.01

0.1 Loading level a ( g) Figure 5.43 Median amplification factors at EL + 25 ft.

1

2

0.14 0.12

0.5Hz 25Hz

0.1

100Hz

0.08 1Hz

0.06 0.04 0.02 0.01

10Hz

2.5 Hz

5Hz

0.1 Loading level a ( g) Figure 5.44 Logarithmic standard deviations at EL + 25 ft.

1

2

5.6 site response analysis

221

SAs ( F ) at frequency 100 Hz on the free surface with an AEP of 1×10−4 is determined. From PSHA reports, SAhr ( F ) = 0.58g In order to better demonstrate the approach,

at 100 Hz with an AEP of 1×10−4 can be obtained. The amplification functions for 100 Hz at eleven loading levels are evaluated and are shown in Figure 5.39. Therefore,

SAs ( F ) can be determined according to equations (5.6.45) and (5.6.47). More details are given in Table 5.19. Finally, solving z in equation (5.6.45) gives

SAs ( F ) on free surface corresponding to

AEP of 1×10−4 , i.e., 1×10−4 =

11 

P af >

I=1

z  ! hr aI  aI pA (aI ).

In this example, SAs ( F ) at free surface and foundation levels with an AEP of 1×10−4 are developed for frequencies 0.25, 0.5, 1, 2.5, 5, 10, 15, 25, 33, and 100 Hz, respectively. If the median seismic hazard curves are used, one obtains the median GMRS and FIRS, whereas if the mean seismic hazard curves, one obtains the mean GMRS and FIRS. The mean and median horizontal GMRS and FIRS are shown in Figure 5.45. Vertical FIRS V/H ratios from GMPEs in the WUS is applied to calculate the site-specific V/H ratios. Three GMPEs are usually used, e.g., Abrahamson and Silva (1997), Bozorgnia and Campbell (2004), and Gulerce and Abrahamson (2011). An example smooth function as shown in Figure 5.46 is used. The vertical FIRS is obtained from the horizontal FIRS and site-specific V/H ratios and is shown in Figure 5.45. HCSCP Based on the strain-compatible soil properties obtained from SHAKE91, μln and σln of shear modulus G and damping D are computed for the six epistemic branches (BE, UB, LB three soil cases each with two curve sets, as shown in Figure 5.37) at eleven ground motion levels. They are functions of

SAhr ( F ) at 100 Hz and 1 Hz, respectively.

The μln and σln of the first soil layer obtained at 100 Hz are shown in Figures 5.48 to 5.51, in which the values corresponding to

SAhr ( F = 100Hz) are marked,

whereas the

results obtained at 1 Hz are shown in Figures 5.52 to 5.55. The weight for each epistemic branch is the same as that in determining the amplification function, i.e., 0.2 for BE with Curve 1 or Curve 2, and 0.15 for UB or LB with Curve 1 or Curve 2. By using equation (5.6.41), μln and σln considering epistemic uncertainty are calculated for 100 Hz and 1 Hz as shown in Table 5.20. For example,

222 1.6

Mean horizontal GMRS at surface EL+45 ft

Spectral acceleration (g)

1.4

Median horizontal GMRS at surface EL+45 ft

1.2 1 Mean horizontal FIRS at foundation EL+25 ft

0.8

Median horizontal FIRS at foundation EL+25 ft

0.6

Mean vertical FIRS at fWoundation EL+25 ft

0.4 0.2 0

Frequency (Hz) 1

0.25

Figure 5.45

10

100

Horizontal GMRS and FIRS for the hypothetical site.

V/H ratio

0.95 0.90 0.85 0.80 Frequency (Hz) 40 1 10 Figure 5.46 V/H ratios for the hypothetical site.

Annual frequency of exceedance

0.75 0.1

100

10−1 10−2 10−3

1 Hz

100 Hz

10−4 10−5 10−6 0.001

0.1 0.12

0.01 Figure 5.47

0.58

1

PGA (g)

5

Mean hazard curves.

μln and σln for G at 100 Hz in the first layer are determined as μln =

σln

6  I=1

wI · μln, I

= 0.2 · 7.30+0.2 · 7.54+0.15 · 7.48+0.15 · 7.64+0.15 · 6.68+0.15 · 7.14 = 7.31,   6  1/2   2 2 = wI · (μln, I − μln ) + σln, I I=1

=











0.2 · (7.30−7.31)2 +0.592 + · · · + 0.15 · (7.14−7.31)2 +0.452 = 0.62.

5.6 site response analysis

223

The results of μln and σln considering epistemic uncertainty are calculated for 100 Hz and 1 Hz as shown in Table 5.20. For the hypothetical site, the site investigation data is assumed not to be detailed for soil layer and weathered rock layer; σln is no less than 0.833. For rock layer, the CoV should be no less than 0.5, giving σln of 0.472, based on the small variations in this layer. Then the BE, UB, and LB soil properties at 100 Hz and 1 Hz are computed according to Table 5.18; the values of G at these two frequencies are listed in Table 5.21. The corresponding values of shear modulus G at 100 Hz and 1 Hz in Table 5.21 are compared. If the difference between them is less than 10 %, the value at 100 Hz is selected. Otherwise, the average of the two values is used. The damping ratio D is obtained similarly. The shear modulus G is always expressed by Vs by  Vs = G/ρ =⇒ G = ρVs2 ,

(5.6.48)

and Vs is limited to the value of 9200 ft/s in bedrock. The HCSCP of shear-wave velocity Vs and damping ratio D are shown in Figures 5.56 and 5.57.

5.6.5 Conclusions In this Section, a general procedure for site response analysis in ENA is presented. Generally, there are two approaches to perform the analysis: 1. Time-history analysis method • The input ground motions are spectrum-compatible time-histories for eleven ground motion levels. • The outcome of the analyses are the amplification functions af and timehistories at a specific elevation level, and strain-compatible soil properties. 2. RVT method • The input ground motions are FAS for eleven earthquake ground motion levels. • The outcome of the site response analyses are af and FAS at a specific elevation level, and strain-compatible soil properties. Prior to performing site response analysis, input ground motions at reference hard rock and site-specific geotechnical model are determined. For sites in ENA, reference hard rock is defined as bedrock with shear-wave velocity Vs of 2830 m/s (9200 ft/s). There are two available point-source models (single-corner frequency source model and double-corner frequency source model) to develop the input ground motions at reference hard rock in terms of FAS or spectrum-compatible time-histories. The weights for these two source models are based on experience and judgement. For each

Logarithmic mean of G

224 7.8 7.6 7.4 7.2 7.0

7.64 7.54 7.48

0.01

Logarithmic standard deviation of G

0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30

Loading level (g) 0.58

1

2

1

2

Logarithmic mean of shear modulus at 100 Hz.

BE (Curve set 1) BE (Curve set 2) UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2)

0.60

0.59 0.58

0.50 0.49 0.45

0.01

0.1

Figure 5.49 −1.5

−2.0 −3.0 −3.5

Loading level (g) 0.58

Logarithmic standard deviation of shear modulus at 100 Hz.

BE (Curve set 1) BE (Curve set 2) UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2)

−2.5 Logarithmic mean of D

6.68

0.1

Figure 5.48

−2.32 −2.75 −3.13

−2.79 −3.15 −3.47

−4.0 −4.5 −5.0

0.01

0.1

Figure 5.50 Logarithmic standard deviation of D

7.14

BE (Curve set 1) BE (Curve set 2) UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2)

6.8 6.6 6.4 6.2 6.0 5.8

7.30

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

Loading level (g) 0.58

1

2

1

2

Logarithmic mean of damping ratio at 100 Hz. 0.44

BE (Curve set 1) BE (Curve set 2)

0.42

0.42 0.34 0.33

UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2) 0.01

Figure 5.51

0.1

Loading level (g) 0.58

Logarithmic standard deviation of damping ratio at 100 Hz.

Logarithmic mean of G

5.6 site response analysis 7.8 7.6 7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8 0.02

7.67 7.60 7.57 7.45 7.24

Logarithmic standard deviation of G

0.45 0.40 0.35 0.30

6.95

BE (Curve set 1) BE (Curve set 2) UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2)

Figure 5.52 0.80 0.75 0.70 0.65 0.60 0.55 0.50

225

0.1 0.12 Loading level (g) Logarithmic mean of shear modulus at 1 Hz.

BE (Curve set 1) BE (Curve set 2) UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2)

0.5

0.54 0.53 0.53 0.47 0.46 0.41

0.1 0.12

0.02

Figure 5.53

Loading level (g)

0.5

Logarithmic standard deviation of shear modulus at 1 Hz.

Logarithmic mean of D

−1.5 BE (Curve set 1) BE (Curve set 2) LB (Curve set 1) LB (Curve set 2)

−2.0 −2.5 −3.0

−3.41

−3.5

−3.10 −3.42 −3.73

−4.0 −4.5 −5.0

0.1 0.12

0.02

Figure 5.54 Logarithmic standard deviation of D

−2.63 −2.98

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

UB (Curve set 1) UB (Curve set 2)

Loading level (g)

0.5

Logarithmic mean of damping ratio at 1 Hz.

BE (Curve set 1) BE (Curve set 2)

0.42 0.36 0.31

0.40

0.39 0.36

UB (Curve set 1) UB (Curve set 2) LB (Curve set 1) LB (Curve set 2) 0.1 0.12

0.02

Figure 5.55

Loading level (g)

Logarithmic standard deviation of damping ratio at 1 Hz.

0.5

226 Table 5.20

μln and σln at 100 Hz and 1 Hz.

Layer Values at 100 Hz Number Shear Modulus Damping Ratio (I ) μln σln μln σln 1 7.31 0.62 −2.94 0.64 2 10.21 0.77 −3.04 0.56 3 12.26 0.24 −4.55 0.19 4 12.29 0.24 −4.52 0.20 5 12.32 0.24 −4.52 0.21 6 12.37 0.23 −4.72 0.21 7 12.40 0.23 −4.72 0.21 8 12.43 0.23 −4.72 0.21 9 12.45 0.23 −4.72 0.21 10 12.48 0.23 −4.72 0.21 11 12.51 0.22 −4.72 0.21 12.53 0.22 −4.72 0.21 12 13 12.56 0.22 −4.72 0.21 14 12.58 0.22 −4.72 0.21 15 12.67 0.26 −4.72 0.19 16 12.69 0.25 −4.72 0.19 17 12.70 0.25 −4.72 0.19 18 12.72 0.24 −4.72 0.19 19 12.74 0.23 −4.72 0.19 20 12.75 0.22 −4.72 0.19 21 12.83 0.19 −4.72 0.19 Table 5.21

Layer Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

BE, UB, and LB for G at 100 Hz and 1 Hz (unit: psi).

BE At 100 Hz 1517.46 27491.65 211563.14 217406.65 223518.92 236470.09 243021.77 249661.95 256390.66 263207.86 270113.60 277107.82 284190.57 291361.86 317284.79 323001.76 328747.49 334521.79 340324.37 346155.09 374221.42

Values at 1 Hz Shear Modulus Damping Ratio μln σln μln σln 7.42 0.55 −3.21 0.60 10.26 0.74 −3.16 0.53 12.26 0.24 −4.57 0.19 12.29 0.24 −4.55 0.20 12.32 0.24 −4.54 0.20 12.37 0.23 −4.72 0.21 12.40 0.23 −4.72 0.21 12.43 0.23 −4.72 0.21 12.45 0.23 −4.72 0.21 12.48 0.23 −4.72 0.21 12.51 0.22 −4.72 0.21 12.53 0.22 −4.72 0.21 12.56 0.22 −4.72 0.21 12.58 0.22 −4.72 0.21 12.67 0.26 −4.72 0.19 12.69 0.25 −4.72 0.19 12.70 0.25 −4.72 0.19 12.72 0.24 −4.72 0.19 12.74 0.23 −4.72 0.19 12.75 0.22 −4.72 0.19 12.83 0.19 −4.72 0.19

At 1 Hz 1675.60 28480.23 211677.67 217632.78 223721.67 236470.09 243021.77 249661.95 256390.66 263207.86 270113.60 277107.82 284190.57 291361.86 317284.79 323001.76 328747.49 334521.79 340324.37 346155.09 374221.42

UB At 100 Hz At 1 Hz 3490.47 3854.24 63236.54 65510.48 339177.48 339361.09 348545.77 348908.31 358344.95 358669.99 379108.23 379108.23 389611.87 389611.87 400257.38 400257.38 411044.84 411044.84 421974.15 421974.15 433045.42 433045.42 444258.53 444258.53 455613.57 455613.57 467110.57 467110.57 508670.15 508670.15 517835.58 517835.58 527047.12 527047.12 536304.46 536304.46 545607.14 545607.14 554954.94 554954.94 599950.80 599950.80

LB At 100 Hz 659.70 11951.81 131963.25 135608.16 139420.71 147499.05 151585.69 155727.52 159924.58 164176.83 168484.31 172846.98 177264.86 181737.98 197907.50 201473.48 205057.40 208659.14 212278.52 215915.45 233421.93

At 1 Hz 728.46 12381.58 132034.69 135749.21 139547.18 147499.05 151585.69 155727.52 159924.58 164176.83 168484.31 172846.98 177264.86 181737.98 197907.50 201473.48 205057.40 208659.14 212278.52 215915.45 233421.93

Depth (ft)

5.6 site response analysis

0 0

227

2000

Shear-wave velocity (ft/sec) 4000 6000 8000

10000

20 40 0 0

BE

LR 2000

4000

6000

UR 8000

10000

BE

UR

500 1000

Depth (ft)

1500 LR

2000 2500 3000 3500 3820 Figure 5.56

0 0

BE, UB, and LB shear-wave velocities.

0.02

0.04

Damping ratio 0.06 0.08

0. 1

0.12

0. 1

0.12

Depth (ft)

20 40 LB

60

BE

80 100 0 0

UB 0.02

0.04

0.06

0.08

500

Depth (ft)

1000 1500 2000 2500

LB BE UB

3000 3500 3820 Figure 5.57

BE, UB, and LB damping ratios.

228

source model, a minimum of eleven input ground motions ranging from 0.01g to 1.5g are required to account for the influence of ground motion intensities on site responses. A site-specific geotechnical model, which consists of soil profiles, shear-wave velocity profiles, and nonlinear dynamic material properties, is then developed. In order to capture epistemic uncertainty and aleatory randomness in the model, logic trees are usually used. A number of epistemic branches are developed to account for epistemic uncertainty; whereas randomization is applied to capture aleatory randomness. Having obtained the input ground motions and site-specific geotechnical model, site response analysis can be performed using a suitable computer software. af and strain-compatible soil properties are two important products of the analysis. af is used to calculate seismic hazard curves at a specific elevation (e.g., free surface or foundation level). The strain-compatible soil properties are used to conduct SSI analysis. Seismic hazard curves at a specific elevation level are obtained by convolving seismic hazard curves at reference hard rock with af at the specified elevation level. Horizontal GMRS and FIRS are then determined by interpolating the seismic hazard curves at free surface level and foundation level for an AEP level, e.g., 1×10−4 . Vertical GMRS and FIRS can be obtained by multiplying the horizontal GMRS and FIRS, respectively, by the site-specific V/H ratios. ❧

❧

❧ Probabilistic seismic hazard analysis (PSHA) provides a framework, in which aleatory and epistemic uncertainties in the locations, the sizes, and the rates of occurrence of earthquakes, and the variation of ground-motion characteristics with earthquake size and location can be identified, quantified, and combined in a mathematically rigorous manner to describe the seismic hazard at a given site in terms of one ground motion parameter, such as PGA. ❧ PSHA is extended to vector-valued PSHA when multiple ground motion parameters are considered. ❧ Seismic hazard deaggregation (SHD) analysis gives the contributions of parameters (such as earthquake magnitude, source–site distance, and the rate of occurrence) of earthquake scenarios to the seismic hazard at a given site. The results of a SHD in terms of earthquake magnitude and source–site distance are used in the selection of recorded ground motions for seismic analysis. ❧ Uniform hazard spectrum (UHS) is one of the most important products of PSHA.A UHS generally represents an envelope over different earthquakes that contributes to the seismic hazard at the site of interest. Despite the controversy on the UHS for rep-

5.6 site response analysis

229

resenting multiple earthquakes at different frequencies and possessing broadband spectral shape, it is still considered one of the most suitable seismic inputs for seismic analysis and design in the nuclear power industry because it provides sufficient margin in the seismic design and analysis of an NPP due to its low probability level and broadband spectral shape. ❧ PSHA gives the seismic input at the base hardrock underneath the site of interest. The objectives of site response analysis are to assess the effects of soil conditions on the ground shaking due to the propagation of shear waves in the soil deposit, and to obtain UHS at specific elevation levels of interest (such as ground level and foundation level).

C

H

6 A

P

T

E

R

Ground Motions for Seismic Analysis Seismic response history analysis (SRHA) is a major seismic response analysis method for seismic design verification and seismic qualification of structures, systems, and components (SSCs) of a nuclear power plant (NPP; USNRC-0800 SRP 3.7.1, USNRC, 2012b; ASCE 4-98,ASCE, 1998; ASCE 43-05,ASCE/SEI, 2005; CSA N289.3, CSA, 2010a). The SRHA procedure is concerned with the determination of structural response as a function of time when the structural system is subjected to a given ground acceleration. In general, the resulting structural response time-histories are then used: ❧ as direct seismic inputs for seismic design and analysis of secondary SSCs, ❧ for obtaining design quantities, such as internal forces, combined with those induced by other loading cases, for seismic design verification and qualification of primary structures, ❧ for generating floor response spectra (FRS) for seismic design and analysis of secondary SSCs. Hence, representative input earthquake ground motions, which comprehensively reflect local seismic hazard and site conditions, are required for this analysis method.

6.1 Generating Ground Motions for Seismic Analysis There are generally three types of earthquake ground motions for SRHA: ❧ linearly scaled recorded earthquake ground motions, ❧ spectrum-compatible artificial/synthetic earthquake ground motions, 230

6.1 generating ground motions for seismic analysis

231

❧ spectrum-compatible ground motions based on recorded ground motions. A linearly scaled recorded ground motion is obtained by multiplying a constant factor throughout the entire recorded ground-motion time-history without modifying its frequency contents. This type of ground motion is usually applied in design and analysis of civil structures, in which the response spectrum of the ground motions closely matching the seismic design spectrum is not required. An earthquake ground motion, whose response spectrum closely matches a seismic response spectrum over a range of vibration frequencies of engineering concern, is usually called a spectrum-compatible earthquake ground motion. The use of spectrumcompatible ground motions, instead of linearly scaled recorded ground motions, is attractive for multiple reasons (Carballo and Cornell, 2000): ❧ they are able to produce structural responses that present relatively lower dispersion; ❧ there are only a small number of recorded ground motions available for many regions in the world. In seismic design and analysis of nuclear structures, the input earthquake ground motions are required to be compatible with the design response spectra. Furthermore, a number of earthquake and ground-motion parameters of the input earthquake ground motions need to satisfy the acceptance criteria. These acceptance criteria, including spectral compatibility, are (USNRC-0800 SRP 3.7.1, USNRC, 2012b; ASCE 4-98, ASCE, 1998; ASCE 43-05, ASCE/SEI, 2005; CSA N289.3, CSA, 2010a): 1. Critical Damping Ratios. The determination of critical damping ratios, at which the response spectra are calculated from the generated ground motions, depends on the damping ratios of target response spectra. The target response spectra, especially those based on PSHA, are usually defined at a critical damping ratio of 5 %. The 5 % damped response spectra should then be calculated from the generated ground motions when the spectral compatibility is checked. In certain circumstances, a check of spectral compatibility for all damping ratios in the seismic response analysis may be required (USNRC-0800 SRP 3.7.1). 2. Frequency Interval. When the response spectra of the generated ground motions are calculated, the frequency intervals at which the spectral values are determined need to be sufficiently small, such that the frequency contents of the ground motions can be adequately captured. CSA N289.3 requires a minimum of 100 points at the frequency intervals from 0.1 Hz to 34 Hz. More stringent criterion is specified in ASCE 43-05 that spectral accelerations should be computed at 100 points per

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frequency decade, uniformly spaced over the logarithmic frequency scale, from 0.1 Hz to 50 Hz or the Nyquist frequency. 3. Spectral Compatibility. In seismic design and analysis of NPPs, acceleration has been recognized as one of the most relevant seismic damage potential parameters and has been widely used in analyzing SSCs in an NPP. Thus, seismic design acceleration spectra are often used as the seismic design basis. As a result, the generated ground motions are primarily defined by the seismic design acceleration spectra; in other words, the acceleration response spectra of the generated ground motions should be compatible with the target design acceleration spectra. In general, a ground motion is considered spectrum-compatible when ❧ its response spectrum does not fall more than 10 % below the target response spectrum and does not exceed the target response spectrum by more than 30 % at each frequency computed in the frequency range of interest; ❧ no more than nine adjacent frequency points should fall below the target response spectrum to prevent response spectrum in large frequency windows from falling below the target response spectrum; ❧ the average of the ratios of response spectrum of the ground motion to the target spectrum frequency by frequency is equal to or greater than 1 (USNRC0800 SRP 3.7.1; ASCE 43-05). CSA N289.3 has a similar criterion that no more than 6 % of the total number of frequency points should fall below the target spectrum, and no point on the response spectrum should fall below the target spectrum by more than 10 %. 4. Power Spectral Density (PSD). PSD of an earthquake ground motion describes the strength of variations (power or energy) present in the ground motion as a function of frequency. It is required that PSD of the generated ground motion be shown to not have any significant gaps in energy at any frequency computed in the frequency range of interest (USNRC-0800 SRP 3.7.1; ASCE 4-98; ASCE 43-05; CSA N289.3). USNRC-0800 SRP 3.7.1 also provides guidance on obtaining the minimum PSD functions for R.G. 1.60 design spectra and other design spectra. These minimum PSD functions are used as the target PSD, and the averaged one-sided PSD of the generated ground motion should exceed 70 % or 80 % of the target PSD depending on which target PSD is used. Studies (USNRC-0800 SRP 3.7.1) have shown that the generated ground motions produce PSD functions having a quite different appearance (fluctuations) from one individual function to another, even when all these ground motions are generated

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so as to closely match the same seismic design spectra. These fluctuations on the energy of the ground motion indicate its lack of frequency contents, which could result in no resonant amplification on structural responses at certain frequencies when subjected to this ground motion. As a result, it may produce unconservative results for the response of SSCs. While the compatibility of the seismic design spectra is the primary acceptance criterion, a check of PSD is considered a secondary one to guard against unwanted power dips in the input ground motions. 5. Time Interval. The generated ground motion should have a sufficiently small time interval to be able to adequately capture the frequency contents of the motion and to accommodate the maximum frequency of interest involved in the numerical structural response analysis. It is required that the time interval of the ground motion be set to produce a Nyquist frequency above the maximum frequency of interest (USNRC-0800 SRP 3.7.1; ASCE 4-98; ASCE 43-05; CSA N289.3). USNRC0800 SRP 3.7.1 and ASCE 43-05 also requires a minimum Nyquist frequency of 50 Hz, which corresponds to a maximum time interval of 0.01 second. 6. Time Duration. The generated ground motion should have sufficiently long total and strong-motion durations (USNRC-0800 SRP 3.7.1; ASCE 4-98; ASCE 43-05; CSA N289.3). The durations should be consistent with the expected duration based on the site-specific seismic hazard and should also meet the minimum requirements specified by the codes and standards to ensure adequate design margin. CSA N289.3 requires a minimum total duration of 15 seconds and a minimum strong-motion duration of 6 seconds, in which the strong-motion duration is defined as the time span between 5 % and 95 % of the Arias intensity. USNRC-0800 SRP 3.7.1 has a more stringent requirement that the total duration should be at least 20 seconds and the strong-motion duration should be at least 6 seconds, in which the strong-motion duration is defined as the time span between 5 % and 75 % of the Arias Intensity. 7. Statistical Independence. The target response spectra for nuclear structures are usually defined for ground motions in two orthogonal horizontal and one vertical directions. Tridirectional ground motions should then be generated to be compatible with the target response spectrum in the corresponding direction. To preserve the nature of the recorded ground motions that the correlation coefficients between three orthogonal components of one set of ground motions are generally very small, i.e., statistically independent numerically, these correlation coefficients for the generated ground motions should also be kept adequately small. To be considered statistically independent, CSA N289.3 and ASCE 43-05 require that

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the absolute value of the correlation coefficients does not exceed 0.3. USNRC-0800 SRP 3.7.1 has a more stringent requirement that the absolute value of the correlation coefficients does not exceed 0.16. 8. Earthquake Parameters. Real recorded earthquake ground motions are considered complicated; they are influenced by, and consequently reflect, characteristics of the seismic source, the rupture process, the source–site travel path, and local site conditions. Recorded earthquake ground motions contain a wealth of information about the nature of the earthquake, carry all the ground-motion characteristics (amplitude, frequency, energy content, duration, and phase characteristics), and reflect all the factors that influence earthquake motions (characteristics of the source, path, and site). Among those earthquake parameters, earthquake magnitude and distance have been recognized as the primary parameters that influence the characteristics of the ground motions. In engineering practice, it is convenient to characterize them with a small number of parameters; however, such characterizations can never be complete (Kramer, 1996). When the ground motions are generated based on the recorded ground motions, the selection of the seed recorded motions should be based on the identification of dominant earthquake magnitude and distance pairs that impact the site-specific target response spectra (USNRC-0800 SRP 3.7.1; ASCE 43-05). 9. Appearance of Ground-Motion Time-Histories. In the generation of spectrumcompatible ground motions, the majority of the parameters that are taken into consideration is in frequency domain due to high variability in time domain of the ground motions. However, some critical time domain criteria should still be satisfied. USNRC-0800 SRP 3.7.1 requires that the Arias Intensity of the ground motion grow uniformly with time, which is consistent with the nature of most of the recorded ground motions. To ensure that the relations between the acceleration, velocity, and displacement ground motions in time domain are compatible with each other, the ratios V/A and AD/V2 (A, V, and D are peak ground acceleration, velocity, and displacement, respectively) should be consistent with the characteristic values for the earthquake magnitude, distance, and local site conditions of the controlling earthquake events defining the site-specific target response spectra (USNRC-0800 SRP 3.7.1; ASCE 43-05). In addition, it should also be demonstrated that the displacement time-history does not drift. 10. Multiple Sets of Ground Motions. It has been well recognized that the results from the SRHA may vary significantly, mainly due to the substantial variations in

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the input ground motions and the induced responses of the structures. In this case, the use of multiple sets of ground motions for analysis and design of SSCs is acceptable. To obtain statistically meaningful results, an adequate number of input ground-motion sets should be used depending on the types of SRHA, such as linear or nonlinear response analysis (USNRC-0800 SRP 3.7.1; ASCE 43-05). As required in USNRC-0800 SRP 3.7.1 and ASCE 43-05, the criteria for spectral compatibility and PSD check should be satisfied by using the results of the average of these multiple sets of ground motions, and all other criteria should be met for each of the ground motions in these sets. As summarized earlier, the generated ground motions have to satisfy a number of acceptance criteria. Individual code or standard may have slightly different acceptance criteria and different combinations of criteria. The primary purpose of these acceptance criteria is to ensure that (1) the earthquake and ground-motion parameters of the generated motions that are most relevant to seismic design and analysis are properly considered in the ground-motion generation procedure; (2) the values of these parameters of the generated motions comprehensively reflect the local seismic hazard and site conditions; (3) adequate margin is introduced into the generated ground motions from the point of view of engineering. In this chapter, some existing approaches for generating spectrum-compatible earthquake ground motions are introduced briefly. Several spectral matching algorithms, which have been used in seismic design and analysis for NPPs, are presented in detail and illustrated with numerical examples.

6.2 Spectral Matching Algorithms for Artificial Ground Motions Several studies have been conducted for generating spectrum-compatible artificial or synthetic earthquake ground motions. One of the most classical attempts is to use a Fourier series representation to generate ground motion based on random vibration theory. An iterative process is then applied to the Fourier series to refine the result to match the target response spectrum. In these studies, various envelope or shape functions are used to characterize approximately the nonstationary properties of real recorded ground motions (Scanlan and Sachs, 1974; Levy and Wilkinson, 1976; Vanmar-

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cke and Gasparini, 1977; King and Chen, 1977; Preumont, 1980; Preumont, 1984; Spanos and Loli, 1985). Synthetic earthquake ground motions can be generated from seismological source models by accounting for path and site effects. Atkinson and Boore (1998) simulated earthquake ground motions consistent with the earthquake magnitudes and distances that contribute relatively most strongly to hazard at the selected sites and probability level. These simulated motions match the short- and long-period ranges of the target uniform hazard spectra (UHS), respectively. These simulations for local and regional crustal earthquakes are based on a point-source stochastic simulation procedure (Preumont, 1984). Iyengar and Rao (1979) attempted to generate spectrum-compatible artificial ground motions from a random process without resorting to the PSD function. This approach with random phases, amplitudes, and signs ensures that the generated response spectra are at least greater than the target velocity response spectra. Giaralis and Spanos (2009) used a stochastic dynamics solution based on wavelet technique to obtain a family of simulated nonstationary earthquake motions whose response spectrum is on the average in good agreement with the target displacement response spectrum.

6.2.1 Fourier-Based Spectral Matching Algorithms As discussed earlier, a number of studies have been conducted on the topic of generating spectrum-compatible artificial earthquake ground motions. In current engineering practice, commercial or in-house developed softwares on the generation of artificial ground motions usually incorporate different options for certain approach into one algorithm, which provides options for the users to meet different engineering needs. In this section, the basic concept of a popular Fourier-based spectral matching algorithms is presented. Various options involved in the Fourier-based approach on the shape functions, frequencies, determination of amplitudes and phases, signs, iteration process, and derivation of orthogonal components are then discussed. Based on the fact that any periodic function can be expanded into a series of sinusoidal waves, i.e., a Fourier series, an earthquake ground-motion acceleration timehistory can be expressed as X(t) = F(t)

n  I=1

(−1)I AI sin(2π F I t+φI ),

(6.2.1)

where F(t) is the shape function, AI , F I , and φI are the amplitude, frequency, and phase, respectively, of the Ith sinusoidal wave, and n is the total number of frequencies considered in the Fourier expansion (6.2.1).

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In a generation procedure, F(t), F I , φI , and n are usually predefined. The ultimate step in the procedure is to determine the amplitude AI such that the response spectrum of X(t) closely matches the target response spectrum. This step is achieved using an iteration process. A spectrum-compatible ground motion can thus be generated. To simulate in part the nonstationary characteristics of real recorded ground motions, a shape function F(t) is usually used to be multiplied by the stationary motion as shown in equation (6.2.1). The shape function could be the positive envelope over a recorded ground motion, which is positive at all times and piecewise linear (Levy and Wilkinson, 1976). To be more simplified and more representative for seismic analysis and design, a trapezoidal function is often used and has been adopted in ASCE 4-98 (ASCE, 1998). Figure 6.1 shows the shape function where tr , tm , and td depend on earthquake magnitude as specified in ASCE 4-98. F(t) tr

tm

td

1

t (s)

0 Figure 6.1 Trapezoidal shape function.

The term (−1)I in equation (6.2.1) is used to flip the sinusoidal wave about the time axis at every other frequency, equivalent to a phase shift by 180◦ . It has been observed that this term improves the solution in the iteration process (Levy and Wilkinson, 1976). The determination of the frequencies F I and the number of the frequencies n in equation (6.2.1) is dependent on the duration and time interval of the ground motion that is to be simulated. Based on signal processing (Appendix B), for a typical ground motion having a duration of 20 second and time interval of 0.005 second, the lowest frequency F 1 , the highest frequency F n (n = 2000), and the frequency interval for equation (6.2.1) are 0.05 Hz, 100 Hz, and 0.05 Hz, respectively. The phases φI in equation (6.2.1) are usually obtained by assuming that the phases follow a uniform distribution between 0 and 2π (Vanmarcke and Gasparini, 1977). Random numbers are then generated to produce the phase for each sinusoidal wave of each ground-motion component. To preserve the phase information of the recorded ground motions, in the generation procedure, the phases φI can be determined using the phase values of the recorded motions selected. It has been recognized that the phases of the ground motion are key parameters affecting the correlations between the three

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orthogonal components of the ground motion. Using random phases usually produces adequately small correlation coefficients between the three orthogonal ground-motion components (Levy and Wilkinson, 1976); using phases from the real recorded ground motions also results in satisfactory correlations as these are the intrinsic properties of the recorded motions. To determine the amplitudes AI through iteration, the initial values of AI , denoted (0)

(0)

as AI , need to be derived first. One attempt was to take the initial values AI

to

be proportional to the corresponding target response spectral values at frequencies F I (Levy and Wilkinson, 1976). Using the initial trial ground motion, a response spectrum is computed and compared with the target response spectrum. In the (K+1)th iteration, (K+1)

the new amplitudes AI

(K)

are obtained by multiplying the previous Kth values AI

the ratio of the target response spectral values

SA( F I ) at frequencies

F I to that

by

S

(K) A ( F I)

computed for the previous Kth trial ground motion, i.e., (K+1)

AI

(K)

= AI ·

SA( F I ) . SA(K) ( F I )

(6.2.2)

This iteration process is continued until the computed response spectrum is close enough to the target response spectrum. An alternative approach for determining the initial values of AI is to link them to the PSD function (Vanmarcke and Gasparini, 1977). Based on the definition of the PSD in (0)

equation (3.1.15), the initial amplitudes AI is given by  (0) AI = 2 S( F I ) F,

(6.2.3)

where F is the frequency interval and S( F I ) is the PSD function that can be derived based on the target response spectrum and the peak factor as discussed in Section 3.2. (0)

Having obtained the initial amplitudes AI , AI can be finally determined through the same iterative process as presented previously.

Numerical Example To illustrate the Fourier-based spectral matching algorithm introduced in this section, a numerical example is presented in the following. The standard-shape seismic design spectra for rock site with 5 % critical damping specified in CSA N289.3 are used as the target response spectra. The horizontal component of the target spectra has a PGA of 0.25g, and the vertical component is two-thirds of the horizontal component. To simulate the nonstationary characteristics of the ground motions, the trapezoidal function, as shown in Figure 6.1 with tr = 2 s, tm = 15 s, and td = 5 s, is used. The target response spectra and the corresponding response spectra of the generated ground motions are shown in Figures 6.2 and 6.5. The resulting response spectrum

6.3 spectral matching algorithmsbased on recorded ground motions

239

for each direction closely matches the target response spectrum. Figures 6.3, 6.4, and 6.6 show the generated ground-motion time-histories. It is seen that the nonstationary characteristics of the generated motions, i.e., the rise time at the beginning of the motion, the duration of the strong motion, and the decay time in the end of the motion, have been simulated through the shape function. It is noted that baseline correction may be required for this algorithm to ensure that the velocity and displacement time-histories do not unrealistically drift. A check of statistical independence between any two components of the generated motions is also needed if the phases are randomly produced.

6.3 Spectral Matching Algorithms Based on Recorded Ground Motions Various methods have been developed for modifying recorded ground motions such that the resulting response spectra are compatible with the target response spectra. Tsai (1972) selected an existing recorded ground motion, the response spectrum of which generally matches the target spectrum. The recorded motion was then passed successively through certain frequency-suppressing filters to reduce the spectrum wherever necessary. Similarly, sinusoidal motions were superposed over the selected motion to increase the spectrum as required. Several researchers used a very similar technique. However, they found it convenient to work in the frequency domain rather than with the time-histories in the time domain. The technique is to scale the Fourier amplitudes or phases of recorded motions such that the resulting response spectrum is compatible with the target spectrum (Rizzo et al., 1975; Kost et al., 1978; Silva and Lee, 1987). However, Fourier-based techniques do not account for the instantaneous variations in the frequency contents of an earthquake ground motion, which arise due to the arrivals of different types of seismic waves at different time-instants and due to the phenomenon of dispersion in these waves. In order to simulate the nonstationary characteristics of earthquake ground motions, a wavelet-based procedure has been used. It decomposes a recorded earthquake ground motion into a desired number of time-history components with non-overlapping frequency contents; each component is then suitably scaled for matching the response spectrum of the modified recorded motion with a target response spectrum (Mukherjee and Gupta, 2002). An alternative method of preserving the nonstationary characteristics of earthquake ground motions is to adjust recorded ground motions by adding wavelet functions in the time domain to match the target response spectra. It is based on the assumption

240

Spectral acceleration (g)

0.8 0.6 0.4

Target Horizontal motion 1

0.2 0.0

Horizontal motion 2 Frequency (Hz) 0. 1 Figure 6.2

1 10 Response spectra (Horizontal).

100

A(t) (g)

0.3 0.0 −0.3 V(t) (cm/s)

20 0 −20 −40 D(t) (cm)

10 0 −10

Time (s) 0

5

10 15 20 25 30 Figure 6.3 Generated ground motions (Horizontal 1).

35

0

5

10 15 20 25 30 Figure 6.4 Generated ground motions (Horizontal 2).

35

40

A(t) (g)

0.3 0.0

V(t) (cm/s)

−0.3 30 0 −30 D(t) (cm)

15 0 Time (s) −15

40

6.3 spectral matching algorithmsbased on recorded ground motions

241

Spectral acceleration (g)

0.6

0.4

Target Vertical motion

0.2

Frequency (Hz)

0.0 0.1

1 10 Figure 6.5 Response spectra (Vertical).

100

A(t) (g)

0.2 0.0 −0.2 V(t) (cm/s)

20 0

D(t) (cm)

−30 10 0 Time (s) −10

0

5

10 Figure 6.6

15 20 25 30 Generated ground motions (Vertical).

35

40

that the time at which the spectral response of a time-history occurs is not perturbed by a small adjustment of the time-history (Kaul, 1978; Lilhanand and Tseng, 1988; Lee and Kim, 1999; Choi and Lee, 2003; Hancock et al., 2006; Atik and Abrahamson, 2010). To properly preserve the frequency contents and nonstationary characteristics of recorded ground motions, a signal processing method called Hilbert–Huang transform (HHT) has been applied in generating spectrum-compatible earthquake ground motions (Ni et al., 2010; Ni et al., 2011a; Ni et al., 2011b; Ni et al., 2013). To consider a large number of recorded ground motions simultaneously, a neuralnetwork-based technique has been proposed. It uses the decomposing capabilities of Fourier or wavelet packet transform on recorded ground motions and the learning abilities of stochastic neural network to expand the knowledge of the inverse mapping

242

from target response spectrum to the generated earthquake ground motion (Ghaboussi and Lin, 1998; Lin and Ghaboussi, 2001; Lin et al., 2006; Amiri et al., 2009). In Sections 6.3.1 and 6.3.2, two recently developed spectral matching algorithms, which have been used in the seismic analysis and design of NPPs, are presented.

6.3.1 Wavelet-Based Spectral Matching Algorithms in Time Domain A formal procedure for spectral matching, which adjusts the ground motions in the time domain by adding wavelets to the initial ground motions, was first proposed by Kaul (1978). It was extended to simultaneously match the target spectra at multiple damping values by Lilhanand and Tseng (1988). A number of extensions and improvements on this approach were proposed (Lee and Kim, 1999; Choi and Lee, 2003; and Hancock et al., 2006). In this section, the improved algorithm on this approach proposed by Atik and Abrahamson (2010) is introduced. A fundamental assumption of this approach is that the time of the peak response of an SDOF oscillator subjected to a time-history does not change as a result of wavelet adjustment on the time-history. If A(t) is the initial seed ground-motion accelerogram, the objective is to modify A(t) such that the response spectrum of A(t) closely matches the target response spectrum in the entire frequency range of engineering interest. Suppose that there are a total of M spectral values (at frequencies ω1 , ω2 , . . . , ωM ) that are to be matched. For a given damping ratio ζ and at frequency ωI , the difference between the target response spectrum A(t) is SI =



SAt(ωI )

and the response spectrum

SAt(ωI ) − SA(ωI )



ςI ,

SA(ωI )

of

(6.3.1)

where ςI is the polarity of the peak response of the oscillator. ςI = 1 if the maximum oscillator response is positive and ςI = −1 if the maximum oscillator response is negative. The use of ςI is to keep the sign consistent between the spectral difference and the maximum oscillator response. Suppose that the peak response of the SDOF oscillator (with frequency ωI and damping ratio ζ ) under the excitation of A(t) occurs at time τI , and this time will not be perturbed by adding a small adjustment to A(t). The objective of this approach is to determine the adjustment time-history A(t) such that the peak oscillator response due to A(t) at time τI is equal to SI for all I. A(t) can be expressed as A(t) =

M  m=1

Bm

w

m (t),

(6.3.2)

6.3 spectral matching algorithmsbased on recorded ground motions

where

w

m (t),

243

m = 1, 2, . . . , M, is a set of predefined adjustment functions, such as the

wavelet functions, and Bm are the coefficients to be determined. The absolute acceleration response of the SDOF oscillator under the excitation of A(t) is given by, using equation (3.3.17 ),  u¨I =

−ωI2

t

0

  A(τ ) hI (t−τ ) dτ 

t = τI

,

hI (t) = e − ζ ωI t

sinωI t . ωI

(6.3.3)

Substituting equation (6.3.2) into equation (6.3.3) gives M 

u¨I =

m=1

 Bm

t

0

−ωI2

w

m (τ ) hI (t−τ ) dτ.

(6.3.4)

Denote cIm as the response of the SDOF oscillator at time τI under the excitation of the wavelet function

w

m (t):

 cIm =

−ωI2

t 0

w

  (τ ) h (t−τ ) dτ  m I

t = τI

.

(6.3.5)

Substituting equation (6.3.5) into equation (6.3.4) and setting u¨I to SI of equation (6.3.1) result in

M  m=1

cIm Bm = SI ,

I = 1, 2, . . . , M.

(6.3.6)

Equation (6.3.6) gives a system of M linear algebraic equations for the M coefficients B1 , B2 , . . . , BM , which can be readily solved. Having obtained the coefficients Bm , m = 1, 2, . . . , M, the adjustment time-history A(t) is given by equation (6.3.2). The adjusted ground motion is given by ˜ = A(t) + α · A(t), A(t)

(6.3.7)

where 0 < α 0, where



S T-H h1 (x, t), Tm



   





S T-H h1 (x, t), Tm − Sht(Tm ) , Sht(Tm )

(6.3.25)

r = 1, 2, . . . , 2L,

is the spectral acceleration of the first generated horizontal

earthquake ground motion T-H h1 (x, t) at a specific period Tm ,

Sht(Tm ) is the spectral

acceleration of the target response spectrum for horizontal component at Tm , and V is the objective function. By solving the optimization problem (6.3.25) with a suitable optimization algorithm, the first horizontal earthquake ground motion, whose response spectrum closely matches the target response spectrum, can be generated. 2. Generating the Second Horizontal Spectrum-Compatible Ground Motion Having obtained the first horizontal spectrum-compatible earthquake ground motion

T-H h1 (t), it is treated as a new seed motion and decomposed into N2 IMF components as the basis to represent the second horizontal ground motion via EMD.

The time-dependent amplitudes a2, j (t) and the instantaneous frequencies ω2, j (t) of the first horizontal ground motion are then generated through HSA. The phase of each IMF of the first horizontal ground motion is thus shifted to obtain the second

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257

horizontal earthquake ground motion

T-H h2 (x, t) = Re

 N2



j=1

a2, j (t)e

i

.

ω2, j (t) dt + x j



,

(6.3.26)

where x1 , x2 , . . . , xN2 are the phase shifting parameters between 0 an 2π. By merely shifting the phase of each IMF, the second generated horizontal ground motion has the same frequency–time–energy distribution, i.e., HES, as the first generated horizontal ground motion T-H h1 (t) (Wen and Gu, 2004).

A constrained optimization model is then used: minimize: V =

M  m=1

F m (x),

subjected to constraint:

   ρ

F m (x) =

   





S T-H h2 (x, t), Tm − Sht(Tm ) , Sht(Tm )

T-H h1 (t), T-H h2 (x, t)



(6.3.27)

   ε,

where ε is a small prescribed value to ensure that these two generated ground motions are statistically independent in engineering sense. The correlation coefficient between

T-H h1 (t) and T-H h2 (t) is defined as 



ρ T-H h1 (t), T-H h2 (x, t) =

   E[ T-H h1 (t)−μh1 T-H h2 (t)−μh2 ] σh1 · σh2

,

(6.3.28)

where μh1 and μh2 are the mean values, and σh1 and σh2 are the standard deviations of

T-H h1 (t) and T-H h2 (t), respectively (Chen, 1975; ASCE 43-05, ASCE/SEI, 2005).

By minimizing the objective function V subjected to the constraint in optimization model (6.3.27), the second horizontal earthquake ground motion, which is statistically independent of the first horizontal earthquake ground motion and closely matches the target response spectrum, can be generated. For generating the second horizontal spectrum-compatible ground motion, the HES is considered to be a further quantitative generation target. While the seismic response spectrum characterizes the ground motion and effects on structures, the HES represents the time–frequency–energy distribution of the ground motion. Because the target response spectra for two horizontal directions are usually the same, equal Hilbert energy spectra of the two generated horizontal ground motions may be desirable to represent the seismic hazard environment at the same location. 3. Generating Vertical Spectrum-Compatible Ground Motion The last step is to generate vertical spectrum-compatible earthquake ground motion. Each of the selected vertical recorded ground motions is decomposed into a number of IMF components via EMD. The N3 generated IMF components from the selected

258

vertical recorded ground motions are treated as the basis to represent a nonstationary vertical earthquake ground motion. The time-dependent amplitude and the instantaneous frequency of each IMF are then generated through the HSA. The HHT amplitudes av, K (t) and the instantaneous frequencies ωv, K (t) of the vertical recorded ground motions are thus scaled to obtain the vertical earthquake ground motion

T-H v (x, t) = Re



N3   K=1



xK av, K (t) e

.

i xN3 +K ωv, K (t) dt

 ,

(6.3.29)

where x1 , x2 , . . . , xN3 are the amplitude scaling parameters, xN3 +1 , xN3 +2 , . . . , x2N3   are the frequency scaling parameters, and x = x1 , x2 , . . . , x2N3 T . A constrained optimization model is then used: minimize: V =

M  m=1

F m (x),

subjected to constraints:

where

   ρ    ρ

F m (x) =

   





S T-H v (x, t), Tm − Svt (Tm ) , Svt (Tm ) 

T-H v (t), T-H h1 (x, t)   ε,  T-H v (t), T-H h2 (x, t)   ε,

(6.3.30)

Svt (Tm ) is the spectral acceleration of the target response spectrum for vertical

component at period Tm . By solving the optimization problem (6.3.30), the ground motion in the vertical direction, whose response spectrum closely matches the target response spectrum for vertical component, can be generated. The generated spectrum-compatible earthquake ground motion in the vertical direction is thus statistically independent of each of the two generated spectrum-compatible ground motions in the horizontal directions.

Numerical Example For illustration purpose, a numerical example is presented for this algorithm. The CENA UHS is used as the horizontal target response spectrum (Atkinson and Elgohary, 2007). The vertical target response spectrum for CENA is taken as two-thirds of the horizontal response spectra throughout the entire frequency range. The seed recorded ground motions for CENA sites are selected based on the results of SHD (Atkinson and Elgohary, 2007). The results of the SHD and the selected earthquake ground motions are listed in Tables 6.1 and 6.2, respectively. To characterize the large-amplitude high-frequency portion of the CENA UHS, three small near-field earthquakes are selected. One large far-field earthquake is used for the low-frequency portion of the CENA UHS.

6.3 spectral matching algorithmsbased on recorded ground motions Table 6.1

Location CENA Sites

259

Mean values of seismic hazard deaggregation.

Vibration period Short Long

Hypocentral distance 0−50 km 20−150 km

Seismicity Moderate Moderate

Moment magnitude 5.5−6.5 6.5−7.5

Table 6.2 Selected actual earthquake records.

Location CENA Sites

EQ∗ CL SA SA LP

TP§ SN SN SN LF

D† (km) 16.3 51.7 70.5 78.3

M‡ 6.2 5.9 5.9 6.9

Hor-1 ,-2 G01-230,-320 S16-214,-124 S17-000,-270 PJH-045,-315

Ver G01-UP S16-UP S17-UP PJH-UP

∗

Earthquake names: LP, Loma Prieta Earthquake, 1989/10/18 00:05; CL, Coyote Lake Earthquake, 1979/08/06 17:05; SA, Saguenay Earthquake, 1988/11/25 23:46.

§

Earthquake types: SN, Small Near-field Earthquake; LF, Large Far-field Earthquake.

†

D is the closest distance to fault rupture, which is generally the hypocentral distance.

‡

M is the moment magnitude.



Names of two horizontal components and one vertical component of earthquake records, which can be searched from the PEER and NRCAN databases. Table 6.3

Location CENA Sites

Information of generated earthquake ground motions.

PGA-DS∗ (g) H1 = 0.300 H2 = 0.300 V = 0.200

PGA-TH§ (g) H1 = 0.367 H2 = 0.388 V = 0.208

AR† H1 = 1.16 H2 = 1.11 V = 1.02

ρ‡ ρ[H1, H2] = 0.275 ρ[H1, V] = 0.018 ρ[H2, V] = 0.013

∗

PGA of target seismic response spectrum.

§

PGA of generated earthquake ground motion.

†

Average of ratios of resulting response spectrum to target seismic response spectrum.

‡

Correlation coefficient between two ground motion components.



H1, H2, and V are two horizontal and one vertical components of a set of tridirectional ground motions, respectively.

The response spectra of the generated horizontal and vertical ground motions and the selected seed recorded earthquake ground motions are shown in Figures 6.23 and 6.26, respectively. The ratios of response spectra of generated spectrum-compatible ground motions to the target response spectra frequency by frequency, as shown in Figures 6.24 and 6.27, respectively, are within the required range from 0.9 to 1.3. As shown in Table 6.3, the PGA of each generated ground motion is greater than that of the corresponding target response spectrum. The average of the ratios of each resulting response spectrum to the target response spectrum frequency by frequency

260

1

Target CENA UHS

Spectral acceleration (g)

GRS of Horizontal-1 GRS of Horizontal-2 CENA region 5% damping Firm ground

0.1 Large far-field

Small near-field

0.01 0.01

0.1 Figure 6.23

1

Period (s)

10

Horizontal response spectra for ENA.

is greater than 1. The correlation coefficients between the components of each set of tridirectional ground motion are less than 0.3. The generated horizontal and vertical spectrum-compatible ground motions and their corresponding seed recorded earthquake ground motions are shown in Figures 6.25 and 6.28, respectively. Although the two generated horizontal components of each set of the ground motion appear to be similar, they are statistically independent of each other. The generated spectrum-compatible ground motions generally preserve the temporal characteristics of the seed recorded ground motions. To ensure that there are no drifts in the velocity and displacement time-histories, baseline correction may be required for this algorithm. The color mapped Hilbert energy spectra (HES) of the generated tridirectional spectrum-compatible ground motions are shown in Figures 6.29 and 6.30. Two horizontal generated ground motions of each set have the same HES. It can be seen that the generated ground motions have sufficient energy over the entire frequency range.

6.4 6.4.1

Generating Drift-Free and Consistent Time-Histories Using Eigenfunctions Consistent Time-Histories and Drift

Ground motion acceleration, velocity, and displacement time-histories are all widely used in seismic analysis and design, although acceleration time-histories are the most

Ratio of

S to S TH

6.4 generating drift-free and consistent time-histories using eigenfunctions

1.3 1.2

H2 H1

1.1 1.0 0.9

0.1

0.01 Figure 6.24

Period (s) 10

1

Ratio of response spectra (horizontal).

0.2

MW=5.7 D=9.3 km CL/G01230

0.0

Small near-field

−0.2 0.2

MW=5.7 D=9.3 km CL/G01320

0.0

Small near-field

−0.2 0.2

MW=5.9 D=51.7 km SA/S16-214

0.0

Small near-field

−0.2 0.2

MW=5.9 D=51.7 km SA/S16-124

0.0

Small near-field

−0.2 0.2 Acceleration ( g)

261

MW=5.9 D=70.5 km SA/S17-000

0.0 Small near-field

−0.2 0.2

MW=5.9 D=70.5 km SA/S17-270

0.0 Small near-field

−0.2 0.1

MW=6.9 D=78.3 km LP/PJH045

0.0 Large far-field

−0.1 0.1

MW=6.9 D=78.3 km LP/PJH315

0.0 Large far-field

−0.1 0.4

Horizontal-1 for ENA

0.0 −0.4 0.4

Horizontal-2 for ENA

0.0 −0.4

0

5 Figure 6.25

10

15

20 Time (s) 25

Horizontal earthquake ground motions for ENA.

262

0.6 Target CENA UHS

Spectral acceleration (g)

GRS of vertical motion

0.1

CENA region 5% damping Firm ground

Large far-field

Small near-field 0.01 0.01

0.1

1

Period (s)

10

S to S TV

Figure 6.26 Vertical response spectra for CENA.

1.1

Ratio of

1.0 0.9

0.1

0.01 Figure 6.27

1

Period (s) 10

Ratio of response spectra (vertical).

0.10

MW = 5.7 D=9.3 km CL/G01-UP

0.00

Small near-field

−0.10 0.06

MW = 5.9 D=51.7 km SA/S16-UP

Acceleration (g)

0.00 Small near-field

−0.06 0.06

MW = 5.9 D=70.5 km SA/S17-UP

0.00

Small near-field

−0.06 0.03

MW =6.9 D=78.3 km LP/PJH-UP

0.00

Large far-field

−0.03 0.25

Vertical component for CENA

0.00 −0.25

0

5

10

15

20 Time (s) 25

Figure 6.28 Vertical earthquake ground motions for ENA.

6.4 generating drift-free and consistent time-histories using eigenfunctions

263

100

Frequency (Hz)

80

60

40

20

0

0

5

10

15

20

t (s)

25

HES of generated horizontal ground motions for CENA.

Figure 6.29

100

Frequency (Hz)

80

60

40

20

0

0

5 Figure 6.30

10

15

20

t (s)

25

HES of generated vertical ground motion for CENA.

frequently used. For example, acceleration time-histories are used in dynamic analysis of structures (linear or nonlinear) and generating floor response spectra, velocity time-histories are used in sloshing analysis, and displacement time-histories are used in pseudo-kinetic analysis of multiply supported system, such as piping.

264

Consistent Time-Histories ˙ = u(t) Given a displacement time-history D(t) = u(t), velocity V(t) = D(t) ˙ is the deriva˙ = u(t) tive of displacement D(t) = u(t), and acceleration A(t) = V(t) ¨ is the derivative of velocity V(t) = u(t). ˙ Conversely, velocity V(t) = u(t) ˙ is the integration of acceleration A(t) = u(t), ¨ and displacement D(t) = u(t) is the integration of velocity V(t) = u(t). ˙ If a set of acceleration A(t), velocity V(t), and displacement D(t) = u(t) time-histories satisfy these relationships, they are called consistent time-histories. It is important that a set of acceleration, velocity, and displacement time-histories used in seismic analysis and design are consistent; otherwise, responses to unrelated inputs may be obtained. For earthquakes without permanent ground displacement, which is the case for sites of NPPs, it is obvious that acceleration, velocity, and displacement time-histories of earthquake ground motions must satisfy the initial and terminal at-rest conditions, i.e., A(0) = A(T) = V(0) = V(T) = D(0) = D(T) = 0,

(6.4.1a)

u(0) ¨ = u(T) ¨ = u(0) ˙ = u(T) ˙ = u(0) = u(T) = 0,

(6.4.1b)

where T is the duration of the time-histories.

Causes of Drift There are many known problems with integrating an earthquake acceleration timehistory to obtain velocity and displacement time-histories by numerical integration (Berg and Housner, 1961; Boore, 2001): ❧ The velocity and displacement time-histories do not satisfy all initial and terminal at-rest conditions. ❧ Velocity and displacement time-histories are usually made to satisfy the initial atrest conditions when integrating the acceleration time-history. Not satisfying the terminal at-rest conditions will cause the time-histories to drift indefinitely. It is important to understand the cause of drift in velocity and displacement timehistories obtained from integrating accelerogram. Given an accelerogram A(t),  V(t) = 0

t

 A(τ ) dτ + C1 ,

t

D(t) = 0

(t−τ ) A(τ ) dτ + C1 t + C0 .

(6.4.2)

There are four at-rest conditions to be met, i.e., V(0) = V(T) = D(0) = D(T) = 0. No C0 and C1 can be found to satisfy these four independent conditions. Suppose, say, C0 = C1 = 0 are selected so that V(0) = D(0) = 0. Then, V(T) and D(T) may not be zero. For t > T, V(t) and D(t) will drift with t.

6.4 generating drift-free and consistent time-histories using eigenfunctions

265

Baseline Correction To eliminate drift in velocity and displacement time-histories, baseline correction (Brady, 1966; Converse and Brady, 1992) is usually applied to remove the effect of terms with low frequencies (such as linear and quadratic terms) from the time-histories. However, it is observed that ❧ baseline correction by incorporating polynomials into the time-histories will make the adjusted displacement, velocity, and acceleration inconsistent; ❧ the polynomials used in baseline correction will introduce unrealistic low-frequency waves, distorting the energy content and ruining the characteristics of the original ground motions. To illustrate, consider a simple accelerogram given by A(t) = t (1 − t),

0  t  1,

(6.4.3)

which is at rest at both ends. Integrating A(t) gives 2 3 V(t) = t2 − t3 + C1 ,

3 t4 D(t) = t6 − 12 + C1 t + C 0 .

(6.4.4)

There are four initial and terminal at-rest conditions for V(t) and D(t) but only two constants of integration. Letting V(0) = 0 and D(0) = 0 gives C1 = 0 and C0 = 0. Equation (6.4.4) becomes 2 3 V(t) = t2 − t3

=⇒

V(1) = 61 = 0,

3 t4 D(t) = t6 − 12

=⇒

1 D(1) = 12 = 0.

(6.4.5)

Hence, the velocity and displacement time-histories drift. Applying a baseline correction results in 3 2 V(t) = t2 − t3 − 6t

=⇒

˙ = t−t 2 − 1 = A(t); V(t) 6

3 t4 t2 D(t) = t6 − 12 − 12 . (6.4.6)

˙ = A(t), the baseline-corrected time-histories are not consistent. Various Because V(t) time-histories of this example are shown in Figure 6.31. It has been traditionally believed that numerical errors cause drift in velocity and displacement time-histories obtained by integrating acceleration time-history. However, it should be emphasized that, in this example, there is no numerical error involved in integrating the accelerogram to obtain velocity and displacement time-histories. This example demonstrates that it is the inherent problem due to mathematical overdeterminacy that causes drift in velocity and displacement time-histories obtained by integrating accelerogram. Hence, in practice, noise, overdeterminacy, and numerical error cause the drift in velocity and displacement time-histories. Baseline correction unreasonably removes

266 0.25

A(t)= t(1−t)

0.20

Drift 0.15

Integrate A(t) to obtain V(t)

0.10

Drift Integrate V(t) to obtain D(t)

0.05

V(t) after baseline correction 0.00 0.0

0.2 Figure 6.31

D(t) after baseline correction t 1.0

0.4 0.6 0.8 Example of drift from integrating accelerogram.

significant low-frequency components from the original accelerogram and yields unrealistic “adjusted” acceleration time-history.

Consistent and Drift-Free Time-Histories in Practice For engineering analysis, such as SSI analysis, using drifted velocity and displacement time-histories may have a negative effect on results (Yang et al., 2006). Even if an artificial baseline correction is used to “correct” the drifted velocity and displacement time-histories, it will change the characteristics of the original time-history and thus yield unrealistic results. A realistic, non-drifting displacement time-history, consistent with the accelerogram associated with it, is paramount in seismic analysis of structural responses, especially responses of multiply supported systems. For example, piping is a multiply supported system; some supports may be on different parts or different levels of a structure, while some may be on different structures or equipment. Different support points may undergo different displacements. Seismic qualification of a multiply supported system requires consideration of Seismic Anchor Movements (SAM), in addition to the inertia effects. If the calculated displacement processes drift or the baselines are improperly corrected, the results of analysis from using the incorrect displacement inputs will be unrealistic and, worse yet, will lead to an erroneous conclusion about the design. To resolve the problem of drift in generating velocity and displacement timehistories from acceleration time-history, eigenfunctions of a sixth-order ordinary differential eigenvalue problem are used to expand the acceleration time-history.

6.4 generating drift-free and consistent time-histories using eigenfunctions

267

6.4.2 Expansion Using Eigenfunctions 6.4.2.1 Sixth-Order Eigenvalue Problem To obtain consistent acceleration, velocity, and displacement time-histories, it is critical to find consistent basis functions satisfying all initial and terminal at-rest conditions. Because there are six initial and terminal at-rest conditions, six constants in the solution function are needed to satisfy these conditions. Hence, consider the following sixth-order ordinary differential eigenvalue problem d6 ϕ + λ6 ϕ = 0, dt 6

0  t  T,

(6.4.7)

satisfying six boundary conditions, i.e., the six initial and terminal at-rest conditions, ϕ(0) = ϕ(0) ˙ = ϕ(0) ¨ = ϕ(T) = ϕ(T) ˙ = ϕ(T) ¨ = 0.

(6.4.8)

The general solution of equation (6.4.7) is ϕ(t) = C1 cosλt + C2 sinλt + exp

√

3λt 2



λt C3 cos λt 2 + C4 sin 2



 √   λt λt + exp − 3λt C cos + C sin 5 6 2 2 2 .

(6.4.9)

Differentiating with respect to t gives 2 = −2C1 sinλt + 2C2 cosλt λ  √   √    λt λt λt √ λt + exp 3λt C 3 cos − sin + C cos + 3 sin 3 4 2 2 2 2 2

ϕ(t) ˙ ·

 √   √    λt λt λt √ λt C − exp − 3λt 3 cos + sin + C −cos + 3 sin 5 6 2 2 2 2 2 , (6.4.10) 2 = −2C1 cosλt − 2C2 sinλt λ2 √    √  λt √ λt λt λt + exp 3λt C cos − 3 sin + C 3 cos + sin 4 3 2 2 2 2 2

ϕ(t) ¨ ·

 √     √  λt √ λt λt λt + exp − 3λt C cos + 3 sin + C − 3 cos + sin 5 6 2 2 2 2 2 .

(6.4.11)

Substituting equations (6.4.9) to (6.4.11) into the six initial and terminal at-rest conditions (6.4.8) yields a system of six homogeneous linear algebraic equations for the coefficients C1 , C2 , . . . , C6 :

268

⎡

1

⎢ 0 ⎢ ⎢ ⎢ −2 ⎢ ⎢ ⎢. ⎢ cosν ⎢ ⎢ ⎢ ⎢ −2 sinν ⎣ −2 cosν

0

1 √ 3

2 0

0 1 √ 3

1 √

sinν

e

3ν 2

√

cos ν2

3ν 2

2 cosν

e

−2 sinν

−e

√

Cs−

3ν 2

1 √ − 3

Sc−

√

3ν 2

e

√

sin ν2

e−

3ν 2

3ν 2

Sc+

−e −

3ν 2

Cs+

e−

e √

e

1 √

cos ν2

√

3ν 2

√

3ν 2

Cs+

Sc+

⎤⎧ ⎫ ⎪ ⎪ ⎪C 1 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎥⎪ ⎪ ⎪ C2 ⎪ ⎥⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 3 ⎥ ⎥⎪ ⎪ ⎨ C ⎥ 3⎬ √ 3ν ⎥ ν e − 2 sin 2 ⎥⎪ ⎪ C4 ⎪ ⎥⎪ ⎪ √ ⎥⎪ ⎪ ⎪ ⎪ ⎪ 3ν ⎥ ⎪ − 2 − ⎪ ⎪ ⎪ −e Sc ⎥⎪ ⎪ C ⎪ ⎪ 5 ⎪ ⎪ ⎦ ⎪ ⎪ √ ⎪ ⎪ ⎭ ⎩ 3ν − 2 − C −e Cs 6 0

= 0, (6.4.12) where Cs± =

√ 3 cos ν2 ± sin ν2 ,

Sc± =

√ 3 sin ν2 ± cos ν2 ,

ν = λT.

For C1 , C2 , . . . , C6 to have nontrivial solutions, the determinant of the coefficient matrix must be 0, resulting in the eigenequation: √ √ 8 sinν − sin2ν − 16 sin ν2 · cosh 23 ν + 2 sinν · cosh 3ν = 0.

(6.4.13)

The eigenequation (6.4.13) is a transcendental equation and has infinitely many roots or eigenvalues. It can be shown that there are two sets of roots: ❧ The first set of roots are given exactly by sin ν2 = 0: ν = 2Kπ, K = 1, 2, . . . . ❧ The second set of roots are 9.427055571, 15.707953379, 21.991148618, 28.274333882, …, which are given approximately by cos ν2 = 0: ν = (2K+1)π, K = 1, 2, . . . . It seems impossible to find analytically exact solutions; but for ν > 30, the approximation is extremely good with relative error less than 10−12 . In summary, the eigenvalues can be written as νn = λn T = (n+1)π, n = 1, 2, . . . , in which the results are exact when n is odd and approximate when n is even.   For the nth eigenvalue νn , the corresponding eigenvector Cn = Cn1 , Cn2 , . . . , Cn6 T, n = 1, 2, . . . , can be determined from equation (6.4.12) and the eigenfunction ϕn (t) is obtained from equation (6.4.9). Analytical expressions of eigenfunctions can be easily obtained using a symbolic computation software, such as Maple. Referring to equation (6.4.9), the values of C2 are very small; the effect of the sine function is negligible. The middle portions of the eigenfunctions are dominated by the cosine function. The terms with the positive exponential function modify the right end, whereas the terms with the negative exponential function modify the left end. As an example, for T = 25, ν20 = λ20 T ≈ 21π, ϕ20 (t), ϕ˙20 (t), and ϕ¨20 (t) are shown in Figure 6.32. It is clearly seen that, unlike sine and cosine functions used in Fourier series, ϕn (t), ϕ˙n (t), and ϕ¨n (t) satisfy the initial and terminal at-rest conditions.

6.4 generating drift-free and consistent time-histories using eigenfunctions ϕ20(t)

269

C1 cos(λ20 t)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6

ϕ20(t)/λ20

−C1 sin(λ20 t)

ϕ20(t)/λ220

−C1cos(λ20 t)

0

5

10

15

20

25

Example of eigenfunction.

Figure 6.32

The eigenfunctions possess the following orthogonality properties:  T  T ... ... ϕm (t) ϕn (t) dt = ϕ m (t) ϕ n (t) dt = 0, m = n. 0

(6.4.14)

0

The eigenfunctions could be used as basis functions to expand a time-history u(t) of duration T as u(t) =

∞ 

an ϕn (t),

(6.4.15)

n=1

where an are constant coefficients. If the eigenfunctions are normalized such that



0

T

ϕn2 (t)dt = 1, then Parseval’s inequality becomes 

T

0

2

u (t) dt =

∞ ∞   m=1 n=1

 am an

0

T

ϕm (t) ϕn (t) dt =

∞ 

a2n >

n=1

N 

a2n .

(6.4.16)

n=1

The eigenfunction expansion (6.4.15) is convergent uniformly and pointwise. Hence, integration or differentiation can be done term-by-term, and the order of operation (integration, differentiation, and summation) can interchange.

6.4.2.2

Decomposing Time-Histories Using Eigenfunctions

For the nth eigenfunction ϕn (t), the eigen-circular-frequency is λn = 2π F n =

(n+1)π , T

(6.4.17)

270

where T in second is the duration and F n is the eigen-frequency in Hz. From equation (6.4.17), Fn =

n+1 , 2T

n = 2T F n − 1.

(6.4.18)

Suppose that the frequency range of interest of the time-histories A(t), V(t), and D(t) is F min  F

 F max ;

let

Nmin = 2T F min −1,

Nmax = 2T F max −1,

(6.4.19)

where x denotes the ceiling function (the smallest integer larger than or equal to x), and x denotes the floor function (the largest integer smaller than or equal to x). Eigenfunctions ϕn (t) with n < Nmin would not have an effect on frequencies F

 F min ,

whereas eigenfunctions ϕn (t) with n > Nmax would not have an effect on frequencies F

 F max .

Hence, only eigenfunctions ϕn (t) with Nmin  n  Nmax need to be used.

For example, suppose the duration of the time-histories is T = 24 s, and the frequency range of interest is 0.1  F

 100 Hz;

then

Nmin = 2T F min −1 = 2×24×0.1−1 = 3.8 = 4, Nmax = 2T F max −1 = 2×24×100−1 = 4799 = 4799.

Hence, only the eigenfunctions with 4  n  4799 are used. Note that, for n = 3 and n = 4800, the corresponding eigen-frequencies are F3 =

3+1 n+1 = = 0.08 Hz, 2T 2×24

F 4800 =

4800+1 = 100.02 Hz, 2×24

which are out of the frequency range of interest. In the frequency range of interest F min  F

 F max ,

an earthquake accelerogram A(t)

of duration T can be decomposed using a set of eigenfunctions with Nmin  n  Nmax : A(t) =

N max

n=Nmin

an ϕ¨n (t).

(6.4.20)

Multiplying both sides of equation (6.4.20) by ϕ¨m (t), m = Nmin , Nmin +1, . . . , Nmax , and integrating from 0 to T yield   T N max A(t) ϕ¨m (t) dt = an 0

n=Nmin

T 0

ϕ¨n (t) ϕ¨m (t) dt.

(6.4.21)

If the accelerogram A(t) is sampled at K discrete time instances tK = (K−1)t, K = 1, 2, . . . , K, t = T/(K−1), then  T K−1  A(t) ϕ¨m (t) dt = A(tK ) ϕ¨m (tK ) t = Bm . 0

K=1

(6.4.22)

6.4 generating drift-free and consistent time-histories using eigenfunctions

271

Equation (6.4.21) becomes, for m = Nmin , Nmin +1, . . . , Nmax , N max

n=Nmin



mn an = Bm ,

mn =

0

T

ϕ¨m (t) ϕ¨n (t) dt.

(6.4.23)

Equation (6.4.23) gives a system of (Nmax − Nmin +1) linear algebraic equations for unknown coefficients an , n = Nmin , Nmin +1, . . . , Nmax , which can be readily solved. The velocity and displacement time-histories can then be determined as V(t) =

N max

n=Nmin

an ϕ˙n (t),

D(t) =

N max

n=Nmin

an ϕn (t).

(6.4.24)

In equations (6.4.20) and (6.4.24), ϕn (t), ϕ˙n (t), and ϕ¨n (t) are the nth eigenfunction and its derivatives given by equations (6.4.9) to (6.4.11). Because ϕn (t), ϕ˙n (t), and ϕ¨n (t) are consistent and satisfy all initial and terminal at-rest conditions, the acceleration A(t), velocity V(t), and displacement D(t) timehistories given by equations (6.4.20) and (6.4.24) are guaranteed to be consistent and satisfy all initial and terminal at-rest conditions, hence no drift. It should be emphasized that the eigenfunctions can also be used to expand any time series provided that the time series is continuous and is equal to zero at the beginning and at the end. Convergence of the expansion is pointwise.

6.4.3

Generating Time-Histories Using Eigenfunctions

Li et al. (2017) employed the method of optimization to generate consistent, drift-free, and spectrum-compatible time-histories based on the eigenfunctions. In this section, an efficient method of modifying an accelerogram using the influence matrix method that is capable of achieving perfect compatibility with a target response spectrum is presented.

6.4.3.1 Influence Matrix Method Consider an SDOF oscillator with natural circular frequency ωm and damping ratio ζ under the excitation of acceleration A(t). Decompose A(t) using (Nmax − Nmin +1) eigenfunctions as in equation (6.4.20). For small ζ , the absolute acceleration response of the oscillator under the excitation of A(t) is given by, using equation (3.3.17 ), max

N sinωm t 2 2 . u¨m (t) = −ωm hm ∗A = an −ωm hm ∗ ϕ¨n , hm (t) = e − ζ ωm t . (6.4.25) ωm n=N min

272

According to the definition of spectral absolute acceleration in equation (4.1.1c), the response spectral acceleration of the SDOF oscillator is given by    N  max  

 2 Sm = u¨m (t)max =  an −ωm hm ∗ ϕ¨n  , n=Nmin  max

(6.4.26)

which is supposed to occur at time instance t = τm . Let ςm be the sign of u¨m (τm ), i.e.,  1, if u¨m (τm ) > 0; ςm = (6.4.27) −1, if u¨m (τm ) < 0. Because τm is known, the contribution that the nth eigenfunction ϕ¨n to the response

spectral acceleration Sm can be expressed as, from equation (6.4.26),

2 hm ∗ ϕ¨n I mn = −ωm



t = τ

m

,

(6.4.28)

which is called the influence coefficient of the nth eigenfunction ϕ¨n on spectral acceler-

ation Sm at frequency ωm . Using (6.4.28), equation (6.4.26) can be written as    N  N max

  max 2 Sm =  an −ωm hm ∗ ϕ¨n  = ςm I mn an . (6.4.29) t=τ n=Nmin n=Nmin m

If the GRS



S = S1 , S2 , . . . , SM

T

is discretized at M frequencies ωm , m = 1, 2, . . . ,

M, equation (6.4.29) can be written in the matrix form as   a = aNmin , aNmin +1 , . . . , aNmax T ,

S = I a,

(6.4.30)

where I is the influence matrix of dimension M×(Nmax − Nmin +1), the mnth element of which is ςm I mn .

˜ that is compatible with Suppose that it is required to generate a time-history A(t)  T GRS S˜ = S˜1 , S˜2 , . . . , S˜M , also discretized at M frequencies ωm , m = 1, 2, . . . , M.

The corresponding coefficient vector a˜ has to be adjusted following (6.4.29), i.e.,

S˜ = I a˜ .

(6.4.31)

Because I is a rectangular matrix, multiplying equation (6.4.31) by I T from the left gives





I T I a˜ = I T S˜ ,

(6.4.32)

where the square matrix I T I is of dimension (Nmax − Nmin +1)×(Nmax − Nmin +1). Equation (6.4.32) gives a system of linear algebraic equations, from which a˜ can be solved.

6.4 generating drift-free and consistent time-histories using eigenfunctions

273

There is a caveat on using equation (6.4.32). The influence matrix I is determined based on time τm , which is the time instance when the absolute acceleration response of the SDOF oscillator (with frequency ωm and damping ratio ζ ) under the excitation of time-history A(t) reaches its peak value Sm . When ˜ compatible with S˜ will be S˜ is significantly different from S , time-history A(t)

significantly different from A(t), and the time τ˜m when S˜m is reached will be significantly different from τm . Hence, equation (6.4.32) can only be used for ˜ is close to cases when the spectrum S˜ is close to S so that the time-history A(t) A(t), and the time τ˜m is close to τm .

Procedure IMM — Influence Matrix Method

❧ Input: • Accelerogram Ai (t) in eigenfunction expansion   t T • Target response spectrum S t = S1t , S2t , . . . , SM • Required number of eigenfunctions Nmin and Nmax • Parameter ε ❧ Output: • Accelerogram An (t) in eigenfunction expansion The input accelerogram is N max

Ai (t) =

ani ϕ¨n (t).

(6.4.33)

n=Nmin

For an SDOF oscillator (with frequency ωm , m = 1, 2, . . . , M, and damping ratio ζ ), the absolute acceleration response under Ai (t) is, using equation (6.4.25), 2 u¨mi (t) = −ωm

N max



n=Nmin

hm ∗ ϕ¨n ani .

(6.4.34)

i The maximum and minimum absolute acceleration responses, i.e., u¨ m, max > 0 and i i i u¨ m, min < 0, and the time instances t m, max and t m, min when they occur can be easily

determined.   i i   ❧ If u¨ m, max > u¨ m, min , then

i Smi = u¨ m, max ,

  i i   ❧ If u¨ m, max < u¨ m, min , then

i  Smi = u¨ m, min ,





i i τmi = t m, max , ςm = 1. i i τmi = t m, min , ςm = −1.

Using equation (6.4.29), one has

Smi = ςmi

N max

i I mn ani ,

n=Nmin



i 2 I mn = −ωm hm ∗ ϕ¨n

 

t = τmi

.

(6.4.35)

274

In the matrix form, one has

S i = I i ai

=⇒



T



I i T I i ai = I i



S i,

Ii



mn

i = ςmi I mn ,

(6.4.36)

in which I i is the influence matrix of dimension M×(Nmax − Nmin +1). The new response spectrum is taken as

S n = (1−ε) S i + ε S t = S i +  S n , where, for small ε,  S n is small so that

 S n = ε ( S t − S i ),

(6.4.37)

S n is not too different from S i. As a result,

❧ the resulting time-history An (t) is not too different from Ai (t); ❧ the absolute acceleration response u¨mn (t) of the SDOF oscillator under the excitation of An (t) is not too different from the response u¨mi (t) under the excitation of Ai (t); ❧ the time instance τmn , when the spectral acceleration occurs, is close to τmi , and the sign ςmn is the same as ςmi . Hence, the approximation I n ≈ I i is applicable. Letting an = ai + an ,

(6.4.38)

one has, similar to equation (6.4.36), 

T



I i T I i an = I i

S n.

(6.4.39)

Using equations (6.4.36) and (6.4.37), equation (6.4.39) yields 





I i T I i an = I i T  S n .

(6.4.40)

Because  S n is small, each element of an is small. Equation (6.4.40) can be readily solved for an . The new coefficient vector an is then obtained from equation (6.4.38), and the new accelerogram is given by An (t) =

N max

ann ϕ¨n (t).

(6.4.41)

n=Nmin

☞

Equation (6.4.37) indicates that

S n is closer to the target spectrum S t than S i

is; hence, each iteration improves the spectrum compatibility.

6.4 generating drift-free and consistent time-histories using eigenfunctions

275

The following iterative procedure for generating spectrum-compatible, drift-free, and consistent time-history A(t), V(t), and D(t) in one direction can be applied. Procedure 1D-TH — Generating Time-Histories in One Direction

Generate a set of drift-free, consistent acceleration A(t), velocity V(t), and displacement D(t) time-histories of duration T, which is compatible with the   t T, discretized at M frequencies ω , target spectrum S t = S1t , S2t , . . . , SM m m = 1, 2, . . . , M. A suitable recorded accelerogram A(0) (t) of duration T is used as the seed motion.

☞

ω1 and ωM correspond, respectively, to the lower-bound F min and upper-bound

☞

For the frequency range of interest F min  F  F max , the required eigenfunctions

F max of the frequency range of interest.

are Nmin  n  Nmax . 1. Expand the seed motion A(0) (t) in eigenfunctions as in Section 6.4.2.2, i.e., N max

A(0)(t) =

a(0) n ϕ¨n (t).

(6.4.42)

n=Nmin

2. After the (I−1)th iteration, the (I−1)th accelerogram is N max

A(I−1)(t) =

a(I−1) ϕ¨n (t). n

(6.4.43)

n=Nmin

In the Ith iteration, apply Procedure IMM with input accelerogram A(I−1)(t) and target spectrum

S t to obtain the Ith accelerogram N max

A(I)(t) =

a(I) n ϕ¨n (t).

(6.4.44)

n=Nmin

3. Step 2 is repeated until spectrum compatibility is achieved, i.e., the difference

S (I) of the generated time-history A(t) = A(I) (t) and the target response spectrum S t is acceptable.

between the response spectrum

4. The spectrum-compatible, drift-free, and consistent time-histories are given by . A(t) =

N max

an ϕ¨n (t), V(t) =

n=Nmin

N max

an ϕ˙n (t), D(t) =

n=Nmin

N max

an ϕn (t).

n=Nmin

(6.4.45)

276

6.4.3.2 Numerical Examples Example 1: CENA UHS For CENA UHS, one horizontal ground motion recorded at the WAHO station during the Loma Prieta Earthquake (United States) in 1989 is selected as the seed motion. For the seed time-history, a total of 4703 discrete time points are included, with a time interval of s = 0.005 s and duration T = 23.51 s. The lower-bound and upper-bound of the frequency range of interest for CENA UHS are F min = 0.2 Hz and F max = 100 Hz. Correspondingly, the required eigenfunctions are Nmin = 9 and Nmax = 4701. The response spectra of the target, the seed timehistory, and the generated time-histories are shown in Figure 6.33. The response spectra for selected intermediate steps demonstrate that the intermediate spectra are getting closer to the target spectra in each iteration. The relative error defined as

 E( F ) =







RS of the Generated Time-History S( F ) − Target S t( F )



Target S t( F )



×100 % (6.4.46)

is also plotted in Figure 6.33; it is seen that the approach gives excellent compatibility with absolute value of the relative error less than 1 % throughout the frequency range 



except at 100 Hz; in particular in the frequency band of 0.6 Hz, 50 Hz , which covers almost all frequencies of engineering interest, the spectral match is nearly perfect. The generated drift-free and consistent acceleration, velocity, and displacement timehistories are shown in Figure 6.34. Example 2: NUREG R.G. 1.60 For the design response spectra from NUREC R.G. 1.60, the second horizontal ground motion recorded at Norcia-Zona Industriale station during the Umbria Marche (foreshock) Earthquake (Italy) in 1997 is used. In order to demonstrate the remarkable capability of the influence matrix method in spectral matching with target spectrum, the generated time-history in Li et al. (2017) using the optimization method is selected as the seed motion A(0) (t). For the seed time-history, a total of 4404 discrete time points are included, with a time interval s = 0.005 s and duration T = 22.015 s. The lower-bound and upper-bound of the frequency range of interest for R.G. 1.60 are F min = 0.1 Hz and F max = 100 Hz. The required eigenfunctions are Nmin = 4 and Nmax = 4402. After several iterations, very good spectral matching can be achieved in 



the low-medium frequency range, especially in the frequency band of 0.5 Hz, 20 Hz , as seen from the plot of relative error. However, it is seen that the relative errors in the high frequency range are larger than those in the low-medium frequency range. In this

6.4 generating drift-free and consistent time-histories using eigenfunctions

277

Spectral acceleration (g)

1.0 0.9

CENA UHS

0.8

Seed time history Generated time history

0.7

Intermediate results

0.6 0.5 0.4 0.3 0.2 0.1 Frequency (Hz)

Relative error (%)

0.0 0.2 3

1

10

100

10

100

2 1

0

Frequency (Hz)

−1 0.2

1

Figure 6.33

Spectral Matching of CENA UHS.

A(t) ( g)

0.5

0 Seed motion

Generated time history

Intermediate iteration results

V(t) (cm/sec)

−0.5 20 10 0 −10 −20 −30 8

D(t) (cm)

6 4 2 0 −2 −4

t (s) 0

5

10

15

Figure 6.34 Time-histories compatible to CENA UHS.

20

23.51

278

Spectral acceleration (g)

1.0 0.9

R.G. 1.60 design spectrum

0.8

Seed time history

0.6

Generated time history with perfect compatibility in low-medium frequency

0.5

Intermediate results

0.7

0.4 0.3 0.2 0.1 Frequency (Hz)

Spectral acceleration (g)

0.0 0.1 1.0

1

0.9

R.G. 1.60 design spectrum

0.8

Generated time history

10

100

10

100

0.7 0.6 0.5 0.4 0.3 0.2 0.1 Frequency (Hz)

0.0 0.1 3

1

Relative error (%)

2

1

0

−1

Generated time history with perfect compatibility in low-medium frequency

Generated time history

−2 Frequency (Hz)

−3 0.1

1

Figure 6.35

10

25

Spectral Matching of R.G. 1.60 design spectrum.

100

6.4 generating drift-free and consistent time-histories using eigenfunctions 279 Higher-frequency eigenfunctions have no influence 2820 − 4402 eigenfunctions (64.070 −100 Hz) on lower-frequency spectrum

1234 − 2819 eigenfunctions (28.049 − 64.047 Hz)

440 −1233 eigenfunctions (10.016 − 28.026 Hz) 44 − 439 eigenfunctions (1.022 − 9.993 Hz)

Lower-frequency eigenfunctions have influence on higher-frequency spectrum

0.1 1.0

1

4 − 43 eigenfunctions (0.114 − 0.999 Hz) 10

100

R.G. 1.60 design spectrum

0.8 0.6 0.4 0.2 0.0

Frequency (Hz)

−0.2 0.1

Figure 6.36

1 10 Contributions of eigenfunctions to R.G. 1.60 design spectrum.

100

0.5

A(t) (g)

0.25 0

−0.25 −0.5 60 Generated time history

V(t) (cm/sec)

Seed motion 30 0 −30

Intermediate iteration results −60 50

D(t) (cm)

25 0 −25 −50

0

5

10

15

Figure 6.37 Time-histories compatible to R.G. 1.60 design spectrum.

20

22.015

280

case, it seems rather challenging to improve spectral matching in both the low-medium frequency range and high frequency range using all the required eigenfunctions. Note the following observations: ❧ Each eigenvalue behaves essentially as a harmonic function except at the two ends as illustrated in Figure 6.32. ❧ Referring to the DMF of an SDOF oscillator under harmonic ground excitation as shown in Figure 3.11, DMF is very small when the frequency ratio r 1; whereas DMF is large when r ≈ 1 and approaches 1 when r > 1. Therefore, all eigenfunctions with frequencies less than F would have a significant influence on response spectrum

S( F );

whereas eigenfunctions with frequencies higher than F would have little

influence on S( F ). Figure 6.36 demonstrates the contribution of each eigenfunction to the target response spectrum, which is the signed response spectrum of the corresponding eigenfunction (with associated amplitude and sign). Hence, a strategy of two-stage spectral matching can be employed based on these observations. 1. Spectral Matching in Low-Medium Frequency Range Use the full set of required eigenfunctions (Nmin  N  Nmax ) to achieve excellent spectral matching in the low-medium frequency range. The dimension of the influence matrix I i in equation (6.4.40) is M×(Nmax −Nmin +1). M is the total number of frequencies to be spectrally matched. In ASCE 43-05 it is required that spectral accelerations be computed at 100 points per frequency decade, uniformly 



spaced over the logarithmic frequency scale; for frequency range 0.1 Hz, 100 Hz , M = 301. The dimension of I i is 301×(4402−4+1) = 301×4399. 2. Spectral Matching in High Frequency Range To improve spectral matching in the high frequency range, use a subset of eigen  functions with high frequencies in, say, 25 Hz, 100 Hz , i.e., F˜min = 25 Hz and N˜ min = 2T F˜min −1 = 2×22.015×25−1 = 1099.75 = 1100.

As a result, the equation (6.4.40) for improving the compatibility of the highfrequency portion becomes  i i i I˜ T I˜ a˜ n = I˜ T  S˜ n ,

(6.4.47)

˜ min +1)×(Nmax − N˜ min +1). where I˜ is the influence matrix of dimension (M− M ˜ min is the number of frequency point for spectral matching with F ˜  F˜min . M Mmin i

Because log10 F M = −1 + 0.01 (M−1),

1  M  301,

(6.4.48)

6.4 generating drift-free and consistent time-histories using eigenfunctions

281

the minimum value M for F M  F is given by 0 / (6.4.49) M = 100( log10 F + 1) + 1 . Hence, / 0 / 0 ˜ min = 100( log F˜min +1)+1 = 100( log 25+1)+1 = 240.79 = 241. M 10 10 i n The dimension of I˜ is (301−241+1)×(4402−1100+1) = 61×3303.  S˜ is ˜ min +1) = 61 that contains the desired change of a vector of dimension (M− M

response spectrum for F  F˜min . Solving equation (6.4.47) gives a˜ n .

The final response spectrum of the generated time-history and the relative error after performing spectral matching in the high frequency range only are also shown in Figure 6.35. As seen in the plot of relative error, spectral matching for F  F˜min = 25 Hz is improved significantly, with very small changes for 10 Hz < F < 25 Hz and no changes for F < 10 Hz. By performing the two-stage spectral matching, excellent spectrum compatibility is achieved over the entire frequency range of interest. Although the seed motion is already spectrum-compatible satisfying the compatibility requirements according to USNRC SRP 3.7.1 using the method of optimization, the method of influence matrix is capable of achieving perfect spectrum compatibility, which has never been achieved. The generated drift-free and consistent time-histories are shown in Figure 6.37. It is observed that, although the changes in acceleration are not significant, there are remarkable changes in velocity and displacement when perfect compatibility is reached. As a result, the velocity and displacement of the “compatible” accelerogram defined by current codes and standards may still be far from convergence. A consequence is that the result obtained for SAM can be erroneous. ❧

❧

Time-history analysis has been widely used in the nuclear industry, from nonlinear dynamic analysis to analysis of multiply-supported systems such as piping systems. Codes and standards have specific requirements for time-histories to be acceptable in seismic design, seismic qualification, and seismic assessment. ❧ In the past few decades, various methods have been developed to generate spectrumcompatible time-histories. Some methods presented in this chapter include • Fourier-based spectral matching algorithms for artificial ground motions, • spectral matching algorithms based on real earthquake ground motions using • wavelets,

282

• Hilbert–Huang transforms (HHT). ❧ Realistic ground motion time-histories that are drift-free and consistent are critical for the results of these analyses to be acceptable. Drift time-histories lead to erroneous results, and inconsistent time-histories result in responses to unrelated inputs. ❧ When integrating an acceleration time-history to obtain velocity and displacement time-histories, they would drift due to mathematical overdeterminacy. Baseline correction has been applied to remove drift in velocity and displacement. However, blindly applying baseline correction to accelerograms may ruin characteristics of the original time-histories; hence, it is not an appropriate method. ❧ Recently, the method of eigenfunctions using the eigenfunctions of a sixth-order eigenvalue problem has been developed. It can generate a set of consistent timehistories satisfying initial and terminal at-rest conditions, i.e., drift free. Furthermore, when coupled with the influence matrix method, it is capable of achieving perfect compatibility with target spectrum, which has never been accomplished.

C

H

7 A

P

T

E

R

Modelling of Structures 7.1 Introduction To perform seismic design and analysis of safety-related structures, systems, and components (SSCs) in a nuclear power plant (NPP), the responses of a structure need to be determined by developing a mathematical model of the structure and calculating its responses to the prescribed loads. A dynamic analysis is generally required to obtain seismic responses. Finite element method (FEM) is extensively used to develop mathematical models for structural analysis. FEM assumes that any continuous function over a global domain can be approximated by a series of functions operating over a finite number of simple and small subdomains called elements. These series of functions are piecewise continuous and should approach the exact solution as the number of subdomains approaches infinity. The methodologies and computation tools for finite element (FE) modelling and analysis have evolved tremendously during the past decades, which allow an engineer to develop more realistic and refined three-dimensional (3D) FE models. However, this does not mean that a very complex and detailed model is always necessary, and selecting a crude model does not mean that the engineer is not familiar with the more sophisticated methods. With a good understanding of the dynamic behavior of a structure, economical considerations in general also demand the use of a simple model for dynamic analysis, as the computational effort of a dynamic analysis can be an order of magnitude larger than that of a static analysis with the same discretization. The amount of details used to represent a structure also depends on the purpose and the desired output responses of the analysis. Different response parameters, which 283

284 Table 7.1

Responses parameters of structural analysis

Response Parameter Member force/stress Floor response spectra Displacement time-histories

Analysis Purpose Structural design Seismic qualification of equipment Input to piping analysis

Analysis Type Static/dynamic Dynamic Dynamic

may be of interest to designers and analysts, can be obtained from static analysis or dynamic analysis of an FE model. For example, the geometry and material property of a slab with openings should be explicitly modelled to obtain allowable maximum stress for slab design considering the stress concentration effect. In contrast, the slab may be represented by a rigid diaphragm in a dynamic model for generating floor response spectra. Table 7.1 lists some common response parameters from structural analysis and their usage. For instance, maximum internal forces and stresses under different loading cases can be utilized for design of structure components. Many textbooks on FEM provide excellent mathematical fundamentals and formulations. The objective of this chapter is to bridge the gap between the FEM and its engineering applications. Methodologies and considerations in developing a linear FE model will be discussed. A 3D FE model of a containment structure in NPP is established by the commercial software ANsys.

7.2 Structural Modelling A general procedure of developing a dynamic model of structure is illustrated in Figure 7.1. The objectives of analysis and the associated response parameters are identified first. The selection of element type, mesh size, and element shape is critical to the accuracy of the modelling. The mass, stiffness, and damping of structural components should be reflected appropriately in the model.

7.2.1 Element Type For three-dimensional FE modelling of civil engineering structure, typical element types include beam, plate/shell, and solid. Other specialty elements include concentrated mass, spring/damper, and contact elements. Two major factors to be considered in the selection of FE types for analysis are: • Type of structural elements to be modelled • Response parameters of interest Elements commonly used in structural modelling of NPPs will be discussed.

7.2 structural modelling

285

Drawings

Geometry Member dimensions Section shape Equipment layout Material Property Elastic modules Density Possion’s ratio

Mesh Size Accuracy Numerical cost Element Type 1-Node mass 2-Node beam 4-Node shell 8-Node solid

Analysis Objectives Floor response spectra Displacement histories Structural design Component deformation

Discretization Structural Parameters Mass, stiffness & damping Idealized Mathmatical Model

M x¨ (t) + C x˙ (t) + Kx(t) = −M

3

I=1

I I u¨gI (t)

Response Parameters Nodal responses: displacement x acceleration x ¨ Member force Section stress and strain

Seismic Input u¨gI (t)

Figure 7.1

General procedure and consideration of structural modelling.

Beam Element A three-dimensional beam element is a straight slender member connected by two nodes at each end, which transfers lateral loads, axial loads, and moments. Each node has six degrees-of-freedom (DOF), including three translations and three rotations in the element coordinate system. Beam elements assume constant or linearly varying cross-sectional properties. Properties, such as cross-sectional geometry or area and area moments of inertia, must be input for beam elements because the beam’s geometry cannot be determined from the two nodes. A 3D beam element formulation includes the effects of biaxial bending, torsion, axial deformation, and biaxial shear deformations. Beam elements are used to model beams, columns, trusses, and bracing in NPPs.

286

Shell Element A shell element is normally used to model thin structures, in which one dimension is very small compared with another two dimensions. A shell element may have either a quadrilateral shape or a triangular shape. Shell elements have only one node at their vertices so that the thickness of the plate must be specified either as a constant or linear variation. Each node has three translational DOF and three rotational DOF. The aspect ratio of a rectangular shell element is defined as length over width. Many FE programs have a restriction on the aspect ratio. The reason for this restriction is that, if the element stiffness in two directions is very different, the structural stiffness matrix has both very large numbers and almost zero numbers on the main diagonal. Consequently, the computed displacements and stresses may not be accurate. In some modern software, this may not be a problem because high-accuracy algorithms are used. Shell elements are usually employed to model walls, slabs, roofs, and containments.

Solid Element Solid elements can be considered to be the most general of all solid FE because solid elements can have arbitrary geometric shapes, material properties, and boundary conditions in space. A 3D solid element can be a tetrahedron or hexahedron in shape with either flat or curved surfaces. Each node of the element will have three translational DOF. The element can thus deform in all three directions. Theoretically, solid elements can be used to model all kinds of structural components, including trusses, beams, plates, and shells. However, it can result in tedious geometry creation and meshing, and the computational effort is demanding. Solid elements can be used to model the foundation medium in NPPs.

Mass Element The mass element is defined by a single node, concentrated mass components in three translational directions that define the element coordinate axes and rotary inertias about the element coordinate axes. Mass elements can model point masses or concentrated loads. The mass element has no effect on the static analysis solution unless acceleration or rotation is present. In static analysis, a concentrated mass acts as concentrated loading for dead weight. In dynamic analysis, a mass gives inertial resistances to accelerations of the structure. Mass elements are commonly used to model structure, systems, and components not included in the structural model but may affect dynamic behavior of structure, such as dead load, appropriate part of live load, stationary equipment, and siding partitions.

7.2 structural modelling

287

Shape Function In FEM, continuous models are approximated using information at a finite number of discrete nodes. The shape function is the function that interpolates the solution between the discrete values obtained at the mesh nodes. Appropriate functions have to be chosen. Usually, the shape functions can be given in the form of low-order polynomials, which may achieve sufficient accuracy. Higher-order shape functions generally increase the convergence rate of the solution. Typical shape functions include linear, quadratic, and cubic shape functions.

7.2.2 Mesh One major step in FEM is discretizing a structural body into elements. For a given model and a set of loading conditions, the accuracy of results is dependent on the mesh and element type. Two important factors define the mesh: element density and element distortion.

Element Density In general, the accuracy of the model increases with the number of elements. ❧ The structure’s geometry can be more accurately defined with more points. This is especially true when a curved boundary is modelled with linear elements having nodes only at their vertices. These elements must represent the curved boundary as a series of straight lines. ❧ Stress gradients can be more accurately defined by having more elements, thereby minimizing stress gradients within elements. This is especially important for linear, constant strain elements. ❧ More elements generally means smaller elements close to a fillet or notch surface where a maximum stress may occur. Smaller elements means that the centroid of the surface elements will be closer to the structure’s surface simply because of the element size. More elements in these regions give more definition of stress gradients. Element density is generally a trade-off between accuracy and computational expense. The goal in developing a mesh is to have as uniform a change as possible in stress between adjacent elements throughout the model in a stress analysis. However, this does not mean that a uniform mesh for the entire model is always necessary. A rule of thumb is to use a relatively fine discretization in regions where a high gradient of strains and/or stresses is expected. Regions to watch out for high gradients are • near entrant corners or sharply curved edges,

288

• in the vicinity of concentrated (point) loads, concentrated reactions, cracks, and cutouts. • in the interior of structures with abrupt changes in thickness, material properties, or cross-sectional areas. In practice, the selection of element density is essentially an exercise in engineering judgement because the analytical solutions, to which FE analysis results can be compared, are unavailable for a complex structure. Alternatively, the response parameters of interest are acceptable if a convergence study is performed for a small structure with dynamic similarity to demonstrate that the results are not significantly affected by further refinement of elements.

Element Distortion FE models of geometrically complex structures have the potential problem of element distortion. It is known that elements give the best results when they are used in their ideal shape, i.e., equilateral triangles, square quadrilaterals, and three-dimensional solid elements that are perfect cubes. In practice, however, it is difficult to have all elements with perfect shapes. Elements will tolerate a modest amount of deviation from their ideal shape without any noticeable decrease in accuracy. Element distortion can be caused by high aspect ratios, large parallel deviations, wide maximum corner angles, and warping (Figure 7.2). Elements with considerable distortion, such as elongated or “skinny” elements, should be avoided in discretization. For example, elements with quadrilateral aspect ratios exceeding 20 will be warned in ANsys.

However, elements with high aspect ratios warning will not necessarily produce

bad results but will depend on the loading and boundary conditions of the problem.

7.2.3

Material Property

For a static stress analysis, the only material properties required are modulus of elasticity and Poisson’s ratio because only the stiffness of the structure needs to be calculated. The values of modulus of elasticity and Poisson’s ratio are obtained from existing codes and standards. For a dynamic analysis, the material mass density must also be input. FE program user’s manuals do not specify a set of units; care must be taken to ensure that the unit used for mass density is consistent with the units of length, time, acceleration, and force. Material damping coefficient may also be needed. The determination of material damping coefficient is discussed in Section 7.2.6.

7.2 structural modelling

289

1

20 (a) Aspect ratio

0

70

100

150

170 (b) Parallel deviation

90

140 (c) Maxiumum corner angle

180

(d) Wraping Figure 7.2

Element distortion.

7.2.4 Modelling of Stiffness For a 3D FE model, the stiffness matrix of a structure is constructed by assembling element stiffness matrices. An element stiffness matrix is generally determined by: ❧ Material properties, including elasticity modulus, Poisson’s ratio, shear modulus. Reinforced concrete elements are modelled using best-estimate stiffness properties depending on the stress state of the concrete due to the most critical seismic load combinations. ❧ Geometry properties, such as cross-sectional area, area moment of inertia, and length of beam, or thickness of plate. For a beam element, the geometry properties can be either input directly or calculated by the FE software based on the userdefined cross-section shape.

7.2.5 Modelling of Mass As a general rule, the construction of the mass matrix M pairs with the stiffness matrix K. Mass matrices for individual elements are formed in local coordinates, transformed to the global coordinates using exactly the same techniques as for the stiffness matrices.

290

ASCE 4-98 (ASCE, 1998) specifies that the inertial mass properties of a structure may be modelled by assuming that the structural mass and associated rotational inertia are discretized and lumped at nodal points of the model. Alternatively, the consistent mass formulation may be used. A bar under axial load is used as an example to illustrate matrix formulation for these two types of mass.

Lumped-Mass Matrix If lumped-mass approximation is considered for a bar element, the distributed mass of the bar is replaced by statically equivalent point masses at the ends as shown in Figure 7.3. This yields a diagonal mass matrix with zero elements corresponding to the rotational coordinates. The diagonal elements are m1 =

B m, a+B

a m. a+B

m2 =

(7.2.1)

If the bar has a uniform density ρ and the length is L, the lumped-mass matrix can be expressed as Me = ρ m1

a

m

1 L/2 0

0

2

L/2

. b

(7.2.2) m2

G Figure 7.3

Lumped-mass approximation for a beam.

A diagonal mass matrix can be easily inverted because the inverse of a diagonal matrix is also diagonal. Therefore, a lumped-mass matrix has significant advantages for computations that involve the inverse of mass matrix M−1 . This is balanced by some negative aspects that are examined in some detail later. However, some limitations need to be noted when using lumped-mass approximation. Elements containing both translational and rotational DOF will have mass contributions only for the translational DOF. Furthermore, lumping, by its very nature, eliminates mass coupling between DOF.

Consistent-Mass Matrix The kinetic energy method is used to construct the consistent mass matrix of a bar element for axial motion. The kinetic energy of the bar element can be expressed by  L 1 Te = 2 ρ(x) u˙2 (x, t)dx, (7.2.3) 0

where ρ(x) denotes the mass density per unit length and u(x, ˙ t) is the displacement of location x at time instant t.

7.2 structural modelling

291

Following the FEM philosophy, the element displacement field is interpolated by shape functions as u(x, t) = NT(x) ue ,

(7.2.4)

where N(x) is the shape function vector and ue is the nodal displacement vector. Substituting equation (7.2.4) into (7.2.3) yields, Te = 12 u˙ Te Me u˙ e ,

(7.2.5)

where the element mass matrix is defined as  L Me = ρ(x) NT N dx.

(7.2.6)

0

If the shape functions N are the same as those used to obtain the stiffness matrix, then the matrix Me obtained is known as the consistent-mass matrix.

 T For a uniform bar using linear shape functions, with N = 1−x/L, x/L and

ρ(x) = ρ, the consistent-mass matrix of a bar element for axial motion is given by 11 12 Me = ρL

3 1 6

6 1 3

.

(7.2.7)

It can be easily verified that the consistent-mass matrix defined by equation (7.2.7) preserves linear momentum. In many FEM software, point mass element is applied at the nodes to model the lumped-mass, which can represent the effects of dead load, stationary equipment, and the appropriate part of the live load and snow load. The mass inertia, which defines the mass element, can be input in three translational and three rotational DOF and is determined based on dynamic properties of the SSCs. The advantage of using lumped-mass in a dynamic analysis is that it can reduce the number of DOF, thus saving computational effort without compromising accuracy when the lumped-mass can rationally represent the dynamic behavior of the SSCs. In FEM software, consistent mass can be considered by defining the mass density of materials. The mass effect of main structural components, such as concrete or steel beams, columns, walls, and slabs, is normally treated as consistent mass. Additional distributed mass can be added on specified elements.

7.2.6 Modelling of Damping The effect of damping, which represents the energy dissipation capability of materials, should be considered in a dynamic analysis. The global damping matrix can be formulated by different forms of damping depending on the dynamic analysis methods,

292

including time-history analysis and response spectrum analysis. More than one form of damping can be specified in a model. A FEM program will formulate the damping matrix C as the sum of all the specified forms of damping.

Proportional Damping (Rayleigh Damping) A damping matrix formed by a linear combination of the mass matrix M and stiffness matrix K may be used: C = αM + β K.

(7.2.8)

α and β can be evaluated if ζm and ζn are specified for two frequencies ωm and ωn ⎧   ⎧ 2ωm ωn 1 α ⎪ ⎪ ⎪ ⎨ ζm = 2 ωm + βωm , ⎨ α = ω2 − ω2 (ωn ζm − ωm ζn ), n m =⇒ (7.2.9)   ζ ⎪ ⎪ 2ω ω ζm  m n n ⎩ ζ = 1 α + βω , ⎪ ⎩β = n n 2 ωn 2 ωm − ωn . ωn2 − ωm If ζm = ζn = ζ , then α=

2ωm ωn ζ , ωm +ωn

β=

2ζ . ωm +ωn

(7.2.10)

ωm is generally taken as the fundamental frequency, whereas ωn is selected from higher frequencies of modes contributing significantly to the dynamic response.

Composite Damping For structural systems that consist of substructures with different damping properties, the composite global damping matrix may be obtained by appropriate superposition of damping matrices for individual substructures C=

Ns  n=1

Cn ,

(7.2.11)

where Ns is the number of substructures being assembled and Cn is damping matrix for the nth substructure.

Modal Damping For response spectrum analysis and time-history analysis based on mode superposition, modal damping is needed. Either constant modal damping or composite modal damping can be applied in the analysis, depending on the damping composition of the structure. The damping coefficients recommended in ASCE 4-98 can be used for structure composed of the same material or with similar damping characteristics. For structure with different damping composition, modal damping is given by ζK =

ϕ TK C ϕ K , 2ωK

(7.2.12)

7.2 structural modelling

293

where ζK is the damping ratio of the Kth mode, ωK and ϕ K are the frequency and mode shape normalized to mass matrix of the Kth mode, respectively.

7.2.7

Boundary Conditions

Appropriate boundary conditions should be applied to the FE model in seismic analysis of NPP structures. For structure founded on hard rock when the effect of soil–structure interaction (SSI) is negligible, fixed-base boundary conditions can be applied to the foundation of structure. For SSI analysis, different types of boundary conditions are used for different analysis methods including substructure method and direct method. ❧ For substructure method, the boundary conditions are represented by impedance functions, which can be interpreted as frequency-dependent generalized soil springs. The impedance functions can be developed on the basis of continuum mechanics, FEM, tables of data, or other methods. For the simplified soil spring method, frequency-independent spring stiffnesses and dashpots are most frequently used. ❧ For direct method, the soil is explicitly modelled as a soil cake supporting the structure model. The lower boundary of the soil cake are located far enough from the structure that the seismic response at points of interest is not significantly affected. The lower boundary of the model may be placed at a soil layer with a modulus of at least 10 times the shear modulus of the layer immediately below the structure foundation level. However, the lower boundary needs not be placed more than three times the maximum foundation dimension below the foundation. The lower boundary may be assumed to be rigid. The location and type of lateral boundaries are selected so as not to significantly affect the structural response at points of interest. Elementary, viscous, or transmitting boundaries may be selected in the FEM program used to perform the SSI analyses.

7.2.8 Applied Load Regardless of how forces are input in the FEM program, all forces are converted into point loads applied at the nodal points. For a structural analysis, applied forces can be generally categorized into the following groups: • Concentrated load • Surface load • Body load • Inertia load

294

Concentrated Load Concentrated loads are specified according to the DOF so that each DOF of a node may have a different force amplitude. The direction of force is specified by the sign of the force amplitude relative to the global coordinate system. Forces not coincident with the global axes are represented by the vector sum of the individual force components. Surface Load Surface loads such as pressure are almost always input to the program by specifying a range of elements over which the pressure acts, the direction, and the amplitude of the pressure. The program converts the pressure into the appropriate nodal forces. When pressure is applied to a surface, the equivalent nodal forces are calculated by the product of the node’s share of the area and the pressure over that area. Body Load Another class of applied forces is body loads, including forces such as thermal forces that act throughout the structure and are applied to every element. These forces are often dependent on element properties such as thermal expansion coefficient. Body loads are always applied by specifying the global input variable, such as temperature for thermal forces, and allowing the program to calculate nodal forces on the element level. Inertia Load Inertia loads are effective only if the model has mass, which is usually supplied by a material density specification or a mass element. The inertia loads, such as acceleration, angular velocity, and angular acceleration, can be applied to the structure or any components in the global Cartesian directions.

7.3 Numerical Example Three dimensional linear FE models of a typical containment structure in NPP are established by the commercial software ANsys version 14.0. The procedure of modelling and validation of the model is introduced step by step. The purpose of this example is to demonstrate the modelling theory introduced in Section 7.2.

7.3.1

Structure Information

The containment structure, which houses the internal structures and equipment, plays an important role to prevent the release of radioactive materials into the environment. The structural layout of the selected CANDU-type containment is shown in Figure 7.4 (Radulescu et al., 1997). The containment structure consists of a cylindrical perimeter

1.68m

42.19m

4.27m

38 cm

41.45m

Figure 7.4

Base slab

2.82m

2

1.07m

1.91m

Ring beam

4

Y

X

4(21.265, 0, 0) 3(24, 0, −0.84)

4

Y

1

M1

2

M2

3

M3

Z

1

2

3

X

4 12.35 m

M4

Stick model

10.55 m

10.55 m

10.55 m

10.55 m

5(21.265, 0, 42.19)

7(21.265, 0, 46.46) 6(21.265, 0, 46.155)

2(21.265, 0, −0.84)

Keypoints and lines in 2D

1(0, 0, −0.84)

Z

8(6.249, 0, 47.465)

9(5.029, 0, 47.631)

10(6.249, 0, 48.875)

9 8

10

11(0, 0, 50.871)

CANDU 6 containment structure.

Perimeter wall

20.73m

Inner dome

Upper dome

Configuration

Subbase

41.45m

61cm

7.3 numerical example 295

296 Material Properties of Containment Structure.

Table 7.2

Material Prestressed concrete Concrete Massless prestressed concrete Table 7.3

Keypoint Number 1 2 3 4 5 6 7 8 9 10 11 Table 7.4

Component Upper dome Inner dome Ring beam Perimeter wall Base slab Massless part Total

Density (kg/m3 ) 2.4 ×103 2.4 ×103 0

Elastic Modulus (MPa) 3.2 ×104 2.8 ×104 3.2 ×104

Poisson’s Ratio 0.15 0.15 0.15

Keypoint Descriptions.

Positions Base slab centre Perimeter wall and base slab intersection Base slab edge Perimeter wall at base slab top surface Perimeter wall top Ring beam and upper dome intersection Ring beam top Parapet bottom Inner dome edge Parapet bottom Upper dome top

Finite element numbers of structural components.

Mesh 1 48 160 32 80 96 16 432

Mesh 2 192 448 64 320 288 32 1344

Mesh 3 768 1408 192 1280 960 64 4672

Mesh 4 3072 4864 640 5120 3456 128 17280

wall, a ring beam, an inner dome, and an upper dome. The perimeter wall sits on a circular base slab. The exterior portion of the structure, including the perimeter wall, ring beam, and upper dome, is prestressed to maintain the structural integrity under internal and external loads. The inner dome and base slab are made of reinforced concrete. For simplicity, only these five major components are represented in the model, and structural openings are not considered. SHELL181 elements are used to model the structural components based on the geometrical features of the structure. The material properties are listed in Table 7.2. It should be noted that massless prestressed concrete is defined to address the overlapping portion between the perimeter wall and base slab. The application of the massless material will be discussed in the discretization of model.

7.3 numerical example

297

7.3.2 Geometry An FE model is formed by nodes and elements. In ANsys, nodes and elements can be generated by two methods. In the first method, nodes are first defined in terms of coordinates in an active coordinate system, elements are then defined by assigning the connectivity between the nodes. This method is quite straightforward but lacks flexibility. For example, nodes and elements have to be redefined when a convergence analysis is required where element sizes need to be changed. This procedure is time-consuming and tedious. A more efficient approach is to automatically generate the nodes and elements by meshing the geometry entities, such as lines, areas, and volumes. The numbers or sizes of elements can be easily controlled by meshing the geometry entities. Element properties are then assigned to nodes and elements but cannot be assigned to the geometry entities. The geometry entities in ANsys consist of keypoints, lines, areas, and volumes. The keypoints are defined in terms of their coordinates, and lines are defined by the keypoints that bound them. Keypoints may be specified in global Cartesian, cylindrical, spherical, or local coordinate systems. It should be noted that an FE model is nondimensional, and all dimensions and values must be checked by the user for consistency. Any unit of measure may be used as long as it is consistent with the units of force, modulus of elasticity, and mass density. The geometrical configuration of the containment structure is shown in Figure 7.4. Because the containment structure is axisymmetric, it is convenient to define the geometry in terms of keypoints and lines in two dimension, then to extrude the lines into areas by rotating about the vertical axis in the 3D space. Keypoints and Lines Keypoints are defined at corners and critical positions from the drawing in Figure 7.4. Descriptions of the selected keypoints are listed in Table 7.3. When data are input interactively in FE software, it is recommended that, as a good practice, plots be made frequently; plots of geometric entities should be made at each major step in the modelling process. Once one is satisfied that the keypoint pattern is correct, the line connectivity may be entered. If the lines appear to be improper, then there is an error in the keypoint connectivity. Figure 7.4 shows that the keypoints are connected by straight and arc lines in ANsys, giving an initial view of the model in two dimensions. Local spherical coordinate systems are defined to sketch the arc lines.

298

Area Cylindrical areas are generated by rotating all lines about the global Z-axis. The areas, which compose an outline of the model, are shown in Figure 7.5. The geometry entities only provides a general picture of the structure. Nodes and elements, which form the FE model, are generated by meshing on these geometry entities. A number of elements attributed by material and section properties are assembled on the areas in the next step. Based on the locations of the keypoints, the areas represent the midsurface of the containment structure components except the ring beam. In the FE model, the ring beam and perimeter wall share the same line, which is the midplane of the perimeter wall. An offset is applied in the assembly of ring beam elements to account for the eccentricity so that the midplanes of the ring beam and perimeter wall are different.

7.3.3 Assembly and Mesh Element Type SHELL181 elements, which are suitable to simulate thin to moderately thick shell structure with considerably large dimensions as compared to its thickness, are used to model the containment structure. SHELL181 element is a four-node element with three translational and three rotational DOF at each node as shown in Figure 7.6. The accuracy of SHELL181 modelling is governed by Mindlin-Reissner shell theory. The Mindlin-Reissner shell theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one-tenth of the planar dimensions. All structural components in this model satisfy this criteria. Assembly and Mesh The geometric areas are meshed into a number of small quadrilateral-shaped areas and assembled by the defined shell elements attributed with corresponding section and material properties. It should be noted that massless prestressed concrete should be assigned to the bottom portion of the perimeter wall inserted into the base slab. The massless portion is represented by the belt area at the bottom of the containment wall in Figure 7.5. The purpose of this treatment is to avoid double-counting of the concrete mass in the intersection region. Material properties must be checked by printout of the input data. The two most common mistakes are ❧ Modulus of elasticity is incorrect by a power of 10. ❧ Inconsistent units are assumed for material density. Because the FEM is nondimensional, one must ensure that the units assumed for density are consistent with the units used for stiffness, time, force, and acceleration in a dynamic analysis.

7.3 numerical example

299

Figure 7.5

Geometric areas of containment structure.

Z Y

X

8 L y4

z0 z 5

y0

6

I 1

x x0

5

7 K 3 4

2

3

6 J 1 2

x0 = Element x-axis if ESYS is not provided x = Element x-axis if ESYS is provided Figure 7.6

SHELL181 element.

When more than one material is used in a model, elements with different materials can be plotted in different colours for easy checking. To investigate the effect of meshing, four mesh schemes as shown in Table 7.4 are applied to the model. Structural analyses will be conducted for the models subject to different loads to perform convergence studies.

300

7.3.4 Finite Element Model of Containment Structure An overview of the complete and half of the FE model of the containment structure with Mesh Scheme 3 is shown in Figure 7.7, with 4926 nodes connected by 4672 shell elements. Because the structure is assumed to be founded on hard rock, all DOF of the nodes on the base slab are restrained to model the fixed-base boundary condition. The FE models of each structural component, including the upper dome, inner dome, ring beam, perimeter wall, and base slab, are shown in Figure 7.8. It can be seen that the mesh at the interfaces between the structural components are consistent to ensure compatibility. All shell elements have regular shapes, and the aspect ratios are less than 3.

7.3.5 Model Validation An essential task of FE modelling is the validation of the model before the solution stage because the quality of analysis results relies on the accuracy of the modelling. The model validation should include but not be limited to: • Geometry • Material properties • Connectivity of elements • Consistency of element local coordinate systems • Validity of boundary conditions • Adequacy of mesh density • Correctness of applied loads • Rationality of deflected shapes and force distribution The model can also be validated by comparing the numerical results of the structural responses subject to simple load cases with analytical solutions or numerical benchmarks. To examine the validity of the FE model of the containment structure, three loading cases are applied to the structure with different mesh sizes. The loading cases to be performed are: • Self-weight • 100 kPa internal pressure • Seismic response spectrum analysis

7.3.6 Self-Weight Load An inertia load of gravity acceleration 9.806 m/s2 is applied to the structure with different meshes. Structural responses, including the nodal displacement, element

7.3 numerical example

301

V

NSYS

Z Y

X

Z Y

X

Figure 7.7

Finite element model of containment structure.

Upper dome

Z X

Inner dome

Ring beam Figure 7.8

Y

Perimeter wall

Base slab

Finite element model of containment structure components.

302

stresses, and reaction forces, are obtained. In this analysis, it is assumed that the mass of an element is located at its centroid. The mass of each structural component, total structure mass, centre of mass, and mass moments of inertia for Mesh Scheme 4 (17280 elements) are presented in Tables 7.5 and 7.6 as an example. It can be seen that: ❧ The moments of inertia about two horizontal axes are equal, and the product of inertia is close to zero, which agrees with the symmetry of the structure. ❧ The mass of component with regular shape, such as the base slab, can be easily verified by hand calculations. ❧ The output total reaction forces are zeros in the two horizonal directions and 2.7377×109 N in the vertical direction, which is consistent with the total mass of the structure. From these observations, the geometry and materials of the model are readily verified. Figure 7.9 plots displacement contours of the containment structure under selfweight load for Mesh 1 and Mesh 3. It can be seen that the contours of the two mesh schemes are very similar, even though the numbers of elements are different by more than 10 times. The maximum displacement occurs at the top of the upper dome, and the relative difference of the maximum displacement for the two mesh schemes is 4.2 %. ASCE 4-98 specifies that the response parameters of interest should not be underestimated by more than 10 %. Therefore, a coarse mesh (Mesh 1) may be sufficient to capture the structure displacement under static load. To examine the convergence of other response parameters, Figure 7.10 compares Von Mises stress contours of the containment structure under self-weight load for Mesh 1 and Mesh 3. It is observed that there are apparent differences in Von Mises stress contours for the two mesh schemes. The maximum stress occurs at the edge of the upper dome, which is connected with the ring beam, and the maximum values are 1.63 MPa and 2.57 MPa for Mesh 1 and Mesh 3, respectively. The relative difference is 36.6 %. Table 7.7 gives the maximum displacement and Von Mises stress for the four mesh schemes. Figure 7.11 presents the relationship of maximum displacement and stress, respectively, with the number of elements. Compare to the nodal displacement, the element stress is more sensitive to the mesh density. Therefore, more refined mesh is required to perform stress analysis of the structure under static loads. However, it is noted that only element size is discussed in this example; other factors, such as element shape and shape functions, can also affect the convergence rate of structural responses. It should be noted that the element used in this model is a linear three-dimensional element. There is a linear variation in displacement within the element due to the fact

7.3 numerical example

Figure 7.9

303

Displacement vector contour under self-weight load.

304

Figure 7.10 Von Mises contour under self-weight load.

305

Maximum displacement (mm)

5.0

3.0 Maximum displacement

2.8

4.9

2.6 2.4

4.8

Maximum Von Mises stress

2.2 2.0

4.7

1.8 4.6 102

103 Number of elements 104 Figure 7.11 Convergence under self-weight.

105

1.6

Maximum Von Mises stress (MPa)

7.3 numerical example

Table 7.5 Structure mass.

Component Upper dome Inner dome Ring beam Perimeter wall Base slab Total

Mass (kg) 2.1832 ×106 1.3661 ×106 2.6142 ×106 14.4702 ×106 7.2844 ×106 27.9188 ×106

Table 7.6 Mass moments of inertia.

Moments of Inertia (kg · m2 ) IXX IYY IZZ IXY IYZ IZX

About Origin (0, 0, 0) 0.2703 ×1011 0.2703 ×1011 0.1062 ×1011 − 0.6817× 10−6 − 1.351 − 1.351

About Mass Centre (0, 0, 20.89 m) 0.1484 ×1011 0.1484 ×1011 0.1062 ×1011 − 0.6817× 10−6 − 0.8718 − 0.8718

Table 7.7 Convergence analysis under self-weight.

Mesh No. 1 2 3 4

Maximum Displacement (mm) 4.636 4.874 4.943 4.966

Maximum Stress (MPa) 1.63 2.12 2.57 2.86

that displacements are calculated only at the vertices. The first derivative of displacement, i.e., strain, is therefore assumed to be constant over the element. Theoretically, there is continuity of displacement between elements along the boundaries. However, There is no continuity of strain at the element boundaries. The strain varies across

306

the global model as a series of step functions. For many commercial FEM software packages, stress averaging is done to smooth the stress contour plot.

7.3.7 Internal Pressure Load One major function of containment structure is to resist the internal pressure caused by radioactive steam or gas in a severe accident. For a linear analysis, 100 kPa internal pressure, which is considered as an unit internal pressure load case, is applied on the internal surface of the containment structure. The analysis results can be easily scaled to obtained results under any magnitude of internal pressure. The internal pressure, which is a surface load, must be applied on the element with the proper magnitude, location, and direction. One common mistake is applying pressure on the wrong faces of the shell element. The normal directions of the shell elements can be identified by displaying the element coordinate systems. Figure 7.12 shows the element coordinate systems for a selected slice of elements. The element coordinate system is displayed at the centroid of each element. It can be observed that the local X-axes of the perimeter wall elements are in circumferential (hoop) directions, while that of the upper dome elements are in meridional directions. Identification of the element coordinate systems is critical to interpret the output results. Applied forces may be checked either from the printout or from a node or element plot with the appropriate force display option selected as shown in Figure 7.12. In many design applications governed by structural codes, stress values are not needed. Instead, forces and moments are required input for concrete containment structure design, for example. The preferred format in this case is in-plane forces and out-of-plane moments typically defined as force or moment per unit length. The forces will be used for the design of the prestressing tendons, which are placed in meridional and circumferential directions. Figure 7.13 shows the force/stress output of SHELL181 element. N11, N22, N12, and N21 represent in-plane forces per unit length; M11, M22, M12, and M21 denote out-of-plane moments per unit length; Q13 and Q23 are transverse forces per unit length. ANsys computes M11, M22, and M12 with respect to the shell reference plane. Figure 7.14 plots typical internal forces under 100 kPa internal pressure load versus the height of perimeter wall for different mesh schemes. Based on the element coordinate systems shown in Figure 7.12, N11, M11, M22, and Q23 represent the tensile forces in hoop directions, moment about the hoop directions, moment about meridional directions, and shear forces in radial directions, respectively. It can be seen that the change of all internal force parameters are relatively small in the intermediate portion

7.3 numerical example

307

V

NSYS

Z X

Y Z Y X

Element local coordinate system Figure 7.12

Internal pressure applied on containment structure

Internal pressure applied on containment structure.

SY SX

SX (TOP) SX (MID) SX (BOT)

y, y0 Q13 M12

M11

Q23

N11 M22

z0

N12

z

x, x0

N22

L P

M21

N12

y0

y

K x x0

I J

x0 = Element x-axis if ESYS is not provided, x = Element x-axis if ESYS is provided Figure 7.13

SHELL181 element force and stress output.

308

of the perimeter wall (height of 5 m to 30 m) compared to those at the top and bottom portions of the perimeter wall where the forces increase or decrease drastically. Another observation is that the result variability is not significantly affected by the mesh density, particularly in the intermediate portion. Therefore, a coarse mesh may be sufficient to obtain the internal forces of the containment structure, and high mesh densities should be used at regions close to the interfaces of different structural components.

7.3.8 Response Spectrum Analysis As discussed in Section 3.6.4, response spectrum analysis, based on the principle of mode superposition, is frequently applied for seismic design. A response spectrum analysis must be preceded by a modal analysis. The seismic input, load directions, load distributions, and ground response spectra need to be specified to determine modal responses, which will then be combined through an appropriate combination method.

Modal Analysis Natural frequencies, mode shapes, participation factors, and modal mass ratios are retrieved by performing modal analyses for the four models with different mesh schemes. The Block Lanczos method is employed to solve the eigenvalue problem. Because the total number of the natural modes for a structure model is related to the number of DOF, which may be considerably large for a detail 3D FE model, it is desirable to extract a sufficient number of modes to characterize the structural responses in the frequency range of interest rather than to include all the structural modes. In the modal analysis, either the number of modes to be extracted or a cut-off frequency can be specified before solving the eigenvalue problem. In this example, a cut-off frequency of 100 Hz is defined to cover the frequency range of the seismic input response spectrum. However, a high cut-off frequency may not always be necessary depending on both the dynamic characteristics of structure and seismic input, and engineers need to judge or perform some trial runs to examine the convergence of the results. According to ASCE/SEI 4-16 (ASCE/SEI, 2017), a sufficient number of modes should be included in the analysis to ensure that the inclusion of the remaining modes does not result in more than a 10 % increase in the responses of interest. Figure 7.15 shows the natural frequencies versus the mode number for the four mesh schemes; the natural frequencies of the corresponding modes are represented by different type of markers. It can be observed that the natural frequencies of the first mode, which will be found as the dominant mode, are almost identical for the four types of mesh density. The differences in the natural frequency values are very small

309

45

40

40

35

35

30

30

Height (m)

45

25 20

Mesh 1 Mesh 2 Mesh 3 Mesh 4

10

10

5

5

1

2

N11 (N/m)

0

3 ×105

45

45

40

40

35

35

30

30

25 Mesh 1 Mesh 2 Mesh 3 Mesh 4

20

10

10

5

5

0

5

M22 (N/m) Figure 7.14

10 ×104

0

5

10 ×103

M11 (N . m/m)

Mesh 1 Mesh 2 Mesh 3 Mesh 4

20 15

−5

−5

25

15

0

Mesh 1 Mesh 2 Mesh 3 Mesh 4

20 15

0 0

Height (m)

25

15

Height (m)

Height (m)

7.3 numerical example

0 −1

0

1

Q23 (N/m)

Internal forces of containment structure.

2

3 ×104

310

Frequency (Hz)

100

Mesh Mesh Mesh Mesh

10

5

10

1

Figure 7.15 Table 7.8

Mode 1 3 7 9 12

Frequency 5.455 7.628 8.699 9.321 10.767

100 Number of mode Convergence of natural frequencies.

1 2 3 4

1000

Descriptions of selected modes.

Mode shape Overall sway Perimeter wall vibration Inner dome vibration Upper dome vibration Overall torsion

Participation Factor (X-Dir) 3897.5 3.2454 ×10−2 1.0813 ×10−7 2.7735 ×10−5 − 9.3497× 10−7

for the lower modes with frequencies less than 10 Hz, and the differences become large as the frequencies increase. However, it can be seen that the frequency curves for Mesh 3 and Mesh 4 are very close, and the differences between Mesh 2 and Mesh 3 are small except in high-frequency range with frequencies larger than 33 Hz. Therefore, Mesh 2 may be sufficient to obtain the responses of interest if the total structural response is not contributed considerably by the higher modes. Because the FE model is symmetric and possesses a large number of local vibration modes, some typical mode shapes of Mesh 3 are shown in Figures 7.16 to 7.20. The mode shapes are normalized to the mass matrix. The modal information and descriptions are given in Table 7.8. Based on the output modal information, it is observed that the modal participation factor of the first mode is far larger than those of the other modes; the containment structure can be considered as a one-mode dominant structure. The modal response is determined by the spectral value at the corresponding frequency multiplied by the mode coefficient, which is the product of the participation factor and the mode shape. In ANsys, it is possible to consider only those modes with mode coefficients exceeding a significant threshold value (e.g., larger than 0.01). Only those modes having a certain amplitude of mode coefficient are chosen for mode combination. This option can save significant computational cost and disk storage, particularly when the element stresses are required.

7.3 numerical example

Figure 7.16

311

First mode shape, overall horizontal sway vibration.

Figure 7.17 Third mode shape, perimeter wall vibration.

312

Figure 7.18

Seventh mode shape, inner dome vibration.

Figure 7.19

Ninth mode shape, vertical vibration.

7.3 numerical example

313

Figure 7.20 Twelfth mode, overall torsional vibration.

Figure 7.21

Displacement contour under seismic input.

314

Seismic Responses USNRC R.G. 1.60 (USNRC, 2014) 5 % damped response spectrum anchored to 0.3g PGA is selected as seismic input in the horizontal X-direction. It should be noted that the unit of spectral acceleration defined in a response spectrum is usually the gravity acceleration g. To be consistent with the units used in the FE modelling and analysis, the spectral acceleration should be converted into m/s2 by multiplying a factor of 9.806 in the analysis. The seismic input at the foundation level of the structure is considered to be uniform. The complete quadratic combination (CQC) modal combination method is used to combine the modal responses. The response displacement contours are plotted in Figure 7.21 for Mesh 3. The maximum horizontal displacement occurs around the top of the upper dome, and its magnitude is 9.251 mm. From the deformed shape of the structure, it can be seen that the structure behaves like a cantilever beam with uniform mass and stiffness distribution under horizontal seismic input. The convergence of the response parameter is examined. The deformation shapes for the four mesh schemes are nearly identical; the maximum nodal displacement is 9.208 mm, 9.241 mm, 9.251 mm, and 9.301 mm for Mesh 1 to 4, respectively. The convergence rate agrees with the observations in the dead load case, where a coarse model may be sufficient to capture the response parameters when nodal responses are of interest. However, it should be noted that when a coarse mesh is used, the responses at the location of interest may not be given by the output results directly. Liner interpolation can be used to estimate the response at the location between nodes. Additional errors may be introduced depending on the structural dynamic characteristics. Because the containment structure is one-mode dominant, the seismic response is calculated by considering only the first mode for Mesh 3. It turns out that the maximum displacement is 9.204 mm, which is close to 9.251 mm when hundreds of modes are considered. Therefore, with a good understanding of the structure dynamic characteristics, engineers may use a simplified method or model to estimate the structural responses for design purpose.

Missing-Mass Effect The response spectrum analysis is based on the modal superposition approach, in which responses of higher modes may be neglected; therefore, part of the mass of the structure is missing in the dynamic analysis. Clause 6.4.3.1.2 of CSA N289.3 (CSA, 2010a) requires that the mass accounted for in the analysis should be more than 90 % of the total mass. However, this criteria may not be satisfied in practice; it actually

7.3 numerical example

315

depends on the considerations in the modelling. In this numerical example, the accumulative participation mass is about 72 % and 70 % in the horizontal and vertical directions, respectively, even though a large number of modes are considered and the cut-off frequency is set to 100 Hz. The reason is that the fixed-base boundary condition is assumed in this example, which means that the base slab mass is fixed. It can be seen from Table 7.5 that base slab mass is 26 % of the total mass of the structure, which agrees with the amount of missing mass. To investigate the effect of neglecting higher modes, referring to Section 3.6, the equations of motion for the relative displacements of a 3D structure with N nodes (each having 6 DOF) are given by (3.6.3). The relative displacement of node n in direction j of mode K under ground motion excitation in direction I is I I xn, j; K = ϕn, j; K Q K ,

Q IK = KI qKI ,

(7.3.1)

and its maximum value is      I   I I I I      maxxn, j; K = max ϕn, j; K K qK = ϕn, j; K K · max qK   = ϕn, j; K KI  SDI (ζK , ωK ),

(7.3.2)

in which SDI (ζK , ωK ) is the displacement response spectrum in direction I. The relative displacement of node n in direction j under ground motion excitation in direction I is xn,I j =

6N  K=1

I xn, j; K =

6N  K=1

ϕn, j; K KI qKI .

(7.3.3)

  From equations (7.3.2) and (7.3.3), max xn,I j  can de determined using a suitable modal combination method. For example, if SRSS method is applied, then   6N  6N  2 2   I    I I maxϕn, j; K K qK  = ϕn, j; K KI · SDI (ζK , ωK ) max xn, j  = K=1

 =

r  

K=1

K=1

2

ϕn, j; K KI · SDI (ζK , ωK ) ,

(7.3.4)

in which it is assumed that the structure has r flexible modes. For all rigid modes with K > r,

SDI (ζK , ωK ) = 0.

The absolute acceleration of node n in direction j under ground motion excitation in direction I is I I u¨ n, ¨gI = j = x¨ n, j + u

6N  K=1

I

ϕn, j; K Q¨ K + u¨gI

6N  K=1

ϕn, j; K KI =

6N  K=1

I Q¨ K + KI u¨gI ϕn, j; K

316

= =

r 

6N 

I KI ϕn, j; K u¨gI Q¨ K + KI u¨gI ϕn, j; K +

K=1 r  K=1

K=r+1

I I u¨ n, j; K + u¨ n, j; rigid ,

(7.3.5)

I in which u¨ n, j; rigid includes the absolute acceleration response of all rigid modes and

I I I I ¨ ¨g ϕn, j; K = −ωK2 ϕn, j; K Q IK , (7.3.6a) u¨ n, j; K = Q K + K u I u¨ n, j; rigid =

6N  K=r+1

  r  KI ϕn, j; K u¨gI = u¨gI 1− KI ϕn, j; K .

(7.3.6b)

K=1

From equations (7.3.6), the maximum absolute acceleration of node n in direction j of mode K (K  r) and rigid modes under ground motion excitation in direction I are     I  2 I  I 2 I     maxu¨ n, j; K = −ωK ϕn, j; K Q K = K ϕn, j; K ωK S D (ζK , ωK )   (7.3.7a) = KI ϕn, j; K  SAI (ζK , ωK ), K  r,      r r   I  I      I I  = u¨ 1 −  = S I 1−  maxu¨ n,  ϕ (7.3.7b)  ϕ K n, j; K  K n, j; K , pga  j; rigid  g K=1

where

S

I pga

K=1

is the PGA or zero period acceleration (ZPA, which is the acceleration

response of the SDOF oscillator with period 0) of ground motion in direction I.   From equations (7.3.5) and (7.3.7), maxu¨ I  can be obtained using an appropriate n, j

modal combination method. For example, if SRSS is applied, then  r  2  2  I   I    + maxu¨ I  , maxu¨ n, maxu¨ n, j  = j; K n, j; rigid K=1

( ) r  ) I ϕ =* K=1

K n, j; K

·

2 I A (ζK , ωK )

S

 +

S

I pga

 1−

r  K=1

KI ϕn, j; K

 2 . (7.3.8)

Equation (7.3.7b) or the shaded term in equation (7.3.8) is the maximum absolution acceleration response of node n in direction j under ground motion excitation in direction I contributed from all residual (rigid) modes, which has often been omitted in the common practice by neglecting higher modes. On the other hand, to characterize the missing mass due to neglecting the higher modes, consider a rigid structure. Because the equivalent earthquake force in direction I is −M I I u¨gI , the inertia force due to ground acceleration in direction I is I . F I = −M I I Spga

(7.3.9)

Using the method of modal superposition, the modal inertia force for mode K is I I . FKI = −M ϕ K Q¨ K = −M ϕ K KI Spga

(7.3.10)

7.3 numerical example

317

The missing inertia force vector is then the difference between the total inertia force and the sum of the modal inertia forces: FmI = F I − where

6N  K=1

I FKI = M i I Spga ,

iI =

6N  K=1

ϕ K KI − I I ,

(7.3.11)

i I is the fraction of mass missing. The missing-mass displacement response Dm

is the static deflection due to the missing inertia forces given by DmI = K−1 FmI .

(7.3.12)

The application of these equations can be extended to flexible structures because the higher truncated modes are supposed to be mostly rigid and exhibit pseudo-static responses to a base acceleration excitation. Because the missing-mass response is a pseudo-static response, it is in phase with the imposed acceleration but out of phase with the modal responses; hence, the missingmass response given in equation (7.3.12) and the modal responses are combined using the SRSS method.

7.3.9 Lumped-Mass Stick Model It has been shown that a simplified model may be capable of providing sufficiently accurate results. In the past decades, model idealization, such as using lumped-mass stick models, has been applied to predict seismic responses of structures. Because the containment structure exhibits simple dynamic behaviour, model idealization can be employed. For a lumped-mass stick model, all shear walls of the structure contributing to the stiffness are modelled by massless beam elements. The stiffness of the beam elements is taken into account by using the cross-sectional areas, moments of inertia, torsional properties, effective shear areas of the structural walls, and material properties. The mass of the building is lumped at a number of representative elevations. Some general assumptions are applied in developing a lumped-mass stick model: ❧ The floor slab is assumed to be rigid in the in-plane horizontal direction, but flexible in the out-of-plane direction. ❧ The lateral loads are mainly resisted by shear walls, and very little load is taken by the frame action of the beam-column frame system. The flexural rigidities of the beam-column frames are considered to be low so that they are neglected. The columns support mainly the vertical load, for which they are sized.

318 aud =aid =20.73

tud = 0.61 tid = 0.38

M5

5

ud kud

rb hrb =4.27

M4

rb id ud id pw

rrb2 =22.64

4’

M4

pw 10.55

rud =41.45

4

4

12.35

4

rpw1 = rrb1 =20.73 M3

M3

3

3

rpw2 =21.8 rid =41.45

10.55

3

3 2

M2 hpw = 42.19

10.55

2

2

M1

M1

1

Z Y

Z X

Stick model

Figure 7.22

1

1

10.55

All dimensions in meters

2

M2

Stick models.

Y

X

Stick model flexible UD

❧ Masses are lumped at the floor nodes, and the beams are taken as massless because, at any floor level, the floor mass typically accounts for more than 90 % of the total mass and the wall mass is only 10 %. Referring to Figure 7.22, the parameters of the stick model are determined.

Equivalent Stiffness In a stick model, the shear deformation of the massless beam elements has to be considered because the beam elements are of the type of thick beams. The transverse shear stiffness is defined by GAs , where G is the shear modulus of the beam and As is the effective shear area. For an annulus cross-section, As = 12 A. The flexural stiffness of the containment is provided by the perimeter wall. Because the cross-section of the perimeter wall is an annulus, the area moments of inertia are IXX = IYY = IZZ =

π 4 4 (r − rpw1 ) = 3.234×104 m2 , 4 pw2

π 4 4 (r − rpw1 ) = 6.469×104 m2 , 2 pw2

(7.3.13) (7.3.14)

where rpw1 and rpw2 are the inner and outer radii of the perimeter wall, respectively. The area of the cross-section is 2 2 A = π(rpw2 − rpw1 ) = 142.965 m2 .

(7.3.15)

7.3 numerical example

319

Translational Lumped-Mass The lumped-masses are usually assigned at the elevations of floor slabs. In this example, the perimeter wall is divided into four equal segments. Because the perimeter wall is uniformly distributed in the vertical direction, the elevations of lumped-masses M1 , M2 , and M3 for the perimeter wall are located at 10.55m, 21.1m, and 31.65m, respectively, as shown in Figure 7.22. The upper dome, inner dome, ring beam, and a portion of the perimeter wall are integrated as one part with mass M4 located at elevation 44m. The stick model has fixed base at the base slab; hence, the geometry and mass of the base slab are not considered further. The volume of each structural component is determined first. ❧ Volume of the Upper Dome The upper dome is modelled as a spherical cap with thin thickness.    2 2 Vud = 2πrud rud − rud −aud tud 

= 2π ×41.45× 41.45 − 41.452 −20.732 ×0.61 = 882.687 m3, (7.3.16) where rud is the radius of the upper dome sphere, aud is the radius of the circle of the projection, and tud is the thickness of the upper dome. ❧ Volume of the Inner Dome The inner dome is approximately modelled as a spherical cap with thin thickness because the complex geometric dimensions are not available.    2 2 −aid tid Vid = 2πrid rid − rid 

= 2π ×41.45× 41.45 − 41.452 −20.732 ×0.38 = 549.871 m3,

(7.3.17)

where rid is the radius of the inner dome sphere, aid is the radius of the circle of the projection, and tid is the thickness of the inner dome. ❧ Volume of the Ring Beam 2 2 Vrb = π(rrb2 −rrb1 )Hrb = π (22.642 −20.732 )×4.27 = 1111.221 m3 , (7.3.18)

where rrb1 and rrb2 are the inner and outer radii of the ring beam, respectively, and Hrb is the height of the ring beam. ❧ Volume of the Perimeter Wall 2 2 Vpw = π(rpw2 −rpw1 )Hpw = π (21.82 −20.732 )×42.19 = 6031.684 m3 , (7.3.19)

where rpw1 and rpw2 are the inner and outer radii of the perimeter wall, respectively, and Hpw is the height of the perimeter wall.

320

Multiplying the volume of each component by the density of the concrete (2400 kg/m3 ) yields the component mass. Table 7.9 lists the mass of each structure component; it can be seen that the masses obtained by hand calculations are comparable to those output from the 3D FE model given in Table 7.5. Table 7.9 Structure component mass by hand calculations.

Component Upper dome Inner dome Ring beam Perimeter wall

Mud Mid Mrb Mpw

Mass (kg) 2.1184 ×106 1.3197 ×106 2.6669 ×106 14.4760 ×106

Mass Moment of Inertia The mass moment of inertia measures the extent to which an object resists rotational acceleration about a particular axis. It should not be confused with the second moment of area, which is used in beam stiffness calculations. The mass moment of inertia is often also known as the rotational inertia and sometimes as the angular mass. For simple objects with geometric symmetry, one can often determine the moment of inertia in exact closed-form expression when the mass density is constant. For simplicity, approximate the upper dome and inner dome as flat solid disks and the ring beam and perimeter wall as thin tubes. ❧ Mass Moment of Inertia of the Upper Dome Mud, XX = Mud,YY = Mud, ZZ =

2 Mud aud = 2.276×108 kg · m2 , 4

2 Mud aud = 4.552×108 kg · m2 . 2

(7.3.20) (7.3.21)

❧ Mass Moment of Inertia of the Inner Dome Mid, XX = Mid,YY = Mid, ZZ =

2 Mid aid = 1.418×108 kg · m2 , 4

2 Mid rid = 2.836×108 kg · m2 . 2

(7.3.22) (7.3.23)

❧ Mass Moment of Inertia of the Ring Beam Note that the X- and Y-axes are located at the bottom of the ring beam. Mrb, XX = Mrb,YY =

2 +2H 2 ) Mrb (3rrb rb = 6.433×108 kg · m2 , 6

2 Mrb, ZZ = Mrb rrb = 12.541×108 kg · m2 ,

(7.3.24) (7.3.25)

where rrb = 12 (rrb1 +rrb2 ) = 21.685 m is the radius of the thin-walled ring beam.

7.3 numerical example

321

❧ Mass Moment of Inertia of the Perimeter Wall (1/8 of the Wall) As illustrated in Figure 7.22, the perimeter wall is divided into four equal segments. For each segment, the top half of the mass is lumped to the top end and the bottom half to the bottom end. Hence, each small lumped mass shown in 7.22 is one-eighth of the perimeter wall lumped to its end; its mass moments of inertia are ˆ pw, XX = M ˆ pw,YY M

2 2 +2 1 H Mpw 3rpw 8 pw = · = 4.259×108 kg · m2 , 8 6

2 ˆ pw, ZZ = Mpw · rpw = 8.183×108 kg · m2 , M

8

(7.3.26) (7.3.27)

where rpw = 12 (rpw1 +rpw2 ) = 21.265 m is the radius of the thin perimeter wall.

Stick Model Referring to Figures 7.4 and 7.22, there are four lumped masses in the stick model because the perimeter wall is divided into four segments, and the structure is fixedbase. Each segment of the perimeter wall is modelled by a two-node massless beam element, using BEAM44 element. The section properties are defined by real constants in ANsys, given by equations (7.3.13) and (7.3.14). The element stiffness matrix is calculated and assembled to obtain the global stiffness matrix. The shear areas of the beams need to be specified. For example, only the area of walls parallel to the X-axis, which are defined as shear walls, is counted as shear area in the X-direction. Because the containment perimeter wall is a thin-wall ring, only half of the area obtained in equation (7.3.15) is taken as the shear area in each horizontal translational direction. In ANsys or STARDYNE, the shear area is defined in the beam property by assigning a shear coefficient, which is multiplied with the actual section area to determine the shear area. As illustrated in Figure 7.22, each of the lumped-masses, M1 , M2 , and M3 , is obtained from two of one-eighth perimeter wall segments, one above and one below. Hence, MX = MY = MZ = 2×

Mpw 1.448×107 = = 3.619×106 kg, 8 4

(7.3.28)

ˆ pw, XX = 2×4.259×108 = 8.518×108 kg · m2 , MXX = MYY = 2× M

(7.3.29)

ˆ pw, ZZ = 2×8.183×108 = 16.365×108 kg · m2 . MZZ = 2× M

(7.3.30)

The lumped-mass M4 is obtained by combining the upper dome, inner dome, ring beam, and one-eighth of the perimeter wall as a single lumped-mass located at elevation of 44 m. The mass and mass moment of inertia can be determined by summing the

322 Table 7.10 Comparison of modes between 3D FE model and stick models.

Mode

3D FE Model Participation Factor F (Hz) X Y Z Mode

1 2

5.455 5.455

9

9.321

15 16 17 18 19 20 25 28 29

11.653 11.653 12.011 12.948 12.948 13.137 14.717 14.930 14.930 15.188 16.118 16.118 17.197 17.197 18.108 20.839 22.376 23.399 23.399 24.103 24.103

36 37 44 45 50 59 68 77 78 83 84

0.29 0.29 0.11

Stick Model F (Hz) Mode Shape

1 2

5.685 5.685

3F

10.822

3 4 5

15.297 16.869 16.869

Overall sway in X Overall sway in Y Vertical mode (Flexible UD)

0.03 0.03 0.11 0.01 0.01

0.01 0.01 0.04 0.08

0.04 0.04 UD is set as rigid 0.11 0.11 0.06 0.06 0.17 0.05 0.05 0.04 0.04 0.01 0.01

Vertical mode Overall sway in X Overall sway in Y

corresponding values of the components: MX = MY = MZ = =

Mpw + Mrb + Mud + Mid 8

1.448×107 +(2.667+2.118+1.320)×106 = 7.915×106 kg, 8

(7.3.31)

ˆ pw, XX + Mrb, XX + Mud, XX + Mid, XX MXX = MYY = M = (4.259 + 6.433 + 2.276 + 1.418)×108 = 14.385×108 kg · m2 ,

(7.3.32)

ˆ pw, ZZ + Mrb, ZZ + Mud, ZZ + Mid, ZZ MZZ = M = (8.183 + 12.541 + 4.552 + 2.836)×108 = 28.111×108 kg · m2 .

(7.3.33)

Modal Analysis A modal analysis is performed to obtain the modal information of both the 3D FE model and the stick model. Table 7.10 lists the modes with modal frequencies less than 25 Hz and at least 1 % of the normalized modal participation factors. The normalized

7.3 numerical example ANSYS 14.0 NODAL SOLUTION STEP=1 SUB =1 FREQ=5.45545 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.337E-03 SMX =.337E-03

323 MX

MX

YV =-1 DIST=28.5524 ZF =25.1168 VUP =Z PRECISE HIDDEN 0 .374E-04 .748E-04 .112E-03 .150E-03 .187E-03 .224E-03 .262E-03 .299E-03 .337E-03

Z Y

X

MN

MN

Z Y

0 .333E-04 .666E-04 .999E-04 .133E-03 .167E-03 .200E-03 .233E-03 .267E-03 .300E-03

X

ANSYS 14.0 NODAL SOLUTION STEP=1 SUB =36 FREQ=16.1176 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.288E-03 SMX =.288E-03 YV =-1 DIST=26.8761 ZF =23.5928 VUP =Z PRECISE HIDDEN 0 .320E-04 .640E-04 .960E-04 .128E-03 .160E-03 .192E-03 .224E-03 .256E-03 .288E-03

MX

MX

ZZ

Y

Figure 7.23

X

Z MN

Y

NODAL SOLUTION STEP=1 SUB =1 FREQ=5.68467 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.300E-03 SMX =.300E-03 YV =-1 DIST=24.255 XF =1.09351 YF =.14049 ZF =22.05 VUP =Z

X MN

NODAL SOLUTION STEP=1 SUB =6 FREQ=16.8698 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.260E-03 SMX =.260E-03 XV =-1 DIST=24.2 XF =.106135 YF =.807316 ZF =22 VUP =Z PRECISE HIDDEN 0 .289E-04 .578E-04 .866E-04 .116E-03 .144E-03 .173E-03 .202E-03 .231E-03 .260E-03

Comparison of horizontal sway modes.

modal participation factor is the absolute values of the modal participation factor divided by the sum of the modal participation factors, which indicates the percentage contribution of the mode to the total response. It is seen that the first and second overall sway modes in the horizontal directions of the stick model agree well with the results of the 3D FE model in both the modal frequencies (Table 7.10) and in mode shapes (Figure 7.23). For mode 36 of the 3D FE model, there is significant local vibration in the upper dome, which is not shown for clearer presentation.

324

Figure 7.24 ANSYS 14.0 NODAL SOLUTION STEP=1 SUB =9 FREQ=9.32066 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.00184 SMX =.00184 YV =-1 DIST=29.774 ZF =26.2273 VUP =Z PRECISE HIDDEN 0 .204E-03 .409E-03 .613E-03 .818E-03 .001022 .001226 .001431 .001635 .00184

Figure 7.25

Comparison of vertical modes with rigid upper dome. MX

Z Y

ANSYS 14.0 NODAL SOLUTION STEP=1 SUB =3 FREQ=10.8219 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.002733 SMX =.002733

MX

X

Z MN

MN

Y

X

YV =-1 DIST=25.4678 ZF =23.1525 VUP =Z PRECISE HIDDEN 0 .304E-03 .607E-03 .911E-03 .001215 .001519 .001822 .002126 .00243 .002733

Comparison of vertical modes with flexible rigid dome.

However, the vertical mode with frequency of 15.297 Hz does not agree with any of the vertical modes of the 3D FE model. This is because the upper dome is considered rigid in the vertical direction in the stick model, while it is flexible in the 3D FE model. If the Young’s modulus of the upper dome is increased so that it can be regarded as rigid in the 3D FE model, then the vertical mode has frequency of 15.188 Hz, agreeing with that of the stick model well. The mode shapes are shown in Figure 7.24.

7.3 numerical example

325

Stick Model with Flexible Upper Dome To capture the vertical mode (mode 9 of the 3D FE model with frequency 9.321 Hz), the upper dome must be considered flexible. For simplicity of illustration, the upper dome is considered as a flat circular plate of radius rud clamped at the circular edge. The mass of a circular portion of radius

1 4 rud ,

M5, Z =

or

1 M , 16 ud

(7.3.34)

is lumped at the centre of the plate (node 5) that has only one DOF in the vertical Z-direction, as shown in Figure 7.22. The mass of node 4 in the Z-direction becomes M4, Z =

Mpw 15 + Mrb + Mud + Mid , 8 16

(7.3.35)

while all other values are the same as those given by equations (7.3.31) to (7.3.33). Node 5 and node 4 are connected by a linear spring in the Z-direction; the stiffness of the spring Kud can be determined from the deflection of the centre of the plate Zmax under a point load W applied at the centre: Kud =

W 16π D2 = a , Zmax ud

D=

3 E tud . 12(1−ν 2 )

(7.3.36)

The vertical mode of the stick model with flexible upper dome (mode 3F in Table 7.10) has a frequency of 10.822 Hz, which is reasonably close to that of mode 9 of the 3D FE model. The mode shapes are compared in Figure 7.25.

Summary A stick model captures the global dynamic behaviour of the structure, while a 3D FE model includes a large number of local modes besides the global modes. From Figures 7.23 to 7.25, it is seen that the dominant modes of the stick models in the horizontal and vertical directions are comparable with the results from the 3D FE model. However, there are some discrepancies between the modal frequencies obtained from these two models. This can be explained by the limitation of the stick model which is mainly used for capturing the overall dynamic properties of the structure while local vibrations are ignored. Because the perimeter wall is a thin-tube structure and the upper dome is a thin shell, there are various local vibrations. Typical local modes of the 3D containment structure are plotted in Figures 7.17 and 7.18. However, limitations in the stick models do not mean that they are not useful. For simple and regular structures, such as the containment structure, a stick model can be used to determine the overall response parameters. Furthermore, stick models can be applied in a multiple-step analysis, in which overall structure responses are obtained in stick models first and are then used as input in subsequent analyses.

326 6 Node 4

Spectral acceleration (g)

5 4

Dashed line Solid line

3D model Stick model

3 2 1 0 0.1

Node 2

1

Frequency (Hz)

Figure 7.26

10

100

Comparison of FRS.

Comparison of Floor Response Spectra (FRS) from 3D FE Model and Stick Model To further investigate the effect of modelling methods on structural responses, the direct spectra-to-spectra method (see Chapter 8 for details on generating FRS) is employed to generate FRS at Node 2 and Node 4 (Figure 7.4) for both the 3D FE model and stick model. For simplicity of presentation, only seismic input in the X-direction is considered, which is taken as the 5 % USNRC R.G. 1.60 GRS anchored at 0.3g PGA. Figure 7.26 compares the FRS with 5 % modal damping in the X-direction at Nodes 2 and 4. It is seen that the FRS obtained from these two models are comparable for both one-mode dominant level (single FRS peak, Node 4) and multiple-mode dominant level (multiple FRS peaks, Node 2). Therefore, if FRS are the response parameters of interest, a stick model could produce results with acceptable accuracy. ❧

❧

In this chapter, the fundamentals of modelling of engineering structures are introduced, and numerical examples using different modelling methods are presented. ❧ In general, FE modelling includes the determination or selection of structural geometry, material properties, element type, element properties, meshing, boundary conditions, and load conditions. ❧ An FE model should be validated before the solution stage. The details of the modelling can be inspected between the modelling steps, and the rationality of the whole model can be examined by performing analyses for simple loading cases.

7.3 numerical example

327

❧ The mesh density of the model depends on the purpose of the structural analyses. A coarse mesh may be sufficient to capture the nodal responses of the structure. More refined meshes are needed when the stress-level responses are required, or responses at certain locations with more complex geometry, such as structural openings and component interfaces. ❧ A stick model could accurately represent the overall structural responses, such as floor response spectra. A numerical example shows that FRS obtained by a stick model are comparable to those obtained from a detailed 3D FE model.

C

H

8 A

P

T

E

R

Floor Response Spectra 8.1 Introduction Secondary systems are structures, systems, and components (SSCs) supported by the primary structures, such as reactor buildings and internal structures. These secondary systems play various functions to maintain operational activities and safe shutdown of nuclear power plants. Secondary systems are usually attached to the floors or walls of primary systems; as a result, they are subject to the vibrational motion of the floor to which they are attached rather than the ground motion excitation directly. The vibration transmitted by primary structures could be amplified serval times and may damage secondary systems. Hence, the seismic input for secondary systems is not only determined by a ground motion input to the primary structure, but also significantly affected by the dynamic characteristics of the supporting primary structure. Seismic analysis, design, and qualification for some secondary systems in nuclear power plants are mandatory, e.g., ASCE 4-98 (ASCE, 1998) and ASCE 43-05 (ASCE/SEI, 2005). The determination of seismic input for secondary systems is essential in seismic margin assessment (SMA) and seismic probabilistic safety assessment (seismic PSA) for nuclear facilities. It is therefore important to develop accurate, reliable, and practical approaches to determine the seismic input for secondary systems and to study the seismic behavior of secondary systems.

8.1.1

Seismic Analysis Methods for Secondary Systems

Two approaches are employed for the determination of seismic response of secondary systems, i.e., the floor response spectrum (FRS) approach and the combined primarysecondary system approach, as shown in Figure 8.1. 328

8.1 introduction

329

Response of floor un(t) Secondary system

un un(t) Secondary system Floor response spectrum

GRS-compatible time-history ug(t)

Primary system

Primary system

ug(t)

ug(t) Floor response spectrum (FRS) approach

Figure 8.1

Ground response spectrum

Combined primary-secondary system approach

Seismic analysis methods for secondary systems.

The floor response spectrum approach is a decoupled analysis method, in which the primary and secondary systems are analyzed separately. A dynamic analysis, using time-history analysis or modal analysis, is performed for the primary structure. The input for the primary structure can be a prescribed ground motion response spectrum or a set of spectrum-compatible time-histories. Without considering the effect of the secondary systems, the responses of the primary structural system on the desired floors (at which the secondary systems are attached) are obtained, from which response spectra of the floors, called floor response spectra (FRS), are generated and are used as input to the secondary systems. In the combined primary-secondary system approach, second systems are modelled as an integral part of the primary-secondary structural system. Either spectral analysis or time-history analysis can be applied to determine the seismic responses of secondary systems. Although this approach can give theoretically accurate responses of secondary systems, there are some challenges and difficulties: ❧ The large differences between the characteristics of primary structure and secondary systems, such as mass and stiffness, may cause serious numerical problems in modal analysis or time-history analysis and give inaccurate solutions.

330

❧ There are usually an excessive number of degrees-of-freedom in a combined primarysecondary system, the analysis is not efficient because only seismic responses of secondary systems are of interest. ❧ Although a combined primary-secondary system analysis can give reliable seismic response for a secondary system at a certain location, there are a large number of secondary systems, and the locations of secondary systems are varied in a nuclear power plant, and the recalculation of each case is a tedious and expensive process. As a result, the combined primary-secondary system approach is not widely used in practice. For secondary systems whose masses, stiffnesses, and resulting frequency ranges should be considered, a combined primary-secondary system can be established to account for possible dynamic interaction effects. 4.0 Damping = 5%

Frequency ratio Rf

3.5 3.0

NUREG-0800

2.5

Dynamic analysis

2.0

0.0

0.001

0.01

15%

Coupling required

Coupling not required

1.5 1.25 1.0 0.8 0.5

10%

0.1

1

Mass ratio Rm Figure 8.2

Decoupled and coupled analysis criteria.

Criteria on decoupled and coupled analyses are recommended in some studies (Gupta and Tembulkar, 1984; Hadjian and Ellison, 1986) and USNRC Standard Review Plan (NUREG-0800, USNRC, 2013, page 3.7.2-10). Typical rules are shown in Figure 8.2, in which the solid lines are suggested in Hadjian and Ellison (1986) based on results of parametric studies of different models, and the dashed lines represent the criteria specified in NUREG-0800. The criteria of decoupling analysis depend on the

8.1 introduction

331

mass ratio and the frequency ratio between the secondary and the primary systems Rm =

Total mass of secondary system , Total mass of the primary structure

Fundamental frequency of secondary system RF = . Dominant frequency of the primary structure

(8.1.1)

Although there are problems associated with the assumption of decoupled analysis in some special applications, this decoupling assumption is widely accepted in practice because the majority of secondary systems have relative small masses compared to the masses of the supporting primary structure; the effect of interaction between the primary and secondary systems is negligible. For such secondary systems, a separate analysis is performed using time-history of the floor response or floor response spectra as the input. In seismic analysis and design of secondary systems, floor response spectra are more familiar to engineers and are more convenient to use in practice.

8.1.2

Floor Response Spectrum

ASCE Standard 4-98 (ASCE, 1998) recommends that floor response spectra (FRS) be generated by time-history analyses or a direct spectra-to-spectra method (Figure 8.3).

Figure 8.3 Two methods of generating floor response spectra.

332

Time-History Method A dynamic analysis for primary structure is conduct by using step-by-step time integration. The time-histories of responses at the floors (nodes) to which secondary systems are attached are obtained and are used to generate FRS.A time-history analysis can give accurate responses for the given ground motion record. However, recorded ground motion time-histories representative of the site of interest are often not available; ground motions compatible with a reference ground motion response spectrum are generated as input for the primary structure. It has been recognized that there is significant variability in the FRS generated by the timehistory method, in the sense that two spectrum-compatible time-histories may give significantly different FRS.

Direct Spectra-to-Spectra Method The direct spectra-to-spectra method can avoid the deficiencies of time-history method; floor response spectra are calculated directly from ground motion spectra. A modal analysis of the primary structure is performed to obtain the basic modal information of the structure, including modal frequencies, modal shapes, and modal participation factors. Response spectra of desired floors are then obtained in terms of the modal information and the prescribed ground response spectrum. A direct method based on Duhamel’s integral is developed in Section 8.4.

8.2 Floor Response Spectra 8.2.1 Ground Response Spectrum In seismic analysis of nuclear power plants, seismic excitations in two orthogonal horizontal directions H1 and H2 , and vertical direction V are usually applied. Suppose 2 H2 3 V 1 that ugI (t), I = 1, 2, 3, where ug1 (t) = uH g (t), ug (t) = ug (t), and ug (t) = ug (t), is the

displacement of the ground motion in direction I. When a single degree-of-freedom (SDOF) oscillator with circular frequency ω0 and damping coefficient ζ0 is subjected to this ground motion, as shown in Figure 8.5, the equation of motion is x¨GI + 2ζ0 ω0 x˙GI + ω02 xGI = − u¨gI (t),

(8.2.1)

where xGI (t) = uGI (t)−ugI (t) is the relative displacement of the oscillator and uGI (t) is the absolute displacement. The subscript “G” denotes that the oscillator is mounted on the ground. The absolute acceleration is u¨GI (t) = x¨GI (t)+ u¨gI (t) = −(2ζ0 ω0 x˙GI +ω02 xGI ).

8.2 floor response spectra

333

Floor response spectrum

SF(ω0, ζ0)=max |uF(t)|

uF(t) (Absolute)

xF(t)=uF(t)−u(t) (Relative) ω 0, ζ 0

u SDOF primary structure

u(t) (Absolute)

Floor response ω, ζ

ug(t) Figure 8.4

FRS of SDOF primary structure.

Floor response spectrum

N

uF,n, j (t) (Absolute)

xF,n, j (t)= uF,n, j (t)− uF,n, j (t) (Relative)

un,6 un,3 un,2 n

un,5 ω0, ζ0

un,1 un,4 Floor response

un,j (t) (Absolute) Ground response spectrum

3

2

SAi (ω0, ζ0)=max |uGi (t) |

uGi (t) (Absolute)

Multiple DOF primary structure

xGi (t)= uGi (t)− ugi (t) (Relative)

1

ω0, ζ0 SDOF oscillator

Ground motion

ug3(t)

Sn,j (ω0, ζ0)=max |uF,n, j (t)|

ug2(t) ug1(t)

ugi (t)

Tridirectional ground excitations Figure 8.5

Response spectra.

The maximum absolute acceleration of the oscillator 



SAI (ω0 , ζ0 ) = maxu¨GI (t) is the ground (acceleration) response spectrum (GRS) in direction I.

(8.2.2)

334

8.2.2 FRS of SDOF Primary Structure For the special case when the primary structure is SDOF with circular frequency ω and damping coefficient ζ , u(t) and x(t) = u(t)−ug(t) are the absolute and relative displacements of the structure, respectively, satisfying x(t) ¨ + 2ζ ω x(t) ˙ + ω2 x(t) = −u¨g(t),

(8.2.3)

˙ − ω2 x(t). u(t) ¨ = x(t) ¨ + u¨g(t) = −2ζ ω x(t)

(8.2.4)

The motion of an SDOF oscillator with circular natural frequency ω0 and damping coefficient ζ0 mounted on the primary structure (Figure 8.4) is governed by x¨F + 2ζ0 ω0 x˙F + ω02 xF = − u(t), ¨

(8.2.5)

¨ = −2ζ0 ω0 x˙F (t) − ω02 xF (t), u¨F (t) = x¨F (t) + u(t)

(8.2.6)

where xF (t) = uF (t)−u(t) and uF (t) are the relative and absolute displacements of the oscillator. The maximum absolute acceleration of the oscillator 



SF (ω0 , ζ0 ) = maxu¨F (t)

(8.2.7)

is the floor (acceleration) response spectrum (FRS) of the SDOF primary structure.

8.2.3 FRS of Multiple Degrees-of-Freedom (MDOF) Primary Structure Consider a three-dimensional model of a structure with N nodes studied in Section 3.6.4. A typical node n has six DOF: three translational DOF un,1 , un,2 , un,3 , and three rotational DOF un,4 , un,5 , un,6 . The structure is subjected to tridirectional seismic excitations (Figure 8.5). The relative displacement vector x of dimension 6N is governed by M x¨ (t) + C x˙ (t) + Kx(t) = −M

3  I=1

I I u¨gI (t).

(8.2.8)

Applying modal analysis presented in Section 3.6.4, equation (8.2.8) is decoupled into 6N SDOF systems: q¨KI (t) + 2ζK ωK q˙KI (t) + ωK2 qKI (t) = − u¨gI (t), K = 1, 2, . . . , 6N, I = 1, 2, 3.

(8.2.9)

The absolute acceleration of the nth node in direction j due to earthquake excitation in direction I can be obtained using equations (3.6.9) and (8.2.9) I I ¨gI (t)δIj = u¨ n, j (t) = x¨ n, j (t) + u

6N  K=1

ϕn, j; K KI q¨KI (t) + u¨gI (t)δIj

8.3 time-history method for generating frs

=

6N 

=−





ϕn, j; K KI − u¨gI (t) − (2ζK ωK q˙KI + ωK2 qKI ) + u¨gI (t)δIj ,

K=1 6N 

=−

335

K=1 6N  K=1

6N  K=1

ϕn, j; K KI = δIj ,

ϕn, j; K KI (2ζK ωK q˙KI + ωK2 qKI ) I u¨ n, j; K ,

I 2 I I I u¨ n, j; K = ϕn, j; K K (2ζK ωK q˙K + ωK qK ),

(8.2.10)

I I in which u¨ n, j; K is the contribution from the Kth mode and ϕn, j; K K is the contribution

factor.

Floor Response Spectrum If the absolute response un,I j (t) of the nth node in direction j due to earthquake excitation in direction I is input to an SDOF oscillator with circular frequency ω0 and damping coefficient ζ0 , as shown in Figure 8.5, the governing equation of motion is I x¨ FI , n, j + 2ζ0 ω0 x˙ FI , n, j + ω02 x FI , n, j = − u¨ n, j (t), I I 2 I u¨ FI , n, j (t) = x¨ FI , n, j (t) + u¨ n, j (t) = −2ζ0 ω0 x˙ F, n, j − ω0 x F, n, j ,

(8.2.11) (8.2.12)

I (t) the displacement of the oscillator relative to the where x FI , n, j (t) = u FI , n, j (t)−un, j

nth node in direction j and u FI , n, j (t) is the absolute displacement of the oscillator. The subscript “F” denotes that the oscillator is mounted on the floor. The maximum absolute acceleration of the oscillator 



Sn,I j (ω0 , ζ0 ) = u¨ FI , n, j (t)max

(8.2.13)

is the floor (acceleration) response spectrum (FRS) of the nth node (floor) in direction j subjected to earthquake excitation in direction I. It is specified in ASCE 4-98 (ASCE, 1998) that, for direct spectra-to-spectra method, when the response spectrum at a given location and in a given direction has contributions from more than one spatial component of earthquake, these contributions shall be combined by the square root of sum of squares (SRSS) rule. Hence, combining contributions from tridirectional earthquake excitations, FRS of the nth node in direction j is given by



Sn, j (ω0 , ζ0 ) =

3  

I=1

Sn,I j (ω0 , ζ0 )

2

.

(8.2.14)

8.3 Time-History Method for Generating FRS One of the most commonly used methods for generating FRS is the time-history method. As discussed in Section 8.1.2 and illustrated in Figure 8.3, the structural

336

response time-histories at a specified location of the structure are obtained first using the time-history method, as presented in Section 3.6. The response spectra of the structural response time-histories are then calculated and postprocessed to determine the FRS at the specified location. This section focuses on the FRS generation procedure subsequent to that the structural response time-histories have been obtained.

8.3.1 Treatment of Spatial Components in Generating FRS In the process of time-history analysis for a structure, the tridirectional ground-motion time-histories, as presented in Chapter 6, may be applied simultaneously or individually depending on the time-history analysis technique used. For instance, the structural dynamic analysis using direct integration requires the tridirectional ground-motion time-histories being input simultaneously, while SSI analysis using the substructuring method needs the time-history to be applied for each direction individually. ASCE 4-98 (ASCE, 1998) and ASCE 4-16 (ASCE/SEI, 2017) outline three cases for the treatment of spatial components in generating the FRS. 1. When the supporting structure is subjected to the simultaneous action of three statistically independent spatial components of seismic input ground-motion (as discussed in Chapter 6), two horizontal translational components and one vertical translational component of the floor acceleration shall be used to compute the corresponding FRS. In this case, the interaction between the effects of spatial components of the input ground motion has been implicitly incorporated in the analysis. 2. When the supporting structure is subjected individually to the action of the three statistically independent spatial components of seismic input ground motion, the floor acceleration for each direction shall be obtained by the algebraic summation of the codirectional accelerations from the three individual analyses. The resulting floor accelerations shall then be used to compute the corresponding FRS. In this case, the interaction between the effects of spatial components of the input ground motion has been considered and the phases in the floor accelerations have been preserved, by the algebraic summation. 3. When the time-history analysis of the supporting structure is performed individually for each of two horizontal spatial components and one vertical spatial component of the seismic input ground motion, and these spatial components are not statistically independent, the floor accelerations from each individual analysis shall be used to generate floor response spectra. The resulting FRS shall then be obtained

8.3 time-history method for generating frs

337

by combining the codirectional spectral amplitudes from the three individual analyses using the SRSS rule. This case could be very rare for building structures in the nuclear industrial practice because statistical independence is an intrinsic property of the recorded ground motions and is one of the acceptance criteria for generating the input ground-motion time-histories.

8.3.2 Frequency Interval for Generating FRS Similar to the requirement of frequency interval for generating the spectrum-compatible ground-motion time-histories, as presented in Section 6.1, the FRS should also be computed at sufficiently small frequency interval. The frequencies, at which the FRS is computed, should include: 1. the fundamental frequencies of the supporting structure as significant spectral peaks are normally expected at those frequencies; 2. frequencies of all supporting substructures in cases involving a subsystem mounted on substructures; 3. predominant frequencies of input ground motions. CSA N289.3 (CSA, 2010a) requires that the FRS values should be calculated following the frequency increments specified for generating the spectrum-compatible groundmotion time-histories. ASCE 4-98 and ASCE/SEI 4-16 provide an extra option to establish a set of frequencies that each frequency is within 5 % or less of the previous one. ASCE 4-98 and ASCE/SEI 4-16 also recommend that the frequency interval be increased in the frequency range above twice the dominant SSI frequency to capture the spectral peaks due to higher modes effects.

8.3.3 Treatment of Uncertainties in Generating FRS FRS are used as seismic inputs for analyzing the SSCs mounted on the support structures. The generation of FRS should account for uncertainties in response due to the uncertainties in the supporting structure frequencies and SSI analysis, including uncertainties in material properties of structure and soil, uncertainties in damping values of structure, and uncertainties in modelling techniques for soil, structure, and SSI. In the nuclear power industrial practice, two acceptable approaches can be used to account for uncertainties in the dynamic behaviour of soil and structures for generating the FRS: peak broadening and reduction, and peak shifting, as specified in many codes and standards (CSA N289.3, ASCE 4-98, ASCE 4-16).

338

Spectral acceleration (g)

7 6 5

0.15 f1

Best-estimate soil case

+0.15 f1

S1

0.15 S1 Raw FRS

4

Broadened FRS

3

Broadened-and-reduced FRS

2 1 f1 1 10 Figure 8.6 Peak broadening and reduction of FRS.

0 0.1

Frequency (Hz) 100

6 Spectral acceleration (g)

5.70 5 4

f0.8

5.70× 80%=4.56 f0.8 =5.00−4.20=0.8

3

fc =(4.2+5)/2 = 4.6

2

f0.8 = 0.17 < 0.3 fc

1

fc 10 4.20 5.00 Figure 8.7 Bandwidth-to-central-frequency ratio.

0 0.1

−0.15f +0.15f FRS-1

f1 f f2

Freq

f1 f f2

Freq Figure 8.8

Freq

+0.15f

Freq

Peak shifting of FRS.

FRS-5

Acceleration

FRS-4

Acceleration

FRS-3

FRS-2

f 1 f f2

Freq

f2 − f

f − f1

f1 f f2

Acceleration

Acceleration

−0.15 f

Acceleration

f1 f f2

Acceleration

Frequency (Hz) 100

1

f1 f f2

Freq

8.4 direct method for generating frs

339

1. Peak Broadening and Reduction. The minimum broadening is ❧

±15 % at each spectral peak for the best-estimate soil shear modulus case,

❧

+15 % at each spectral peak for the upper bound soil shear modulus case,

❧

−15 % at each spectral peak for the lower bound soil shear modulus case.

Figure 8.6 shows the peak broadening for the best-estimate soil case. In conjunction with the peak broadening, a 15 % reduction may be applied to the narrow frequency peaks of the unbroadened FRS for each soil case if the subsystem damping ratio is less than 10 %. Narrow frequency peaks has a bandwidth-to-central-frequency ratio less than 0.30, as defined in Figure 8.7, i.e., F0.8 < 0.3, Fc in which F0.8 is the total frequency range over spectral amplitudes that exceed 80 % of the peak spectral amplitude, and F c is the central frequency for the frequencies that exceed 80 % of the peak amplitude. Figure 8.6 also shows the peak reduction for the best-estimate soil case. The final FRS shall be an envelope of the peak broadened-and-reduced spectra for best-estimate, upper-, and lower-bound soil cases. This approach is simple and economical but may introduce substantial conservatism in the subsystem seismic analysis. 2. Peak Shifting. If there are N subsystem natural frequencies, F n , n = 1, 2, . . . , N, satisfying the inequality 0.85 F < Fn < 1.15 F, where F is the FRS peak frequency, (N+2) peak shifting shall be performed, i.e, shifting spectral frequencies by   ±0.15 F and  F − F  . As a result, (N+3) FRS are obtained, including the raw n

FRS and (N+2) shifted FRS, for the subsystem seismic analysis. Figure 8.8 shows a case with two subsystem natural frequencies within the specified frequency range and five FRS generated by shifting. This peak shifting procedure should be applied independently to the best-estimate, upper-bound, and lower-bound soil cases. The envelope of the resulting responses of the subsystem seismic analysis for all the FRS generated should be used for design and evaluations.

8.4 Direct Method for Generating FRS In this section, a direct spectra-to-spectra method for generating FRS is developed based on Duhamel’s integral (Jiang et al., 2015).

340

8.4.1 SDOF Oscillator Mounted on SDOF Structure Consider an SDOF oscillator mounted on an SDOF structure, as shown in Figure 8.4. Adopt the notations h(t) = e−ζ ωt

sinωd t , ωd

h0 (t) = e−ζ0 ω0 t

sinω0,d t , ω0,d

hc (t) = e−ζ ωt hc0 (t) = e−ζ0 ω0 t

cos ωd t , ωd cos ω0,d t , ω0,d

 ωd = ω 1−ζ 2 ,

(8.4.1)

 ω0,d = ω0 1−ζ02 . (8.4.2)

Motion of Structure For an SDOF system (8.2.3) with zero initial conditions, using Duhamel’s integral, the relative displacement x(t) and the relative velocity x(t) ˙ can be expressed as x(t) = h(t) ∗ u¨g(t),

˙ (t) ∗ u¨ (t), x(t) ˙ =h g

(8.4.3)

where h(t) is the unit impulse response function with respect to base excitation of the structure defined by equation (8.4.1) (see Section 3.3.3). The derivative of h(t) is ˙ (t) = −  ζ h e−ζ ωt sinωd t + e−ζ ωt cos ωd t = −ζ ω h(t) + e−ζ ωt cos ωd t. (8.4.4) 2 1−ζ Substituting equation (8.4.3) into (8.2.4), the absolute floor acceleration of the structure is given by ˙ (t) ∗ u¨ (t) − ω2 h(t) ∗ u¨ (t). u(t) ¨ = −2ζ ω h g g

(8.4.5)

Motion of Oscillator The motion of the structure, to which the oscillator is attached, defines the input to the SDOF oscillator with circular natural frequency ω0 and damping coefficient ζ0 ; the relative and absolute motions of the oscillator are governed by equations (8.2.5) and (8.2.6), respectively. Using Duhamel’s integral and equation (8.4.5), the relative displacement xF (t) and velocity x˙F (t) between the structure and the oscillator are ˙ (t) ∗ u¨ (t) − ω2 h (t) ∗ h(t) ∗ u¨ (t), xF (t) = h0 (t) ∗ u(t) ¨ = −2ζ ω h0 (t) ∗ h g 0 g ˙ (t) ∗ u(t) ˙ (t) ∗ h ˙ (t) ∗ u¨ (t) − ω2 h ˙ (t) ∗ h(t) ∗ u¨ (t), ¨ = −2ζ ω h x˙F (t) = h 0 0 g 0 g

(8.4.6)

where the unit impulse response function h0 (t) is defined by equation (8.4.2). Substituting (8.4.6) into (8.2.6) yields the absolute acceleration of the oscillator u¨F (t) = −2ζ0 ω0 x˙F (t) − ω02 xF (t) 

˙ (t) ∗ h ˙ (t) ∗ h(t) ˙ (t) + 2ζ ω ω2 · h = 4ζ0 ζ ω0 ω · h 0 0 0 0 

˙ (t) + ω2 ω2 · h (t) ∗ h(t) ∗ u¨ (t), + 2ζ ω02 ω · h0 (t) ∗ h 0 0 g

(8.4.7)

8.4 direct method for generating frs

341

which can be simplified to 

u¨F (t) = (1−2ζ02 −2ζ 2 +4ζ02 ζ 2 )ω02 ω2 · h0 (t) ∗ h(t)  + 4ζ0 ζ (1−ζ 2 )(1−ζ02 ) ω02 ω2 · hc0 (t) ∗ hc (t)  + 2ζ0 1−ζ02 (1−2ζ 2 ) ω02 ω2 · h(t) ∗ hc0 (t)   + 2(1−2ζ02 )ζ 1−ζ 2 ω02 ω2 · h0 (t) ∗ hc (t) ∗ u¨g(t).

(8.4.8)

For most SSCs in nuclear power plants, the damping coefficients ζ, ζ0 < 0.2 (EPRITR-103959, EPRI, 1994). When t is sufficiently long, it is reasonable to assume that   c   h (t) ∗ hc (t) ∗ u¨ (t) h (t) ∗ h(t) ∗ u¨ (t) ≈ 0 0 g g max max     ≈ h(t) ∗ hc0 (t) ∗ u¨g(t)max ≈ h0 (t) ∗ hc (t) ∗ u¨g(t)max .

(8.4.9)

In general, the maximum values of the terms in (8.4.8) do not occur simultaneously because of the phase differences between the sine and cosine terms. The SRSS combination rule is used to calculate the maximum response. For lightly damped systems, the values of ζ 2 , ζ02 , and ζ0 ζ are very small compared to 1, so that the corresponding terms are negligible. The maximum response of the oscillator is then reduced to     2 ω2 h (t) ∗ h(t) ∗ u¨ (t) u¨ (t) , ≈ ω g 0 F 0 max max

(8.4.10)

which is expressed analytically as a double convolution. Note that, if the SDOF oscillator is mounted directly on the ground, the term ω2 h(t) is removed from equation (8.4.10) and FRS reduces to GRS, i.e., 







SA(ω0 , ζ0 ) = ω02 h0 (t) ∗ u¨g(t)max = ω0 e−ζ0 ωt sinω0 t ∗ u¨g (t) Denote that

max

.

(8.4.11)

C (t) = h0 (t) ∗ h(t). From the definition of Duhamel’s integral, it is obvious

C (t) is the response of an oscillator with the circular frequency ω0

and damping

coefficient ζ0 under the excitation of h(t). The equation of motion is given by

C¨ (t) + 2ζ0 ω0 C˙ (t) + ω02 C (t) = h(t) =

1 −ζω sinωd t. ωd e

(8.4.12)

The general solution for this differential equation is C (t) = C C (t)+ C P (t), where

C C (t) = e−ζ0 ω0 t



C1 cos ω0, d t + C2 sinω0, d t ,

for ζ0 < 1,

(8.4.13)

is the complementary solution with coefficients C1 and C2 determined by the initial

conditions, and C P (t) is a particular solution determined in the following.

342

8.4.2 Non-tuning Case If ω = ω0 and ζ  = ζ0 , the right-hand side of equation (8.4.12) is not contained in the complementary solution. A particular solution C P (t) is given by

C P (t) = e−ζ ωt where



P1 cos ωd t + P2 sinωd t ,

 r 1−ζ 2 · A P1 = − 2 , ω0 ω d · 

P2 =

(1−ζ 2 ) · B , ω02 ωd · 

r=

(8.4.14) ω , ω0

and A = 2(ζ0 −ζ r), B = 1−r 2 −ζ r · A,  = r 2 · A+(1−ζ 2 ) · B2 . For zero initial conditions C (0) = 0 and C˙ (0) = 0, the coefficients C and C of the complementary solution 1

are given by C1 = −P1 ,

2

 r 1−ζ 2 · P2 A · P1 −  . C2 = −  2 1−ζ02 1−ζ02

Having obtained C (t) = H0 (t) ∗ H(t), the maximum absolute acceleration of the oscillator given by equation (8.4.10) is     u¨ (t) =  C1 ω02 ω2 e−ζ0 ω0 t cos ω0, d t + C2 ω02 ω2 e−ζ0 ω0 t sinω0, d t F max 

 + P1 ω02 ω2 e−ζ ωt cos ωd t + P2 ω02 ω2 e−ζ ωt sinωd t ∗ u¨g(t)

max

. (8.4.15)

Floor Response Spectra For lightly damped systems, ω0, d ≈ ω0 and ωd ≈ ω. Equation (8.4.15) reduces to     u¨ (t) =  C1 ω02 ω2 e−ζ0 ω0 t cos ω0 t + C2 ω02 ω2 e−ζ0 ω0 t sinω0 t F max 

 + P1 ω02 ω2 e−ζ ωt cos ωt + P2 ω02 ω2 e−ζ ωt sinωt ∗ u¨g(t)

max

      ˙ (t) ∗ u¨ (t) + C ω ω2 · ω2 h (t) ∗ u¨ (t) = C1 ω0 ω2 · ω0 h 0 g 2 0 g 0 0     ˙ (t) ∗ u¨ (t) + P ω2 ω · ω2 h(t) ∗ u¨ (t)  . (8.4.16) + P1 ω02 ω · ω h g 2 0 g max

  The maximum response u¨F (t)max may be overestimated if it is calculated by the sum of the maximum values of each term in equation (8.4.16) because the maximum       ˙ (t) ∗ u¨ (t) , or ω2 h (t) ∗ u¨ (t) and values of ω2 h(t) ∗ u¨g (t)max and ω h g g 0 0 max max   ˙  ω h 0 0 (t) ∗ u¨g (t) max , do not occur simultaneously. Because there is π/2 phase difference between the sine and cosine functions, it is appropriate to employ the SRSS combination rule to calculate the maximum absolute acceleration. Therefore, in non-tuning cases (when the frequencies of the structure and equipment are well separated), the FRS is obtained from equation (8.4.16) as

S2F (ω0 , ζ0 ) = AF02 · SA2 (ω0 , ζ0 ) + AF 2 · SA2 (ω, ζ ), -

-

(8.4.17)

8.4 direct method for generating frs

343 ug(t)

ug(t) x0(t)

x(t)

ω0, ζ0

ug(t) x0(t)

ug(t)

x(t)

ω0, ζ0

ω0, ζ0

ω, ζ Very stiff

ω, ζ

ω, ζ Very flexible

ω > ω0

Figure 8.9 Two extreme cases of motion amplification.

in which





SF (ω0 , ζ0 ) = u¨F (t)max

is the FRS or the spectral acceleration of an oscillator

with the circular frequency ω0 and damping ratio ζ0 mounted on the SDOF structure with circular frequency ω and damping ratio ζ ,

SA (ω0 , ζ0 ) is the GRS or the spec-

and AF tral acceleration of the oscillator mounted on the ground, and AF - 0 are the amplification factors discussed in the following section.

8.4.3 Amplification Factors For non-tuning cases, the amplification factors AF F are given by - 0 and A -

AF0 =  -

AF =  -

r2 (1−r 2 )2 +4(ζ02 +ζ 2 )r 2 −4ζ0 ζ r(1+r 2 ) 1 (1−r 2 )2 +4(ζ02 +ζ 2 )r 2 −4ζ0 ζ r(1+r 2 )

, (8.4.18) .

If damping is light and the effect of damping is neglected, the amplification factors are approximately

AF0 ≈ -

r2 , 1−r 2

AF ≈ -

1 , 1−r 2

ω r= ω . 0

(8.4.19)

From equation (8.4.17), the FRS SF (ω0 , ζ0 ) can be interpreted as a combination of • amplified spectral acceleration AF - 0 · SA (ω0 , ζ0 ) of the oscillator, and • amplified spectral acceleration AF - · SA (ω, ζ ) of the structure. To illustrate the physical meaning of equation (8.4.17), consider two extreme cases as shown in Figure 8.9: ❧ Frequency ratio r→∞ (ω  ω0 ): The structure is very stiff compared to the oscillator, so that the structure and the ground can be considered as an integral rigid body. The frequency components in ground motion, to which the oscillator is sensitive, are transmitted by the structure without modification. Therefore, the equipment

344

behaves as if it is directly mounted on the ground. When r→∞, the amplification

F factors AF - 0 = 1 and A - = 0 agree with this case. ❧ Frequency ratio r→0 (ω ω0 ): The oscillator is very stiff compared to the structure or the structure is very flexible compared to the oscillator, so that the response of the oscillator is the same as that of the structure. When r = 0, the amplification

F factors AF - 0 = 0 and A - = 1, and the maximum response of the oscillator is equal to

the spectral acceleration of the structure.

F given by equations (8.4.18) and (8.4.19) are for The amplification factors AF - 0 and A non-tuning cases. To extend the concept of amplification factors to perfect-tuning and near-tuning cases, the behavior of the amplification factors given by equation (8.4.19) is investigated by plotting them in Figure 8.10(a). The amplification factor of ground motion AF F are similar to the dynamic - 0 and A magnification factors (DMF) of an SDOF oscillator subjected to harmonic loading and under harmonic base excitation, respectively,

DMF0 = 

r2

-

(1−r 2 )2

+ (2ζ r)2

,

DMF =  -

1 (1−r 2 )2

+ (2ζ r)2

.

(8.4.20)

DMF0 and DMF are shown in Figure 8.10(b) and (c). It is seen that damping has little -

-

effect on the response amplification in non-tuning cases (when r is not close to 1), but has a significant effect on the response in perfect-tuning or near-tuning cases (when r approaches 1). Based on the expressions of

DMF0 -

and

DMF given by equation (8.4.20), -

when

the effect of damping is considered, it is appropriate to assume that the amplification

F are of the form, for both tuning and non-tuning cases, factors AF - 0 and A -

AF0 =  -

r2 (1−r 2 )2

+ (2ζe

r)2

,

AF =  -

1 (1−r 2 )2

+ (2ζe r)2

,

(8.4.21)

in which ζe is the equivalent damping coefficient for the amplification factors. In the non-tuning cases, the amplification factors given by equations (8.4.18) and (8.4.19) can be used directly; it is not necessary to specify the equivalent damping coefficient ζe in equation (8.4.21). In the following subsection, the tuning case is investigated to quantify the equivalent damping coefficient ζe .

8.4 direct method for generating frs

Figure 8.10 Amplification factors.

345

346

8.4.4 Perfect-Tuning Case When ω0 = ω and for small damping ζ0 , ζ 1, C (t) = h0 (t) ∗ h(t) becomes  t 1 −ζ ωτ 1 −ζ0 ω0 (t − τ ) e h0 (t) ∗ h(t) = e sinω0 (t−τ ) · sinωτ dτ ω ω 0 0  1 −ζ ωt t −(ζ −ζ ) ωτ 0 0 e sinω(t−τ ) sinωτ dτ = 2e ω 0   2 1 −ζ ωt −ζ ωt −ζ ωt −ζ ωt  (e −e 0 ) cos ωt + (e +e 0 ) sinωt , = 3 ω 4+(ζ −ζ0 )2 ζ −ζ0 which can be simplified to, for small damping (ζ −ζ0 )→0, h0 (t) ∗ h(t) =

=

1 2ω3 (ζ −ζ0 )

(e−ζ ωt − e−ζ0 ωt ) cos ωt +

1 (e−ζ ωt + e−ζ0 ωt ) sinωt 4 ω3

˙ (t) ˙ (t)− h h h(t)+ h0 (t) 0 + . 3 2ω (ζ −ζ0 ) 4 ω2

(8.4.22)

Substituting (8.4.22) into (8.4.10) yields the maximum response of the oscillator  ˙ ˙ (t) ∗ u¨ (t) ω2 h(t) ∗ u¨ (t)+ω2 h (t) ∗ u¨ (t)     ω h(t) ∗ u¨g(t)−ω h 0 g g g 0 u¨ (t)   + = F  max 2(ζ −ζ0 ) 4 max     1 u(t)− ¨ u¨0 (t) u(t)+ ¨ u¨0 (t)  =  · (8.4.23) +  , 2 ζ − ζ0 4 max in which the following relationships have been used ˙ (t) ∗ u¨ (t), u(t) ˙ (t) ∗ u¨ (t). (8.4.24) u(t) = h(t) ∗ u¨g(t), u(t) ˙ =h ¨ = ω2 h(t) ∗ u¨g(t) = ω h g g ¨ = u¨0 (t); the first term in equation (8.4.23), which is dominant, is When ζ0 = ζ , u(t) undefined. For (ζ −ζ0 )→0, equation (8.4.23) becomes      ¨ 1  ∂ u(t)   + u(t) ¨  . SF (ω, ζ ) = u¨F (t) max =  2 ∂ζ max

(8.4.25)

˙ (t) ∗ u¨ (t) = ω e−ζ ωt cos ωt with respect to ζ gives Differentiating u(t) ¨ =ωh g 



˙ (t) ∗ u¨ (t) ∂ ωh ∂ u(t) ¨ g = = −ω2 t e−ζ ωt cos ωt ∗ u¨g(t). ∂ζ ∂ζ

(8.4.26)

Note that u(t) ¨ can also be written as u(t) ¨ = ω2 h(t) ∗ u¨g(t) = ω e−ζ ωt sinωt ∗ u¨g(t). Hence, in the perfect-tuning case with ω0 = ω, ζ0 = ζ, the FRS given by equation (8.4.25) becomes   SF (ω, ζ ) = 21 −ω2 te−ζ ωt cos ωt ∗ u¨g(t) + ω e−ζ ωt sinωt ∗ u¨g(t)

max

= SAt (ω, ζ ), (8.4.27)

8.4 direct method for generating frs

347

where SAt (ω, ζ ) is the t-response spectrum (tRS) studied in Section 4.2. FRS given by equation (8.4.27) can also be expressed in the form of equation (8.4.17). Note that, in the perfect-tuning case, ω0 = ω, r = 1, ζ0 = ζ, and

AF0 = AF. Equation (8.4.17) can be written as -

SA(ω0 , ζ0 ) = SA(ω, ζ ),

-

SF (ω0 , ζ0 ) =

√

t 2 · AF - 0 · SA(ω, ζ ) = SA (ω, ζ ),

which gives

AF0 = AF = -

-

1 √ · 2

(8.4.28)

SAt (ω, ζ ) . SA(ω, ζ )

(8.4.29)

From equation (8.4.21), when r = 1, one has  1   = , 2 2 2  2ζ (1−r ) + (2ζe r) r=1 e

AF0 =  -

r2

AF = -

1 . 2ζe

(8.4.30)

From equations (8.4.29) and (8.4.30), the equivalent damping coefficient is given by 1 ζe = √ · 2

SA(ω, ζ ) . SAt (ω, ζ )

(8.4.31)

8.4.5 SDOF Oscillator Mounted on MDOF Structure Because all engineering structures have MDOF, the formulation in Section 8.4.1 is extended to an oscillator mounted on an MDOF structure in this section. Consider a three-dimensional model of a structure with N nodes (each node having six DOF) subjected to tridirectional earthquake ground excitations, as shown in Figure 8.5. The equation of motion in the matrix form is given by equation (8.2.8). Applying modal analysis as presented in Sections 3.6.4 and 8.2.3, the 6N-DOF system (3.6.3) is reduced to a series of 6N SDOF systems, in which the modal displacement qKI (t) of the Kth mode (SDOF system) under earthquake excitation in direction I is I (t) of node n in direction governed by equation (8.2.9). The absolute acceleration u¨ n, j

j under earthquake excitation in direction I is given by equation (8.2.10), which is a linear combination of all 6N modal responses and the contribution factor of the Kth modal response qKI is ϕn, j; K KI .

Maximum Modal Response Contribution As derived in Section 8.4.3, for an SDOF oscillator mounted on an SDOF structure, the maximum response of the oscillator is given by equation (8.4.17). Therefore, from equations (8.2.9) to (8.2.13), for the maximum absolute acceleration

Sn,I j (ω0 , ζ0 )

in

direction j of an oscillator (with frequency ω0 and damping ratio ζ0 ) mounted at node

348

n under earthquake excitation in direction I, the maximum contribution by the Kth mode, i.e., the maximum absolute acceleration of the oscillator under the excitation I of equation (8.2.10), is given by u¨n, j; K I I Rn, j; K = ϕn, j; K K

where





AF 0,2 K · SAI (ω0 , ζ0 )



-

2

2 + AF - K ·



SAI (ωK , ζK )



2



,

(8.4.32)

SAI (ω, ζ ) is the GRS of earthquake excitation in direction I, and AF0, K and AF K -

-

are the amplification factors of the Kth mode given by, rK = ωK /ω0 ,

AF. K =  -

1 (1−rK2 )2 +(2ζK,e rK )2

,

AF0, K = rK2 AFK , -

-

1 ζK,e = √ · 2

SA(ωK , ζK ) . SAt (ωK , ζK )

(8.4.33)

Modal Combination: FRS-CQC (Complete Quadratic Combinations) I Because the maximum responses Rn, j; K of the oscillator contributed to

Sn,I j (ω0 , ζ0 )

by each of the K modes (K = 1, 2, . . . , 6N) do no occur at the same time, they have to be combined following an appropriate combination rule. Comparing equations (8.2.3) to (8.2.6) with equations (8.2.9) to (8.2.12), and using equation (8.4.10), the contribution from the Kth mode to the response of the oscillator (with frequency ω0 and damping ratio ζ0 ) mounted on the MDOF structure under earthquake excitation in direction I is approximately given by Q IK (t) = ω02 ωK2 · C K (t) ∗ u¨gI (t),

C K (t) = h0 (t) ∗ hK (t).

(8.4.34)

The covariance between Q IK (t) and Q KI (t) of modes K and K is given by I E[ Q IK (t) Q K (t+τ ) ]

=

= ω04 ωK2 ω2K =



ω04 ωK2 ω2K · E

∞

C



−∞ ∞  ∞



−∞ −∞ ∞  ∞

ω04 ωK2 ω2K

−∞

−∞

¨gI (t−τ1 ) dτ1 K (τ1 ) u



∞ −∞

C

¨gI (t+τ −τ2 ) dτ2 K (τ2 ) u



C K (τ1 ) C K (τ2 ) E[ u¨gI (t−τ1 ) u¨gI (t+τ −τ2 ) ] dτ1 dτ2 C K (τ1 ) C K (τ2 ) Ru¨gI u¨gI (τ +τ1 −τ2 ) dτ1 dτ2 .

(8.4.35)

Taking Fourier transform of both sides yields  ∞ I E[ Q IK (t) Q K (t+τ ) ] · e − i ωτ dτ SQ I Q I (ω) = K

K

= ω04 ωK2 ω2K

−∞ ∞  ∞





∞

C K (τ1 ) C K (τ2 ) Ru¨gI u¨gI (τ +τ1 −τ2 ) · e − i ωτ dτ1 dτ2 dτ.

−∞ −∞ −∞

Setting τ3 = τ +τ1 −τ2 , equation (8.4.34) can be written as

SQ IK Q KI (ω) =



ω04 ωK2 ω2K

∞

C K (τ1 )e

−∞

i ωτ1



dτ1

∞

C K (τ2 )e

−∞

− i ωτ2

 dτ2

∞

(8.4.36)

Ru¨gI u¨gI (τ3 )e − i ωτ3 dτ

−∞

8.4 direct method for generating frs

349

= ωK2 ω2K · C K∗ (ω) · C K (ω) · Su¨gI u¨gI (ω),

C

H

H

K (ω) = 0 (ω) K (ω) is ∗ (ω) is the complex conjugate of K (PSD) of the excitation u¨gI (t).

where

C

(8.4.37)

the Fourier transform of the convolution

C K (t),

C K (ω), and Su¨gI u¨gI (ω) is the power spectral density

Taking the inverse Fourier transform of equation (8.4.37) yields  ∞ 1 I I E[ Q K (t) Q K (t+τ ) ] = S I I (ω) ei ωτ dω 2π − ∞ Q K Q K  ∞ ω4 ω2 ω2 C K∗ (ω) · C K (ω) · Su¨gI u¨gI (ω) ei ωτ dω. = 0 K K 2π −∞

(8.4.38)

Setting τ = 0 results in I E[ Q IK (t) Q K (t) ] =

ω4 ω2 ω2 = 0 K K 2π

ω04 ωK2 ω2K 2π  ∞ −∞



∞ −∞

C K∗ (ω) · C K (ω) · Su¨gI u¨gI (ω) dω

H0∗ (ω) HK∗ (ω) · H0 (ω) HK (ω) · Su¨gI u¨gI (ω) dω.

(8.4.39)

Because ground motions can be generally modelled as wide-band noises, it is reason-

able to assume the seismic input u¨gI (t) as a white noise by letting the PSD Su¨ I u¨ I (ω) = S I . g g

Therefore, equation (8.4.39) can be written as I E[ Q IK (t) Q K (t) ] =

where  IKK = 

= =

∞ −∞ ∞

ω04 ωK2 ω2K · S I · IKK , 2π

H0∗ (ω) HK∗ (ω) · H0 (ω) HK (ω) dω 1

·

1

2 2 2 2 (ω2 −ω2 )− i2ζ ω ω − ∞ (ω0 −ω ) +(2ζ0 ω0 ω) K K K     ∞  2 2 2 2 (ωK −ω )+ i2ζK ωK ω · (ωK −ω )− i2ζK ωK ω −∞ 

(8.4.40)

KK

 

·

1 (ω2K −ω2 )+ i2ζK ωK ω

dω

dω = Re (IKK ) + i Im (IKK ),

 



KK = (ω02 −ω2 )2 +(2ζ0 ω0 ω)2 · (ωK2 −ω2 )2 +(2ζK ωK ω)2 · (ω2K −ω2 )2 +(2ζK ωK ω)2 ,

Re (IKK ) and Im (IKK ) are the real and imaginary parts of IKK , respectively and can be evaluated by the method of residue to yield  ∞ (ωK2 −ω2 ) · (ω2K −ω2 ) + (2ζK ωK ω) · (2ζK ωK ω) π · αKK dω = , Re (IKK ) = K K 2ζ0 ω03 · ω04 −∞  ∞ 2ζK ωK ω · (ω2K −ω2 ) − 2ζK ωK ω · (ωK2 −ω2 ) Im (IKK ) = dω = 0, K K −∞

350

in which αK K =

3 3 L=0

D−1 · KK , L

4  L=0

CKK , L ζ0L ,

(8.4.41)

CKK , L and DKK , L are constants in terms of ζ0 , ζK , ζK , rK = ωK /ω0 , and rK = ωK /ω0 , given by DKK , 1 = 1 − 2rK2 + rK4 + 4ζ0 ζK rK + 4ζ0 ζK rK3 + 4ζ02 rK2 + 4ζK2 rK2 , DKK , 2 = 1 − 2rK2 + rK4 + 4ζ0 ζK rK + 4ζ0 ζK rK3 + 4ζ02 rK2 + 4ζK2 rK2 , DKK , 3 = (rK2 −rK2 )2 + 4ζK ζK rK rK (rK2 +rK2 ) + 4rK2 rK2 (ζK2 +ζK2 ), CKK , 0 = (1 − rK2 − rK2 + rK2 rK2 + 4ζK ζK rK rK ) · DKK , 3 , CKK , 1 /4 = 2ζK rK + 2ζK rK + 8ζK ζK rK rK (ζK rK +ζK rK ) − 4(ζK rK3 +ζK rK3 ) + 8ζK3 rK3 + 8ζK3 rK3 − 2rK rK (ζK rK3 +ζK rK3 ) + 8ζK ζK rK rK (ζK rK3 +ζK rK3 ) + 4rK2 rK2 (ζK rK +ζK rK ) − 8rK2 rK2 (ζK3 rK +ζK3 rK ) − 8ζK ζK rK2 rK2 (ζK rK +ζK rK ) + 32ζK2 ζK2 rK2 rK2 (ζK rK +ζK rK ) + rK rK (ζK rK5 +ζK rK5 ) + rK2 rK2 (ζK rK3 +ζK rK3 ) + 4ζK ζK rK2 rK2 (ζK rK3 +ζK rK3 ) − 2rK3 rK3 (ζK rK +ζK rK ) + 4rK3 rK3 (ζK3 rK +ζK3 rK ) + 8ζK ζK rK3 rK3 (ζK rK +ζK rK ), CKK , 2 /4 = 8ζK2 rK2 + 8ζK2 rK2 + 16ζK ζK rK rK + 64ζK2 ζK2 rK2 rK2 − 4ζK ζK rK rK (rK2 + rK2 ) + 32ζK ζK rK rK (ζK2 rK2 +ζK2 rK2 ) + 6rK2 rK2 − 12rK2 rK2 (ζK2 +ζK2 ) − 3(rK4 +rK4 ) + 8ζK ζK rK rK (rK4 +rK4 ) − rK2 rK2 (rK2 +rK2 ) + 8ζK2 rK4 + 8ζK2 rK4 + 4rK2 rK2 (ζK2 +ζK2 )(rK2 +rK2 ) + 16ζK2 ζK2 rK2 rK2 (rK2 +rK2 ) + 16ζK ζK rK3 rK3 (ζK2 +ζK2 ) + rK6 + rK6 , CKK , 3 /16 = 8ζK ζK rK rK (ζK rK +ζK rK ) + 2ζK rK3 + 2ζK rK3 + rK rK (ζK rK3 +ζK rK3 ) + 4ζK ζK rK rK (ζK rK3 +ζK rK3 ) − 2rK2 rK2 (ζK rK +ζK rK ) + 4rK2 rK2 (ζK3 rK +ζK3 rK ) + 8ζK ζK rK2 rK2 (ζK rK +ζK rK ) + ζK rK5 + ζK rK5 , CKK , 4 /16 = DKK , 3 . Therefore, equation (8.4.40) can be expressed as I E[ Q IK (t) Q K (t) ] =

When

K = K,

ω04 ωK2 ω2K π · αKK ω0 S I · SI · = · αKK · rK2 rK2 . 2π 4ζ0 2ζ0 ω03 · ω04

(8.4.42)

equation (8.4.42) becomes  2 ω SI E[ Q IK (t) ] = 0 · βK · rK4 , 4ζ0

(8.4.43)

8.4 direct method for generating frs

351

where βK =

ζ0 +4ζ02 ζK rK +4ζ0 ζK2 rK2 +ζK rK3

. ζK rK3 1−2rK2 +rK4 +4ζ0 ζK rK +4ζK2 rK2 +4ζ02 rK2 +4ζ0 ζK rK3

(8.4.44)

Hence, the correlation coefficient between the contributions to the response of an oscillation mounted on the structure under earthquake excitation in direction I by Kth and K th modes is obtained as ω0 S I · αKK · rK2 rK2 4ζ0

I (t) E[ Q IK (t) Q K α ] ρKI K =   = =  KK ,    2 βK β K I (t) 2 ω0 S I ω SI E[ Q IK (t) ] · E[ Q K ] · βK · rK4 × 0 · βK · rK4 4ζ0 4ζ0

which is independent of the direction I of earthquake excitation and can be written as α ρKK =  KK . βK β K

(8.4.45)

I Combining the maximum absolute acceleration Rn, j; K of the oscillator contributed

by mode K, given by equation (8.4.32), for all 6N modes gives the FRS of node n in direction j under earthquake excitation in direction I defined by equation (8.2.13):

S

I n, j (ω0 , ζ0 )

 = u¨ I

F, n, j

 (t)

 max

=

6N  6N 

K=0 K =0

ρKK RKI RKI .

(8.4.46)

Sn, j (ω0 , ζ0 ) of the nth node in direction j under tridirectional earthquake excitaI tions is then obtained from FRS Sn, j (ω0 , ζ0 ), I = 1, 2, 3, using the SRSS combination FRS

rule given by equation (8.2.14).

Comments on Modal Combination Because the modal combination in equation (8.4.46) is a complete quadrature for maximum responses of the oscillator contributed by all 6N modes, it is therefore called FRS-CQC to differentiate from CQC (complete quadratic combination), which combines maximum responses of the 6N modes. To visualize the correlation coefficient of FRS-CQC, for given damping ratios ζ0 , ζK , and ζK , the correlation coefficient ρKK is a function of frequency ratios rK and rK and can be plotted as a surface. Figure 8.11 shows the plot of ρKK with ζ0 = ζK = ζK = 5 %, rK and rK ranging from 0 to 2.5. Some remarkable features of FRS-CQC can be observed: ❧ Similar to the correlation curve of the conventional CQC, which is symmetric about ωK = ωK , the correlation surface of FRS-CQC is symmetric about the plane rK = rK

352

Figure 8.11

3D-view of FRS-CQC correlation coefficients with 0  r  2.5.

Figure 8.12

2D-view of FRS-CQC correlation coefficients with 0  r  2.5.

(ωK = ωK ). The correlation coefficient ρKK = 1 for rK = rK , meaning that responses of closely spaced modes are fully correlated. ❧ Different from the correlation coefficient in conventional CQC, which is uniformly positive, the correlation coefficient of FRS-CQC is negative inside the areas approximately for rK < 1 < rK and rK < 1 < rK as shown in Figure 8.12. In other words, negative correlation generally occurs when the equipment frequency is located between the structural frequencies of two not–closely spaced modes, which usually results in a valley between the FRS peaks.

8.4 direct method for generating frs

Figure 8.13

353

3D-view of FRS-CQC correlation coefficients with 0  r  0.02.

Figure 8.14

3D-view of FRS-CQC cut by rK = 0.01.

❧ For the extreme case when the equipment frequency is significantly higher than the structural frequency with rK →0 and rK →0, FRS-CQC is reduced to the conventional CQC. Figure 8.13 shows the correlation surface of FRC-CQC for rK and rK ranging from 0 to 0.02, which is an enlarged view of the tiny portion of the surface close to the origin in Figure 8.11. The intersection between the surface and a plane defined by rK = a or rK = a (a is an arbitrary positive value that approaches zero) can provide a correlation curve of the conventional CQC. For instance, the correlation surface is cut by a plane rK = 0.01 as shown in Figure 8.14. It can be observed that the correlation coefficient ρKK = 1 at rK = 0.01, when two structural frequencies are coincident ωK = ωK . Furthermore, the correlation curve is positive and symmetric about rK = rK = 1. ❧ To determine responses of MDOF structures under earthquake excitations using a response spectrum method, the correlation coefficient between two modal re-

354

sponses is determined for CQC (Der Kiureghian, 1981), i.e., E[ q K (t) q K (t) ] ρKcqc K =   2 2 ,  E[ q K (t) ] · E[ q K (t) ]

(8.4.47)

where q K (t) = HK (ω) ∗ u¨g(t) is the response of the Kth mode. ❧ To determine FRS, the response of an oscillator (with frequency ω0 and damping ratio ζ0 ) mounted on the MDOF structure is required. The correlation coefficient between the responses of the oscillator contributed by two modal responses is determined for FRS-CQC, i.e., E[ Q K (t) Q K (t) ] ρKfrs-cqc =  K 2 , 2  E[ Q K (t) ] · E[ Q K (t) ]

(8.4.48)

where Q K (t) = ω02 ωK2 H0 (t) ∗ HK (t) ∗ u¨g(t) is the response of the oscillator contributed by the Kth mode. ❧ When ω0 →∞, i.e., when the oscillator is very rigid, Q K (t)→q K (t). Therefore, includes ρKfrs-cqc K • the correlation between Q K (t) and Q K (t), • the correlation between q K (t) and q K (t), • the correlations between Q K (t) and q K (t) and between Q K (t) and q K (t). ❧ It is important to note that CQC was derived for responses of MDOF structures (Der Kiureghian, 1981), considering only the correlation between two modal responses q K (t) and q K (t). Applying CQC (with ρKcqc K ) or SRSS in modal combination to generate FRS may lead to large errors, especially for structures with closely spaced modes.

Generation of Floor Response Spectra For an SDOF oscillator mounted on an MDOF structure, the procedure of the direct spectra-to-spectra method of generating FRS is illustrated in Figure 8.15. A modal analysis is performed first to obtain the modal information of the structure. The amplification factors and FRS-CQC coefficients are determined from the modal information along with tRS that corresponds to the prescribed GRS. Multiplying the amplification factors to the target GRS results in the modal responses, which are then combined by FRS-CQC rule to generate FRS.

8.4 direct method for generating frs

Figure 8.15

355

Procedure of the direct method for generating FRS.

8.4.6 Numerical Examples The accuracy and efficiency of the direct spectra-to-spectra method developed in Section 8.4.5 for generating FRS is demonstrated through numerical examples by comparing results from the direct method with those from time-history analyses. The primary source of variability in time-history analysis stems from the inherent uncertainties and randomness of the time-histories, reflected in the rugged spectral shapes of FRS.Although there are large variations in individual FRS as will be seen later in the numerical results, the statistical results (such as mean FRS, median FRS, or FRS with 84.1 % nonexceedance probability [NEP]) from a large number of time-history analyses converge to smooth spectra; such statistical results are used as benchmarks for verifying the accuracy of the direct method.

Model Information A service building of a nuclear power plant is selected as the primary structure. A three-dimensional finite element model of the building, as shown in Figure 8.16, is established using the commercial finite element analysis software STARDYNE. The superstructure of the building consists of steel frames and concrete floor slabs, and the basement is constructed using concrete. The elevation of each floor and the

356

24.95 m

Node 1

21.00 m

Node 2

18.00 m

7.50 m 7.50 m 7.50 m 7.50 m

15.00 m 12.00 m

13.30 m

5.75 m

0.00 m Elevation −5.00 m

ug3(t) 8.00 m

8.00 m

8.00 m

Figure 8.16

ug2(t) 8.00 m

ug1(t)

8.00 m 3D finite element model of a service building.

Table 8.1 Information of finite element model.

Number

Node

Lumped Mass

1351

120

Beam Element Section 1740 31

Shell Element Section 830 8

Table 8.2 Modal information at Node 1.

Mode 2 20 21 31 103 106

Frequency (Hz) 2.676 5.838 5.918 7.212 22.95 23.96

Participation Factor − 7.413 − 2.945 2.943 − 8.883 − 100.8 − 337.3

Modal Shape − 0.05082 − 0.02603 0.06409 − 0.01942 0.00088 0.00024

Contribution Factor 0.38 0.08 0.19 0.17 − 0.09 − 0.08

Table 8.3 Modal information at Node 2.

Mode 2 20 21 31 105 106 107

Frequency (Hz) 2.676 5.838 5.918 7.212 23.34 23.96 23.98

Participation Factor − 7.413 − 2.945 2.943 − 8.883 − 96.07 − 337.3 − 50.65

Modal Shape − 0.14630 − 0.01904 0.04151 0.03847 − 0.00045 − 0.00011 − 0.00092

Contribution Factor 1.08 0.06 0.12 − 0.34 0.04 0.04 0.05

8.4 direct method for generating frs

357

dimensions of the building are shown in Figure 8.16. Some information of the finite element model is listed in Table 8.1. A modal analysis is performed to obtain modal frequencies, modal participation factors, and modal shapes of the model. Modal information of 145 modes, in which the modal frequencies are less than 33 Hz, is extracted. FRS at two nodes located on the second and third floors of the building are considered; Node 1 is on an edge of the second floor, and Node 2 is on the third floor. Modal information of the significant modes at these two typical nodes is listed in Tables 8.2 and 8.3. The participation factors and modal shapes in these two tables are for direction 2 shown in Figure 8.16. The contribution factor is the product of the participation factor and the modal shape, quantifying the contribution of the corresponding mode in the response of the node; all other modes that are not listed in Tables 8.2 and 8.3 have absolute values of the contribution factors less than 0.04. The summation of the 145 mode contribution factors at each node is close to 1. It is seen that there are closely spaced modes with considerable contributions to the responses at both Nodes 1 and 2. For example, modes 20 and 21 are closely spaced for Node 1; modes 20 and 21, modes 105 to 107 are closely spaced for Node 2.

Input GRS Two types of response spectra are selected as input GRS in the numerical examples. ❦ GRS of USNRC R.G. 1.60 The 5 % horizontal and vertical design spectra in USNRC R.G. 1.60 (USNRC, 2014) are taken as GRS in this example. The horizontal GRS are anchored at 0.3g PGA, and the vertical PGA is taken as 2/3 of the horizontal PGA. Thirty sets of tridirectional time-histories compatible with GRS are generated following the Approach 2 of USNRC SRP 3.7.1 (USNRC, 2012b), as shown in Figure 8.17. ❦ Standard UHS for CENA The 5 % standard CENA UHS (Atkinson and Elgohary, 2007) anchored at 0.3g is chosen as the horizontal GRS; the vertical input GRS is taken as two-thirds of the horizontal GRS. Thirty sets of tridirectional spectra-compatible time-histories are generated following the requirements of CSA N289.3 (CSA, 2010a), as shown in Figure 8.18.

☞

For both USNRC R.G. 1.60 GRS and CENA UHS, all time-histories are generated

using the Hilbert–Huang transform method (Ni et al., 2011b; Ni et al., 2013).

358

1.4

Spectral acceleration (g)

1.2

Upper bound: +30%

1.0

Bound: +10%

0.8

R.G. 1.60 5%-damping

0.6 Lower bound: −10%

0.4

PGA  0.3 g

0.2 0

0.2

1 Figure 8.17

Frequency (Hz)

10

100

Ground response spectrum.

Spectral acceleration (g)

0.9 0.8

Mean of 30 time-histories

0.7

Target Horizontal UHS

0.6

Bound: +10%

0.5 0.4

Bound: −10%

0.3

PGA  0.3 g

0.2 0.1 0

0.2 Figure 8.18

1

Frequency (Hz)

10

100

Response spectra of compatible time-histories.

Comparison of FRS ❦ FRS under the Excitation of GRS of USNRC R.G. 1.60 FRS at Node 1 and Node 2 obtained from both time-history analyses and the direct spectra-to-spectra method are plotted in Figures 8.19 and 8.20, respectively. These FRS are calculated over 200 frequencies including natural frequencies of the dominant modes of the structure. The benchmark mean FRS obtained from time-history analyses, are highlighted by bold dashed lines; the FRS generated by the direct method are shown as bold solid lines. It is seen that FRS generated by the direct method agree extremely well with the benchmark FRS over the entire frequency range. The relative errors are less than 5 % at the peaks of FRS, whereas there are large variabilities in FRS from time-history

8.4 direct method for generating frs +28%

359 +37%

3.0 2.5

Relative error 0.2%

Spectral acceleration (g)

2.0

Relative error −0.2%

1.5 Relative error 1%

1.0

−20%

C22% 0.5 Time-history Direct method 0 1

Frequency (Hz)

Figure 8.19

10

100

FRS for Node 1 (USNRC R.G. 1.60 GRS).

8

+32%

+29%

7 Relative error 2% Spectral acceleration (g)

6 Time-history Direct method

5 4 3

Relative error 4%

2

−24%

−14% 1 0

0.1

1

Figure 8.20

Frequency (Hz)

10

100

FRS for Node 2 (USNRC R.G. 1.60 GRS).

analyses. This example demonstrates that time-history analysis can lead to approximately 30 % overestimation or 20 % underestimation at the FRS peaks, even though the time-histories are well compatible with the target GRS (within 10 % of the target GRS). Hence, FRS from a single time-history analysis may be overconservative at some peaks but significantly underestimate at other peaks. The primary source of variability in time-history-analysis stems from the inherent uncertainties and randomness of the spectrum-compatible time-histories, which are

360

3.0

Exact FRS

21%

FRS-CQC CQC

2.5 Spectral acceleration (g)

0.2%

29%

2.0

1.5

–0.2% 44%

1.0

–2%

0.5

0 0.3

1

Frequency (Hz)

10

100

Errors in modal combination rules for FRS at Node 1.

Figure 8.21

8

Exact FRS FRS-CQC

7

CQC

Spectral acceleration (g)

6 5 4 3 4% 2 –30%

1 0 0.3

1 Figure 8.22

Frequency (Hz)

10

100

Errors in modal combination rules for FRS at Node 2.

reflected from their rugged spectral shapes. As seen in Figures 8.17 and 8.18, there is an apparent difference between the response spectrum of a spectrum-compatible timehistory and the target GRS, which has a smooth spectral shape. From equation (8.4.17), it is clear that FRS are amplified GRS. Therefore, this difference is also amplified by an amplification factor, which can range from 5.5 to 7 in tuning cases. For an oscillator

8.4 direct method for generating frs

361

mounted on an SDOF structure, a 5 % difference in GRS can result in approximately 30 % difference at FRS peaks. For an oscillator mounted on an MDOF structure, modal responses are multiplied by the contribution factors and combined through equations (8.4.32) and (8.4.46). As a result, variabilities in FRS are combinations of the differences in all modal responses and are also significant. To further verify the significance of modal combination on the determination of FRS, Figures 8.21 and 8.22 show the FRS generated by the direct method using the conventional CQC combination rule. For FRS at Node 1, there are 21 %, 29 %, and 44 % relative errors at the first and second peaks and the valley between the two peaks, respectively. For Node 2, CQC gives results close to the benchmark FRS around the first peak. However, there is 30 % underestimation at the second peak. Figure 8.23 illustrates the seismic responses of the first four significant modes at Node 1. The effect of modal combination can be analyzed qualitatively as follows. ❧ For low frequencies F < 2 Hz, the response of the oscillator is contributed mainly by the amplified ground motion. Because the oscillator is an SDOF system, it does not involve modal combination. ❧ For frequencies from 2 to 20 Hz, which cover the dominant modal frequencies of the structure, each mode has considerable contribution so that the effect of modal combination becomes significant. Conventional CQC combination rule, developed to combine structural responses, cannot fully account for the correlation between the responses of the oscillator contributed by different modal responses and the correlation between response of the oscillator contributed by a modal response and the response of a structure mode. ❧ For high frequencies F > 20 Hz, because the oscillator is sufficiently rigid, its response is close to the structural response. The formula of FRS-CQC can be reduced to CQC in this case; hence, the resultant FRS given by these two combination rules are close. Because structures in nuclear power plants have multiple dominant modes, and some of them are closely spaced, modal combination rules significantly influence the resultant FRS. The numerical example demonstrates that FRS-CQC combination rule is valid and accurate to combine modal responses. ❦ FRS under the Excitation of Standard UHS for CENA For Standard UHS for CENA, the direct method is validated by comparing FRS obtained from the direct method with the benchmark FRS, as shown in Figures 8.24 and 8.25. It is observed that FRS given by the direct method agree extremely well with the “exact” FRS over the entire frequency range, and the relative errors at peaks are

362

mostly less than 5 %. Compared to Figures 8.19 and 8.20, there are some differences in the spectral shapes of FRS, particularly over the higher-frequency range from 10 to 40 Hz. FRS peak up in this range because the spectral acceleration of UHS reaches the maximum value while the spectral acceleration of R.G. 1.60 GRS decreases. Peak FRS generated from UHS are generally lower than those from R.G. 1.60 GRS. The reason is that the spectral accelerations of UHS are apparently lower than those of R.G. 1.60 over the frequency range from 2 to 8 Hz, where the dominant modes of the structure contribute most.

Probabilistic Descriptions of FRS Peaks Another major advantage of the direct method is that it can give probabilistic descriptions for FRS peak values. Using the probabilistic description of tRS, FRS with any desired level of NEP p can be determined from the given GRS and the corresponding tRS with NEP p. FRS with 84.1 % NEP at Node 1 obtained by time-history analyses and the direct spectra-to-spectra method are compared in Figure 8.26 for USNRC R.G. 1.60 GRS and in Figure 8.27 for Standard UHS for CENA. The relative errors are all less than 5 %. This excellent agreement further demonstrates the accuracy of the direct method. It is noted that the mean FRS and FRS with 84.1 % NEP given by the direct method are almost the same for non-tuning cases, as shown in Figures 8.26 and 8.27. This can be explained by equation (8.4.21) and Figure 8.10. The amplification factors given by equation (8.4.21) depend on the equivalent damping ratio ζe , which is determined by tRS and GRS in equation (8.4.31) and have a significant effect on the amplification factors in the perfect-tuning and near-tuning cases but have a negligible effect in non-tuning cases, as shown in Figure 8.10. As a result, for non-tuning cases, the amplification factors are almost the same for all values of the equivalent damping ratio, leading to that FRS for all levels of NEP p are almost the same. Because of the large amplification factors in the tuning cases, small deviations of the response spectrum of a time-history from the target GRS are significantly amplified. Although the compatibility of the time-histories is good by satisfying code requirements, there are large variabilities in the FRS from time-history analyses, particularly in the tuning cases. Peak responses can be overestimated and underestimated by as much as 35 % and 25 %, respectively. Hence, results from time-history analyses using a single set or a small number of sets of spectrum-compatible ground motions are not adequate to give accurate FRS. This observation further highlights the advantage of the direct method, which uses the target GRS as input directly, without generating spectrum-compatible time-histories that are the primary source of variabilities.

8.4 direct method for generating frs

363

Modal response

Mode 2

Closely spaced modal 21 frequencies 31

20 0.5

1

2

4

Frequency (Hz)

10

40

20

Figure 8.23 Analysis of modal combination of FRS. 1.4

Time-history Direct method

Relative error 5.1%

Spectral acceleration (g)

1.2

Relative error 3.1%

Relative error 1.7%

1.0

Relative error 2.0%

0.8 0.6 0.4 0.2 0

0.1

1

Frequency (Hz)

35%

24%

10

100

25%

22%

Relative error 5.1%

Relative error 1.7%

17%

15% 24%

25% Relative Relative error error 3.1% 2.0%

Figure 8.24

FRS for Node 1 (UHS for CENA).

364 3.0

Time-history Direct method

Spectral acceleration (g)

2.5

Relative error 0.2%

2.0 Relative error % 1.5

1.0

Relative error 3.7%

0.5

0 0.1

1

Frequency (Hz)

24%

10

100 26%

18%

Relative error %

Relative error 0.2%

25%

20% 19%

Relative error 3.7%

Figure 8.25

FRS for Node 2 (UHS for CENA).

Conclusions FRS determined by time-history analyses have large variabilities, particularly in tuning cases or at FRS peaks; hence, FRS determined by time-history methods using a single set or a small number of sets of spectrum-compatible tridirectional time-histories are not reliable. Modal combination methods significantly affect the results; there will be large errors if the conventional CQC or SRSS modal combination methods are applied to determine FRS. The direct method can avoid these deficiencies and give accurate FRS because of the following three significant features:

8.5 scaling method for generating frs +28%

3.0

365 +37%

Relative error 4.5%

2.5 Relative error 1% Spectral acceleration (g)

2.0

1.5 Relative error 1%

1.0

20%

22% 0.5 Time-history: 0 1

Figure 8.26

84.1% 84.1%

Direct method: Frequency (Hz)

Mean Mean

10

100

Probabilistic description of FRS for Node 1 (USNRC R.G. 1.60 GRS).

1. Using the statistical relationships between tRS and GRS, FRS in the tuning cases can be determined accurately. 2. The correlations of responses between equipment and its supporting structure can be fully accounted for through FRS-CQC combination rule. As a result, the direct method can generate accurate FRS for complex three-dimensional finite element structural models with closely spaced modes. 3. From the complete probabilistic descriptions of tRS for given GRS, the direct method can give complete probabilistic descriptions of FRS peaks.

8.5 Scaling Method for Generating FRS 8.5.1

Introduction

In many practical situations, scaling methods are efficient and economical approaches to obtain FRS: Scaling Problem 1: Knowing FRS it is required to determine

SF ( F, ζ0 ) with one or a few values of damping ratio,

SF ( F, ζ0 ) for a number of different damping ratios ζ0 .

Scaling Problem 2: Knowing FRS-I ratio for GRS-I

SG-I ( F, ζ ),

SF-I ( F, ζ0 )

with one or a few values of damping

it is required to determine FRS-II

number of different damping ratios

ζ0

under different GRS-II

SF-II ( F, ζ0 )

SG-II ( F, ζ ).

for a

366 1.4

Relative error 1.3%

Spectral acceleration (g)

1.2

Relative error 0.7%

Relative error 0.3%

1.0 0.8 0.6

Time-history Mean

84.1%

Direct method Mean

84.1%

Relative error 3.8%

0.4 0.2 0

0.1

1 +35%

Frequency (Hz) +24%

10

100

+25%

+22%

Relative error 1.3%

Relative error 0.3%

15% 25%

24% 17% Relative Relative error error 0.7% 3.8%

Figure 8.27

Probabilistic description of FRS for Node 1 (UHS for CENA).

Scaling Problem 1 Scaling Problem 1 arises quite frequently in practice. FRS corresponding to one or only a few damping ratios are usually available. However, FRS for various damping ratios, which may range from 2 % to 15 %, are required. For example, for many existing NPPs, low structural damping ratios were usually used in the original dynamic models. The final FRS results were presented for low equipment damping ratios up to 5 % or 7 %. In seismic margin assessment, median damping ratios for structures are required, which are larger than those used in the original dynamic analyses. FRS with higher equipment damping ratios are also required. Engineering activities, driven by schedule

8.5 scaling method for generating frs

367

and budget, call for a prompt and economical approach to generate the updated FRS for high equipment damping ratios with the high (median) structural damping ratios. However, existing scaling approaches (ASCE 4-98 Clause 3.4.2.4,ASCE, 1998; SQUG GIP Section 4.2.2, SQUG, 2001) are essentially simple scaling with a uniform scaling factor for all frequencies, or linear interpolation based on various assumptions between

SF ( F, ζ ) and ζ or F, which are not valid when the equipment damping ratios are out of the range, or when only one FRS with 5 % equipment damping ratio is available. Scaling Problem 2 An accurate and reliable method for Scaling Problem 2 is important in many engineering projects. For example, in a life-extension project of an existing nuclear power plant,

SF-I ( F, ζ ) are usually available for design basis earthquake (DBE) SG-I ( F, ζ ). SF-II ( F, ζ ) are required for site-specific ground motion response spectra (GMRS) or review-level earthquakes (RLE)

SG-II ( F, ζ ) in seismic margin analysis.

Project scope and budget

may not warrant a complete seismic structural analysis to obtain SF-II ( F, ζ ). In refurbishment projects, sometimes structures need to be strengthened due to a higher seismicity

SG-II ( F, ζ ) than the original design SG-I ( F, ζ ).

It is tricky to decide

which strengthening scheme is the most economical from the seismic point of view. A quick yet accurate approach to determine

SF-II ( F, ζ )

from

SG-II ( F, ζ )

will assist

engineers to decide which scheme is optimal. Similarly, in a new-build,

SF-I ( F, ζ ) are available for a generic design based on a

SG-I ( F, ζ ), such as those in CSA N289.3 (CSA, 2010a) or USNRC R.G. 1.60 (USNRC, 2014). An efficient and good estimate of SF-II ( F, ζ ) for site-specific GRS SG-II ( F, ζ ) is critical for feasibility analysis, budgeting, scheduling, bidding and tenderstandard GRS

ing, and procurement of important equipment, which may take years to manufacture, before the site-specific design is finalized and a complete seismic analysis is performed. It is obviously desirable for engineers to use as much of the available information and results of previous analyses as possible without performing a complete dynamic analysis, which is time consuming and introduces extra costs. However, the existing scaling methods recommended in EPRI NP-6041-SL (EPRI, 1991a) basically give approximate estimates with an uniform scaling factor and are restricted to some special cases. Because of their crude approximations, they are not widely used in the nuclear industry. In this section, a scaling method for solving the two scaling problems based on the direct spectra-to-spectra method presented in Section 8.4 is presented.

368

1

Spectral acceleration

F-I( f, ζ0)

Raw FRS

ζ0

Broadened-and-smoothed FRS

2

3

=

4

f1 Figure 8.28

f2

f3

f4

Frequency (Hz)

Broadened-and-smoothed FRS

8.5.2 System Identification An essential task in a scaling method for generating FRS is system identification: to recover the most significant dynamic characteristics of the underlying structure from

SG-I ( F, ζ ) and available SF-I ( F, ζ ). It is known that FRS is contributed primarily by a number of significant modes of the structure, and FRS peaks occur at the frequencies of these modes. Therefore, for the mth DOF (corresponding to the nth node in direction j), the first step is to extract the significant equivalent modal information (frequencies and the corresponding spectral accelerations) from the available FRS-I. It should be noted that the available FRS have usually been broadened and smoothed, which means some spectral values may have been modified artificially and thus are inappropriate to be used for identifying the structural information. Nevertheless, because the plateaus of FRS result from broadening (normally by ±15 %) the peaks of raw FRS, it is reasonable to use the middle point at an FRS plateau for the natural frequency of a significant structural mode and the corresponding spectral acceleration. If there is a wide plateau, it may be assumed that it is the result of broadening and smoothing from more than one peak, as shown in Figure 8.28. In this case, two or more significant modes may be taken considering that the corners are usually the results of broadening from a peak by ±15 %; however, it is understood that the broadened-andsmoothed FRS may not accurately reflect the underlying raw FRS. The number of the significant structural modes can be larger than the number of FRS plateaus due to the possible existence of closely spaced modes. However, a cluster of closely spaced modes can be treated as one equivalent mode with the same frequency

8.5 scaling method for generating frs

369

and an equivalent modal contribution factor. This assumption may not be able to reproduce exactly the same dynamical information as the original structure, but it simplifies the calculation for generating FRS without compromising the accuracy. In general, the available FRS-I in direction I is obtained under tridirectional excitations. In system identification, the available FRS-I and GRS-I in direction I are used to obtain the equivalent significant modes of the underlying structure. Hence, the equivalent system contains the significant dynamic characteristics of generating FRS in direction I under tridirectional seismic excitations from GRS in direction I. As a result, even though only GRS in direction I is used in generating FRS in direction I in the scaling method, the generated FRS contains the effect of tridirectional seismic excitations. Suppose that GRS-I SG-I ( F, ζ0 ) and FRS-I SF-I ( F, ζ0 ) for the mth DOF (corresponding to the nth node in direction j) of the original structure are available. For clarity of presentation, the subscript m signifying the mth DOF is dropped. It is assumed that there are N significant modes in the underlying structure, where N may be slightly larger than the number of plateaus in FRS-I. As an illustration, for a given FRS-I as shown in Figure 8.28, the frequencies of the four significant modes F K , K = 1, 2, 3, 4, and the corresponding spectra accelerations

SK = SF-I ( F K , ζ0 ) can be easily obtained by inspection and simple calculation.

The maximum value of the contribution of the Kth significant mode to the absolute acceleration of the oscillator mounted in the mth DOF is, from equation (8.4.32),     AF- 0,K SG-I ( F 0 , ζ0 ) 2 + AF- K SG-I ( F K , ζK ) 2 , (8.5.1) RK = ϕK  K 4 56 7 4 56 7 XK aK where ζ0 is the damping ratio of the FRS-I, ζK is the damping ratio of the significant

mode K of the underlying structure, and the amplification factors AF F - 0,K and A - K can be evaluated from equation (8.4.33). Note that the superscript I is dropped because only the direction corresponding to the mth DOF is considered. Hence, the value of aK can be easily determined. The unknown quantity XK characterizes the contribution factor of significant mode K in the response of the mth DOF. From equation (8.4.46), the FRS-I value of the mth DOF at frequency F 0 is given by 



SF-I ( F 0 , ζ0 ) 2 =

N  N 

K=1 K =1

ρKK RK RK .

(8.5.2)

370

Setting F 0 = F s , s = 1, 2, . . . , N , where F s is the frequency of the sth significant mode,  = a give substituting equation (8.5.1) into (8.5.2), and denoting a  K; s

K F 0= F s

N  N 

ρKK aK; s aK ; s XK XK =

K=1 K =1





SF-I ( F s , ζ0 ) 2 = Ss2 ,

s = 1, 2, . . . , N ,

(8.5.3)

where, with F 0 = F s , aK; s =

AFK =  -









AF0,K SG-I ( F 0 , ζ0 ) 2 + AFK SG-I ( F K , ζK ) 2 , -

-

1

(1−rK2 )2 + (2 ζK,e rK )2

,

AF0,K = rK2 AFK , -

-

rK =

ζK,e = √

FK , F0

SG-I ( F K , ζK ) . t 2 · SG-I ( F K , ζK )

For a damping ratio ζ0 , there are N spectral accelerations at the frequencies F s , s = 1, 2, . . . , N , of the significant modes. Hence, there are N quadratic equations in (8.5.3) for N unknowns XK , which can be readily solved numerically. It is noted that the solution sets of the quadratic system are generally non-unique. For instance, the number of possible solution sets may be up to four when N = 2 because the solutions can be graphically represented as the intersections of two ellipses. An effective way to find the most realistic solutions is by taking advantage of the modal property N 

K=1

K ϕK =

N 

K=1

XK →1.

(8.5.4)

It should be emphasized that XK denotes the equivalent modal contribution factors, which may not represent the underlying system exactly. Therefore, the summation of XK is expected to approach 1 rather than equal to 1 exactly; the problem can be interpreted as an optimization problem of minimizing the objective function  N    F(X) =  XK − 1,

(8.5.5)

subject to nonlinear constraints N N     2  ρ a a X X − S s K K K; s K ; s K K 

(8.5.6)

K=1

K=1 K =1



εs ·

Ss2 ,

s = 1, 2, . . . , N ,

where εs are error tolerances usually set as small as 10 − 2 to 10 − 3 . This optimization process can be easily implemented by many mathematical software packages, such as Excel. An efficient method of identifying significant equivalent modal information of

the underlying structure is summarized in Figure 8.29.

8.5 scaling method for generating frs

371 Gound Response Spectra-I

Floor Response Spectra-I

Statistical Relationship

Inspection Max Modal Responses

t-Reponse Spectrum fk

Frequencies Spectral peaks

Sk

FRS-CQC ρkκ AF0,k and AFk Nonlinear Optimation Objective: minimize f(X) Constraints: | gs(X)  Ss2 | ≤ εs Ss2 , s=1, 2, ..., N

System Identification

Frequencies fk Modal contribution factors Xk

Figure 8.29

Procedure of system identification.

8.5.3 Scaling of FRS Scaling GRS to Different Damping Ratios In contrast to the primary structures in nuclear power plants, whose modal damping ratios are usually from 5 % to 7 %, components and various types of equipment are generally made of different materials so that their damping ratios can range from 2 % to 15 %. In order to assess the seismic demands for different types of equipment accurately, GRS and FRS with the corresponding damping ratios are needed. Based on equation (8.4.32) for the modal response    u¨ 2 = AF - 0,K 0,K max







SG (ω0 , ζ0 ) 2 + AFK SG (ωK , ζK ) 2 , -

(8.5.7)

it can be anticipated that the change of the equipment damping ratio ζ0 will affect the

F the amplification factors AF - 0,K and A - K , as well as the ground input

SG (ω0 , ζ0 ).

In

Section 8.4, it is demonstrated that the damping effect on the amplification factors

372

are negligible for non-tuning cases. When the equipment is relatively much stiffer than the structure, the modal response of the structure-equipment system is reduced to the structural modal response

SG (ωK , ζK ).

As a result, the equipment damping

ratio has no effect in this case. When the equipment is relatively much more flexible than the structure, the modal response of the structure-equipment system is reduced to the response of the equipment supported directly on the ground, i.e.,

SG (ω0 , ζ0 ).

Consequently, the effect of damping on FRS is the same as that on GRS. However, the most-common standards and codes, such as ASCE 43-05 (ASCE/SEI, 2005), NUREG CR-0098 (USNRC, 1978), and CSA N289.3 (CSA, 2010a), provide GRS for only 5 % damping. Therefore, Damping Correction Factors (DCF) defined as D(ω; ζ0 , ζ0 ) =

is used to adjust GRS

S

G (ω, ζ0 ) of

SG (ω, ζ0 )

SG (ω, ζ0 ) SG (ω, ζ0 )

(8.5.8)

corresponding to ζ0 = 5 % damping ratio to GRS

another damping level ζ0 . A comprehensive study on DCF for horizontal

GRS was conducted by Cameron and Green (2007), in which DCF is tabulated for various damping ratios, site conditions, and earthquake magnitudes.

Generating FRS for Different Damping Ratios Consider the underlying structure with the significant modes identified in Section 8.5.2 under the excitation of

SG (ω, ζ0 ).

F In the perfect tuning case, ω0 = ωK , AF - 0,K = A - K , and equation (8.5.7) becomes  u¨.

 

0,K max

=

1 2ζK,e









SG (ωK , ζ0 ) 2 + SG (ωK , ζK ) 2 ,

ζK,e = √

SG (ωK , ζK ) . 2 · SGt (ωK , ζK )

Using equation (8.5.8), equation (8.5.9) can be written as  D(ωK ; ζK , ζ0 )2 + 1   u¨  SG (ωK , ζK ). 0,K max = 2ζK,e

(8.5.9)

(8.5.10)

For ζ0 = ζK , D(ωK ; ζK , ζ0 ) = 1, equation (8.5.10) reduces to   u¨  0,K max =

SG (ωK , ζK ) √

2 ζK,e

.

(8.5.11)

From equation (8.5.10), the modal responses with equipment damping ratio ζ0 is  D(ωK ; ζK , ζ0 )2 + 1   u¨  = SG (ωK , ζK ). (8.5.12) 0,K max 2ζK,e

8.5 scaling method for generating frs

uk(ζ)

373

SGt(ωk, ζ0 = ζk)

A

SGt(ωk,ζ) C

SG(ωk,ζ0 = ζk) B

ζ0 = ζk

ζ0 = ζk ζ

O

Figure 8.30

SGt(ωk, ζ0 ) ζ

tRS correction factor.

Because the physical meaning of tRS is the modal response of an equipment-structure system in perfect-tuning, equations (8.5.11) and (8.5.12) result in      u ¨ SG (ωK , ζ0 = ζK )  0,Kmax D(ωK ; ζK , ζ0 )2 + 1 ζK,e · , = = 2 ζK,e SGt (ω , ζ = ζ ) u¨ 0,K max K

0

(8.5.13)

K

is the equivalent damping coefficient corresponding to modal damping ζ where ζK,e K

and equipment damping ζ0 = ζK , and

   ∂ u˙K (ζK )    , · S ∂ζK max   1  u˙K (ζ0 ) − u˙K (ζK )  SG (ωK , ζ0 = ζK ) = ·  .  2 ζ0 − ζK max t G (ωK , ζ0 = ζK )

To determine

1 = 2

(8.5.14a) (8.5.14b)

SG (ωK , ζ0 = ζK ), consider the maximum modal velocity u˙K (ζ ), which

decreases monotonically with the modal damping ratio ζK , as illustrated in Figure 8.30 (without loss of generality, the case of ζ0 > ζK is shown): ❧ From equation (8.5.14a),

SGt (ωK , ζ0 = ζK ) and SGt (ωK , ζ0 = ζK ) equal to half of the

slopes of the tangent line at points A (with ζ0 = ζK ) and B (with ζ0 = ζK ), respectively. ❧ From equation (8.5.14b),

SG (ωK , ζ0 = ζK ) is equal to half of the slope of the secant

connecting points A and B. ❧ From the Mean Value Theorem, there exists ζa¯ between ζK and ζ0 such that

SGt (ωK , ζa¯ ) = SG (ωK , ζ0 = ζK ),

(8.5.15)

where ζa¯ = α · ζ0 +(1−α) · ζK , 0 < α < 1, i.e., the slope of the tangent line at some point C (with ζa¯ ) is equal to the slope of the secant connecting points A and B. Parametric study shows that when α = 0.5, in which ζa¯ represents the average damping ratio of equipment and the Kth structural mode, equation (8.5.15) gives sufficiently

374

accurate approximation for

SG (ωK , ζ0 = ζK ) over the frequency range from 0.1 Hz to

100 Hz. The accuracy of this approximation is affected by the damping ratio difference   ζ = ζ −ζ ; a correction factor is hence introduced in equation (8.5.15) to yield 0

K

SG (ωK , ζ0 = ζK ) = SGt (ωK , ζa¯ ) ·

  1 + ζ0 −ζK  ,

ζa¯ = 12 (ζ0 +ζK ).

(8.5.16)

It has been shown through numerical simulations that equation (8.5.16) provides excellent approximations over the entire frequency range and for various equipment damping ratios. The responses are more sensitive for lower equipment damping ratios, say ζ0 < 5 %.

for modal From equations (8.5.13) and (8.5.16) the equivalent damping ratio ζK,e

damping ζK and any equipment damping ratio ζ0 can be obtained as  t SG (ωK , ζK ) D(ωK ; ζK , ζ0 )2 +1 ζK,e . = ζK,e · ·  

2 S t ω , 1 (ζ +ζ ) · 1+ ζ −ζ  K

G

tRS

0

2

K

0

(8.5.17)

K

SGt (ω, ζ ) for any frequencies and damping ratios is given in Section 4.2, whereas

DCF D(ω; ζ , ζ0 ) is tabulated in Cameron and Green (2007).

FRS of the mth DOF of the original structure for damping ratio ζ0 can then be

obtained using equations (8.4.32), (8.4.33), and (8.4.46): 

where RK = XK

AFK =  -



SF (ω0 , ζ0 ) 2 =





N  N 

K=1 K =1

ρKK RK RK ,



(8.5.18)



AF0,K SG (ω0 , ζ0 ) 2 + AFK SG (ωK , ζK ) 2 , -

-

1 (1−rK2 )2 + (2 ζK,e r K )2

,

AF0,K = rK2 AFK , -

-

rK =

ωK , ω0

(8.5.19)

is given by equation (8.5.17). and the equivalent damping ratio ζK,e

Scaling of FRS

SG-I ( F, ζ0 ) and SF-I ( F, ζ0 ) are available. 1. System Identification: Identify significant modes from

SG-I ( F, ζ0 ) and SF-I ( F, ζ0 )

and obtain frequencies F K and modal contribution factors XK , K = 1, 2, . . . , N .

Scaling Problem 1 2. Direct Method: Using the direct method, equations (8.5.17) to (8.5.19), determine D D SF-I ( F, ζ0 ) and SF-I ( F, ζ0 ) from SG-I ( F, ζ0 ) for the desired damping ratio ζ0 , where

the superscript “D” stands for “Direct Method” .

8.5 scaling method for generating frs

375

3. Scaling FRS ❧ If

SF-I ( F, ζ0 )

is raw FRS, the scaled FRS-I for damping ratio ζ0 is obtained

through the scaling factor

S

( F, .ζ0 ) F-I ❧ If ❧

=

R

FRS-I

R FRS-I( F, ζ0, ζ0 ):

S

( F, ζ0 , ζ0 ) · F-I ( F, ζ0 ),

R

FRS-I

D SF-I ( F, ζ0 ) . = D SF-I ( F, ζ0 )

( F, ζ0 , ζ0 )

(8.5.20)

D SF-I ( F, ζ0 ) has been broadened and smoothed, SF-I ( F, ζ0 ) = SF-I ( F, ζ0 ).

SF-I ( F, ζ0 ) is then broadened and smoothed as needed.

Scaling Problem 2 2. Direct Method: Using the direct method, equations (8.5.17) to (8.5.19), determine ❧ ❧

D SF-I ( F, ζ0 ) from SG-I ( F, ζ0 ), D SF-II ( F, ζ0 ) for the desired damping ratio ζ0 from SG-II ( F, ζ0 ).

3. Scaling FRS ❧ If

SF-II ( F, ζ0 ) is raw FRS, the scaled FRS-II for damping ratio ζ0 under GRS-II

SG-II ( F, ζ0 ) is obtained through the scaling factor R FRS-II( F, ζ0 , ζ0 ):

S

( F, ζ0. ) F-II ❧ If ❧

☞

If

=

R

S

( F, ζ0 , ζ0 ) · F-I ( F, ζ0 ),

FRS-II

R

( F, ζ0 , ζ0 )

FRS-II

D SF-II ( F, ζ0 ) . = D SF-I ( F, ζ0 )

(8.5.21)

D SF-I ( F, ζ0 ) has been broadened and smoothed, SF-II ( F, ζ0 ) = SF-II ( F, ζ0 ).

SF-II ( F, ζ0 ) is then broadened and smoothed as needed. SF-I ( F, ζ0 )

has been broadened and smoothed, the scaling factors in (8.5.20)

and (8.5.21) are not used because the broadened-and-smoothed FRS-I contains a large amount of artificially modified information, which is inappropriate to use for scaling.

8.5.4

Numerical Examples

To verify the accuracy and demonstrate the efficiency of the scaling method for generating FRS, numerical examples are presented for the two scaling problems. A typical service building in nuclear power plants, as shown in Figure 8.16, is considered the primary structure. USNRC R.G. 1.60 GRS (USNRC, 2014) is selected as GRS-I and Standard UHS for CENA (Atkinson and Elgohary, 2007) is selected as GRS-II. Figures 8.17 and 8.18 illustrate the two GRS with 5 % damping ratio anchored at 0.3g PGA, along with the response spectra of 30 sets of tridirectional spectrum-compatible time-histories generated using the Hilbert–Huang transform method.

376

FRS at Node 1 in Figure 8.16 are obtained through numerical time-history analyses of the structure, and the mean FRS from the 30 sets of simulations are ued as benchmark FRS. Herein, only the mean FRS with 5 % damping ratio produced by time-histories compatible with USNRC R.G. 1.60 GRS are treated as available FRS-I; all other mean FRS will be used as benchmark for validating the scaling method.

Equivalent Modal Information Modal information of the equivalent structural modes is identified from the existing FRS-I using the method developed in Section 8.5.2. FRS-I at Node 1 with 5 % damping ratio is shown as the dashed line in Figure 8.31, which is considered as the available

SF-I ( F, ζ0 = 5 %).

There are three peaks located

around 2.5 Hz, 5.8 Hz, and 18 Hz, where significant modes exist. It is observed that the second peak has a wider band, indicating that there may exist multiple closely spaced modes in the range from 5.5 to 7.5 Hz. It should be noted that, although the third peak is relatively lower, flat, and wide, some significant modes may exist in the higher frequency range (15 to 30 Hz) because GRS-I has lower spectral values and decreases drastically in this range. The number of equivalent modes should be equal to or larger than the number of FRS peaks. To study the effect of the number of equivalent modes, FRS-I is approximated by 3, 4, 5, and 6 equivalent modes using the method for system identification. The coordinates of the critical points and modal information of the identified equivalent modes are listed in Table 8.4, where F K ,

SK ,

and XK denote the frequencies, spectral

accelerations, and contribution factors of the equivalent modes, respectively. By comparing FRS-I generated through the direct method using the equivalent modal information with the available FRS-I in Figure 8.31, some remarkable features can be observed: 1. All the reproduced FRS-I agree well with the benchmark FRS-I. FRS at other points generally converge to the benchmark values as the number of modes increases, and five-equivalent-mode approximation can give sufficient accuracy. 2. Assuming multiple closely spaced modes at the second peak, where the relatively wider peak occurs, can give better approximation as anticipated. 3. One exception is that the four-mode approximation does not produce a better result than the three-mode approximation. This phenomenon can be explained by the equivalent modal information listed in Table 8.4. From the sum of the contribution factors, it is seen that the optimal solutions for the four-mode approximation are weakly satisfactory to model the real physical structural system in this case. How-

8.5 scaling method for generating frs

377

Table 8.4 Equivalent modal information at Node 1.

Mode 1 2 3 4

Mode 1 2 3 4 5 6

3-Mode Approximation F K (Hz) SK (g) XK 2.6 2.33 0.4105 5.8 1.82 0.3543 17.5 0.72 0.2092 

XK

0.974

5-Mode Approximation F K (Hz) SK (g) XK 2.6 2.33 0.4091 5.8 1.82 0.3441 7.2 1.44 0.2199 17.5 0.72 − 0.1789 26.0 0.62 0.1704 

XK

0.965

4-Mode Approximation F K (Hz) SK (g) XK 2.6 2.33 0.4205 5.8 1.82 0.3685 17.5 0.72 0.1834 26.0 0.62 − 0.1825  XK 0.790 6-Mode Approximation F K (Hz) SK (g) XK 2.6 2.33 0.4082 5.8 1.82 0.3182 6.6 1.50 0.1283 7.2 1.45 0.1215 17.0 0.72 − 0.1721 26.0 0.62 0.1732  XK 0.977

ever, it will be seen that these discrepancies do not have a significant effect when scaling factors are employed to generate FRS-II. 4. Despite lower spectral acceleration in higher frequency range (15 to 30 Hz), the contribution factors of these modes are considerably large. It will be seen that these modes have a significant effect when scaling FRS-I to FRS-II, which corresponds to a GRS-II with rich high-frequency content.

Scaling Problem 1 – Scaling FRS to Various Damping Ratios Based on the validation of identified equivalent modal information, 5-mode approximation is applied to generate the scaling factors in equation (8.5.20) for scaling FRS-I with 5 % damping ratio to FRS-I with other damping ratios at Node 1. Figure 8.32 shows the comparison of the benchmark FRS-I and FRS-I obtained from the scaling method, for various equipment damping ratios; it is seen that the scaled FRS-I agree excellently with the benchmark FRS-I over the entire frequency range. Scaling factors are plotted in Figures 8.33. It is important to note that the shapes of the scaling factors are quite similar to the shapes of the FRS, which are functions of frequency and damping ratio. Furthermore, peaks emerge at the natural frequencies of the equivalent modes, indicating that scaling of FRS depends on the modal information of the structure. Therefore, using a constant scaling factor to scale FRS will lead to inconsistent conservatism or underestimation in any situations. At low frequencies, the scaling factors are nearly constant because FRS are close to GRS based on the physical

378

interpretation of the formula of the direct method. In very high frequency range, scaling factors converge to 1 as equipment responses approach the structural responses at the node, which are independent of the equipment damping ratios.

Scaling Problem 2 – Scaling FRS for Different GRS The scaling factors given in equation (8.5.21) are calculated using the modal information approximated by three, four, five, and six equivalent-modes; the reproduced FRS-II using the equivalent modal information in the scaling method are compared with the benchmark FRS-II in Figure 8.34. All approximations lead to excellent agreement with the benchmark results, except in the high-frequency range for the three-mode approximation, which is caused by ignoring contributions from high-frequency modes. Although these modes may not have a pronounced effect on FRS-I under GRS-I that lacks high-frequency content, their effect will be significantly amplified when the input GRS-II is rich in high-frequency content. Consequently, it is necessary to consider a few modes in the higher frequency range where GRS-II possesses abundant high-frequency content. It is also found that even though there may be significant discrepancy between the identified equivalent modal information and that of the real structure, the scaling method can still generate FRS with sufficient accuracy. FRS-II obtained using the scaling method (with five-equivalent-mode approximation) are shown in Figure 8.35. It is seen that FRS-II obtained using the scaling method agree very well with the benchmark FRS. The scaling factor given by equation (8.5.21) is shown in Figure 8.36. Note that the peaks of the scaling factors may not occur at equivalent modal frequencies because the scaling factors depend on not only the modal information but also on the differences in the spectral shapes.

Scaling Broadened FRS In practice, the available FRS-I are usually broadened and smoothed, and raw FRS may not be available. As shown in Figure 8.37, the piecewise straight solid line represents the broadened-and-smoothed FRS-I at Node 1. The dash curve is the raw FRS-I; it is shown for reference only, and its information is not used in the analysis. Different from the raw FRS-I where the locations of FRS-I peaks can be accurately identified, peaks of broadened FRS-I are generally assumed at the middle points of the plateaus. Thus, three critical points are selected from the middle point of the plateau or 15 % from the corner of the plateau. Other critical points are selected approximately based on the shape of the broadened FRS-I. It should be noted that these critical points are not necessarily on the original raw FRS-I, which is assumed to be unknown.

8.5 scaling method for generating frs

379

2.5

3-mode approximation 4-mode approximation 5-mode approximation 6-mode approximation Benchmark FRS-I

Spectral acceleration (g)

2.0

1.5

1.0

0.5

0

0.2

1

Figure 8.31

Frequency (Hz)

10

100

Equivalent-mode approximations of FRS-I at Node 1.

4.0

Scaled FRS-I

Spectral acceleration (g)

3.5

Exact FRS-I

3.0

ζ0 =2%

2.5

ζ0 =4% ζ0 =7%

2.0

ζ0 =10%

1.5

ζ0 =15%

1.0 0.5 0

0.2

1

Frequency (Hz)

Figure 8.32

10

100

Scaled FRS-I at Node 1.

1.8

R FRS-I =

1.6

D ( f, ζ0 ) SF-I D SF-I ( f, 5%)

1.4

ζ0 =2% Scaling factor

1.2

ζ0 =4%

1.0

5%

0.8

ζ0 =7% ζ0 =10% ζ0 =15%

0.6 0.4

0.2

1

Figure 8.33

Frequency (Hz)

10

Scaling factors at Node 1.

100

380 1.2

3-mode approximation 4-mode approximation 5-mode approximation 6-mode approximation Exact FRS-II

Spectral acceleration (g)

1.0 0.8 0.6 0.4 0.2 0

0.2

1

Figure 8.34

Frequency (Hz)

10

100

Equivalent-mode approximations of FRS-II at Node 1.

2.0

Scaled FRS-II

1.8

Exact FRS-II

Spectral acceleration (g)

1.6 1.4

ζ0 =2%

1.2

ζ0 =4%

1.0

ζ0 =7%

0.8

ζ0 =10%

0.6

ζ0 =15%

0.4 0.2 0

0.2

1

Frequency (Hz)

Figure 8.35 1.8

R

Scaling factor

1.6

FRS-II

=

10

100

Scaled FRS-II at Node 1.

SFD-II( f, ζ0 ) SFD-I ( f, 5%)

1.4

ζ0 =2%

1.2

ζ0 =4% ζ0 =7%

1.0

ζ0 =10% ζ0 = 15%

0.8 0.6 0.4

0.2

1

Figure 8.36

Frequency (Hz)

10

Scaling factors at Node 1.

100

8.5 scaling method for generating frs

381

2.5

Raw FRS-I Broadened-and-smoothed FRS-I 6-mode approximation

Spectral acceleration ( g)

2.0

1.5

1.0

0.5

0

0.2

1

Frequency (Hz)

10

10 0

Figure 8.37 Verification of identified modal information. 2.0

6-mode approximation Exact raw FRS-II

1.8

Spectral acceleration ( g)

1.6 1.4

ζ0 =2%

1.2

ζ0 =4%

1.0

ζ0 =7%

0.8

ζ0 =10%

0.6

ζ0 =15%

0.4 0.2 0

0.2

1

Frequency (Hz)

10

Figure 8.38

Comparison of FRS-II.

100

Table 8.5 Equivalent modal information of six-mode approximation for broadened FRS.

Mode

F K (Hz)

SK (g)

XK

1 2 3 4 5 6

2.6 5.5 6.5 7.5 17.0 25.0

2.35 1.50 1.85 1.50 0.75 0.65

0.40907 0.16376 0.34813 0.10277 − 0.18680 0.18178

382

Because the second plateau is wide, it is assumed that there are three closely spaced modes. In addition, a high-frequency mode is assumed in the higher-frequency range. Therefore, the available FRS-I is approximated by six equivalent-modes. The equivalent modal information obtained by applying the system identification technique and the coordinates of the selected critical points are listed in Table 8.5. FRS-I reproduced by using the identified equivalent modal information in the direct method are plotted as the solid curve in Figure 8.37. It can be seen that there are certain shifts at FRS-I peaks compared to the original raw FRS-I due to the bias in selecting critical points; however, these differences are not significant after broadening and smoothing. The equivalent modal information is then employed in the direct method to generate FRS-II (Figure 8.38). FRS-II obtained from the direct method can match the benchmark FRS-II very well after both are broadened and smoothed. It is worthy to emphasize that scaling factor is not used in this case because the raw FRS-I is assumed unavailable, and the broadened-and-smoothed FRS-I contains a large amount of artificially modified information, which is inappropriate to use for scaling. Nevertheless, the direct method can procedure adequately accurate FRS-II when an appropriate number of equivalent modes are included.

Conclusions A scaling method for generating FRS based on the direct spectra-to-spectra method for generating FRS is presented. The analytical formulation of the direct method provides a strong physical insight into FRS, which allows the identification of dynamical information of the significant equivalent modes of the underlying structure from GRS-I and the available FRS-I. Scaling factors are then determined in terms of the dynamical information (including modal frequencies, damping ratios, and contribution factors) and the input GRS-I and GRS-II. The method is efficient, accurate, and convenient to implement. It allows engineers to generate accurate FRS for different GRS and for various damping ratios by using as much of the available results as possible without performing a complete dynamic analysis, which introduces extra costs and is time consuming. However, it should be noted that the accuracy of scaled FRS-I or FRS-II obviously depends on the accuracy of the available FRS-I; for example, if the available FRS-I contains excessive conservatism, the scaled FRS-I or FRS-II would contain the same level of conservatism. In Appendix A: Benchmark Studies to Verify an Approximate Method for Spectra Scaling of EPRI 1002988 (EPRI, 2002, p.A-1), it is commented that

8.6 generating frs considering ssi

383

More sophisticated scaling procedures can be applied providing that the eigensolutions for the original models are available. These scaling procedures can utilize random vibration theory, direct generation computer programs, also based on random vibration theory, or time-history solutions. In some cases, the eigensolution outputs in the analysis reports are only partially complete . . . spectra are scaled . . . by more simplified procedures using only frequencies and participation factors. It should be emphasized that this scaling method does not require any information on the underlying structure yet still yields excellent FRS results. The dynamic information of the equivalent significant modes of the underlying structure are recovered by using system identification based on the direct method, which has been demonstrated to be very accurate as long as the available FRS are reasonable. If eigensolutions are available, then there is no need to use scaling methods. The direct method in Section 8.4 can be applied to generate FRS with accuracy matching those obtained from a large number of time-history analyses and with complete probabilistic descriptions of FRS peaks (any level of NEP p). On the other hand, if partial modal information (modal frequencies) is available, it can be useful in system identification in helping to locate significant modes, especially high-frequency modes; this is particularly important when available FRS-I has been broadened and smoothed.

8.6 Generating FRS Considering SSI 8.6.1 Introduction When a structure is founded on soil, the effect of interaction between the structure and its surrounding soil is not negligible (Wolf, 1985; 1987): ❧ Seismic responses at the foundation of the structure are different from the free-field responses at the site due to the presence of the structure. ❧ The structure will interact with the surrounding soil, leading to a further change of the seismic motion at the base. The typical myth about the effect of soil–structure interaction (SSI) is that considering SSI will reduce the overall seismic responses of the structure because it elongates the fundamental period of the structure, which usually corresponds to a lower spectral acceleration in a GRS. Furthermore, the effective damping of a soil–structure system, which consists of structural damping, soil material damping, and soil radiation damping, is considerably higher than that of the structure, leading to more energy dissipation

384

2.5

Spectral acceleration (g)

FRS with fixed-base 2.0

Difference 1.5

FRS with SSI 1.0 0.5 0 0.2

Equipment frequency 1 Figure 8.39

Frequency shift

Frequency (Hz)

10

50

Effect of SSI on FRS.

and further reduction of the responses. However, it is well understood that FRS peaks occur at the frequencies of the dominant modes of the structure. Considering the SSI effect results in shifting of structural natural frequencies, and thus leads to shifting of FRS peaks, which could approach the resonant frequencies of equipment mounted on the structure. Consequently, the seismic input to equipment could be significantly increased. Figure 8.39 illustrates this effect: Rhe frequency of the dominant mode of a structure reduces from 5.8 to 4.5 Hz, and the FRS peak shifts from 5.8 to 4.5 Hz when the SSI effect is considered. Although the FRS peak value for the soil–structure system is less than that for the fixed-base structure, the increase of the seismic input can be as large as 40 % (from 1.5g to 2.1g) for an equipment with a natural frequency of 4.5 Hz. Therefore, seismic input and structural analysis should not be considered independently when a structure is founded on relatively soft soil. The effect of soil will be considered in two major steps of SSI analysis: 1. Because response spectra are normally prescribed at the bedrock or ground surface, a site response analysis is performed to determine the foundation input response spectra (FIRS) base on wave propagation theory. The free-field can be generally modelled as a series of soil layers resting on the bedrock, which is usually regarded as an elastic homogeneous half-space as shown in Figure 8.40. 2. A dynamic analysis of the structure is conduct using FIRS, considering the interaction between the structure and the surrounding soil. The most straightforward approach for considering SSI effect is to model the soil–structure system as an integral part, then perform dynamic analysis for the entire system. This method is referred to as the complete method of SSI analysis. However, in contrast

8.6 generating frs considering ssi

385 Foundation input response spectra (FIRS)

Soil surface Soil layer 1 Soil type 1 Soil layer 2 Soil type 2 Site response analysis Soil layer m Soil type m Bedrock Seismic wave propagation

Seismic source

Figure 8.40

Response spectra at bedrock

Soil–structure interaction.

Finite-element structure model

Finite-element soil model

Artificial boundaries

Figure 8.41

Complete method for SSI analysis.

to the structure, which can be modelled with sufficient accuracy by a system with a finite number of DOF, the soil medium is essentially an unbounded domain. Therefore, modelling of the soil is accomplished by a truncated soil medium with so-called artificial boundaries, as shown in Figure 8.41. Conceptually, the artificial boundary conditions are capable of representing the dynamic properties of the missing soil and perfectly absorbing the incoming waves. However, the complete method requires solving a large system of coupled equations with excessive DOF, which is not only computationally expensive but also inefficient because only the responses of the structure are of interest. Moreover, when the properties of the structure or soil are changed, the entire analysis has to be repeated.

386

For these reasons, the substructure method for SSI analysis (Gutierrez and Chopra, 1978), which is theoretically equivalent to the complete method, yet allows to divide the systems into more manageable parts and to analyze these parts separately using appropriate methods, has been developed. Some commercial finite element analysis software packages, such as

SASSI

(Lysmer et al., 1983) and ACS SASSI (Ghiocel, 2015),

were developed on the basis of the substructure method and are currently employed in practice to perform dynamic analysis for soil–structure systems. However, the seismic inputs required by ACS SASSI are spectrum-compatible time-histories, which means that the deficiencies of time-history analysis for generating FRS are inevitable. Furthermore, when soil is involved in analysis, it is important to consider uncertainties in soil, resulting in more time-consuming analyses.

8.6.2 Substructure Method Dynamic Stiffness Matrix For an MDOF linear system, the equation of motion is of the form M x(t) ¨ + C x(t) ˙ + K x(t) = p(t),

(8.6.1)

where M, C, K are the mass, damping, and stiffness matrices, respectively, p(t) is the load vector, and x(t) is the response vector. Under harmonic excitation p(t) = Pe i ωt, the response x(t) can be expressed as x(t) = X e i ωt, and equation (8.6.1) becomes SX = P,

S = −ω2 M + i ω C + K,

(8.6.2)

where S is the frequency-dependent dynamic stiffness matrix. In terms of the dynamic stiffness matrix, equation of motion (8.6.1) can be expressed as an equation of dynamic equilibrium (8.6.2).

Substructure Model for Flexible Foundation A coupled soil–structure model is shown in Figure 8.42. Let Us and Ub be amplitudes of the absolute displacement vectors of the structure and foundation, respectively, where the subscripts “s” and “b” stand for the DOF of “structure” and “base” (or boundary of soil–structure interface), and the superscript “s” stands for “structure” . The equation of dynamic equilibrium of the structure is given by 2    1 s Us Ps Ssss Ssb = , s s S bs S bb Ub Pb

(8.6.3)

where Ps is the amplitude vector of the loads applied on the nodes of the structure, and Pb is the amplitude vector of the interaction forces between the structure and soil. For

8.6 generating frs considering ssi

387

Finite-element model Ssss

Us

Sssb, Ssbs U gb

Ssbb

S gbb

Ub

Seismic input at bedrock

Seismic input at bedrock

Free-field soil model

Excavated soil ⇒ “Structure”

Foundation input response spectra

~s Sbs =0 ~s Sbb =S ebb

FIRS

U fb

S fbb

S ebb ~ Ub =U fb

U gb S gbb

Free field

Soil with excavation

Site response analysis Seismic input at bedrock

Seismic input at bedrock

Figure 8.42

Coupled soil–structure model.

Finite-element model Ssss

Us

SssO , SsOs SsOO

O

UO

Seismic input at bedrock

Foundation input response spectra

f

SOO

O f

UO

Free field

O

UOg

Seismic input at bedrock

Free-field soil model FIRS

SgOO

~s SOs =0 ~s SOO =SeOO

Excavated soil ⇒ “Structure”

SeOO ~ UO =UOf

SgOO

O

UOg

Soil with excavation

Site response analysis Seismic input at bedrock

Figure 8.43

Seismic input at bedrock

Coupled soil–structure model with rigid foundation.

388

earthquake excitation, the nodes of the structure not in contact with the soil are not loaded, i.e., Ps = 0, and hence s Ssss Us + Ssb Ub = 0.

(8.6.4)

Let Sgbb be the dynamic stiffness matrix of the soil with excavation, and Ubg be the amplitudes of absolute displacement vector of the soil with excavation under the earthquake excitation. The superscript “g” stands for ground or the soil with excavation. The interaction forces of the soil depend on the relative motion between the foundation (base) and the soil at the interface, i.e., Pb = Sgbb (Ub − Ubg ). Equation (8.6.3) becomes 1 Ssss S sbs

s Ssb

2

S sbb + Sgbb

Us Ub



 =

(8.6.5)

0 Sgbb Ubg

 .

(8.6.6)

The earthquake excitation is characterized by Ubg , which is the motion of the nodes on the soil–structure interface of the soil with excavation. It is desirable to replace Ubg by the free-field motion Ubf that does not depend on the excavation.

Free-Field Soil Model The free-field soil can be divided into the excavated soil and the soil with excavation as shown in Figure 8.42. Regarding the excavated soil as a “structure” and referring s = 0, and hence S s = S e , which is the ˜ bb ˜ b = Ubf , S˜ bs to equation (8.6.6), one has U bb dynamic stiffness matrix of the excavated soil. The superscript “e” stands for excavated soil. The second block-row of equation (8.6.6) gives   × 8 s  9  s + Sg = Sgbb Ubg =⇒ (Sebb + Sgbb )Ubf = Sgbb Ubg . S˜ bs S˜ bb bb f Ub

(8.6.7)

Adding the excavated soil to the soil with excavation leads to the free-field system, i.e., Sgbb + Sebb = S fbb,

or

Sgbb = S fbb − Sebb .

(8.6.8)

Hence, equation (8.6.7) can be written as S fbb Ubf = Sgbb Ubg ,

(8.6.9)

in which S fbb is the dynamic stiffness matrix of the free-field discretized at the nodes where the structure is inserted, and Ubf is the free-field motion at the nodes of the

8.6 generating frs considering ssi

389

soil–structure interface. Hence, Ubf is the free-field response of the soil at the foundation level; the acceleration response spectra of u¨ fb are the FIRS, which can be obtained from a site response analysis of the free-field. Using (8.6.9), equation (8.6.6) becomes 1 2  s Ssss Ssb Us S sbs

S sbb + Sgbb

Ub

 =



0 S fbb Ubf

.

(8.6.10)

This is the equation of motion of the structure supported on a generalized soil spring characterized by the dynamic stiffness matrix Sgbb , and the other end of the spring is subjected to earthquake excitation Ubf , which is free-field response at the foundation level. Using (8.6.8), equation (8.6.10) can also be written as 1 s 2    s Ssb Sss 0 Us = . Ub S sbs (S sbb − Sebb )+ S fbb S fbb Ubf

☞

(8.6.11)

A generalized soil spring, characterized by the dynamic stiffness matrix Sgbb , is not an elastic spring in the ordinary sense characterized by spring constant K.

Substructure Model for Rigid Foundation In many engineering applications, such as in NPPs, the foundations can be assumed to be rigid. In this case, the free-field earthquake excitation is applied at only one node O on the foundation (Figure 8.43). Hence, referring to the general case of flexible s , Ss foundation, one has S sbb =⇒ SOO bs

Ub

=⇒

UO , Ubg

=⇒

UOg , Ubf 1 Ssss s SOs

=⇒

s , Ss SOs sb

=⇒

s , Sg SsO bb

=⇒

g , Sf SOO bb

=⇒

f , SOO

=⇒

UOf . Equation (8.6.10) then becomes 2    s 0 SsO Us = . s + Sg f Uf SOO UO SOO OO O

(8.6.12)

This is the equation of motion of the structure supported on a generalized spring g at node O, and the other end of characterized by the dynamic stiffness matrix SOO

the spring is subjected to earthquake excitation UOf , which is free-field response at the foundation level (node O as shown in Figure 8.44). Using (8.6.8), equation (8.6.12) can also be written as

1

Ssss

s SsO

s s − S e )+ S f SOs (SOO OO OO

2

Us UO



 =

0 f Uf SOO O

 .

(8.6.13)

For a structure with N nodes (not including the rigid foundation), each node has six DOF (three translational and three rotational). The rigid foundation has one node O with six DOF. The dimensions of the vectors Us , UO , and UOf are 6N, 6, and 6,

390

Finite-element model

Ssss

Us

SssO , SOs s

O

SsOO

SgOO

SfOO

UOf

Seismic input at bedrock

Figure 8.44

Soil-spring model of SSI with rigid foundation.

Finite-element model

Ssss

Us

SssO O Fixed-base fb

Tridirectional UO Foundation level input response spectra

FLIRS Seismic input at bedrock

Figure 8.45

Fixed-base model with rigid foundation.

s , Ss , Ss respectively. The dimensions of the dynamic stiffness submatrices Ssss , SsO Os OO f , Sg , Se of the structure are 6N×6N, 6N×6, 6×6N, and 6×6, respectively, and SOO OO OO

of the soil are all of dimension 6×6.

Fixed-Base Model for Rigid Foundation If the soil is firm enough so that the structure can be considered as fixed-base as shown in Figure 8.45, the motion of point O of the basemat is the earthquake input to the structure. From the first block-row of equation (8.6.12), one has s UO = 0 Ssss Us + SsO

=⇒

Us = Sfb UO ,

s , Sfb = − Ssss −1 SsO

(8.6.14)

where Sfb is the dynamic stiffness matrix for fixed-base analysis, the superscript “fb” stands for fixed-base. In seismic analysis and design, only translational ground motions are considered, while rotational ground motions are not considered. Reorganize vector Us and rewrite

8.6 generating frs considering ssi

UO as

 Us =

Us,T Us, R

391



 UO =

,

UOfb



0

6N×1

,

(8.6.15)

6×1

in which the subscripts “T ” and “R ” stand for translational and rotational DOF, respectively. Rearranging and partitioning Sfb accordingly, one has 1 fb 2 fb S S TT TR Sfb = , fb fb SRT SRR 6N×6

(8.6.16)

in which each submatrix is of dimension 3N×3. Equation (8.6.14) can be written as   1 fb 2  fb  ⎧ fb fb ⎫ fb ⎨ STT UO ⎬ Us,T STT STR UO . (8.6.17) = = ⎩ Sfb U fb ⎭ fb fb Us, R 0 SRT SRR RT O fb T Multiplying the first block-row of equation (8.6.17) by STT yields

fb STT

T

Us,T =



fb STT

T



fb STT UOfb .

The reason for performing this manipulation is to make



fb STT

(8.6.18)

T



fb a square matrix STT

of dimension 3×3, the purpose of which will be clear in Section 8.6.3. The tridirectional (translational) acceleration response spectra UOfb applied at the foundation of a fixed-base structure are called foundation level input response spectra (FLIRS), as shown in Figure 8.45. It is important to note that FLIRS are different from FIRS, which are the acceleration response spectra at the elevation of the foundation of the free-field, as illustrated in Figure 8.43. The concept of FLIRS, which are the seismic input to fixed-base structures, is important in seismic design and assessment of NPPs. Generic design of an NPP is based on fixed-base analysis under the tridirectional seismic excitations represented by standard GRS, such as those in CSA N289.3 or USNRC R.G. 1.60, anchored at a specific PGA. By comparing the site-specific FLIRS with the standard GRS, based on which the generic NPP is designed, initial feasibility of the generic design at the desired site can be assessed and SSCs that may be vulnerable can be identified. Because the dynamic stiffness submatrix Ssss is of dimension 6N×6N, evaluating its inverse in equation (8.6.14) could be numerically challenging when N is large. To take advantage of the modal properties of the structure, a modal analysis is conducted. For a three-dimensional model of a structure with N nodes (not including rigid foundation), the relative displacement vector x of dimension 6N is governed by equation (8.2.8). For a structure with rigid foundation resting on soil, the base excitations

392

may also contain rotational components, equation (8.2.8) can be extended to M x(t) ¨ + C x(t) ˙ + K x(t) = −M I u¨ O (t),

(8.6.19)

where node O is at the rigid foundation, and 



I = I1 I2 I3 I4 I5 I6 ,

!T

u¨ O (t) = u¨O1 (t), u¨O2 (t), u¨O3 (t), θ¨O1 (t), θ¨O2 (t), θ¨O3 (t) .

Here I I are defined in equation (3.6.2) for I = 1, 2, 3, and ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ 0 ⎪ − y¯n ⎪ z¯n ⎪ r11 ⎪ r12 ⎪ r13 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−z¯n ⎪ ⎬ ⎬ ⎨ r2 ⎪ ⎬ ⎨ r1 ⎪ ⎨0⎪ ⎬ ⎨ x¯ n ⎪ ⎬ ⎨ r3 ⎪ ⎬ 2 2 2 I 4 = . , r1n = 01 , I 5 = . , rn2 = 00 , I 6 = . , rn3 = 00 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ .. ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ 2⎭ ⎩ 1⎭ ⎭ ⎩ ⎩ ⎭ ⎩ 3⎭ rN rN rN 0 0 1 in which x¯ n , y¯n , and z¯n represent the coordinates of the nth node in a Cartesian coordinate system with its origin located at Node O. Letting x(t) = Xe i ωt and uO (t) = UO e i ωt, equation (8.6.19) becomes (−ω2 M + iω C + K)X = ω2 M I UO .

(8.6.20)

Applying the modal transformation X = Q, where  is the modal matrix, substituting into equation (8.6.20), and multiplying T from the left yield (−ω2 T M + iω T C + T K)Q = ω2 T MI UO .

(8.6.21)

Employing the orthogonality gives !

diag −ω2 + i2ζn ωn ω + ωn2 Q = ω2 UO ,

=

T MI , T M

(8.6.22)

where is a 6N×6 matrix of the modal participation factors. Hence, X = ω2 H UO ,

(8.6.23)

where H is a 6N×6N diagonal matrix of the complex frequency response functions, i.e.,



 1 H = diag . ωn2 − ω2 + i2ζn ωn ω

(8.6.24)

Because the relative displacement x = u−I uO , substituting into equation (8.6.23) gives U = (ω2 H + I ) UO .

(8.6.25)

Comparing equations (8.6.25) and (8.6.14), one obtains Sfb = ω2 H + I.

(8.6.26)

8.6 generating frs considering ssi

393 N

un,5 Mn,5 =mn,5 un,5 n

Multiple DOF structure

un,1 Fn,1 =mn,1 un,1

3

2

ug3(t)

ug2(t) ug1(t)

Figure 8.46

1

MO,5

O

FO,1

Dynamic equilibrium of structure-foundation system.

Based on Newton’s second law, the dynamic force equilibrium of the structurefoundation system in Direction 1, as illustrated in Figure 8.46, is given by  −ω

2

N  n=1

 mn,1 Un,1 + mO,1 UO,1

= FO,1 ,

(8.6.27)

in which FO,1 is the interaction force in Direction 1. Equation (8.6.27) can be written in the matrix form as 



−ω2 (I 1 )T MU + mO,1 UO,1 = FO,1 .

(8.6.28)

Similarly, taking moment about Node O, the dynamic moment equilibrium of the structure-foundation system in Direction 5 is given by  −ω

2

N  n=1

 (mn,5 Un,5 + mn,1 Un,1 z¯n ) + mO,5 UO,5 = MO,5 ,

(8.6.29)

or in the matrix form 



−ω2 (I 5 )T MU + mO,5 UO,5 = MO,5 .

(8.6.30)

Equations of dynamic equilibrium in other directions can be derived similarly. Hence, the dynamic equilibrium equation of the entire structure-foundation system is −ω2 (I T MU + MO UO ) = FO ,

(8.6.31)

where the first term represents the resultant of motion of the structure about the foundation at Node O, MO is a 6×6 mass matrix of the foundation, and FO is the f (U f − U ) vector of SSI forces acting on the foundation, which are given by FO = SOO O O

from equation (8.6.5).

394

Therefore, equation (8.6.31) can be rewritten as 



f f −ω2 I T MU + −ω2 MO + SOO UO = SOO UOf .

(8.6.32)

Comparing with the second block-row of equation (8.6.13), a structure founded on the e = 0; hence ground surface implies SOO s SOs = −ω2 I T M,

8.6.3

s SOO = −ω2 MO .

(8.6.33)

Foundation Level Input Response Spectra (FLIRS)

It is desirable to determine the equivalent FLIRS for a structure with rigid foundation in seismic design and assessment. In SSI analysis, a fixed-base analysis can be performed using the equivalent FLIRS as the seismic input, instead of a coupled soil–structure analysis using FIRS as the seismic input. From the first block-row of equation (8.6.13), one obtains s UO = Sfb UO . Us = − Ssss −1 SsO

(8.6.34)

From the second block-row of equation (8.6.13), one has  

s s e f f SOs Us + SOO − SOO UO = SOO UOf . + SOO

(8.6.35)

Substituting equation (8.6.34) into (8.6.35) yields  

s s e f f SOs Sfb UO + SOO − SOO UO = SOO UOf , + SOO

which gives f UOf , UO = S −1 SOO 4567 4567 4567 4567

6×1

6×6

6×6 6×1

s s e f + SOO S = SOs Sfb + SOO − SOO . 4567 4567 4 56 7 6×6N 6N×6

(8.6.36)

6×6

f is a square matrix of dimension 6×6; partition it as follows: Note that S −1 SOO 1 2 TTT TTR −1 f S SOO = T = , (8.6.37) TRT TRR 6×6

in which each submatrix is of dimension 3×3. Substituting equation (8.6.33) into (8.6.36) yields

f S = −ω2 I T M Sfb + MO + SOO .

(8.6.38)

Because the earthquake influence matrix I and the fixed-base structural response f denotes the dynamic stiffness of the transfer matrix Sfb are dimensionless, and SOO

soil springs, equation (8.6.38) can be expressed in terms of a standard dynamic stiffness matrix as ˜ + iω C f +Kf , S = −ω2 M

˜ = I T M Sfb + MO , M

f SOO = iω C f +Kf,

(8.6.39)

8.6 generating frs considering ssi

395

˜ is a 6×6 mass matrix determined by the structure and foundation mass where M matrices, influence matrix, and the fixed-base structure transfer matrix Sfb ; Kf and C f are the stiffness and damping matrices of the generalized soil springs, respectively. Therefore, the problem can be interpreted as a synthesized 6-DOF mass, which is frequency-dependent, supported by generalized soil springs. With this understanding of the physical behaviour of the soil–structure system, the advantage of the direct spectra-to-spectra method becomes evident: When the properties of a structure or soil are changed, only the synthesized mass or the stiffnesses of the generalized soil springs need to change; as a result, a reanalysis of the entire system, which is timeconsuming, can be avoided. Furthermore, the required computational effort is reduced significantly because it is needed to evaluate the inverse of a 6×6 matrix rather than a 6N×6N matrix, which may lead to numerical difficulties for a large-scale system. In a site response analysis, the soil medium is modelled as a series of infinite layers on a half-space, and the rotational responses of free-field should be very small under the translational excitation at bedrock. Hence, the rotational input at foundation level is negligible compared to the translational input; the rotational input is usually not given by a site response analysis and is taken as 0. From equations (8.6.34) and (8.6.36), one has Us = Sfb T UOf , i.e., 

1



Us,T

=

Us, R

=

1

TTT

TTR

2



f UO, T



fb TRT TRR 6×6 0 6×1 SRR 6N×6 ⎫ ⎤ ⎧ 2⎡ f fb T U f + Sfb T U f ⎬ fb ⎨ STT TTT UO, STR TT O,T TR RT O,T T ⎣ ⎦= . (8.6.40) f ⎩ Sfb T U f + Sfb T U f ⎭ fb TRT UO,T SRR RT TT O,T RR RT O,T fb SRT

6N×1

1

2 fb STR

fb STT

fb STT fb SRT

Note that it is not possible to have a single set of tridirectional translational FLIRS in a fixed-base analysis to give both correct translational responses Us,T and rotational responses Us, R . In generating FRS, only translational responses are needed; hence, from the first block-row of equation (8.6.40), one has fb f fb f Us,T = STT TTT UO, T + STR TRT UO,T .

(8.6.41)

fb T Multiplying STT from the left yields

fb STT

T

Us,T =

= Because



fb STT



T



fb STT

fb STT

T

T

! fb T fb fb f STT TTT + STT STR TRT UO, T

fb STT



TTT +



fb STT

T

fb STT



−1



fb STT

T

! fb f STR TRT UO, (8.6.42) T.



fb is a square matrix of dimension 3×3, it is easy to determine its STT

inverse. Thus, the purpose of the transformation in equation (8.6.18) becomes evident.

396

Comparing equations (8.6.42) and (8.6.18), one obtains the equivalent FLIRS as  

f fb T fb −1 fb T fb UOfb = T UO, T = TTT + STT STT STT STR TRT . (8.6.43) T, 4567 4567 4 56 7 4 56 7 4567 4 56 7 3×3

3×3

3×3

3×3N 3N×3 3×3

The first and second terms of T denote the contributions from the translational and rotational motions of the foundation in the soil–structure system, respectively. It is important to emphasize that, although the FLIRS given by equation (8.6.43) would not give correct rotational responses Us, R of a structure, it gives exact translational responses and hence exact FRS because only translational responses are required to generate FRS. Therefore, the fixed-base analysis of the structure under the excitation of FLIRS UOfb given by equation (8.6.43) gives exactly the same FRS as a full coupled f . soil–structure analysis under the excitation of FIRS UO, T

Based on the theory of random vibration, the relation between the PSD functions of UOfb

f can be determined by, using equation (3.5.5), and UO, T    SUfb¨ U¨ (ω) = T(ω)2 SUf¨ U¨ (ω),

where

(8.6.44)

SUfb¨ U¨ (ω) and SUf¨ U¨ (ω) are 3×1 vectors of the PSD functions of

respectively.

f , UOfb and UO, T  2 T(ω) denotes the matrix in which each element is equal to the

squared modulus of the corresponding element in T. For a complex number a+iB, its   √ modulus is defined as a+iB = a2 +B 2 . It is found that, for structures in NPPs, the off-diagonal terms of T are relatively small compared to the diagonal terms, and thus may be neglected. It means that the motion of the foundation in one direction is only induced by the earthquake excitation in the same direction. f can The mean-square response of an SDOF oscillator under a base excitation UO, T

be obtained by, using equation (3.5.7),  ∞  2  1 2 ω H(ω)2 S f (ω) dω , ¨ E[ X0 (t) ] = 0 U¨ U¨ 2π − ∞ where

(8.6.45)

H(ω) is the complex frequency response function with respect to base excitation

of the SDOF oscillator (with circular frequency ω0 and damping ratio ζ0 ) given by equation (3.3.24). For excitations with wide-band PSD functions, approximated by constant

S

f . U¨ U¨

SUf¨ U¨ (ω)

can be

From equations (8.6.44) and (8.6.45), the ratios

between the mean square responses of an SDOF oscillator under base excitation UOfb f can be calculated by and those under base excitation UO, T

R2 (ω0 , ζ0 ) =

 ∞      2 ω H(ω)2 T(ω)2 −∞

0

SUf¨ U¨ (ω)dω

 ∞   2 ω H(ω)2 S f (ω)dω 0 U¨ U¨ −∞

=

 ∞       H(ω)2 T(ω)2 1dω −∞

 ∞ −∞

| H(ω)|2 dω

,

(8.6.46)

8.6 generating frs considering ssi

397

where 1 is the 3×1 vector with all elements being 1. Equation (8.6.46) can be easily evaluated numerically. The maximum response of an SDOF oscillator, which is by definition the response spectrum, is usually related to its root-mean-square response through a peak factor as (see Sections 3.2.3 and 3.2.4) 



SA(ω0 , ζ0 ) = X0 (t)max = Pf ·

 E[ X20 (t) ].

(8.6.47)

Combining equations (8.6.46) and (8.6.47) yields the tridirectional fixed-base FLIRS

SAfb(ω0 , ζ0 ) =

Pf fb Pf f

SAf(ω0 , ζ0 ).

R(ω0 , ζ0 )

(8.6.48)

For responses in earthquake engineering, the values of peak factors Pf fb and Pf f do

not differ significantly, i.e., Pf fb ≈ Pf f. Hence

SAfb(ω0 , ζ0 ) ≈ R(ω0 , ζ0 ) SAf(ω0 , ζ0 ),

(8.6.49)

in which R(ω0 , ζ0 ) can be interpreted as the response spectrum modification factor from FIRS to FLIRS.

Procedure – Generating FRS Considering SSI For a structure in an NPP plant with its rigid foundation embedded in layered soil, a procedure for generating FRS considering SSI is illustrated in Figure 8.47 and is summarized as follows: 1. Consider the layered soil as a free-field. With seismic input applied at the bedrock, f at the elevation a site response analysis is performed to obtain the FIRS UOf or UO, T

of the foundation (Section 5.6). 2. Establish a model of the layered soil. Determined the dynamic stiffness matrices e and the soil with excavation S g . The dynamic stiffness of the excavated soil SOO OO f = Sg + Se . matrix of the free-field is SOO OO OO

3. Set up a finite element model of the structure. Determine the dynamic stiffness s , S s , S s . Perform a modal analysis to determine the modal matrices Ssss , SsO Os OO

frequencies ωn , modal damping coefficients ζn , modal matrix , and matrix of modal contribution factors . 4. Determine the FLIRS: 1 ❧ Partition matrix S

fb

=

ω2 H

+I=

fb STT

2 fb STR

fb SRT

fb SRR

6N×6

398

3

Seismic input at bedrock

Finite-element model of structure s Sss

Dynamic stiffness matrices

FIRS O

O

1 UOf

Ss , SOs s s O s SOO

Free-field soil model

Free field Site response analysis

f SOO

Seismic input at bedrock

Excavated soil e SOO

O g SOO

Modal information

ωn , ζn , , H

Foundation input response spectra

FIRS

4

Procedure for generating FRS considering SSI.

Soil with excavation

Dynamic stiffness matrices

Soil model

2

Figure 8.47

5

un,6

un,3

n

un,2

un,5

un,1 un,4

O

fb

Fixed-base

Tridirectional U

O

Direct method for generating FRS

FLIRS

Foundation level input response spectra

8.6 generating frs considering ssi

 H = diag =

399

1 2 2 ωn − ω + i2ζn ωn ω

6N×6N

T M I is a 6N×6 matrix of the modal participation factors T M 

I = I1 I2 I3 I4 I5 I6 ❧







6N×6

s Sfb + S s − S e f S = SOs OO OO + SOO , dimension 6×6

Determine the inverse S −1 ❧ Partition matrix

f S −1 SOO

1

=

❧ Transfer matrix: T = TTT +

TTT

TTR

TRT

TRR



fb STT

T

2

fb STT

6×6



−1

 ❧ FLIRS modification factor: R2 (ω0 , ζ0 ) =



fb STT

∞ −∞

T

fb T STR RT

      H(ω)2 T(ω)2 1 dω 

∞ −∞

❧ FLIRS: 5. The FLIRS

   H(ω)2 dω

SAfb(ω0 , ζ0 ) = R(ω0 , ζ0 ) SAf(ω0 , ζ0 )

SAfb(ω0 , ζ0 )

are input to the fixed-base finite element model of the

structure to generate the required FRS, which are exactly the same as the FRS obtained from a full coupled soil–structure analysis under the excitation of FIRS. Therefore, when the direct spectra-to-spectra method presented in Section 8.4 is applied to the fixed-base structure under the excitation of FLIRS

SAfb(ω0 , ζ0 ), FRS with

complete probabilistic descriptions of FRS peaks (FRS with any desired level of NEP p) can be obtained. If the method of time-history is applied, such a result could only be obtained from a large number of coupled soil–structure analyses using a commercial finite element software, such as ACS SASSI, with a large number of generated time-histories compatible with the FIRS.

8.6.4 Numerical Examples To verify the accuracy and demonstrate the efficiency of the direct method, FRS of a reactor building in NPP founded on the surface of soil medium are generated following the procedure summarized in Section 8.6.3.

400

Figure 8.48

Figure 8.49

Dynamic model of reactor building.

Finite element model of reactor building.

8.6 generating frs considering ssi

401

Finite Element Modelling The reactor building consists of a cylindrical containment and an internal structure, both of which are supported by a circular base slab (radius 19.8 m and thickness 3 m), as illustrated in Figure 8.48. Fixed-base finite element model of the reactor building is first established by commercial software ANsys; the superstructure is modelled as a lumped-parameter system, which can characterize the most significant dynamic properties of the structure. The model is symmetric about X- and Y-axes, and the finite element model information is described in Tables 8.6 and 8.7 (Li et al., 2005). A modal analysis is performed for the fixed-base reactor building model. Basic modal information, including natural frequencies and mode shapes of the total 66 modes, is extracted. Modal information of some significant modes at locations of interest is listed in Table 8.8, and the modal damping ratio of the structure is 5 %. The established ANsys model is then imported into the commercial SSI software ACS SASSI,

in which the underlying soil properties can be defined. In order to compute

the soil impedance for SSI analysis, the base slab, which can be rationally considered as rigid foundation, is discretized into massless shell elements, and a lumped mass that connects to the base of the superstructure is assigned at the center of the circular foundation. There are 11 nodes with lumped masses for the superstructure and 112 shell elements for the foundation. Therefore, the structure-foundation system is modelled as a 6×12 DOF system. For a fixed-base model, the DOF at Node O, which is the center of the circular foundation, are fully constrained. The soil medium is modelled as 25 infinite soil layers resting on a homogeneous half-space. Each soil layer has the following properties: depth 6m, unit weight 25.89 kN/m3 , Poisson’s ratio 0.3, material damping 0.01, and shear wave velocity is 2100 m/s in the top 10 layers, 2150 m/s in the next 10 layers, and 2200 m/s in the lower five layers. The half-space has the same unit weight and Poisson’s ratio as the soil layers, material damping is 0.07, and shear wave velocity is 2804 m/s. Note that even though there are only three sets of soil properties, it is still necessary to divide the soil into a number of thinner layers, because the thickness of a layer should not exceed one-fifth of the wave length. A finite element model of the structure-foundation-soil system is presented in Figure 8.49, in which a truncated soil medium is shown for purpose of illustration.

Foundation Level Input Response Spectra For the purpose of illustration, USNRC R.G. 1.60 response spectra (USNRC, 2014) are assumed as the FIRS obtained from a site response analysis. The PGA are anchored at

402 Table 8.6

Nodal information of reactor building model.

Node Elevation (m) Mass ( ×106 kg) 0 1 2 3 4 5 6 7 8 9 10 11

− 10 − 4.5 4 10.32 19.15 29 − 0.585 9.875 20 30 39.15 50.02 Table 8.7

Section 1 2 3 4 5 6 7

8425 13420 5710 5970 6750 1270

Beam 0 1 2 3 4 5 6 −10

2288 3033 2960 2960 3068 6271

Area (m2 ) 1204 50 110 140 60 176 107

2 4 7 12

Frequency (Hz) 4.393 5.449 12.721 18.753

424 568 554 554 562 910

824 1087 1063 1063 1081 1727

Beam element properties of reactor building model.

Table 8.8

Mode

Moment of Inertia (× 106 kg · m2 ) Ixx = Iyy Izz 843 1643 1260 1931 370 0 394 0 500 0 110 0

Shear Area (m2 ) 1084.7 19 70 70 30 88 53.5

Second Moment of Area (m4 ) 115436 5720 8160 8160 325 30570 19241

Modal information of significant modes.

Participation Factor 1.279 1.336 − 0.511 0.114

Modal Contribution Factor Node 2 Node 3 Node 4 Node 5 0.04 0.05 0.06 0.08 0.64 0.82 1.05 1.34 0.19 0.14 − 0.01 − 0.51 0.11 0.05 − 0.10 0.06

0.3g and 0.2g for the horizontal and vertical directions, respectively. 30 sets of tridirectional compatible time-histories generated in Section 8.4.6 are used for performing time-history analyses to provide the benchmark FRS. Following Step 4 in the procedure in Section 8.6.3, the mass matrix and earthquake influence matrix can be readily determined from the information in Table 8.6. The dimensionless transfer matrix of the fixed-base model Sfb is calculated for different values of ω. Each element in matrix Sfb is complex and can be regarded as a transfer

8.6 generating frs considering ssi

403

fb corresponding to the translational DOF function. The modulus of the elements STT

versus frequency are plotted in Figure 8.50 for Nodes 2 to 5. It can be observed that the modulus of the transfer functions peak at the frequencies of the significant modes. The transfer matrix T defined by equation (8.6.43) is determined frequency by frequency; Figure 8.51 compares the modulus of the translation component at the foundation Node O, which is characterized by the modulus of the first term TTT in equation (8.6.43), with the transfer function of the horizontal motion at Node O given by ACS SASSI. It is seen that the transfer function given by the direct method is in excellent agreement with the result from ACS SASSI. To distinguish the contributions to FLIRS from the translational and rotational   motions of the foundation, the FLIRS transfer function T , which includes contri11

butions from both the translational and rotational motions, and the contribution of    are plotted in Figure 8.52. It can be observed that the translational component  T TT , 11

rotational components have a pronounced effect on the total equivalent base excitation to the fixed-base model in the frequency range from 2 Hz to 10 Hz, which covers the frequencies of the dominant structural modes. Therefore, the rotational motion of foundation cannot be neglected in this case. Analogous to the modulus of the transfer matrix of the fixed-base structure shown in Figure 8.50, where peaks emerge at the frequencies of the significant structural modes, the frequencies corresponding to the peaks in Figure 8.52 can be interpreted as the natural frequencies of the soil–structure system (or the equivalent synthesized mass-spring-damper system). For instance, the first two peaks of the soil–structure system, located at 4.1 Hz and 5.1 Hz, can be explained as a result of frequency shift due to the SSI effect from 4.4 Hz and 5.4 Hz of the fixed-base model. On the other hand, the significant modal frequencies of the fixed-base model correspond to the bottom of the valley between the peaks, implying considerable reductions of the responses of the structure around those frequencies. The FLIRS modification factors R(ω0 , ζ0 ) are then used to generate FLIRS from FIRS; FLIRS are used in the direct method for generating FRS from the fixed-base model. The horizontal FLIRS is shown in Figure 8.53. It can be seen that FLIRS decreases around the dominant frequency (between 5 Hz and 8 Hz), but increases at some other frequencies, especially around 4 Hz. Therefore, it is anticipated that FRS may increase when the effect of SSI is taken into account.

Floor Response Spectra The direct method (Section 8.4) is applied to generate FRS in the internal structure of the reactor building. The FLIRS are used as the input response spectra to the fixed-base

404 14 Node 2 Dimensionless complex modulus

12

Node 3 Node 4

10

Node 5

8 6 4 2 0

0.2

Figure 8.50 1.2

1

Frequency (Hz)

Modulus of fixed-base model transfer function

10

100

fb for Nodes 2 to 5 in X-direction. STT

Soil-structure model natural frequencies

1.1

Amplification factor

1.0 0.9 0.8 0.7 SASSI Direct method

0.6

Fixed-base model natural frequencies

0.5 0.4 0.3 0.2

0.5 1 10 Frequency (Hz) Figure 8.51 Modulus of horizontal component in transfer function of foundation.

70

1.8

Dimensionless modulus

1.6

FLIRS transfer function |T11|

1.4

Soil-structure model natural frequencies

1.2 1.0 0.8

Translational contribution |TTT,11|

0.6 0.4

Fixed-base model natural frequencies

0.2 0 0.5 Figure 8.52

Frequency (Hz) 1 10 70 Effect of soil properties on modulus of transfer matrix horizontal component.

8.6 generating frs considering ssi

1.2 Spectral acceleration (g)

405

Horizontal FLIRS

1.0 0.8

Horizontal FIRS

0.6

Vertical FIRS

0.4

Vertical FLIRS

0.2 0.2

1

10 Frequency (Hz) Figure 8.53 Horizontal FLIRS.

6

+27%

Spectral acceleration (g)

5

−23%

100

Time-history Direct method Relative error 1%

4

3

2

1 0 1 10 Frequency (Hz) Figure 8.54 Comparison of FRS with 50 % NEP at Node 4.

+24%

8

100

Time-history Direct method

7 Relative error 2.1%

−24%

Spectral acceleration (g)

6 5 4 3

Relative error 4%

2 1

0.2

0 1 10 Frequency (Hz) Figure 8.55 Comparison of FRS with 50 % NEP at Node 5.

100

406

model. FRS with 50 % NEP at Node 4 and Node 5, along with FRS generated by the 30 sets of time-history analyses, are plotted in Figures 8.54 and 8.55, respectively. The mean FRS of the time-history analyses, which is regarded as the benchmark FRS, is shown in bold dashed line. It is seen that the FRS obtained by the direct method generally agree very well with the benchmark FRS over the entire frequency range, whereas individual FRS from time-history analyses exhibit large variability. Particularly, FRS peak values, which are of main interest to engineers, can be overestimated by more than 24 % or underestimated by more than 23 %. However, the differences at the FRS peaks between the direct method and the benchmark FRS are generally less than 5 %, well within the range of acceptable errors. As discussed in Section 8.4, a remarkable feature of the direct method is that it is capable of providing complete probabilistic descriptions of FRS peaks. Figures 8.56 and 8.57 demonstrate the accuracy of the direct method by comparing FRS with 84.1 % NEP at Node 4 and Node 5, respectively.

Effect of Soil–Structure Interaction on FRS To understand the effect of SSI on FRS, the mean FRS at Node 5, for both fixed-base and soil–structure models, are plotted in Figure 8.58. Although the soil is sufficiently firm, it is observed that the peak value of FRS is reduced by 10.4 % when the SSI effect is accounted, and the peak floor acceleration (representing the structural response) is reduced by 7 %. This implies that the effect of SSI is more significant on FRS. However, FRS of the soil–structure model are not always lower than those of the fixed-base model. It can be seen that there is a peak emerging on the left of the main peak (around 4 Hz), which leads to a 16 % increase in FRS. Furthermore, the spectral value at the second FRS peak is increased by 11 %. This phenomenon can be explained by Figure 8.59. For the fixed-base model, the vibration of the containment is independent of the vibration of the internal structure. For the system of containment and internal structure, the first mode shape is dominated by the vibration of the containment, while the second mode is mostly contributed by the vibration of internal structure. When SSI is taken into account, the movement of these two parts is no longer uncoupled because they are supported by the same foundation. For instance, the motion of the containment under earthquake excitation leads to considerable rotational inertia forces to the foundation because the containment has large mass and is nearly 50 m tall. The rotation of the foundation will subsequently result in the movement of the internal structure. This also explains that FRS of the internal structure increases in the first mode, which is the dominant mode of the containment, when the SSI effect is considered.

8.6 generating frs considering ssi

6

+27%

Time-history Direct method

Spectral acceleration (g)

5

−23%

407

Relative error 0.2%

4

3

2

1 0 1 10 Frequency (Hz) Figure 8.56 Comparison of FRS with 84 % NEP at Node 4. 24%

8

Time-history Direct method

7

Spectral acceleration (g)

6

−24%

100

Relative error 1%

5 4 Relative error 2% 3 2 1

0.2

0 1 10 Frequency (Hz) Figure 8.57 Comparison of FRS with 84 % NEP at Node 5.

Spectral acceleration (g) 6

Fixed-base

100

10.4%

5 4 3 2

16%

11%

SSI

1 0.2

7% 1 Frequency (Hz) 10 Figure 8.58 Illustration of SSI effect on FRS.

100

408

First mode for fixed-base model (4.4 Hz)

Containment

Internal structure

Displacement due to rotation of foundation

Displacement due to structural vibration

Second mode for fixed-base model (5.4 Hz) Foundation

SSI effect on structural response

Figure 8.59

Rotational effect of SSI.

Conclusions The efficient and accurate direct spectra-to-spectra method for generating FRS in Section 8.4 is extended to consider SSI using the substructure technique. The tridirectional FIRS, obtained from a site response analysis of the free field, are modified by multiplying a vector of modification factors, which depend on the properties of both the structure and soil. The modified response spectra, called FLIRS, are then used as the input to the fixed-base structure to generate FRS using the direct method. The concept of FLIRS has great practical significance in seismic risk assessment. FRS obtained by the direct method agree very well with the resultant FRS (such as mean, median, and 84.1 % NEP) from a large number of time-history analyses; whereas FRS obtained from time-history analyses exhibit large variability at FRS peaks. It is also demonstrated that the effect of SSI may increase FRS at certain frequencies, which leads to higher seismic demands for SSCs mounted on the supporting structure. ❧

❧

Floor response spectra (FRS) are the most important seismic input to structures, systems, and components (SSCs) in seismic design, qualification, and assessment. ❧ There are two types of methods for determining FRS • time-history method, • direct spectra-to-spectra method.

8.6 generating frs considering ssi

409

❧ The time-history method is easy to apply, and there are a number of commercial finite element packages to perform this task. For a given set of tridirectional timehistories, the FRS obtained are numerically exact. However, it is observed that, for time-histories that satisfy code requirements for compatibility, there are large variabilities in FRS, especially at FRS peaks. Numerical examples show that such variabilities can be as large as from −30 % to +30 %. Hence, different spectrumcompatible time-histories give inconsistent FRS results, and FRS obtained from a single set of tridirectional spectrum-compatible time-histories could be very unreliable. ❧ The recently developed direct spectra-to-spectra method overcomes the deficiencies of other existing direct method and time-histories, by using the t-response spectra (tRS) and the empirical relationship between tRS and GRS to deal with tuning cases and using the new FRS-CQC combination rule to deal with closely spaced modes. The direct method is capable of giving FRS results that are comparable to those obtained from a large number of time-history analyses. Furthermore, it can also provide a complete probabilistic description of FRS peaks. ❧ A scaling method is to generate FRS from available FRS and GRS without performing dynamic analysis. By combining a system identification technique to uncover the dynamic information of the equivalent significant modes of the underlying structure with the direct method for generating FRS, the scaling method can give satisfactorily accurate FRS for various damping ratios and for different GRS. It is an accurate, efficient, and economical method for generating FRS, which is important to refurbishment projects of existing NPPs and critical for new builds in feasibility analysis, budgeting, scheduling, bidding and tendering, and procurement of important equipment. ❧ The direct method for generating FRS was formulated for fixed-base structures. Applying the substructure method, the effect of soil–structure interaction (SSI) is accounted for by using the modified response spectra, called foundation level input response spectra (FLIRS), which depend on the foundation input response spectra (FIRS) and the dynamic properties of both the soil and structure. FLIRS can then be used as the seismic input in the direct method for generating FRS. Accounting for SSI in generating FRS is important in seismic analysis and design of nuclear power facilities; continued efforts are being made to develop efficient and accurate direct method for generating FRS.

C

H

9 A

P

T

E

R

Seismic Fragility Analysis 9.1 Seismic Fragility Definition of Seismic Fragility In general, fragility is the conditional probability that the damage exceeds a specified limit state (or damage state) D, for a given level of hazard:    Fragility = P Damage > D  Hazard .

(9.1.1)

In seismic probability safety assessment (PSA), seismic hazard is expressed in terms of a GMP a, such as PGA, spectral acceleration at a specified natural period (or frequency), or the average spectral acceleration over a range of natural periods (or frequencies), which is consistent with the ground motion parameter (GMP) a used in the seismic hazard curve (see Section 5.2.4). Ground acceleration capacity A is often used to measure the capability of a systems, structures, and components (SSC) to withstand seismic hazard, i.e., the ground motion level beyond which the seismic response of an SSC would result in its structural or functional failure. Failure of an SSC implies that its ground acceleration capacity A is less than the given GMP level a. Seismic fragility is defined as conditional probability of failure (e.g., structural failure or functional failure) of an SSC on a given seismic hazard in terms of a selected GMP. Hence, seismic fragility of a SSC is expressed as the conditional probability that its ground acceleration capacity A is less than a given GMP level a, i.e.,    pF (a) = P A < a  GMP = a .

410

(9.1.2)

9.1 seismic fragility

411

For simplicity of notation, the condition of given the “GMP = a” may sometimes be dropped in equation (9.1.2) without losing the implied meaning that seismic hazard is specified. In seismic PSA, the GMP in probabilistic seismic hazard analysis (PSHA) and fragility analysis should be used consistently, because PSHA provides the probability distribution of the GMP that is the condition in seismic fragility analysis.

Fragility Model Ground acceleration capacity A is often expressed as the product of the best-estimate or median ground acceleration capacity Am and two random variables εR and εU : A = Am εR εU ,

(9.1.3)

where εR represents inherent randomness (aleatory uncertainty) about the median value, and εU represents the uncertainty (epistemic uncertainty) in estimating the median value due to lack of knowledge. The random variables εR and εU are usually taken to be lognormal with unit median (zero logarithmic mean) and logarithmic standard deviations of βR and βU , respectively. With perfect knowledge, i.e., if there is no uncertainty in estimating Am , then εU = 1. Considering only the inherent randomness in Am leads to the ground accelera-

tion capacity AR = Am εR . Because εR ∼ LN (0, βR2 ), then AR is lognormally distributed 

with AR ∼ L

2 N ( lnAm , βR ). 

fragility, is pF (a) = P



Hence, the condition probability of failure, or the seismic

    ln(a/Am ) lna − lnAm AR < a =

=

. βR βR 

(9.1.4)

However, due to lack of knowledge, there is uncertainty in estimating Am . Let U AU m = Am εU be the estimated median when uncertainty is considered. Am is log-

2 2 normally distributed with AU m ∼ LN ( lnAm , βU ) because εU ∼ LN (0, βU ). 



The confidence level Q = q is expressed as the probability of exceedance of εU equal to q, i.e., 

P εU > εU,q



  lnεU,q − 0 =1−

= q. βU

(9.1.5)

εU,q can be determined from equation (9.1.5) −1 (1−q)

εU,q = e βU

−1 (q)

= e− βU

.

(9.1.6)

U Using equation (9.1.6), the estimated median capacity Am , q at the confidence level Q = q can be expressed as   U −βU −1 (q) Am , P AUm > AmU , q = q. (9.1.7) , q = Am εU, q = Am e

412 U obtained in equation (9.1.7) yields the Replacing Am in equation (9.1.4) by Am ,q seismic fragility, or the conditional probability of failure given a ground motion level

a, at confidence level Q = q (Kennedy et al., 1980; Kennedy and Ravindra, 1984)      ln(a/Am ) + βU −1 (q) . pF, q (a) = P A < a  GMP = a, Q = q =

βR

(9.1.8)

The confidence levels Q are often taken as several discrete values, such as 5 %, 50 %, and 95 %; equation (9.1.8) gives a family of fragility curves for various levels of confidence.

Alternative Derivation of Seismic Fragility The seismic fragility can also be derived directly with the random variables εR and εU in equation (9.1.3). (1) Probability of failure considering randomness Considering inherent randomness, the ground acceleration capacity of an SSC is A = Am εR . Associate εRpF with a limit state; e.g., equate the ground acceleration capac-

ity represented by Am εRpF to the ground motion level a. The SSC will fail when εR is smaller than εRpF, which gives pF = P {εR < εRpF}. Because εR ∼ LN (0, βR2 ), one has 

    lnεRpF − 0 . pF = P εR < εRpF =

βR

(9.1.9)

Solving for εRpF from equation (9.1.9) gives εRpF = e βR

−1 ( p ) F

.

(9.1.10)

(2) Confidence level in the presence of uncertainty While εR describes the probability of failure due to inherent randomness, εU represents the uncertainty due to lack of knowledge. Similar to εR , one has q=P



q εU > εU

=1−P



q εU < εU

  q lnεU − 0 . =1−

βU

 q q For a given value εU , the event εU > εU means success;



(9.1.11)



P εU > εUQ = q is therefore

the probability of success, or the confidence level Q = q in the presence of uncertainty. q

Solving for εU from equation (9.1.11) gives q

−1 (1 − q )

εU = e βU

.

(9.1.12)

9.1 seismic fragility

413

(3) Ground acceleration capacity A From equation (9.1.3), the ground acceleration capacity A can be written as A = Am εR εU . The probability that A < a, given the GMP level a, gives the probability of failure pF at the confidence level Q = q, i.e., 





P A < a  GMP = a, Q = q = pF, q (a) ,

(9.1.13)

where q

−1 ( p ) F

a = Am εRpF εU = Am e βR

−1 (1 − q )

e βU

.

Therefore, the ground acceleration capacity corresponding to probability of failure pF at the confidence level Q = q is q

q

−1 ( p ) F

C pF = ApF = Am e βR

+ βU −1 (1 − q )

.

(9.1.14) q

Noting that −1 (1−q ) = − −1 (q), setting the ground acceleration capacity ApF equal to the GMP level a, and solving for pF from equation (9.1.14) recover equation (9.1.8):

 ln(a/Am ) + βU −1 (q) . pF, q (a) = pF =

βR 

Composite Variability The separation of εR and εU , based on which the seismic fragility was derived, is sometimes judgmental. Since the 1990s, a composite variability εC = εR εU has been commonly used in the nuclear energy industry. εC is a lognormally distributed random variable with unit median (zero logarithmic mean) and logarithmic standard deviation  βC = βR2 + βU2 . (9.1.15) Ground acceleration capacity A is then expressed as A = Am εC . Seismic fragility in terms of the composite variability is thus given by      ln(a/Am ) , pF (a) = P A < a  GMP = a =

βC

(9.1.16)

which provides a single “best-estimate” fragility curve, called the composite fragility curve or mean fragility curve, without explicitly separating uncertainty from inherent randomness. If the conditional probability of failure pF (a) for a specific GMP a is known, the median capacity is given by A m = a e βC

−1 ( p ) F

.

(9.1.17)

414

Solving for a in equation (9.1.16) yields the ground acceleration capacity corresponding to probability of failure pF on the composite fragility curve −1 ( p ) F

C pF = ApF = Am e βC

.

(9.1.18)

After the 2011 Fukushima Daiichi nuclear accident, USNRC issued the “Guidance on Performing a Seismic Margin Assessment in Response to the March 2012 Request for Information Letter” (USNRC/JLD-ISG-2012-04, USNRC, 2012a). It states that For the important accident sequences (i.e., direct core damage, large early release, and low seismic margins), the full family of fragility curves (Am , βR , and βU) and probability distributions on random failure rates and operator error rates should be used in the convolution procedure to obtain the accident sequence fragility curves and plant level fragility curves.

Example As an example, suppose that the fragility parameters for a component are Am = 0.87g, βR = 0.25, and βU = 0.35, a family of fragility curves with different confidence levels and a composite fragility curve, with βC = 0.43, are determined using equations (9.1.8) and (9.1.16) and are shown in Figure 9.1. ❧ For a given probability of failure pF , solving equation (9.1.14) gives Q=P



 −1    lna − ln Am e βR ( pF ) , A > a  pF = 1 −

βU 

(9.1.19)

−1 ( p )  F

indicating that A is lognormal with logarithmic mean ln Am e βR

and loga rithmic standard deviation βU . The probability density functions FA (a  pF ) are also

shown in Figure 9.1 for pF = 0.2, 0.5, and 0.8. ❧ For a given confidence level Q = q, equation (9.1.8) can be written as

  −1    lna − ln Am e−βU (q)  pF, q (a) = P A < a Q = q =

, βR 

indicating that A is lognormal with logarithmic mean ln Am e−βU

−1 (q) 

(9.1.20) and log-

arithmic standard deviation βR . The fragility curve with 95 % confidence level is  shown again in Figure 9.2. The probability density function FA (a  Q = 95 %) is also shown in Figure 9.2. ❧ For a given GMP = a, the probability that the probability of failure PF is less than pF is given by, using equation (9.1.19),

9.1 seismic fragility

415

1.0

Conditional probability of failure pF

0.9

fA(a | pF = 0.8)

Q = 95%

Q= 50%

0.8 0.7

Fragility with 95% confidence

0.6

fA(a | pF = 0.5)

Q = 5%

Mean fragility with composite variability Q = 50%

Q = 95%

Q =5%

0.5 Fragility with 50% confidence

0.4 Q = 95%

f (a p = 0.2) 0.3 A | F

Fragility with 5% confidence

Q = 50%

Am = 0.87 g βR = 0.25 βU = 0.35

Q = 5%

0.2 0.1 0.0 0.0

Ground motion parameter a (g) 0.2

0.4

0.6

0.8

1.0

Figure 9.1

1.2

1.4

1.6

1.8

2.0

2.2

Fragility curves.

1.0

fA(a | Q = 95%)

Conditional probability of failure pF

0.9

pF =0.8

0.8 0.7

Fragility with 95% confidence

pF (a)= P { A < a | Q = 95 %} fA(a | Q = 95%)

0.6 0.5

pF =0.5 Am = 0.87g

0.4

βR = 0.25

0.3

fA(a | Q = 95%) pF =0.2

0.2

βU = 0.35

0.1 0.0 0.0

Ground motion parameter a (g) 0.1

0.2 Figure 9.2

0.3

0.4

0.5

0.6

0.7

Fragility curve with 95 % confidence level.

0.8

0.9

1.0

416 1.0

0.05

Conditional probability of failure pF

0.9

Fragility with 95% confidence

Probability of exceedance 0.05

0.8 0.7

fPF (pF | a=0.5)

0.6

Am = 0.87g βR = 0.25

0.5

βU = 0.35

0.4

fPF (pF | a=0.7)

0.3

Probability of nonexceedance P {PF < pF | GMP=a} = q=0.95

0.05

0.2 0.1

0.95

0.0 0.0

0.1

fPF (pF | a=0.3) 0.2

0.95 0.3

0.4

0.95 0.5

0.6

0.7

0.8

0.9

Ground motion parameter a (g)

Figure 9.3





Fragility curve with 95 % confidence level.









P PF < pF  GMP = a = P A > a  pF = q



 −1  lna − ln Am e βR ( pF ) . (9.1.21) =1−

βU

 The probability density function FP pF  GMP = a can be determined by differenF

tiating equation (9.1.21) with respect to pF

   d P PF < pF  GMP = a  FP ( pF  GMP = a) = . F d pF

(9.1.22)

The fragility curve with 95 % confidence level is shown again in Figure 9.3 to illus 

trate this case. The probability density functions FP p  GMP = a for a = 0.3, F

F

0.5, and 0.7 are also shown.

9.2 HCLPF Capacity The HCLPF (high confidence of low probability of failure) capacity Chclpf of an SSC is defined as the ground acceleration capacity C595% % corresponding to 5 % probability of failure ( pF = 0.05) on the fragility curve with 95 % confidence (Q = 0.95). Substituting pF = 0.05, and Q = 0.95 in equation (9.1.14) gives the HCLPF capacity Chclpf = C595% % = Am e(βR +βU )

−1 (0.05)

= Am e−1.6449 (βR +βU ) .

(9.2.1)

9.2 hclpf capacity

417

0.06 Probability of failure p F 0.05 Fragility with 95% confidence

0.04 Am = 0.87 g βR = 0.25

0.03

Fragility with composite variability

βU = 0.35

0.02 0.01 0.0

Ground acceleration capacity a (g) 0.0

0.05

0.1

0.15

Figure 9.4

0.2

0.25 0.3 C C1% = 0.320 g

0.35 0.4 0.45 95% C5% =0.324 g

0.5

HCLPF capacity on fragility curves.

For the special case when βR = βU , equation (9.2.1) becomes Chclpf = Am e−1.6449 × 2βR .  √ Noting that βC = βR2 +βU2 = 2βR , equation (9.2.2) can be written as Chclpf = Am e−1.6449 × = Am e

√ √ 2 × 2 βR

−1 (0.01) β C

= Am e−2.3263 ×

√ 2 βR

,

(9.2.2)

−1 (0.01) = −2.3263,

= C1C% .

(9.2.3)

❧ When βR = βU , equation (9.2.3) is exact, i.e., HCLPF capacity can be obtained either by using pF = 0.05 on the fragility curve with 95 % confidence (Q = 0.95) or by using pF = 0.01 on the composite fragility curve, i.e., Chclpf = C595% % = C1C% . ❧ When βR = βU , equation (9.2.3) is not exact; it is acceptable as an approximation in practice when 0.5  βR /βU  2, i.e., Chclpf = C595% % ≈ C1C% . Figure 9.4 shows the fragility curve with 95 % confidence level and the composite fragility curve. The HCLPF capacity obtained using the two fragility curves are, respectively, 0.324 g and 0.320 g, with negligible difference. ❧ HCLPF capacity is determined from the composite fragility curve as Chclpf ≈ C1C% = Am e−2.3263 βC .

(9.2.4)

NUREG/CR-4334 (USNRC,1985) recommended two methods for computing the HCLPF capacity of an SSC: fragility analysis (FA) method and conservative deterministic failure margin (CDFM) method. ❧ Fragility Analysis Method As presented in Section 9.1, fragility analysis is a probabilistic approach, which was first introduced in seismic PSA.

418

In the fragility method, fragility curves of an SSC are determined first. The HCLPF value Chclpf of the SSC is determined from the fragility curve with 95 % confidence using equation (9.2.1) or from the composite fragility curve using equation (9.2.4). ❧ Conservative Deterministic Failure Margin (CDFM) Method EPRI-NP-6041-SL (EPRI, 1991a) recommended the CDFM method as an approximate method for estimating the seismic capacity C1C% corresponding to 1 % probability of failure on the composite fragility curve and thus HCLPF, i.e., C1C% ≈ Ccdfm , in the EPRI seismic margin assessment (SMA) program. This method is deterministic, but has been extensively benchmarked against the fragility method. The CDFM method is presented in Section 9.4. Therefore, the HCLPF capacity Chclpf can be obtained from ❧ the capacity C595% % of 5 % probability of failure on the fragility curve with 95 % confidence, ❧ the capacity C1C% of 1 % probability of failure on the composite fragility curve, ❧ the capacity Ccdfm obtained using the CDFM method: Chclpf = C595% % ≈ C1C% ≈ Ccdfm .

(9.2.5)

9.3 Methodology of Fragility Analysis 9.3.1 Introduction The objective of fragility analysis (FA) is to produce realistic ground acceleration capacities A of SSCs, neither conservative nor optimistic. The product of fragility analysis is a family of seismic fragility curves expressed in terms of fragility parameters: ❧ best-estimate of median ground acceleration capacity Am , ❧ logarithmic standard deviations βR and βU , or composite variability βC . In seismic fragility analysis, a reference earthquake (RE), e.g., safe shutdown earthquake (SSE) specified for design or review level earthquake (RLE) for seismic margin assessment (SMA), may be used as seismic input. Let ARef be the reference ground acceleration capacity that is represented by a GMP value, such as 0.3g PGA from the reference earthquake. Note that ARef is a deterministic value in the fragility analysis. In estimating the fragility parameters, it is more convenient to work with an intermediate variable F, called the factor of safety. F describes the level that the ground

9.3 methodology of fragility analysis

419

acceleration capacity A is above the reference capacity ARef and is defined as A = F · ARef , F= =

(9.3.1)

Actual seismic capacity of SSC Actual response due to RE Actual seismic capacity of SSC Calculated response due to RE × Calculated response due to RE Actual response due to RE

= FC · FRS = Fm εR εU ,

(9.3.2)

where FC is a capacity factor, FRS is a response factor, εR and εU are random variables representing inherent randomness and epistemic uncertainty about Fm , respectively. εR and εU in equation (9.3.2) are essentially identical to those in equation (9.1.3). Therefore, the median factor of safety Fm can be directly related to the median ground acceleration capacity Am as Am = Fm · ARef .

☞

(9.3.3)

The logarithmic standard deviations of dimensionless εR and εU in estimating Fm and Am are identical. Therefore, in practice, logarithmic standard devia-

tions βR and βU of εR and εU in estimating Fm are calculated instead. The procedure of fragility analysis follows the same steps that an engineer would normally perform for conventional design, except that median values should be used for each of the basic variables that affect the seismic capacity. Conservatism in each basic variable should be removed to reflect the realistic case. The following general procedure is usually followed: ❧ Select a reference earthquake input, preferably with site-specific median spectral shape anchored to a reference GMP value, e.g., 0.3g PGA, with a specified probability level. ❧ Through structural dynamic analysis, floor response spectra (FRS) at the location of an SSC of interest are determined for the reference seismic input. ❧ Identify possible failure modes of the SSC, and calculate the response quantities for the calculated FRS combined with normal operating loads. ❧ By comparing the response quantity of the governing failure mode to the allowable response quantity, the factor of safety is determined. For fragility analysis, the factor of safety F against the reference earthquake and logarithmic standard deviations βR and βU need to be reasonably estimated.

420

9.3.2 Structures In seismic PSA, the strength of structures includes the inherent capacity up to the yield strength for steel members and the ultimate capacity for concrete elements. Capacity beyond strength is included in the inelastic energy absorption factor. Hence, the factor of safety can be modelled as the product of three random variables FC = FS · Fμ =⇒ F = FS · Fμ · FRS ,

(9.3.4)

where FS is the strength factor, Fμ is the inelastic energy absorption factor, and FRS is the structure response factor.

9.3.2.1

Strength Factor

The strength factor FS represents the ratio of ultimate strength (or strength at loss-offunction) to the stress calculated for ARef . To determine the strength factor or the elastic scale factor that is applied to the reference earthquake to reach the elastic capacity, equate the total demand to the structure capacity (including any reduction in capacity caused by the earthquake) Demand = FS · DS + DNS ,

Capacity = C − CS · FS ,

Demand = Capacity =⇒ FS =

C − DNS , DS + CS

(9.3.5)

where C is the capacity or the strength of the structural element for the specific failure mode, CS is the reduction in capacity due to concurrent seismic loadings, DS is reference elastic seismic demand, and DNS is the concurrent non-seismic demand or normal operating load (such as dead load and operating temperature load). For higher levels of earthquake, other transients, such as safety relief valve discharge and turbine trip, have a high frequency of occurring simultaneously with the earthquake; in such cases, the loads from these transients should be included in the normal operating load DNS . In equation (9.3.5), it is assumed that no additional reduction in capacity due to concurrent seismic loadings occurs above the elastic level.

9.3.2.2 Inelastic Energy Absorption Factor The inelastic energy absorption factor Fμ considers the fact that an earthquake is a limited energy source, and many SSCs are capable of absorbing energy beyond yield without loss-of-function. Newmark (1977) suggested to use ductility modified response spectra to determine the deamplification effect resulting from the inelastic energy dissipation. The deampli-

9.3 methodology of fragility analysis

421

Inelastic Energy Absorption Factor Fμ .

Table 9.1

Low-Rise Concrete Shear Walls

Ductile Moment Frames

3.0 2.0

8.0 3.2

Median value of μ Median value of Fμ Variability in μ Variability in Fμ

βR = 0.15, βU = 0.45 βR = 0.08, βU = 0.25

fication factor is a function of the ductility ratio μ, defined as the ratio of maximum displacement to displacement at yield, and the system damping. For low-rise concrete shear walls (typical of auxiliary building walls) and ductile moment frames, the median and variability values of μ and Fμ are given in Table 9.1.

9.3.2.3

Structure Response Factor

The structure response factor FRS recognizes that, in the design analyses, the structural response is computed using specific (often conservative) deterministic response parameters for structures. The structure response factor FRS is further modelled as a product of several factors that influence the response variability FRS =

3 I

FRSI =⇒ F = FS · Fμ ·

3 I

FRSI ,

(9.3.6)

where FRSI denotes the Ith response factor. Some basic response factors that influence structure response are ❧ Ground Motion • Earthquake response spectrum shape • Horizontal direction peak response • Vertical component response ❧ Damping ❧ Modelling • Modal frequency • Modal shape • Torsional coupling ❧ Modal Combination ❧ Time-History Simulation ❧ Foundation–Structure Interaction • Ground motion incoherence • Vertical spatial variation of ground motion • SSI analysis ❧ Earthquake Component Combination

422

Details of these response factors are discussed in the following subsections.

9.3.2.4 Ground Motion In fragility analysis, the input consists of three ground response spectra: two horizontal and one vertical. A single GMP, such as PGA or spectral acceleration, is selected. It is assumed that the GMP is the average of the corresponding GMPs from the two horizontal directions, with the associated reference response spectrum shape defined. The vertical input at the ground level is assumed to be two-thirds of the horizontal input. Three basic variables account for the influence of ground motion variability: ❧ Earthquake response spectrum shape ❧ Horizontal direction peak response ❧ Vertical component response 1. Earthquake Response Spectrum Shape A smooth reference response spectrum shape, e.g., NUREG/CR-0098, may be assumed for each of the two horizontal directions and is anchored to the GMP selected for seismic PSA. In general, real earthquakes are different from the smooth reference response spectrum used in the fragility analysis. The peaks and valleys in real response spectra indicate that a future earthquake response spectrum, with the same GMP, will have spectral ordinates which are either higher or lower than the smooth reference spectrum. This peak-and-valley variability is due to randomness. Furthermore, there is uncertainty in the earthquake signature that cannot be predicted exactly, which is reflected in the uncertainty in the smooth response spectrum shape. Figure 9.5 shows an example relationship between the assumed reference response spectrum shape anchored to a PGA parameter, an actual earthquake response spectrum, and the site-specific response spectrum shape. A study of 38 earthquake time-histories (Kipp et al., 1988; Pacific Gas and Electric Company, 1988) indicates that values for both βR and βU for response spectrum shape should be considered as a function of frequency, as shown in Table 9.2 (EPRI-TR103959, EPRI, 1994, Table 3-2).

☞

As the frequency gets farther from the frequency at which the spectra return to the PGA (i.e., decreases from 33 Hz), the uncertainty in response increases.

☞

Because the same reference response spectrum shape is assumed for both horizontal directions, the same values for βR and βU are used for each horizontal direction.

9.3 methodology of fragility analysis

423

Spectral acceleration

Peak-and-valley randomness Spectral shape uncertainty

Real earthquake response spectrum

PGA Reference response spectrum shape

Site-specific response spectrum shape

Figure 9.5

Frequency Peak and valley randomness and spectral shape uncertainty.

Spectral acceleration

Horizontal peak response randomness

N-S component response spectrum

E-W component response spectrum Reference (average) response spectrum

Figure 9.6

Frequency Horizontal component peak response randomness. Table 9.2

Ground motion variables.

βR

Basic Variable

βU

Response Spectrum Shape Anchored to PGA 1 Hz 5 Hz 10 Hz 16 Hz 33 Hz Anchored to Averaged SA 1 Hz 5 Hz 10 Hz 16 Hz 33 Hz

0.18 0.18 0.18 0.15 0.12

to to to to to

0.22 0.22 0.22 0.19 0.15

0.32 0.24 0.16 0.12 0

0.18 0.18 0.18 0.15 0.12

to to to to to

0.22 0.22 0.22 0.18 0.15

0.20 0 0 0.10 0.13

Horizontal Peak Response

0.12 to 0.14

0

Vertical Response Vertical equals 2/3 horizontal Site-specific analysis

0.22 to 0.28 0.22 to 0.28

0.20 to 0.26 −

424 Table 9.3

Ground motion variables.

Basic Variable Horizontal Direction Peak Response Vertical Component Response Vertical equals 2/3 horizontal Site-specific analysis

Logarithmic Standard Deviation βR βU 0.12 to 0.14

0

0.22 to 0.28 0.22 to 0.28

0.20 to 0.26 Less than generic values

Uniform Hazard Spectrum (UHS) as Reference Spectrum Shape Essentially all PSHA studies have included the peak-and-valley randomness βrs as part of the aleatory uncertainty when developing seismic hazard estimates as a function of the annual frequency of exceedance (AFE). Thus, at any AFE, the resulting UHS already fully includes the effect of βrs . Because it does not appear likely to get the seismic hazard estimators to remove βrs from the aleatory uncertainty included in the PSHA, the current recommendation (EPRI-1019200, EPRI, 2009) is to drop βrs from the fragility estimate because it is included in the hazard estimate. When site-specific response spectrum is provided by seismic hazard analysts, the epistemic uncertainty is unnecessary to be considered in the fragility estimate. Typically, in most seismic fragility evaluations, all of spectral shape variability is attributed to βrs (EPRI-1019200). For those HCLPF capacities that include βrs in seismic fragility analyses, a correction factor Frs should be multiplied to obtain modified HCLPF capacities, which can be obtained from (EPRI-1019200)  2 2 Frs = e2.3263 (βC, rs −βC ) , βrs = βC, rs − βC ,

Chclpf = Frs · Chclpf ,

(9.3.7)

where βC is the overall composite variability excluding βrs , while βC, rs includes βrs . For modern SPRA studies, site-specific UHS are used as reference spectrum shape in seismic fragility analysis. Thus, the typical ranges of values for peak-and-valley randomness and spectrum shape uncertainty are removed (EPRI-1019200). The updated Table 3-2 in EPRI-TR-103959 only provides recommendations for horizontal direction peak response variability and for vertical component response variability (Table 9.3). 2. Horizontal Direction Peak Response Because the GMP is assumed to be the average of the two horizontal directions, the smooth reference response spectrum shape for one direction will be higher than the shape for the perpendicular direction, as illustrated in Figure 9.6. This is a randomness variability, and a value of βR between 0.12 and 0.14 is recommended in Table 9.2. The effect of the horizontal direction peak response variability on the final fragility parameter of a structure or equipment depends on how the two horizontal earthquake

9.3 methodology of fragility analysis

425

components individually affect the response of the structure or equipment. The following four example cases demonstrate how this variability works in practical problems. (1) Specific Direction Response A structure or equipment is affected by only one horizontal direction response (e.g., failure of a concrete shear wall due to in-plane response). Because an average parameter is used, the real earthquake response could be either higher or lower. This case corresponds to a specific direction response relative to the average response of two horizontal directions. A median response factor of 1.0 and a βR of 0.13 are recommended for this case. (2) Colinear Vector Response A structure or equipment is affected equally by the two horizontal components; for example, considering an anchor bolt securing a tall square cabinet to the floor, tension is the dominant response, and the total tension is the sum of tension forces from the two horizontal earthquake components. If the tension in a bolt caused by equal input in the two horizontal directions is combined by 100-40 rule, the ratio R of the random response to the deterministically combined response is given by the following equation R= where X ∼

☞

X + 0.4 X , 1.4

(9.3.8)

LN (1.0, 0.132 ). It is found that the median of R is 1.0 and βR = 0.07. 

As the contribution from the two horizontal directions becomes more and more unequal, the value of βR in the combined response increases above 0.07 and approaches 0.13 when only one horizontal direction controls.

(3) General Vector Response This is similar to Case 2, except that the anchor bolt secures a short cabinet so that the dominant response is pure shear where the directional responses are 90◦ apart. If the shear in a bolt caused by equal input in the two horizontal directions is combined by 100-40 rule (NUREG/CR-0098, USNRC, 1978), the ratio R of the random response to the deterministically combined response is given by  2 X2 + 0.4 X R= . (9.3.9) 12 + 0.42 If X ∼

LN (1.0, 0.132 ), then the median of R is 1.0 and βR = 0.10. 

(4) Largest Direction Response An example is a flat-bottom round storage tank; its failure is controlled by simultaneous yielding of the anchor bolts and buckling of the shell. For this case,

426

the largest of the two horizonal components controls because the tank capacity is the same in all directions, and the direction of the largest response dominates. For the analysis of this type of SSC, the median largest response is higher than the average response, but the value of βR is reduced because the largest response is constrained between the average values and the higher values. The ratio of the random response to the deterministically combined response is given by 1 max X, X . R= 1.0 For X ∼

(9.3.10)

LN (1.0, 0.132 ), the median value of R is 1.09 and βR = 0.10. 

Table 9.4 summarizes the median factor and βR values of these four cases. 3. Vertical Component Response For most SSCs, the vertical earthquake component does not have a major effect on capacity. It is usually assumed to be equal to 2/3 times the horizontal component, and the ranges of βR and βU given in Table 9.2 are recommended. If a site-specific analysis is performed, the variability is often assumed to be all randomness for simplicity, because of the small effect that the vertical component has on the fragility curve.

9.3.2.5

Damping

Damping is estimated primarily from observations and is normally considered to be strain dependent. Table 9.5 recommends damping values for median and minus-oneSD for uncertainty depending on whether the stress level is near yield or near 1/2-yield in the main structural elements (EPRI-TR-103959, Table 3-4). Similar to structures, an estimate of median damping and damping at the minus-one-SD level is required. Table 9.5 also summarizes damping values recommended for equipment fragility analyses (EPRI-TR-103959, Table 3-8).

☞

Thevariability in response due to damping probably includes contributions from both randomness and uncertainty. However, for damping of both structures and equipment, it is difficult to determine the exact split; the results should not be significantly affected if damping variability is assumed to be all uncertainty.

9.3.2.6 Modelling Uncertainty in modelling of structure and equipment influences primarily the modal frequencies and the mode shapes, which are considered to be median centred for realistic modelling as discussed in EPRI-TR-103959 (EPRI, 1994, pages 3-15 to 3-18).

9.3 methodology of fragility analysis Table 9.4

427

Response parameters for effect of horizontal direction for peak response.

Case

βR

1.

Specific direction response Average direction response

In-plane shear wall response

1.0

0.13

2.

Colinear vector response Average direction response

Tension response of anchor bolt

1.0

0.07

3.

General vector response Average direction response

Shear response of anchor bolt

1.0

0.10

4.

Largest direction response Average direction response

Compression in flat-bottom tank

1.09

0.10

Median

βR

βU †

3%

−

2%

5% 7% 5%

− − −

3% 5% 3%

7%

−

5%

10 %

−

7%

10 %

−

7%

5%

−

3.5 %

5% 15 % 5% 5%

− − − −

3.5 % 10 % 3% 3.5 %

Table 9.5

Damping values for structures and equipment.

Structures About One-Half Yield Welded steel Prestressed concrete Reinforced concrete (slightly cracking) Reinforced concrete (considerable cracking) Bolted steel Block walls Beyond or Just Below Yield Point Welded steel Prestressed concrete (without complete prestress loss) Prestressed concrete (complete prestress loss) Reinforce concrete Bolted steel Block walls Equipment Electrical cabinets Mechanical components Piping Cable trays Flat-bottom tanks (impulsive mode) Horizontal heat exchanger †

Median .Factor

Example

βU is estimated based on −1σ damping.

1. Modal Frequency The modal frequencies of structures depend on the mass and stiffness of the structure. The variability of the structure mass term is relatively small compared to the stiffness term. The results of dynamic test and study (Hadjian et al., 1977) indicate that the frequencies of NPP steel or concrete structures are unbiased, and the logarithmic

428

standard deviation for frequency is about 0.15. As models become cruder the βU value can be up to about 0.35 for fairly approximate models. For equipment, it is estimated that uncertainty βU in modal frequency varies from 0.10 to 0.30, depending on the complexity of the model and boundary conditions. The lower value should be used for simple models that can be represented by single modes, while the upper value would be appropriate for complex configurations. In practice, the values of βU due to frequency variability for structures and equipment are combined by SRSS, and the result is usually only slightly larger than the largest of the two βU values. 2. Modal Shape Based on experience and judgment, the value of βU for response due to mode shape is estimated to be between 0.05 and 0.15 (Kennedy et al., 1980), with the lower value for a simple SSC and the upper value for more complex SSC. 3. Torsional Coupling When the centres of rigidity and mass at each level are not the same, torsional motion about a vertical axis will occur due to each of the horizontal earthquake components. For a three-dimensional structural dynamic model, torsional coupling and its uncertainties are automatically considered in the modal frequency and shape variables. For structures that are modelled separately for the two horizontal directions, the median factor Fm for torsion and the associated βU value should be consistent. Because torsion will always increase the peak response except at the centre of rigidity, the probability of a median factor less than 1.0 is very small. The 1.0 value should be set at least at the minus-two-standard-deviation level, i.e., 1.0  Fm e−2βU =⇒ βU



1 2 lnFm .

For example, if the median factor Fm is judged to be 1.10, then βU



1 1 2 lnFm = 2 × ln1.10 ≈ 0.05.

9.3.2.7 Modal Combination The combination of response modes is random due to random phasing of the individual modal responses. A square-root-of-sum-of-squares (SRSS) combination of modes, with closely spaced and higher-frequency modes combined by an algebraic summation, is considered median centred, except for very low damping values. The βR value is estimated to be about 0.05 for a simple SSC, and about 0.15 for an SSC with multiple important modes (Kipp et al., 1988).

9.3 methodology of fragility analysis

429

Alternatively, when individual model contribution to a particular response is known, the absolute sum of the modal values is judged to be two or three standard deviations above the median SRSS response (Kipp et al., 1988). If there are only two modes that contribute to the response, then two standard deviations should be used. If there are several modes contributing, then three standard deviations are appropriate. With this assumption the βR value on response can be estimated directly by: βR =

V 1 ln abs , φ V¯

(9.3.11)

where Vabs is the response based on absolute sum of modal contributions, V¯ is the median response based on SRSS combination, and φ is the number of standard deviations.

9.3.2.8 Time-History Simulation Time-history simulation is considered only if a ground motion time-history compatible with the reference ground motion response spectrum is generated. Usually the response spectrum from a generated time-history will envelope the target response spectrum, and there would be a slight bias to the conservative side, and the uncertainty βU would reflect the peaks and valleys. If the bias is a function of frequency, the difference between the two spectra in the vicinity of the structural fundamental frequency should be used to determine βU . The median factor that adjusts for time-history simulation bias and the associated βU value should also be consistent. Referring to Figure 9.7, the number of standard deviations to the lowest valley should be at least 2 to ensure the probability of being lower is realistic, i.e.,

SA, low = SA, median ×e−2 βU

☞

=⇒ βU = 1 ln 2

SA, median . SA, low

(9.3.12)

If a response spectrum analysis is performed to determine structure forces or floor response spectra, this variable does not have to be considered.

9.3.2.9 Foundation–Structure Interaction The interaction between the structure and the supporting foundation includes ❧ Ground motion incoherence ❧ Vertical spatial variation of ground motion ❧ SSI analysis 1. Ground Motion Incoherence At any time instant, the motion at every point under the structure foundation is not the same. For massive rigid foundations, such as those of NPP structures, the overall

430

Spectral acceleration

SA,high SA,median

SA, low

Reference response spectrum Generated response spectrum

Frequency Figure 9.7

Comparison of generated response spectrum to the reference response spectrum. Table 9.6 Reduction factor due to ground motion incoherence.

Frequency (Hz) 5 10 25

Reduction Factor RF150-foot 1.0 0.9 0.8

motion is reduced as high-frequency waves cancel each other across the foundation/soil interface. The amount of reduction is a function of the size of the foundation and the frequency of response. The median reduction factors in Table 9.6 may be conservatively used in a seismic PSA to reduce either ground response spectra (GRS) or FRS to account for the incoherence effect for a 150-foot plan dimension foundation (EPRI-TR-103959, pages 3-22 and 3-23). ❧ For a foundation of plan dimension D (feet), the following equation may be used to determine the median reduction factor Fm,D : 1 − Fm,D

=

1 − Fm,150-foot

. (9.3.13) D 150 For example, at 10 Hz, the median reduction factor is 0.95 for a 75-foot dimension foundation, and 0.8 for a 300-foot dimension foundation. ❧ For frequencies between the values given in Table 9.6, a linear interpolation of the reduction factor in the log-log plane may be used. For frequencies greater than 25 Hz, the reduction at 25 Hz may be used. The values of uncertainty βU should be consistent with the median reduction factors used. The probability of exceeding 1.0 should be very small. In establishing the value for βU , the number of standard deviations to the 1.0 value should be at least 2.0, i.e., 1.0 = Fm,D ×e2 βU =⇒ βU = − 12 lnFm,D .

(9.3.14)

9.3 methodology of fragility analysis

431

2. Vertical Spatial Variation of Ground Motion At soil sites, the ground motion decreases with depth as compared to the ground surface motion in the free field. Because the site hazard curves are usually given for the ground surface in the free field, a deconvolution analysis should be conducted to obtain the reduced motion at the foundation level. In deconvolution analyses, reductions are usually limited to 60 % of the free field response spectrum at the ground surface. ❧ It is assumed that no reduction with depth represents a response at three standard deviations. Denote the median reduction factor at depth H as RFH . The value of βU at depth H is given by 1.0 = RFH ×e3 βU =⇒ βU = − 13 lnRFH ,

(9.3.15)

which is considered to be a conservative upper bound for βU . ❧ A small value of βR = 0.08 is included to account for the inherent randomness that using different time-histories in a deconvolution analysis will produce somewhat different response reductions. 3. SSI Analysis ❦ Three SSI Analyses Three soil–structure interaction (SSI) analyses should be conducted to investigate the effects on response due to uncertainty in the soil properties. It is recommended that one analysis at the median soil shear modulus and additional analyses at the plus- and minus-one-SD soil shear modulus values be conducted: ❧ The coefficient of variation CV of the soil shear modulus should be determined using site-specific data if available. ❧ CV = 0.5 is the minimum value, because of other uncertainties that are not directly considered, such as structure-to-structure interaction, not perfectly vertical wave input, and variable connectivity between the soil and sides of the structure. ❧ CV = 1.0 is a reasonable upper bound, if only minimal data is available. Median soil material damping can be used in all analyses because its variability is small relative to radiation damping variability, which is automatically reflected in the three soil shear modulus cases considered.

432

❦ One SSI Analysis When only one SSI analysis is conducted, an acceptable approach is given in ATC 3-06 (ATC, 1978) to estimate the soil–structure system fundamental frequency and damping at the plus- and minus-one-SD soil shear modulus values. The upper-bound frequency F U and the lower-bound frequency F L can be estimated by:  F U = F 1 1+

1 1+CV

 − 12   − 12 F 1 2 F 1 2 −1 , F L = F 1 1+(1+CV ) , (9.3.16) −1 F ssi F ssi

where CV is the coefficient of variation of the soil shear modulus, F 1 and F ssi are the fundamental frequency of structure fixed on the base and the fundamental frequency of the soil–structure system from the single best-estimate analysis, respectively. The corresponding damping values can be estimated for each frequency: ζ = ζ0 + ζstructure

 F −3 1 , F soil

(9.3.17)

where ζ0 is the soil damping, ζstructure is the damping ratio of the structure with fixed based, and F soil is either F U or F L . Equation (9.3.16) can be used to determine where the structure frequency variability influences the building response: . FU =

F ssi e1 × βU =⇒

βU = − 12 ln

    F 1 −2 F 1 2 1 −1 . (9.3.18) 1+ F ssi F ssi 1+CV

❧ For F 1 / F ssi  2.0 Because CV  0.5, equation (9.3.18) gives βU  0.144. A value of βU  0.15 on structure frequency does not significantly affect building responses. Structure stiffness variability can be neglected. From equation (9.3.17), it is seen that ζ ≈ ζ0 , and structure damping variability can be neglected for most SSI analyses. ❧ For F 1 / F ssi < 2.0 Equations (9.3.18) and (9.3.17) can be used to determine the influence of structure frequency and damping on building response variability directly. ❦ Uncertainty in Soil Modelling Because SSI is three-dimensional, a βU value of 0.10 on response is assumed for a twodimensional model to account for the uncertainty in soil modelling. If all SSI analyses are based on simplified soil-spring models, a higher value of βU would be appropriate.

9.3 methodology of fragility analysis

433

9.3.2.10 Earthquake Component Combination The SRSS procedure for combining the responses due to the two horizontal and vertical earthquake components is considered to be median centred. The 100-40-40 rule requires that 100 % of the response in one direction be combined with 40 % of the responses from the other two directions. Whether the SRSS or 100-40-40 rule is used, a randomness βR for response needs to be included in the fragility analysis because the actual response will be higher or lower. In past seismic PSA, it has been assumed that the absolute-sum of the three directional responses is two to three standard deviations above the median. Similar to equation (9.3.11) for combining modal responses, the value of βR for earthquake component combination is given by βR =

R 1 ln abs , φ R¯

(9.3.19)

where Rabs is the response based on absolute sum (or 100-100-100) combination of direction contributions, R¯ is the median response based on SRSS or 100-40-40 combination, and φ is the number of standard deviations. ❧ It is always conservative to choose the number of standard deviations φ = 2.3. ❧ However, if the lowest two responses are each at least 20 % of the largest, then φ = 3.0 can be used. ❧ An upper bound value of βR = 0.18 can always be used, but may be excessively conservative for cases where the response is primarily from a single direction. Alternatively, if the 100-40-40 rule is used, a separate βR can be assigned to each of the 40 % factors applied to the two responses in the secondary directions. These two randomness values are independent of each other. From Monte Carlo simulation, it is found that βR is between 0.40 and 0.45. It is always conservative to use 0.45; however, if the lowest two responses are each at least 20 % of the largest, then βR = 0.40 can be used. These βR values look large; however, when this variability is properly propagated through the fragility analysis, their contribution is usually significantly reduced.

9.3.2.11 Variabilities For each variable affecting the factor of safety, the randomness βR and uncertainty βU must be estimated separately: ❧ βR represents the variability due to the randomness of the earthquake characteristics for the same acceleration and to the structural response parameters that relate to these characteristics.

434 Table 9.7

Randomness and uncertainty as a fraction of total variability.

Strength Inelastic energy absorption Spectral shape Damping Modelling Combination of earthquake components Modal combination Soil–structure interaction

βU /βC 1 0.71 0.11 0.99 1 0 0 0.99

βR /βC 0 0.71 0.99 0.11 0 1 1 0.11

❧ The dispersion represented by βU is due to such factors as: 1. Lack of understanding of structural material properties such as strength, inelastic energy absorption, and damping; 2. Errors in calculated response due to use of approximate modelling of the structure and inaccuracies in mass and stiffness representations; 3. Usage of engineering judgement in lieu of obtaining complete plant-specific data on fragility levels of equipment capacities, and responses. Table 9.7 summarizes the assignment of variability to randomness and uncertainty for each of the key variables used in estimating the fragilities when a standard shape reference earthquake is used.

9.3.3 Equipment For equipment and components, the factor of safety F is expressed as the product of a capacity factor FC , an equipment response factor FRE (relative to the structure), and a structure response factor FRS : F = FC · FRE · FRS . .

(9.3.20)

The capacity factor FC for an equipment is the ratio of the ground motion level at which the equipment ceases to perform its intended function to the seismic design level. FC can be evaluated as the product of strength factor FS and ductility factor Fμ : FC = FS · Fμ .

9.3.3.1

(9.3.21)

Strength Factor

The strength factor FS is calculated using equation (9.3.5). The strength S of an equipment is a function of the failure mode, which is classified into three categories:

9.3 methodology of fragility analysis

435

❧ Elastic Functional Failures Elastic functional failures involve the loss of intended function while the component is stressed below its yield point. Examples of this type of failure include • elastic buckling in tank walls and component supports, • chatter and trip in electrical components, • excessive blade deflection in fans, • shaft seizure in pumps. The load level at which the functional failure occurs is considered to be the strength of the component. ❧ Brittle Failures Brittle failures are defined as those failure modes that have little or no system inelastic energy absorption capability. Examples of brittle failure include • anchor bolt failures, • component support weld failures, • shear pin failures. Each of these failure modes has the ability to absorb some inelastic energy on the component level, but the plastic zone is very localized and the system ductility for an anchor bolt or a support weld is very small. The strength of a component failing in a brittle mode is calculated using the ultimate strength of the material. ❧ Ductile Failures Ductile failure modes are those in which the structural system can absorb a significant amount of energy through inelastic deformation; for examples • pressure boundary failure of piping, • structural failure of cable trays and ducting, • polar crane failure. For tensile loading, the strength of the component failing in a ductile mode is calculated using the yield strength of the material. For flexural loading, the strength is defined as the limit load or load to develop a plastic hinge. Strength of structural elements involves a strength equation and material properties. Code minimum material allowable values are generally used in the design of structural elements. These values are typically assumed to be at the 95 % confidence level. Table 3-9 of EPRI-TR-103959 (EPRI, 1994, page 3-52) lists nominal and median strengths and logarithmic standard deviations for uncertainty of some common materials (Table 9.8); Table 3-10 (page 3-53) gives the equations for median capacities for several common structural elements and the associated βU values (Table 9.9).

436 Table 9.8 Material strengths for common materials.

Strength (ksi) Yield σy Nominal Median

Material ASTM A36 carbon structural steel ASTM A307 bolts & studs ASTM A325 structural bolts

βU

βU

36

44

0.12

58

64

0.06

36

44

0.12

58 120

64 142

0.06 0.05

ASTM A490 structural bolts Weld ASTM SA240 Type 304LN stainless steel plates †

Ultimate σu Nominal Median

37

150

165

0.04

Fexx †

1.1 Fexx

0.05

84

0.07

0.13

Fexx is the minimum code nominal tensile strength for weld material. Table 9.9 Strength capacity equation and uncertainty for common elements.

βU Equation Material Fabrication SRSS

Median Capacity Bolt Ultimate Strength Tension Nu, m = φ Anet σu, m † Shear Vu, m = 0.62 Anet σu, m

0.11 0.06

0.06 0.06

0.05 0.05

0.13 0.10

0.11 0.11

0.05 0.05

0.15 0.15

0.19 0.19

0.06

0.12

0

0.13

0.06

0.12

0

0.13

Fillet Weld in Shear Pweld, m Longitudinal direction 0.84 Aw Fexx, m Transverse direction 1.26 Aw Fexx, m Plate in Bending My, m Yield point B H2 4 σy, m well defined   Yield point B H2 σy, m + 2 σu, m 4 3 not well defined †

φ is the notch reduction factor and strength reduction due to accidental moment. φ = 0.9 for A307 bolts.

9.3.3.2 Inelastic Energy Absorption Factor The inelastic energy absorption factor Fμ for an equipment is a function of the ductility ratio μ: ❧ For brittle and functional failure modes, the median value Fμ is close to 1.0. ❧ For ductile failure modes of equipment responding in the amplified acceleration region of the design spectrum (2 Hz to 8 Hz): Fμ = ε



2μ − 1,

(9.3.22)

9.3 methodology of fragility analysis

where ε ∼

437

LN (1.0, βU2 ) to reflect the error in equation (9.3.22), with βU ranging 

from 0.02 to 0.10 (increasing with the ductility ratio). ❧ For rigid equipment, Fμ is given by Fμ = ε μ0.13 ,

(9.3.23)

where ε is the same as that in equation (9.3.22).

9.3.3.3 Equipment Response Factor The equipment response factor FRE is the ratio of equipment response calculated in the design to the realistic equipment response. Both responses are calculated for the design floor spectra. FRE is the factor of safety inherent in the computation of equipment response. It depends upon the response characteristics of the equipment and is influenced by some of the variables listed in Section 9.3.2.3. These variables differ according to the seismic qualification procedure. ❧ For equipment qualified by dynamic analysis, the important variables that influence the equipment response and its variability are • qualification method, • spectral shape, including the effects of peak broadening and smoothing, and artificial time-history generation, • modelling (affecting mode shape and frequency results), • damping, • combination of modal responses (for response spectrum method), • combination of earthquake components. ❧ For rigid equipment qualified by static analysis, only the qualification method is significant. The equipment response factor is the ratio of the specified static coefficient divided by the zero period acceleration of the floor level where the equipment is mounted. ❧ For flexible equipment designed by the static coefficient method, the dynamic characteristics of the equipment must be considered. This requires estimating the fundamental frequency and damping, if the equipment responds predominantly in one mode. The equipment response factor is the ratio of the static coefficient to the spectral acceleration at the equipment fundamental frequency. ❧ For equipment qualified by testing, the response factor must take into account • qualification method, • spectral shape, • boundary conditions in the test versus installation,

438

• damping, • spectral test method (sine beat, sine sweep, complex waveform, etc.), • multidirectional effects. The overall equipment response factor is the product of factors of safety corresponding to each of the variables identified above.

9.3.3.4

Structure Response Factor

The structure response factor FRS is based on the response characteristics of the structure at the location of component (equipment) support. Thus, the variables pertinent to the structural response analyses used to generated floor spectra for equipment design are the only variables of interest for equipment fragility, which includes • spectral shape, • damping, • modelling, • SSI. For equipment whose seismic capacity level has been reached while the structure is still within the elastic range, the structural response factors should be calculated using the damping values corresponding to less than yield conditions (e.g., a median damping of about 5 % for reinforced concrete).

☞

The combination of earthquake components is not included in the structural response because this variable is addressed for specific equipment orientation in the analysis of equipment response.

9.3.4 Procedure of Fragility Analysis For each of the parameters affecting capacity and response factors of safety, the median and variabilities are estimated following the guidelines presented in Section 9.3.2 for structures and in Section 9.3.3 for equipment. The estimates of median and variabilities need to be combined to obtain the overall median factor of safety Fm and variabilities βR and βU , required to define the fragility curves for the structure or equipment. Suppose that the factor of safety or scale factor F in equation (9.3.1) can be expressed as a function of the basic demand and capacity variables XI , I = 1, 2, . . . , n, F = F(X) = F(X1 , X2 , . . . , Xn ).

(9.3.24)

An example of equation (9.3.24) is the factor of safety of a structure given in (9.3.6).

9.3 methodology of fragility analysis

439

¯ = (X¯ 1 , X¯ 2 , . . . , X¯ n ) of variExpanding F in Taylor series about the mean values X ables X = (X1 , X2 , . . . , Xn ) gives  K   ∂ ∂ ∂ ¯ ¯ ¯ (X1 − X 1 ) + (X2 − X 2 ) + · · · + (Xn − X n ) F  F(X) = ∂X1 ∂X2 ∂Xn K=0 K! X¯   n ∂F  n  n   ∂ 2 F   (X − X¯ ) + 1 ¯ + ¯ ¯ = F(X) I I   (XI − X I )(Xj − X j ) + · · ·. (9.3.25) 2 I=1 ∂XI X¯ I=1 j=1 ∂XI ∂Xj X¯ ∞ 1 

Taking expected value of equation (9.3.25) yields the mean factor of safety   n ∂F  n  n   ∂ 2 F  1  ¯ ¯ ¯ ¯ + F¯ = F(X)  E[ XI − X I ] + 2  E[ (XI − X I )(Xj − X j ) ] + · · ·. I=1 ∂XI X¯ I=1 j=1 ∂XI ∂Xj X¯ (9.3.26) Assuming that the variables XI , I = 1, 2, . . . , n, are uncorrelated, one has  E[ (XI − X¯ I )2 ] = Var(XI ) = σX2I , I = j, ¯ ¯ E[ (XI − X I )(Xj − X j ) ] = 0, I = j, where σXI is the standard deviation of variable XI , and equation (9.3.26) becomes  n ∂ 2F   1  σ2 + ··· . ¯ + (9.3.27) F¯ = F(X) XI 2 2 ∂X I=1 I X¯ From equations (9.3.25) and (9.3.27), the variance of F can be obtained    2  n ∂F  n  ∂F 2    2   E[(X − X¯ )2 ] + · · ·, ¯ ¯ ]=E E[(F− F) = I I  (XI − X I ) + · · ·  I=1 ∂XI X¯ I=1 ∂XI X¯ i.e.,



σF2

=

2   2  σ + ··· . ∂XI X¯ XI

n  ∂F I=1

(9.3.28)

Variables XI are usually modelled as lognormally distributed random variables, i.e., XI ∼ LN (XI, m , βX2I ). Using equations (A.2.3) and (A.2.4), the mean value X¯ I , median 

XI, m , standard deviation σXI , and logarithmic standard deviation βI are related as β /2 X¯ I = XI, m e XI , 2

βX2

COV2I = e

I

− 1,

σXI = COVI · X¯ I .

(9.3.29)

Similarly, the median Fm and logarithmic standard deviation β of the factor of safety F can be obtained from the mean value F¯ and standard deviation σF : 

σ 2 COVF = F , βF = ln COV2F + 1 , Fm = F¯ e−βF /2 . (9.3.30) F¯ Depending on how many terms are kept in equations (9.3.27) and (9.3.28), one has the following second-moment methods.

440

❦ Second-Moment First-Order Method From equations (9.3.27) and (9.3.28), taking ¯ F¯ = F(X),



σF2 =

2   2  σ , ∂XI X¯ XI

n  ∂F I=1

¯ = (X¯ 1 , X¯ 2 , . . . , X¯ n ), X

(9.3.31)

gives the Second-Moment First-Order method. ❦ Second-Moment First-Order-Mean Method From equations (9.3.27) and (9.3.28), taking  n ∂ 2F    σ2 , ¯ + 1 F¯ = F(X) XI 2 2 I=1 ∂XI X¯



σF2 =

2   2  σ , ∂XI X¯ XI

n  ∂F I=1

(9.3.32)

gives the Second-Moment First-Order-Mean method.

☞

A second-order formulation can be derived for the standard deviation, but it offers very little benefit and is unreasonably complex. Therefore, the secondorder procedure is recommended only when calculating the mean.

❦ Approximate Second-Moment Method In the approximate second-moment method, the median capacity Fm is obtained by using median values XI, m for all the basic variables XI in a deterministic analysis:

Fm = F(Xm ) = F X1, m , X2, m , . . . , Xn, m .

(9.3.33)

The logarithmic standard deviation for randomness βR and uncertainty βU are obtained by using the square-root-of-sum-of-squares (SRSS) rule:  βF =

n 

I=1

βI2 ,

(9.3.34)

where βI represents the part of the final β-value due to the effect of variability in the Ith basic variable and is obtained from   1 Fm βI =   ln , φ  Fφσ

(9.3.35)

I

where Fφσ is the scale factor applied to the reference earthquake input to reach failure I

when the Ith variable XI is set at the φ-standard-deviation level, while all other variables are kept at their median levels. The parameter φ is usually set at 1 or −1 on the side of the median that leads to the lower capacity:

9.4 conservative deterministic failure margin (cdfm) method

441

❧ Demand variables are increased by evaluating at median-plus-one-SD level, i.e., φ =1

=⇒

XI = XI, m e

βX

I

.

❧ Capacity variables are decreased by setting at median-minus-one-SD level, i.e., φ = −1

=⇒

XI = XI, m e

−βX

I

.

This may be slightly conservative, depending on the effect on the median capacity from the relative variability of the underlying basic variable.

9.4 Conservative Deterministic Failure Margin (CDFM) Method The CDFM method is a deterministic method for estimating seismic capacity and is aimed at achieving a seismic capacity corresponding to about 1 % nonexceedance probability (NEP).

9.4.1 CDFM Method The CDFM evaluation follows the procedure: 1. Select a reference seismic margin earthquake (SME) level, called RLE. 2. A linear elastic seismic demand DS for the RLE is determined. 3. The CDFM capacity C is evaluated in accordance with the guidelines recommended in EPRI-NP-6041-SL (EPRI, 1991a). The process of strength capacity evaluation in CDFM, developed based on a design analysis procedure, is similar to that of the fragility analysis formulated based on the probabilistic concept as presented in Section 9.3. The difference between the two methodologies is the technique used to quantify the conservatism. The general criteria (EPRI-NP-6041) for the CDFM approach are outlined in Table 9.10. Essentially, the CDFM approach intends to achieve the following: ❧ Seismic Demand For the specified RLE, the computed elastic response (RLE demand) of structures and components mounted thereon should be defined at the 84 % NEP. ❧ CDFM Strength For most components, the CDFM strengths should be defined at about the level of 98 % exceedance probability, so that even if the RLE demand slightly exceeds this CDFM strength by more than a permissible conservatively specified inelastic energy absorption capability, the probability of failure is still very low.

442 Table 9.10 Summary of CDFM approach.

Load Combination Ground response spectrum Damping Structural model Soil–structure interaction Material strength Static capacity equations

Inelastic energy absorption

Floor response spectra

Normal + RLE Conservatively specified (84 % NEP) Conservative estimate of median damping Best estimate (Median) + Uncertainty variation in frequency Best estimate (Median) + Parameter variation Code specified minimum strength or 95 % exceedance actual strength if test data are available. Code ultimate strength (ACI), maximum strength (AISC), service level D (ASME), or functional limits. If test data are available to demonstrate excessive conservatism of code equations, then use 84 % exceedance of test data for capacity equation. For non-brittle failure modes and linear analysis, use 80 % of computed seismic stress in capacity evaluation to account for ductility benefits, or perform nonlinear analysis and go to 95 % exceedance ductility levels. Use frequency shifting rather than peak broadening to account for uncertainty plus use median damping.

For very brittle failure modes, such as weld failure and relay chatter, that have no inelastic energy absorption capability so that this capability cannot be conservatively underestimated, the CDFM strength should be defined at about the level of 99 % exceedance probability to increase the conservatism. ❧ Inelastic Distortion The permissible level of inelastic distortion should be specified at the level of about 5 % failure probability. For this permissible level of inelastic distortion, the inelastic energy absorption capability Fμ should be conservatively estimated at the level of about 84 % NEP. It is required that RLE Demand CDFM Strength

 Fμ .

(9.4.1)

The CDFM strength satisfying equation (9.4.1) is a HCLPF capacity, because of the conservatism introduced at the various steps. Any seismic evaluation that introduces approximately the same level of conservatism as defined earlier meets the intent of the CDFM approach and would be expected to achieve a HCLPF capacity. The traditional capacity/demand ratio for elastic response is (C/D)E =

C − CS , DS + DNS

(9.4.2)

9.4 conservative deterministic failure margin (cdfm) method

443

where DS is the elastic seismic demand under the specified RLE, DNS is the concurrent non-seismic demand (loading), C is the capacity of the SSC for the specific failure mode, and CS is the reduction in the capacity due to concurrent seismic loadings. For a permissible level of inelastic response, the inelastic capacity/demand ratio is (C/D)I =

C − Kμ · CS , Kμ · DS + DNS

(9.4.3)

where Kμ = 1/Fμ is the ductility reduction factor. ❧ The HCLPF capacity Ccdfm > RLE when (C/D)I > 1, and Ccdfm < RLE when (C/D)I < 1. If the purpose of analysis is to demonstrate Ccdfm > RLE by setting RLE at the highest level of seismic excitation of interest, it is sufficient to show that (C/D)I > 1. ❧ However, (C/D)I does not define the scale factor to obtain the HCLPF capacity by scaling RLE. The elastic scale factor FE and the inelastic scale factor F I by which the RLE can be scaled for elastic and permissible inelastic responses are C − DNS , DS + CS

FE =

F I = FE · Fμ =

FE , Kμ

(9.4.4)

and the HCLPF capacity is Ccdfm = F I · RLE.

☞

(9.4.5)

Comparing with equations (9.3.4) and (9.3.5), the elastic scale factor FE is similar to the strength factor FS , and the inelastic scale factor F I is similar to the capacity factor FC in the fragility analysis method. However, the levels of probability of exceedance or conservatism in the two methods are different.

When equation (9.4.5) is used to determine the HCLPF capacity level, the RLE may be set at any level including the plant SSE for which elastic demand analyses are likely to already exist. If one is willing to accept the conservatism that exists in these SSE demand analyses, then equation (9.4.5) may be used to determine the HCLPF capacity without performing any new seismic analyses or scaling any existing analysis results.

9.4.2 Estimation of the Conservatism Introduced by the CDFM Method C with composite variability of an SSC is estimated as The median seismic capacity C50 % C C50 % =

S50% F · RLE, D50% N, 50%

(9.4.6)

444

where S50% , D50% , and FN, 50% are median estimates of the seismic strength, seismic demand for the specified RLE input, and inelastic energy absorption (nonlinear) factor, respectively. Similarly, the CDFM seismic capacity Ccdfm is given by Ccdfm =

Scdfm F · RLE, Dcdfm N, cdfm

(9.4.7)

where Scdfm , Dcdfm , and FN, cdfm are the deterministic strength, demand, and nonlinear factor defined in accordance with the CDFM method. Defining RS , RD , and RN as the median conservatism ratios associated with the CDFM method, i.e., RS =

S50% Scdfm

D50% 1 = Dcdfm RD RN =

FN, 50% FN, cdfm

Scdfm =

=⇒

S50% = RS · Scdfm ,

Dcdfm = RD · D50% ,

=⇒

=⇒

S50% , RS

FN, cdfm =

FN, 50% RN

,

D50% =

(9.4.8a)

Dcdfm RD

(9.4.8b)

FN, 50% = RN · FN, 50% ,

(9.4.8c)

then C C50 % = RC · Ccdfm ,

RC = RS · RD · RN ,

(9.4.9)

where RC is the overall median conservatism ratio associated with the CDFM method. The ratios RS , RD , and RN are estimated in the following.

Median Strength Conservatism Ratio RS The CDFM strength is usually computed using the allowable ultimate (maximum) strengths specified by design codes. Based on a review of median capacities from past seismic probabilistic risk assessment studies versus U.S. code-specified ultimate strengths for a number of failure modes, the following conclusions are drawn: ❧ For ductile failure modes The code-specified ultimate strengths have at least a 98 % probability of exceedance, when the conservatism of material strengths, code strength equations, and seismic strain-rate effects are considered. −1 (0.98) β S

Ductile failure modes: RS = e

= e2.0537 βS .

(9.4.10a)

❧ For low ductility failure modes An additional factor of conservatism of about 1.33 is typically introduced. Low Ductility failure modes: RS = 1.33 e2.0537 βS .

(9.4.10b)

In equations (9.4.10), βS is the strength logarithmic standard deviation (typically in the range of 0.2 to 0.4).

9.4 conservative deterministic failure margin (cdfm) method

445

Median Demand Conservatism Ratio RD Seismic demands for CDFM evaluation are typically computed in accordance with the requirements of ASCE 4-98, except that median response spectrum is used in ASCE 4-98 instead of mean-plus-one-SD response spectrum required in the CDFM method. When both response spectra are anchored to the same PGA, the ratio of meanplus-one-SD to median response spectra is equal to the ratio of mean-plus-one-SD to median spectral acceleration amplification factors, over a broad frequency range of interest, such as 3 to 8 Hz. The spectral acceleration amplification factors αA are αA = 4.38−1.04 lnζ ,

for mean-plus-one-SD,

αA = 3.21−0.68 lnζ ,

for median,

(9.4.11)

as given in Table 4.2. Hence, for damping ratio ζ ranging from 0 to 20 %, the average ratio of mean-plus-one-SD to median spectral acceleration amplification factors is   1 20 αA, mean-plus-one-SD 1 20 4.38−1.04 lnζ Ratio = dζ = dζ = 1.22. (9.4.12) αA, median 20 ζ =0 20 ζ =0 3.21−0.68 lnζ Furthermore, as noted in its Preface, ASCE 4-98 (ASCE, 1998) is aimed at achieving seismic responses that have about a 90 % probability of nonexceedance for an input response spectrum specified at the 84th percentile (mean-plus-one-SD) nonexceedance level. Thus the median demand ratio RD can be estimated as RD =

−1 (0.90) β

e

D

1.22

=

e1.2816 βD , 1.22

(9.4.13)

where βD is the seismic demand logarithmic standard deviation for a specified seismic input (typically in the 0.2 to 0.4 range).

Median Nonlinear Conservatism Ratio RN ❧ Ductile failure modes In the CDFM method, the nonlinear factor is expected to be specified at level of about 95 % NEP. Thus for ductile failure modes, the median nonlinear factor ratio RN is Ductile failure modes: RN = e

−1 (0.95) β N

= e1.6449 βN ,

(9.4.14a)

where βN is the logarithmic standard deviation for the nonlinear factor (typically in the 0.2 to 0.4 range).

446

❧ Low ductility failure modes For low ductility (brittle) failure modes, no credit is taken for a nonlinear factor, i.e., FN, 50% ≈ FN, cdfm ≈ 1.0. Thus, Low ductility failure modes: RN = 1.0.

(9.4.14b)

CDFM Capacity Conservatism Combining equations (9.4.9), (9.4.10), (9.4.13), and (9.4.14), the CDFM capacity ratio RC is given by Ductile failure modes:

RC = 0.82 e2.0537 βS +1.2816 βD +1.6449 βN ,

Low ductility failure modes: RC = 1.09 e2.0537 βS +1.2816 βD .

(9.4.15a) (9.4.15b)

From the composite fragility curve and equation (9.4.9), C

C1C% = C50 %e

−1 (0.01) β C

= RC · Ccdfm e−2.3263 βC ,

βC2 = βS2 + βD2 + βN2 . (9.4.16)

From equations (9.4.15) and (9.4.16), one has for different failure modes  0.82 e2.0537 βS +1.2816 βD +1.6449 βN − 2.3263 βC , ductile, C1C% = Ccdfm 1.09 e2.0537 βS +1.2816 βD − 2.3263 βC , low ductility.

(9.4.17)

The values of C1C% /Ccdfm , given by equation (9.4.17), are listed in Table 9.11 for typical values of βS , βD , and βN . It can be seen that, over these variability ranges, the ratio C1C% /Ccdfm ranges from 0.93 to 1.20. Hence, as an approximation C1C% ≈1 Ccdfm

=⇒

C1C% ≈ Ccdfm .

(9.4.18)

9.5 Case Study − Horizontal Heat Exchanger 9.5.1 Background In nuclear power plants, heat exchanger is used to transfer heat produced by nuclear reaction to drive steam turbines for electricity production. The anchorage of heat exchanger has been identified as one of the governing components for overall plant risk (EPRI-TR-1000895). In this section, the determination of seismic capacity of the heat exchanger is demonstrated using both the FA method and the CDFM method. ❦ Construction Details and Potential Failure Modes Details of the horizontal heat exchanger is shown in Figure 9.8, and the properties are listed in Table 9.12. It has a diameter of 8 ft = 96 in, length of 30 ft = 360 in, and is supported by three equally spaced saddles. Each saddle is secured to the concrete

9.5 case study − horizontal heat exchanger Table 9.11

Strength Variability βS 0.2

0.3

0.4

Demand Variability βD 0.2 0.3 0.4 0.2 0.3 0.4 0.2 0.3 0.4

Table 9.12

447

Ratio of C1C% /Ccdfm .

Ductile Failure Modes βN = 0.2 βN = 0.3 βN = 0.4 0.99 0.97 0.92 1.04 1.04 1.01 1.07 1.09 1.07

1.00 1.00 0.97 1.08 1.09 1.07 1.13 1.16 1.15

Low Ductility Failure Modes

0.99 1.00 0.99 1.08 1.11 1.10 1.15 1.19 1.20

1.10 1.04 0.97 1.13 1.10 1.05 1.13 1.14 1.11

Deterministic properties of heat exchanger.

Property Heat Exchanger Tank Diameter Length Floor to bottom tank Height to centre of gravity Shell thickness Weight Saddle Supports (ASTM A36) Base plate thickness Anchor bolt hole diameter Slotted anchor hole dimension Saddle plate to edge of base plate Distance between outside bolts in saddle base plate Weld length Weld leg dimension Stiffener width Stiffener height (outside pair) Stiffener height (inside pair) Stiffener thickness Number of supports Anchor Bolts (ASTM A307) Area through bolt Area through threads Embedment length Bold diameter Head diameter Eccentricity from anchor bolt centreline to saddle plate Number of anchor bolt locations at each saddle Number of anchor bolts at each location

Variable

Value

D L H Hcg t W

96 in 360 in 24 in 72 in 3/8 in 110 kip

tb Db Ds Lb Db Lw tw Ls H1 H2 ts NS

0.5 in 1-1/8 in 3-1/8 in 6 in 72 in 6 in 1/4 in 12-1/2 in 60 in 26 in 0.5 in 3

Agross Anet Le Do Dh es NL NB

0.7854 in2 0.6057 in2 16 in 1 in 1-1/2 in 3 in 3 2

448

Figure 9.8

Heat exchanger.

9.5 case study − horizontal heat exchanger

449

floor by three sets of two cast-in-place anchor bolts. Two of the saddle base plates (Support S1 ) have slotted holes, which allow the thermal expansion of the tank in the longitudinal direction. Each saddle has four stiffener plates to increase the rigidity of the heat exchanger in the longitudinal direction. A total weight of W = 110 kips is estimated for the exchanger. The connecting piping is relatively light, and its weight is included in W = 110 kips. The heat exchanger is located at the ground surface on a rock site and will be subjected to tridirectional excitations during seismic events. The first important step in both FA and CDFM methods is to identify the potential failure modes of an SSC in seismic events. This is usually given by the seismic review team (SRT) members of the plant based on walkdown inspection or experience. From seismic PSA fragility estimates, it has been concluded that the lowest capacity failure modes are from anchorage or component supports (EPRI-NP-6041-SL). In all cases, nozzles, vessel walls, and tube bundles have capacity significantly higher than the governing failure mode. This is typical of pressure vessel-type equipment. The supports and anchorage design are dominated by seismic loading and may fail in a brittle mode, whereas pressure boundary designs are controlled by a combination of pressure, seismic, and piping reactions and are almost always ductile. Furthermore, earthquake experience data on pressurized tanks have shown that any failures are associated with anchorage and not the pressure boundaries. ❧ For PGA < 0.3g, only the anchorage and supports need to be evaluated. ❧ For 0.3  PGA  0.5g, if the heat exchangers have been designed by dynamic analysis, or by a static coefficient method that results in loading that envelopes inertia and nozzle loading, only the anchorage and supports need to be evaluated. ❧ For PGA > 0.5g or for equipment not meeting these criteria, all failure modes must be assessed. It is assumed that the heat exchanger itself was designed to be seismically robust. The capacity of the connection of the saddles to the heat exchanger is relatively high and this potential failure mode is not considered. Only the following failure modes regarding the anchorage and support are considered: • steel failure of the anchor bolt, • anchorage failure of the anchor bolt in the concrete, • bending failure of the support base plate, • weld connection failure between base plate and saddle plate.

450 Table 9.13

Material capacity properties.

Property Steel (ASTM A36, A307) Yield strength Ultimate strength Concrete Compressive strength Weld Tensile strength of electrode (Fexx = 60 ksi) Anchor Bolt Tension Shear Coefficient of friction for shear friction capacity of concrete

Variable

Median

βU

σy σu

44 ksi 64 ksi

0.12 0.06

Fc

6120 psi

0.12

Fexx

1.1 Fexx

0.05

Ntension Vshear

0.9 Anet σu 0.62 Anet σu

0.13 0.10

μ

1.0

0.24

❦ Strength Variables The basic variables for static strength analysis for this heat exchanger are given in Table 9.13. For both steel and concrete, the nominal values Xn , median values Xm , and the associated βU satisfy Xn = Xm e−1.645βU , which implies the nominal strength is defined at the 95 % confidence level.

9.5.2 Fragility Analysis Method 9.5.2.1 RLE Selection The reference GRS in each of the horizontal directions is chosen as the NUREG/CR0098 (USNRC, 1978) median (50 %) rock spectrum with 5 % damping and anchored to 0.3g PGA, as shown in Figure 4.14. The vertical GRS is assumed to be two-thirds of the horizontal GRS over the entire frequency range.

9.5.2.2 Fundamental Frequencies of the Heat Exchanger The fundamental frequency and the associated spectral acceleration of the heat exchanger in each of the three earthquake directions need to be determined first. Because this component is relatively simple, the heat exchanger responds primarily in the first mode in each earthquake direction. (1) Longitudinal direction of the heat exchanger The fundamental frequency F L is controlled by bending in the end saddle support S2 about the weak axis. This is the only support that resists shear force because the bolt holes for the other two supports S1 are elongated in the longitudinal direction to allow thermal expansion of the heat exchanger. Due to the squat configuration of this component and the long distance between outside bolts, it is unlikely that

9.5 case study − horizontal heat exchanger

Figure 9.9

451

Stiffness of a stiffener plate.

stretching in the anchor bolts will significantly affect the frequency in this direction. The fundamental frequency of this SDOF system is given by   1 1 Ks Ks g FL = = , (9.5.1) 2π m 2π W where Ks is the stiffness of one saddle support (S2 ) about the weak axis, and W is the weight (m is the mass) of the heat exchanger. For simplicity of analysis without considering partial fixity, assume that the stiffeners have the equivalent boundary condition of a fixed connection at the tank and a pinned connection at the base plate. The equivalent boundary conditions are likely to be slightly conservative. Because the stiffener is a deep beam, it is modelled as a Timoshenko beam to take into account shear deformation. (Figure 9.9). The equations of equilibrium are 2 d  dϕ  = q(x), EI dx2 dx

dw 1 d  dϕ  EI , =ϕ− κAG dx dx dx

(9.5.2)

where A is the cross-section area of the beam, w(x) is the translational displacement, ϕ(x) is the angular displacement, E is the elastic or Young’s modulus, G is the shear modulus, I is the second moment of area, q(x) is a distributed load (force per unit length), and κ is the Timoshenko shear coefficient depending on the geometry of the

452

beam cross-section. For a rectangular cross-section, κ=

10(1+ν) ≈ 56 , 12+11ν

for ν = 0.3.

In terms of the shear force and bending moment, the equations of equilibrium can be written as dϕ 1 M(x), =− EI dx

(9.5.3a)

dw V(x) . =ϕ+ κAG dx

(9.5.3b)

For a Timoshenko beam with sliding support at x = 0 and pinned support at x = L, and subjected to a concentrated load P at x = 0, the shear force and bending moment at x are V(x) = −P,

M(x) = M0 − Px.

(9.5.4)

Using the boundary condition w (L) = 0 gives M0 = PL. Integrating equation (9.5.3a) yields M P 2 x − 0 x. 2EI EI Substituting into equation (9.5.3b), integrating, and using the boundary condition ϕ=

w(L) = 0 give w(x) =

P P PL 2 2 (L−x) − (L3 −x3 ) + (L −x ). κAG 6EI 2EI

(9.5.5)

Applying a unit load P = 1, the flexibility of the Timoshenko beam is equal to the displacement at x = 0, i.e., Flexibility =

3

L L 1 + . = w(0) = K κAG 3EI

(9.5.6)

For the heat exchanger, As = ts L s = 0.5×12.5 = 6.25 in2 ,

Is =

0.5×12.53 ts L3s = = 81.38 in4 . 12 12

Because there are two stiffeners of height H1 and two stiffeners of height H2 , the stiffness of support S2 is Ks =

2 H31

H1 + κAs G 3EIs

+

2

2

=

603

5 6

(9.5.7)

H3 H2 + 2 κAs G 3EIs

60 + 3 · 29 ·103 · 81.38 · 6.25 ·11.2 ·103

2

+ 5 6

263 26 + 3 3 · 29 ·103 · 81.38 · 6.25 ·11.2 ·10

9.5 case study − horizontal heat exchanger

453

d

Hcg = 72

kbolt . Db . θ D kbolt . 2b . θ θ

Db

Db 2

M Axis of rotation

D b = 72

Figure 9.10

= 746.4

Rotational vibration in the transverse direction.

kips . in

(9.5.8)

The fundamental frequency in the longitudinal direction is, from equation (9.5.1),  746.4×386.4 1 FL = = 8.15 Hz. (9.5.9) 2π 110

☞

In EPRI-TR-103959 (EPRI, 1994, page 8-7), the value of κ is taken as 1.0.

(2) Transverse direction of the heat exchanger In the transverse direction, the heat exchanger is much stiffer and the flexibility is controlled principally by the flexibility in the bolts. The frequency contribution from the in-plane stiffness of the saddle due to bending and shear is very high (> 50 Hz). Suppose that the tank rocks back and forth about an effective length equal to distance between the outside saddle anchor bolts, the fundamental frequency F T in the transverse direction, due to the flexibility in the anchor bolts, is  1 Kθ g , FT = 2π Iθ

(9.5.10)

where Kθ is the transverse rotational stiffness, and Iθ is the rotational inertia about the axis passing through the outside saddle anchor bolts, as shown in Figure 9.10. Introducing a rotation θ (Figure 9.10), the moment required to balance the resisting forces from the anchor bolts is  D    D M = NS · NB · (Kbolt · Db · θ) · Db + Kbolt · b · θ · b , 2 2

Kbolt =

EAgross . (9.5.11) Le

454

The transverse rotational stiffness is M = NS · NB · Kbolt · 54 D2b Kθ = θ 29×103 ×0.7854 5 × 4 ×722 = 5.53×107 kips-in. = 3×2× (9.5.12) 16 Note that the mass moment of inertia of a circular cylinder about its central axis is 1 2 8 mD , where m is the mass and D is the diameter. The mass (weight) moment of

inertia of the heat exchanger tank about the axis of rotation is, applying the parallel axis theorem (see Figure 9.10),



Iθ = I¯θ + W · D2 = 18 WD2 + W H2cg +

 D 2  b

2

2 = 18 ×110×962 + 110(722 + 362 ) = 8.40×105 kips-in .

(9.5.13)

Hence, the fundamental frequency in the transverse direction due to the flexibility in the anchor bolts is, from equation (9.5.10),  1 5.53×107 ×386.4 FT = = 25.4 Hz. (9.5.14) 2π 8.40×105 ☞ The rising of the centre of gravity of the heat exchanger as it rocks about the

outside bolts is not significant and is not included in the analysis. (3) Vertical direction In the vertical direction, the heat exchanger is very stiff. The frequency of the shell translating vertically between supports exceeds 100 Hz and the frequency of the system, where only the bolt and saddle flexibility are considered, exceeds 33 Hz. Thus, the frequency at which the response spectrum returns to the PGA is used for the vertical direction, i.e., F V = 33 Hz.

(9.5.15)

Response spectrum method is used to calculate peak equipment responses in three directions; hence, only spectral accelerations at fundamental frequencies in three directions of the heat exchanger are needed in the calculation. From the horizontal and vertical reference earthquake components at 5 % damping, the spectral accelerations at the three fundamental frequencies in the three directions are given in Table 9.14.

9.5.2.3 Demand Analysis An important task in the fragility analysis it to estimate the median demand in the highest stressed anchor bolt. Because the heat exchanger is located on the ground, GRS instead of FRS can be used. For the critical anchor bolts, the tension and shear forces due to tridirectional seismic excitations are determined as follows.

9.5 case study − horizontal heat exchanger

455

Table 9.14 Fundamental frequencies of heat exchanger.

Direction Longitudinal Transverse

Frequency (Hz)

SA (g)

8.15 25.4

0.630 0.344

33

2 3 × 0.30 = 0.20

Vertical

Figure 9.11

Figure 9.12

Forces due to longitudinal excitation.

Forces due to transverse rocking.

(1) Longitudinal direction of the heat exchanger In the longitudinal direction, under seismic excitation, the tank is subjected to an inertia force equal to the product of its weight W and the spectral acceleration SA( F L ), as shown in Figure 9.11. The inertia force is then transferred to the supports, exerting tension and shear force on anchors bolts. Because Supports S1 have slotted holes to allow for longitudinal movement, the shear force will be evenly distributed in the anchor bolts of Support 2 only. The shear force in a single bolt is given by VL =

W · aL NL · NB

(9.5.16)

456

110×0.630 = 11.55 kips, 3×2

=

where aL = SA( F L ) is the acceleration in the longitudinal direction. Tension forces in

the two Supports 1 are due to the moment W SA( F L ) · Hbg , as shown in Figure 9.11. For the critical anchor bolts, the tension force is given by NL = =

W · aL · Hcg

NL · NB · 2S + 12 S 110×0.630×72 3×2× 52 ×120

(9.5.17)

= 2.77 kips.

(2) Transverse direction of the heat exchanger In the transverse direction, under seismic excitation, the seismic loading due to transverse excitation is also transferred to the supports, exerting tension and shear forces in the anchor bolts, as shown in Figure 9.12. Shear force is induced in all the bolts in all the supports evenly. For a single bolt, the shear force is W · aT (9.5.18) NL · NB · NS 110×0.344 = 2.10 kips, = 3×2×3 where aT = SA( F T ) is the acceleration in the transverse direction. The moment induces VT =

tension forces in the anchor bolts at two locations in all three supports, as shown in Figure 9.12. For the critical anchor bolts, the tension is NT = =

☞

W · aT · Hcg

NB · NS · Db + 14 Db 110×0.344×72 2×3× 54 ×72

(9.5.19)

= 5.05 kips.

In the determination of the bolt tension forces from the two horizontal directions, it is assumed that the tension forces in the anchor bolts vary linearly from the axis of rotation (i.e., the outside anchor bolts as shown in Figures 9.11 and 9.12). This assumption is consistent with the result that the anchor bolt shear capacity (brittle failure) dominates the total seismic capacity of the heat exchanger; in this case it is unlikely that redistribution of the tension forces will occur. This assumption is slightly on the conservative side.

☞

For general cases where the tension force controls the analysis, the forces in the tension bolts would be assumed to all reach the yield capacity before failure occurs; this would increase the tension capacity of the system.

9.5 case study − horizontal heat exchanger

457

(3) Vertical direction In the vertical direction, under seismic excitation, the inertia force of the tank due to seismic vertical acceleration is transferred to the support as pure tension force, without shear force. All anchor bolts share the seismic load evenly so that the tension force is W · aV NL · NB · NS 110×0.2 = 1.22 kips, = 3×2×3

NV =

(9.5.20)

where aV = SA( F V ) is the acceleration in the vertical direction. When the bolts are in tension, the dead load of the heat exchanger also exerts forces in the anchor bolts. All the bolts share the dead load evenly as NDL =

−110 −W = = −6.11 kips. NL · NB · NS 3×2×3

(9.5.21)

(4) Combination of responses due to three earthquake components When the response spectra method is used, the maximum earthquake-induced response of interest in an SSC should be obtained by the SRSS combination or the 10040-40 percent combination of the maximum responses from the three earthquake components calculated separately (USNRC-RG-1.92, USNRC, 2006). Suppose RI , I = 1, 2, 3 are the responses of an SSC, such as the tension force in an anchor bolt, due to the three earthquake components. ❧ SRSS Combination: The maximum response R obtained by this rule is  R = R21 + R22 + R23 .

(9.5.22)

❧ 100-40-40 Combination: The maximum response R obtained by this rule is             R = 1.0× R1  + 0.4× R2  + 0.4× R3 , suppose R1   R2   R3 . (9.5.23) To combine the effect of the three earthquake components on the critical anchor bolt, the 100-40-40 combination rule is used, first assuming that the longitudinal direction controls and then assuming that the transverse direction controls. It is obvious that the vertical direction will not control; thus this case is not considered further. ❧ Longitudinal direction controls • Tension force in the critical anchor bolt is Nlong = 1.0×NL + 0.4×NT + 0.4×NV = 1.0×2.77 + 0.4×5.05 + 0.4×1.22 = 5.28 kips. • Shear force in the longitudinal direction in the critical anchor bolt is

(9.5.24)

458

VL,long = 1.0×VL,L + 0.4×VL,T + 0.4×VL,V

(9.5.25)

= 1.0×11.55 + 0.4×0 + 0.4×0 = 11.55 kips, where VL,L , VL,T , VL,V denote shear forces in the longitudinal direction due to earthquake components in the longitudinal, transverse, and vertical directions, respectively, and VL,long is the maximum shear forces in the longitudinal direction assuming that the longitudinal direction controls. • Shear force in the transverse direction in the critical anchor bolt is VT,long = 1.0×VT,L + 0.4×VT,T + 0.4×VT,V

(9.5.26)

= 1.0×0 + 0.4×2.10 + 0.4×0 = 0.84 kips, where VT,L , VT,T , VT,V denote shear forces in the transverse direction due to earthquake components in the longitudinal, transverse, and vertical directions, respectively, and VT,long is the maximum shear forces in the longitudinal direction assuming that the longitudinal direction controls. • Because the shear forces in the longitudinal direction VL,long and in the transverse direction VT,long are orthogonal, the maximum shear force is their resultant and can be combined using the formula for rectangular components  2 2 + VT,long Vlong = VL,long  = 11.552 + 0.842 = 11.58 kips.

(9.5.27)

❧ Transverse direction controls • Tension force in the critical anchor bolt is Ntran = 1.0×NT + 0.4×NL + 0.4×NV

(9.5.28)

= 1.0×5.05 + 0.4×2.77 + 0.4×1.22 = 6.64 kips. • Shear force in the longitudinal direction in the critical anchor bolt is VL,tran = 1.0×VL,T + 0.4×VL,L + 0.4×VL,V

(9.5.29)

= 1.0×0 + 0.4×11.55 + 0.4×0 = 4.62 kips. • Shear force in the transverse direction in the critical anchor bolt is VT,tran = 1.0×VT,T + 0.4×VT,L + 0.4×VT,V

(9.5.30)

= 1.0×2.10 + 0.4×0 + 0.4×0 = 2.10 kips. • The maximum shear force is the resultant of the shear force components in the longitudinal and transverse directions:

9.5 case study − horizontal heat exchanger Table 9.15

459

Seismic demand to the heat exchanger.

Controlling Direction

Shear Force (kips)

Tension Force (kips)

Longitudinal

11.58

5.28

Transverse

5.07

6.64

Table 9.16

Median capacity factor.

β

0.80

0.8413

0.85

Q 0.90

0.95

0.98

0.99

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

1.0430 1.0878 1.1346 1.1833 1.2342 1.2872 1.3425 1.4002 1.4604 1.5232 1.5887 1.6569

1.0513 1.1052 1.1618 1.2214 1.2840 1.3498 1.4190 1.4917 1.5682 1.6486 1.7331 1.8219

1.0532 1.1092 1.1682 1.2303 1.2958 1.3647 1.4373 1.5137 1.5942 1.6790 1.7683 1.8624

1.0662 1.1367 1.2120 1.2922 1.3777 1.4688 1.5660 1.6697 1.7802 1.8980 2.0235 2.1575

1.0857 1.1788 1.2798 1.3895 1.5086 1.6380 1.7784 1.9308 2.0963 2.2760 2.4711 2.6829

1.1081 1.2280 1.3608 1.5079 1.6710 1.8517 2.0520 2.2739 2.5198 2.7923 3.0943 3.4289

1.1234 1.2619 1.4176 1.5924 1.7889 2.0095 2.2574 2.5359 2.8487 3.2001 3.5948 4.0382

 2 2 VL,tran + VT,tran  = 4.622 + 2.102 = 5.07 kips.

Vtran =

(9.5.31)

The seismic tension and shear forces are summarized in Table 9.15.

9.5.2.4

Capacity Analysis

The objective of capacity analysis is to determine the median capacity for each potential failure mode, using static strength formulas from design codes or textbooks. However, most design codes present the formulas for nominal strength, with a degree of built-in conservatism (significantly higher than 50 % probability of exceedance). For example, as illustrated on page 467 for fillet weld, the nominal weld strength given by equation (J2-4) of ANSI/AISC 360-05 (AISC, 2005) has a 92.5 % probability of exceedance if βU = 0.3, implying that 92.5 % of the welds will have strength exceeding the nominal value. The design strength is obtained from the nominal strength by further multiplying with reduction factors to achieve a certain level of probability of exceedance for a given level of variability. For example, the design weld strength of a fillet weld is obtained as 75 % of the nominal weld strength (ANSI/AISC 360-05, AISC, 2005), which gives a

460

99.2 % probability of exceedance if βU = 0.3, implying that 99.2 % of the welds will have strength exceeding the design value. Suppose the capacity is lognormally distributed with logarithmic standard deviation β. Using equation (A.2.9), the ratio of the median capacity and the nominal capacity (with probability of exceedance Q) is Feqn =

−1 Sm = e β (Q) SQ

=⇒

Q=





lnFeqn . β

(9.5.32)

The values of Feqn are list in Table 9.16. Feqn can be used as a median capacity factor: if the probability of exceedance Q and variability β of the nominal capacity are known, Feqn can be multiplied to the nominal capacity to obtain the median capacity. However, the challenge is that the target probability of exceedance Q and the level of variability β of the nominal capacity are not explicitly specified in design codes. In many cases, further research is required to determine the actual bias between nominal capacities and median capacities. In cases when there is no information, it is generally conservative to use nominal capacity as median capacity, implying that Feqn = 1.0.

Capacities of Anchor Bolts Typical failure mechanisms of concrete anchor bolts are illustrated in Figure 9.13.

Figure 9.13 Anchorage failure modes.

9.5 case study − horizontal heat exchanger

461

Anchorage Failure Modes under Tensile Loading ❧ Anchor steel failure: An anchor tensile failure occurs when the ultimate tensile load is achieved by reaching the anchor’s ultimate tensile strength prior to a concrete cone failure or the anchor slipping out of the hole, as shown in Figure 9.13-1(a). Anchor tensile failures are considered to be ductile failures. The anchor material itself must be ductile and must have adequate embedment depth and sufficient spacing between anchors to achieve a tensile failure. ❧ Anchor pullout failure: An anchor pullout (slip) failure occurs when the maximum tensile load of the anchor is reached by slipping out of the hole, as shown in Figure 9.13-1(b). Anchor slip failures, like anchor tensile failures, are nonlinear and nonbrittle. Anchor pullout failure occurs because the applied force has exceeded the frictional or compressive force between the anchor and the concrete. ❧ Concrete cone failure: A concrete cone failure occurs when an anchor subjected to tension pulls out a concrete cone, as shown in Figure 9.13-1(c). This is a non-ductile or brittle failure mechanism. Concrete anchors with sufficient embedment depth and adequate spacing between anchors do not exhibit this failure mode. Anchorage Failure Modes under Shear Loading For anchor bolts loaded in shear, similar failure behaviour to that under tensile loading can be observed. ❧ Anchor steel failure: Anchor steel failure, often proceeded by a local concrete spall in front of the anchor, occurs when the ultimate shear strength is reached for anchor bolts sufficiently far away from the edge (Figure 9.13-2(a)). ❧ Concrete pryout failure: A concrete pryout failure (Figure 9.13-2(b)) of anchorage located quite far away from the edge may occur for single anchors, and especially for groups of anchors, with a small ratio of embedment depth to anchor diameter and high tensile capacity. ❧ Concrete breakout failure: In tensile loading, concrete breakout failure may occur only if the ratio of anchorage depth to anchor diameter is small, whereas, under shear loading, a brittle concrete failure will occur for anchor bolts located close to the edge, as shown in Figure 9.13-2(c), and cannot be avoided by increasing anchorage depth. The American Concrete Institute (ACI) provides guidelines for the design of anchorage to concrete in Appendix D of ACI 349-06 (ACI, 2007a). ACI 349.2R-07 (ACI, 2007b) provides examples of the application of ACI 349-06 to the design of steel anchorage and embedment, in terms of the bolt effective area and embedment depth. In general, if

462

proper embedment length and spacing between anchor bolts are chosen, the concrete cone failure can be effectively avoided, allowing the steel failure to be dominant.

Steel Failure in Tension For steel strength of anchor in tension, EPRI-TR-103959 (EPRI, 1994, page 8-11, see also Table 9.9) gives the median tensile strength as Ntension, m = 0.90 Anet σu

(9.5.33)

= 0.90×0.6057×64 = 34.89 kips.

(9.5.34)

Clause D.5.1.2 of ACI 349-06 (ACI, 2007a) states that the nominal strength Nsa of a single anchor in tension is Ntension, n = Anet σu .

(D-3, ACI 349-06)

If the strength reduction factor φ = 0.80 (given by Clause D.4.5) is applied to equation (D-3) in ACI 349-06, the nominal tensile strength becomes Ntension, n = 0.80 Anet σu = 0.80×0.6057×58 = 28.10 kips.

(9.5.35)

The median and nominal ultimate strengths of 64 ksi and 58 ksi, respectively, of ASTM A307 bolts and studs are given in Table 9.8.

☞

Clause B.6.5.1 of ACI 349-97 (ACI, 1997) gives the nominal tensile strength of an anchor as Anet σy , in terms of the yield strength not the ultimate strength. This is significantly smaller than the value given by equation (D-3) in ACI 349-06 because σy < σu for A307 steel (σu = 58 ksi, σy = 36 ksi).

Steel Failure in Shear Provision D.6.1.2 of ACI 349-06 (ACI, 2007a) states that the nominal strength Vsa of a single anchor in shear is for cast-in headed stud anchors :

Vshear, n = Anet σu ,

(D-18, ACI 349-06)

for cast-in headed bolts :

Vshear, n = 0.6 Anet σu .

(D-19, ACI 349-06)

Hence, the nominal shear strength of a single A307 cast-in headed bolt in the heat exchanger, given by ACI 349-06, is Vshear, n = 0.60×0.6057×58 = 21.08 kips.

(9.5.36)

EPRI-TR-103959 (page 8-11, see also Table 9.9) gives the median shear strength: Vshear, m = 0.62 Anet σu

(9.5.37)

9.5 case study − horizontal heat exchanger

463

= 0.62×0.6057×64 = 24.03 kips.

☞

(9.5.38)

Clause B.6.5.2.1 of ACI 349-97 (ACI, 1997) gives the nominal shear strength of an anchor as 0.70 Anet σy , in terms of theyield strength not the ultimate strength, for connections with the contact surface of the baseplate flush with the surface of the concrete.

Pullout Strength of Anchor in Tension From Clause D.5.3 of ACI 349-06 (ACI, 2007a), the nominal pullout strength of a single anchor in tension is Npullout, n = ψc,P Np ,

(D-14, ACI 349-06)

where ψc,P = 1.0 for cracked concrete. Letting Abearing be the bearing area, the pullout strength in tension of a single headed stud or headed bolt, Np , is Np = 8 Abearing Fc .

(D-15, ACI 349-06)

Hence, for a cast-in headed bolt with bolt diameter Do = 1 in and head diameter D h = 1.5 in,

π 2 2 2 2 2 Abearing = π 4 (D h − Do ) = 4 (1.5 − 1.0 ) = 0.9817 in ,

Npullout, n = 1.0×8×0.9817×6.12 = 48.06 kips.

(9.5.39)

Concrete Breakout Strength for Anchors in Tension The nominal compressive strength of concrete Fc = 6120 psi is used to calculate strengths of potential failure modes. From Clause D.5.2 of ACI 349-06 (ACI, 2007a), the nominal concrete breakout strength of a single anchor in tension is Nbreakout, n =

ANc ψ ψ ψ N , ANc0 ed,N c,N cp,N b

(D-4, ACI 349-06)

where • for a single stud away from edge

ANc = 1.0, ANc0

• ψed,N = 1.0 is the modification factor for edge (Clause D.5.2.5), • ψc,N = 1.0 is the modification factor for concrete cracking (Clause D.5.2.6), • ψcp,N is the modification factor for splitting control applicable to post-installed anchors only, ψcp,N = 1 is taken for cast-in anchors (Clause D.5.2.7), • Nb is the basic concrete breakout strength of a single anchor in tension in cracked concrete given by Nb = Kc



Fc H1.5 ef ,

(D-7, ACI 349-06)

464

in which Kc = 24 for cast-in headed stud, and Hef is the embedment length. Hence, the concrete breakout strength of a single anchor in tension is √ Nb = 24× 6120×161.5 = 120162 lb = 120.0 kips, Nbreakout, n = 1.0×1.0×1.0×1×120.0 = 120.0 kips.

☞

(9.5.40) (9.5.41)

Provision B.5.1.1 of ACI 349-97 (ACI, 1997) gives the nominal breakout strength of concrete as 

Fc π Le (Le +D h ) √ = 4× 6120 ×π ×16×(16+1.5) = 275.26 kips.

Nbreakout, n = 4

(9.5.42) (9.5.43)

which is significantly higher than the value given by ACI 349-06 (ACI, 2007a).

☞

In ACI 349-97, the projected concrete failure area of a single anchor is a circle; whereas, in ACI 349-06, it is a square.

EPRI-TR-103959 (page 8-11) gives the median breakout strength of concrete 

Fc π Le (Le +D h ) · FEQN √ = 4× 6120 ×π ×16×(16+1.5)×1.4 = 385.4 kips,

Nbreakout, m = 4

(9.5.44) (9.5.45)

in which a median capacity factor of Feqn = 1.4 is used to compensate for the bias between the ACI 349-97 capacity and the median capacity.

☞

The nominal breakout strength of concrete given by ACI 349-97 is the same as that given in EPRI-TR-103959.

Concrete Pryout Strength for Anchors Far from a Free Edge in Shear From Clause D.6.3.1 of ACI 349-06 (ACI, 2007a), the nominal pryout strength of a single anchor in shear is Vpryout, n = Kcp Nbreakout, n ,

(D-28, ACI 349-06)

where Kcp = 2.0 when the effective embedment depth Hef  2.5 in, and Nbreakout, n is the nominal concrete breakout strength for a single anchor in tension, given in equation (D-4, ACI 349-06). Hence, for the anchor bolts of the heat exchanger, the concrete pryout strength of a single anchor in shear is, using equation (9.5.41), Vpryout,n = 2.0×120.0 = 240.0 kips.

(9.5.46)

9.5 case study − horizontal heat exchanger

☞

465

Clause RD.6.3 ACI 349-06 states that the pryout shear resistance can be approximated as one to two times the anchor tensile resistance with the lower value appropriate for Hef less than 2.5 in. This can be seen by comparing equations (9.5.41) and (9.5.46).

Shear-Friction Failure The nominal and median yield strength σy of 36 ksi and 44 ksi, respectively, of ASTM A307 bolts and studs are given in Table 9.8. Provisions 11.7 of ACI 349-06 (ACI, 2007a) are applicable to consider shear transfer across a given plane, such as an existing or potential crack. A cast-in anchor bolt may be considered as a shear-friction reinforcement. If it is perpendicular to the shear plane, the nominal shear-friction strength is Vshear-friction, n = μ Avf σy ,

(11-25, ACI 349-06)

where Avf = Anet is the area of shear-friction reinforcement, and μ is the coefficient of friction. For the anchor bolts in the heat exchanger, the nominal shear strength is Vshear-friction, n = 1.0×0.6057×36 = 21.81 kips.

(9.5.47)

EPRI-TR-103959 (EPRI, 1994, page 8-11) gives a formula for evaluating the median shear-friction strength in terms of the ultimate stress Vshear-friction, m = 0.9μ Anet σu = 0.9×1.0×0.6057×64 = 34.89 kips. The ratio between these two values is Vshear-friction, m Vshear-friction, n

=

34.89 = 1.60. 21.81

(9.5.48) (9.5.49)

(9.5.50)

If β = 0.3, then Vshear-friction, n has an 81.6 % probability of exceedance using equation (9.5.32).

Failure of Support Base Plate due to Bending During earthquakes, the base plate might move away from the floor. However, anchor bolts would resist its movement. Hence, the base plate is in bending in the area surrounding anchor bolts. For this heat exchanger support, the base support plate is connected to saddle plates and stiffener plates. A typical portion of base plate is constrained on three sides and free on one side (Figure 9.8). The plate bending capacity can be realistically estimated using yield line theory. A postulated yield line pattern for the steel base plate is shown in Figure 9.14. Because of symmetry, x is the only unknown dimension to be determined so that the minimum capacity is obtained.

466

Npb A Yield line lb

x es

ds δ lb

B

D δ x

δ

tb x

C

y

Figure 9.14 Yield line pattern of the base plate.

Give point C of the plate a small downward virtual displacement δ. The external work done by force Npb from the anchor bolt is es δ. Npb · Lb Denote My = 14 (1 · t b2 )σy as the plastic (yield) moment of resistance per unit length. The internal work done is summarized in the following table Segment

Components of Rotation θx θy

Components of Work My, x · θx · y0 My, y · θy · x0

ABC

δ x

δ Lb

My ·

δ · (2L b − D s ) x

My ·

δ ·x Lb

ADC

δ x

δ Lb

My ·

δ · (2L b − D s ) x

My ·

δ ·x Lb

and is given by

 δ

My 2

x

(2L b −D s ) + 2



δ x , Lb

in which the slotted bolt hole length D s is used because it is the critical case. From the Principle of Virtual Work Npb · which gives Npb =





es δ δ (2L b −D s ) + δ = 2My x , Lb x Lb

2L b My  2L b −D s x , + es x Lb

My =

t b2 σ . 4 y

(9.5.51)

For the minimal value of Npb , 2L My  2L b −D s 1  − =0 + = b es x2 Lb dx

dNpb

 =⇒

x = Lb

2−

Ds . Lb

(9.5.52)

Substituting equation (9.5.52) into (9.5.51) gives the median capacity of the base plate  L t2 D Npb,m = 2− s · b b σy (9.5.53) Lb es

9.5 case study − horizontal heat exchanger

467

Saddle plate lw

Fillet weld lb

es tw Throat = 0.707tw Base plate

tw Figure 9.15

 =

2−

Fillet weld failure.

3.125 6×0.52 × ×44 = 26.76 kips. 6 3

(9.5.54)

Failure of Fillet Weld between Saddle Plate and Base Plate Fillet welds are commonly used in structures. A fillet weld can be loaded in any direction in shear, compression, or tension. However, it always fails in shear. The weld area Aw resisting these applied loads is given by an effective length Lw times the effective throat √ thickness, which is equal to tw / 2 = 0.707tw , where tw is the weld leg size, as shown in Figure 9.15; hence Aw = 0.707Lw tw . Formulas for evaluating the capacities of fillet-weld connections are given in Appendix P of EPRI-NP-6041-SL (EPRI, 1991a) and summarized in Table 3-10 of EPRITR-103959 (EPRI, 1994); see Table 9.9, in which Fexx, m and Fexx are the median and nominal tensile strength of electrode. EPRI-TR-103959 recommends Fexx, m = 1.1Fexx . In Table 9.9, the median capacity Pweld, m can be written in the general form as Pweld, m = 0.84 Aw Fexx, m (1.0 + 0.50 sin1.5 θ ),

(9.5.55)

where θ is the angle of loading measured from the weld longitudinal axis. In longitudinal direction, θ = 0, and equation (9.5.55) reduces to Pweld, m = 0.84 Aw Fexx, m (1.0 + 0.50 sin1.5 0◦ ) = 0.84 Aw Fexx, m ; in transverse direction, θ = 90◦ , and equation (9.5.55) reduces to Pweld, m = 0.84 Aw Fexx, m (1.0 + 0.50 sin1.5 90◦ ) = 0.84 Aw Fexx, m × 1.5 = 1.26 Aw Fexx, m . For the heat exchanger, the equivalent median tension capacity of the bolt based on the median capacity of the weld in the transverse direction is given by Aw = 0.707Lw tw = 0.707×6×0.25 = 1.0605 in2 , Pweld, m = 1.26 Aw Fexx, m = 1.26×1.0605×(1.1×60) = 88.19 kips.

(9.5.56)

468 Table 9.17

Summary of equivalent anchor bolt capacities.

Failure Mode Anchor bolt steel Pullout Concrete capacity Cone failure Shear friction Base plate bending Fillet weld base plate to saddle plate

Equivalent Anchor Bolt Capacities (kips) Shear Equation Tension Equation 24.03 (9.5.38) 34.89 (9.5.34) 48.06 (9.5.39) 240.0 34.89

(9.5.46) (9.5.49)

385.4

(9.5.45)

26.76 88.19

(9.5.54) (9.5.56)

ANSI/AISC 360-05 (AISC, 2005) gives the nominal strength Pweld, n for fillet welds loaded in plane as Pweld, n = Aw Fw ,

(J2-4, ANSI/AISC 360-05)

where Fw = 0.60 Fexx (1.0 + 0.50 sin1.5 θ ), in which a factor of 0.60 is applied to the nominal tensile strength of electrode Fexx to introduce extra conservatism. Thus, the equivalent nominal tension capacity of the bolt based on the nominal capacity of the weld in the transverse direction is Pweld, n = 1.0605×(0.60×60×1.5) = 57.27 kips.

(9.5.57)

The ratio of the median capacity Pw, m given by equation (9.5.55) and the nominal strength Pw, n given by equation (9.5.57) is Pweld, m Pweld, n



=

0.84 Aw (1.1 Fexx ) (1.0+0.50 sin1.5 θ ) 

Aw 0.60 Fexx

(1.0+0.50 sin1.5 θ )





=

0.84×1.1 = 1.54. (9.5.58) 0.60

If β = 0.3, then Pw, n has 92.5 % probability of exceedance using equation (9.5.32).

Controlling Failure Modes The capacities for the four failure mode categories determined in this section are summarized in Table 9.17. Because of the inconsistency in the strength equations, the equation numbers are also listed. Based on the capacities in Table 9.17 and the seismic tension and shear demand forces in Table 9.15, it is clear that the anchor bolt steel failure (in shear) will control VST = 24.03 kips,

NST = 34.89 kips,

(9.5.59)

and that the earthquake forces in the longitudinal direction Vlong = 11.58 kips,

N long = 5.28 kips,

(9.5.60)

will control the factor of safety by which the reference earthquake can be scaled to reach failure.

9.5 case study − horizontal heat exchanger

469

9.5.2.5 Scale Factor and Median Capacity To determine the strength factor, it is necessary to understand the failure mechanism of the anchor bolts under seismic loading. In general, a tension-shear interaction relationship for the bolt failure is required.

Bilinear Approximation of Shear-Tension Interaction Relation Based on a large number of shear-tension test data, EPRI-NP-5228-SL (EPRI, 1991b, page 2-95) recommends a shear-tension-interaction formulation for expansion bolts and cast-in bolts. The results are plotted in terms of N/Nm and V/Vm in a bilinear form as shown in Figure 9.16(a), where Nm and Vm are the bolt tension and shear capacities in the absence of combined loading: N = 1.0, Nm

0.7

N V + = 1.0, Nm Vm

0.3
aK = H(aK ). Hence, the mean annual frequency of occurrence that the GMP a is between aK and aK+1 is given by 



P aK  a < aK+1 = H(aK ) − H(aK+1 ) =−

H(aK+1 ) − H(aK )

aK+1 − aK

(aK+1 − aK ) = −

H(aK ) · aK , (10.3.1) aK

where H(aK ) = H(aK+1 )− H(aK ) and aK = aK+1 −aK . Combining the seismic hazard curve and the plant fragility using the total probability theorem and employing equation (10.3.1) give   ∞      H(aK )  FE = P A< aK aK  a < aK+1 · P aK  a < aK+1 = pF(aK )· − ·aK , aK K=0 K=0 (10.3.2) in which FE represents seismic risk, i.e., annual frequency of plant damage state. ∞  

Taking the limit aK →0, equation (10.3.2) becomes  ∞ d H(a) pF(a) da. FE = − da 0

10.3 seismic risk quantification

537

Integrating by parts yields    ∞  ∞ ∞  FE = − pF(a) d H(a) = − pF(a) H(a)  − H(a) d pF(a) a=0 0 0    ∞ = − pF(∞) H(∞) − pF(0) H(0) + H(a) d pF(a) 0



∞

= −(1 · 0 − 0 · 1) +

H(a) d pF(a) =

0



∞

H(a)

d pF(a)

0

da

da.

Hence, the mean annual frequency of plant damage state is given by  FE = −

∞

0

d H(a) da = pF(a) da



∞

H(a)

0

d pF(a) da

da.

(10.3.3)

It is generally impossible to evaluate equation (10.3.3) analytically, and numerical evaluation must be applied. If earthquakes with GMP < al have no effect on structures and SSCs; whereas the probability of earthquakes with GMP > au occurring at the site of interest is negligible. The domain of integration can be trun



cated to al , au . Discretizing the domain of integration into N small intervals as a1 = al , a2 , a3 , . . . , aN+1 = au , equation (10.3.2) can then be approximated as FE =

N 



K=1



pF(aK ) H(aK ) − H(aK+1 ) .

(10.3.4)

10.3.2 Approximate Mean Annual Frequency of Failure If the seismic hazard curve approximated by equation (5.2.18) in Section 5.2.4, i.e., 



P GMP > a = H(a) = Ki a−Kh ,

(10.3.5)

is used, an approximation of the mean annual frequency of failure can be determined. From equation (9.1.16), the composite fragility curve of an SSC or system is FE(a) = P



   β −1 ln (a/C ) z2 50% −   1 ln(a/Am )  e 2 dz, (10.3.6) A< a GMP = a =

=√ β 2π 0

in which the script “C” for Composite fragility is dropped for simplicity of notations. From equation (10.3.3), the mean annual frequency of failure is  FE =

∞

H(a)

d pF(a) da

0

Ki =√ 2π β



∞

a 0



∞

da =

−Kh −1

Ki a

−Kh

0 −

e

1 2β 2

[ ln (a/C50% )]

2   1

1 − 2β 2 [ ln (a/C50% )] 1 1 · · da √ e β a 2π

2

da.

(10.3.7)

538

Letting lna = x =⇒ a = ex , da = a dx and denoting lnC50% = M yield  ∞ ! Ki 1 2 dx FE = √ exp − Kh x − (x − M) 2β 2 2π β −∞  ∞ !  Ki 1  =√ exp − 2 x − (β 2 Kh −M) 2 − Kh M + 12 (β Kh )2 dx 2β 2π β −∞  ∞ !  ! 1 1  1 2 exp − 2 x − (β 2 Kh −M) 2 dx = Ki exp − Kh M + 2 (β Kh ) · √ 2β 2π β −∞ −Kh M+ 12 (β Kh )2

= Ki · e

1

−Kh 2 = Ki · C50 % ·e

(β Kh )2

.

(10.3.8)

Let H be a reference annual frequency of exceedance (AFE) and aH be the corresponding GMP level. Hence, from the seismic hazard curve (10.3.5), H = Ki aH−Kh

Ki = H · aHKh .

=⇒

(10.3.9)

The mean annual frequency of failure given by equation (10.3.8) becomes 1

−Kh 2 FE = H · aHKh · C50 % ·e

(β Kh )2

,

which can be written as 1

−Kh 2 FE = H · F50 % ·e

(β Kh )2

,

C F50% = a50% . H

(10.3.10)

Note that equation (10.3.10) can be applied at the plant level if the plant level fragility curve is used or at the SSC level if the fragility curve is for SSC.

10.4 Seismic Margin Assessment There is substantial conservatism in the seismic design of nuclear power plants to ensure that the plants can be safely shut down in the event of the safe shutdown earthquake (SSE) or design basis earthquake (DBE) for which the plant was licensed. The objectives of seismic margin assessment (SMA) are ❧ to demonstrate that the plant has additional seismic margin to withstand a higher earthquake level, called seismic margin earthquake (SME) or review level earthquake (RLE) with high confidence; ❧ to identify any “weaker link” components that reduce the HCLPF capacity of the plant below the desired SME or RLE earthquake level. The seismic margin is defined in terms of HCLPF capacity of critical SSCs and the overall HCLPF of the plant. For the specified SME or RLE earthquake, there is approximately 95 % confidence of not exceeding 5 % probability of failure. SSCs with HCLPF values lower than the RLE are flagged, and recommendations to improve their HCLPF

10.4 seismic margin assessment

539

values will be made if they do not meet their safety targets. This process ensures that the plant has sufficient seismic “margin”. The principal products of an SMA are a list of component capacities and an estimate of the HCLPF capacity of the plant. Because a large seismic margin generally exists for ground motion levels well above the DBE, SMA should be treated as safety reevaluations and not as design evaluations. The general procedure of SMA is shown schematically in Figure 10.17. There are two distinct SMA approaches: USNRC SMA and EPRI SMA; the primary difference between these two approaches is in the system analysis. The EPRI approach is deterministic and applies success path modelling, whereas the USNRC SMA employs fault tree and event tree modelling. Table 10.4 summarizes a comparison of seismic PSA, USNRC and EPRI SMA.

❦ USNRC SMA The so-called USNRC SMA approach was recommended by the U.S.Nuclear Regulatory Commission (NUREG-CR-4334, USNRC, 1985). HCLPF capacities of SSCs can be determined using either the FA method or the CDFM method. In probabilistic safety assessment (seismic or otherwise), the following two groups of plant safety functions are generally considered. Group A 1. Reactor Subcriticality: shutting down the nuclear reaction such that the only heat being generated is decay heat. 2. Normal Cooldown: providing cooling to the reactor core through the use of the normal power conversion system, normally defined as the main steam, turbine bypass, condenser, condensate, and main feedwater subsystems. 3. Emergency Core Cooling (Early): providing cooling to the reactor core in the early (transient) phase of an event sequence by the use of one or more emergency systems designed for this purpose. Group B 1. Emergency Core Cooling (Late): providing cooling to the reactor core in the late (stabilized) phase of an event sequence by the use of one or more emergency systems designed for this purpose. 2. Containment Heat Removal: removing heat from the containment to the ultimate heat sink during the late (stabilized) phase of an event sequence by the use of one or more emergency systems designed for this purpose.

540

min HCLPF

System Analysis

⇒

max HCLPF

Component HCLPF Capacity

“OR” gate

⇒

HCLPF Max/Min Method

“AND” gate

Plant damage state

SSC

Fault & event trees from internal event with passive equipment and structures added

Plant damage state HCLPF DS

Methodology for seismic margin assessment.

Plant SSCs

Eliminate SSCs with HCLPF exceeding the screening level

Success path 2

Success path 1

Two success paths

for remaining SSCi

USNRC SMA SSC

Fragility Analysis Method i =C 95% ≈C C CHCLPF 5% 1%

Conditional probability of failure

SSC

SSC

SSC

i CCDFM

i ≈C i CHCLPF CDFM

SSCi ⇒ C CDFM

Ground motion parameter a

OR

CDFM Method

EPRI SMA ⇒

SSC

i ≈C i CHCLPF CDFM

Set screening level

Screening Table

CDFM Method

Remaining SSCs

Figure 10.17

Uniform hazard spectra

Seismic Input Motion

Output

Seismic Capacity Evaluation

Screening

Plant Walkdown

Plant level HCLPF Risk insights Dominant risk contributors

Risk insights

Dominant risk contributors

or CDFM method

either fragility analysis method

HCLPF capacity using

Seismic risk (CDF or LERF)

Median seismic capacity Randomness & uncertainty

Fragility analysis method

based on seismic capacity and seismic risk considerations

Controlling SSCs

Plant level HCLPF

CDFM method

HCLPF capacity using

EPRI NP-6041-SL screening tables

Two success paths

(four alternative methods in EPRI-NP-6041-SL)

Review level earthquake

Seismic margin earthquake

EPRI NP-6041-SL procedure Seismic spatial interaction, and seismically induced internal fires and floods

EPRI NP-6041-SL screening tables

EPRI SMA

Plant level HCLPF capacity

from internal event with passive equipment and structures added

Fault trees and event trees

Site-specific seismic hazard curves

Seismic Hazard

System Analysis

Seismic risk (CDF or LERF)

Purpose

USNRC SMA

Comparison of seismic PSA, USNRC SMA, and EPRI SMA.

Seismic PSA

Table 10.4

10.4 seismic margin assessment 541

542

3. Containment Overpressure Protection: controlling the buildup of pressure in the containment due to the evolution of steam by condensing this steam during an event sequence by the use of one or more emergency systems designed for this purpose. Results of PSA studies have shown that failure of Group B is virtually assured if failure of Group A occurs, and that failure of Group B is unlikely to contribute to dominant plant damage states given success of Group A. Hence, as one of the screening criteria, it is only necessary to consider the functions in Group A. This greatly simplifies the analysis by eliminating a large number of systems and parts of systems from analysis. In the system analysis of USNRC SMA, complete fault trees and event trees are developed for Group A functions. Boolean algebra is then applied to these trees to remove redundant events for the plant damage state. By incorporating PSA methodology into SMA, it is possible to systematically consider many potential accident sequences.

❦ EPRI SMA This purely deterministic approach was developed by the Electric Power Research Institute (EPRI-NP-6041-SL, EPRI, 1991a). The CDFM method is recommended to determine the HCLPF capacities of SSCs. Instead of developing detailed fault trees and event trees for Group A functions, EPRI-NP-6041-SL recommended to define components on a “success path”, which is an operational sequence of plant systems that will bring the plant to a stable condition (either hot or cold shutdown) and maintain that condition for at least 72 hours. Several possible success paths may exist. EPRI-NP-6041-SL recommended to select one primary and one alternate success path for which it will be easiest to demonstrate an adequate seismic margin. Only SSCs within these success paths need to be reviewed in the SMA. The key benefit of this approach is to reduce the amount of system modelling required for two success paths and to reduce the number of SSCs investigated.

Plant Damage State HCLPF Capacity Having obtained the fault trees and event trees for the plant damage state in USNRC SMA approach or the success paths in EPRI SMA approach, the HCLPF max/min method is used to estimate the HCLPF capacity of a damage state given the HCLPF capacities of every SSC in the fault/event trees or success paths of the damage state. This HCLPF max/min method consists of the following two rules: ❧ The HCLPF capacity of SSCs connected in series or combined by a “OR” gate is equal to the minimum HCLPF capacity of the SSCs being combined.

10.4 seismic margin assessment

543

❧ The HCLPF capacity of SSCs connected in parallel or combined by an “AND” gate is equal to the maximum HCLPF capacity of the SSCs being combined. The damage state HCLPF capacity can be readily estimated from the individual SSC HCLPF capacities and the plant damage state fault/event trees or success paths.

Screening Tables for Screening out SSCs from HCLPF Computations In order to be cost-efficient, an SMA should incorporate a step where SSCs are quickly screened from further review, or a detailed SMA, based on experience and judgement concerning their seismic ruggedness to withstand the specified SME or RLE level. The advantage of screening out SSCs from further review is that a great amount of unnecessary HCLPF capacity computations are eliminated for SSCs whose HCLPF capacities clearly exceed the screening level, so that efforts can be quickly focused on those SSCs for which there is a legitimate concern about seismic ruggedness. EPRI-NP-6041-SL (EPRI, 1991a, Tables 2-3 and 2-4) provides screening tables that can be used as part of a seismic walkdown to screen out many SSCs from further HCLPF capacity calculations. These screening tables are set at 0.8g, 5 % damped peak spectral acceleration (approximately 0.33g PGA) and 1.2g, 5 % damped peak spectral acceleration (approximately 0.5g PGA). It is important to set an appropriate screening level. If it is set too low and all computed HCLPF capacities for the non-screened out components either nearly equal or exceed the screening level, then no weaker link components that govern the seismic risk will be determined from the SMA. Hence, the screening level should be set sufficiently high that it is unnecessary to determine which components are the weaker link components when all components have HCLPF capacities exceeding this screening level. It is important to note that the screening step implies that the plant HCLPF capacity cannot exceed the capacity level on which the screening guidance is based. For example, the success path approach in EPRI SMA treats the safe shutdown equipment list (SSEL) as a series of components, like the links in a chain. If any one link fails then the chain fails. Thus, analogous to that the strength of the chain is equal to the strength of the weakest link, the SSEL is only as strong as the weakest component. When a component is screened out it is based on the assumption that its capacity is greater than the screening level, but it is not known how much stronger. Even if all the calculated HCLPF capacities are greater than the screening level, the plant HCLPF capacity can only be stated as exceeding the screening level.

544

Selection of a Review Level Earthquake (RLE) The RLE is an engineering representation of earthquake ground motion chosen to have a lower AFE than that of the DBE. An RLE is established at a reasonably high and achievable level based on site seismicity and plant-specific design features. The AFE of the RLE is generally agreed upon by the owner/licensee and the regulatory authority. An AFE of 10–4 per year or less is typically selected for the RLE. The RLE is chosen to challenge seismic design of the plant over the DBE. In principle, the RLE level should be set sufficiently high so that some SSCs are found to have HCLPF capacities less than this RLE level; both the SSCs that control the HCLPF capacities of the plant can be identified and the HCLPF capacity of the plant can be established. If the RLE level is set too low, and if all SSCs pass the review, then only a lower bound on the HCLPF can be established at this low RLE level. It is desirable to find SSCs that are the “weakest links” or to demonstrate a greater seismic margin. Weakest-link SSCs are only found if one or more SSCs do not pass the review procedure at the selected RLE level. Increased HCLPF seismic margin is shown only by selecting a higher RLE level. On the other hand, the RLE level should not be set so high as to result in a substantial increase in the workload for the SMA. When the RLE level is set at or below approximately 0.3g PGA (with a corresponding 5 %-damped peak spectral acceleration less than 0.8g), screening tables enable a large number of SSCs being screened out of further HCLPF evaluation based on earthquake experience and judgement. RLE is typically specified in terms of a smooth broad-frequency response spectrum shape. Regardless how the response spectrum shape is developed, the HCLPF capacities of SSCs and the plant are estimated based on the assumption that, at each natural frequency and in each direction, there is 15.9 % probability that the RLE response spectrum ordinate will be exceeded, or the nonexceedance probability (NEP) is 84.1 %. Therefore, the selected response spectrum shape should be consistent with the conservatism relative to this assumption. Generally, one response spectrum shape is specified for both orthogonal horizontal directions, and the vertical response spectrum is specified by multiplying the horizontal response spectrum by a frequency-dependent or frequency-independent ratio. EPRI-NP-6041-SL (pages 2-5 to 2-10) presented four alternative approaches for the determination of the smooth broad-frequency response spectrum shape. Alternative 1. The RLE is specified in terms of the horizontal PGA. The variability in the ratio of response spectral acceleration and PGA is taken into account by using 84.1 % NEP response spectral amplification factors anchored to the PGA (see Section 4.1). The spectral amplification factors may be determined as follows:

10.4 seismic margin assessment

545

• Preferably, realistic site-specific amplification factors should be used. A set of ground motion time-histories are normalized to the same PGA, and 84.1 % NEP amplification factors are then determined. This amounts to generating a site-specific mean-plus-one-SD Newmark response spectrum. • Alternatively, it may also be reasonable and often conservative to use the 84.1 % NEP amplification factors from NUREG/CR-0098 (USNRC, 1978) or USNRC Regulatory Guide 1.60 (USNRC, 2014). The primary advantage is that the RLE is simply specified in terms of the PGA. Thus, the resultant HCLPF statement is only conditional on the PGA value not being exceeded. This alternative is ideal for deterministically specified ground motion or when the seismic hazard for a given AFE is only specified in terms of PGA. Alternative 2. The RLE is specified in terms of a UHS shape. • Select a RLE level in terms of PGA, such as 0.3g. • By varying the AFE, select a UHS that has the selected PGA. The shape corresponding to either the 84.1 % NEP or the mean UHS can be selected as the spectrum shape for RLE, depending on the objectives of the review. • Determine the HCLPF capacity, defined in terms of PGA, for SSCs and the plant by performing the SMA. • Using the 84.1 % NEP or the mean UHS PGA, determine the AFE, which gives the AFE corresponding to the HCLPF capacity. Using this approach, the resultant HCLPF statement is conditional on the AFE, at the 84.1 % NEP or the mean level, rather than on PGA as for Alternative 1. Alternative 3. Specify the site-specific seismic hazard in terms of a specific earthquake magnitude range (such as 5.8  M  6.8) with a specified epicentral distance range (such as less than 25 km from the site). The magnitude range reflects the uncertainty in the size estimate of earthquakes local to the site. Using a combination of either scaled or unscaled real-time-histories and/or numerically generated time-histories appropriate for the site conditions, the hazard predictor can provide both 50 % NEP and 84.1 % NEP site-specific RLE spectra for these specified conditions. The 84.1 % NEP spectrum is used for the purposes of HCLPF capacity estimations. The HCLPF statement is then expressed as a ratio of this 84.1 % NEP site-specific spectrum for the specified earthquake conditions. Alternative 4. A standard (non-site-specific) RLE spectrum may be negotiated with the regulator. For instance, the median NUREG/CR-0098 (USNRC, 1978) spectrum anchored to 0.3g might be specified as the RLE. This approach does not require

546

seismic hazard information at the time when SMA is performed. However, it has the disadvantage of using a potentially inappropriate spectral shape.

☞

NUREG/CR-0098 provides both 50 % and 84.1 % NEP amplification factors; whereas USNRC R.G. 1.60 provides only 84.1 % NEP amplification factors.

10.5 Seismic Probabilistic Safety Assessment with Screening Tables The screening tables in EPRI-NP-6041-SL (EPRI, 1991a, Tables 2-3 and 2-4) have also been used in seismic PSA reviews to screen out relatively strong elements.

Surrogate Element Unlike SMA, in which screened-out SSCs are removed from further consideration, when screening tables are used in seismic PSA, a surrogate element must be added to the plant damage state fault/event trees to replace all of the components that have been screened out to account for the capacity level assumed in the screening tables. Based on EPRI-TR-103959 (EPRI, 1994), this surrogate element should have the following median capacity C50% and variability β: Surrogate Element:

C50% = 2× SL,

β = 0.3,

(10.5.1)

where SL is the screening level. This surrogate element produces a HCLPF capacity Chclpf = C1% = C50% e−2.3263β = (2× SL) e−2.3263 × 0.3 = SL.

(10.5.2)

If the surrogate element is not added, then the damage state seismic risk coming from the screened out components will have been ignored. If all SSCs were to be screened out, and no surrogate element added to the seismic PSA, it might be falsely concluded that the frequency of failure is zero.

Screening Level By using the screening tables in a seismic PSA, computation of fragilities for a large number of seismically rugged SSCs can be avoided. However, the replacement surrogate element may be a significant contribution to the computed seismic risk, which can mask the actual risk contributions. To overcome this problem, the screening level can be selected as follows: ❧ Establish a permissible level of seismic risk FE0 , which can be contributed by the surrogate element.

10.6 hybrid method for seismic risk assessment

547

Table 10.5 Ratio of FE / FE0 .

AR

Kh

FE / FE0

4.00 3.75 3.50 3.00 2.50 2.00 1.50

1.66 1.74 1.84 2.10 2.51 3.32 5.68

1.03 1.00 0.97 0.90 0.81 0.68 0.58

❧ From the seismic hazard curve H(a), determine the ground motion level aH corresponding to exceedance frequency H = 2 FE0 , i.e., H(aH) = H = 2× FE0 .

(10.5.3)

❧ The screening level SL is then set at SL  0.8×aH .

(10.5.4)

The screening level satisfying (10.5.4) will ensure that the surrogate element will not contribute a seismic risk greater than FE0 , which can be shown as follows. From equations (10.5.4) and (10.5.2), one has aH



SL

0.8

=

C1 % . 0.8

(10.5.5)

The mean annual frequency of failure of the surrogate element is given by (10.3.10)  −Kh 1 (β K )2   1 (β Kh )2 C50% −Kh C h  H · 0.8× 50% ·e2 ·e2 FE = H · aH C1% 1

= 2 FE0 · (0.8×e 2.3263 β )−Kh · e 2

(β Kh )2

,

which gives FE FE0



2×(0.8×e 2.3263 × 0.3 )−Kh ×e0.5 × (0.3 × Kh ) . 2

(10.5.6)

The values of the right-hand side of (10.5.6) are listed in Table 10.5 for values of Kh . It is clear that, for AR  3.75 or Kh  1.74, the seismic risk contribution from the surrogate element FE  FE0 . The procedure of seismic PSA with screening table is shown in Figure 10.18.

10.6 Hybrid Method for Seismic Risk Assessment The hybrid method is similar to seismic PSA with screening tables presented in Section 10.5, except that the fragility curve for each remaining SSC is estimated from the

548

Screening Table 1. Set a permissible level of seismic risk from the surrogate element fE0 2. Seismic hazard curve H(aH) = 2 × fE0 ⇒ aH 3. Screening level SL < 0.8×aH

pFSSCi(a)

Plant damage state fault & event trees from internal event with passive equipment and structures added

System Analysis

a Ground motion parameter a

Mean seismic hazard curve

Annual frequency of exceedance

Seismic Hazard Analysis

H(a)

a Ground motion parameter a

Conditional probability of failure

for remaining SSCi

Fragility Analysis

Variability β = 0.3

SE = 2 ×SL Median capacity C50%

Surrogate Element

Screened out SSCs

Remaining SSCs

Plant SSCs

Conditional probability of failure at GMP level a

∞ ∞

dpF(a) (a) dp F da da da da

∞ ∞ H (a) d (a) dH(a) da da ppFF(a) da da

H(a) H(a)

00

Seismic Risk Quantification

00

Conditional probability of failure

= =

=− − pfFE =

pF(a)

Plant DS fragility a Ground motion parameter a

Plant DS conditional probability of failure at GMP level a

pF(a) = P {Plant DS failure | a }

Methodology for seismic probabilistic safety assessment with screening table.

pFSSCi (a) = P {Capacity of SSCi < a | a } Figure 10.18

10.6 hybrid method for seismic risk assessment

549

HCLPF value determined using the CDFM method. The hybrid method is briefly discussed in EPRI-TR-103959 (EPRI, 1994) and presented in more detail in Kennedy (1999). The procedure of the hybrid method is shown in Figure 10.19. The median capacity and variability β of the surrogate element are given by equation (10.5.1). The steps of estimating the fragility curve for each remaining SSC are as follows: 1. For each SSC, the capacity Ccdfm is determined by the CDFM method (Section 9.4). The HCLPF capacity C1% is approximately equal to the CDFM capacity, i.e., C1% ≈ Ccdfm .

(10.6.1)

2. For each SSC, the composite variability β is estimated by engineering judgement and the following guidance. • For structures and major passive mechanical components mounted on the ground or at low elevations within structures, β is typically from 0.3 to 0.5. • For active components mounted at high elevations in structures, β typically ranges from 0.4 to 0.6. • When in doubt, use β = 0.4. 3. The median capacity is given by, from equation (9.2.4), C1% = C50% e−2.3263 β =⇒ C50% = C1% e 2.3263 β .

(10.6.2)

It should be noted that, from equation (10.6.2), overestimating β (larger β) is unconservative because it increases the median capacity C50% . The basis of the hybrid method is from the observation that the mean annual frequency of failure FE for any SSC is relatively insensitive to β. This mean annual frequency of failure (seismic risk) can be computed with adequate precision from the CDFM capacity Ccdfm and a crude estimate of β. The hybrid method combines the advantages of seismic PSA and SMA methods: ❧ It retains all essential steps of the seismic PSA method and addresses all six questions of the seismic PSA method (listed in Section 10.1) with only a small and tolerable loss of precision relative to the seismic PSA method. ❧ It retains the fundamental simplicity of the EPRI SMA method in only requiring HCLPF capacities to be computed by the CDFM method, as opposed to the seismic PSA method of developing seismic fragilities.

550

Screening Table 1. Set a permissible level of seismic risk from the surrogate element fE0 2. Seismic hazard curve H(aH) = 2× fE0 ⇒ aH 3. Screening level SL < 0.8 × aH

Screened out SSCs Surrogate Element

C1%

Plant damage state fault & event trees from internal event with passive equipment and structures added

System Analysis

a Ground motion parameter a

Mean seismic hazard curve

Annual frequency of exceedance

Seismic Hazard Analysis

H(a)

⇒

Fragility SSC

i CCDFM

SSCi

SE = 2 ×SL Median capacity C50%

⇒

Variability β = 0.3

CDFM

⇒

for remaining SSCi 1. CDFM

2. Estimate variability β

Remaining SSC 2.326β SSC SSCs 3. C50% i = C1% i e

Plant SSCs

Mean conditional probability of failure at GMP level a

Hybrid Method for seismic safety assessment.

pSSCi (a) = P {Capacity of SSCi < a | a } F Figure 10.19

00

00

∞ ∞

(a) dpF(a) dp F da da da da

∞ ∞ d H (a) (a) dH(a) da da ppFF(a) da da

H(a) H(a)

Mean plant damage state fragility

Conditional probability of failure

= =

=− − pfFE =

Seismic Risk Quantification

F

p (a)

a Ground motion parameter a

Mean plant DS conditional prob. of failure at GMP level a

pF(a) = P {Plant DS failure | a }

10.7 estimation of seismic risk from hclpf capacity Table 10.6

AR

Kh

7.00 6.50 6.00 5.00 4.50 4.00 3.50 3.00 2.50 2.00 1.95 1.90 1.75 1.70 1.60 1.55 1.50

1.18 1.23 1.29 1.43 1.53 1.66 1.84 2.10 2.51 3.32 3.45 3.59 4.11 4.34 4.90 5.25 5.68

551

Ratio of FE /H10% .

β 0.3 0.68 0.67 0.66 0.63 0.62 0.60 0.57 0.54 0.51 0.46 0.45 0.45 0.44 0.44 0.45 0.46 0.48

0.4 0.61 0.60 0.59 0.57 0.55 0.53 0.51 0.49 0.46 0.44 0.44 0.45 0.47 0.49 0.55 0.62 0.72

0.5 0.56 0.55 0.54 0.52 0.50 0.49 0.47 0.45 0.44 0.47 0.49 0.50 0.59 0.65 0.87 1.09 1.48

0.6 0.52 0.51 0.50 0.48 0.47 0.46 0.45 0.44 0.45 0.57 0.60 0.64 0.89 1.05 1.74 2.53 4.21

10.7 Estimation of Seismic Risk from HCLPF Capacity In seismic margin assessment (SMA) as presented in Section 10.4, the HCLPF capacities of remaining SSCs are determined using the CDFM method. The plant damage state HCLPF capacity is determined through system analysis using the HCLPF max/min method. The mean annual probabilities of failure (or seismic risks) for SSCs and for plant damage state can be estimated using the following formulation, which can be applied at both the component level and the plant level. Having obtained the HCLPF capacity for an SSC or for the plant damage state, the variability β is estimated. For an individual SSC, β is estimated by engineering judgement and guidance as in Section 10.6. For plant damage state, its fragility curve has a lower β than the individual component fragility curves, it is recommended that β = 0.3 be used for the plant damage state variability. From equations (10.6.1) and (10.6.2), the median capacity is given by C50% = C1% e 2.3263β , C1% ≈ Chclpf . Using equation (9.1.18), let aH = C10% = C50% e β

−1(0.1)

= C1% e2.3263β · e−1.2816β = C1% e1.0447β ,

(10.7.1)

be the ground motion capacity corresponding to 10 % probability of failure on the composite fragility curve of the SSC or plant damage state. From the seismic hazard curve (10.3.5), the corresponding seismic hazard exceedance frequency is −Kh H10% = Ki · C10 % .

(10.7.2)

552

Substituting into equation (10.3.10), the mean annual frequency of failure of the plant damage state is

−Kh 2 (β Kh ) , FE = H10% · F50 % ·e 1

2

(10.7.3)

where, using equation (9.1.18), F50% =

C50% C50% = = e1.2816β . −1 C10% C50% e β (0.1)

Hence, equation (10.7.3) becomes FE 1 1 2 2 −Kh 2 (β Kh ) = e−1.2816β Kh + 2 (β Kh ) . = F50 % ·e H10%

(10.7.4)

The ratio FE / H10% is listed in Table 10.6 over a range of AR (or Kh ) and β values. Over the most common AR (or Kh ) range, it is seen that FE ≈ 0.5× H10% .

(10.7.5)

The use of equation (10.7.5), along with equations (10.7.2) and (10.7.3), to estimate mean annual frequency of failure should be limited to the range of AR (or Kh ) values for which FE / H10% is less than about 0.6, in the shaded cells in Table 10.6, in order to avoid significant error. The so-called simplified hybrid method, presented in Kennedy (1999), is actually an SMA with estimation of seismic risk (at both the component level and plant level) based on HCLPF capacities, as shown in Figure 10.20. This method is intended to provide a quick estimation of the seismic risk due to design changes or modifications to existing plants, and to provide a sanity check on seismic risk results obtained by either the seismic PSA or the hybrid method.

10.8 Numerical Examples − ECI System This example aims to quantitatively demonstrate the procedures of USNRC SMA and seismic PSA by determining the HCLPF capacity and annual frequency of failure of the ECI system described in Section 10.2. The ECI system is assumed to be located on the ground level of the hypothetical site shown in Figure 5.6. The reduced fault tree of the ECI system is shown in Figure 10.11.

10.8.1

Fragility Parameters of Components

For the ECI system, failures of water tank, pump, and control panel are assumed to be governed by anchorage failure, while manual valve, check valve, and motor-operated valve are assumed to be controlled by functional failure during the earthquake.

⇒

SSC

i CCDFM

2. Estimate variability β

1. CDFM

⇒

SSCi

Figure 10.20

SSC

⇒H

SSC 10%

SSC fESSC = 0.5 ×H10%

2. Seismic risk

C10%

SSC

1. Seismic hazard curve

for SSC

Seismic Risk fE

Ground motion parameter a

SSC

C10%

Mean seismic hazard curve

Simplified hybrid method for seismic safety assessment.

pFSSCi (a) = P {Capacity of SSCi < a | a }

Mean conditional probability of failure at GMP level a

C1%

Fragility

for remaining SSCi

CDFM

⇒

Variability β = 0.3

SE = 2 ×SL Median capacity C50%

Surrogate Element

Screened out SSCs

Remaining SSC SSC 2.326β SSCs 3. C50% i = C1% i e

Plant SSCs

3. Screening level SL < 0.8×aH

2. Seismic hazard curve H(aH) = 2× fE0 ⇒ aH

1. Set a permissible level of seismic risk from the surrogate element fE0

Screening Table

SSC H10%

Annual frequency of exceedance

Seismic Hazard Analysis

2

3

4

5

6

Plant damage state fault & event trees from internal event with passive equipment and structures added

1

Damage State

Seismic risk fE for plant damage state

System Analysis

10.8 numerical examples − eci system 553

554 Table 10.7

Component Water tank Manual valve Pump Check valve MOV ECI signal

Fragility parameters of components.

Am ( g) 0.85 1.26 7 1.26 1.2 2

βR 0.15

βU 0.24

0.65

0.65

0.2 0.38

0.35 0.34

βC 0.28 0.4 0.92 0.4 0.4 0.51

Chclpf ( g) 0.45 0.5 0.82 0.5 0.49 0.61

Fragility parameters in terms of PGA for each component in the ECI system, listed in Table 10.7, can be determined using the methods presented in Chapter 9. Based on the fragility parameters, the HCLPF capacity of each component, listed in Table 10.7, can be determined using equation (9.2.1) with Am , βR , βU , or using (9.2.4) with Am , βC .

10.8.2

USNRC SMA

Screening Table Table 10.7 shows that the water tank has the minimum HCLPF seismic capacity of 0.45g PGA. To identify the “weak link” components that limit the seismic capacity of the ECI system, the screening level should be set greater than 0.45g PGA. Based on the screening tables in EPRI-NP-6041-SL (EPRI, 1991a, Tables 2-3 and 2-4), the screening level of the ECI system can be taken as 0.5g PGA.

Review Level Earthquake Based on Alternative 3 of EPRI-NP-6041-SL (EPRI, 1991a; see also page 545), the sitespecific UHS of the hypothetical site shown in Figure 5.6 is chosen as RLE. An AFE of 1×10−4 or less is typically selected for the RLE. PSHA is performed for the hypothetical site. UHS with AFE of 1×10−4 and 1×10−5 are determined and shown in Figure 10.21. Spectral acceleration at 100 Hz is taken as PGA. Figure 10.21 shows that the PGA value from UHS with 1×10−5 AFE is much closer to the screening level of 0.5g PGA; UHS with AFE of 1×10−5 is taken as the spectral shape of the RLE. The horizontal RLE is obtained by anchoring the UHS to PGA at screening level of 0.5g. The vertical RLE is assumed to be two-thirds of the horizontal RLE over the entire frequency range.

HCLPF Seismic Capacity of ECI System Using the screening table, the pumps and ECI signal are eliminated because their HCLPF seismic capacities exceed 0.5g PGA. The fault tree is then further reduced as shown in Figure 10.22. HCLPF seismic capacities of these components are summarized in Table 10.8.

10.8 numerical examples − eci system

555

Spectral acceleration (g)

2

1×10−5

1

1×10−4

0.5g

0.5g

0.3g

0.3g

0.1

0.03 0.2

1

Frequency (Hz)

Figure 10.21

T

10

100

UHS at the hypothetical site.

V

Tank T fails

M

Manual valve V1 fails to stay open

C C1

M1 MV1 fails to open

C2

Check valve CV1 fails to open

M2 MV2 fails to open

M3 MV3 fails to open

Check valve CV2 fails to open

Figure 10.22 The reduced fault tree of ECI system in USNRC SMA. Table 10.8

Component Chclpf ( g)

HCLPF seismic capacities of components.

Water Tank 0.45

Manual Valve 0.5

Check Valve 0.5

MOV 0.49

Annual frequency of exceedance

10 −1 10 −2 Hazard curve 10

−3

10

−4

Linear approximation

1×10−4

−5 10 −5 1×10

2 ×10−6 10

−6

0.01

0.1 Figure 10.23

PGA (g)

0.334g 0.513 g 0.66 g 1

Seismic hazard curve for PGA at hypothetical site.

556 Table 10.9

Fragility Parameters of Components

Component Water tank Manual valve Pump Check valve MOV ECI signal Table 10.10

Am ( g) 0.85 1.26 1.0 1.26 1.2 1.0

βR 0.15

βU 0.24

0.2

0.35

βC 0.28 0.4 0.3 0.4 0.4 0.3

Conditional Probabilities of Failure at PGA = 0.75 g

Component Water Tank Manual Valve Pump Check Valve P (T) P (V) P (P) P (C) pF (0.75g) 0.327 0.097 0.169 0.097

MOV

P (M) 0.12

ECI Signal P (S ) 0.169

Referring to Figure 10.22, HCLPF max/min method is used to calculate HCLPF seismic capacity of the ECI system: ❧ Event C consists of two check valves connected by an “AND” gate; thus HCLPF seismic capacity of event C is taken as the greater one of these two valves. Because HCLPF seismic capacities of two check valves are both equal to 0.5g PGA, HCLPF seismic capacity of event C is taken as 0.5g PGA. ❧ Event M consists of three MOV connected by an “AND” gate; thus HCLPF seismic capacity of event M is taken as the maximum one of these three valves. Because HCLPF seismic capacities of three MOV are all equal to 0.49g PGA, HCLPF seismic capacity of event M is taken as 0.49g PGA. ❧ Events T, V, C, and M are connected by an “OR” gate; thus HCLPF seismic capacity of the ECI system is equal to the minimum one of these four events. From Table 10.8, HCLPF seismic capacities of the water tank and manual valve are 0.45g PGA and 0.5g PGA, respectively. Therefore, HCLPF seismic capacity of the ECI system is equal to 0.45g PGA. The results show that the water tank and MOV are the “weak link” components in the ECI system. To ensure that the ECI system meets the screening level, efforts should be made primarily on improving the water tank and MOV.

10.8.3

Seismic PSA

Seismic Hazard Curve PSHA is performed to determine the seismic hazard curve for PGA, as shown in Figure 10.23, from which PGA value corresponding to any AFE can be obtained.

10.8 numerical examples − eci system

557

Selection of Review Level Earthquake Screening level can be taken following the procedure in Section 10.5: 1. Establish permissible level of seismic risk FE0 = 1×10−6 , which is contributed by the surrogate element. 2. Determine the PGA value corresponding to AFE of H = 2 FE0 = 2×10−6 from seismic hazard curve in Figure 10.23, i.e., aH = 0.669g PGA. 3. Based on aH, screening level SL should satisfy (10.5.4), i.e., SL  0.8×aH = 0.8×0.669g PGA = 0.535g PGA.

(10.8.1)

Hence, SL = 0.5g PGA is selected. UHS with AFE of 1×10−5, as shown in Figure 10.21, is taken as the spectral shape of the RLE. Anchoring the UHS to PGA at 0.5g gives the horizontal RLE; the vertical RLE is assumed to be two-thirds of the horizontal RLE over the entire frequency range.

Surrogate Elements In seismic PSA, the screened-out elements need to be replaced by surrogate elements accounting for seismic risk contributed from these elements. Based on the screening level and from equation (10.5.1), pump and control panel are replaced by surrogate elements with median seismic capacity Am and logarithmic standard deviation β Am = C50% = 2× SL = 2×0.5g PGA = 1.0g PGA,

β = 0.3.

(10.8.2)

Seismic Fragility Curve of ECI System Seismic Fragilities of Components Fragility parameters of the ECI system, considering surrogate elements (in shaded cells), are listed in Table 10.9. Using these parameters, composite seismic fragility of each component is calculated using equation (9.1.16); the fragility curves are shown in dashed lines in Figure 10.24. Seismic Fragility of ECI System For a given PGA = a value, conditional probability of failure pF (a) of each component can be obtained from the corresponding fragility curve (Figure 10.24) or using equation (9.1.16). Taking a = 0.75g as an example, pF (a) values are given in Table 10.10. Seismic fragility pFeci (a) of the ECI system is then determined by equation (10.2.9). Given PGA = 0.75g, pFeci (0.75g) is calculated as    pFeci (0.75g) = P E  PGA = 0.75

558

1 ECI system

0.9

MOV

Conditional probability of failure

0.8 Manual valve & check valve 0.7 Surrogate element 0.6 0.526

Water tank

0.5 0.4 0.327

0.3

0.2 0.169 0.12 0.1 0.097 0

0.5

0

0.75 0.549

Figure 10.24

1

1.5 PGA ( g)

2

2.5

3

Seismic fragility curves of components in the ECI system. 

= 1 − (1−0.327)(1−0.097)(1−0.169)× 1 − (0.169+0.097−0.169×0.097) 

×(0.169+0.097−0.169×0.097) ×(1−0.12×0.12×0.12) = 0.526.

(10.8.3)

Repeating the analysis for PGA values from lower bound to upper bound results in the mean seismic fragility curve of the ECI system, as shown in solid line in Figure 10.24.

Annual Frequency of Failure of ECI System Having obtained the seismic hazard curve of the site and the system seismic fragility curve, annual frequency of failure of the ECI system can be calculated by equation (10.3.4). Taking the lower bound of 0.1g and upper bound of 5g and uniformly discretizing the domain into 100 intervals in the base-10 logarithmic scale, annual frequency of failure of the ECI system is calculated as FEeci =

100  K=1





pFeci(aK ) H(aK ) − H(aK+1 ) = 3.079×10−6 .

(10.8.4)

10.8 numerical examples − eci system

559

Estimation of Seismic Risk from HCLPF Capacity From the seismic hazard curve in Figure 10.23, the PGA values corresponding to H(a1 ) = 10−4 and H(a2 ) = 10−5 AFE are a1 = 0.334g and a2 = 0.513g, respectively.

The hazard curve can be reasonably approximated by a straight line. From equations (5.2.19) and (5.2.20), one has Kh =

1 1 = = 5.372, log10 a2 − log10 a1 log10 0.513− log10 0.334

Ki = H(a1 ) a1Kh = 10−4 ×0.3345.372 = 2.772×10−7 .

(10.8.5) (10.8.6)

From the fragility curve of the ECI system (Figure 10.24), the PGA capacity with eci = 0.549g. Using equation (10.7.2), 10 % probability of failure is C10 % eci −Kh H10% = Ki · (C10 = 2.772×10−7 ×0.549−5.372 = 6.926× ×10−6 . %)

(10.8.7)

The annual frequency of failure of the ECI system can be estimated using equation (10.7.4) as 1

FEeci = H10% · e−1.2816β Kh + 2 (β Kh )

2

1

= 6.926×10−6 ×e−1.2816 × 0.3 × 5.372+ 2 (0.3 × 5.372) = 3.217×10−6 . 2

(10.8.8)

The relative error compared with the frequency of failure 3.079×10−6 obtained from integrating fragility and hazard using equation (10.8.4) is 4.5 %. If the approximation (10.7.5) is used, then FEeci ≈ 0.5× H10% = 0.5×6.926×10−6 = 3.463×10−6 .

(10.8.9)

Lognormal Approximation Note that the fragility curve of the ECI system does not follow exactly the lognormal distribution. Suppose that the fragility given by equation (10.8.3), i.e., a = 0.75g, pFeci (a) = 0.526, is known. The median capacity given by equation (9.1.17) is −1 ( peci ) F

βC

Aeci m = a·e

−1 (0.526)

= 0.75×e0.3×

= 0.735 (g),

(10.8.10)

eci is, from equation (9.1.18), and C10 % eci eci βC

C10 % = Am · e

−1 ( peci ) F

= 0.735×e0.3×

−1 (0.1)

= 0.501 (g),

(10.8.11)

eci = 0.549g) which is different from the value obtained from the fragility curve (C10 %

with a relative error of 8.7 %. From equation (10.7.2), eci −Kh H10% = Ki · (C10 = 2.772×10−7 ×0.501−5.372 = 1.139× ×10−5 . (10.8.12) %)

560

Using equation (10.7.4), the annual frequency of failure of the ECI system is 1

FEeci = 1.139×10−5 ×e−1.2816 × 0.3 × 5.372+ 2 (0.3 × 5.372) = 5.291×10−6 , 2

(10.8.13)

or, if the approximation (10.7.5) is used, FEeci ≈ 0.5× H10% = 0.5×1.139×10−5 = 5.695×10−6 .

10.8.4

(10.8.14)

Summary

In this section, USNRC SMA and seismic PSA are performed separately for a simplified ECI system. Site-specific UHS with AFE of 1×10−5 anchoring to PGA at 0.5g PGA is chosen as review level earthquake for both assessment methods. USNRC SMA results show that HCLPF capacity of the ECI system cannot meet seismic margin requirement. Water tank and motor-operated valves (MOV) are found to be “weak link” components. Modification of the tank and MOV are required to ensure their HCLPF seismic capacities exceed the screening level. Seismic PSA results indicate that mean annual frequency of failure FEeci of the ECI

system is equal to 3.08×10−6 . For a large early release accident, permissible FE lies in

the range between 1×10−6 and 1×10−5 . Therefore, FEeci of the ECI system satisfies

the requirement. It can be seen that USNRC SMA and seismic PSA provide opposite conclusions for the ECI system. ❧

❧

In this chapter, methods of system analysis and various methods of seismic risk quantification are presented. ❧ Engineering problems often involve multiple failure modes. Occurrence of any one of the potential failure modes will constitute failure or non-performance of SSCs. In a complex multicomponent engineering system, the possibilities of failure of the system are so involved that a systematic scheme is necessary. ❧ A fault tree is a systematic approach of identifying various faults and their interactive effects on a failure event. A fault tree diagram decomposes the main failure event (top event) into unions and intersections of subevents. The process of decomposition continues until the probabilities of the subevents can be evaluated as single-mode failure probabilities. A fault tree analysis includes a quantitative evaluation of the probabilities of the various faults or failure events to yield the probability of the main failure event (top event). It may serve to identify critical events that contribute significantly to system failure and reveal weak links in the system.

10.8 numerical examples − eci system

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❧ Event trees is a systematic scheme for identifying the possible consequences of various failure events. ❧ In seismic margin assessment (SMA) of an NPP, it is required to demonstrate that the plant can withstand the RLE with high confidence and to identify seismic vulnerabilities and any potential weak links. Seismic capacity of SSCs on the success path is estimated in terms of HCLPF values based on FRS. SSCs with HCLPF values lower than the RLE are flagged, and recommendations to improve their HCLPF values are made if they do not meet their safety targets. This process ensures that the NPP has sufficient seismic margin. ❧ Seismic probabilistic safety assessment (Seismic PSA) or seismic probabilistic risk assessment (SPRA) is the formal process in which the randomness and uncertainty in seismic ground motion input, structure responses, and material capacity variables are propagated through an engineering model leading to a probability distribution or frequency of occurrence of failure or other adverse consequences, such as CDF or LERF. Seismic PSA is a useful tool to identify weak links in a system or facility, which can guide the efficient allocation of funds to strengthen or modify an existing NPP. It can be used as part of the design process to size members to comply with a performance standard.

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Basics of Normal and Lognormal Distributions

A.1 Normal Distribution Normal distribution is the most important distribution in statistical analysis. Many physical quantities, which result from the superimposition of a large number of random factors, are often considered to be normally distributed. This is a consequence of the result known as the central limit theorem, which states that, under very general conditions, a random variable that occurs as the sum of many smaller independent random variables, each of which is almost negligible in itself, is approximately normally distributed, whatever the distributions of the component variables are. A normally distributed random variable with expected value (mean) μ and standard deviation σ is denoted as X ∼ N (μ, σ 2 ). Its probability density function is given by   (x − μ)2 1 exp − FX (x) = √ , −∞ < x < +∞. (A.1.1) 2σ 2 2π σ As shown in Figure A.1, the normal probability density function is ‘‘bell-shaped’’, symmetrical about the mean μ, and approaches zero at large deviations from μ. The probability distribution function is  x   FX (x) = P X  x =

−∞

√

  (x − μ)2 exp − dx. 2σ 2 2π σ 1

(A.1.2)

It is not possible to integrate equation (A.1.2) for any values of x. To avoid this difficulty, standard normal distribution is used. 562

a.1 normal distribution

563

fX(x)

0.4

μ=0, σ=1 0.3

μ=–4, σ=1.5

μ=0, σ =2 0.2

μ=2, σ=2.5

μ=0, σ=3

0.1

x –10

–8

–6

–4

0

–2

2

4

6

8

10

Figure A.1 Probability density functions of normal distribution. fZ(z)

fZ(z)

P {z1 0, 2β 2 FX (x) = 2π β x ⎪ ⎩ 0, x  0.

(A.2.1)

Probability distribution function of X is given by 



 lnx − α  . β

P X< x =

(A.2.2)

The mean value X¯ and standard deviation σX can be obtained from the logarithmic mean and logarithmic standard deviation 2 X¯ = E[ X ] = e α+β /2 ,

2 2 2 2 σX2 = Var(X) = e 2 α+β e β − 1 = X¯ e β − 1 .

(A.2.3)

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appendix a. basics of normal and lognormal distributions

Given the mean X¯ and standard deviation σX , the logarithmic mean and logarithmic standard deviation can be determined from the following equations β 2 = ln(1 + CoV2 ),

CoV =

σX , X¯

(A.2.4)

α = ln X¯ − 12 β 2 .

For a lognormal distribution, the median characterizes the central tendency, while the logarithmic standard deviation β is a measure of the dispersion of the distribution. Let Xm = X50% be the median, then from equation (A.2.2) one has 







P X < Xm = P X < X50% =

 lnX − α  m = 0.5, β

(A.2.5)

which yields α = μln = lnXm ,

or

Xm = X50% = e α = e μln .

(A.2.6)

In practice, there are cases that the lower limit of a random variable, such as the strength of material strength, or the capacity of a flood channel, or the acceleration of ground motion, is of engineering interest. A lower limit above which the variable may lie, properly determined on a probabilistic basis, is often used as a critical parameter. The probability that the variable will lie above the lower limit is called confidence level. Denote Q (shaded area in Figure A.4) as the confidence level and XQ the corresponding critical value of the lognormal random variable X. Letting ZQ =

lnXQ − α

(A.2.7)

β

be the critical value of the standard normal random variable Z corresponding to the confidence level Q, i.e., 







P X > XQ = P Z > ZQ = 1 − (ZQ ) = Q,

(A.2.8)

one has, using equations (A.2.7) and (A.2.6), XQ = eα+βZQ = Xm e βZQ , or, using equations (A.2.8), XQ = Xm e β

−1 (1 − Q )

= Xm e−β

−1 (Q )

=⇒

−1 (Q )

XQ = e μln − σln

.

(A.2.9)

Therefore, for confidence levels Q = 1 %, 5 %, 50 %, 95 %, and 99 %, the lower limits of random variable XQ are given by, respectively, X1% = Xm e−β

−1 (0.01)

= Xm e2.3263 β ,

X99% = Xm e−β

−1 (0.99)

= Xm e−2.3263 β ,

a.2 lognormal distribution −1 (0.05)

X5% = Xm e−β

−1 (0.50)

X50% = Xm e−β

567

= Xm e1.6449 β ,

X95% = Xm e−β

−1 (0.95)

= Xm e−1.6449 β ,

= Xm .

(A.2.10)

The product of lognormally distributed random variables is also lognormally distributed. For example, if X1 , X2 , and X3 are independent lognormally distributed random variables, and X=C

X1a · X2B , X3c

(A.2.11)

where a, B, c, and C are constants, then X is also a lognormal random variable, whose logarithmic mean αX , median Xm , and logarithmic standard deviation βX are αX = lnC + a αX1 + B αX2 − c αX3 , βX2

=

2 a2 βX1

2 + B2 βX2

Xm = C

a · XB X1m 2m , c X3m

(A.2.12)

2 + c2 βX3 ,

where αX1 , αX2 , and αX3 are the logarithmic means, X1m , X2m , and X3m are the median values, and βX1 , βX2 , and βX3 are the logarithmic standard deviations of X1 , X2 , and X3 , respectively.

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Digital Signal Processing Some important topics in digital signal processing that are particularly relevant to earthquake engineering are briefly introduced in this Appendix. The objective is to highlight the important concepts, features, and challenges that may be encountered in processing real earthquake records and generating artificial earthquake time-histories. Software packages, such as Matlab, are available to perform many of the essential tasks in digital signal processing.

B.1 Sampling Consider a continuous signal X(t) with Fourier transform (spectrum) X( F ). Discretize it at regular interval of s , i.e., at time instances tn = ns , −∞ < n < +∞. Fs = 1/s is called the sampling rate. The subscript “s” stands for “sampling”. Hence, a sequence of discrete values is obtained Xn = X(tn ) = X(ns ),

−∞ < n < +∞.

The discrete-time Fourier transform (spectrum) of the discrete sequence is Xs ( F ). The most important question on sampling is how fast must a given continuous signal be sampled (discretized) in order to preserve its desired information characteristics. Consider a harmonic function X(t) = sin(2π F0 t) of frequency F0 (Hz). Its sampled value at tn = n s is Xn = X(tn ) = sin(2π F0 · n s ) = sin(2π F0 · n s + 2mπ ),    m  ns . = sin 2π F0 + ns 568

m = integer,

b.1 sampling

569

X(t)

10 Hz

1.0

130 Hz

f0 = 70 Hz

0.5

t

0.0

0.02

0.04

0.06

0.08

0.1

−0.5 −1.0

s

s

1 = 0.01667 s = 60

s

s

s

Figure B.1 Aliasing in sampling (time domain). Xs( f ) 60Hz

60Hz

60Hz

f −75

−50

−30Hz

−15

60Hz

10

45

30 Hz

70

105

130

60Hz

60 Hz

Figure B.2 Aliasing in sampling (frequency domain).

Letting m be an integer multiple of n, i.e., m = Kn, yields 



Xn = sin 2π( F0 +K Fs )ns ,

K = 0, ±1, ±2, . . . ,

(B.1.1)

which implies that if Xn is a sampled value of a harmonic function of frequency F0 , then it is also exactly a sample value of harmonic functions of frequencies F0 +K Fs , K = ±1, ±2, . . . . Therefore, equation (B.1.1) means that a sequence of sampled values X0 , X1 , . . . , XN representing a harmonic function of frequency F0 also represents exactly harmonic functions of frequencies F0 +K Fs , K = ±1, ±2, . . . . As a result, when sampling at rate Fs , it is impossible to distinguish between a harmonic function of frequency F0 and a harmonic of frequency F0 +K Fs , K = ±1, ±2, . . . . As an illustration, consider a sinewave with frequency F0 = 70 Hz as shown in Figure B.1. If it is sampled at rate Fs = 60 Hz, then the sampled values could come from the sinewave with frequency F0 − Fs = 70−60 = 10 Hz, or from the sinewave with frequency F0 + Fs = 70+60 = 130 Hz. That means, when sampled at 60 Hz, it is impossible to distinguish sinewaves with frequencies 10 Hz, 70 Hz, 130 Hz. If this discrete sequence X1 , X2 , . . . , XN is applied to excite an SDOF oscillator with frequency 10 Hz, then the sinewave would be recognized by the oscillator to have a frequency of 10 Hz, and resonance will occur. In other words, a sinewave of 70 Hz with the real name “70 Hz” is recognized by its alias of “10 Hz”. In fact, all sinewaves with frequencies 70+60K, K = 0, ±1, ±2, . . . , will be recognized as having a frequency of 10 Hz when sampled at 60 Hz.

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appendix b. digital signal processing X( f ) Continuous spectrum

f f0 Xs( f ) Discrete spectrum fs

fs

fs

fs

f f0 − f s

f0 −2 fs fs

fs 2 fs

f0

fs 2

f0 + f s

fs

f0 +2 fs fs

Figure B.3 Continuous and discrete spectra.

In general, the phenomenon that a signal appears to have a lower frequency than it actually has is called aliasing. Figure B.2 shows aliasing in the frequency domain. It is clearly seen that, when sampled at 60 Hz, a 10 Hz harmonic is alias of −50 Hz, 70 Hz, and 130 Hz harmonics, and a −15 Hz harmonic is alias of −75 Hz, 45 Hz, and 100 Hz harmonics. Equation (B.1.1) further implies that the spectrum of a discrete sequence of sampled values contains periodic replications of the original continuous spectrum. The period between these spectral replicas in the frequency domain is Fs , and the spectral replicas repeat throughout the frequency domain. For example, consider a signal with continuous spectrum as shown in Figure B.3. Suppose that it is sampled at rate Fs . A harmonic component with frequency F0 will also appear at frequencies . . . , F0 −2 Fs , F0 − Fs , F0 + Fs , F0 +2 Fs , . . . as shown, because sampling at rate Fs cannot distinguish harmonics with frequencies F0 ±K Fs , K = 0, ±1, ±2, . . . . As a result, the continuous spectrum repeats periodically with period Fs in the discrete spectrum.   If the frequency range of interest is restricted in  F   F max , it is important to know what harmonic components will be aliased into this frequency band.

Nyquist-Shannon Sampling Theorem If a continuous signal X(t) is band-limited with its highest frequency component being B, then X(t) can be completely recovered from its sampled values if the sampling rate Fs is greater than the Nyquist rate (frequency) F Nyquist = 2B, i.e., Fs  F Nyquist or Fs  2B. Referring to Figure B.4, for a continuous signal with frequency band −B  F  B, if the signal is sampled at rate Fs  2B, then adjacent spectral replicas are separated in

b.1 sampling

571 X( f )

Continuous spectrum

f −B

B

Xs( f )

Discrete spectrum fs

2B

f fs −B 2 fs

fs

B f s 2 fs

Xs( f ) Aliasing

Aliasing

Aliasing

fs

Discrete spectrum fs Aliasing

Aliasing

2B

Aliasing

f −B fs

fs 2 fs

fs 2 fs

B fs

Figure B.4 Sampling of band-limited signal. X( f )

Continuous spectrum

Noise

Noise

f −B Xs( f ) Aliasing

B Discrete spectrum Aliasing

f

fs

fs 2 fs

−B

B f s 2 fs

fs

Figure B.5 Sampling of band-limited signal with high-frequency noise.

the discrete spectrum, and there is no distortion or corruption in the frequency band −B  F  B. However, if Fs < 2B, adjacent spectral replicas overlap: portions of the spectral replicas are combined with the original spectrum, resulting in aliasing errors. According to the Nyquist-Shannon Sampling Theorem, if the highest frequency component of interest of a continuous signal is F max , then the signal must be sampled at rate Fs  2 F max . However, in the following two cases • the signal is not band-limited having components with frequency beyond F max ,

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appendix b. digital signal processing

• the signal is band-limited but there are high-frequency noises beyond the bandlimit F max = B, even when the signal is sampled at rate Fs  2 F max , there will be aliasing errors due to these high-frequency components or noises, as shown in Figure B.5. To resolve this problem, the signal should be passed through a low-pass filter to remove components with frequencies higher than F max before sampling (see Section B.4 for digital filters).

B.2 Fourier Series and Fourier Transforms B.2.1 Fourier Series A periodic function X(t) of period T can be expressed in Fourier series in the complex form X(t) =

∞  K=−∞

CK e i Kωt ,

ω=

2π . T

(B.2.1)

Using the Euler’s formula e i θ = cosθ + i sinθ, equation (B.2.1) can be written as X(t) = C0 + = C0 + = C0 + =

1 2 a0

∞  K=1 ∞   K=1 ∞  

CK e i Kωt + C−K e−i Kωt

CK ( cosKωt + i sinKωt) + C−K ( cosKωt − i sinKωt) (CK +C−K ) cosKωt + i (CK −C−K ) sinKωt

K=1 ∞ 

+

K=1





aK cosKωt + BK sinKωt ,

(B.2.2)

which is Fourier series in the real form, and a0 = 2C0 ,

aK = CK +C−K ,

BK = i (CK −C−K ),

CK =

1 2

(aK − iBK ).

¯ = X(t) and On the other hand, because X(t) is real, X(t) ¯ = C¯ 0 + X(t)

∞  K=1

C¯ K e−i Kωt + C¯ −K e i Kωt .

(B.2.3)

Comparing the coefficients leads to C¯ 0 = C0 ,

C¯ K = C−K ,

C¯ −K = CK .

To determine the coefficients of the Fourier series, multiply equation (B.2.1) by e−i nωt

and integrate with respect to t from 0 to T = 2π/ω:  T  T ∞  −i nωt X(t) e dt = CK e i (K−n)ωt dt = Cn T. 0

K=−∞

0

(B.2.4)

b.2 fourier series and fourier transforms

573

Hence, CK =

1 T

aK =

2 T

BK =

2 T



T

X(t) e−i Kωt dt,

(B.2.5)

0



T

X(t) cosKωt dt =

0



T

X(t) sinKωt dt =

0

2 T 2 T



T

2πK t dt, T

(B.2.6)

2πK t dt. T

(B.2.7)

X(t) cos 

0 T

X(t) sin 0

B.2.2 Fourier Transform Consider function X(t), which can be real or complex and is not necessarily periodic. Select a time 2T that is large compared to the duration of X(t), and construct an artificially periodic function: X2T (t) = X(t),

0  t  2T, (B.2.8)

X2T (t±2T) = X2T (t),

which is periodic with period 2T. Then the periodic function F2T (t) can be represented as a Fourier series X2T (t) =

∞  K=−∞

ˆ CK e i K ωt =

  ∞ 1  2π ˆ , (2TCK ) e i (K ω)t 2π K=−∞ 2T

where 1 CK = 2T

Let ω = K ω, ˆ ω = ωˆ =



T

−T

ωˆ =

2π , 2T

ˆ X2T (t) e−i K ωt dt.

(B.2.9)

(B.2.10)

2π 2π . When T→∞, ω = →0. The discrete spectrum 2T 2T

becomes continuous, and ∞  K=−∞

 →

+∞

and

−∞

X2T (t) → X(t).

Equations (B.2.9) and (B.2.10) become  +∞   1 lim (2TCK ) e i ωt dω, T→∞ 2π −∞ T→∞  ∞ X(t) e−i ωt dt = X(ω), lim (2TCK ) =

X(t) = lim X2T (t) =

T→∞

−∞

which result in the Fourier transform (FT) and the inverse Fourier transform (IFT) FT

X(ω) = F X(t) = 





∞

−∞

X(t) e−i ωt dt,

(B.2.11)

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appendix b. digital signal processing

X(t) =

IFT

F −1





X(ω) =

1 2π



+∞

−∞

X(ω)e i ωt dω.

(B.2.12)

  Because X( F ) is generally complex, a plot of the modulus X( F ) as a function of frequency F is called a Fourier amplitude spectrum (FAS). Using equation (B.2.12), the total energy e of X(t) is given by    +∞  +∞  +∞  +∞  2 1 i ωt   ¯ X(ω)e dω dt X(t) dt = e= X(t) X(t)dt = X(t) 2π −∞ −∞ −∞ −∞  +∞   +∞  +∞ 1 1 i ωt = X(ω) X(t) e dt dω = X(ω) X¯ (ω)dω, 2π −∞ 2π −∞ −∞ which gives Parseval’s theorem

e=



+∞ 

 X(t)2 dt = 1 2π

−∞



+∞ 

−∞

  X(ω) 2 dω.

Parseval’s theorem (B.2.13) states that the total energy

(B.2.13)

e in a continuous signal is the

same whether it is evaluated in the time domain, in terms of X(t), or in the frequency domain, in terms of X(ω). Parseval’s theorem also describes how the energy in the 2  signal is distributed over the frequency range by the function  X(ω) , which is also called the energy spectral density (ESD) function.

Convolution Integral Consider the convolution of functions X(t) and Y(t) defined as  +∞ X(τ ) Y(t−τ ) dτ. C(t) = X(t) ∗ Y(t) = −∞

Its FT is

C (ω) = F



C(t) =

Letting s = t−τ yields  +∞  C (ω) =  =

−∞

+∞ −∞





+∞

−∞

+∞  +∞

−∞

−∞

(B.2.14)



X(τ ) Y(t−τ ) dτ e−i ωt dt.

(B.2.15)

X(τ ) Y(s) e−i ω(s+τ ) dτ ds −i ωτ

X(τ ) e

 dτ

+∞ −∞

Y(s) e−i ωs ds = X(ω) Y(ω).

Similarly, consider the convolution of the FT X(ω) and Y(ω):  +∞ 1 1 X(ω) ∗ Y(ω) = C (ω) = X(ν) Y(ω −ν) dν. 2π 2π −∞

(B.2.16)

(B.2.17)

b.2 fourier series and fourier transforms III − s(t)

575

i− IIFs( F ) δ(t −nS)

δ( f −n fS )

⇒⇒ f

t S

fS =1/S

Figure B.6 Fourier transform pair of the Dirac comb. WT ( f ) T

WTB(t) 1

⇒⇒

T sinc(πT f ) f

t −T/2

T/2 −1/T 1/T WB(t) WBB( f )

B B sinc(πBt)

1

⇒⇒

f

t −B/2

B/2

−1/B 1/B

Figure B.7 Fourier transform pair of the “box-car” .

Its IFT is

C (t) = F

−1





1 C (ω) = 2π



+∞ 

−∞

1 2π



+∞

−∞



X(ν) Y(ω −ν) dν e i ωt dω. (B.2.18)

Letting ω = ν + yields     1 2 +∞ +∞ C (t) = X(ν) Y() e i (ν + )t dν d 2π −∞ −∞  +∞  +∞ 1 1 = X(ν) e i ν t dν Y() e i t d = X(t) Y(t). 2π −∞ 2π −∞

(B.2.19)

Hence, there are the following FT and IFT pairs for the convolution integrals X(t) ∗ Y(t)

⇐==⇒

X(ω) Y(ω),

X(t) Y(t)

⇐==⇒

1 X(ω) ∗ Y(ω). 2π

(B.2.20)

δ(t−n),

(B.2.21)

The Dirac Comb The Dirac comb is defined as III −(t) = 

∞ 

n=−∞

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appendix b. digital signal processing

in which III − is the Cyrillic letter “Shah” used for its likeness to a comb. III −(t) is a periodic function with period ; in one period −/2 < t < /2, the functions is III −(t) = δ(t). III −(t) can be expanded in Fourier series III −(t) = where 1 

CK =

∞ 

2π , 

(B.2.22)

· δ(t) e−i Kωt dt = 1.

(B.2.23)

K=−∞



/2

−/2

CK e i Kωt ,

ω=

Hence, the Dirac comb can be rewritten as III −(t) = Using

F



∞ 

e i 2π K t/ .

K=−∞

(B.2.24)



e i Bt = 2π δ(ω −B), the FT of III −(t) is   +∞  +∞  ∞  i 2πK t/ −i ωt −i ωt (t) e dt = i− II(ω) = III e dt e − =

−∞

−∞

∞ 

  2πK . δ ω−  K=−∞

K=−∞

F





e i 2π K t/ = 2π

K=−∞

∞ 

(B.2.25)

Hence, the FT of a Dirac comb is another Dirac comb as shown in Figure B.6, with Fs = 1/s , III −s(t) = s

∞ 

δ(t−ns )

⇐==⇒

n=−∞

i− IIs( F ) = 2π

∞ 

δ( F −n Fs ).

(B.2.26)

n=−∞

The “Box-Car” Function Consider the “box-car” function in the time domain  T T 1, − < t < , b 2 2 WT (t) = 0, otherwise, in which the superscript “b” stands for “box-car” . Its FT is  +∞  +T/2 b −i ωt WT (ω) = WT (t) e dt = e−i ωt dt = −∞

−T/2

 T 1  −i ω T 2 ωT 2 − ei ω 2 = = ω sin e , 2 −i ω

(B.2.27)

1 −i ωt +T/2 e  t=−T/2 −i ω (B.2.28)

which can be written as

WT ( F ) = T

sinπ T F = T sinc(π T F ), πT F

(B.2.29)

b.2 fourier series and fourier transforms

577

sinx where sinc(x) = x is the “sinc” function. Similarly, consider the “box-car” function in the frequency domain  B B 1, − < F < , b 2 2 WB ( F ) = 0, otherwise.

Its IFT is WB (t) =

1 2π



+∞ −∞

WBb ( F ) e i 2π F t d(2πF ) =



+B/2 −B/2

(B.2.30)

e i 2π F t dF = B sinc(πBt). (B.2.31)

In summary, as shown in Figure B.7, the FT and IFT pairs of the “box-car” are:  WTb (t)

=

1,

−

T T Fc ,   i.e., frequency components with  F  > Fc are removed. H( F ) given by equation (B.4.5) is the “box-car” function given by equation (B.2.30); from equation (B.2.32), its IFT H(t) and the discrete values of Hn are H(t) = 2 Fc sinc(2π Fc t), Hn = 2 Fc sinc(2π Fc · ns ),

(B.4.6) −∞ < n < +∞.

(B.4.7)

b.4 digital filters

585 100

hnm

tn −0.20

−0.15

−0.125

−0.10

0

−0.05

0.05

−20

0.125

0.10

0.15

0.20

wnm-blackman

1.0 0.8 0.6 0.4

n= −M= −25 −0.20

−0.15

−0.125

n = M=25

0.2

−0.10

−0.05

tn 0

0.05

100

0.10

0.125

0.15

0.20

0.10

0.125

0.15

0.20

hnm-blackman = hnm wnm-blackman

tn −0.20

−0.15

−0.125

−0.10

0

−0.05

0.05

−20

H m ( fk ) H( f )

Gibbs phenomenon 1.0

Transition region

0.8

Transition region

0.6 0.4

H m-blackman ( fk )

0.2 −100

−80

−60

−50

−40

−20

0

Figure B.10

20

40

50

60

fk

80

100

FIR low-pass filter.

The “sinc” function is symmetric about n = 0 and infinite in extent. Suppose only a finite-length section of Hn is selected as  Hn , −M  n  M, m Hn = 0, otherwise,

(B.4.8)

which is equivalent to multiplying the sequence Hn by a rectangular or “box-car” window extending from −M to M. The frequency response of the filter is given by the DFT of equation (B.2.47)

HK =

N/2  n=−N/2+1

Hn e−i 2π Kn/N =

H( F K ) = H(K F ) = s HK ,

M  n=−M

F =

Hnm e−i 2πKn/N ,

1 , Ns

K =−

N N N +1, . . . , −1, . 2 2 2

(B.4.9) (B.4.10)

The results of Hnm and H m ( F K ) are shown in Figure B.10 for sampling interval s = 0.005 s, duration T = 40 s, N = 8000, F = 1/T = 0.025 Hz, Fc = 50 Hz, M = 25. It is seen that there are large ripple-like oscillatory errors in H( F K ), known as Gibbs   phenomenon, which increase in magnitude close to the discontinuities at  Fc  = 50 Hz.

586

appendix b. digital signal processing

Window functions wn are usually multiplied to Hn to reduce Gibbs phenomenon. For example, the Blackman window function is given by ⎧     ⎨0.42 − 0.5 cos 2π n+M + 0.08 cos 4π n+M , −M  n  M, m-blackman 2M 2M wn = ⎩ 0, otherwise, which is shown in Figure B.10. The integer n corresponds to discrete time tn = ns . The resulting filter becomes Hnm-blackman = Hnm wnm-blackman .

(B.4.11)

In the time domain, the integer n corresponds to discrete time tn = ns . The frequency response of the filter H m-blackman ( F K ) is also shown in Figure B.10. In the frequency domain, the integer K corresponds to discrete frequency F K = KF . It is seen that Gibbs phenomenon has been significantly reduced by the Blackman window. However, a side effect of reducing Gibbs phenomenon by applying a window function is an increased transition region. There are also other popular window functions, such as the Chebyshev and the Kaiser window functions.

B.5 Resampling Consider a continuous signal X(t) with FT X( F ). It has been sampled with interval s (sampling rate Fs ) to yield a discrete sequence Xn = X(tn ) = X(n s ),

−∞ < n < +∞,

(B.5.1)

and the DTFT is Xs ( F ). Sometimes, it is necessary to change the sampling interval to  s to yield

Xn = X(tn ) = X(n  s ),

☞

−∞ < n < +∞.

(B.5.2)

It should be noted that the new sampling rate Fs = 1/ s must still satisfy the Nyquist sampling requirement Fs  2 F max .

Sampling Rate Reduction Suppose the sampling rate is reduced by an integer factor D, i.e., Fsd = Fs /D or ds = Ds to yield

Xnd = X d(n ds ) = X(n · Ds ) =

∞ 

X(r s ) δ(r−n D),

(B.5.3)

r=−∞

where the superscript “d” stands for “downsampling” or “decimation”. The case of downsampling with D = 3 is illustrated in Figure B.11.

b.5 resampling

587

X(ns)

Original sampling s

Continuous signal X(t) 0

1

2

3

n

4

X d(nds )

Downsampling D =3

ds = Ds n

0

1

2

X u(nus )

Upsampling U = 2

us = s /U n

0

1

2

3

4

Resampling.

Figure B.11

The DTFT of the downsampled signal Xnd , −∞ < n < +∞, is  +∞ ∞  d Xsd( F ) = X d(t) e−i 2π F t dt = X d(n ds ) e−i 2πF · ns ds = =

−∞ ∞ 

n=−∞



n=−∞ ∞ 

∞ 

X(r s ) δ(r−n D) e−i 2π F · ns ds d

r=−∞



X(r s )

r=−∞

Note that



∞ 



δ(r−n D) e−i 2π F · (r/D)s ds . d

(B.5.4)

n=−∞ ∞ 

δ(r−n D) =

n=−∞

D−1 1 

D

e i 2π mr/D ,

(B.5.5)

m=0

because it is a periodic function of period D and can be expressed in Fourier series. Substituting equation (B.5.5) into equation (B.5.4) gives

Xsd( F ) = s



∞ 

=

X(r s )

r=−∞ ∞ 

r=−∞



D−1

Xr

D−1 1 

D

 e

i 2π mr/D

e−i 2πF · (r/D)s (Ds ) d

m=0

e−i 2π [−m/(Ds)+ F ] (r s) =

m=0



D−1

 s

m=0

∞  r=−∞

 Xr e−i 2π [ F−m Fs /D] (r s) ,

which yields

Xsd( F ) =



D−1

m=0

  D −1 F Xs F −m Ds = Xs ( F −m Fsd ).

(B.5.6)

m=0

Xsd( F ) results from superimposing replicas of Xs ( F ) at m( Fs /D), m = 0, 1, . . . , D −1, as shown in Figure B.12.

588

appendix b. digital signal processing X( f )

Continuous spectrum

f −B

B Xs( f )

Discrete spectrum fs

2B

f −B

B fs

fs Xsd( f )

D (2B)

Downsampling fs

D=3

f −B fsd

fsd

B fsd Xsu( f )

/

fsd = fs D

fsd

Upsampling U=2

f −B

B fs

fs

Figure B.12

Resampling.

Xs( f )

Discrete spectrum fs

D (2B)

f −B

B fs

fs Xsd( f )

D =3 Aliasing

Aliasing

Aliasing

Aliasing

Aliasing

Aliasing

Aliasing

Aliasing

f fsd

fsd

fsd

−B fsd

B

/

fsd

Xs( f )

fsd = fs D

fsd

Discrete spectrum fs

fsd

D (2B )

Low-pass filter −B 

f

B

fs

fs Xsd( f )

D =3

f fsd

fsd

fsd

−B  fsd

Figure B.13

B fsd

/

fsd = fs D

Downsampling.

fsd

fsd

b.6 numerical example − gaussian white noise

589

As illustrated in Figure B.13, there is aliasing if Fsd = Fs /D is less than the Nyquist rate; in other words, the sampling rate can be reduced by a factor of D without aliasing if the original sampling rate Fs is at least D times the Nyquist rate F Nyquist = 2B, i.e., Fs  D (2B). If this condition is not satisfied, a low pass filter can be applied to the original signal to reduce its bandwidth to B so that Fs  D (2B ) before downsampling.

Sampling Rate Increase Suppose the sampling rate is increased by an integer factor U, i.e., Fsu = U Fs or us = s /U to yield

Xnu = X u(n us ) = ∞ 

=

    X n · s , n = 0, ± U, ±2U, . . . , U

0,

otherwise,

X(r s ) δ(n−r U),

(B.5.7)

r=−∞

where the superscript “u” stands for “upsampling” . The case of upsampling for U = 2 is illustrated in Figure B.11, in which an extra sampling point is added between two adjacent original sampling points and 0 is assigned at the new sampling points.

☞

In the digital signal processing literature, the upsampling operation is also called “interpolation” . However, only 0 is filled at the new sampling points, and no attempt is made to “interpolate” the missing data values.

The DTFT of the upsampled signal Xnu , −∞ < n < +∞, is  +∞ ∞  u u Xs ( F ) = X u(t) e−i 2πF t dt = X u(n us ) e−i 2πF · ns us −∞

=



∞  n=−∞

=

∞ 

n=−∞

∞ 

 u X(r s ) δ(n−r U) e−i 2πF · ns us

r=−∞

X(r s ) e−i 2πF · r U (s/U) (s /U ) =

r=−∞

1 U

· s

∞  r=−∞

Xr e−i 2πF · rs,

which yields

Xsu( F ) = U1 Xs ( F ).

(B.5.8)

As shown in Figure B.12, there is no aliasing in upsampling a digital signal.

B.6 Numerical Example − Gaussian White Noise Gaussian white noise (GWN) is a stationary random process with constant FAS or PSD, and its PDF is normal distribution.

590

appendix b. digital signal processing

Matlab is a software package widely used in digital signal processing. Function wgn(m,n,p) can generate an m×n matrix of GWN. p specifies the power PdBW of

GWN in decibel-watts (dBW), which is related to the power P W in watts (W) as 



PdBW = 10 log10 P W /(1W) .

(B.6.1)

As an example, consider a GWN time-history X(t) with duration T = 50000 s and time interval s = 0.005 s. The length of X(t) is N=

T

s

=

50000 = 107 . 0.005

(B.6.2)

The sampling frequency Fs = 1/s = 200 Hz, and the frequency resolution is given by F = 1/T = 0.00002 Hz. X(t) is generated using Matlab function wgn(m,n,p) with

m = 107 , n = 1, and p = 0 (implying PdBW = 0 or P W = 1 W). Because there are too

many points in the time-history, only a small portion is plotted in Figure B.14(a).

X( F ) of X(t) is computed based on DFT. Matlab function fft(x), where X(t) is input as x, is employed to obtain DFT XK of X(t) first. X( F K ) is then determined by multiplying XK by time interval s = 0.005 s using equation (B.2.42). Because F = 1/T, the longer the duration T of the time-history, the higher the resolution (or the smaller the value of F ) of the corresponding FAS.Although GWN is   a stationary random process with constant X( F ), X(t) is only one realization of GW; hence, the corresponding FAS will be quite scattered. For duration T = 50000 s, FAS is able to distinguish very small frequency difference of F = 0.00002 Hz, far smaller than frequency resolution range (e.g., F = 0.02 Hz) of engineering interest. Figure B.14(b) shows a small portion of FAS, which seemingly does not provide much information.   Therefore, in engineering applications, it is necessary to smooth the FAS X( F ) by reducing its frequency resolution. Taking  F = 0.02 Hz as an example, the length of the   smoothed FAS is N = 2×100/0.02 = 10000. The smoothed FAS X  are evaluated   at frequencies F n = n  F , − 5000  n  5000. X  is determined as  2  X ( F ) = n

2 1   X( F n +0.00002K) , 1000 K=0 999

(B.6.3)

i.e., the frequency components F n +0.00002K, K = 0, 1, . . . , 999 are all considered as frequency component F n . Equation (B.6.3) ensures that the energy of X(t) would not change during the smoothing (frequency resolution reduction) process. The smoothed FAS is shown in Figure B.14(c). It is seen that the smoothed FAS is much closer to that of a white noise process. In engineering practice, it is common to plot the one-sided FAS, i.e., F  0, as shown in Figure B.14(d).

b.6 numerical example − gaussian white noise

591

  Figure B.14(e) presents the FAS X( F ) with frequency resolution of  F = 0.05 Hz, with number of frequency points N = 2×100/0.05 = 4000. It is seen that, with the reduction of frequency resolution (increase of F ), a smoother FAS is obtained. Figure B.14(f) shows the one-sided FAS. The energy of X(t) is given by

e = T · P W = 50000×1 = 50000 W · s = 50000 N · m.

(B.6.4)

On the other hand, the total energy of X(t) is given by the Parseval’s theorem in the discrete form

e=

N−1  n=0

N−1    X 2 s = X( F )2 F = 50000 N · m. n K=0

(B.6.5)

  Because X(t) is a GWN process, X( F ) is a constant over the entire frequency range. Equation (B.6.5) becomes   e = N X( F )2 F

=⇒

  X( F ) =



e N F

.

Equation (B.6.6) is independent of how the frequency resolution is reduced:      e e e X( F ) = = = N F N F N  F    50000 50000 50000 = = = 15.81. = 7 10 ×0.00002 10000×0.02 4000×0.05

(B.6.6)

(B.6.7)

Note that for one-sided FAS, 1-side = 2-side and N 1-side = 12 N 2-side . Hence, F F   √ √   1-side   e e X ( F ) = = 2· = 2 · X 2-side ( F ). (B.6.8) N 1-side 1-side N 2-side 2-side F F  2 that, because FAS is related to energy through X( F ) , ☞ It should be emphasized √ one-sided FAS is 2 times of two-sided FAS, not twice of two-sided FAS.    √  For this GWN process, X 2-side ( F ) = 15.811 and X 1-side ( F ) = 2×15.811 = 22.36. Digital Filters Figure B.14(g) shows the filtered-GWN by passing GWN through a low-pass filter with cut-off frequency Fc = 30 Hz. The digital filtering is performed using Matlab function fftfilt(b,x), where the GWN X(t) is input as x. b=fir1(n,Wn,ftype,window) is

used to determine window-based filter coefficients, in which n is the order of the filter; n = 2M in equation (B.4.3). Wn is the normalized cut-off frequency defined as the

592

appendix b. digital signal processing 4

(a)

X(t)

2 0 −2 −4

t (s) 0

|X( f )|

50 40

2

3

4

(b)

5

6

7

8

9

10

Frequency resolution = 0.00002 Hz, two-sided FAS (partial)

30 20 10 0

0.002

0

20

|X( f )|

1

0.004

0.008

0.006

0.014

0.012

0.016

0.018

0.02

f (Hz)

(c)

15 10

0.01

Frequency resolution = 0.02 Hz, two-sided FAS f (Hz) −100

−80

−60

−40

−20

0

20

40

60

80

100

|X( f )|

25

Frequency resolution = 0.02 Hz, one-sided FAS 15

(d)

|X( f )|

20

f (Hz) 10

0

20

30

40

60

70

80

90

100

(e)

15 10

50

Frequency resolution = 0.05 Hz, two-sided FAS f (Hz) −100

−80

−60

−40

−20

0

20

40

60

80

100

|X( f )|

25 20

Frequency resolution = 0.05 Hz, one-sided FAS

15 0 3

f (Hz) 10

20

30

40

1

2

3

4

50

60

70

80

90

100

(g)

2

X(t)

(f)

1 0 −1 −2 −3

Filtered-GWN, f c =30 Hz, Δ s =0.005 s, M =25 0

|X( f )|

20

(h)

6

7

M =25

15 10

5

t (s) 9

10

Frequency resolution = 0.02 Hz two-sided FAS

M =15

Filtered-GWN, f c =30 Hz

8

5 0

f (Hz) −100

−80

−60

−40

−30

−20

0

20

30

40

60

80

100

|X( f )|

30

M =25

20

M =15

Filtered-GWN, f c =30 Hz

10

Frequency resolution = 0.02 Hz one-sided FAS

(i) 0

0

f (Hz) 10

20

Figure B.14

30

40

50

60

70

80

GWN time-history and corresponding FAS.

90

100

b.6 numerical example − gaussian white noise 1.5

(a)

X(t)

1.0

Filtered-GWN, f c = 30 Hz, Δ s =0.005 s, M = 25, upsample U =5, Δus =0.001 s

0.5 0.0

−0.5 −1.0 6.20 3 2

X(t)

593

t (s) 6.22

6.24

6.26

6.28

6.30

6.32

6.34

6.36

6.38

9

9.5

6.40

(b)

1 0 −1 −2 −3

Filtered-GWN, f c = 30 Hz, Δ s =0.005 s, M = 25, upsample U =5, Δus =0.001 s 5

5.5

6

6.5

7

7.5

8

8.5

t (s) 10

|X( f )|

30

Filtered-GWN, f c =30 Hz, M =25

20 10 0

(c) 0

Filtered-GWN, f c = 30 Hz, M = 25, upsample U =5 10

20

30

Figure B.15 3

40

50

f (Hz) 60

70

80

90

8

9

100

Upsampling of filtered-GWN.

(a)

2

X(t)

Frequency resolution = 0.02 Hz one-sided spectrum

1 0 −1 −2 −3

Filtered-GWN, f c =50 Hz, Δ s =0.005 s, M =25 0

3

2

1

2

3

4

5

6

t (s) 7

10

(b)

2

X(t)

1

1 0 −1 −2 −3

Filtered-GWN, f c =50 Hz, Δ s =0.005 s, M =25, downsample D =2, Δds =0.01 s 0

3

4

5

6

7

8

t (s) 9

10

|X( f )|

30 Aliasing

20 10 0

(c) 0

3

10

20

1

2

1

2

30

Filtered-GWN, f c =50 Hz, M =25 40

50

60

f (Hz)

70

80

90

7

8

9

100

(d)

2

X(t)

Filtered-GWN, f c =50 Hz, M =25, downsample D = 2

Frequency resolution = 0.02 Hz one-sided FAS

1 0 −1 −2 −3

Filtered-GWN, f c =40 Hz, Δ s =0.005 s, M =25 0

3

4

5

6

t (s) 10

(e)

2

X(t)

3

1 0 −1 −2 −3

Filtered-GWN, f c =40 Hz, Δ s =0.005 s, M =25, downsample D =2, Δds = 0.01 s 0

3

4

5

6

7

8

t (s) 9

10

|X( f )|

30

Filtered-GWN, f c =40 Hz, M =25, downsample D =2

20 10 0

(f) 0

Frequency resolution = 0.02 Hz one-sided FAS 10

20

30

Figure B.16

Filtered-GWN, f c =40 Hz, M =25 40

50

60

70

Downsampling of filtered-GWN.

f (Hz) 80

90

100

594

appendix b. digital signal processing

ratio of cut-off frequency Fc to Nyquist frequency. ftype indicates the type of filter, such as ’low’ for low-pass and ’high’ for high-pass filters. window includes a number of commonly used window functions. b=fir1(50,0.3,’low’,blackman(51))   is used in Figure B.14(g) to remove frequency contents with  F  > Fc ( Fc = 30 Hz and F Nyquist = 100 Hz giving Wn = 30/100 = 0.3). The Blackman window function is applied with M = 25. The two-sided and one-sided FAS of the filtered process are shown in Figure B.14(h) and (i), respectively. The filtered process is a “band-limited GWN” with a constant   FAS for  F  < Fc . However, as discussed in Section B.4, due to the use of a window function in digital filter, there is a significant transition region between 20 and 40 Hz. The transition region can be reduced by increasing the order of the filter, i.e., the value of M; as seen in Figure B.14(h) and (i), the transition region with M = 25 is smaller than that with M = 15. Upsampling Based on the filtered-GWN with Fc = 30 Hz and M = 25, as shown in Figure B.14(g), Figure B.15(a) and (b) shows portions of the time-history from upsampling with U = 5 (giving the new sampling interval us = 0.001 s). The resulting FAS of the time-history from upsampling is shown in Figure B.15(c) along with the FAS of the filtered-GWN. It is clearly seen that the two FAS satisfy equation (B.5.8) with U = 5. Downsampling A portion of the time-history of filtered-GWN with Fc = 50 Hz and M = 25 is shown in Figure B.16(a), and the corresponding downsampled time-history with D = 2 (leading to ds = 0.01 s) is shown in Figure B.16(b). For the downsampled time-history, Fsd = 1/ds = 100 Hz and the maximum frequency range is F max  Fsd /2 = 50 Hz. The resulting FAS from downsampling is shown in Figure B.15(c) along with the FAS of the original filtered-GWN. Even though F max = Fc = 50 Hz = Fsd /2 satisfying the NyquistShannon sampling requirement, there is still aliasing error near 50 Hz due to the existence of transit region from digital filtering. Similar results are shown in Figure B.16(d) to (f) for filtered-GWN with Fc = 40 Hz and M = 25. In this case, F max = 40 Hz and Fsd = 100 Hz, resulting in Fs = 2.5× F max . From Figure B.16(f), it is seen that, for F  50 Hz, FAS of the downsampled signal is the same as the original signal and there is no aliasing error.

☞

Therefore, to avoid aliasing errors, it is recommended that Fs  2.5× F max .

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Index A Acceleration response spectra (ARS), 181 Accelerograph, 37 Aleatory randomness, 174, 197, 411 Amplification factor, 343 Amplification function, 210 Arias intensity, 39, 233 Autocorrelation function, 60

B Balance of plant (BOP), 5 Baseline correction, 265 Boiling water reactor (BWR), 1 Boolean algebra, 526

C CANDU, 2 Central limit theorem, 64, 89, 562 Complete quadratic combination (CQC), 94 Composite variability, 413 Conditional mean spectrum, 180 Conservative deterministic failure margin (CDFM), 417, 441 Consistent-mass, 290 Consistent time-histories, 264 Continental drift, 18 Controlling earthquake, 171 Conventional island (CI), 5 Core damage frequency (CDF), 16, 517 Correlation coefficient of spectral accelerations, 50

D Damping, 291 Damping ratio, 231, 371 Design basis earthquake (DBE), 8, 538 Design response spectrum (DRS), 122, 133 Deterministic seismic hazard analysis (DSHA), 153 Digital filter, 584 Direct spectra-to-spectra method, 137, 332, 339 Direct time integration method, 98 Discrete-time Fourier transform, 577

Discrete Fourier transform, 578 Drift, 264 Duhamel integral, 80 Dynamic magnification factor (DMF), 82 Dynamic stiffness matrix, 386

E Earthquake excitation factor, 85, 97 Eigenfunction, 268 Eigenvalue problem, 267 El Centro Earthquake, 117 Elastic rebound theory, 22 Empirical mode decomposition (EMD), 248 Energy spectral density (ESD), 63 Epistemic uncertainty, 174, 197, 411 Equipment response factor, 437 Equivalent damping coefficient, 347 Ergodic process, 63 Event tree, 534

F Factor of safety, 418 Failure event, 522 Fault, 22 Fault tree, 524 Finite element model, 283, 300 First-passage problem, 75 Flexural stiffness, 318 Floor response spectra (FRS), 15, 328, 331 Forced vibration, 82 Foundation input response spectra (FIRS), 210, 384 Foundation level input response spectra (FLIRS), 394 Fourier amplitude spectra (FAS), 62, 181 Fourier displacement spectrum, 188 Fourier series, 572 Fourier transform, 87, 573 Fragility analysis (FA), 417 Fragility curve, 15 Free vibration, 78 Frequency-domain dynamic response analysis, 191 609

610 FRS-CQC, 348 Fukushima, 34

G Gaussian excitation, 86 Gaussian process, 64, 89 Gaussian white noise, 589 Great East Japan Earthquake, 32 Ground-motion prediction equation (GMPE), 44, 153, 179 Ground acceleration capacity, 410, 413 Ground motion, 184, 230 Ground motion model, 184 Ground motion parameter (GMP), 410 Ground response spectra (GRS), 13, 116, 131 Gutenberg-Richter recurrence relationship, 160

H Harmonic load, 82 Hazard-consistent strain-compatible material properties (HCSCP), 212 HCLPF, 15 HCLPF capacity, 416 Heat exchanger, 446 Hilbert amplitude spectrum (HAS), 251 Hilbert energy spectrum (HES), 251 Hilbert spectral analysis (HSA), 248 Hilbert–Huang transform (HHT), 247

I Inelastic energy absorption factor, 420, 436 Influence matrix method, 271 Instantaneous frequency, 251 Interplate earthquake, 22 Intraplate earthquake, 24 Intrinsic mode function (IMF), 248

L Large early release frequency (LERF), 16, 517 Logic tree, 175 Lognormal distribution, 564 Love wave, 28 Lumped-mass, 290

M Main control room (MCR), 11 Masonry block wall, 483 Mean-square response, 88, 92--93, 105 Mesh, 287, 298 Method of residue, 108

index Minimal cut set, 522 Missing-mass effect, 314 Modal combination, 351 Modal combination for FRS, 348 Modal participation factor, 97 Modified Mercalli intensity (MMI), 28 Moment magnitude, 29 Multiple degrees-of-freedom (MDOF), 84

N Nonexceedance probability (NEP), 122 Nonstationary, 237 Nonstationary process, 62 Normal distribution, 562 Nuclear island (NI), 5 Nuclear power plant (NPP), 1 Nuclear steam plant (NSP), 5

P P-wave, 24 Peak broadening, 339 Peak factor, 72, 76, 94 Peak ground acceleration (PGA), 39 Peak ground displacement (PGD), 39 Peak ground velocity (PGV), 39 Peak shifting, 339 Plate boundaries, 20 Plate tectonics, 18 Poisson distribution, 160 Poisson process, 75, 163 Power spectral density (PSD), 42, 87, 92, 232 Predicted spectrum, 179 Pressurized water reactor (PWR), 1 Probabilistic seismic hazard analysis (PSHA), 15, 54, 154 Probability density function (PDF), 58 Probability distribution function, 57 Probability mass function (PMF), 168, 176 Pseudo-acceleration response spectrum, 116, 120 Pseudo-velocity response spectrum, 116, 120

R Random process, 57 Random variable, 57 Random vibration theory (RVT), 181, 189 Rayleigh wave, 28 Reactor building (RB), 5, 7 Reference earthquake, 418 Reference hard rock, 184, 189, 214 Reliability block diagram, 520 Resampling, 586

index

Response spectrum, 115 Review level earthquake (RLE), 16, 418, 544 Richter local magnitude, 29

S S-wave, 24 Safe shutdown earthquake (SSE), 418, 538 Safety functions, 8 Safety objectives, 11 Sampling, 568 Scaling method, 365 Screening table, 543, 546 Second-moment method, 440 Secondary control area (SCA), 10 Seismic design spectrum, 176 Seismic energy, 30 Seismic fragility, 15, 410 Seismic fragility analysis, 518 Seismic hazard, 13, 230, 410 Seismic hazard analysis, 518 Seismic hazard curve, 163, 184, 410, 536 Seismic hazard deaggregation (SHD), 167 Seismic levels, 8 Seismic margin assessment (SMA), 16, 328, 418, 538 Seismic probabilistic risk assessment (SPRA), 16, 517 Seismic probabilistic safety assessment (seismic PSA), 16, 328, 517 Seismic response history analysis (SRHA), 97, 230 Seismic response spectrum analysis (SRSA), 99 Seismic risk, 13, 16 Seismic risk quantification, 518 Seismicity, 8 Seismograph, 37 Shear-wave velocity, 183, 198 Shear area, 317 Shear modulus, 183 Single degree-of-freedom (SDOF), 77 Site condition, 230 Site design earthquake (SDE), 8 Site response analysis, 181 Soil condition, 181

611

Soil–structure interaction (SSI), 15, 383, 431 Spectral shape, 419 Spectrum-compatible ground motion, 230, 232, 235, 256, 271 Spectrum amplification factor, 125 Spectrum shape, 422 Square root of sum of squares (SRSS), 94 SSC, 8, 15, 328 Standard deviation, 60 Standard deviation of prediction equation, 44 Stationary Gaussian process, 88 Stationary process, 59, 89 Stationary response, 86 Statistical independence of ground motions, 233 Stick model, 317 Strength factor, 420 Strong-motion duration, 233 Strong ground motion, 37 Structure response factor, 421, 438 Substructure method, 386 Surrogate element, 546 System analysis, 518 System identification, 368

T t-response spectrum (tRS), 137, 346 Time-domain dynamic response analysis, 193 Time-history, 13 Time-history analysis, 15, 332 Transient response, 79 Tripartite, 120 Tsunami, 31

U Uniform hazard spectrum (UHS), 154, 169, 177, 214

V Vector-valued PSHA, 166 Vector-valued SHD, 172

W Wavelet, 242