Table Of ContentEndurance Time
Excitation Functions
Seismic assessment and earthquake-resistant design are essential applications of
earthquake engineering for achieving seismic safety for buildings, bridges, infra-
structure, and many other components of the built environment. The Endurance
Time Method (ETM) is used for seismic analysis of simple and complex structural
systems and civil engineering infrastructure as well as producing optimal and cost-
effective structural and detail designs. ETM is a relatively new approach to seismic
assessment and design of structures. It has developed into a versatile tool in the field,
and its practical applications are expected to increase greatly in the near future.
Taylor and Francis Series in Resilience
and Sustainability in Civil, Mechanical,
Aerospace and Manufacturing Engineering
Systems
Series Editor
Mohammad Noori
Cal Poly San Luis Obispo
PUBLISHED TITLES
Sustainable Development for the Americas
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Honorary Editor: E. William Colglazier
Hassan A. Vafai and Kevin E. Lansey
With the administrative assistance of Molli D. Bryson
Endurance Time Excitation Functions
Intensifying Dynamic Loads for Seismic Analysis and Design
Homayoon E. Estekanchi
Mohammadreza Mashayekhi
Hassan A. Vafai
SERIES EDITOR
Mohammad Noori is a Professor of Mechanical Engineering at California
Polytechnic State University, San Luis Obispo. He received his B.S. (1977), M.S.
(1980), and Ph.D. (1984) from the University of Illinois, Oklahoma State University,
and the University of Virginia, respectively, all degrees in Civil Engineering with
a focus on Applied Mechanics. His research interests are in stochastic mechanics,
nonlinear random vibrations, earthquake engineering and structural health monitor-
ing, AI-based techniques for damage detection, stochastic mechanics, and seismic
isolation. He serves as the executive editor, associate editor, the technical editor, or
a member of editorial boards of eight international journals. He has published over
250 refereed papers, has been an invited guest editor of over 20 technical books, has
authored/co-authored six books, and has presented over 100 keynote and invited pre-
sentations. He is a Fellow of ASME and has received the Japan Society for Promotion
of Science Fellowship.
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Endurance Time
Excitation Functions
Intensifying Dynamic Loads for Seismic
Analysis and Design
Homayoon E. Estekanchi, Mohammadreza
Mashayekhi, and Hassan A. Vafai
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First edition published 2023
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ISBN: 9781032107127 (hbk)
ISBN: 9781032107141 (pbk)
ISBN: 9781003216681 (ebk)
DOI: 10.1201/9781003216681
Typeset in Times New Roman
by codeMantra
Contents
List of Figures ...........................................................................................................ix
List of Tables ............................................................................................................xv
Foreword ................................................................................................................xvii
Preface.....................................................................................................................xix
Authors ....................................................................................................................xxi
Nomenclature .......................................................................................................xxiii
Chapter 1 Introduction to the Endurance Time Method and Intensifying
Dynamic Load Functions .....................................................................1
Review ..................................................................................................1
Concept of Endurance Time Method ...................................................1
Applications of Endurance Time Method ............................................4
Implementation of the Endurance Time Method .................................5
Intensifying Dynamic Load Functions ................................................5
Simulated Endurance Time Excitations ...............................................7
References ............................................................................................9
Chapter 2 Objective Functions for Generating Endurance
Time Excitation Functions .................................................................11
Review ................................................................................................11
Second Generation of Endurance Time Excitations ..........................11
Third Generation of Endurance Time Excitations .............................13
Fourth Generation of Endurance Time Excitations ...........................13
Fifth Generation of Endurance Time Excitations ..............................17
References ..........................................................................................21
Chapter 3 Optimization Variable Spaces ............................................................23
Review ................................................................................................23
Optimization Variable Space for Simulating Endurance
Time Excitations .................................................................................23
Time Domain .....................................................................................23
Wavelet Decomposition ......................................................................24
Basic Theory of Wavelet Analysis .................................................24
Simulation Method ........................................................................26
Simulating in Wavelet Transform Space ..................................26
Simulating in Filtered Transformed Space ...............................28
The Significance of the Proposed Method in Simulating ET
Excitations .....................................................................................28
Conformance of Generated Excitations ........................................28
v
vi Contents
Sensitivity of Generated Excitations to Initial Optimization
Points .............................................................................................29
Optimum Level of Filtering ......................................................31
Results ............................................................................................31
Increasing Sine Functions ..................................................................32
Parameter Tuning ...............................................................................35
Accuracy of Excitations Simulated with the Proposed Method ........38
References ..........................................................................................40
Chapter 4 Generating ETEFs Based on Linear Spectra .....................................41
Review ................................................................................................41
Simulation of “a” Series .....................................................................41
Simulation of “lc” Series ....................................................................43
Problem Definition ........................................................................43
Problem Solution ...........................................................................45
Discretization ............................................................................45
Signal Representation ...............................................................46
Optimization Algorithm ...........................................................46
Simulated Excitations ....................................................................46
20-Second Excitations ..............................................................46
40-Second Excitations ..............................................................48
Calculation of Residuals ................................................................51
References ..........................................................................................52
Chapter 5 Nonlinear Analysis-Based Endurance Time Excitation Generation ........53
Review ................................................................................................53
Procedure of Simulating Third Generation ........................................53
Objective Function Definition .......................................................54
Discretization.................................................................................54
Signal Representation and Optimization Algorithm ..........................55
Results ................................................................................................55
Comparison of Dynamic Characteristics of Generated
Endurance Time Excitations and Targets ...........................................56
Acceleration Spectra ......................................................................56
Residuals of Generated Endurance Time Excitations ...................57
References ..........................................................................................58
Chapter 6 Generating ETEFs Considering Spectral Energy Content .................59
Review ................................................................................................59
Procedure of Simulating Fifth Generation .........................................59
Signal Representation and Optimization Algorithm ..........................60
Results ................................................................................................60
Comparison of Dynamic Characteristics of Generated
Endurance Time Excitations and Targets ...........................................61
Acceleration Spectra ......................................................................61
Contents vii
Hysteretic Energy Demands ..........................................................62
Residuals of Generated Endurance Time Excitations ...................63
Damage Spectra of Generated Endurance Time Excitations
vs. Ground Motions ............................................................................64
References ..........................................................................................66
Chapter 7 Application of Meta-Heuristic Optimization Methods in
Generating ETEFs ..............................................................................67
Review ................................................................................................67
Imperialist Competitive Algorithm ....................................................67
Parameter Tuning of ICA ...................................................................74
Significance of Optimization Space and Initial Population
Generating Approach .........................................................................76
Significance of Optimal Parameter in Simulating Endurance
Time Excitations .................................................................................77
Comparison of Simulated Endurance Time Excitations with Targets ...79
References ..........................................................................................80
Chapter 8 Modifying ETEFs as an Alternative to Their Generation .................81
Review ................................................................................................81
Endurance Time Excitations Spectral Matching ................................81
Step-by-Step Application ...................................................................86
Accuracy of Spectral Matched Endurance Time Excitations .............88
Results ................................................................................................90
Reference ............................................................................................91
Chapter 9 Generating ETEFs for Direct Response Variability Estimation ........93
Review ................................................................................................93
Problem Formulation ..........................................................................93
Problem Solving .................................................................................95
Simulated Excitations .........................................................................95
Seismic Response Probability Distribution Calculations ...................97
Application .........................................................................................99
References ........................................................................................105
Chapter 10 Optimal Objective Functions for Generating ETEFs .......................107
Review ..............................................................................................107
Generalized Objective Function .......................................................107
Finding Optimal Objective Function ...............................................109
Application .......................................................................................111
Results ..............................................................................................114
Reference ..........................................................................................118
Index ......................................................................................................................119
List of Figures
Figure 1.1 The concept of the endurance time method for determining the
seismic demand of three structures. .....................................................2
Figure 1.2 Increasing response plot of three structures: A, B, and C. ...................3
Figure 1.3 D ifferent increasing profiles: (a) linear, (b) ascending rate, and
(c) descending rate. ...............................................................................6
Figure 1.4 First-generation endurance time excitation. .........................................8
Figure 2.1 (a) Nonlinear single degree-of-freedom configuration and
(b) behavior model. .............................................................................18
Figure 2.2 Y ield strengths used in this study for generating ETEFs
compared to those proposed by Miranda (1993). ...............................19
Figure 2.3 Time-dependent variations of target displacement demands:
(a) T = 0.5 second, μ = 8; (b) T = 2.0 seconds, μ = 8; (c) T = 2.0
seconds, μ = 4; and (d) T = 2.0 seconds, μ = 2.....................................19
Figure 2.4 Time-dependent variation of target hysteretic energy demands:
(a) T = 0.5 second, μ = 8; (b) T = 2.0 seconds, μ = 8; (c) T = 2.0
seconds, μ = 4; and (d) T = 2.0 seconds, μ = 2. ....................................20
Figure 3.1 Definition of decision variables in time-domain optimization
space. ..................................................................................................24
Figure 3.2 S caling function ϕ(t) and mother wavelet function ψ(t) of db12. .......26
Figure 3.3 Algorithm of generating endurance time excitations in the
DWT space. ........................................................................................27
Figure 3.4 O ptimization convergence history for the time-domain, DWT,
and filtered DWT spaces. ...................................................................29
Figure 3.5 Normalized objective functions vs. filtering levels in the
DWT space. ........................................................................................30
Figure 3.6 ETA20FD01~03 acceleration time histories. .....................................31
Figure 3.7 Comparison of the ETA20FD01~03 acceleration spectra and the
targets at t = 5, 10, 15, and 20 seconds. ...............................................32
Figure 3.8 Comparison of the ETA20FD01~03 maximum absolute
acceleration responses with the targets at periods (a) T = 0.03
second, (b) T = 0.8 second, (c) T = 2.0 seconds, and (d) T = 3.0
seconds. ..............................................................................................32
Figure 3.9 Increasing sine basis function. ...........................................................33
ix
