thesis,209, nethar

ABSTRACT

The propagation of fault-rupture towards a site at a velocity close to the shear wave

velocity causes most of the seismic energy from the rupture to arrive in single/multiple

large long-period pulse/s of motion that generally occurs at the beginning of the record.

The radiation pattern of the shear dislocation on a fault causes this large pulse of motion

to be oriented in the direction perpendicular to the fault, which is known as the directivity

effect. This became evident from the ground records of numerous recent major

earthquake events, which have caused extreme structural destructions in near-fault zones

that are within 10 miles of the fault-rupture planes.

A methodology has been proposed to identify pulse-effects by the use of a discrete-time

signal processing method in which a low-pass filter with a suitable cut-off frequency is

applied to the Fourier transforms of the processed acceleration or velocity time history

records. An important difference between the proposed approach and those of the other

researchers is that no predetermined pulse shape is assumed in the analysis. The extracted

pulses are then used to determine their effects on linearly-elastic or inelastic single

degree-of-freedom (SDF) systems.

Quantitatively, these pulse-types are identified through a modified displacement response

factor, Rd. A statistical study of the effects of dynamic magnification due to near-fault

ground motions, using the modified displacement response factor, Rd, is presented. A new

method is introduced to allow engineers to construct analogous displacement response

spectra for an undamped linearly elastic SDF system subjected to long-period ground

motions; these spectra can be constructed directly from the Fourier amplitudes of velocity

time history.

Finally, this dissertation examines the displacement ductility requirements for a typical

bridge bent subjected to such extreme loadings. The ductility requirement, as stated by

prevailing design codes, may not be valid for such structures in near-fault zones. Due to

the dominance of pulses, medium- to long-period structures are significantly affected,

often resulting in high residual displacements (permanent deformations) after the

cessation of seismic ground motions. A suitable balance between ductility and residual

displacement should be a goal in the design of such structures.

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v

TABLE OF CONTENTS

Abstract ……………… i

Table of Contents ……………… v

List of Figures ……………… viii

List of Tables ………………. x

Acknowledgments ..……………. xii

Chapter 1: Introduction …………….. 1

1.1 Preview …………….. 1

1.2 Pulse-Type Near-Fault Ground Motion (NFGM) ……………… 1

1.3 Historical Perspective ………………. 8

1.4 Objectives and Scope ……………… 21

Chapter 2: Pulse Identification and Displacement Response Evaluation …………….. 28

2.1 Introduction …………….. 28

2.2 Excitation and its Response in the Frequency Domain …………….. 31

2.3 Low-pass Filter Response Function, Hlp(ω) ……………. 32

2.3.1 Digital Filters ……………. 32

2.3.2 Ideal Low-pass Filter ……………. 34

2.3.3 Butterworth Low-pass Filter …………… 35

2.4 Pulse Identification and Cut-off Frequency Determination …………….. 36

2.5 Pulse Response Evaluation …………….. 42

2.6 Pulse Effects …………….. 45

2.7 Discussion and Observations ……………… 52

2.8 Parseval’s Theorem and Pulse Energy ……………… 54

vi

2.9 Summary and Conclusions ……………… 55

Chapter 3: Modified Displacement Response Factor, Rd ………………… 58

3.1 Introduction and Overview …………………. 58

3.2 Brief Review …………………. 60

3.3 Modified Displacement Response Factor, Rd ………………… 61

3.4 Beat Phenomena versus “Simple” Harmonic Excitation ……………… 77

3.5 Evaluation of the Mean Rd for Different Events ……………….. 81

3.5.1 Imperial Valley-06 1979 Earthquake, Mw = 6.53 ………………. 82

3.5.2 Northridge-01 1994 Earthquake, Mw = 6.69 ……………….. 84

3.5.3 Kobe 1995 Earthquake, Mw = 6.90 ……………….. 87

3.5.4 Loma Prieta 1989 Earthquake, Mw = 6.93 ……………….. 90

3.5.5 Kocaeli 1999 Earthquake, Mw = 7.51 ………………... 93

3.5.6 Chi-Chi 1999 Earthquake, Mw = 7.62 ……………….. 96

3.5.7 Other Earthquakes, Mw > 7.0 ……………… 98

3.5.8 Summary ………………. 99

3.6 Observations ……………….. 100

3.6.1 Comparison of Seismic Rd with that due to Harmonic Excitation..100

3.6.2 Response and Damping …………………104

3.7 Summary and Conclusions …………………. 105

Chapter 4: Analogous Displacement Response Spectra …………………. 109

4.1 Introduction …………………. 109

4.2 Brief Review …………………. 110

4.3 Spectral Displacements and Fourier Amplitudes ………………… 111

vii

4.4 Discussion ………………… 118

4.5 Conclusions ……………….. 120

Chapter 5: Pulse Effects on Displacement Ductility Requirement for Bridge Bent … 121

5.1 Preview ………………. 121

5.2 Introduction ………………. 122

5.3 Existing Design Code Provisions for the Displacement

Ductility Demand Value, µc ……………….. 124

5.3.1 CALTRANS Seismic Design Criteria, 2006, v 1:4 …………... 124

5.3.2 AASHTO (2009) Spec. for LRFD Seismic Bridge Design ..… 125

5.4 Brief Background …………………. 127

5.5 Evaluation Procedures …………………. 131

5.6 Parametric Study …………………. 135

5.7 Discussion ………………… 148

5.8 Summary and Conclusions ………………… 150

Chapter 6: Summary and Conclusions ……………….. 152

6.1 Extraction of Acceleration and Velocity Pulse/s ………………. 152

6.2 Displacement Response Factor, Rd ………………. 154

6.3 Analogous Displacement Response Spectrum ………………. 156

6.4 Displacement Ductility Requirement for a SDF System ………………. 157

6.5 Scope for Future Research ………………... 159

APPENDIX: MATLAB Programming Applications ………………. 162

REFERENCES ………………. 192

VITA ………………. 196

viii

LIST OF FIGURES

Figure 1.1. Map of the Landers region showing the location of the rupture

of the 1992 Landers earthquake [Somerville et al., 1997] ……….. 3

Figure 1.2. Schematic diagram of rupture directivity effect for a vertical

strike-slip fault. [Somerville et al., 1997] ………... 4

Figure 1.3. Schematic illustration of the directivity effect on ground motions

[Kramer 1996] ……….. 5

Figure 1.4. 1992 Landers earthquake, ATH, VTH & DTH for fault-normal

and fault-parallel direction [Somerville et al., 1997] ………. 7

Figure 1.5. Velocity pulses idealized in triangular form [Hall et al., 1995] ………10

Figure 1.6. Triangular velocity pulses from Alavi and Krawinkler [2001]:

Pulses P1 & P2 ….. 11, 12


Pulses P4, P5 and P3 ...13, 14, 15

Figure 1.8. Idealized trigonometric pulses by Makris and Chang [2000] ……… 17

Figure 1.9. Triangular and sinusoidal simple velocity pulses [Lili et al., 2005] ……... 18

Figure 1.10. Pulses from Mavroeidis and Papageorgiou [2003, 2004] …….. 19

Figure 2.1. Solution of linear displacement response to earthquake pulse/s ……. 30

Figure 2.2. For low-pass filter, specifications for the effective frequency

response of the overall system. [Oppenheim et al., 1999] ……. 34

Figure 2.3. Effect of NB on Butterworth low-pass filter magnitude ……… 36

Figure 2.4. Fourier spectra of VTH, Nishi-Akashi, 140-FN, Kobe 1995 ……… 39

Figure 2.5. Fourier spectra of VTH, Lucerne, 239-FN, Landers 1992 ……… 40

Figure 2.6. VTH and DTH, Lucerne, 239-FN, Landers 1992 ………. 40

Figure 2.7. Fourier spectra of VTH, Westmorland Fire Station (HWSM),

233-FN, Imperial Valley-6 1979 ……….. 41

Figure 2.8. Original and pulse VTH, Lucerne, 239-FN, Landers 1992 ……….. 46

Figure 2.9. Original and pulse ATH, Lucerne, 239-FN, Landers 1992 ……….. 46

Figure 2.10. Original, pulse and residual DTH, Lucerne, 239-FN, Landers 1992 …… 47

Figure 2.11. Displacement Response Spectra (DRS),

Lucerne, 239-FN, Landers 1992 ………… 48

Figure 2.12. Original and pulse ATH, Westmorland Fire Station (HWSM),

233FN, Imperial Valley-6 1979 ………… 49

Figure 2.13. DRS, Westmorland Fire Station (HWSM),

233FN, Imperial Valley-6 1979 …………. 50

Figure 2.14. Original and pulse ATH, Oakland - Outer Harbor Wharf

(CH1), 038-FN, Loma Prieta 1989 …………. 51

Figure 2.15. DRS, Oakland - Outer Harbor Wharf (CH1),

038-FN, Loma Prieta 1989 …………. 51

Figure 3.1. Velocity Fourier spectra and pulse-ATH, TCU052,

N322E, Chi – Chi 1999 ………….. 65

Figure 3.2. DRS and Rd spectra, TCU052, N322E, Chi – Chi 1999 ………….. 66

Figure 3.3. Velocity Fourier spectra and pulse-ATH, Yarimca-180FN, Kocaeli 1999 .. 69

Figure 3.4. DRS and Rd spectra, Yarimca-180FN, Kocaeli 1999 ………….. 70

ix

Figure 3.5. Velocity Fourier spectra, pulse-ATH and DRS,

Yarimca-090FP, Kocaeli 1999 …………… 72

Figure 3.6. Velocity Fourier spectra and pulse-ATH, Saratoga-W Valley

College (WVC), 038FN, Loma Prieta 1989 …………… 75

Figure 3.7. DRS and Rd spectra, Saratoga-W Valley College (WVC),

038FN, Loma Prieta 1989 ……………. 76

Figure 3.8. VFA, pulse-ATH and Rd spectra, TCU068-N320E, Chi Chi 1999 ……… 78

Figure 3.9. VFA and pulse-ATH, Osaj (OSA-140FN), Kobe 1995 …………… 79

Figure 3.10. Rd spectra, Imperial Valley-6 1979 earthquake event …………… 83

Figure 3.11. Rd spectra, Northridge 1994 earthquake event …………… 86

Figure 3.12. Rd spectra, Kobe 1995 earthquake event …………… 89

Figure 3.13. Rd spectra, Loma Prieta 1989 earthquake event ……………. 92

Figure 3.14. Rd spectra, North Anatolian Fault, Turkey ……………. 95

Figure 3.15. Rd spectra, Chi-Chi 1999 earthquake event ……………. 97

Figure 3.16. Rd spectra, Other Pulse-like Ground Motions, Mw > 7 ……………. 98

Figure 3.17. Concept of equivalent harmonic forcing frequency (ω/ωn) ratio ……….. 102

Figure 3.18. Concept of factors k & α comparing with equiv harmonic excitation ……102

Figure 3.19. Rd spectra for each major event & harmonic excitation …………… 103

Figure 4.1. Velocity Fourier spectra, TCU052, N322E, Chi Chi 1999 …………… 116

Figure 4.2. Transformed-VFA, TCU052, N322E, Chi Chi 1999 …………… 117

Figure 4.3. Velocity Fourier spectra, TCU054, N142E, Chi Chi 1999 …………… 117

Figure 4.4. Transformed-VFA, TCU054, N142E, Chi Chi 1999 ……………. 118

Figure 5.1. Local Displacement Capacity [Caltrans SDC 2006] ……………. 125

Figure 5.2. Idealized Elasto-Plastic behavior model [Clough 1966] ……………. 128

Figure 5.3. Idealized Stiffness Degrading (SD) behavior model [Clough 1966] …… 128

Figure 5.4. CR values of SDF system EPP model [Sec. C.2.1 of ATC-55] ………… 129

Figure 5.5. CR values of SDF system SD model [Sec. C.2.2 of ATC-55] ………….. 129

Figure 5.6. Displacement response of SDF system to Takatori, Kobe 1995………… 130

Figure 5.7. SD Model for Group 1: Mw = 6.5 ± 0.25, Soil Type A, B

(a) DRS, (b) Dres/D spectra …………. 137

Figure 5.8. SD model for Group 2: Mw = 6.5 ± 0.25, Soil Type D

(a) DRS, (b) Dres/D spectra ………….. 138

Figure 5.9. SD model for Group 3: Mw =7.25 ± 0.25, Soil Type A, B


Figure 5.10. SD model for Group 3: Mw = 7.25 ± 0.25, Soil Type D


Figure 5.11. Takatori, Kobe 1995 (a) DRS, (b) Dres/D spectra for SD model,

(c) Residual Displacements for SD model ……… 143,144

Figure 5.12. ATC – 55 (20+2) Group (a) DRS, (b) SD model, Dres/D spectra,

(c) EPP model, Dres/D spectra ……… 146,147

Figure 5.13. ATC-55 (20+2) Group, Residual Displacements (SD) model ………… 147

Figure 5.14. ATC-55 (20+2) Group, Residual Displacements (EPP) model ………. 148

x

LIST OF TABLES

Table 1.1. Directivity and fling directions summarized [Abrahamson 2001] …………. 7

Table 1.2. Summary of research on velocity pulse identification …………. 20

Table 3.1. Pulse-like Ground Motions of Imperial Valley-6 1979 Earthquake ………. 84

Table 3.2. Pulse-like Ground Motions of Northridge 1994 Earthquake ………….. 87

Table 3.3. Pulse-like Ground Motions of Kobe1995 Earthquake ………….. 89

Table 3.4. Pulse-like Ground Motions of Loma Prieta 1989 Earthquake …………. 93

Table 3.5. Pulse-like Ground Motions near the North Anatolian Fault, Turkey ………. 95

Table 3.6. Pulse-like Ground Motions of Chi-Chi 1999 Earthquake …………. 97

Table 3.7. Other Pulse-like Ground Motions, Mw > 7 ………….. 99

Table 3.8. Summary of Rd for different events ………….. 99

Table 5.1. Pulse-like records of Mw = 6.5 ± 0.25, Soil = A & B ………….. 132

Table 5.2. Pulse-like records of Mw = 6.5 ± 0.25, Soil = D …………. 132

Table 5.3. Pulse-like records of Mw = 7.25 ± 0.25, Soil = A & B ………….. 133

Table 5.4. Pulse-like records of Mw = 7.25 ± 0.25, Soil = D …………… 133

Table 5.5. Pulse-like Near-fault (20+2) records used in ATC – 55 …………… 134

xi

This dissertation is dedicated to the author’s former advisor

Dr. Chu-Kia Wang,

Professor Emeritus of the University of Wisconsin-Madison,

who still remembers his student even after a long span of thirty-five years.

xii

ACKNOWLEDGMENTS

I graciously acknowledge the support given to me by Syracuse University in the form of

a University Fellowship during my first- and third-year of doctoral study. The financial

support for the fourth-year was made possible by the “Wen-Hsiung and Kuan-Ming Li

Graduate Fellowship” from the Department of Civil and Environmental Engineering at

Syracuse University.

I express my sincere gratitude towards the faculty members in Civil and Environmental

Engineering at Syracuse University, who gave me a unique opportunity to return to

school after a span of thirty years. My advisor, Professor Eric M. Lui, has always been

there, when I needed him. His continued encouragement, support and guidance not only

helped me in my academic fulfillment, but also in my career development. Dr. Shobha

K. Bhatia gracefully offered me her expertise in geotechnical engineering, which I lacked

initially. Dr. James Mandel, Dr. Samuel P. Clemence and Dr. Riyad S. Aboutaha

fostered intellectual curiosity in me from their teachings. Special thanks go to Professor

Eugene Poletsky, the Chair of the Mathematics Department who made understanding of

the functions of complex variables much simpler to me.

I thank Dr. Shailesh Ozarker (former doctoral student in the Department of Chemical

Engineering, Syracuse University), who spent countless hours teaching me the relevant

programming aspects of MATLAB. The staff at the department, Ms. Linda Lowe, Ms.

Mickey Hunter and Ms. Elizabeth Buchanan, was very supportive of my shortcomings.

I have no words to express my gratitude towards my wife, “children” and grandson.

1

CHAPTER 1

INTRODUCTION

1.1 PREVIEW

The propagation of fault rupture at a velocity close to the shear wave velocity of the site

causes most of the seismic energy from the rupture to arrive at the site in a single large

long-period pulse of motion that occurs at the beginning of the record [Archuleta and

Hartzell 1981; Somerville et. al., 1997a; Somerville 2003]. The term “pulse” has been

used with reference to the acceleration, velocity and displacement of ground motion, i.e.,

an “acceleration pulse” or “velocity pulse”. This chapter first briefly discusses the

seismological concepts in rupture propagation and its effects on the occurrence of

directivity effects. Many researchers have made efforts to identify the pulse-effects that

may occur in near-fault ground motions when certain conditions are met. A brief

historical perspective is presented on research works that have tried to characterize pulse-

like near-fault ground motions and their effects on elastic or inelastic structures. After

this background, the salient features of the objectives and the scope of this dissertation

are briefly discussed.

1.2 PULSE-LIKE NEAR-FAULT GROUND MOTIONS [Somerville et al., 1997]

The earthquake events of Imperial Valley-6 1979 (Mw = 6.53), Loma Prieta 1989 (Mw =

6.93), Landers 1992 (Mw = 7.28), Northridge 1994 (Mw = 6.69), Kobe 1995 (Mw =

2

6.90), Kocaeli 1999 (Mw = 7.51), Chi-Chi 1999 (Mw = 7.62), etc., have caused extreme

structural destruction including land-slides and liquefaction. Furthermore, over the past

five years, the world has experienced several large earthquake events such as the 2003

Tokachi-oki, Japan earthquake (Mw = 8.0); the 2004 South-East off Kii peninsula

earthquake (Mw = 7.4); the 2004 Niigata-ken Chuetsu, Japan earthquake (Mw = 6.6); the

2007 Niigata-ken Chuetsu-oki, Japan earthquake (Mw = 6.7), and the 2008 Wenchuan,

China earthquake (M = 7.9), etc. Ground motion records from these quakes have

provided engineers with useful data for studying the nature of rupture directivity or path

effects embedded in seismic ground motions. These effects are often characterized by the

existence of velocity or acceleration pulses in long-period ground motions. They are

important design considerations because they often impose large demands on medium- to

long-period structures.

Rupture directivity effects cause spatial variation in ground motion amplitude and

duration around faults, and cause differences between the strike-normal and strike-

parallel components of horizontal ground motion amplitudes, which also exhibit spatial

variation around a fault. These variations become significant around a period of 1 second

and generally grow in size as the period increases. In Fig. 1.1, the directivity effect in

strike-slip faulting is illustrated using the strike-normal components of ground velocity

from two near-fault recordings of the magnitude Mw = 7.28 Landers earthquake of 1992.

The Lucerne record, which was taken 1.1 km from the surface rupture and 45 km from

the epicenter of the Landers earthquake, consists of a large, brief velocity-pulse of motion

of high amplitude (due to forward directivity effects); while the Joshua Tree record, taken

3

at 11.03 km from the fault-rupture and 13.67 km from the epicenter, consists of a long

duration, low amplitude record (due to backward directivity effects).

Figure 1.1. Map of the Landers region showing the location of the rupture of the 1992

Landers earthquake (which occurred on three fault segments), the epicenter, and the

recording stations at Lucerne and Joshua Tree. The fault-normal velocity time histories at

Lucerne and Joshua Tree exhibit forward and backward rupture directivity effects,

respectively. [Somerville et al., 1997]

4

The propagation of rupture towards a site at a velocity that is almost as large as the shear

wave velocity at the site causes most of the seismic energy from the rupture to arrive in a

single large pulse of motion, which occurs at the beginning of the record. This pulse of

motion represents the cumulative effect of almost all of the seismic radiation from the

fault, as illustrated in Fig. 1.2.

Figure 1.2. Schematic diagram of the rupture directivity effect for a vertical strike-slip

fault. The rupture begins at the hypocenter and spreads circularly at a speed about 80% of

the shear wave velocity. The figure shows a snapshot of the rupture front at a given

instant. The resulting time histories close to and away from the hypocenter are

represented by strike-normal velocity recordings of the 1992 Landers earthquake at

Joshua Tree and Lucerne, respectively; their locations are shown in Figure 1.1.

[Somerville et al., 1997]

slipping

5

The radiation pattern of the shear dislocation on a fault causes this large pulse of motion

to be oriented in the direction perpendicular to the fault. As the rupture progressed across

the fault as a series of dislocations towards the Lucerne station, waves emanating from

the fault caused constructive interference of SH waves generated from the epicenter. This

directivity effect can be observed in Fig. 1.3.

Figure 1.3. Schematic illustration of the directivity effect on ground motions at sites

toward and away from the direction of the fault ruptures in the case of a strike-slip fault.

The overlapping of pulses can lead to a strong fling pulse at the site towards which the

fault ruptures. [Kramer1996]

Forward rupture directivity effects occur when two conditions are met [Somerville et al.,

1997]: (1) the rupture front propagates toward the site; and (2) the direction of slip on the

fault is aligned with the site. This condition for generating forward rupture directivity

effects are readily met in strike-slip faulting, where the fault slip direction is oriented

6

horizontally along the strike of the fault, and the rupture propagates horizontally along

the strike, either unilaterally or bilaterally. Backward directivity effects, which occur

when the rupture propagates away from the site, give rise to the opposite effect: long-

duration motions with low amplitudes at long periods, as shown in Figs. 1.1 and 1.2.

The conditions required for forward directivity are also met in dip-slip faulting, involving

both reverse and normal faults. The alignment of both the rupture direction and the slip

direction up the fault plane produces rupture directivity effects at sites located around the

surface exposure of the fault (or its up-dip projection, if it does not break the surface).

Consequently, it is generally the case that all sites located near the surface exposure of a

dip-slip fault experience forward rupture directivity when an earthquake occurs on that

fault. Unlike the case of strike-slip faulting, where we expect forward rupture directivity

effects to be most concentrated away from the hypocenter, dip-slip faulting produces

directivity effects on the ground surface that are most concentrated up-dip from the

hypocenter.

The acceleration, velocity and displacement time histories recorded at Lucerne are shown

in the left part of Fig. 1.4. There is a large difference between the strike-normal and

strike-parallel motions at long periods (velocity and displacement), but this difference

vanishes at short periods (acceleration). The displacement response spectrum of the

strike-normal component greatly exceeds that of strike-parallel component for periods

longer than 1 second, as seen at the right of Fig. 1.4.

7

Figure 1.4. Left: acceleration, velocity and displacement time histories of the strike-

normal and strike-parallel components of horizontal motion recorded at Lucerne during

1992 Landers earthquake. Right: strike-normal (solid line) and strike-parallel (dashed

line) displacement response spectra of the Lucerne record. [Somerville et al., 1997]

Table 1.1. Directivity and fling directions summarized [Abrahamson 2001]

SENSE OF SLIP DIRECTIVITY FLING

Strike-slip

e.g., Imperial Valley 1979 Fault normal Fault parallel

Dip-Slip

e.g., Chi-Chi 1999 Fault Normal Fault Normal

The response due to most pulse-type seismic ground motions falls in the displacement-

sensitive part of the response spectrum. Because of the use of logarithmic scaling in the

construction of the response spectrum, the displacement characteristics are usually not

adequately represented in the spectrum. Furthermore, when the period of pulse is in the

vicinity of the period of the structure, resonance occurs but the resonance characteristics

are not often visible on the spectrum. Thus, the use of response spectrum for the design

of structures subjected to pulse-type ground motion is not always advisable.

8

An important reason why the study of system response to pulse-like near-fault ground

motion (NFGM) is warranted is that these long-period pulses often impose large demands

on structures with relatively long natural periods. When the input time history is a near-

fault pulse, small modifications to this time history can have a major effect on structural

response even though they do not necessarily manifest themselves in the response

spectrum. The response spectrum alone does not provide an adequate characterization of

these pulses because they are relatively simple long-period events with a relatively brief

duration, as opposed to being a stochastic process of relatively long duration.

Furthermore, for Performance Based Design it is crucial that realistic ground motion

inputs and representative models be used to evaluate structure response [Somerville

1998]. A normalized 5%-damped elastic response spectrum for ground motions in fault-

rupture zone has been recently proposed [Goel and Chopra 2008].

1.3 HISTORICAL PERSPECTIVE

The advent of modern digital seismographs has made the recording of seismic ground

motions of relatively low frequencies (i.e., long periods) possible. Such ground motion

records of quakes from the late eighties have provided engineers with useful data to study

the nature of rupture directivity embedded in these ground motions. The effects caused

by rupture directivity are often characterized by the existence of velocity- or acceleration-

pulses in the near-fault ground motions. They are important design considerations

9

because they often impose large displacement demands on medium- to long-period

structures.

Broadband directivity models that can be used to modify conventional ground motions to

account for the amplitude and duration effects of rupture directivity have been developed

[Somerville et al., 1997; Abrahamson 2000]. Using strong motion records for a number

of earthquakes in Taiwan and Turkey, Somerville [2003] observed that near-fault rupture

directivity is a narrowband pulse whose period increases with magnitude. Using this

observation, he proposed the use of a one-cycle triangular forcing function to represent

the directivity velocity pulse. On the other hand, Abrahamson [2001] proposed the use of

a single sine-wave forcing function to model the fling effect in acceleration.

Over the years, different velocity pulse shapes such as triangular (Somerville 2003; Hall

et al., 1995; Alavi and Krawinkler 2001), trigonometric (Makris and Chang 2000),

sinusoidal (Sasani and Bertero 2000; Rodriguez-Marek 2000), and modified Gabor

wavelet (Mavroeidis and Papageorgiou 2003; Mavroeidis et al., 2004) have been

proposed to represent the directivity and/or fling effects.

In one of the first papers presented on the effects of near-fault ground motion (NFGM) on

flexible buildings, Hall et al. [1995] presented the displacement function as the integral

of Brune’s [1970] far-field time function. “It was chosen for its smooth shape in

frequency domain.” The paper presented two types of simple ground pulses. One is a

forward-only (non-reversing) displacement, denoted by ground motion A in Fig. 1.5.a;

10

the other is a forward-and-back (reversing) displacement, denoted by ground motion B in

Fig. 1.5.b. The duration of displacement-pulse B is Tp, while that of pulse A is Tp/2. The

peak ground displacements and the peak ground velocities were calculated at 121 stations

for a simulated Mw 7.0 earthquake. The period of pulse was related to the fundamental

period of two shear buildings. Their velocity pulses were idealized in triangular form.

Figure 1.5. Velocity pulses idealized in triangular form: Simple pulse-type ground

motions A (forward motion only) and B (forward-and-back motion) [Hall et. al., 1995]

Alavi and Krawinkler [2001] adopted five velocity pulse forms that were best fitted onto

14 different velocity time-histories. These triangular velocity pulses have forms with 0.5,

1, 1.5, 2 and 2.5 cycles of motion.

12


Pulse P1 and P2 ground acceleration, velocity and displacement time histories.

15

Figure 1.7. Triangular velocity pulses from Alavi and Krawinkler [2001]: P4, P5 and P3

ground acceleration, velocity and displacement time histories.

16

Pulse type was based on an inspection of the time history trace, and on a comparison

between the ground motion and pulse spectral shapes (primarily velocity and

displacement spectra). The period of pulse, Tp was identified from the location of a global

and clear peak in the velocity response spectrum. The velocity and displacement spectra

of these pulses were superimposed on the velocity and displacement spectra of near-fault

records. The best-fit was adopted. “The objective of the work was not to develop pulses

that can accurately replicate recorded ground motion, but to develop pulses that can

reasonably simulate predominant response characteristics of structures located in the

near-fault regions.”

Makris and Chang [2000] introduced “physically realizable” cycloidal pulses, and

illustrated their resemblance to recorded near-source ground motions. A type-A cycloidal

pulse approximates a forward motion, a type-B cycloid pulse approximates a forward-

and-back motion, whereas, a type-Cn pulse approximates a recorded motion that exhibits

n main pulses in its displacement history. The velocity histories of all type-A, type-B and

type-Cn pulses are differentiable signals that result in finite acceleration values. The best-

fit method of pulse identification was adopted by Makris and Chang (Fig. 1.8).

17

Figure 1.8. Idealized trigonometric pulses by Makris and Chang [2000]. Fault-normal

components of the acceleration, velocity and displacement time histories recorded at the

Rinaldi station during the 1994 Nothridge, earthquake (left), a cycloidal type-A pulse

(center) and a cycloidal type-B (right)

Sasani and Bertero [2000] adopted the best-fit method with simple sinusoidal shapes for

velocity pulses. Rodriguez-Marek [2000] also proposed that near-fault forward-

directivity motions can be adequately represented by simplified time-histories consisting

of one or a few sine-pulses. Later, Lili et al. [2005] proposed four Fling-step pulses

(FSP) and four forward-directivity pulses (FDP), as shown in Fig. 1.9.

18

Figure 1.9. Triangular and sinusoidal simple velocity pulses: Description and

classification [Lili et al., 2005]

Sasani [2006] later proposed an equivalent rectangular acceleration pulse, called the

Significant Peak Ground Acceleration (SPGA). The SPGA is defined as the maximum

ratio of the significant variation of the ground velocity (SVGV) and its duration.

In recent work, Jalali et al. [2007] adopted a very simplified approach, and the pulse is

uniquely adopted as Brune’s pulse, [Brune 1970] which is compared to and calibrated

against fault-slip and recorded ground motions in terms of their peak amplitudes in time

and their spectral contents.

19

In well designed work, Mavroeidis and Papageorgiou [2003] and Mavroeidis et al. [2004]

have tried to represent near-fault a velocity pulse by Gabor (1946) wavelet. Twenty

records were analyzed. They have submitted that the idealized pulse captures the time

history and response spectrum characteristics of the actual near-fault records.

Figure 1.10. Pulses from Mavroeidis and Papageorgiou [2003]

Sample of synthetic waveform (black trace) fitted to actual NF records (gray trace).

Ground motion time histories (displacement, velocity and acceleration series), as well as

the corresponding 5% damping elastic response spectra are illustrated.

20

Using the Haskell source model and fifty-two NFGM records, Fu and Menon [2004]

proposed a velocity and its corresponding acceleration pulse model based on the synthetic

ground motions generated.

Table 1.2. Summary of research on velocity pulse identification:

No Article Vel. Pulse Shape Characteristics

1 Hall et al. 1995 Triangular A:for forward

displacement.

B: for forward &

backward displacement

2 Somerville 1998 Random -

3 Somerville 2003 Triangular

Narrow band rupture

directivity, 1 cycle

4 Alavi and Krawinkler 2001 Triangular ½, 1, 1.5, 2 & 2.5 cycles

5 Makris and Chang 2000 Trigonometric/

cycloidal

A, B & Cn types

6 Sasani and Bertero 2000 Sinusoidal ½ & 1 cycle, modified

normalized response

spectra

7 Rodrigue-Marek 2000 Sinusoidal 1 or a few cycles

8 Lili et al., 2005 Triangular &

sinusoidal

4 fling & 4 forward

directivity, 1 or ½ cycles

9 Sasani 2006 Rectangular Acceleration, SPGA

10 Jalali et al., 2007 Brune’s pulse -

11 Mavroeidis and Papageorgiou

2003, 04

Modified Gabor

wavelet

Misc.

12 Baker 2007 Daubechies

wavelet of 4th

order

Misc.

A quantitative classification of NFGM using wavelet analysis has been proposed by

Baker [2007]. He proposed the use of a Daubechies wavelet of the fourth order as the

mother wavelet in the wavelet analysis for pulse extraction, since “it approximates the

shape of many velocity-pulses.”

21

The Wave Propagation Method has been proposed by a few researchers to estimate the

displacement demand of structures in near-fault areas. One of the first proposals is by

Hall et al. [1995]. Iwan, W. D. [1997] proposed the use of a Drift Spectrum to measure

the demand for buildings in pure shear subjected to pulse-like NFGM. Miranda and

Akkar [2006] recently developed a generalized interstory drift spectrum that can be

extended to buildings that may deform laterally in flexure, as well as like a shear beams.

There have been case studies on the behavior of 10 to 20-story regular, as well as

irregular steel moment frame buildings subjected to pulse-like near-fault ground motions.

[Alavi and Krawinkler 2001; Kalkan and Kunnath 2006; Krishnan 2007]

1.4 OBJECTIVES AND SCOPE

The major objective of this study was to develop a basic understanding of the

characteristics of pulse-like near-fault ground motions (NFGM) & its impact on response

characteristics of SDF system. The excitation properties intrinsic to pulses are inherently

different from non-pulse type seismic ground motions.

The response due to most pulse-type seismic ground motions falls in the displacement-

sensitive part of the response spectrum. Because of the use of logarithmic scaling in the

construction of the response spectrum, the displacement characteristics are usually not

adequately represented. Furthermore, when the period of the excitation-pulse is close to

the period of the structure, the resonance characteristics are not often visible on the

spectrum even though the displacement dynamic magnification may be quite high. Over

22

the last two decades, very large displacements have been observed for bridge

superstructures, piers and abutments in the near fault area. Recent research has

attributed these large displacements to pulse-like seismic ground motions.

In a seismic design, design for displacement is just as important as the design for forces.

Because different criteria for displacement, strength, and ductility requirements may be

different for pulse-like or non-pulse-like seismic ground motions, in-depth study of

response characteristics of structures subjected to pulse-like ground motion is warranted.

The excitation properties intrinsic to pulses are inherently different from non-pulse type

seismic ground motions. For example, while structural damping does not have an

appreciable effect on the response characteristics of long-period structures when

subjected to non-pulse type ground motions, it greatly affects the response of long-period

structures under pulse-type ground motions. Such pulse-like excitations affect medium-

to long-period structures, often resulting in excessive displacements during the

earthquake or residual displacements after the cessation of earthquake.

In view of this, the study of SDF system subjected to pulse-like NFGM is the subject of

this research work. Because of the limited number of pulse-like seismic ground motion

samples available world-wide (around seventy-five) when compared to the thousands of

seismic ground motion time histories of non-pulse type (regular) seismic ground motions,

a deterministic (as opposed to a probabilistic) approach is used in the present study.

The principal objectives of this dissertation are four-fold:

23

a) MORE REALISTIC PULSE-SHAPES: Most other researchers have assumed a

predetermined pulse shape in the analysis. The principal objective here was to

explore if there is a better way to extract the near-fault velocity pulse/s by using

signal processing techniques that would yield more realistic pulse-shapes.

b) DYNAMIC MAGNIFICATION: The pulse effect of forward rupture directivity

becomes significant at a period of 0.67 sec. and generally grows in size with

increasing period [Somerville et al., 1997]. These responses generally fall within

the displacement-sensitive region of the response spectrum. One of our objectives

was to statistically evaluate the effect of pulses on the dynamic magnification of a

linearly elastic SDF system.

c) ANALOGOUS DISPLACEMENT RESPONSE SPECTRUM: For rupture

directivity and path effects, the lowermost frequency contents of ground motions -

i.e., frequencies between the lowest usable frequency (LUF) and 1.67 Hz - have

been found to play an important role in the displacement response of a single

degree-of-freedom (SDF) system. Because of limitations in instrumentation and

ground motion processing technology, not all frequency records are reliable. The

lowest usable frequency is defined as the lowest recorded frequency that can be

used with confidence for seismic analysis and design. They can be determined

using methods outlined in http://peer.berkeley.edu/nga/NGA_Documentation.pdf.

LUF value for each earthquake record can be found in the PEER NGA Ground

Motion Library. In the time domain, the pulses are visible in the velocity time

history. However, in the frequency domain, the question we would like to have

an answer for is: what information do the Fourier amplitudes convey within this

24

frequency range? This inquiry resulted in the introduction of an analogous

displacement response spectrum for an undamped linearly elastic SDF system.

d) DISPLACEMENT DUCTILITY REQUIREMENT: Herein, we would like to

explore whether the ductility and strength requirements for structures subjected to

pulse-like NFGM are different from those structures subjected to non-pulse type

(regular) seismic ground motions. This objective resulted in the last chapter of

the dissertation, which examines the displacement ductility requirements for a

typical bridge bent subjected to such extreme loadings.

For rupture directivity and path effects, the lowermost frequency contents of ground

motions - i.e., frequencies between the lowest usable frequency (LUF) and 1.67 Hz - of

have been found to play an important role in the displacement response of a single

degree-of-freedom (SDF) system. Using this information, a methodology has been

proposed (see Chapter 2) to identify these effects. An important difference between the

proposed approach and those of the other researchers is that no predetermined pulse

shape is assumed in the analysis. The natural shape and duration of the pulses are

extracted directly from the processed ground motion records. These extracted pulses are

then used to determine their effects on linearly elastic SDF systems.

The pulses can be identified by the use of a discrete-time signal processing method, in

which a low-pass filter with a suitable cut-off frequency is applied to the Fourier

transforms of the processed acceleration or velocity time history records. Chapter 2

discusses the pulse-effects by plotting displacement response spectra computed from the

25

original, pulse and residual acceleration time histories (ATH). By using just the

acceleration pulse as the excitation force, it has been shown that the displacement

response of a linearly elastic SDF system with a natural period exceeding a certain value-

referred to as the cut-off period Tc (the reciprocal of the cut-off frequency fc) –is quite

comparable to the one caused by the original ground excitation.

Qualitatively, the effects of three types of pulses -monotonically increasing, ripple and

resonance -on the system displacement response are identified. Quantitatively, these

three pulse types are identified though a modified displacement response factor Rd (see

Chapter 3). Because the values of Rd can be as high as 10 to 25 for resonant pulses acting

on an undamped SDF system, it can be concluded that such pulses are the most

devastating. A statistical study of the effects of dynamic magnification caused by near-

fault ground motions, using a modified displacement response factor Rd, is presented.

This Rd is used in conjunction with the acceleration pulses extracted event-wise from

pulse-like ground motions to quantify the elastic response characteristics of a system

having damping ratios ζ = 0, 0.02 & 0.05.

The evaluation and examination of the displacement response spectra (DRS) for a

number of near-fault ground motions have led to the observation that they bear a strong

resemblance to the ground velocity Fourier amplitudes. In the low-frequency range (i.e.,

LUF ≤ f ≤ fc < 1.67 Hz), it is observed that the maximum displacement of a linearly

elastic undamped SDF system having a natural period of Tn ≈ Tj (where Tj is the period

corresponding to the jth

harmonic of the earthquake ground motion) is somewhat related

26

to the amplitude of the jth

harmonic ground velocity. Using this information, a method is

introduced in Chapter 4 to allow engineers to construct analogous displacement response

spectra for an undamped linearly elastic SDF system subjected to long-period ground

motions. This spectrum, which is obtained through the transformation of the velocity

Fourier amplitudes (VFA) of the ground velocity time history (VTH), may be used to

determine design ground motions or may be used in Performance-Based Seismic Design

by relating spectral displacements directly to ground velocities via Fourier amplitudes.

In Chapter 5, the dissertation finally examines the ductility requirements for a typical

bridge bent subjected to such pulse-like near-fault ground motions. The ductility

requirement stated in prevailing design codes may not be valid for such structures in

near-fault zones (of less than 10 miles distance). Due to the dominance of the pulses,

medium- to long-period structures are highly affected, which often results in high

residual or permanent deformations after the cessation of seismic ground motions. As per

most design codes, the magnitude of displacements associated with P-Delta effects is

required to be captured using non-linear time history analysis. The higher the value of

the target displacement ductility demand or the P-Delta ratio, the larger is the magnitude

of the residual displacements. Because permanent deformations in a bridge bent would

lead to higher eccentric loading from the superstructure, the bent would be subjected to

higher secondary moments from its own design load. A suitable balance should therefore

be made between ductility and residual displacements in the design of such structures.

27

The scope of this research is limited by the availability of pulse-like acceleration-time-

history samples worldwide as of to-date. At present, only around seventy-five are

available for use in research. This is a very small number when compared to the

availability of non-pulse type (regular) acceleration-time-history samples, which are in

the thousands. These seventy-five records are arranged event-wise in Chapter 3, and

arranged as per seismic moment magnitude and the soil-types in Chapter 5.

28

CHAPTER 2

PULSE IDENTIFICATION AND DISPLACEMENT RESPONSE EVALUATION

2.1 INTRODUCTION

Directivity effects in ground motions can usually be detected in signal processing using

either Wavelet theory or Fourier analysis. While Wavelet theory is more useful when

transitory characteristics or time information is more crucial in the analysis, Fourier

analysis is more appropriate when the frequency contents of the signals are of greater

importance, as in the case of directivity, when a pulse might not be apparent in the

velocity time history (VTH).

In this chapter, an analytical approach for pulse identification and linear displacement

response evaluation in the frequency domain is proposed. The method makes use of

structural dynamics theory [Clough and Penzien 1992; Chopra 2007; Humar 2002], in

conjunction with digital signal-processing techniques [Oppenheim et al. 1999; Mitra

2006], to develop the necessary filter and displacement frequency response functions for

the analysis. An important difference between the proposed approach and those of the

other researchers is that no predetermined pulse shape is assumed. The natural shape and

duration of the pulses are extracted directly from processed ground motion records.

These extracted pulses are then applied to single degree-of-freedom (SDF) systems, so

that their effects can be determined.

29

Although digital filters have been applied to raw records for some time, the proposed

application of low-pass digital filters to processed records for identifying velocity and/or

acceleration pulses is relatively new. In addition, in order to avoid aliasing in the

reconstructed signals, a suitable type and order of filter with an appropriate cut-off

frequency must be used. A procedure for selecting such a filter is also proposed.

According to Somerville et al. [1997], the pulse effect of forward rupture directivity

becomes significant at a period of 0.6 seconds (or at a frequency of 1.67 Hz) and

generally grows in size with the increasing period. For a pulse-like near-fault ground

motion (NFGM), this observation suggests that the cut-off frequency ( fc or c

ω = 2π fc),

defined herein as the frequency above which the Fourier amplitudes are significantly and

continuously lower, can be identified through visual examination of the Fourier

transforms of the ground velocity time history (VTH), measured in the direction of

maximum velocity.

In some cases (Lucerne, TCU068, etc.), applying Fourier analysis to the VTH will yield a

single peak in the Fourier spectra within a frequency range that is defined by the range

from the lowest usable frequency (LUF) to the cut-off frequencyc

f (which should not

exceed 1.67 Hz). This single peak signifies the presence of a dominant single-period

pulse. However, in other cases (such as KJMA, Kobe 1995; HWSM, Imperial Valley-6

1979; CH1, the Loma Prieta 1989 earthquake, etc.) multiple peaks are observed in the

velocity Fourier spectra in the frequency range LUF ≤ f ≤ fc ≤ 1.67 Hz. The presence of

multiple peaks signifies the existence of multiple-period pulses. These pulses will affect

30

the displacement response characteristics of a linearly elastic SDF system at different

natural vibration periods n

T .

In this dissertation, velocity and acceleration pulses are identified by applying a low-pass

discrete-time (digital) filter ( )lp

H ω at a suitable cut-off frequency, fc, to the Fourier

transforms of the ground velocity and acceleration time histories. The linear

displacement response of the extracted acceleration pulse can then be simultaneously

evaluated as the Fourier integral subjected to an arbitrary excitation in the frequency

domain. The proposed procedure is shown schematically in Fig. 2.1.

System Input System Output

( )

uH ω ( ) ( )

uH Pω ω displacement response, ( )u t Eq. (2.4)

Excitation, p(t) ( )lp

H ω ( ) ( )lp

H Pω ω pulse excitation, ( )p

p t Eq. (2.11)

( ) ( )u lp

H Hω ω ( ) ( ) ( )u lp

H H Pω ω ω displacement due to pulse

excitation, ( )p

u t Eq. (2.14)

Figure 2.1. Solution of the linear displacement response to earthquake pulse/s

As shown in the topmost path of Fig. 2.1, the displacement response of a single degree-

of-freedom (SDF) system can be obtained in the frequency domain through modulation

of ( )P ω , defined as the Fourier transform of an excitation function p(t), by a

displacement response function ( )uH ω . Herein, as shown in the middle path of the

figure, it is proposed that a low-pass filter response function ( )lp

H ω be used to modulate

31

( )P ω to obtain an excitation function pp(t) due to the pulse/s. ( )lp

H ω is also used to

identify the acceleration pulse/s ( )p

a t and velocity pulse/s ( )p

v t . Finally, as per the

lowermost path of Fig. 2.1, the displacement response due to the pulse excitation is

obtained by the simultaneous application of ( )uH ω and ( )lp

H ω . In what follows, a more

detailed discussion of each of the aforementioned paths will be given.

By using only the pulse component of the ground motion as the excitation force, it is

shown that the displacement response of a SDF system with a natural period exceeding a

certain value, referred to as the cut-off period c

T , is quite comparable to that caused by

the original ground excitation. Also, for ground motions that contain multiple-period

pulses, it is shown that the displacement response spectrum of the SDF system exhibits

multiple peaks at different natural system periodsn

T .

2.2 EXCITATION AND ITS RESPONSE IN THE FREQUENCY DOMAIN

The equation of motion for a SDF damped system subjected to an excitation force ( )p t is

given by [Clough and Penzien 1992; Chopra 2007; Humar 2002]

( )mu cu ku p t+ + =&& & (2.1)

where m is the mass, c is the viscous damping coefficient, k is the stiffness, and ( )u t ,

( )u t& and ( )u t&& are the resulting time-dependent displacement, velocity and acceleration of

the system, respectively.

If we express the excitation force p(t) as

32

1

( ) ( )2

i tp t P e d

ωω ωπ

∞

−∞

= ∫ (2.2)

where

( ) ( ) -i tP p t e dt

ωω∞

−∞

= ∫ (2.3)

in which 1= −i and ω is the forcing frequency in radians/second, the response of the

system can be obtained as

1

( ) ( )2

i tu t U e d

ωω ωπ

∞

−∞

= ∫ (2.4)

where

( ) ( ) ( )u

U H Pω ω ω= (2.5)

and

( ) ( )

2

1 1( ) ( / )

1 / 2 /u u n

n n

H Hk

ω ω ωω ω ζ ω ω

= = × − +

i (2.6)

in which /n

k mω = is the natural frequency, and / 2n

c mζ ω= is the damping ratio of

the SDF system. It may be observed that ( )u

H ω modulates the frequencies by

convoluting with the input ( )p t to yield the linear displacement response ( )u t .

2.3 LOW-PASS FILTER RESPONSE FUNCTION, ( )lp

H ω

2.3.1 DIGITAL FILTERS

Frequency-selective filter implies a system that passes certain frequency components and

totally rejects all others, however, broadly it is any system that modulates certain

frequencies relative to others [Oppenheim et al., 1999].

33

Digital filters may be broadly classified into:

1. Discrete-time Infinite Impulse Response (IIR) filters designed from continuous-

time filters, e.g., the Butterworth filter, the Chebyshev filter, etc.

2. Finite Impulse Response (FIR) filters (linear phase) designed by ‘windowing’,

e.g., rectangular, Bartlet, Hanning, Hamming, Blackman, Kaiser etc.

3. FIR filters (nonlinear phase) designed by optimization algorithms like the Parks-

McClellan algorithm, e. g., Equiripple filters.

Let the sampling interval be t∆ and N the total number of sampling points of the record.

Then the sampling ratio, SR, is equal to 1/ t∆ . To avoid ‘aliasing’, the number of Fourier

Transform (FT) points should be equal to or more than the number of sampling points. If

these are fewer than the sampling points, the original time history is not recoverable;

there is a distortion referred to as aliasing distortion, or simply, aliasing. Typical required

low-pass specifications of ( )lp

H ω are depicted in Fig. 2.2., where the limits of tolerable

approximation errors are 1δ and 2δ for passband and stopband, respectively. p

ω and

sω are the passband frequency and the stopband frequency, respectively. Passband is that

band of a spectrum, which frequencies are allowed to pass through (retained). The

frequencies in stopband are discarded. Only in the case of an ideal low-pass filter is the

transition band width zero, as stated in the following section 2.3.2. The transition band is

required to avoid aliasing distortions while reconstructing the modulated signals. The

width of the transition band depends upon the type and order of the filter used.

34

The purpose of following two sections is to introduce two simple filters and to show how

the frequency response function of filter ( )lp

H ω plays an important role in modulating

frequencies.

Figure 2.2. For the low-pass filter, specifications for the effective frequency response of

the overall system. [Oppenheim et al., 1999]

2.3.2 IDEAL LOW-PASS FILTER:

Let the system response to ideal low-pass filter be

1

( )0

c

lp

c

Hω ω

ωω ω π

<=

< ≤

where c

ω is the lowpass cut-off frequency in radians per second. In this case,

p s cω ω ω= =

The corresponding discrete impulse response is, [ ]sin

- clp

nh n n

n

ω

π= ∞ < < ∞

35

An ideal low-pass filter should not be applied to “real” input systems, since it gives rise

to aliasing distortions.

2.3.3 BUTTERWORTH LOW-PASS FILTER [Oppenheim et al., 1999]

Although numerous low-pass filters are available in the literature, Butterworth low-pass

filters will be used for the present study. Butterworth low-pass filters are characterized

by the property that the magnitude response is maximally flat in the passband, and that

the magnitude response is monotonic in both the passband and stopband. In particular,

the continuous-time Butterworth low-pass filter is expressed as

( )

1( )

1 /iB

lp N

c

H ωω ω

=+

(2.7)

where 1= −i , c

ω is the desired cut-off frequency in radians/second, and B

N is the

order of Butterworth filter. The corresponding magnitude-squared function is,

( )

2

2

1( )

1 / Blp N

c

H ωω ω

=+

(2.8)

A plot of ( )lp

H ω versus /c

ω ω is given in Fig. 2.3. Note that asB

N increases, the filter

characteristics become sharper, although the magnitude-squared functions at the cut-off

frequency c

ω will always equal one-half because of the nature of Eq. (2.8). AnB

N value

of 32 is used in the current study because, as indicated in the figure, the transition band

becomes extremely narrow at this value. It should also be noted that while “low-cut” and

“high-cut” Butterworth filters are often used to process raw earthquake records to

36

eliminate the lowest and uppermost unwanted frequencies, this low-pass Butterworth

filter retains all frequencies lower than c

ω , but eliminates all frequencies higher than c

ω .

Figure 2.3. Effect of NB on Butterworth low-pass filter magnitude.

2.4 PULSE IDENTIFICATION AND

CUT-OFF FREQUENCY DETERMINATION

For directivity or path effect to be dominant, the peak ground velocity (PGV) in the

direction of maximum velocity is usually larger than 30 cm/s [Baker 2007]. In this study,

digital filters are applied to ground motion time history responses to identify the pulse/s.

If we denote ( )A ω and ( )V ω as the Fourier transforms of the ground acceleration ( )a t

and velocity ( )v t time histories, respectively, the following equations are proposed for

the computation of time histories for acceleration and velocity pulses:

1

( ) ( ) ( )2

i t

p lpa t H A e d

ωω ω ωπ

∞

−∞

= ∫ (2.9)

NB = 4

NB = 8

NB = 16

NB = 32

( )lp

H ω

ω/ωc

37

1

( ) ( ) ( )2

i t

p lpv t H V e d

ωω ω ωπ

∞

−∞

= ∫ (2.10)

In the above equations, ( )lp

H ω is the low-pass Butterworth filter function given by Eq.

(2.7). Similarly, the time history for pulse excitation due to ground motion ( ) ( )p t ma t= −

can be obtained using

1

( ) ( ) ( )2

i t

p lpp t H P e d

ωω ω ωπ

∞

−∞

= ∫ (2.11)

where ( )P ω is the Fourier transform of ( )p t given by Eq. (2.3).

Because directivity effects are most significant for frequencies less than 1.67 Hz (i.e., a

period longer than 0.6 seconds), this criterion is used in the present study to identify the

pulse characteristics. Based on a study of about seventy pulse-like ground motion

records, the following procedure is proposed to determine the cut-off frequency to be

used in Eq. (2.7).

1. For a given set of horizontal ground motion acceleration data recorded at a specified

station, compute the component of the quake in the direction of maximum velocity.

For a strike-slip faulting mechanism, this component is generally in the fault-

normal (FN) direction.

2. Numerically integrate the ground acceleration time history obtained in (1) to obtain

the velocity time history.

3. Perform a Fourier analysis on the velocity time history to obtain the velocity

Fourier spectrum.

38

4. Identify high amplitude spikes in the velocity Fourier spectrum within the window

of frequencies ranging from the Lowest Usable Frequency (LUF), as given in the

ground motion data source, to 1.67 Hz.

5. Determine the cut-off frequency (fc or c

ω ) inside the aforementioned frequency

window as the frequency above which the velocity Fourier amplitudes will become

significantly and continuously lower.

The following examples are used to demonstrate the application of the above procedure.

Example 1 – The Fault-normal component of Nishi-Akashi, Kobe 1995 earthquake

For this record, no velocity pulse is present in the velocity time history (VTH). To verify

this, the velocity Fourier spectrum for the fault-normal (FN) component of this

earthquake at the Nishi-Akashi station was computed and plotted and is shown in Fig.

2.4. Because of the existence of high amplitude spikes beyond the 1.67 Hz frequency

range (i.e., outside of the frequency range of interest), it can be concluded that no pulse

exists for this ground motion. The absence of directivity has been confirmed by

Mavroeidis and Papageorgiou [2003] and Mavroeidis et al. [2004] although Nishi-Akashi

is only 7.08 km away from the fault rupture.

Although directivity effects might not be present, this record (soil type B) has dominant

site effects, as is evident from the presence of high, steep and narrow spikes in the

velocity Fourier spectra for frequencies lower than 2.5 Hz.

39

Figure 2.4. Fourier spectra of the velocity time history

(Neutral directivity at Nishi-Akashi, 140-FN, Kobe 1995)

Example 2 – The Fault-normal component of Lucerne, Landers 1992 earthquake

The velocity Fourier spectra for the FN component of a 1992 earthquake at the Lucerne

station is shown in Fig. 2.5. It can be seen that only one spike (at f = 0.083 Hz

0.1HzLUF≈ = ) exists within the frequency range of interest (i.e., LUF≤ f ≤1.67 Hz).

As a result, it can be concluded that directivity is present. This observation is confirmed

in the velocity time history plot shown in Fig. 2.6. Also shown in the figure are the PGV

and PGD values, as well as the displacement time history plot to indicate that a fling of

170.9 cm is present. The lowermost frequency beyond which amplitudes of the velocity

Fourier transforms start to attenuate is 0.395 Hz. Thus, this frequency is determined to be

the cut-off frequency fc. This example typifies the occurrence of a “single dominant

period pulse” that sometimes occurs in near-fault ground motions (NFGM).

f (Hz)

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

40

Ve

locity (

cm

/s),

Dis

pla

ce

me

nt (c

m)

Time (sec)

Figure 2.5. Fourier spectra of the velocity time history showing a single peak

(Single-period pulse at Lucerne, 239-FN, Landers 1992)

Figure 2.6. Velocity and displacement time histories

(Single-period pulse at Lucerne, 239-FN, Landers 1992)

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

f (Hz)

f = 0.083 Hz

fc =0.395 Hz

PGD= 236.0 cm

Fling = 170.9 cm

PGV= 136.1 cm/s

Velo

city (

cm

/s),

Dis

pla

ce

ment

(cm

)

41

Example 3 – The Fault-normal component of the Westmorland Fire Station,

Imperial Valley-6 1979 earthquake

The velocity Fourier spectra for the FN component of this earthquake at the Westmorland

Fire Station is shown in Fig. 2.7. It can be seen that two spikes (at f = 0.1 Hz and 0.225

Hz) exist within the frequency range of interest (i.e., LUF=0.1 Hz ≤ f ≤ 1.67 Hz). The

lowermost frequency beyond which the velocity Fourier amplitudes start to attenuate is

0.475 Hz. This frequency is therefore identified as the cut-off frequency fc. This

example is used to show the existence of two dominant spikes in the velocity Fourier

spectra, which correspond to the two distinct peaks in the displacement response spectra

at Tn ≈ 4.7 sec and 9.5 sec (see Fig. 2.13).

Figure 2.7. Fourier spectra of velocity time history showing multiple peaks

(Multiple-period pulses at the Westmorland Fire Station (HWSM),

233-FN, Imperial Valley-6 1979)

f (Hz)

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

f = 0.225 Hz

fc = 0.475 Hz

42

2.5 PULSE RESPONSE EVALUATION

If we decompose the ground motion into its pulse and non-pulse components, i.e.,

)()()()()( tmatmatptptp nppnpp −−=+= (2.12)

where ( )p

a t is the acceleration associated with the pulse given by Eq. (2.9), and ( )np

a t is

the residual acceleration (i.e., the acceleration excluding the pulse), the displacement

response that corresponds to the pulse component ( ) ( )p p

p t ma t= − can be evaluated

using

1

( ) ( ) ( )2

i t

p u pu t H P e d

ωω ω ωπ

∞

−∞

= ∫ (2.13)

Since ( ) ( ) ( )p lp

P H Pω ω ω= and realizing that ( )u

H ω and ( )lp

H ω are essentially

functions of ( / )n

ω ω and ( / )c

ω ω , respectively, Eq. (2.13) can be written as

( ) ( )1

( ) / / ( )2

i t

p u n lp cu t H H P e d

ωω ω ω ω ω ωπ

∞

−∞

= ∫ (2.14)

where ( / )u n

H ω ω , ( / )lp c

H ω ω , and ( )P ω are given by Eqs. (2.6), (2.7), and (2.3),

respectively.

On the other hand, the displacement response that corresponds to the non-pulse

component of the excitation )()( tmatp npnp −= can be evaluated using

( ) ( )1

( ) / / ( )2

i t

np u n hp cu t H H P e d

ωω ω ω ω ω ωπ

∞

−∞

= ∫ (2.15)

43

where ( / )u n

H ω ω and ( )P ω are defined as before, and ( )hp

H ω is the Butterworth high-pass

filter given by

( ) ( )

( / )1( ) 1 ( ) 1

1 / /i i

B

B B

N

chp lp N N

c c

H Hω ω

ω ωω ω ω ω

= − = − =+ −

(2.16)

It should be noted that this high-pass filter retains frequencies higher than the cut-off

frequency (fc or c

ω ) and discards frequencies lower than the cut-off frequency. Thus, the

application of this filter to the original system response would yield system response

excluding the pulse-effect.

By evaluating and plotting ( )u t , ( )p

u t and ( )np

u t on the same graph, one can compare

and observe the displacement response characteristics for SDF systems having different

natural periods. This comparison will be performed in the following section.

Because ground motion is intrinsically random in nature, all the aforementioned Fourier

integrals have to be performed numerically. For instance, to compute displacement

response given in Eq. (2.14), the following discretized form of the equation is used

( ) ( ) ( ) ( ) ( )1 1

i( ) i(2 / )

0 0

j n

N Nt nj N

p u lp j u lp jj jn j jj j

u H H P e H H P eω π

− −

= =

= =∑ ∑ (2.17)

where N is the total number of samples (i.e., the number of recorded data points), and

( ) ( )2

1 1( )

1 / 2 /u j

j n j n

Hk

ωω ω ζ ω ω

= × − +

i (2.18)

is the discretized form of Eq. (2.6).

( )1

( )1 /

Blp j N

j c

H ωω ω

=+ i

(2.19)

44

is the discretized form of Eq. (2.7), and

1 1

( ) (2 / )

0 00

1 1j n

N Nt nj N

j n n

n n

P p e t p eT N

ω π− −

− −

= =

= ∆ =∑ ∑i i (2.20)

is the discretized form of Eq. (2.3).

The ( )j

H ω values in Eqs. (2.18) and (2.19) are determined with the following

interpretation ofj

ω :

0

0

0 /2

-( - ) /2 -1j

j j N

N j N j N

ωω

ω

≤ ≤=

≤ ≤ (2.21)

In the above equation, 0 02 /Tω π= is the fundamental frequency and j

ω is the frequency

of the jth harmonic. To avoid aliasing, the highest frequency that needs to be used in a

Fourier analysis, known as the Nyquist or folding frequency, is denoted as

max 0 ( / 2) /N tω ω π= = ∆ , where 0 /t T N∆ = . The shortest and longest periods of the

harmonics included in the Fourier expansion are thus determined to be 2 t∆ and T0,

respectively.

Similarly, Eqs. (2.9) and (2.10) are evaluated using the following discretized forms:

( ) ( ) ( )1 1

2 /

0 0

( ) ( )j n

N Nt nj N

p lp j j lp j jnj j

a H A e H A eω πω ω

− −

= =

= =∑ ∑i i

(2.22)

( ) ( ) ( )1 1

2 /

0 0

( ) ( )j n

N Nt nj N

p lp j j lp j jnj j

v H V e H V eω πω ω

− −

= =

= =∑ ∑i i

(2.23)

with the complex Fourier coefficients Aj and Vj computed from

1 1

( ) (2 / )

0 00

1 1j n

N Nt nj N

j n n

n n

A a e t a eT N

ω π− −

− −

= =

= ∆ =∑ ∑i i (2.24)

and

45

1 1

( ) (2 / )

0 00

1 1j n

N Nt nj N

j n n

n n

V v e t v eT N

ω π− −

− −

= =

= ∆ =∑ ∑i i (2.25)

2.6 PULSE EFFECTS

From a study of more than seventy pulse-like ground motions, it has been observed that

the so-called velocity and acceleration pulses are merely dominant low-frequency

sinusoidal waves that occur in the lower-most range of the Fourier spectra. Although

these pulses are sometimes identifiable from the ground velocity time history, they are

not so apparent in the ground acceleration time history. In near-fault areas, they manifest

themselves as directivity effect, whereas in the far-fault areas they manifest as path effect.

In Fig. 2.8, the velocity pulse from the Lucerne 239-FN, Landers 1992 earthquake

(Mw=7.28) obtained using Eq. (2.23) is superimposed on the original ground velocity

time history to show that it consists of a single dominant wave, as confirmed by other

researchers. However, in Fig. 2.9 it can be seen that the acceleration pulse obtained using

Eq. (2.22) is not so visible in the ground acceleration time history.

46

Time (sec)

Time (sec)

Figure 2.8. Original and pulse velocity time histories

(Single dominant period velocity pulse at Lucerne, 239-FN, Landers 1992)

Figure 2.9. Original and pulse acceleration time histories

(Lucerne, 239-FN, Landers 1992)

105.3 cm/s

PGV= 136.1 cm/s

Velo

city (

cm

/s)

PGA= 0.691g

0.120 g

Accele

ratio

n (

g)

47

Time (sec)

It is important to note that the presence of pulse/s can strongly affect the displacement

time history (DTH) of the earthquake. In the case of Lucerne, it can be seen from Fig.

2.10 that the DTH computed using just the pulse component of the quake almost overlaps

that computed using the original ground motion record (i.e., both the pulse and non-pulse

components), signifying the predominant effect of the pulse on the displacement

response, and the relatively insignificant effect of the non-pulse component of the

excitation. Note that the proposed method is able to extract the fling effect of the quake

as well. Fling is a permanent ground displacement induced by the earthquake and is an

important design consideration, especially for long-span structures such as bridges. The

oscillatory pattern seen at the beginning and end of the computed time history of the

pulse is due to the Gibbs phenomenon, which results when a finite number of terms are

used in the Fourier analysis to represent a highly discontinuous or singularity function.

Figure 2.10. Original, pulse and residual displacement time histories

(Lucerne, 239-FN, Landers 1992)

Fling = 170.9 cm

PGD = 236 cm

Note the Gibbs phenomenon at the ends

Dis

pla

cem

ent

(cm

)

48

Natural period, Tn (sec)

Fig. 2.11 shows the spectral displacements for a linearly elastic SDF system computed

using the original, pulse and non-pulse components of the ground acceleration time

history (ATH), for two damping ratios (ζ=0 and 0.05). Although the peak amplitude of

the acceleration pulse (0.120 g), as indicated in Fig. 2.9, seems rather small when

compared to that of the original ATH, it can be seen that, regardless of the value of ζ, the

spectral displacements computed using just the pulse ATH and the original ATH are

quite comparable when the period of the SDF system is longer than the cut-off period Tc,

which is equal to 2.5 sec. for this record. This indicates that the displacement response of

a SDF system can be determined with a relatively high degree of accuracy by using just

the extracted acceleration pulse provided that the system period is longer than Tc.

Figure 2.11. Displacement response spectra computed from the original, pulse and

residual acceleration time histories (Lucerne, 239-FN, Landers 1992, ζ = 0 and 0.05)

Because no predetermined pulse duration or wave form is used to represent the pulse, an

important feature of the proposed method is its ability to extract multiple-period pulses

ζ=0

ζ=0

ζ=0

ζ=0.05

ζ=0.05

ζ=0.05

pulse Tc = 2.5 s

Spectr

al d

ispla

cem

ents

(cm

) ζ = 0

ζ=0.05

resid, ζ = 0 & 0.05

49

Time (sec)

from long-period ground motions. An example is the Westmorland Fire Station

(HWSM) 233-FN Imperial Valley-6 1979 earthquake (Mw=6.53). Using a cut-off

frequency of fc=0.475 Hz, as shown in Fig. 2.7, the acceleration pulses are extracted as

depicted in Fig. 2.12, and the corresponding spectral displacements for two damping

ratios (ζ = 0 and 0.05) are plotted, as shown in Fig. 2.13. It can readily be seen from the

figure that the spectral displacements dominantly peaked at two locations at n

T ≈ 4.7 sec

and 9.5 sec, which correspond to the two pulses. Furthermore, the spectral displacements

are very comparable regardless of whether they are computed from the original or the

pulse ATH, again signifying the predominant effect of the pulses on the system response.

Figure 2.12. Original and multiple-period pulse acceleration time histories

(Westmorland Fire Station (HWSM), 233FN, Imperial Valley-6 1979 earthquake)

Accele

ratio

n (

g)

(cm

/s)

50

Natural Period, Tn (sec)


residual acceleration time histories (Westmorland Fire Station (HWSM),

233-FN, Imperial Valley-6 1979 earthquake, ζ = 0 and 0.05.)

Finally, as an example of the path effect, the ground motion record for the Oakland -

Outer Harbor Wharf (CH1), 038-FN, Loma Prieta 1989 earthquake (Mw = 6.93) is shown

in Fig. 2.14, along with the extracted acceleration pulses. The corresponding spectral

displacements, calculated using two damping ratios, are given in Fig. 2.15. The

similarity of the spectral displacements computed using just the pulse ATH and the

original ATH is again observed. This station is located at a distance of 74.26 km (46.14

miles) from the fault-rupture. Far-source ground motions consist of surface waves that

sometimes have longer durations than those of the near-fault ones. They can be very

damaging in certain circumstances. This example shows that the proposed approach for

acceleration pulse extraction can be used for both directivity and path effects in

evaluating response characteristics, since long-period ground motions are dominant in

both the cases.

ζ=0

ζ=0

ζ=0

ζ=.05

ζ=.05

ζ=.05

Spectr

al d

ispla

cem

ents

(cm

)

resid, ζ = 0 & 0.05

ζ = 0

ζ = 0.05

51

Figure 2.14. Original and multiple-period pulse acceleration time histories

(Oakland - Outer Harbor Wharf (CH1), 038-FN, Loma Prieta 1989 earthquake)


residual Acceleration time histories, (Oakland - Outer Harbor Wharf (CH1), 038-FN,

Loma Prieta 1989 earthquake, ζ = 0 and 0.05)

0.106 g

Accele

ratio

n (

g)

Time (sec)

Natural Period, Tn (sec)

Spectr

al d

ispla

cem

ents

(cm

)

ζ=0

ζ=0

ζ=0

ζ=0.05

ζ=0.05

ζ=0.05

resid, ζ = 0 & 0.05

ζ = 0

ζ = 0.05

52

2.7 DISCUSSION AND OBSERVATIONS

Based on the above examples, the investigation of the response of an elastic SDF system

to long-period ground motions has led to the following observations:

1. The spectral displacement of a SDF system due to acceleration pulse excitation is

very comparable to that caused by the original acceleration time history excitation

when the period of the SDF system is equal to or higher than the cut-off period c

T ,

which is equal to the reciprocal of the cut-off frequency c

f .

2. This study reaffirms the common observation that long-period structures are

strongly affected by pulse-like long-period ground motions. This is because large

displacements are observed in all these displacement spectra.

3. A single peak in the velocity Fourier spectrum corresponds to the presence of a

single-period pulse, whereas multiple peaks in the lower frequency range of Fourier

spectrum correspond to the existence of multiple-period pulses. Multiple-period

pulses have the potential to significantly affect the displacement response of a

linearly elastic SDF system at different natural periods n

T , or a multi-degree-of-

freedom system with multiple natural periods.

4. From Figs. 2.11, 2.13 and 2.15 in which spectral displacements are shown for two

values of damping ratios (ζ = 0 and 0.05), it can be seen that damping plays an

important role in extenuating the displacement response of the system, even in the

53

displacement-sensitive part of the spectra (i.e., for the natural period of vibration

nT > 0.63 s).

5. From Figs. 2.11, 2.13 and 2.15, it can be seen that the spectral displacement

response for the non-pulse component of the ground excitations does not appear to

change noticeably with damping. This observation supports the general notion that

the effect of damping is not apparent for long-period structures subjected to short-

period (high frequency) excitations.

6. While some NFGM records with a predominant single-period pulse – in cases such

as Lucerne, Landers 1992 – have the distinct characteristic that the spectral

displacements (Fig. 2.11) increase more or less monotonically from the cut-off

period c

T almost to the Highest Usable Period (equal to the reciprocal of Lowest

Usable Frequency as given in the ground motion data source), most records with

multiple periods – such as the Westmorland Fire Station, Imperial Valley-6 1979

(Fig. 2.13) and Oakland – Outer Harbor Wharf, Loma Prieta 1989 (Fig. 2.15) –

show spectral displacements that often exhibit a ripple effect over a wider range of

natural periods that are higher than c

T . Since c

T in some cases may be from 0.6 to

1.0 second, even medium-period structures may be affected by such ground

motions. It is important to note that these two characterizations are not apparent

when conventional tripartite response spectra are used.

54

2.8 PARSEVAL’S THEOREM AND PULSE ENERGY

In this section, the energy associated with the pulses is computed using Parseval’s

theorem [Mitra 2006] to demonstrate that these pulses contribute a considerable amount

of energy to the total seismic energy of a long-period ground motion.

If we denote x(t) as the amplitude of a signal in the time domain and X(ω) as the Fourier

transform of x(t), the simplified version of Parseval’s theorem states that the total energy

of the signal can be computed from the following relationship

( ) ( )2 21

2x t dt X dω ω

π

∞ ∞

−∞ −∞

=∫ ∫ (2.26)

In discretized form, the above equation can be written as

[ ] [ ]1 1

2 2

0 0

1N N

n k

x n X kN

− −

= =

=∑ ∑ (2.27)

This equation can be used to evaluate the energy of the pulses if we set [ ] [ ]x n v n=

and [ ]0[ ]X k V kω= , where [ ]v n and [ ]0V kω are the velocity time history and its Fourier

transforms, respectively. The calculated ratios of energy generated by the velocity

pulse/s to that generated by the entire velocity time history are 0.952, 0.952 and 0.701 for

the Lucerne 239-FN (near-fault), Westmorland Fire Station 233-FN (near-fault), and

Oakland –Outer Harbor Wharf 038-FN (far-fault) ground motions, respectively. Thus, it

is observed that these velocity pulses carry quite a large percentage of the total seismic

energy generated from the rupture.

55

2.9 SUMMARY AND CONCLUSIONS

In this chapter a procedure for identifying velocity and acceleration pulses for near-fault

ground motion records was proposed. The procedure involves the application a Fourier

analysis to the velocity time history in the direction of maximum velocity, which is

generally fault-normal (FN) for a strike-slip faulting mechanism.

Single- and multiple-period pulses are identified as the existence of one or more large

amplitude spikes in the velocity Fourier spectrum within the frequency range of LUF to

1.67 Hz, where LUF is the lowest usable frequency as specified in the ground motion

data source. For the present research, all the ground motion records were accessed from

PEER NGA data source. Once these pulses are identified, the cut-off frequency is

determined as the lowest frequency on the Fourier spectrum above which the Fourier

amplitudes will attenuate. The cut-off period is then computed as the reciprocal of this

cut-off frequency.

Using this cut-off frequency as input to a Butterworth low-pass filter, Fourier integral and

discretized equations capable of extracting acceleration and velocity pulses from the

original ground motion records were proposed. An analytical equation for computing the

displacement response of a SDF system subjected to these pulses was also presented.

Examples were given to demonstrate the validity of the proposed procedure.

Furthermore, the Parseval Theorem was used to compute the amount of seismic energy

generated by these pulses.

56

Based on the results of this study, the following conclusions can be drawn:

1. Frequencies less than 1.67 Hz play a vital role in pulse-like near-fault ground

motions (NFGM). Because for most pulse-type NFGM records, the velocity

Fourier amplitudes for frequencies lower than 1.67 Hz are often more dominant

than those for frequencies higher than 1.67 Hz, this observation has been used in

the proposed procedure for pulse identification.

2. Long-period (low-frequency contents) pulse-like ground motions can significantly

affect the relevant displacement characteristics of SDF systems.

3. When the pulse displacement time history (DTH) that corresponds to the pulse/s

is plotted with the original DTH, they almost overlap. This demonstrates the

predominant effect of long-period ground motions on DTH.

4. Because most pulse-like NFGM consist of multiple period pulses, representation

of the pulse by characterizing it as a single-period wave-form with a specific

duration may be unrealistic in most cases.

5. Spectral displacement plots of pulse-like NFGM at different damping ratios have

shown that the displacement responses due to the identified pulse/s are very

comparable to those caused by the original excitations when the fundamental

period of the system is longer than its cut-off period c

T , which is defined as the

reciprocal of the cut-off frequency fc.

6. The energy associated with the pulse/s computed using Parseval’s theorem shows

that a considerable amount of the total seismic energy is due to this/these pulse/s.

57

7. Since the displacement response has been evaluated in the frequency-domain, the

application is theoretically restricted only to linearly elastic systems. However,

the identified and extracted pulse/s can be used in the time-domain for both linear

and nonlinear seismic demand computations.

Thus, in this chapter we have extracted acceleration/velocity pulse/s from the original

acceleration time history by discrete-time signaling process using low-pass Butterworth

filter with suitable a cut-off frequency. We have further demonstrated that by using only

the pulse component of the ground motion as the excitation force, the displacement

response of the SDF system with a natural period exceeding a certain value referred to as

the cut-off period c

T , is quite comparable to that caused by the original ground excitation.

Because most NFGM consist of multiple-period pulses, representation of the pulse by

characterizing it as a single-period wave-form with a specific duration may be unrealistic

in most cases. In the next chapter we will proceed to demonstrate this. We will also

proceed to study the qualitative and quantitative nature of pulses, and evaluate the

modified displacement response factor event-wise.

58

CHAPTER 3

Modified Displacement Response Factor, Rd (Dynamic Magnification Factor)

3.1 INTRODUCTION AND OVERVIEW

Long-period near-fault ground motions (NFGMs) occur as a result of the source effects

of rupture directivity. Because of path effects, large subduction-zone earthquakes, as

well as moderate to large crustal earthquakes, can generate far-source long-period ground

motions in distant sedimentary basins. These effects are important design considerations

because they often impose large demands on medium- to long-period structures.

As elaborated in Chapter 2, a discrete-time signal processing method was proposed to

isolate the low-frequency contents of processed ground motions, from which velocity and

acceleration pulses were identified after application of low-pass discrete-time (digital)

filter ( )lpH ω , at a suitable cut-off frequency, fc, to the Fourier transforms of their

respective time histories. By using the acceleration pulse as the excitation force, it was

also shown that the displacement response of a linearly elastic single degree-of-freedom

(SDF) system with natural period exceeding a certain value, referred to as the cut-off

period, is quite comparable with that caused by the original ground excitation. A brief

review of the procedure will be given in Sec. 3.2.

59

In this chapter 3, the displacement response characteristics of a linearly elastic SDF

system subjected to pulse-like excitation is studied qualitatively and quantitatively.

Qualitatively, three types of pulse effects are identified:

(1) Monotonically increasing displacement: The spectral displacements increase

more or less monotonically with the natural period n

T of the structure. This

generally occurs when the acceleration pulse has a single predominant period. (i.

e., a single predominant frequency peak in the Fourier spectrum, in the frequency

range LUF ≤ f ≤ fc ≤ 1.67 Hz).

(2) Ripple effect: The spectral displacement peaks at different natural periods. This

generally occurs for multiple-period pulses (multiple frequency peaks in Fourier

spectrum in the said frequency range)

(3) Resonance effect: This is a special case of multiple-period pulses in which the

spectral displacements are very high compared to their static counterparts. In the

velocity Fourier spectrum, these are apparent as high and steep (high gradient)

Fourier amplitudes.

To quantitatively identify these three types of pulse effects, the concept of a modified

displacement response factor d

R is introduced. Based on the study presented in this

chapter, it can be concluded that out of these three pulse types, resonant pulses are the

most devastating ones because d

R for these pulses can be as high as 10 to 25 for an

undamped system (i.e., for 0ζ = ).

60

3.2 BRIEF REVIEW



( ) ( )mu cu ku p t ma t+ + = = −&& & (3.1)

where m is the mass, c is the viscous damping coefficient, k is the stiffness, a(t) is the

ground acceleration and ( )u t , ( )u t& and ( )u t&& are the resulting time-dependent

displacement, velocity and acceleration of the system, respectively. For a given seismic

ground acceleration time history a(t), the corresponding velocity time history v(t) and

displacement time history s(t) can be readily obtained through successive integration. If

we denote any of these response time histories as x(t), and upon decomposing x(t) into its

pulse and non-pulse components:

)()()( tnpxtpxtx += (3.2)

The time history can be separated into its harmonic components by Fourier integral

1

( ) ( )2

tx t X e d

ωω ωπ

∞

−∞

= ∫i (3.3)

where

( ) ( )t

X x t e dtωω

∞

−∞

= ∫-i (3.4)

is the Fourier transform of x(t), 1= −i , and ω is the forcing frequency in radians/sec.

Butterworth low-pass filter is given by [Oppenheim et al., 1999]

61

( )

1( )

1 / Blp N

c

H ωω ω

=+ i

(3.5)

where ωc is the cut-off frequency in radians/sec., and NB is the order of the Butterworth

filter. By using a suitable cut-off frequency fc=ωc/2π, the pulse component can be

evaluated using the equation

1( ) ( ) ( )

2

t

p lpx t H X e d

ωω ω ωπ

∞

−∞

= ∫i (3.6)

The pulse xp(t) obtained from Eq. (3.6) can be used in a time-domain analysis to compute

spectral displacements. In the following study, Eq. (3.6) is used to compute acceleration

and velocity pulse histories. It should be noted that the pulse component of a time history

is considered to be synonymous with the dominant low-frequency contents of that time

history. The non-pulse component of the time history is obtained by applying a high-pass

filter Hhp(ω) = 1 – Hlp(ω) to the original processed time history.

3.3 MODIFIED DISPLACEMENT RESPONSE FACTOR, d

R

The Displacement Response Factor Rd [Chopra 2007], also called the Dynamic

Magnification Factor [Clough and Penzien 1992], is the ratio of the amplitude 0u of the

maximum dynamic deformation to the maximum static deformation 0( )st

u when a

linearly elastic SDF system is subjected to a harmonic excitation 0( ) sinp t p tω= or

0 cosp tω , where 0p is the amplitude or maximum value of the excitation andω is the

excitation frequency.

62

Here, the modified displacement response factor is defined as the ratio of the maximum

amplitude 0u D≡ of the dynamic deformation to the maximum static deformation 0( )st

u

when a linearly elastic SDF system is subjected to a pulse excitation ( )pp t (obtained

from Eq. (3.6)) by replacing ( )px t by ( )pp t ) in the dynamic analysis, and to a static load

0 0p ma= − (where 0p is the amplitude or maximum value of the excitation having

excitation frequency cω ω≤ ) in the static analysis. This pulse-ATH of a typical NFGM

is obtained when the original ATH is subjected to low-pass filter ( )lpH ω as described

above in Sec. 3.2. Despite the modification, Rd is still used to denote this modified

displacement response factor. The procedure to evaluate Rd is explained in the following

discussion, using the TCU052-N322E, Chi Chi 1999 earthquake as an example.

Example 1 – N322E Component of TCU052, Chi-Chi 1999 Earthquake

The procedure to calculate Rd is given as follows:

(1) Compute the fault-normal component of the acceleration time history (ATH) for the

case of a strike-slip faulting mechanism, or the component of ATH in the direction of

maximum velocity for the case of a reverse or reverse-oblique faulting mechanism. For

TCU052, the direction of maximum velocity is N322E [Shabestari and Yamazaki 2003].

(2) From the velocity Fourier spectrum, identify the cut-off frequency (fc or c

ω ) as the

frequency above which the velocity Fourier amplitudes will become significantly and

continuously lower. From Fig. 3.1a, c

f = 0.167 Hz is identified. (It should be noted that

63

for this record the response characteristics do not appreciably change even if c

f = 0.278

Hz is used.) Also in Fig. 3.1a, it can be seen that within the frequency

rangec

LUF f f≤ ≤ (where LUF is referred to as the lowest usable frequency given as

0.05 Hz in the ground motion record), only a single peak at f = 0.055 Hz is observed.

This suggests that the pulse is of the monotonic type. Sincec

f in most such cases has

been observed to be less than 1 Hz, generally speaking structures with longer periods are

affected by such pulses.

(3) By applying the Butterworth low-pass filter ( )lpH ω to the ATH, as in Eq. (3.6), we

can extract the pulse-ATH as shown in Fig. 3.1.b. From this figure, the peak amplitude

ao of the pulse-ATH is obtained as 0.087g, and the value of the adjacent high peak is

0.070g. The corresponding period of the pulse (i.e., the time needed for the pulse to go

through one cycle as indicated in the figure) is Tp = 9.04 sec. If a SDF system is

subjected to a static load of 0p , the maximum static displacement 0( )st

u is given as

2

0 0 00 2 2

( )4

nst

n

p ma a Tu

k mω π= = = (3.7)

(4) Evaluate and plot the displacement response spectra (DRS) for the original ATH,

pulse-ATH, and a simulated single sine-pulse with 0ζ = , along with the pulse-ATH for

0.05ζ = (Fig. 3.2.a). From the DRS, it can be observed that this pulse-record

exemplifies the first type of pulse response, in that the spectral displacement increases

more or less monotonically with the natural period n

T of the system, from the cut-off

period to Tn = 18 sec. The latter corresponds to the frequency f = 0.055 Hz, associated

with the first spike, as shown in Fig. 3.1.a.

64

Close examination of Fig. 3.2.a indicates that, if this SDF system is subjected to an

excitation force consisting of just the acceleration pulse (having frequencies less than

0.167 Hz), the displacement response of the system with a natural period exceeding the

cut-off period of c

T = 6 sec. (reciprocal of c

f = 0.167 Hz) is quite comparable with that

caused by the original ground excitation. If the displacement response due to a single

sine-pulse with ao = 0.087g and Tp = 9.04 sec. is superimposed on the spectral

displacement plot, it can be seen that it shows a similar trend, from the cut-off period c

T

= 6 sec. to almost n

T =16 sec.

(5) If we denote D as the spectral displacement due to the pulse-ATH at a natural period

of Tn, the modified displacement response factor is computed as

2

2

0 0

4

( )d

st n

D DR

u a T

π= = (3.8)

65

Time (sec)

Figure 3.1. TCU052, N322E, Chi – Chi 1999 earthquake:

(a) velocity Fourier spectra (VFA), (b) pulse-ATH.

fc

f =0.055 Hz

Velo

city F

ourier

Am

plit

ud

es

f (Hz)

Pu

lse A

cce

lera

tion

(g)

0.087g

Tp = 9.04 s

0.070 g

f =0.122 Hz

(a)

(b)

66

Natural period, n

T (sec)

Figure 3.2.a TCU052, N322E, Chi – Chi 1999 earthquake: (a) displacement response

spectra

Figure 3.2.b TCU052, N322E, Chi – Chi 1999 earthquake: (b) d

R spectra.

ζ=0.0

ζ=0.02

ζ=0.05

Tn/Tp

Rd

Spectr

al D

ispla

ce

me

nt

(cm

)

ATH,ζ= 0

p-ATH,ζ=0

sine-pulse,ζ=0

p-ATH,ζ=0.05

(a)

(b)

67

Fig. 3.2.b shows plots of Rd for different values of Tn normalized by Tp = 9.04 sec. in the

range 0 / 2n pT T≤ ≤ for three damping ratios, 0ζ = , 0.02 and 0.05. Note that the

maximum Rd values correspond to the amplitude spike that occurs around f = 0.122 Hz as

indicated in Fig. 3.1a. The reciprocal of 0.122 Hz is 8.2 sec, which is close to the

measured value of Tp = 9.04 sec. This demonstrates that the velocity Fourier spectrum (in

the rangec

LUF f f≤ ≤ ) is the key to understanding the displacement response of long-

period structures. The maximum d

R values for 0ζ = , 0.02 and 0.05 are 4.89, 3.73 and

3.14, respectively. Other examples of such single predominant frequency pulses are the

TCU68-N320E, Chi-Chi 1999 earthquake; Lucerne-239FN, the Landers 1992

earthquake; and the Gebze-184FN, Sakarya-090, Yarimca-090FP, Kocaeli 1999

earthquake.

Example 2 – Fault-Normal Component of Yarimca (YPT), Kocaeli 1999 earthquake

This example demonstrates the second type of displacement response characteristics in

which a ripple effect in the displacement response spectra (DRS) is observed. Because of

the presence of multiple predominant frequency pulses, large spectral displacements are

observed to occur at different natural periods. Such pulses are identifiable in the velocity

Fourier spectra as multiple spikes occurring in the frequency range c

LUF f f< ≤ . Since

cf may be as high as 1.6 Hz, even medium-period structures may be affected by such

pulses. An example is the KJM-140FN, Kobe 1999 earthquake.

68

As seen from the VF spectra shown in Fig. 3.3.a, three amplitude spikes at f = 0.371,

0.286 and 0.2 Hz are observed within the range LUF= 0.088 Hz toc

f = 0.4 Hz. Also, in

Fig. 3.3.b, it can be seen that 0a = 0.064g and pT = 3.5 s. The DRS given in Fig. 3.4.a

shows the presence of multiple spikes that correspond to f = 0.371, 0.286 and 0.2 Hz (i.e.,

at n

T = 2.7, 3.6 and 5.1 sec.). This again demonstrates that the VF spectra (in the range

cLUF f f≤ ≤ ) is the key to understanding the displacement response of long-period

structures. Also seen in Fig. 3.4.a is that the response due to a single sine-pulse does not

truly represent the actual response. HUP, as indicated on the abscissa of the figure, is the

Highest Usable Period, which is equal to the reciprocal of LUF, which in the present case

is equal to 0.088 Hz, as marked in Fig. 3.3.a.

69

Figure 3.3. Yarimca-180FN, Kocaeli earthquake 1999:

(a) velocity Fourier spectra, (b) pulse-ATH.

Plots of the modified displacement response factors computed for different Tn values and

normalized by Tp = 3.5 sec. are shown in Fig. 3.4.b. The maximum Rd values for 0ζ = ,

Velo

city F

ourier

Am

plit

ud

es

f (Hz)

Puls

e A

ccele

ration (

g)

Time (sec)

f =0.20 Hz

f =0.286 Hz

f =0.371 Hz

fc

Tp = 3.5 s

(b)

(a)

0.064 g

70


0.02 and 0.05 are determined to be 8.42, 5.60 and 3.77, respectively. The response

characteristics of such multiple predominant frequency pulses could be more devastating

than the single-pulse type as described in Example 1.

Figure 3.4. Yarimca-180FN, Kocaeli earthquake 1999:

(a) displacement response spectra, and (b) d

R spectra

Tn/Tp

Rd

Spectr

al D

ispla

ce

me

nt

(cm

)

ATH,ζ= 0

p-ATH,ζ=0

sine-pulse,ζ=0

p-ATH,ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(a)

(b)

71

Example 3 – Fault-parallel Component of Yarimca (YPT), Kocaeli 1999 earthquake

The curvature of the basin in which softer alluvial soils have been deposited can trap

body waves and cause some incident body waves to propagate through the alluvium as

surface waves. These waves can produce stronger shaking and longer durations than

would be predicted by one-dimensional analyses that consider only vertically propagating

s-waves. One dimensional ground response analysis can predict the average response of

sediments near the center of valley but not at the edges. There are significant differences

between the amplification functions at the center and edges of valleys. These are

attributed to ‘Basin Effect’. The potential for significant differential motion across

alluvial valleys has important implications for the design of long-span structures, such as

bridges and pipelines that often cross valleys. Differential movements can induce large

loads and cause heavy damage to these types of structures. [Kramer 1996].

This example on fault-parallel component of Yarimca (YPT), Kocaeli 1999 earthquake

demonstrates that in certain cases (e.g., as a result of Basin Effect) spectral displacements

caused by pulses in the fault-parallel direction in a strike-slip faulting mechanism may be

higher than those from the fault-normal direction. As observed from the velocity Fourier

spectra shown in Fig. 3.5.a and the DRS given in Figure 3.5.b, this pulse is of the single

predominant frequency type. After a single peak displacement of 220 cm occurring at

nT = 3.6 sec. (which corresponds to the velocity amplitude spike at f = 0.278 Hz), the

displacement steadily decreases with n

T toward the peak ground displacement at gou =

56 cm.

72

Figure 3.5. Yarimca-090FP, Kocaeli 1999 earthquake: (a) velocity Fourier spectra,

(b) displacement response spectra, and (c) d

R spectra

Velo

city F

ourier

Am

plit

ud

es

f (Hz)

Tn/Tp

Rd


Spectr

al D

ispla

ce

me

nt

(cm

) ATH,ζ= 0

p-ATH,ζ=0

sine-pulse,ζ=0

p-ATH,ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(c)

(b)

(a)

0.278 Hz

73

Fig. 3.5.c displays plots of d

R for 0a = 0.109g for different values of Tn when normalized

by Tp = 4.0 sec (pulse-ATH not shown). The maximum d

R values for 0ζ = , 0.02 and

0.05 are 6.37, 4.47 and 3.49, respectively. It should be noted that these maximum Rd

values occur around n

T = 3.6 sec., which corresponds to the velocity Fourier amplitude

spike at f = 0.278 Hz.

Example 4 – Fault-Normal Component of the Saratoga - W Valley College (WVC),

Loma Prieta 1989 earthquake

The resonant frequency is defined as the forcing frequency ω at which d

R is at

maximum. For an undamped system the resonant frequency is n

ω , and d

R is theoretically

unbounded at this frequency. If /n

ω ω is near 1, d

R will assume a large value, implying

that the deformation amplitude is much larger than the static deformation. There are

acceleration pulses in some near-fault records that can cause d

R to be as high as 10 to 25

for 0ζ = . Such pulses may be designated as resonant pulses due to their extremely high

dynamic magnifications. Seven out of twelve records (identified by ‘R’ after each

individual file name) listed in Table 3.4 for the Loma Prieta 1989 earthquake contain

resonance pulses. Almost all the pulse-like near-fault records of the Kobe 1999

earthquake contain such resonant pulses in the fault-normal and fault-parallel directions.

A single record of the Bolu, 175FN at Duzce 1999 earthquake also contains resonant

pulses at four different frequencies. The effects of the earthquake at this location have

been reported to be quite devastating [Park et al., 2004].

74

Resonant pulses can be identified in the velocity Fourier spectra as high, steep and

comparatively narrower (higher gradient) spikes. As seen in the VF spectra of Fig. 3.6.a,

within the range of LUF=0.125 Hz toc

f = 1 Hz, there are two dominant amplitude spikes

at f = 0.901 and 0.350 Hz. From Fig. 3.6.b, one can obtain 0a = 0.160g and Tp = 1.59 sec.

The multiple spikes at n

T =1.1 and 2.8 sec. (which correspond to spike frequencies of f =

0.901 and 0.350 Hz) are evident in the DRS shown in Fig. 3.7.a. This again demonstrates

that the velocity Fourier spectrum (c

LUF f f≤ ≤ ) is the key to understanding the

displacement response of long-period structures. It can also be seen in Fig. 3.7.a that the

response due to the single sine-pulse does not adequately represent the actual response.

From Fig. 3.7.b, the peak d

R values evaluated for different Tn normalized by Tp = 1.59

sec. for 0ζ = , 0.02 and 0.05 are determined to be 13.4, 7.88 and 5.1, respectively. The

response characteristics of these resonant pulses could be more devastating than the first

two types of pulses discussed in Examples 1 and 2.

75

Figure 3.6. Saratoga-W Valley College (WVC), 038FN, Loma Prieta earthquake 1989:

(a) velocity Fourier spectra, (b) pulse-ATH

Velo

city F

ourier

Am

plit

ud

es

f (Hz)

Puls

e A

ccele

ration (

g)

Time (sec)

Tp = 1.59 s

f =0.350 Hz

f =0.901 Hz

(b)

(a)

fc

76

Figure 3.7. Saratoga-W Valley College (WVC), 038FN, Loma Prieta earthquake 1989:

(a) displacement response spectra, and (b) d

R spectra

Rd

Tn/Tp

Natural period, n

T (sec)

Spectr

al D

ispla

ce

me

nt

(cm

)

ATH,ζ= 0

p-ATH,ζ=0

sine-pulse,ζ=0

p-ATH,ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(a)

(b)

77

3.4 BEAT PHENOMENA VERSUS “SIMPLE” HARMONIC EXCITATION

In this section, we will discuss the differences in response characteristics due to the low-

frequency contents of ground motions such as the TCU068-N320E, Chi Chi 1999

earthquake and the OSA-140FN, Kobe 1995 earthquake.

For the TCU068-N320E, Chi Chi 1999 earthquake (Fig. 3.8), c

f = 0.30 Hz as determined

from the velocity Fourier spectra (Fig. 3.8.a). This value of c

f is used to evaluate the

pulse acceleration from Eq. (3.6). As marked in Fig. 3.8.b, the peak amplitude of the

extracted acceleration pulse 0a = 0.202g. The adjacent peaks have values of 0.107g and

0.104g. The corresponding period of the pulse (i.e., the time needed for the pulse to go

through one cycle, as indicated in Fig. 3.8.b) that has the next higher peak value of

0.107g, is p

T = 6.87 seconds.

For the OSA-140FN, Kobe 1995 earthquake (Fig. 3.9), the peak amplitude of the

extracted pulse acceleration 0a = 0.027 g. The adjacent peak has a value of 0.023g, and

pT = 2.2 seconds, as marked on Fig. 3.9.a.

D and corresponding d

R can be numerically evaluated at different values of n

T for the

known value of 0a . Figures 3.8.c and 3.9.b show plots of d

R for different n

T normalized

by corresponding p

T for three damping ratios, ζ = 0, 0.02 and 0.05.

78

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

Figure 3.8 TCU068-N320E, Chi Chi 1999 earthquake:

(a) VF spectra (b) p-ATH (c) Rd spectra

ζ=0.0

ζ=0.02

ζ=0.05

T =6.87s

Pu

lse A

cce

lera

tion

(g)

Time (sec)

(a) (b)

(c)

Rd

0.104g

Tn/Tp

0.202g

f (Hz)

0.107g

Tp =9.1 s

79

Figure 3.9 Osaj (OSA-140FN), Kobe 1995 (a) p-ATH (b) Rd spectra.

A brief discussion of the contrasting causes of devastation due to two such seismic

ground motions is warranted. The velocity Fourier spectra (Fig. 3.8.a) of TCU068-N320E

are characterized by very high amplitudes, with most of the seismic energy concentrated

in a much narrower band (LUF<f < 0.3 Hz). The extracted acceleration pulse is a smooth

ζ=0.0

ζ=0.02

ζ=0.05

Tp = 2.2 s

Pu

lse A

cce

lera

tion

(g)

Time (sec)

(b)

(c)

Rd

Tn/Tp

80

distinct sinusoidal pulse of one and a half cycles, having the maximum amplitude 0a =

0.202 g, with a maximum distinct period p

T = 9.1 s. It has the effect of monotonically

increasing the displacement response as the natural period of vibration n

T increases.

According to the ground displacement time history (DTH), the maximum ground

displacement go

u = 475 cm. For an undamped linearly elastic SDF system Dmax = 1209

cm when n

T = 12.5 s. Thus, max /go

D u = 2.55. It has high maximum static deformation

0( )st

u in Eq. (3.7), therefore, the values of d

R = 3.0, 2.7 and 2.3 for ζ = 0, 0.02 and 0.05,

respectively, are much lower compared to those of OSA-140FN. The Rd values of

TCU068-N320E may be compared with those due to suddenly applied step force, for

which Rd = 2, 1.95 & 1.85 for ζ = 0, 0.02 & 0.05, respectively.

For the velocity Fourier spectra of OSA-140FN, the magnitude of the velocity Fourier

amplitudes are much less compared to those of TCU068-N320E, and that these

amplitudes occupy a relatively broader band (LUF<,f < 0.68 Hz.) in low-frequencies of

VF spectra. The majority of the spikes are high, steep and narrow. The striking feature of

the extracted pulse-acceleration (Fig. 3.9.b) is the beat-phenomena, which may not be

visible from the original ATH plot. This may cause an amplification in amplitudes of

almost 200% [Mitra 2006]. Furthermore, the pulse-acceleration has a maximum

amplitude 0a = 0.027 g with the corresponding period p

T = 2.2 s, which is considerably

less than the previous example of ChiChi-TCU068-N320E. It has a maximum ground

displacement go

u = 9.2 cm. For an undamped linearly elastic SDF system Dmax = 86 cm

occurs at n

T = 4.4 s (corresponding to the highest peak at f = 0.225 Hz in VF spectra).

81

Thus, max /go

D u = 9.2, which is considerably high. It has a low maximum static

deformation 0( )st

u in Eq. (3.7), therefore, the values of d

R = 21.3, 9.2 and 5.3 for ζ = 0,

0.02 and 0.05, respectively, are very high compared to those of TCU068-N320E.

3.5 EVALUATION OF THE MEAN d

R FOR DIFFERENT EVENTS

This section presents a statistical study of the effects of dynamic magnification caused by

the long-period ground motions, via the application of the modified displacement

response factor d

R . The acceleration pulses of about seventy-two pulse-like NFGM

records were extracted event-wise to quantitatively study elastic response characteristics,

by using d

R for 0ζ = , 0.02 and 0.05.

The pulse records of different earthquake events with known directivity effects

[Mavroeidis & Papageorgiou 2003; Toothong & Cornell 2007; Baker 2007] are listed

under the File column in Tables 3.1 to 3.7. We have depended on these references, as

well as on the references mentioned in Mavroeidis and Papageorgiou [2003] for the fault-

normal directions for a strike-slip and reverse faulting mechanism. The direction of

maximum velocity might be different than the fault-normal direction (as observed for the

Kobe 1995 earthquake event or the Yarimca (YPT) station, Kocaeli 1999 earthquake). In

the case of the Chi-Chi 1999 earthquake, the directions of maximum velocity were

obtained from Shabestari and Yamazaki [2003]. This paper also discusses the important

fault characteristics for the Chi-Chi 1999 earthquake.

82

The paper by Mavroeidis & Papageorgiou [2003] should be referred to for detailed

information on the important fault characteristics and variation in orientation of fault-line

for all the earthquake events discussed in Sec. 3.5.1 to 3.5.7.

3.5.1 Mean d

R for the Imperial Valley-06 1979 Earthquake, Mw = 6.53 (strike-slip

faulting)

This event involves about 16 pulse-like records, all on D soil-type. The majority of pulse-

like ground motions in this event are dominated by single-period velocity pulses. The

mean peak period of vibration ( 0T ), corresponding to the maximum spectral displacement

( 0D ), is around 4.6 seconds. Since the mean c

f is around 0.5 Hz, structures having a n

T

of more than 2 seconds are affected.

In Fig. 3.10.a and 3.10.b, the mean and the mean+one stdv d

R spectra for the Imperial

Valley 1979 earthquake are plotted for three damping ratios, ζ =0, 0.02 and 0.05. From

these figures, it can be seen that the maximum mean d

R values are 4.85, 3.66 and 3.05

for ζ = 0, 0.02 and 0.05, respectively; and the maximum mean+one stdv d

R values for

the same event are 5.91, 4.29 and 3.45 for ζ = 0, 0.02 and 0.05, respectively. These are

lower than the d

R values of all the other events, which have higher moment magnitudes.

83

Rd

0

1

2

3

4

5

0 0.5 1 1.5 2

0

1

2

3

4

5

6

0 0.5 1 1.5 2

Figure 3.10. Modified displacement response factor for the Imperial Valley-06 1979

earthquake: Rd spectra for (a) the mean values (b) the mean+one stdv values

Tn/Tp

Tn/Tp

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(a)

(b)

Rd

Rd

84

TABLE 3.1. Pulse-like Ground Motions of the Imperial Valley 1979-6 Earthquake, Mw

= 6.53

No S File NGA R (km)

fc

(Hz) PGA (g)

PGV (cm/s)

PGD (cm)

T0

(sec) D0

(cm)

1 D IMPVALL_HEMO_233FN 171 0.07 0.73 0.38 115 40.0 3.0 146.0

2 D IMPVALL_HAEP_233FN 158 0.34 0.99 0.36 44.2 10.1 1.6 55.0

3 D IMPVALL_HE07_233FN 181 0.56 0.46 0.46 108.8 45.6 3.8 180.0

4 D IMPVALL_HAGR_233FN 159 0.65 1.45 0.31 53.6 14.9 1.9 57.7

5 D IMPVALL_HE06_233FN 181 1.25 0.46 0.44 111.8 66.7 4.0 212.2

6 D IMPVALL_HE08_233FN 183 3.86 0.51 0.47 48.3 36.8 4.6 154.0

7 D IMPVALL_HE05_233FN 180 3.95 0.43 0.38 91.5 62.0 4.7 205.7

8 D IMPVALL_HEDA_233FN 184 5.09 0.46 0.42 60.0 38.8 6.0 149.2

9 D IMPVALL_HE10_233FN 173 6.17 0.43 0.18 46.9 31.4 6.3 135.6

10 D IMPVALL_HE04_233FN 179 7.05 0.44 0.36 77.9 58.6 4.6 179.8

11 D IMPVALL_HECC_233FN 170 7.31 0.40 0.18 54.5 38.4 6.3 134.2

12 D IMPVALL_HHVP_233FN 185 7.65 0.48 0.26 55.0 33.0 4.2 128.9

13 D IMPVALL_HBRA_233FN 161 10.42 0.50 0.16 36.1 22.7 4.5 89.4

14 D IMPVALL_HE11_233FN R 174 12.45 0.87 0.37 40.8 18.5 7.5 108.3

15 D IMPVALL_HE03_233FN 178 12.85 0.48 0.23 40.0 23.0 5.7 90.8

16 D IMPVALL_HWSM_233FN 192 15.25 0.48 0.08 26.8 19.2 4.7 105.8

The notations used in Tables 3.1 to 3.7 are: S = soil type, NGA = PEER Next Generation

Attenuation record number, R = closest distance from the fault rupture, fc = cut-off

frequency, PGA = peak ground acceleration, PGV = peak ground velocity, PGD = peak

ground displacement, Do = maximum spectral displacement for ζ =0.0, and 0T = n

T for

the corresponding maximum spectral displacement. The letter R in italics after the name

of the file indicates the presence of a resonance pulse in the record, if there is one.

3.5.2 Mean d

R for the Northridge-01 1994 Earthquake, Mw = 6.69 (reverse faulting)

Similar to the Imperial Valley-06 1979 earthquake event, the majority of pulse-like

ground motions in this event are dominated by single-period velocity pulses. As observed

from the tenth column of Table 3.2, the mean of 0T values is around 2.5 seconds. The

85

structures with a natural period of vibration between 2.5 to 4 seconds are dominantly

affected. Four records are found to contain resonant pulses at a natural period of

vibrations: (1) Pacoima Dam – upper left, (PUL) 1.0n

T ≈ sec., (2) Sylmar - Converter

Station (SCS) 1.1n

T ≈ sec., (3) Newhall - Fire Station (NWH) 1.3n

T ≈ , and (4) LA -

Sepulveda VA Hospital (0637) 0.8n

T ≈ .


R spectra for the Northridge

1994 earthquake are plotted for three damping ratios, ζ =0, 0.02 and 0.05. From these

figures, it can be seen that the maximum mean d

R values are 6.18, 4.39 and 3.41 for ζ =

0, 0.02 and 0.05, respectively, and the maximum mean+one stdv d

R values for the same

event are 8.37, 5.55 and 4.17 for ζ = 0, 0.02 and 0.05, respectively.

86

0

1

2

3

4

5

6

0 0.5 1 1.5 2

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2

Figure 3.11 Modified displacement response factor for the Northridge 1994 earthquake,

Rd spectra for (a) the mean values (b) the mean+one stdv values

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

Tn/Tp

Tn/Tp

Rd

Rd

(b)

(a)

87

TABLE 3.2. Pulse-like Ground Motions of the Northridge 1994 Earthquake, Mw = 6.69

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV

(cm/s)

PGD

(cm)

T0

(sec)

D0

(cm)

1 A NORTHR_0655_032FN 0983 5.43 0.56 0.52 67.3 42.6 2.6 155.8

2 A NORTHR_LDM_032FN 1013 5.92 1.2 0.58 77.1 20.0 2.1 64.5

3 A NORTHR_PAC_032FN 1050 7.01 1.6 0.50 49.0 6.4 3.1 30.6

4 A NORTHR_PUL_032FN R 1051 7.01 1.65 1.38 107.0 23.0 1.1 57.6

5 B NORTHR_SCE_032FN 1085 5.19 0.45 0.84 116.5 39.4 2.9 159.5

6 B NORTHR_JEN_032FN 0982 5.43 0.56 0.52 67.4 42.0 2.6 155.5

7 B NORTHR_WPI_032FN 1045 5.48 0.76 0.43 87.7 55.0 2.5 125.0

8 B NORTHR_RSS_032FN 1063 6.5 1.5 0.87 167.0 28.0 1.4 118.3

9 C NORTHR_LOS_032FN 0960 12.44 1.2 0.47 53.1 10.6 1.9 41.3

10 D NORTHR_SYL_032FN 1086 5.3 0.78 0.73 122.7 31.4 2.4 154.6

11 D NORTHR_SCS_032FN R 1084 5.35 1.00 0.59 130.3 54.2 2.9 230.4

12 D NORTHR_NWH_032FNR 1044 5.92 0.93 0.72 120.0 34.8 3.7 100.0

13 D NORTHR_0637_032FN R 1004 8.44 1.34 0.73 63.2 18.4 2.7 103.0

14 D NORTHR_CNP_032FN 0959 14.7 0.8 0.38 53.5 21.4 2.3 94.7

15 D NORTHR_5082A_032FN 1009 23.6 1.01 0.27 32.4 10.4 2.5 62.9

3.5.3 Mean d

R for the Kobe 1995 Earthquake, Mw = 6.90 (strike-slip faulting)

Almost all the ground motion records of this event are dominated by multiple low-

frequency Fourier amplitudes with high, narrow and steep spikes denoting the presence

of a ripple effect in the displacement response. Since c

f ≈ 1.6 Hz for most of the records,

medium- as well as long-period structures are affected. Since the directivity effect is

dominant in fault-normal as well as fault-parallel directions, relevant details for both the

directions are included in Table 3.3, as well as in Rd spectra. In most cases, the peak

ground displacements 0gu are very low compared to the peak spectral displacement 0D

of an undamped SDF system. Therefore, the mean of ratio 0 0/g

D u of the records from

Table 3.3 is relatively higher at 5.8. Except for the record at Takarazuka (TAZ), all of the

other records contain more than one resonant pulse. The Osaj (OSA) record contains

88

resonant pulses at four frequencies, affecting displacement response spectra (DRS) at

corresponding n

T = 1.9, 3.2, 3.6 and 4.4 seconds.

It has been observed that low-velocity soft and near-surface sediments are effective in

amplifying and elongating long-period ground motions [Hatayama et al., 2007; Miyake

and Koketsu 2007]. At stations Takarazuka (TAZ) and Shin-Osaka (SHI), the spectral

displacements due to fault-parallel components are higher than those due to fault-normal

ones, denoting that the site effects may be more dominant than the directivity effect.


R spectra for the Kobe 1995

earthquake are plotted for three damping ratios, ζ =0, 0.02 and 0.05. In these figures, it

can be seen that the maximum mean d

R values are 9.41, 5.58 and 4.04 for ζ = 0, 0.02

and 0.05, respectively, and the maximum mean+one stdv d

R values for the same event

are 14.20, 6.92 and 4.80 for ζ = 0, 0.02 and 0.05, respectively. This event has the highest

Rd values of all the events considered in this study (Fig. 3.19).

In a similar manner, the mean displacement response factor may be evaluated for (a) the

2003 Tokachi-oki, Japan, earthquake, (b) the 2004 South-East off Kii peninsula

earthquake, (c) the 2004 Niigata-ken Chuetsu, Japan earthquake, and (d) the 2007

Niigata-ken Chuetsu-oki, Japan earthquake. This would probably establish a range of

mean ± one stdv factors for Japan. This Rd value may be useful in the design of dampers

for the structures in Japan.

89

0

2

4

6

8

10

0 0.5 1 1.5 2

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

Figure 3.12 Modified displacement response factor for the Kobe 1995 earthquake


TABLE 3.3. Pulse-like Ground Motions of the Kobe1995 Earthquake, Mw = 6.9

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV

(cm/s)

PGD

(cm)

T0

(sec)

D0

(cm)

1 B KOBE_KJM_140FN R 1106 0.96 1.56 0.85 95.8 24.6 1.02 101

1a KOBE_KJM_050FP R - - 1.67 0.55 53.4 10.3 1.3 55.8

2 D KOBE_TAZ_140FN 1119 0.27 1.64 0.65 72.5 20.8 2.0 74.5

2a KOBE_TAZ_050FP - - 0.85 0.70 83.0 26.5 1.6 87.6

3 E KOBE_TAK_140FN R 1120 1.47 0.98 0.68 169.2 45.0 2.1 273.6

3a KOBE_TAK_050FP R - - 1.00 0.61 169.6 22.9 3.1 130.0

4 E KOBE_SHI_140FN R 1116 19.15 1.66 0.19 30.0 8.8 2.7 38.6

4a KOBE_SHI_050FP R - - 1.64 0.27 41.8 7.4 1.2 41.4

5 E KOBE_ OSA_140FN R 1113 21.35 0.68 0.07 20.0 9.8 4.4 86.1

5a KOBE_ OSA_050FP R - - 0.64 0.08 17.1 7.9 3.4 78.8

The PGA, PGV and maximum spectral displacement D0’s for the OSA_140FN station

are 0.073g, 20 cm/s and 86.1 cm, respectively. Although KOBE_TAZ_050FP

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

Tn/Tp

Tn/Tp

Rd

Rd

(a)

(b)

90

(Takarazuka) has a high PGA (0.70g) and PGV (83 cm/sec), its maximum spectral

displacement D0 = 87.6 cm/s ≈ D0 = 86.1 cm/s of Osaj. Similarly, KOBE_SHI_050FP

(Shin-Osaka) has a high PGA (0.27g) and PGV (42 cm/sec), its maximum spectral

displacement D0 = 41.4 cm/s << D0 = 86.1 cm/s of Osaj. This shows that the maximum

spectral displacement, D0, may not be directly related to the PGA or PGV. Similar

observations may be made from the records of other events.

3.5.4 Mean d

R for the Loma Prieta 1989 Earthquake, Mw = 6.93 (reverse/oblique

faulting)

The Loma Prieta 1989 event in North America may be considered the most destructive

one among other events of N. America. Seven out of the thirteen pulse-like records

contain resonant pulses, therefore the mean and mean+one stdv are comparatively higher.

The velocity Fourier amplitudes in 1.67 c

LUF f f Hz≤ ≤ < are dominated by multiple,

steep and high spikes, characterizing a ripple effect. Multiple high-spectral displacements

are observed at multiple periods of vibrations in the displacement response spectra.

Therefore, both medium- as well as long-period structures are affected. This can also be

observed in Table 3.4.

From the seventh column of Table 3.4, it can be observed that the PGA varies from 0.94g

to 0.12g, as the distance from the fault-rupture R varies from 3.88 km to 77.4 km;

however, the displacement response factor d

R is independent of this change. The mean of

the maximum values of d

R for station # 1 to 5 is around 6.1 for ζ =0; however, at each

91

station from # 6 to #12, d

R is more than 10.0 for the same damping ratio. Thus, d

R may

not be dependent upon the PGA. This important observation may be borne in mind when

designing structures in near-fault areas or on distant sedimentary basins.


R spectra for the Loma

Prieta 1989 earthquake are plotted for three damping ratios, ζ =0, 0.02 and 0.05. From

these figures, it can be seen that the maximum mean d

R values are 7.05, 4.97 and 3.72

for ζ = 0, 0.02 and 0.05, respectively; and the maximum mean+one stdv d

R values for

the same event are 10.13, 6.51 and 4.71 for ζ = 0, 0.02 and 0.05, respectively.

92

Rd

0

2

4

6

0 0.5 1 1.5 2

0

2

4

6

8

10

0 0.5 1 1.5 2

Figure 3.13 Modified displacement response factor for the Loma Prieta 1989

Earthquake: Rd Spectra for (a) the mean values (b) the mean+one stdv values

Tn/Tp

Tn/Tp

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(a)

(b)

Rd

Rd

93

TABLE 3.4. Pulse-like Ground Motions of the Loma Prieta 1989 Earthquake, Mw = 6.93

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV (cm/s)

PGD

(cm)

T0

(sec)

D0

(cm) Rd

1 A LOMAP_LGP_038FN 0779 3.88 0.84 0.94 96.5 62.5 5.6 168. 5.4

2 A LOMAP_LEX_038FN 3548 5.02 1.28 0.52 102. 34.0 1.1 76.7 5.8

3 A LOMAP_G01_038FN 0765 6.93 0.92 0.43 40.0 7.2 3.0 29.0 6.6

4 B LOMAP_GIL_038FN 0763 6.93 0.92 0.29 31.0 6.5 3.0 25.3 6.3

5 D LOMAP_STG_038FN 0802 8.5 1.15 0.36 56.0 29.4 6.5 91.5 6.4

6 D LOMAP_WVC_038FN,R 0803 9.31 1.00 0.40 71.0 21.0 2.8 101. 13.4

7 D LOMAP_GOF_038FN, R 0764 10.97 1.13 0.25 22.0 4.5 1.0 27.5 15.2

8 D LOMAP_G02_038FN, R 0766 11.07 0.85 0.41 46.0 13.0 1.5 63.8 10.6

9 D LOMAP_G03_038FN, R 0767 12.82 0.95 0.53 49.0 11.0 1.5 44.5 10.3

10 D LOMAP_G04_038FN, R 0768 14.34 0.85 0.35 36.0 11.5 1.4 52.0 12.

11 D LOMAP_CH1_038FN, R 0783 74.26 0.80 0.33 49.0 12.8 1.5 50.8 10.

12 E LOMAP_NAS_038FN, R 0738 71.00 0.54 0.22 32.0 10.3 2.4 54.0 10.3

13 E LOMAP_TRI_038FN 0808 77.42 1.25 0.12 27.2 9.2 2.5 40.5 5.5

Rd values are calculated for the record at each station, and entered in the last column. The

last three examples, numbered 11, 12 and 13 in Table 3.4 for the Oakland - Outer Harbor

Wharf, the Alameda Naval Air Station Hanger stations and Treasure Island, may be noted

as examples of the path-effect, since these stations are located at distances of 74.26 km

(46.14 miles), 71 km (44.12 miles) and 77.42 km (48.4 miles) from the fault-rupture,

respectively. Far-source ground motions consist of surface waves which may have a

longer duration than those of near-fault motions. They can be very damaging in some

circumstances. An example is the 1985 Michoacan earthquake, which occurred in the

sedimentary basins of Mexico City, which was 350 km away from the epicenter.

3.5.5 Mean d

R for the Kocaeli 1999 Earthquake, Mw = 7.51 with the Duzce-Bolu

1999, Mw = 7.14 and the Erzican-Erz 1992, Mw = 6.69 (strike-slip faulting)

Although pulse-effect is not apparent [Park et al., 2004] in the velocity time history, the

DUZCE_BOL_175FN 1999 earthquake record has resonant pulses at n

T = 1.2, 1.9, 2.8

94

and 4.5 seconds, which is evident from its DRS (not shown), as well as in its velocity

Fourier spectra (not shown). This record at Bolu has high d

R = 13.8, 7.6 and 5.0, for ζ =

0, 0.02 and 0.05, respectively. The Duzce November 1999 earthquake caused a surface

rupture across the Bolu viaduct, which resulted in excessive superstructure movement

and widespread failure of the seismic isolation system. The displacement of the

superstructure relative to the piers exceeded the capacity of the bearings at an early stage

of the earthquake, causing damage to the bearings as well as to the energy dissipation

units [Park et al., 2004].

At Yarimca (YPT), the spectral displacements in fault-parallel directions are more

dominant. The c

f for the majority of records of Kocaeli 1999 event are less than 0.4 Hz.

Large spectral displacements are observed at larger periods of vibration. The mean period

of vibration with maximum spectral displacements is around 9 seconds for the Kocaeli

August 1999 event.


R spectra for pulse-like

ground motions near the North Anatolian Fault, Turkey, are plotted for three damping

ratios, ζ =0, 0.02 and 0.05. From these figures, it can be seen that the maximum mean

dR values are 6.03, 4.43 and 3.50 for ζ = 0, 0.02 and 0.05, respectively; and the

maximum mean+one stdv d

R values for the same event are 9.10, 5.80 and 4.23 for ζ = 0,

0.02 and 0.05, respectively.

95

0

1

2

3

4

5

6

0 0.5 1 1.5 2

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2

Figure 3.14 Modified displacement response factor for the North Anatolian Fault,

Turkey: Rd spectra for (a) the mean values (b) the mean+one stdv values

TABLE 3.5. Pulse-like Ground Motions near the North Anatolian Fault, Turkey

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV

(cm/s)

PGD

(cm)

T0

(sec)

D0

(cm)

1 A KOCAELI_GBZ_184FN 1161 10.92 0.39 0.24 51.9 44.0 9.5 155.4

2 B KOCAELI_SKR_090 1171 3.12 0.28 0.38 79.5 72.3 14.0 212

3 B KOCAELI_ARC_184FN 1148 13.49 0.40 0.17 20.3 16.0 8.9 60.8

4 D KOCAELI_YPT_180FN 1176 4.43 0.40 0.28 48.2 44.0 5.1 179.2

4a D KOCAELI_YPT_090FP - - 0.37 0.31 72.9 56.2 3.6 220.0

5 D KOCAELI_DZC_163FN 1158 15.37 0.37 0.28 52.2 38.0 6.8 144.0

5a D KOCAELI_DZC_073FP - - 0.74 0.38 53.0 26.5 6.0 130.0

6 D ERZIKAN_ERZ_032FN 0821 4.38 0.67 0.49 95.4 32.9 3.0 133.8

6a D ERZIKAN_ERZ_122FP - - 0.82 0.42 45.3 16.5 2.7 90.5

7 D DUZCE_BOL_175FN R 1602 12.04 0.59 0.68 57.5 23.0 1.9 85.0

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

Tn/Tp

Tn/Tp

Rd

Rd

(b)

(a)

96

3.5.6 Mean d

R for the Chi-Chi 1999 Earthquake, Mw=7.62 (reverse/oblique faulting)

This event is characterized by extreme ground displacements. Designing for high ground

displacements (fling) in most cases would probably control for such types of event. d

R

factor for this event is relatively less than most other events.

As seen in the 5th

column of Table 3.6, all the eight records are very close (< 5.3 km) to

the fault-rupture. Stations TCU052 and TCU068 are at the Hanging wall sides. The

spectral displacements at these two stations more or less monotonically increase as the

period of vibration n

T increases (Fig. 3.2.a for TCU052). The remaining six stations are

situated at the foot-wall of the fault-rupture. As seen in Table 3.6, extreme PGVs, PGDs,

as well as spectral displacements, are observed.

The peak ground displacements (PGDs) are very high, the average of the eight stations

being 150 cm. The mean ratio of 0 0/g

D u of the records from Table 3.6 is comparatively

lower at 3.5. Recall from Sec. 3.5.3 that this value for the Kobe 1995 event is 5.8.


R spectra for the Chi-Chi

1999 earthquake are plotted for three damping ratios, ζ =0, 0.02 and 0.05. From these

figures, it can be seen that the maximum mean d

R values are 5.26, 3.78 and 3.02 for ζ =

0, 0.02 and 0.05, respectively; and the maximum mean+one stdv d

R values for the same

event are 7.40, 4.92 and 3.69 for ζ = 0, 0.02 and 0.05, respectively.

97

Rd

0

1

2

3

4

5

0 0.5 1 1.5 2

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2

Figure 3.15 Modified displacement response factor for the Chi-Chi 1999 earthquake:


TABLE 3.6. Pulse-like Ground Motions of the Chi-Chi 1999 Earthquake, Mw = 7.62

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV

(cm/s)

PGD

(cm)

T0

(sec)

D0

(cm)

1 A CHICHI_TCU068_N320E 1505 0.32 0.3 0.57 290.0 475.0 12.4 1209.

2 A CHICHI_TCU052_N322E 1492 0.66 0.28 0.50 167.0 306.0 17.3 814.0

3 A CHICHI_TCU102_N232E 1529 1.51 0.47 0.24 110.0 91.3 7.6 338.0

4 D CHICHI_TCU065_N123E 1503 0.59 0.28 0.82 130.0 93.0 5.3 495.0

5 D CHICHI_TCU075_N094E 1510 0.91 0.36 0.33 88.6 86.0 3.9 247.0

6 D CHICHI_TCU129_N136E 1549 1.84 0.29 0.83 57.0 52.0 5.8 187.0

7 D CHICHI_TCU076_N128E 1511 2.76 0.49 0.35 88.0 41.0 7.9 151

8 D CHICHI_TCU054_N142E 1494 5.30 0.26 0.18 59.1 63.0 9.2 232

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

Tn/Tp

Tn/Tp

(b)

(a)

Rd

Rd

98

3.5.7 Mean d

R for Other Earthquakes, Mw > 7.0


R spectra for the Tabas 1978

earthquake, Mw = 7.35, the Lander 1992 earthquake, Mw = 7.28 and the Cape Mendocino

1992 earthquake, Mw = 7.01 are plotted for three damping ratios, ζ = 0, 0.02 and 0.05.

From these figures, it can be seen that the maximum mean d

R values are 7.13, 5.00 and

3.74 for ζ = 0, 0.02 and 0.05, respectively; and the maximum mean+one stdv d

R values

for the same event are 10.71, 6.29 and 4.50 for ζ = 0, 0.02 and 0.05, respectively.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2

0

2

4

6

8

10

0 0.5 1 1.5 2

Figure 3.16 Modified displacement response factor for other earthquakes as stated:


Rd

Rd

Tn/Tp

Tn/Tp

ζ=0.0

ζ=0.02

ζ=0.05

ζ=0.0

ζ=0.02

ζ=0.05

(a)

(b)

99

TABLE 3.7. Other Pulse-like Ground Motions, Mw > 7

No S File NGA R

(km)

fc

(Hz)

PGA

(g)

PGV

(cm/s)

PGD

(cm)

T0

(sec)

D0

(cm)

1 A TABAS_TAB_TR 143 2.05 0.27 0.85 121.1 95 4.7 419

2 A LANDERS_LCN_239FN 879 2.19 0.40 0.69 136.1 236 11.2 222

3 D LANDERS_YER_240FN 900 23.6 0.86 0.23 56.2 48 7.8 141

4 D LANDERS_BRS_240FN 838 34.9 0.40 0.14 30.9 27 8.4 80

5 B CAPEMEND_PET260FN 828 8.18 1.28 0.62 81.9 25 2.2 67

3.5.8 Summary

TABLE 3.8. Summary of d

R for different events

mean values mean+one stdv values

No EVENT ζ=0.0 ζ=0.02 ζ=0.05 ζ=0.0 ζ=0.02 ζζζζ=0.05

1 Imperial Valley 1979, Mw = 6.53 4.85 3.66 3.05 5.91 4.29 3.44

2 Northridge 1994, Mw = 6.69 6.18 4.39 3.41 8.37 5.55 4.17

3 Kobe 1995, Mw = 6.90 9.41 5.58 4.04 14.20 6.92 4.80

4 Loma Prieta 1989, Mw = 6.93 6.8 4.7 3.6 10.0 6.2 4.5

5 Kocaeli 1999, Mw = 7.51 6.03 4.43 3.50 9.10 5.80 4.23

6 Chi-Chi 1999, Mw = 7.62 5.26 3.78 3.02 7.40 4.92 3.69

7 Other earthquakes, Mw > 7 7.14 5.00 3.74 10.71 6.29 4.50

If we assume ζ = 0.02 for most long-period structures, an appropriate and recommended

Rd value for the design of such structures is between five to seven (the mean+one stdv

values), signifying that these structures would demand very high strength. For ζ = 0.05,

the (m+σ) values of Rd vary from 3.44 to 4.8. The summary of values Rd in Table 3.8

shows that when considering different events all over the world, the value of the

displacement response factor Rd may not always be directly proportional to the moment

magnitude of the event. These may depend more upon directivity, and the site, as well as

upon the path effects.

100

3.6 OBSERVATIONS

3.6.1 Comparison of seismic Rd with that due to simple harmonic excitation

The equation of motion for a SDF damped system subjected to harmonic force ( )p t is


0 sinmu cu ku p tω+ + =&& & (3.9)

The displacement response factor of the steady-state deformation of the system due to

harmonic force is given by

( ) [ ]

0

2 220

1

1 ( / ) 2 ( / )d

stn n

uR

u ω ω ζ ω ω= =

− +

(3.10)

In Eq. (3.10), the Rd = 10 for ζ = 0.05 and (ω/ωn) = 1. In Table 3.8, the (m+σ) values of

Rd vary from 3.44 to 4.8 for ζ = 0.05 for different earthquake events. If we substitute the

value of Rd = 3.44 in Eq. (3.10), we get an equivalent (ω/ωn) = 0.85 and 1.125 for the

Imperial Valley-6 1979 event. Similarly, if we substitute Rd = 4.8 in Eq. (3.10), we get an

equivalent (ω/ωn) = 0.9 and 1.085 for the Kobe 1995 event. Therefore, for pulse-like

complex seismic ground motions, we may equivalently adopt 0.85 ≤ (ω/ωn) ≤ 0.9 or

1.125 ≥ (ω/ωn) ≥ 1.085.

Fig. 3.17. shows the (m+σ) Rd values of the Imperial Valley-6 1979 event plotted against

Tn/Tp for ζ = 0.05 in blue. The (m+ σ) Rd values of the Kobe 1995 event are also plotted

against Tn/Tp for ζ = 0.05 in red. Rd values for harmonic excitation as given by Eq. (3.10)

for ζ = 0.05 is also plotted in black for comparison. Observe that the concept of

101

equivalent forcing frequency (ω/ωn) ratio holds true for the events with the lowest (the

Imperial Valley-6 1979) and the highest (the Kobe 1995) Rd.

To compare Rd spectra of each event with that from continuous harmonic excitation, it is

proposed to modify Eq. (3.10) as follows,

( ) [ ]

0

2 220

1

1 ( / ) 2 ( / )d

stn n

uR

ukω αω ζ ω αω

= = − +

(3.11)

where, n

n p

T

T

αω

ω=

(αTn) is the effective natural period of vibration of the system. α is a period shift factor

and k is a damping modification factor. The Imperial Valley-6 1979 event has the lowest

Rd = 3.44. For α = 0.9 and k = 2.95 (obtained by iteration), the modified (m+σ) Rd spectra

obtained from Eq. (3.11) is plotted in Fig. 3.18 in blue line. It is comparable for Rd and

phase values with the (m+σ) Rd spectra of the same event (from Fig. 3.10.b) plotted in

blue dashed line.

Modified (m+σ) Rd spectra (red line) obtained from Eq. (3.11) for α = 0.77 and k = 2.1

(obtained by iteration) for the Kobe 1995 event is superimposed on the original Kobe

1995 (m+σ) Rd spectra (from Fig. 3.12.b) as shown in red dashed line. From Fig. 3.18, it

is observed that both have a comparable Rd values, as well as periods. Thus, for 0.77

(Kobe) ≤ α ≤ 0.9 (Imperial V) and for 2.1 (Kobe) ≤ k ≤ 2.95 (Imperial V), we can

equivalently represent Rd for all the six major events under consideration.

102

Figure 3.17. Concept of equivalent harmonic forcing frequency (ω/ωn) ratio illustrated

for Imperial Valley-6 1979 and Kobe 1995 events, comparing with harmonic excitation.

Figure 3.18. Concept of combined factors k & α illustrated for Imperial Valley-6 1979

and Kobe 1995 events, comparing with equivalent harmonic excitation, Eq. (3.11)

Imperial V 1979

Kobe 1995

Harmonic

Tn/Tp

Rd

Eqv. Harmonic

Imperial V 1979

Eqv. Harmonic

Kobe 1995

Tn/Tp

Rd

(ω/ωn) = 0.9

(ω/ωn) = 0.85

103

Figure 3.19. Displacement Response Factor, Rd for each major earthquake events

compared with that due to simple continuous harmonic excitation, ζ = 0.05

Thus, for pulse-like seismic ground motions, the concept of an equivalent harmonic

forcing frequency ratio (ω/ωn) or a ‘damping modification factor’ may be introduced.

This concept helps us to compare the values of dynamic magnification among different

earthquake events, as well as those due to simple harmonic excitation. Since damping

Dis

pla

cem

ent

Response

Facto

r, R

d

Tn/Tp

Harmonic, Rd = 10

Imperial V, Rd =3.44

Chi-Chi, Rd = 3.69

NorthR, Rd = 4.17

Kocaeli, Rd = 4.23

LomaP, Rd = 4.50

Kobe 1995, Rd = 4.80

104

controls the response characteristics of such structures, these Rd values can help us to

design dampers that can sustain pulse-like excitations.

The concept of equivalent viscous damping is not new, it was first proposed by Jacobsen

in 1930 to obtain approximate solutions for the steady forced vibration of damped SDF

systems with linear force-displacement relationships but with damping forces

proportional to the nth power of the velocity of motion when subjected to sinusoidal

forces. In 1960, he extended the concept of equivalent viscous damping to yielding SDF

systems. Since then, many methods have been proposed in the literature [Sec. 3.3.1 of

FEMA 440, 2005].

3.6.2 RESPONSE AND DAMPING

In the preceding section, it was seen that 0.85 ≤ (ω/ωn) ≤ 0.9, i.e., the forcing frequency is

close to the natural frequency of the system. Therefore, it is known [Clough and Penzien

1992; Chopra 2007; Humar 1992] that Rd is very sensitive to damping. When (ω/ωn) ≈ 1,

and k is such that 2.1 ≤ k ≤ 2.95, then Eq. (3.10) becomes

0 00

( )

2. . . .

st

n

u pu

k k cζ ω= = (3.12)

where c is the damping coefficient. Thus, damping controls the response of systems

subjected to such pulse-like near-fault ground motions. The Rd values calculated from

different events can assist us with the design of dampers that can sustain pulse-like

excitations.

105


The modified displacement response factor is defined as the ratio of the maximum

amplitude 0u D≡ of the dynamic deformation to the maximum static deformation 0( )st

u ,

when a linearly elastic SDF system is subjected to a pulse excitation ( )p

p t in the

dynamic analysis, and to a static load 0 0p ma= − (where 0p is the amplitude or

maximum value of the excitation having excitation frequency cω ω≤ ) in the static

analysis. This is a simple but very useful tool that helps in the determination of dynamic

amplification. The statistical study for the mean and the (mean+one stdv) values of d

R

for different earthquake events were evaluated for comparison with each other.

Rd values obtained for each earthquake events were compared with that caused due to

simple harmonic excitation. The concepts of equivalent forcing frequency ratio (ω/ωn), a

period shift factor α and damping modification factor k were introduced to facilitate the

comparison with simple harmonic excitation. The Rd values calculated from different

events can assist us in the design of dampers, which would be able to sustain pulse-like

excitations.


1. A simple sine-pulse (single-period pulse) does not always give a good

representation of ground motions in response identification. The use of a sine-

pulse should be limited to those response characteristics in which the spectral

displacement more or less increases monotonically with a natural period n

T from

106

the cut-off frequency to the frequency that corresponds to the peak of the spike

nearest the lowest usable frequency.

2. In most cases (e.g., Figs. 3.2.a, 3.4.a, 3.5.b & 3.7.a), it has been observed that

displacement response cannot be simplistically obtained from a single sine pulse.

A full sinusoidal pulse gives the lower bound value of Rd for any pulse-like

record; whereas a continuous harmonic excitation gives the upper bound value of

Rd. It is the succession of sinusoidal pulses for a finite duration that give rise to

such high value of Rd.

3. In the present study, not a single velocity-pulse extracted from the 72 ground

motions by the proposed signal processing technique was found to have a half-

cycle velocity pulse. Around ten records have a predominant single period

velocity pulse, but most other pulses are multiple-period pulses. That is one of the

reasons why Rd exceeds 3.14.

4. A number of NFGMs exhibit a ripple effect in which maximum spectral

displacements are observed at different natural periods. This is the result of the

presence of multiple predominant frequency pulses in these earthquake records.

5. Resonance pulses are special cases of multiple frequency pulses when the

dynamic displacements are very high compared to their static counterparts. These

pulses are identified as high, steep and narrow amplitude spikes in the velocity

Fourier spectrum. Their magnitudes can be quantified by the use ofd

R .

6. Velocity Fourier spectrum is an important tool for design engineers to use, to

characterize the displacement response of SDF systems.

107

7. A structure subjected to long-period ground motion may undergo very high

spectral displacements, as in the example of the TCU068-N320E, Chi Chi 1999

earthquake, when it has a high maximum static deformation 0( )st

u compared to its

maximum dynamic deformation 0u D≡ .

8. Alternatively, there might be high dynamic amplification as determined from

value of d

R , as in the example of the Kobe 1995 earthquake event, which exhibits

low maximum static deformation 0( )st

u compared to its high maximum dynamic

deformation 0u D≡ . Damping frequently controls in such cases.

9. The displacement response factor d

R may be independent of the magnitude of the

PGA (Sec. 3.5.4), i.e., the d

R for a low PGA may be higher than that for a higher

PGA.

10. When considering different events, all over the world, the value of the

displacement response factor may not always be directly proportional to the

moment magnitude. These depend more upon the directivity and the site, as well

as upon path effects.

11. The directivity and path effects are similar in the dynamic responses of SDF

systems, since both result from dominant long-period ground motions.

12. For ζ = 0.05, the (m+σ) values of Rd vary from 3.44 to 4.8 for different earthquake

events.

13. The concepts of equivalent forcing frequency ratio (ω/ωn), a period shift factor

and “damping modification factor” were introduced to facilitate the comparison

with simple continuous harmonic excitation.

108

14. The Rd values calculated from different events may assist us in designing

dampers, which can sustain pulse-like excitations.

During the evaluation of responses, we have observed very important properties of

velocity Fourier spectrum. We have seen that the Fourier spectrum effectively helps us to

determine spectral displacements. This observation will be used in the next chapter to

propose an analogous displacement response spectrum.

109

CHAPTER 4

ANALOGOUS DISPLACEMENT RESPONSE SPECTRA

4.1 INTRODUCTION

In chapter 2, we detected directivity effects in ground motions by signal processing using

Fourier analysis. A structural engineer is generally more interested in the natural period

of vibration of a system and the period of excitation. Therefore, Fourier analysis is

applied when the frequency contents of the excitation are of greater importance. In the

previous two chapters, we have also seen that the velocity Fourier spectrum is the key to

understanding spectral displacements, as well as dynamic magnification.

The evaluation and plotting of the displacement response spectra (DRS) for a number of

pulse-like ground motions have shown that they bear a strong mirror image resemblance

to ground velocity Fourier amplitudes. The peak spectral displacements in the DRS occur

at natural periods of vibration Tn, which correspond to the frequencies at which the

magnitudes of the Fourier amplitudes are at their peaks. The values of frequencies on the

abscissa of the VFA closely correspond to the reciprocal values of system periods Tn on

the abscissa of the DR spectrum, while the values on the y-axis (the velocity Fourier

amplitudes in g-s) of the VFA nearly correspond to those on the y-axis (spectral

displacements in cm) of the DR spectrum.

110

4.2 BRIEF REVIEW


given by [Clough and Penzien 1992; Chopra 2007; Humar 1992] as

( ) ( )mu cu ku p t ma t+ + = = −&& & (4.1)

where m is the mass, c is the viscous damping coefficient, k is the stiffness, a(t) is the

ground acceleration; and ( )u t , ( )u t& and ( )u t&& are the resulting time-dependent

displacement, velocity and acceleration of the system, respectively. For a given seismic

ground acceleration time history a(t), the corresponding velocity time history v(t) and the

displacement time history s(t) can be readily obtained through successive integration. If

we denote any of these response time histories as x(t), upon decomposing x(t) into its

pulse and non-pulse components

)()()( tnpxtpxtx += (4.2)

The time history can be separated into its harmonic components by Fourier integral

1

( ) ( )2

tx t X e d

ωω ωπ

∞

−∞

= ∫i (4.3)

where

( ) ( )t

X x t e dtωω

∞

−∞

= ∫-i (4.4)

is the Fourier Transform of x(t), 1= −i , and ω is the forcing frequency in radians/sec.

By using a Butterworth low-pass filter [Oppenheim et al., 1997] given by

( )

1( )

1 / Blp N

c

H ωω ω

=+ i

(4.5)

111

where ωc is the cut-off frequency in radians/sec., and NB is the order of the Butterworth

filter. By using a suitable cut-off frequency fc=ωc/2π, the pulse component can be

evaluated using the equation

1( ) ( ) ( )

2

t

p lpx t H X e d

ωω ω ωπ

∞

−∞

= ∫i (4.6)

If we replace ( ) ( )lp

H Xω ω by ( )p

X ω , we can rewrite Eq. (4.6) as

1

( ) ( )2

t

p px t X e d

ωω ωπ

∞

−∞

= ∫i (4.7)

The pulse xp(t) obtained from Eq. (4.7) can be used in a time-domain analysis to compute

spectral displacements. In the following study, Eq. (4.7) is used to compute acceleration

and velocity pulse histories. It should be noted that in this chapter, the pulse component

of a time history is considered to be synonymous with the dominant low-frequency

contents of that time history.

4.3 SPECTRAL DISPLACEMENTS AND FOURIER AMPLITUDES

In this section, a method is proposed for expressing spectral displacements for an

undamped linearly elastic single degree-of-freedom (SDF) system, in terms of the

velocity Fourier amplitudes from the lowermost frequency range of ground motions. This

spectrum, referred to as analogous displacement response spectrum (DRS), is quite

comparable to that obtained when the system is subjected to the original processed time

history or pulse-excitation. The term ‘analogous’ is used here because this spectrum is

not identical to the usual displacement response spectrum used in earthquake engineering

design. It is essentially an “exact” ground velocity Fourier amplitude plot in the lower-

112

most frequency range (LUF ≤ f ≤ fc < 1.67 Hz), rotated about a vertical axis, that passes

through the cut-off frequency fc. Because of this, the discussion in this section is

restricted to velocity Fourier amplitudes (VFAs) in this range.

By replacing xp(t) and Xp(ω) in Eq. (4.7) by the velocity-pulse time history vp(t) and its

Fourier transform Vp(ω), respectively, we obtain

1

( ) ( )2

t

p pt e dv V

ωω ωπ

∞

−∞

= ∫i (4.8)

where

( ) ( )t

p pt e dtV v

ωω∞

−∞

= ∫-i (4.9)

In what follows, we have deleted the subscript p. After division by T0 (defined on page 37

in Sec. 2.5), the above equations can be expressed in discrete form as

1 1

( ) (2 / )

0 00

1 1j n

N Nt nj N

j n n

n n

V v e t v eT N

ω π− −

− −

= =

= ∆ =∑ ∑i i (4.10)

where,

( ) ( )

1 12 /

0 0

j n

N Nt nj N

n j j

j j

v V e V eω π

− −

= =

= =∑ ∑i i

(4.11)

ω0 = 2π/T0 is the fundamental frequency and ωj (j>0) is the frequency of the jth

harmonic.

Vj is a complex valued coefficient that defines the amplitude and phase of the jth

harmonic

of VTH vn. Eq. (4.10) is an expression for computing Vj from the sequence vn, i.e., for

analyzing the sequence vn to determine how much of each frequency component is

required to synthesize vn using Eq. (4.11). It may be noted that the continuous form V(ω)

has a unit of displacement (cm), whereas the discrete form Vj has a unit of velocity

113

(cm/sec).

From the analyses of a number of earthquake records with different sampling intervals ∆t,

it was observed that when the velocity Fourier amplitudes were multiplied by ∆t, an

approximate spectral displacement curve that resembles the exact spectral displacement

curve was obtained, i.e.,

0n jnD u V t≡ ≈ × ∆ (4.12)

When the natural period Tn of a linearly elastic undamped SDF system is equal to Tj, the

jth

harmonic ground velocity is probably exerting the most dominant effect on the

response of the system. Therefore, a proportionality should exist between the amplitude

of the jth

harmonic ground velocity and the spectral displacement in the range under

consideration.

It may be noted that when the amplitude of Vj is low (valley points) in the Fourier

spectra, the other harmonics (≠ jth

) of ground velocity may have some effect on the

spectral displacements. This may be one of the reasons that the values at the valley points

of the transformed–VFA are relatively lower than the response obtained from the original

ATH (Fig. 4.4).

Example 1. Transformed–VFA for TCU052, N322E, Chi-Chi 1999

Figs. 4.1 and 4.2 illustrate an example of a ground motion record, TCU052, N322E, Chi-

Chi 1999, which has a relatively simple velocity Fourier spectrum (Fig. 4.1) dominated

by a single predominant frequency with a peak at f = 0.055 Hz. The Lowest Usable

114

Frequency (LUF), as given in the ground motion data source (PEER NGA in the present

study), is 0.05 Hz and is marked by a dotted vertical line in the figure. The cut-off

frequency, identified as fc = 0.28 Hz, is also marked. Fig. 4.2 illustrates the DRS plot for a

linearly elastic undamped single SDF system subjected to the original and the pulse

acceleration time histories for ζ=0, superimposed on the Transformed–Velocity Fourier

Amplitudes (Trf-VFA). It has been plotted for Tc = 3.6 ≤ Tn ≤ 20 sec., which corresponds

to fc = 0.28 ≥ f ≥ LUF = 0.05 Hz for the present case. Tc is the cut-off period, which is the

reciprocal of the cut-off frequency fc.

The values on the abscissa of the velocity Fourier spectrum are plotted for the sampling

interval of f0=1/(N×∆t). Since N = 18000 and ∆t = 0.005 sec. in the present case, f0 =

0.0111 Hz. Therefore, in the range fc = 0.28 Hz and LUF = 0.05 Hz, the total number of

points, n = [(0.2778 – 0.05)/f0 + 1] = 21.5. If we take the reciprocal of the values for each

of these frequencies, we obtain the corresponding natural period of vibration of the

system, Tn. Thus, a total of 21 values of the Trf-VFA plot are shown.

The spectral displacements are obtained from the corresponding discrete values of the

Fourier amplitudes multiplied by the sampling interval ∆t. For example, corresponding to

f = 0.055 Hz, which happens to be the peak value on the velocity Fourier spectra, we have

Tn = 1/0.055 = 18 sec. At this point, the Fourier amplitude is 161 g-s (Fig. 4.1), i.e.,

161×981=157,940 cm/sec. Multiplying this by the sampling interval ∆t = 0.005 sec., we

obtain the corresponding spectral displacement as 790 cm (Fig. 4.2). At Tn = 18 sec, the

“exact” value of the spectral displacement computed when the system is subjected to the

115

original ATH is 806 cm. Therefore, it can be seen that the two values are very

comparable, with an error of only -2.0 %. By using a similar technique, 21 points from

the VFA were evaluated; they are plotted with the symbol * in Fig. 4.2. Note that the

values 161, 129 & 60 g-s, as marked on the y-axis of the VFA (Fig. 4.1), correspond to

790, 632 & 294 cm, as marked on the y-axis of the Trf–VFA (Fig. 4.2). It can be observed

that this transformed plot represents the spectral displacements reasonably well for the

range under consideration, with the peak points showing better agreement than the valley

points.

In doing the computations, the response plots due to the original and pulse-ATH were

evaluated and plotted at an interval of Tn = 0.1 second. Therefore, they appear smooth

and continuous. However, in constructing the transformed-VFA the number of sample

points was considerably less (only 21, as illustrated above) in the range LUF ≤ f ≤ fc. As

a result, the plot appears somewhat disjointed.

Example 2. Transformed–VFA for TCU054, N142E, Chi-Chi 1999

In Fig. 4.3, the Fourier spectrum for the TCU054, N142E, Chi Chi 1999 record is shown.

It appears more erratic because of the existence of multiple period pulses. In Fig. 4.4 the

transformed–VFA is compared with the DRS obtained from the original and pulse ATH.

Again, except at the valley points, it can be seen that the transformed–VFA resembles the

“exact” DRS computed from the original and pulse ATHs reasonably well. The 15 points

used to construct the Trf–VFA are marked in the figure. Note that the values of 46, 35 and

29 g-s, as marked on the y-axis of the VFA (Fig. 4.3), correspond to 226, 172 & 141 cm

116

as marked on the y-axis of the Trf–VFA (Fig. 4.4).

Because the analogous DRS give good approximations for the actual spectral

displacements, they can be used for the preliminary design of long-period structures.

Nevertheless, the exact DRS should be used for verification of the final design. One

important advantage of constructing these analogous DRS is that for a given set of design

ground motions, one can study the response characteristics from their respective

transformed-VFAs relatively easily. These analogous DRS could be used to determine

design ground motions and soil-structure interactions, or could be used in Performance-

Based Design by relating spectral displacements directly to ground velocities via Fourier

amplitudes.

Figure 4.1 TCU052, N322E, Chi Chi 1999, velocity Fourier spectrum

f (Hz)

fc

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

f = 0.055 Hz

117


Figure 4.2 TCU052, N322E, Chi Chi 1999,

DRS computed from the original and pulse acceleration time histories for ζ=0

superimposed on the “Transformed-Velocity Fourier Amplitudes”

Figure 4.3 TCU054, N142E, Chi Chi 1999, velocity Fourier spectrum

Spectr

al D

ispla

ce

me

nt

(cm

)

f (Hz)

fc

Velo

city F

ourier

Am

plit

ud

es (

g-s

)

f = 0.111 Hz

f = 0.156 Hz

f = 0.211 Hz

Orig ATH

pulse ATH

Trf-VFA

118


Figure 4.4 TCU054, N142E, Chi Chi 1999

DRS computed from the original and pulse acceleration time histories for ζ=0

superimposed on the “Transformed–Velocity Fourier Amplitudes”

4.4 DISCUSSION

A typical Fourier spectrum in the frequency range (LUF ≤ f ≤ fc < 1.67 Hz) has the

following characteristic information:

1. Frequency – abscissa: It has been demonstrated that the reciprocal of the ground

velocity frequency can be compared with the period of vibration of an elastic

undamped SDF system.

2. Fourier amplitudes – ordinates: It has been shown that the values of velocity

Fourier amplitudes can be correlated with the spectral displacements of an elastic

undamped SDF system.

Spectr

al D

ispla

ce

me

nt

(cm

)

119

3. Gradient: It has been observed in chapter 3 that the higher the gradient of velocity

Fourier amplitudes, the greater is the dynamic amplification (Rd) of an elastic

SDF system. Steep and near-vertical velocity Fourier amplitudes relate to a higher

Rd of a SDF system. The author has not been able to establish this observation

analytically.

4. The angle of phase: Each point of the Fourier transforms (not absolute values)

defines the angle of phase. The author has not been able to uncover any relation

between this and the spectral response due to pulse.

The very important distinction between the analogous DRS and the conventional DRS is

that the former is the representation of the ground (pulse) velocity time history, whereas

the latter is the response of an elastic SDF system due to ground acceleration time history.

Analogous DRS is applicable only for the natural period of vibration, Tn greater than the

cut-off period, Tc. It has been observed that the concept presented in the present chapter

no longer holds true for periods less than the cut-off period. The transformed–Fourier

amplitudes digress rapidly from actual spectral displacements.

The analogous DRS presented in this chapter is applicable to an undamped elastic SDF

system. Can we propose any modification or a filter to the pulse-like ground velocity

time history, so that it would compute and closely represent the spectral displacements of

a SDF system with a 5% damping ratio? Since the ground cannot be dampened (recall

that the Fourier spectrum represents the frequency contents of the ground velocity time

120

history), hypothetically it may be possible to devise a synthetic filter that can be applied

to the original Fourier amplitudes, so that they subsequently have less value and have a

more or less smooth and continuous curve. We have already observed in many DRS of

ground motions with multiple period pulses that the spectral displacements for an

undamped elastic system are very uneven and erratic, whereas those for a 5% damped

system are smooth and continuous.

4.5 CONCLUSIONS


(1) Analogous displacement response spectrum (DRS) is a fast and efficient instrument

for understanding the spectral displacement characteristics of undamped SDF

systems. It gives the upper-bound response of structures subjected to pulse-like long-

period ground motions.

(2) Due to its inherent simplicity, the analogous DR spectrum can be used in the

preliminary stage of structural design for seismic loading.

(3) The analogous DRS could also be useful in Performance Based Seismic Design by

directly relating spectral displacements to the Fourier amplitudes of the given ground

velocity.

In chapters 2, 3 and 4, we have mostly dealt with elastic SDF systems. In the next

chapter, we will examine pulse effects on the displacement ductility requirement of an

inelastic SDF system, like a bridge bent.

121

CHAPTER 5

PULSE EFFECTS ON THE DISPLACEMENT DUCTILITY REQUIREMENT OF BRIDGE BENTS

5.1 PREVIEW

This chapter evaluates the displacement ductility requirement for a typical bridge bent

subjected to extreme loads, such as those from pulse-like near-fault ground motions

(NFGMs). The ductility requirement, as stated by prevailing design codes, may not be

valid for such structures in near-fault (less than 10 miles) zones. Due to the dominance

of the pulses, medium- to long-period structures are strongly affected, often resulting in

high residual or permanent deformations after the cessation of seismic ground motions.

As per most design codes, the magnitude of displacements associated with P-Delta

effects is required to be captured using non-linear time history analysis. The higher the

value of the target displacement ductility demand or the P-Delta ratio, the larger is the

magnitude of the residual displacements. Because permanent deformations in a bridge

bent would lead to a higher eccentric loading from the superstructure, the bent would be

subjected to higher secondary moments from it own design load. Even if the bridge

survives the earthquake, subsequent failure may occur due to its own design loads. A

suitable balance should be made between ductility and residual displacements in the

design of such structures using Performance-Based Seismic Design procedures.

122

5.2 INTRODUCTION

Performance-Based Seismic Design (PBSD) is a process that permits the design of new

structures or the upgrade of existing structures with a realistic understanding of the risk of

life, as well as occupancy and economic losses that may occur as a result of earthquakes.

The performance of a structure designed using PBSD is studied after it is subjected to

various earthquake events. Each study provides valuable information on the level of

damage, which consequently allows us to estimate the amount of life, occupancy and

economic losses that may occur. The design can be adjusted until the projected risks or

losses are acceptable. The Applied Technology Council (ATC), under FEMA

sponsorship, is currently engaged in a project (ATC-58) to develop the next-generation

PBSD procedures and guidelines. [FEMA-445 2006]

Long-period ground motions that give rise to directivity or path effects are important

seismic design considerations because they often impose large demands on medium to

long-period structures. The inherent properties that cause these effects, which often

manifest themselves as pulses, have been identified as dominant low-frequency contents

embedded in typical long-period seismic ground motions [Khanse and Lui 2008a; 2008b;

2009].

As per CALTRANS (2006) Seismic Design Criteria (SDC), v 1:4 the target global

displacement ductility demand value µD for single column bents supported on fixed

foundation should be ≤ 4. Further, CALTRANS SDC requires that the dynamic effects of

gravity loads acting through lateral displacements shall be included in the design; and

123

that the magnitude of displacements associated with P-Delta effects can only be

accurately captured with non-linear time history analysis.

This chapter examines the existing provisions for ductility requirements for a typical

bridge bent subjected to seismic ground motions, and demonstrates that these ductility

requirements may not be valid for structures in near-fault (< 10 miles) zones. Due to

dominance of the pulses, structures having a natural period of vibration Tn between 1 and

2 seconds are highly affected, often resulting in high residual displacements (or

permanent deformations) even after the cessation of seismic ground motions. Although

the existing design codes do not specify the extent of permissible permanent

displacements/ deformations, it is clear that a higher target local displacement ductility

demand value µC will result in larger residual displacements, which may not be desirable.

A suitable balance should therefore be made for the design of structures, like single

column bents in near-fault zones using Performance-Based Seismic Design

methodologies.

Appendix C of ATC-55 [FEMA-440 2005] gives plots of CR values (i.e., the ratio of

maximum nonlinear to linear displacements) versus periods of vibration in Sec. C.2.1 and

C.2.2 for the elastic perfectly plastic (EPP) and the stiffness-degrading (SD) hysteretic

models, respectively. Although this critical parameter is important for the design of a

system undergoing seismic excitation (i.e., during an earthquake), the effect of permanent

deformations after cessation of the earthquake is another criterion that needs to be

124

addressed, especially if the permanent deformations are excessive. This chapter intends

to shed light on this aspect of seismic design.

5.3 EXISTING DESIGN CODE PROVISIONS FOR

DISPLACEMENT DUCTILITY DEMAND VALUE, µC

5.3.1 CALTRANS Seismic Design Criteria, 2006, v 1:4 [CALTRANS 2006]

CALSTRANS SDC is focused on concrete bridges. An individual bent or column has

been referred to as a local system. Displacement ductility demand is a measure of the

imposed post-elastic deformation on a member. As per Sec. 2.2.4, the target global

displacement ductility demand value for Single Column Bents supported on fixed

foundation is µD ≤ 4. The target value may range between 1.5 and 3.5 where specific

values cannot be defined. Local displacement ductility capacity for a cantilever column

fixed at the support (Fig. 5.1) is defined by Eq. 3.6 of SDC as:

CC col

y

∆µ

∆= (5.1)

col

y∆ is shown in Fig. 5.1. Local member displacement capacity C

∆ is defined as a

member’s displacement capacity attributed to its elastic and plastic flexibility as defined

in SDC Sec. 3.1.4. The structural system’s displacement capacity C

∆ is the reliable lateral

capacity of the bridge or subsystem as it approaches its collapse limit state. Further, SDC

Sec. # 3.1.4.1 states that each ductile member shall have a minimum local displacement

ductility capacity of µC = 3 to ensure dependable rotational capacity in the plastic hinge

125

regions regardless of the displacement demand imparted to that member. If µC < 3,

approval from competent authority is required.

Figure 5.1. Local Displacement Capacity - Cantilever Column w/ Fixed Base (left)

P-Delta Effects on Bridge Columns [Fig. 3.1 & 4.2 from CALTRANS 2006]

Sec. 4.2 of SDC on P-Delta Effects states that (1) The dynamic effects of gravity loads

acting through lateral displacements shall be included in the design, and (2) The

magnitude of displacements associated with P-Delta effects can only be accurately

captured with non-linear time history analysis.

Sec. 6.1.2.1 of SDC states that for preliminary design of structures within 10 miles of an

active fault, the spectral acceleration on the Acceleration Response Spectrum (ARS)

curves shall be magnified by 20% for Tn≥1.0 sec., and a linear interpolation from 0 to

20% be used for 0.5≤Tn≤1.0 sec.

5.3.2 AASHTO (2007, 2009) Guide Specifications for LRFD Seismic Bridge Design

126

AASHTO (2007) Sec. C4.2.2 states that essential or critical bridges within 6 miles of an

active fault require a site-specific study and inclusion of vertical ground motion in the

seismic analysis. As per Sec. 4.7.1 on ‘Design Methods for Lateral Seismic Displacement

Demands’, each structure shall be categorized according to its intended structural seismic

response in terms of damage levels:

a) Conventional Ductile Response (i.e., Full-Ductility Structures): 4.0 ≤ µD ≤ 6.0,

(see Article 4.9 of AASHTO 2007). This response is anticipated for a bridge in

Seismic Design Category D designed for the Life Safety Criteria.

b) Limited-Ductility Response for horizontal loading, local µD < 4.

c) Limited-Ductility Response in concert with added protective systems. In this case

a structure has limited ductility with the additional seismic isolation, passive

energy dissipating devices, and/or other mechanical devices to control seismic

response. Using this strategy, a plastic mechanism may or may not form. µD is the

local displacement ductility demand in AASHTO 2007.

AASHTO 2007 Sec. C4.7.1 states that design displacements and minimum flexural

capacities of columns are generally governed by P-Delta effects. Sec. 4.9 on Member

ductility requirement for Seismic Design Category D states that for single column bents,

local member ductility demand µD ≤ 5. Sec. 4.11.5 on ‘P-Delta Capacity Requirement

for Seismic Design Category C and D’ requires that the dynamic effects of gravity loads

acting through lateral displacements shall be included in the design. The magnitude of

displacements associated with P-Delta effects can only be accurately captured with non-

linear time history analysis.

127

5.4 BRIEF BACKGROUND

After a system yields, it goes into the inelastic stage. Behavior of a typical steel single

degree-of-freedom (SDF) system may be represented by an Elasto-plastic (EP) force-

deformation model as shown in Fig. 5.2. In the present analysis using BISPEC, the

Elasto-perfectly-plastic (EPP) model (Fig. 5.2.a) is used. However, a more general

version is the bi-linear model shown in Fig. 5.2.b in which the post-yield stiffness k1 is

not zero.

On the other hand, the behavior of a typical concrete SDF system in the inelastic stage is

usually idealized by the stiffness degrading (SD) model as shown in Fig. 5.3. In this

model, Initial Yield Point (IYP) and Current Yield Point (CYP) as marked on Fig. 5.3 are

required to determine subsequent stiffness after unloading or reloading. In the present

analysis using BISPEC, the ordinary stiffness degrading (SD) model is used for the

response evaluation.

The governing equation for a system that experiences inelastic behavior is

( , ) ( )s

mu cu f u u ma t+ + = −&& & &

For a given ground acceleration time history a(t), u(t) depends on ωn, ζ and uy in addition

to the form of the hysteretic force-deformation relation used as illustrated in Fig. 5.2 or

5.3.

As defined on Fig. 5.2.a, the ductility factor is given by

m

y

u

uµ =

128

Another factor that is of interest in an inelastic analysis is the yield strength ratio, given

by β = fy/(mg), where fy is the yield force, m is the mass of the system, and g is the

acceleration due to gravity.

Figure 5.2. Idealized Elasto-Plastic behavior:

(a) Elasto-perfectly-plastic (EPP) model, (b) Bilinear Elasto-plastic model [Clough 1966]

Figure 5.3. Idealized Stiffness Degrading (SD) behavior:

(a) Ordinary SD model, (b) Bilinear SD model [Clough 1966]

As per ATC-55 (FEMA 440), CR is a modification factor to relate the expected maximum

displacement of an inelastic SDF system with EPP or SD hysteretic properties to

displacements calculated for the linear elastic response. A plot of CR against period of

vibration for an EPP model is shown in Fig. 5.4 for first 20 NFGM as listed in Table 5.5.

Fig. 5.5 exhibits a plot of CR for the SD model. The plots are given for different values of

fy

fs fs

fy

fs fs

fy

129

normalized lateral strength, R = (mSa)/fy; where m is the mass of SDF system, and Sa is

the spectral acceleration. (mSa) is the elastic strength demand, and fy is the yield force.

Figure 5.4. CR values of SDF system having ζ = 0.05 with Elasto-perfectly Plastic (EPP)

hysteretic behavior for a set of 20 NFGM as given in Table 5.5 [Sec. C.2.1 of ATC-55]

Figure 5.5. CR values of SDF system having ζ = 0.05 with stiffness degrading (SD)

hysteretic behavior for a set of 20 NFGM as given in Table 5.5 [Sec. C.2.2 of ATC-55]

130

Time (sec)

While the importance of CR values in the analyses cannot be overemphasized, the

residual displacements (permanent deformation) of an inelastic system after cessation of

pulse-like near-fault ground motions (NFGMs) is another critical parameter that needs to

be addressed.

For example, consider a bridge bent modeled as an ordinary stiffness degrading (SD)

SDF system with ζ = 0.05 having height L = 30 ft, m = 5 kip-sec2/in and a natural period

of vibration, Tn = 1.2 sec. (This value of Tn is selected because the period of one of the

pulses of ground ATH used in this example is 1.2 sec.). From k = mω2, we have k = 137

kip/in. To obtain µ = 4, fy was (iteratively) determined to be 1050 kips. Thus, uy = fy/k =

7.6 in ≈ L/50 = 7.2 in. When such a system is subjected to a pulse-like ground motion

such as Takatori, Kobe 1995, a displacement response plot as shown in Fig. 5.8 is

obtained. From the figure it can be seen that the maximum inelastic displacement is, um =

31.5 in, and that after cessation of ground motions a permanent deformation (residual

displacement) of 11.7 inches remains. This residual displacement is considered quite high

for the 30-ft bridge bent. 31.5 137

41050

m m

y y

u u k

u fµ

×= = = ≈

Figure 5.6. Displacement response of a SDF system subjected to Takatori, Kobe 1995

Dis

pla

cem

ent

(in

)

131

The present research work addresses the probable (mean + one standard deviation) values

of residual displacements for SDF system for different groups of pulse-like ground

motions. This area has not been addressed in prevailing codes including CALTRANS

SDC 2006.

5.5 EVALUATION PROCEDURES

This chapter examines the effect of variation in residual displacements (permanent

deformations) on a single degree-of-freedom system after cessation of near-fault pulse-

like ground motions. A bridge bent or column can be modeled as a single degree-of-

freedom system since most of its mass is concentrated at the top free end (Fig. 5.1). The

support at the bottom is assumed fixed, i.e., ground rotation is neglected. Further, the

effect of near-fault vertical ground motions is neglected. These two would likely

aggravate the values of residual displacements after cessation of ground motions. Five

groups of pulse-like near-fault ground motion records with forward directivity are

considered:

(1) Mw = 6.5 ± 0.25, Soil = A & B, 8 records (Table 5.1)

(2) Mw = 6.5 ± 0.25, Soil = D, 19 records (Table 5.2)

(3) Mw = 7.25 ± 0.25, Soil = A & B, 12 records (Table 5.3)

(4) Mw = 7.25 ± 0.25, Soil = D, 15 records (Table 5.4)

(5) ATC-55 (2005), 6.19 ≤ Mw ≤ 7.62, Soil = A, B, D & E, (20 + 2) records (Table 5.5)

132

Table 5.1. Mw = 6.5 ± 0.25, Soil = A & B

No Soil File NGA R

(miles)

PGA

(g)

PGV

(in/s)

PGD

(in)

1 A NORTHR_655_032FN 0983 3.37 0.52 26.5 16.8

2 A NORTHR_LDM_032FN 1013 3.67 0.58 30.3 7.9

3 A NORTHR_PUL_032FN 1051 4.35 1.38 42.1 8.9

4 A NORTHR_PAC_032FN 1050 4.35 0.50 19.3 2.5

5 B NORTHR_SCE_032FN 1085 3.22 0.84 45.8 15.5

6 B NORTHR_JEN_032FN 0982 3.37 0.52 26.5 16.5

7 B NORTHR_WPI_032FN 1045 3.4 0.43 34.5 21.7

8 B NORTHR_RRS_032FN 1063 4.03 0.87 65.7 11.3

Table 5.2. Mw = 6.5 ± 0.25, Soil = D

No Soil File NGA R

(miles)

PGA

(g)

PGV

(in/s)

PGD

(in)

1 D IMPVALL_HEMO_233FN 171 0.04 0.38 45.3 15.8

2 D IMPVALL_HAEP_233FN 158 0.21 0.36 17.4 4.1

3 D IMPVALL_HE07_233FN 181 0.35 0.46 42.83 18.0

4 D IMPVALL_HAGR_233FN 159 0.40 0.31 21.2 6.0

5 D IMPVALL_HE06_233FN 181 0.77 0.44 44.0 26.2

6 D IMPVALL_HE08_233FN 183 2.39 0.47 19.1 14.5

7 D IMPVALL_HE05_233FN 180 2.45 0.38 36.0 24.4

8 D IMPVALL_HEDA_233FN 184 3.15 0.42 23.5 15.3

9 D NORTHR_SYL_032FN 1086 3.29 0.73 48.3 12.4

10 D NORTHR_SCS_032FN 1084 3.32 0.59 51.3 21.3

11 D NORTHR_NWH_032FN 1044 3.67 0.72 47.3 13.8

12 D IMPVALL_HE10_233FN 173 3.83 0.18 18.5 12.4

13 D IMPVALL_HE04_233FN 179 4.37 0.36 30.6 23.1

14 D IMPVALL_HECC_233FN 170 4.53 0.18 21.5 15.1

15 D IMPVALL_HHVP_233FN 185 4.74 0.26 21.7 13.0

16 D NORTHR_0637_032FN 1004 5.23 0.73 24.9 7.4

17 D IMPVALL_HE11_233FN 174 7.72 0.37 16.2 7.3

18 D IMPVALL_HE03_233FN 178 7.97 0.23 16.2 9.3

19 D NORTHR_CNP_032FN 0959 9.11 0.38 21.1 8.5

133

Table 5.3. Mw = 7.25 ± 0.25, Soil = A & B

N S File NGA R

(miles)

PGA

(g)

PGV

(in/s)

PGD

(in)

1 A CHICHI_TCU068_N320E 1505 0.20 0.57 113.9 187

2 A CHICHI_TCU052_N322E 1492 0.41 0.50 65.8 120.3

3 A CHICHI_TCU102_N232E 1529 0.94 0.24 43.3 36.0

4 A TABAS_TAB_TR 143 1.27 0.85 47.7 37.4

5 A LANDERS_LCN_239FN 879 1.36 0.69 53.6 92.9

6 A LOMAP_LGP_038FN 0779 2.41 0.94 38.2 24.6

7 A LOMAP_LEX_038FN 3548 3.11 0.52 39.2 13.5

8 A LOMAP_G01_038FN 0765 4.30 0.43 15.2 2.8

9 A KOCAELI_GBZ_184FN 1161 6.77 0.24 20.5 17.3

10 B KOBE_KJM_140FN 1106 0.6 0.85 37.7 9.7

11 B KOCAELI_SKR_090 1171 1.93 0.38 31.3 27.8

12 B CAPEMEND_PET260FN 828 5.07 0.62 32.2 10.0

Table 5.4. Mw = 7.25 ± 0.25, Soil = D

No S File NGA R

(miles)

PGA

(g)

PGV

(in/s)

PGD

(in)

1 D KOBE_TAZ_140FN 1119 0.17 0.65 28.5 8.2

2 D CHICHI_TCU065_N123E 1503 0.37 0.82 51.1 36.6

3 D CHICHI_TCU075_N094E 1510 0.56 0.33 34.9 34.2

4 D CHICHI_TCU129_N136E 1549 1.14 0.83 22.4 20.3

5 D CHICHI_TCU076_N128E 1511 1.71 0.35 34.6 16.1

6 D ERZIKAN_ERZ_032FN 0821 2.73 0.49 37.6 12.6

7 D KOCAELI_YPT_180FN 1176 2.75 0.28 19.0 17.0

8 D CHICHI_TCU054_N142E 1494 3.3 0.18 23.3 24.8

9 D LOMAP_STG_038FN 0802 5.3 0.36 21.9 11.6

10 D LOMAP_WVC_038FN 0803 5.77 0.40 28.1 8.2

11 D LOMAP_G02_038FN 0766 6.88 0.41 18.0 5.0

12 D DUZCE_BOL_175FN 1602 7.44 0.68 22.6 9.0

13 D LOMAP_G03_038FN 0767 7.94 0.53 19.4 4.3

14 D LOMAP_G04_038FN 0768 8.87 0.35 14.1 4.5

15 D KOCAELI_DZC_163FN 1158 9.55 0.28 20.6 15.0

134

Table 5.5. Near-fault (20+2) records with forward directivity used in ATC – 55

No Soil File Mw NGA R

(miles)

PGA

(g)

PGV

(in/s)

PGD

(in)

1 A LOMAP_LGP_038FN 6.93 0779 2.41 0.94 38.2 24.6

2 A LOMAP_LEX_038FN 6.93 3548 3.11 0.52 39.2 13.5

3 E KOBE_TAK_140FN 6.90 1120 2.37 0.68 66.6 17.7

4 B KOBE_KJM_140FN 6.90 1106 0.6 0.85 37.7 9.7

5 D KOBE_TAZ_140FN 6.90 1119 0.17 0.65 28.5 8.2

6 D ERZIKAN_ERZ_032FN 6.69 0821 2.73 0.49 37.6 12.6

7 B NORTHR_RRS_032FN 6.69 1063 4.03 0.87 65.7 11.3

8 D NORTHR_0637_032FN 6.69 1004 5.23 0.73 24.9 7.4

9 D NORTHR_SYL_032FN 6.69 1086 3.29 0.73 48.3 12.4

10 D NORTHR_NWH_032FN 6.69 1044 3.67 0.72 47.3 13.8

11 D IMPVALL_HEMO_233FN 6.53 171 0.04 0.38 45.3 15.8

12 D IMPVALL_HE06_233FN 6.53 181 0.77 0.44 44.0 26.2

13 A MORGAN_CYC_FN 6.19 451 0.33 0.81 24.4 4.0

14 D MORGAN_AND_FN 6.19 448 2.0 0.45 11.4 2.2

15 D KOCAELI_YPT_180FN 7.51 1176 2.75 0.28 19.0 17.0

16 D KOCAELI_DZC_163FN 7.51 1158 9.55 0.28 20.6 15.0

17 D DUZCE_DZC_175FN 7.14 1605 4.1 0.34 22.8 18.6

18 A KOCAELI_IZT_163FN 7.51 1165 4.5 0.15 9.0 5.8

19 D DUZCE_BOL_175FN 7.14 1602 7.44 0.68 22.6 9.0

20 A KOCAELI_GBZ_184FN 7.51 1161 6.77 0.24 20.5 17.3

21 A CHICHI_TCU052_N322E 7.62 1492 0.41 0.50 65.8 120.3

22 A CHICHI_TCU068_N320E 7.62 1505 0.20 0.57 113.9 187.0

The initial damping ratio ζ was assumed to be equal to 5% for all systems. The natural

periods of vibration Tn for the single degree-of-freedom system were varied from 0.5 to 4

seconds. The values of the (mean+σ) spectral displacements, the (mean+σ) ratio of

residual displacements to spectral displacements (Dres/D) and the (mean+σ) residual

displacements (Dres) are plotted against natural period of vibration Tn for local

displacement ductility values µC = 1, 2 & 4.

BISPEC [BiSpec v. 2.0 Beta] and MATLAB were used for the computations and

plotting. In BiSpec, the stiffness reduction due to P-Delta effect is accounted for by a P-

135

Delta (pd) ratio, defined as (P/L)/K that varies from 0 to 1. In the equation K is the gross

stiffness of the system, P in this case is Pdl, and L is the height of the SDF system as

shown in Fig. 5.1. The Stiffness Degradation (SD) model with two different pd ratios (0

and 0.05) were used in the analysis for the first four sets (i.e., Groups 1, 2, 3 & 4) of

ground motions. BISPEC gives data output in US systems, therefore US system of

measurements are used in this chapter.

The first twenty ground motion records of group No. 5 were taken from Appendix C of

ATC-55 [FEMA-440 2005]. Since the original list did not contain any record from the

Chi-Chi earthquake, two records have been added to make the total of twenty-two. Two

different hysteretic models, the stiffness-degrading (SD) model and the elastic perfectly

plastic (EPP) model, and three pd ratios (0, 0.05, 0.1) were used in the analysis for this

set of ground motions.

5.6 PARAMETRIC STUDY

In Appendix B of the Seismic Design Criteria (SDC), Acceleration Response Spectra

(ARS) curves have been drawn as per the earthquake magnitude (Mw) and soil profile

type for SDF system having damping ratio 5%. Similar sub-divisions are made from

known pulse-like ground motion records having forward directivity. The fault-normal

components of these records are given in Tables 5.1 to 5.4, which have been used in the

parametric computations. An effort is made to determine if there is a relationship between

the residual displacement Dres and the spectral displacement D at a given natural period

of vibration Tn.

136

For Groups 1 and 2, it is observed from Fig. 5.7.a and 5.8.a that the mean+σ spectral

displacements for µc = 1, 2 & 4 begin to deviate from one another when Tn = 2.5 sec.

From Fig. 5.7.b, it is observed that the mean+σ Dres/D ratios remain more or less

constant for Tn = 0.5 to 4 seconds. However, a small variation in P-Delta (pd) ratio from

0 to 0.05 results in a considerable increase (from an average of 0.36 to 0.75) in Dres/D

for µC = 4, although the effect is less pronounced (from an average of 0.21 to 0.38) for µC

= 2. Similar trend and values for Dres/D are observed for ground motions in Group 2 for

soil profile type D, as seen from Fig. 5.8.b. Thus, a change in soil profile type does not

seem to have an appreciable effect on Dres/D.

137

Natural period, n

T (sec)

Natural period, n

T (sec)

Figure 5.7. Group 1, SD Model, Mw = 6.5 ± 0.25, Soil Type A, B

(a) the (mean+σ) Displacement Response Spectrum

(b) the (mean+σ) Residual Displacement/Spectral Displacement Spectrum

Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

µC = 2, pd=0

µC = 2, pd=.05

µC = 4, pd=0

µC = 4, pd=.05

(a)

(b)

138

Natural period, n

T (sec)

Natural period, n

T (sec)

Figure 5.8. Group 2, SD model, Mw = 6.5 ± 0.25, Soil Type D



Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

µC = 2, pd=0

µC = 2, pd=.05

µC = 4, pd=0

µC = 4, pd=.05

(b)

(a)

139

For Group 3 and 4, it is observed from Fig. 5.9.a and 5.10.a that the mean+σ spectral

displacements for µC = 1, 2 & 4 are quite comparable. From Fig. 5.9.b, it is observed that

the mean+σ Dres/D ratios remain more or less constant for µC = 2 and 4. A small

variation in pd ratio from 0 to 0.05 results in a considerable increase (from an average of

0.30 to 0.74) in Dres/D for µC = 4, although the effect is less pronounced (from an

average of 0.22 to 0.39) for µC = 2. Similar trends and values for Dres/D are observed for

ground motions in Group 4 for soil profile type D, as seen from Fig. 5.10.b. Thus, a

change in soil profile type does not seem to have an appreciable effect on Dres/D.

140

Natural period, n

T (sec)

Natural period, n

T (sec)

Figure 5.9. Group 3, SD model, Mw =7.25 ± 0.25, Soil Type A, B

(a) the (mean+σ) Displacement Response Spectrum,


Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

µC = 2, pd=0

µC = 2, pd=.05

µC = 4, pd=0

µC = 4, pd=.05

(b)

(a)

141

Natural period, n

T (sec)

Natural period, n

T (sec)

Natural period, n

T (sec)

Figure 5.10. Group 4, SD model, Mw = 7.25 ± 0.25, Soil Type D



Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

µC = 2, pd=0

µC = 2, pd=.05

µC = 4, pd=0

µC = 4, pd=.05

(b)

(a)

142

It is pertinent here to observe the effect of the record Takatori, Kobe 1995 earthquake

(KOBE_TAK_140FN, soil-type E) on the spectral displacements and residual

displacements. We do not have any other pulse-like record of stations on E-type soil. Fig.

5.11.a illustrates the spectral distances plotted against the period of vibration, Tn. From

the figure it may be observed that the spectral displacements are very high for a period of

vibration 1 to 2.5 sec. From Fig. 5.11.b it may be observed that for µC = 2, Dres/D are

high around 0.4 to 0.5, for pd = 0.05 and 0.1. For µC = 4, they are very high around 0.8

for pd=0.05 and 0.1. High ductile ratio of µC = 4 has a disastrous effect on structures near

this station. Similar observation can be made from Fig. 5.11.c, which shows the residual

displacements for stiffness degrading (SD) model, for pd = 0, 0.05 and 0.1.

143

(a)

(b)

Natural period, n

T (sec)

Natural period, n

T (sec)

Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

144

Figure 5.11. The Takatori, Kobe 1995, (a) Displacement Response Spectrum;

(b) Residual Displacement/Spectral displacement Spectrum for the SD model; and

(c) Residual Displacements for the SD model.

(Line style: Solid, pd=0; Dashed, pd=0.05; Dotted, pd=0.10.

Line color: Blue for µC=2, Red for µC=4)

Group 5 Near-fault (20+2) records from ATC – 55 have 6.19 ≤ Mw ≤ 7.62, and Soil

Profile Types A, B, D & E. Two different hysteretic models – the stiffness-degrading

(SD) model and the elastic perfectly plastic (EPP) model – were used for the SDF system

analysis. Further, three values of P-Delta (pd) ratios (0, 0.05 and 0.10) were used with

each of the models.

It is observed from Fig. 5.12.a that the mean+σ spectral displacements for µC = 1, 2 & 4

are quite comparable. For the SD model shown in Fig. 5.12.b, it is observed that the

(c)

Natural period, n

T (sec)

Res

idu

al D

isp

lace

men

t, D

res

(in

)

µC = 2, pd=0

µC = 2, pd=0.05

µC = 2, pd=0.1

µC = 4, pd=0

µC = 4, pd=0.05

µC = 4, pd=0.1

145

mean+σ Dres/D ratios remain more or less constant for µC = 2 and 4. For µC = 2, the

average Dres/D values are 0.21, 0.38 and 0.58 for pd ratios of 0, 0.05 and 0.10,

respectively. As has previously been observed, a small variation in pd ratio from 0 to

0.05 results in a considerable increase (from an average of 0.30 to 0.74) in Dres/D for µC

= 4, although the effect is less drastic (from an average of 0.74 to 0.82) for µC = 2.

For the EPP model (Fig. 5.12.c) the variation in Dres/D is rather erratic; however, since

the trend is more or less horizontal, the average values of Dres/D for µC = 2 can be

computed as 0.38, 0.44 and 0.5 for pd ratios of 0, 0.05 and 0.10, respectively. The

average values of Dres/D for µC = 4 are 0.51, 0.7 and 0.8 for pd ratios of 0, 0.05 and 0.10.

Because residual displacements computed using the EPP model are higher than those

computed using the SD model, the use of materials that can be modeled using the SD

model (such as concrete) is more preferable for use as bridge bents. This observation is

reinforced in Figures 5.13 and 5.14 in which residual displacement values Dres are

plotted for both the SD and EPP models over a range of Tn from 0.5 to 4.

146

Natural period, n

T (sec)

Natural period, n

T (sec)

Sp

ectr

al D

isp

lace

men

t (i

n)

Dre

s /

Sp

ectr

al D

isp

µC = 1

µC = 2

µC = 4

(b)

(a)

147

Natural period, n

T (sec)

Natural period, n

T (sec)

Figure 5.12. Group 5, ATC – 55, (a) The mean+σ Displacement Response Spectrum; (b)

The mean+σ Residual Displacement/Spectral displacement Spectrum for the SD model;

and (c) The mean+σ Residual Displacement/Spectral Displacement ratio for the EPP

model. (Line style: Solid, pd=0; Dashed, pd=0.05; Dotted, pd=0.10.

Line color: Blue, µC=2, Red, µC=4)

Figure 5.13. Group 5, ATC-55 The (mean+σ) Residual Displacements Dres

using the stiffness degrading (SD) model.

Dre

s /

Sp

ectr

al D

isp

Res

idu

al D

isp

lace

men

t, D

res

(in

)

µC = 2, pd=0

µC = 2, pd=0.05

µC = 2, pd=0.1

µC = 4, pd=0

µC = 4, pd=0.05

µC = 4, pd=0.1

(c)

148

Natural period, n

T (sec)

Figure 5.14. Group 5, ATC – 55 The (mean+σ) Residual Displacements Dres

using the elastic perfectly plastic (EPP) model.

5.7 DISCUSSION

For structures such as a bridge bent or bridge-column in an active near-fault zone (less

than 10 miles from fault-rupture planes), the permissible limit on the target displacement

ductility to ensure that the amount of residual displacements will not be excessive after

cessation of ground motions is currently not adequately addressed in a number of seismic

design provisions.

Using the minimum local displacement ductility requirement of µc = 3 as per SDC Sec.

3.1.4.1 may not be advisable for such structures in near-fault zones. As per Sec. 4.2 of

SDC, the dynamic effects of gravity loads acting through the lateral displacements shall

Res

idu

al D

isp

lace

men

t, D

res

(in

)

µC = 2, pd=0

µC = 2, pd=0.05

µC = 2, pd=0.1

µC = 4, pd=0

µC = 4, pd=0.05

µC = 4, pd=0.1

149

be included in the analysis. From Fig. 5.13, it is observed that for systems that exhibit

the stiffness degrading (SD) behavior (e.g., a concrete bent) with a pd ratio of 0.05 and µc

= 2, the residual displacement, Dres varies from 5 in. (for Tn = 1 sec.) to 10 in. (for Tn = 2

sec.); and for µc = 4 and a pd ratio of 0.05, Dres varies from 12 in. (for Tn = 1 sec.) to 20

in. (for Tn = 2 sec.). The residual displacements will be more pronounced for a higher pd

ratio. Given this observation, the question that needs to be answered is what should the

range of permissible µc be, if residual displacements are to be limited?

In a Performance-Based Seismic Design (PBSD) procedure, structural performance is

communicated by reference to a series of standard performance levels that range from

states of negligible damage and impact on safety, occupancy and use to states of near

complete damage in which there is extensive risk to life, complete loss of economic value

and permanent loss of use and function. Under these procedures, decision makers must

select the desired performance levels (e.g., fully functional, immediate occupancy, life

safety, collapse prevention), and the earthquakes for which these performance levels are

to be achieved. Decision makers should be able to define these risks in terms of the future

life loss, facility repair costs and downtime that could result from design decisions.

[FEMA-445 2006]

The design of structures at fully functional or immediate occupancy level should be more

stringent. In order to avoid permanent deformations, a stronger and stiffer structure

having a lower target displacement ductility value may be desired. Plots similar to Fig.

150

5.13 and 5.14 would be useful in determining the target residual displacements and

displacement ductility.

For life safety or collapse prevention level, adaption of higher displacement ductility may

result in higher permanent deformations. These higher permanent deformations, when

coupled with an increase in the P-Delta ratio due to a possible reduction in gross stiffness

from cracked sections after the quake, may cause the bridge bent to be subjected to a

higher eccentric loading from the superstructure, which is not desirable.


In this chapter, existing provisions for ductility requirements for a typical bridge pier or

bent subjected to seismic loading were briefly discussed. These provisions were applied

to cases in which the structure was subjected to extreme loads such as pulse-like near-

fault ground motions. A bridge bent, modeled as a SDF system, is subjected to different

groups of pulse-like near-fault ground motion records. As expected, higher residual

displacements are observed with an increase in displacement ductility or P-Delta ratio.

The investigation of the residual displacement response of an inelastic SDF system

subjected to different groups of pulse-type near-fault ground motions has led to the

following observations:

1) The spectral displacements do not seem to vary appreciably with different µC

values for Tn = 0.5 to 4 sec. (Figs. 5.8.a, 5.9.a, 5.10.a and 5.12.a)

151

2) The ratio Dres/D (where D is the spectral displacement) remains more or less

constant from Tn = 0.5 to 4 sec. for a constant displacement ductility or P-Delta

ratio (Fig. 5.7.b, 5.8.b, 5.9.b, 5.10.b and 5.12.b)

3) Residual displacements are higher for systems modeled using the EPP hysteretic

model (such as steel) as shown in Fig. 5.14 than for systems modeled by the SD

hysteretic model (such as concrete) as shown in Figure 5.13. Therefore, the use of

concrete is more preferable whenever it is possible.

4) Because of the relatively high residual displacements (Figs. 5.13 & 5.14) for a

nominal P-Delta ratio of 0.05, the target displacement ductility may need to be

restricted between 2 to 3. .

This restriction may not ensure a dependable rotational capacity in regions of plastic

hinge. Failure may be sudden if the actual pulse-like ground motion exceeds the design

excitation. A higher factor of safety may have to be adopted for design of such structures

in near-fault zone.

152

CHAPTER 6

SUMMARY AND

CONCLUSIONS

The major objective of the study presented in this dissertation is to develop a basic

understanding of the characteristics of pulse-like near-fault ground motions (NFGM) and

its impact on the response characteristics of a single degree-of-freedom (SDF) system.

The excitation properties intrinsic to pulses are inherently different from non-pulse type

seismic ground motions. They may affect medium to large period structures often

resulting in excessive displacements during the earthquake or permanent deformations

after the cessation of earthquake.

6.1 EXTRACTION OF ACCELERATION AND VELOCITY PULSE/S

In most research, velocity pulse shapes (triangular, sinusoidal, rectangular, etc.) were

assumed as best-fit to velocity time history (VTH). It is then differentiated to obtain the

acceleration pulse. An important difference between the proposed approach and those of

the other researchers is that no predetermined pulse shape is assumed in this analysis.

The natural shape and duration of the pulses are extracted directly from processed ground

motion records. These extracted pulses are then used to determine their effects on SDF

systems.

153

For the extracted pulses to be close to ‘real’ pulses, it is essential that the response due to

the pulse should be as close to the response due to original acceleration time history

(ATH) as possible. By using only the pulse component of the ground motion as the

excitation force, it is shown in this work that the displacement response of the SDF

system with a natural period exceeding a certain value referred to as the cut-off period Tc

is quite comparable with that due to the original ground excitation.

Directivity effects in ground motions can usually be detected in signal processing using

either Wavelet theory or Fourier analysis. While Wavelet theory is more useful when the

transitory characteristics or time information is more crucial in the analysis, Fourier

analysis is more appropriate when the frequency contents of the signals are of greater

importance; as in the case of directivity when a pulse might not be apparent in the

velocity time history (VTH).

It has been observed that the low-velocity soft and near-surface sediments are found to be

effective on the amplification and elongation of long-period ground motions [Hatayama

et al. 2007; Miyake and Koketsu 2007]. For most long-period ground motions records,

pulses might not be visible in VTH. Beat-phenomenon is intrinsic to the majority of

records from Japan. This phenomenon cannot be detected by Wavelet theory.

In Chapter 2, we have observed from Displacement Response Spectra (DRS) that the

high-frequency (short-period) contents of the ground motion do not appreciable affect the

displacement response of long-period systems. Recall that this non-pulse ATH is

154

obtained after the application of high-pass filter, Hhp(ω). This shows that short-period

content of time history primarily affect the response of short-period structures; whereas

long-period content of time history primarily affect the displacement response of long-

period structures.

Parseval’s Theorem and the principles of signal energy compaction were applied in pulse

extraction. In the present work it has been shown that the extracted pulse(s) indeed

contributes a large percentage of energy to the total seismic energy of near-fault ground

motions.

6.2 DISPLACEMENT RESPONSE FACTOR, Rd

In Chapter 3, it has been established that a simple sine-pulse (single period pulse) does

not always give a good representation of ground motions in response identification. The

use of a sine-pulse should be limited to those response characteristics in which the

spectral displacement more or less increases monotonically with natural period Tn from

the cut-off frequency to the frequency that corresponds to the peak of the spike nearest

the lowest usable frequency.

The pulses were qualitatively determined as (a) Monotonically increasing displacement,

which generally occurs for single period pulses, (b) Ripple effect pulses are generally the

effects of multiple period pulses, and (c) Resonance pulses are a special case of multiple

155

period pulses in which the spectral displacements are very high compared to its static

counterparts.

To quantitatively identify these three types of pulse effects, the concept of modified

Displacement Response Factor Rd was introduced.

Since short-period contents of pulse-like ground motions do not affect the displacement

response of a long-period structure, it is more rational to apply the extracted pulse on an

elastic system to determine its static displacement. In view of this, a modified

Displacement Response Factor was proposed in Chapter 3. Since the amplitude of the

pulses is much less than the Peak Ground Acceleration (PGA), the proposed dynamic

magnification factors are much larger.

A statistical study was presented in Chapter 3 on the effect of dynamic magnification

caused due to the long-period ground motions by application of modified displacement

response factor Rd. The acceleration pulses of around seventy pulse-like NFGM records

are extracted event-wise to study elastic response characteristics quantitatively by using

Rd for ζ = 0, 0.02 & 0.05.

For ζ = 0.05, the (m+σ) values of Rd varies from 3.44 to 4.8 for different earthquake

events. For pulse-like seismic ground motions the concept of equivalent harmonic forcing

frequency ratio (ω/ωn), a phase shift factor and a “damping modification factor” were

introduced in chapter 3. These concepts help us in comparing the values of dynamic

156

magnification with different earthquake events, as well as with that due to simple

harmonic excitation. Since damping controls in response characteristics of such

structures, these Rd values would assist us in the design of dampers, which could sustain

pulse-like excitations.

6.3 ANALOGOUS DISPLACEMENT RESPONSE SPECTRUM

In the fourth chapter a method for expressing spectral displacements for an undamped

linearly elastic single degree-of-freedom (SDF) system in terms of the velocity Fourier

amplitudes from the lowermost frequency range of ground motions is proposed. These

spectra, referred to as analogous displacement response spectrum (DRS), are quite

comparable to those obtained when the system is subjected to the original processed time

history or pulse-excitation. The term ‘analogous’ is used here because these spectra are

not identical to the usual DRS used in earthquake engineering design. It is essentially an

‘exact’ ground velocity Fourier amplitude plot in the lower-most frequency range (LUF ≤

f ≤ fc < 1.67 Hz) rotated about a vertical axis that passes through the cut-off frequency fc.

Because of this, the discussion is restricted to velocity Fourier amplitudes (VFA) in this

range.

The very important distinction between the analogous DRS and the conventional DRS is

that the former is the representation of the ground (pulse) velocity time history, whereas

the latter is the response of an elastic SDF system due to ground acceleration time

history.

157

Analogous displacement response spectra (DRS) are fast and efficient instruments for

understanding spectral displacement characteristics of undamped SDF systems. They

give upper-bound response of structures subjected to pulse-like long-period ground

motions.

Due to its inherent simplicity, the analogous DRS can be used in the preliminary stage of

structural design for seismic loading. The analogous DRS could also be useful in

Performance Based Design by directly relating spectral displacements to the Fourier

amplitudes of the given ground velocity.

6.4 DISPLACEMENT DUCTILITY REQUIREMENT FOR A SDF SYSTEM

The ductility requirement for a typical bridge bent subjected to extreme loads, such as

those from pulse-like near-fault ground motions (NFGMs) was examined in Chapter 5.

The ductility requirement as stated by prevailing design codes may not be valid for such

structures in near-fault (less than 10 miles) zones. Due to dominance of the pulses,

medium- to long-period structures are highly affected; often resulting in high residual or

permanent deformations after the cessation of seismic ground motions.

As per most design codes, the magnitude of displacements associated with P-Delta

effects is required to be captured using non-linear time history analysis. The higher the

158

value of the target displacement ductility demand or the P-Delta ratio, the larger is the

magnitude of the residual displacements.

A parametric study on residual deformations due to five different groups was carried out.

Residual displacements are higher for systems modeled using the EPP hysteretic model

(such as steel) than for systems modeled by the SD hysteretic model (such as concrete).

Therefore, the use of concrete is more preferable whenever it is possible.

Because of the relatively high residual displacements for a nominal P-Delta ratio of 0.05,

the target displacement ductility may need to be restricted to 2. However, this restriction

may not ensure a dependable rotational capacity in regions of plastic hinge. Failure may

be sudden if the actual pulse-like ground motion exceeds the design excitation. A higher

factor of safety may have to be adopted for design of such structures in near-fault zone.

Because of larger residual displacements after cessation of an earthquake, these

permanent deformations in a bridge bent would lead to a higher eccentric loading from

the superstructure. Subsequently, the bent would be subjected to higher secondary

moments from it own design load. Even if the bridge survives the earthquake, subsequent

failure may occur due to its own design loads. A suitable balance should be made

between ductility and residual displacements in the design of such structures.

159

6.5. SCOPE FOR FUTURE RESEARCH

Extensive research is in progress to develop simplified non-linear seismic analysis

procedures for use in performance based seismic design (PBSD). Various plots of

pseudo-spectral acceleration versus spectral displacements are being proposed for

different values of effective period (Te) and effective damping (ηeff) for a range of

ductility ratios (µ). Using Modified Acceleration Displacement Response Spectrum

(MADRS) and from the loci of performance points, the required performance point can

be selected in a performance based seismic design.

The Displacement Demand Ratio (DDR) is defined as the ratio of the maximum inelastic

spectral displacement obtained from a time history analysis to the spectral displacement

calculated from the Capacity Spectrum Method approach using the concept of equivalent

viscous damping. This ratio is used as an indicator of the validity of the equivalent

viscous damping assumption for near-field ground motions [Guyader and Iwan, 2006].

It is well-known that while structural damping does not have an appreciable effect on the

response characteristics of long-period structures subjected to non-pulse type ground

motions, it greatly affects the response of long-period structures subjected to pulse-type

ground motions. It is therefore essential that a method be developed to modify the

effective damping coefficient for structures subjected to pulse-like near-fault ground

motions.

160

Recall that as per ATC-55 (FEMA 440), CR is a modification factor to relate the expected

maximum displacement of an inelastic SDF system with EPP or SD hysteretic properties

to displacements calculated from a linear elastic response. While the importance of CR

values in the analyses cannot be overemphasized, the permanent deformation of an

inelastic system after cessation of pulse-like near-fault ground motions (NFGMs) is

another critical parameter that needs to be adequately addressed.

In essence, it is necessary to create different plots for the determination of performance

points (in PBSD) for dynamic displacements during the earthquake and for permanent

displacements after the cessation of the earthquake.

Using the minimum local displacement ductility requirement of µc = 3 as per SDC Sec.

3.1.4.1 may not be advisable for long-period structures in near-fault zones. As per Sec.

4.2 of SDC, the dynamic effects of gravity loads acting through the lateral displacements

shall be included in the p-delta analysis. In Chapter 5, it has been observed that the

residual displacements are more pronounced for a higher ductility ratio and a higher p-

delta ratio. Figure 5.13 shows that for a very nominal pd ratio of 0.05, Tn = 1 sec and µC

= 2, the residual displacement is around 5 inches. Given this observation, the question

that needs to be answered is what should the range of permissible µc be, if residual

displacements are to be limited?

The present dissertation does not contain the required parametric study to determine the

permissible range of ductility ratios for different Tn and different pd ratios, so that the

161

residual displacement is within the acceptable range. If the residual displacement is

excessive, then one would be required to change the design parameters.

A study on the effect of applied service loads on the deformed state of structures after the

cessation of earthquake should be undertaken. A concept of residual curvature may be

introduced. In p-delta analysis, the service loads (non-seismic) may be applied to the

structure (deformed) with residual curvature (from 0 degree to maybe 10 degrees). The

point of structural failure or the unacceptable deformation may be noted. Backtracking

from that point, one would be able to determine the magnitude of acceptable residual

deformation of a structure.

The most important issues to address would be (a) whether the deformed structure can

sustain the required (non-seismic) service loads, and (b) whether the deformed structure

can withstand another seismic excitation of similar or lesser magnitude? In other words,

is the reduced capacity of the structure after cessation of earthquake acceptable?

The role of suitably designed dampers should be investigated. The dampers would be

required to sustain high dynamic magnification ratio from 3.5 to 4.8 (Figure 3.19),

depending upon the location of the structure, and be able to reduce the p-delta ratio to an

acceptable value during the excitation so that the residual deformation is minimized.

162

APPENDIX

MATLAB PROGRAMMING APPLICATIONS

LIST OF PROGRAMS:

(1) Response due to half-sine pulse. Examples 5.3 & 5.4 from Text-book by Chopra, A.K

(2) Plotting Acceleration, Velocity & Displacement Response of a record

(3) Discrete Fourier Transform – Response evaluation

(4) Effect of NB on Butterworth lowpass filter

(5) Pulse, Non-pulse extraction from ATH. Energy values by Parseval Theorem

(6) Velocity (VRS) & Displacement Response Spectra (DRS) from ATH

(7) Equivalent sine-pulse, and orig-, pulse- Displacement Response Spectra

(8) Displacement Response Factor, Rd: Evaluation & plotting

(9) Analogous Displacement Response Spectra: Evaluation & plotting

(10) Half Sine Pulse – Non-linear EPP model for response evaluation

163

(1) Response due to half-sine pulse. Example 5.3 or 5.4 from Text-book by Chopra

clear all

close all

format short g

% Example 5.3 or 5.4 from Text-book by Prof. A. K. Chopra

td=0.6 % duration of applied force

To=1.2 % duration for which response is to be evaluated

dt=0.01 % time interval in second (shortened)

N=round(To/dt); % N=12

t1=[0:dt:td];

t2=[td+dt:dt:(N-1)*dt];

t=[t1 t2] % check

p1=10*sin(pi*t1/0.6); % applied loading

p2=zeros(1,size(t2,2));

p=[p1 p2]

figure

plot(t,p,'-o'); xlabel('Time (seconds)'); title('Applied Loading'); grid on

ylabel('Discrete force (kips)');

%--------------------------------------------------------------------------

% Average or Linear response analysis of SDF system by Newmark Method

m=0.2533; % kip-sec2/in (force/acceleration)

k=10; % stiffness of SDF system, units in kip/in (force/length)

c=0.1592 % damping coefficient, units in force*time/length, Ccr=2*m*wn

gamma=0.5;

beta=0.25; % beta=1/6 for linear acceleration approach

a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);% uo, vo, ao & po have been initialized to zero.

K=k+(gamma*c)/(beta*dt)+m/(beta*dt*dt)

C1=m/(beta*dt)+gamma*c/beta

C2=m/(2*beta)+c*dt*(gamma/(2*beta)-1)

for n=1:1:N-1

dp(n)=p(n+1)-p(n);

Dp(n)=dp(n)+C1*v(n)+C2*a(n);

du(n)=Dp(n)/K;

dv(n)=gamma*du(n)/(beta*dt)-gamma*v(n)/beta+dt*a(n)*(1-gamma/(2*beta));

da(n)=du(n)/(beta*dt*dt)-v(n)/(beta*dt)-a(n)/(2*beta);

u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

u

164

figure

plot(t,u); title('Response to half sine-pulse'); grid on

xlabel('Time (seconds)');ylabel('Displacement (cm)');

=============================================================

(2) Plotting Acceleration, Velocity & Displacement Response of a record

clc

clear all

format short g

%Accel, Velocity & Displacement Response of Lucerne, N239E, Landers 1992

Y = load('H:\LANDERS_LCN_260.dat');

% Accln TH accessed from PEER NGA is stored in H drive in TEXT format

N=9625 % N is the number of sample points

% Original N=9628. Accessed N points have to be multiples of 5

dt=0.005

t =[0:dt:N*dt-dt]; % time

% row by row

Yr = Y.';

in = 1;

for clm=1:1:N/5

a(in:in+4) = Yr(:,clm);

in = in + 5;

end

figure

plot(t,a);

title('ATH'); grid on

xlabel('Time in seconds');ylabel('Acceleration in g (cm/sec2)');

% Calculation of velocity & disp.

v = zeros(1,N); s = zeros(1,N);

for n = 1:1:N-1

v(n+1) = v(n) + (a(n)+ a(n+1))/2*dt; % vel

s(n+1) = s(n) + (v(n)+ v(n+1))/2*dt; % disp

end

PGV=max(abs(v*981))

PGD=max(abs(s*981))

165

figure % Velocity

plot(t,v*981);

title('VTH'); grid on

xlabel('Time in seconds');ylabel('Velocity in cm/sec');

figure % disp.

plot(t,s*981);

title('DTH'); grid on

xlabel('Time in seconds');ylabel('Displacement in cm');

=============================================================

(3) Discrete Fourier Transform – Response evaluation

clear all

close all

format short g

% Discrete FT Displacement Response to LUCERNE, N239E, LANDERS 1992

N=9625 % total number of samples of the Record - 9628

dt=0.005

SR=1/dt % SR=sampling rate

phi = 239 % Average strike-normal direction at Lucerne station is N239E

alpha = 260 % dirction of first record, N to E

beta = 345 % dirction of second record, N to E

% Above angles are in degrees each measured from N to E-direction



Yr = Y.'; % row by row

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a1=[0 a]; % Initializing acceleration to zero


Yr = Y.';

a = zeros(1,N);

in = 1;

for clm=1:1:N/5;


166

in = in + 5;

end


% -------------------------------------------------------------------------

% At LUCERNE, average strike direction is N329E & strike normal is N239E.

% Rotation OF ATH to fault-normal direction

a = zeros(1,N);

for in=1:1:N;

a(in) = a1(in)*cosd(21) - a2(in)*cosd(74); % FN component of acc.

end

length(a)

% -------------------------------------------------------------------------

% DETERMINATION OF PULSE-ATH. Computation of dis & filter response function

% After visual examination of FT of VTH & ATH, fc is determined = 0.395 Hz.

Tn=4; % Tn is the natural period of SDF system in seconds. fn=wn/(2*pi)

k= 1; % elastic stiffness of the system

zeta=0.05; % zeta is damping coefficient of system

Sn = 19; % corresponding cut-off sample

B=16; % Order of Butterworth filter

fo=SR/N;

fn=1/Tn;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:(N-1)/2

f(j+1)=j*fo; % AKC Eq. A.5.7.a

end

for j=(N-1)/2+1:1:N-1

f(j+1)=-(N-j)*fo; % AKC Eq. A.5.7.b

end

for j=1:1:N

H(j)=(1/k)*1./((1-(f(j)/fn).^2)+i*(2*zeta*f(j)/fn));

end

for j=1:1:N

F(j)=1/(1+i*(f(j)/fc).^B); % Butterworth LP Filter response function

end

%A=fft(a);

for j=0:1:N-1

m1=0;

for n=0:1:N-1

m1=m1+a(n+1)*exp(-i*2*pi*n*j/N);

167

end

A(j+1)=m1/N; % AKC Eq. A.5.4

end

Ap=F.*A;

for n=0:1:N-1

m1=0;

for j=0:1:N-1

m1=m1+Ap(j+1)*exp(i*2*pi*n*j/N);

end

ap(n+1)=m1; % AKC Eq. A.5.4

end

%ap=ifft(Ap); % computes inverse fft of the filtered fft. ap is the p-ATH

% This ap is used for Elastic displacement analysis in Time-domain.

figure

subplot(2,1,1); plot(t,a); hold on

plot(t,ap,'r');

title('Pulse ATH superimposed on original ATH, LANDERS-LCN-239FN'); grid on

xlabel('Time in seconds');ylabel('Acceleration in g (cm/sec2)');

% -------------------------------------------------------------------------

% Displacement response due to pulse-ATH

m=1.; % mass of SDF system in

p=-m*a;

%P=fft(p);

for j=0:1:N-1

m1=0;

for n=0:1:N-1

m1=m1+p(n+1)*exp(-i*2*pi*n*j/N);

end

P(j+1)=m1/N; % AKC Eq. A.5.4

end

%p1=ifft(P); plot(t,p1); grid on % check

U=H.*F.*P;

% u=ifft(U);

for n=0:1:N-1

m1=0;

for j=0:1:N-1

m1=m1+U(j+1)*exp(i*2*pi*n*j/N);

168

end

u(n+1)=m1; % AKC Eq. A.5.4

end

figure

plot(t,u); grid on

title('Uncorrected displacement due to pulse-ATH, Landers-LCN-239FN');

xlabel('Time in seconds'); ylabel('Displacement in cm');

% -------------------------------------------------------------------------

% Veletsos-Ventura Modifications to DFT solution: AKC Section A.7

uo=real(u(1));

U_img=imag(U);

m1=0;

for j=0:1:N/2

m1=m1+j*U_img(j+1);

end

vo=-4*pi/(N*dt)*m1 % AKC Eq. A.7.4 vo=-4*pi/(N*dt+Tn/2)*m1

wn=2*pi*fn;

wd=wn*sqrt(1-zeta^2); % AKC Eq. A.7.3

uc=exp(-zeta*wn*t).*(-uo*cos(wd*t)+(-vo-zeta*wn*uo)/wd*sin(wd*t));%AKC Eq.

A.7.2

u_sum=real(u)+uc; % AKC Eq. A.7.1

figure

plot(t,real(u),'-.'); hold all

plot(t,uc,'--');

plot(t,u_sum);

title('Displacement due to pulse-ATH, Landers-LCN-239FN');

xlabel('Time in seconds'); ylabel('Displacement in cm');

%END

==============================================================

(4) Effect of NB on Butterworth lowpass filter

N=501

B=2

w=[0:0.01:5];

for in=1:1:N

Hlp(in)=1./sqrt(1.+(w(in)).^(2*B));

end

figure

plot(w,Hlp); hold all

B=4

169

w=[0:0.01:5];

for in=1:1:N

Hlp(in)=1./sqrt(1.+(w(in)).^(2*B));

end

plot(w,Hlp)

B=8

w=[0:0.01:5];

for in=1:1:N

Hlp(in)=1./sqrt(1.+(w(in)).^(2*B));

end

plot(w,Hlp)

B=32

w=[0:0.01:5];

for in=1:1:N

Hlp(in)=1./sqrt(1.+(w(in)).^(2*B));

end

plot(w,Hlp)

============================================================

(5) Pulse, Non-pulse extraction from ATH. Energy values by Parseval Theorem

clear all

close all

format short g

% LUCERNE, N239E, LANDERS 1992, LUF = 0.1 Hz

% ROTATION OF ATH to FAULT-NORMAL direction

N=9625

dt=0.005








in = 1;

for clm=1:1:N/5;


in = in + 5;

end

170




a = zeros(1,N);

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a2=[0 a];

a = zeros(1,N);

for in=1:1:N;


a0(in) = a1(in)*cosd(69) + a2(in)*cosd(16); % Fault-parallel component of acc.

end

save LANDERS_LCN_FN.dat a -ascii -double

% -------------------------------------------------------

% VTH & DTH: Linear acceleration method


for n = 1:1:N-1;

v(n+1) = v(n) + (a(n+1)+ a(n))/2*dt;

s(n+1)=s(n)+v(n)*dt+(a(n+1)/6+a(n)/3)*dt*dt;

end

v0 = zeros(1,N); s0 = zeros(1,N);

for n = 1:1:N-1;

v0(n+1) = v0(n) + (a0(n+1)+ a0(n))/2*dt;

s0(n+1)=s0(n)+v0(n)*dt+(a0(n+1)/6+a0(n)/3)*dt*dt;

end

figure

plot(t,v*981);hold on

plot(t,s*981);

title('FLING EFFECT: VTH & DTH, LANDERS-LCN-239 FN'); grid on

xlabel('Time in seconds');ylabel('Velocity in cm/sec');

%--------------------------------------------------------------------------

% FAST FOURIER TRANSFORM (FFT) of FN ACCELERATION, a

A=fft(a);

mA=abs(A);

171

f=(0:length(A)/2-1).'*SR/length(A); % One sided FT, upto 100 Hz

%If frequencies upto 50 cps are only reqd: length(A)/4-1

figure

plot(f,mA(1:length(A)/2)) %To plot upto 50 Hz, replace 2 by 4

title('Acceleration Fourier amplitudes, LANDERS-LCN-239FN'); grid on

xlabel('Frequency f - Hertz'); ylabel('Fourier amplitudes');

% FAST FOURIER TRANSFORM (FFT) of FN VELOCITY, v

V=fft(v);

mV=abs(V);

f=(0:length(V)/2-1).'*SR/length(V); % One sided FT, upto 100 Hz

figure

plot(f,mV(1:length(V)/2)); %to plot upto 50 Hz, replace 2 by 4

title('VFT, Landers-LCN-239FN');

xlabel(' f (Hz)'); ylabel('Fourier amplitudes');

%--------------------------------------------------------------------------

% fc = 0.395 Hz.Sn=20.



fo=SR/N;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:(N-1)/2

f1(j+1)=j*fo;

end

for j=(N-1)/2+1:1:N-1

f1(j+1)=-(N-j)*fo;

end

for j=1:1:N

FL(j)=1/(1+i*(f1(j)/fc).^B); % Butterworth lowpass filter

end

for j=1:1:N

FH(j)=(f1(j)/fc).^B/((f1(j)/fc).^B-i); % Butterworth highpass filter

end

Ap=FL.*A;

ap=ifft(Ap); % computes inverse fft of the filtered fft. ap is the pulse-ATH

figure

plot(t,a); hold on

172

plot(t,ap);

title('Orig & pulse ATH, Landers-LCN-239FN');

xlabel('Time (s)');ylabel('Acceleration (g)');

% Filtering frequencies of velocity

Vp=FL.*V;

vp=ifft(Vp);% computes inverse fft of the filtered fft. vp is the pulse-VTH

figure

plot(t,v*981); hold on

plot(t,vp*981);

title('Orig & pulse VTH, Landers-LCN-239FN');

xlabel('Time (s)');ylabel('Velocity (cm/sec)');

% -------------------------------------------------------------------------

% FAST FOURIER TRANSFORM (FFT) of DISPLACEMENT, s

S=fft(s);

mS=abs(S);

f=(0:length(S)/2-1).'*SR/length(S); % One sided FT, upto 100 Hz

figure

plot(f,mS(1:length(S)/2)); %to plot upto 50 Hz, replace 2 by 4

title('DISPLACEMENT Fourier amplitudes, LANDERS-LCN-239FN'); grid on

xlabel('Frequency f - Hertz'); ylabel('Fourier amplitudes');

% Filtering frequencies of displacement

Sp=FL.*S;

sp=ifft(Sp);% computes inverse fft of the filtered fft. sp is the pulse-DTH

Snp=FH.*S;

snp=ifft(Snp);% computes ifft of filtered fft. snp is the no-pulse-DTH

figure

plot(t,s*981); hold all

plot(t,sp*981);

plot(t,snp*981);

title('Orig & pulse DTH, Landers-LCN-239FN');

xlabel('Time (s)');ylabel('Displacement (cm)');

%--------------------------------------------------------------------------

% PARSEVAL THEOREM, v

E5=0;

for n=1:1:N-1

Et(n)=abs(v(n))*abs(v(n));

173

E5=E5+Et(n);

end

E5

E6=0;

for n=1:1:N-1

Ef(n)=mV(n)*mV(n);

E6=E6+Ef(n);

end

E6=E6/N

filterV = [V(1:194) zeros(1,N-388) V(N-193:N)];

filterv = ifft(filterV);

E7=0;

for n=1:1:N-1

Et(n)=abs(filterv(n))*abs(filterv(n));

E7=E7+Et(n);

end

E7

E8=0;

for n=1:1:N-1

Ef(n)=abs(filterV(n))*abs(filterV(n));

E8=E8+Ef(n);

end

E8=E8/N

%END

===========================================================

(6) Velocity & Displacement Response Spectra from ATH

clear all

close all

format short g

% LUCERNE, N239E, LANDERS 1992, LUF = 0.1 Hz

% ROTATION OF ATH to FAULT-NORMAL direction, Case 2

N=9625 % total number of samples of the Record.

dt=0.005






174



in = 1;

for clm=1:1:N/5;


in = in + 5;

end


%length (a1)



a = zeros(1,N);

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a2=[0 a];

a = zeros(1,N);

for in=1:1:N;


%a0(in) = a1(in)*cosd(69) + a2(in)*cosd(16); % Fault-parallel component of acc.

end

%--------------------------------------------------------------------------

A=fft(a);

% fc = 0.395 Hz.



fo=SR/N;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:(N-1)/2

f(j+1)=j*fo;

end

for j=(N-1)/2+1:1:N-1

f(j+1)=-(N-j)*fo;

end

for j=1:1:N

FL(j)=1/(1+i*(f(j)/fc).^B); % Butterworth lowpass filter

175

end

for j=1:1:N

FH(j)=(f(j)/fc).^B/((f(j)/fc).^B-i); % Butterworth highpass filter

end

Ap=FL.*A;


ap=real(ap);

Anp=FH.*A;

anp=ifft(Anp);

anp=real(anp);

% -------------------------------------------------------------------------

% Linear response analysis of SDF system by Newmark Method

zie=0; % damping ratio

m=1; % kip-sec2/in (force/acceleration)

gamma=1/2; % Average or linear acceleration

%beta=1/4; % Average acceleration

beta=1/6; % Linear acceleration

p=-981*m*a;

pp=-981*m*ap;

pnp=-981*m*anp;

for in = 1:1:100;

Tn(in)=in/10;

wn(in)=2*pi/Tn(in);

k(in)=m*wn(in)*wn(in);

c(in)=zie*2*m*wn(in);

K(in)=k(in)+(gamma*c(in))/(beta*dt)+m/(beta*dt*dt);

C1(in)=m/(beta*dt)+gamma*c(in)/beta;

C2(in)=m/(2*beta)+c(in)*dt*(gamma/(2*beta)-1);

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=p(n+1)-p(n);

Dp(n)=dp(n)+C1(in)*v(n)+C2(in)*a(n);

du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

176

a(n+1)=a(n)+da(n);

end

D(in)=max(abs(u));

V(in)=2*pi*D(in)/Tn(in);

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=pp(n+1)-pp(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

DDp(in)=max(abs(u));

Vp(in)=2*pi*DDp(in)/Tn(in);

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=pnp(n+1)-pnp(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

Dnp(in)=max(abs(u));

Vnp(in)=2*pi*Dnp(in)/Tn(in);

end

figure

plot(Tn,D);hold all

plot(Tn,DDp);plot(Tn,Dnp);

% plot(Tn,V);hold all

% plot(Tn,Vp);plot(Tn,Vnp);

zie=0.05;

for in = 1:1:100;

177

Tn(in)=in/10;

wn(in)=2*pi/Tn(in);






a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=p(n+1)-p(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

D(in)=max(abs(u));

V(in)=2*pi*D(in)/Tn(in);

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;



du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end


Vp(in)=2*pi*DDp(in)/Tn(in);

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=pnp(n+1)-pnp(n);


du(n)=Dp(n)/K(in);

178



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

Dnp(in)=max(abs(u));

Vnp(in)=2*pi*Dnp(in)/Tn(in);

end

plot(Tn,D);hold all

plot(Tn,DDp);plot(Tn,Dnp);

%title('DISP SPECTRA due to O, p & res ATH,Landers-Lucerne-239FN,Tc=2.53');

%xlabel('Natural vibration period, Tn (sec)');ylabel('Spectral Displacement (cm)');

% plot(Tn,V);hold all

% plot(Tn,Vp);

% plot(Tn,Vnp);

% title('Velocity Spectra due to Orig & pulse ATH,Landers-Lucerne-239FN,Tc=2.53');

% xlabel('Natural Vibration Period, Tn(s)');ylabel('Spectral Velocity (cm/s)');

===========================================================

(7) Equivalent sine-pulse and orig-, pulse- Displacement Response Spectra

clear all

close all

format short g

% Chi Chi, TCU052, N322E FN, LUF= 0.05 Hz

N=18000 % total number of samples of the Record.

dt=0.005



Y = load('H:\ChiChi_TCU052E.dat');


in = 1;

for clm=1:1:N/5;


in = in + 5;

end


179

Y = load('H:\ChiChi_TCU052N.dat');


a = zeros(1,N);

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a2=[0 a];

a = zeros(1,N);

for in=1:1:N;

a(in)=-a1(in)*cosd(52) + a2(in)*cosd(38);% Vmax direction component of acc

end

%--------------------------------------------------------------------------

A=fft(a);mA=abs(A);

% fc =0.2778 Hz, Sn=26. For fc=0.1667, Sn=15, fc=1, Sn=90. fc=1.67,Sn=150



fo=SR/N;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:N/2

f1(j+1)=j*fo;

end

for j=N/2+1:1:N-1

f1(j+1)=-(N-j)*fo;

end

for j=1:1:N

FL(j)=1/(1+i*(f1(j)/fc).^B); % Butterworth lowpass filter

end

Ap=FL.*A;


ap=real(ap);

%---------------------------------------------------------------





180


a0=0.087; % pulse amplitude

Tp=9.04;

w0=2*pi/Tp;To=90; % time duration of full sine pulse

N0=round(To/dt);

t1=[0:dt:Tp];

t2=[Tp+dt:dt:(N0-1)*dt];

t=[t1 t2];

p1=a0*981*sin(2*pi*t1/Tp); % Equivalent full sin pulse


p0=[p1 p2];

p=-981*m*a;

pp=-981*m*ap;

for in = 1:1:180;

Tn(in)=in/10;

wn(in)=2*pi/Tn(in);






a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;

dp(n)=p(n+1)-p(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

D(in)=max(abs(u));

a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;


181


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end


a= zeros(1,N0);

v= zeros(1,N0);

u= zeros(1,N0);

for n=1:1:N0-1;

dp(n)=p0(n+1)-p0(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

D0(in)=max(abs(u));

end

figure

plot(Tn,D);hold all

plot(Tn,DDp);plot(Tn,D0);

zie=0.05;

for in = 1:1:180;

Tn(in)=in/10;

wn(in)=2*pi/Tn(in);






a= zeros(1,N);

v= zeros(1,N);


for n=1:1:N-1;


182


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end


end

plot(Tn,DDp);

=============================================================

(8) Displacement Response Factor, Rd: Evaluation & plotting

clear all

close all

format short g

% CHI CHI, Taiwan, TCU068, N320E, LUF= 0.038 Hz

% ROTATION OF ATH to Max Velocity direction

N=18000 % total number of samples of the Record

dt=0.005


phi = 320 % Max Velocity direction at TCU068 station is N320E, Sha-Yam 2003

alpha = 90 % East direction

beta = 0 % North direction




in = 1;

for clm=1:1:N/5;


in = in + 5;

end




a = zeros(1,N);

183

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a2=[0 a];

a = zeros(1,N);

for in=1:1:N;

a(in)=-a1(in)*cosd(50) + a2(in)*cosd(40);% Vmax direction component of acc.

end

%--------------------------------------------------------------------------

A=fft(a);

% % fc = Hz



fo=SR/N;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:N/2

f(j+1)=j*fo;

end

for j=N/2+1:1:N-1

f(j+1)=-(N-j)*fo;

end

for j=1:1:N


end

Ap=FL.*A;


% -------------------------------------------------------------------------




a0=0.202; % pulse amplitude

Tp=6.87;w0=2*pi/Tp;


%beta=1/4; % Average acceleration


pp=-981*m*ap;

184

for in = 1:1:100;

Tr(in)=in/50;

Tn(in)=Tr(in)*Tp;

wn(in)=2*pi/Tn(in);wr(in)=w0/wn(in);






a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

for n=1:1:N-1;



du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end


ust(in)=(981*a0*Tn(in)^2)/(2*m*pi)^2;Rd(in)=DDp(in)/ust(in);% approximation for

ust

end

figure

plot(Tr,Rd,'k-'); hold all

zie=0.02;

for in = 1:1:100;

Tr(in)=in/50;

Tn(in)=Tr(in)*Tp;

wn(in)=2*pi/Tn(in);






a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

for n=1:1:N-1;



185

du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

DDp1(in)=max(abs(u));

ust(in)=(981*a0*Tn(in)^2)/(2*pi)^2;Rd1(in)=DDp1(in)/ust(in);% approximation for

ust

end

plot(Tr,Rd1,'k-.');

zie=0.05;

for in = 1:1:100;

Tr(in)=in/50;

Tn(in)=Tr(in)*Tp;

wn(in)=2*pi/Tn(in);






a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

for n=1:1:N-1;



du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

DDp2(in)=max(abs(u));

ust(in)=(981*a0*Tn(in)^2)/(2*pi)^2;Rd2(in)=DDp2(in)/ust(in);% approximation for

ust

end

plot(Tr,Rd2,'k--');

title('Pulse Deform. Resp. Factor Rd,Lomap-TRI-038FN,ao=0.1226g,Tp=1.57s');

%xlabel('Period ratio, Tn/Tp');ylabel('Rd = D/ust');

=============================================================

186

(9) Analogous Displacement Response Spectra: Evaluation & plotting

clear all

close all

format short g

% Chi Chi, TCU052, N322E FN, luf= 0.05 Hz

% ROTATION OF ATH to FAULT-NORMAL direction, Case 1

N=18000 % total number of samples of the Record

dt=0.005





in = 1;

for clm=1:1:N/5;


in = in + 5;

end




a = zeros(1,N);

in = 1;

for clm=1:1:N/5;


in = in + 5;

end

a2=[0 a];

a = zeros(1,N);

for in=1:1:N;

a(in)=-a1(in)*cosd(52) + a2(in)*cosd(38);% Vmax direction component of acc

end


for n = 1:1:N-1;

v(n+1) = v(n) + (a(n+1)+ a(n))/2*dt;

s(n+1)=s(n)+v(n)*dt+(a(n+1)/6+a(n)/3)*dt*dt;

187

end

%--------------------------------------------------------------------------

% Fast Fourier Transform (FFT) of Velocity, v

V=fft(v);

mV=abs(V);

f=(0:length(V)).'*SR/length(V);

DV=dt*981*mV(5:26);

F=f(5:26);

T=1./F;

figure

plot(T,DV);

%--------------------------------------------------------------------------

A=fft(a);

% % fc= 0.2778 Hz



fo=SR/N;

fc=Sn*fo;

j=[0:1:N-1];

for j=0:1:N/2

f(j+1)=j*fo;

end

for j=N/2+1:1:N-1

f(j+1)=-(N-j)*fo;

end

for j=1:1:N


end

for j=1:1:N

FH(j)=(f(j)/fc).^B/((f(j)/fc).^B-i); % Butterworth highpass filter

end

Ap=FL.*A;

ap=ifft(Ap);

Anp=FH.*A;

anp=ifft(Anp);

% -------------------------------------------------------------------------


188


m=1; % kip-sec2/in (force/acceleration)30*.014



p=-981*m*a;

pp=-981*m*ap;

pnp=-981*m*anp;

for in = 1:1:200;

Tn(in)=in/10;

wn(in)=2*pi/Tn(in);%wr(in)=w0/wn(in);






a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

for n=1:1:N-1;

dp(n)=p(n+1)-p(n);


du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end

D(in)=max(abs(u));

a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

for n=1:1:N-1;



du(n)=Dp(n)/K(in);



u(n+1)=u(n)+du(n);

v(n+1)=v(n)+dv(n);

a(n+1)=a(n)+da(n);

end


189

end

figure

plot(Tn,D,'k-.');hold all

plot(Tn,DDp,'k--');plot(T,DV);

==============================================================

(10) Half Sine Pulse – Non-linear EPP model for response evaluation

clear all

close all

format short g

td=1.6

To=2.

dt=0.01

N=round(To/dt); % N=12

t1=[0:dt:td];

t2=[td+dt:dt:(N-1)*dt];

t=[t1 t2]; % check

p1=10*sin(pi*t1/0.6);


p=[p1 p2];

figure

plot(t,p,'-o'); xlabel('Time (s)'); title('Applied Loading'); grid on

ylabel('Discrete force at time instant');

%[p,t]=resample(x,5,1);

%--------------------------------------------------------------------------

% NONLINEAR RESPONSE ANALYSIS of SDF system by Newmark Method

% Linear acceleration, gamma=1/2 and beta=1/6

fsy=7.5 % in kips

zie=0.05 % damping coefficient

m=0.2533029591; % kip-sec2/in (force/acceleration)

Tn=1 % Natural period

wn=2*pi/Tn % Cyclic frequency, rad/sec

c=2*zie*m*wn % =0.1592 damping coefficient, units in forceXtime/length,

Ccr=2*m*wn

gamma=0.5;

beta=1/6;

C1=m/(beta*dt)+gamma*c/beta

190

C2=m/(2*beta)+c*dt*(gamma/(2*beta)-1)

a= zeros(1,N);

v= zeros(1,N);

u= zeros(1,N);

k=zeros(1,N); fs = zeros(1,N); %initialized to zero.

ko=m*wn*wn;k(1)=ko;%Stiffness of SDF systm, units in (force/length), kip/in

dp(1)=p(2)-p(1);

Dp(1)=dp(1)+C1*v(1)+C2*a(1);

K(1)=k(1)+(gamma*c)/(beta*dt)+m/(beta*dt*dt);

du(1)=Dp(1)/K(1); u(2)=u(1)+du(1);

dv(1)=gamma*du(1)/(beta*dt)-gamma*v(1)/beta+dt*a(1)*(1-gamma/(2*beta));

v(2)=v(1)+dv(1); k(2)=ko; fs(2)= ko*du(1);

a(2)=(p(2)-c*v(2)-fs(2))/m;

bLocation = 1;

for n=2:1:N-1

dp(n)=p(n+1)-p(n);

Dp(n)=dp(n)+C1*v(n)+C2*a(n);

K(n)=k(n)+(gamma*c)/(beta*dt)+m/(beta*dt*dt);

du(n)=Dp(n)/K(n);

u(n+1)=u(n)+du(n);


v(n+1)=v(n)+dv(n);

fs(n+1)=fs(n)+ko*du(n);

if abs(fs(n+1)) > fsy

bLocation = 2;

end

if abs(u(n))>abs(u(n-1));

if abs(u(n))>abs(u(n+1));

bLocation = 3;

end

end

switch bLocation

case 2

k(n+1)=0;

fs(n+1)=sign(fs(n+1))*fsy;

case 3

k(n+1)=ko;

fs(n+1)=fs(n) + k(n+1)*du(n);

otherwise

k(n+1)=ko;

fs(n+1)=fs(n) + k(n+1)*du(n);

end

191

a(n+1)=(p(n+1)-c*v(n+1)-fs(n+1))/m;

end

figure

plot(t,u);xlabel('Time (s)'); ylabel('Disp'); grid on

figure

plot(u,fs,'-x');xlabel('Disp'); ylabel('Applied Force (fs)'); grid on

figure

plot(t,fs);xlabel('Time (s)'); ylabel('Applied force (fs)');grid on

===========================================================

192

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VITA

NAME OF AUTHOR: Ajit Chandrakant Khanse

PLACE OF BIRTH: Pune, India

DATE OF BIRTH: June 16, 1950

GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:

University of Wisconsin-Madison, Wisconsin

University of Bombay, Mumbai, India

DEGREES AWARDED:

Master of Science in Civil & Env. Engineering, 1974, University of Wisconsin-Madison

Bachelor of Engineering in Civil Engineering, 1973, University of Bombay, India

AWARDS AND HONORS:

Syracuse University Fellowship 2005 – 06

Syracuse University Fellowship 2007 – 08

Civil-Env Engg Dept Li Fellowship 2008 – 09

PROFESSIONAL EXPERIENCE:

Worked for Structural Design Industry in India for over thirty years.

URS Corporation, NYC – From April 1, 2009

Course Instructor, 01/2009 - 03/2009

Course Instructor, 05/2007 - 08/2007

ECS 325 – Mechanics of Solids

L. C. Smith College of Engineering and Computer Science

Syracuse University, Syracuse, NY

Teaching Assistant, 05/2006 - 05/2007

ECS 101 – Introduction to Engineering & Computer Science

CIE 331 – Analysis of Structures and Materials

ECS 325 – Mechanics of Solids

L. C. Smith College of Engineering and Computer Science

Syracuse University, Syracuse, NY

thesis,209, nethar

Documents

fault zones

nearfault ground motions

propagation of fault

faultrupture planes

period ground motions

large pulse of motion

period structures

velocity time history