A DATA DRIVEN APPROACH FOR EARTHQUAKE DAMAGE

DETECTION

A MASTER’S THESIS

in

Civil Engineering

Atilim University

by

AHMED S. MOHAMED

JANUARY 2005

A DATA DRIVEN APPROACH FOR EARTHQUAKE DAMAGE

DETECTION

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

ATILIM UNIVERSITY

BY

AHMED S. MOHAMED

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OF

MASTER OF SCIENCE

IN

THE DEPARTMENT OF CIVIL ENGINEERING

JANUARY 2005

Approval of the Graduate School of Civil Engineering.

_____________________

Prof. Dr. brahim Akman

Director

I certify that this thesis satisfies all the requirements as a thesis for the degree ofMaster of Science/Arts.

_____________________

Prof. Dr. S. Erol Ulu

Head of Department

This is to certify that we have read this thesis and that in our opinion it is fullyadequate, in scope and quality, as a thesis for the degree of Master of Science/Arts.

_____________________

Asst. Prof. Dr. Burcu Güne

Supervisor

Examining Committee Members

Prof. Dr. Selçuk Soyupak _____________________

Prof. Dr. S. Erol Ulu _____________________

Asst. Prof. Dr. Tolga Ak _____________________

Asst. Prof. Dr. Burcu Güne _____________________

Asst. Prof. Dr. Ahmet Yakut (METU, C.E.) _____________________

iii

ABSTRACT

A DATA DRIVEN APPROACH FOR EARTHQUAKE DAMAGE

DETECTION

MOHAMED,Ahmed

M.S., Civil Engineering Department

Supervisor: Asst. Prof. Dr. Burcu. GÜNE

JANUARY 2005, 85 pages

In the aftermath of recent major destructive earthquakes, there is increased awareness

for the need to assess the state of the critical and conventional civil structures.

Several methods have been proposed to assess structural health using changes in

vibration characteristics. Most of these methods require a refined finite element

model of the structure and assume that the structure remains linear both before and

after damage. This thesis, however, investigates a data-driven method for post-

earthquake damage detection that is based solely on the recorded seismic data. The

damage is identified by estimating the degree of non-linearity present within the

measured response measurements. The difference between the recorded data and the

predicted response under the premise of linear behavior is used as the detection

criteria for detecting the damage. The methodology adopted here employs the

Eigensystem Realization Algorithm with Observer/Kalman filter approach to identify

and predict the response of the linear system.

The method is applied to a simulated case study of 2-D model of a four-storey shear

building subjected to actual earthquake records with members having a Bouc-Wen

type restoring force-displacement relationship and the method is found promising as

an initial screening process for assessing the health of structural systems after an

extreme event.

iv

Keywords: Non-linearity, Eigensystem Realization Algorithm, Vibration based

damage detection, System identification, Earthquake damage.

v

ÖZ

VER KULLANIMLI DEPREM HASAR TESP T YÖNTEM

MOHAMED, Ahmed

Yüksek Lisans, aat Mühendisli i Bölümü

Tez Yöneticisi: Yrd. Doç. Dr. Burcu GÜNE

Ocak 2005, 85 sayfa

Son zamanlarda meydana gelen y depremlerin neticesinde, özellikle kritik

yap lar için durum tesbiti ihtiyac önem kazanan bir konu olarak ortaya ç kmaktad r.

Bu nedenledir ki titre im karakterlerindeki de iklikleri kullanarak hasar tahmini

yapan birçok metot önerilmi tir. Bu metodlar n bir ço unda yap n detayl sonlu

eleman modeline ihtiyaç duyulmakta ve buna ek olarak da yap n hem hasardan

önce hem de hasardan sonra do rusal elastik davran gösterdi i varsay lmaktad r.

Bu tez çal mas nda ise sadece deprem esnas nda kaydedilen veriler kullan larak

yap labilecek hasar tespit yöntemi incelenmektedir. Yöntem, do rusal elastik

davran önermesi ile sistemin verece i tahmin edilen ivme tepkisi ile hareket

esnas nda al nan ivme kay tlar n kar la lmas ve aradaki fark n lineer

davran tan ne kadar uzakla ld n göstergesi olaca esas na dayanmaktad r. Bu

lineer davran tan sapma derecesi de hasar tespit kriteri olarak incelenmi tir. Bu

incelemede Eigensistem Realizasyon Algoritmas Gözlemci/Kalman Filtresi ile

birlikte lineer sistemin ivme tepkisini tahmin etmek icin kullan lm r.

Önerilen yöntem Bouc-Wen tipi tersinir davran modeline sahip dört katl bir perde

çerçeve sistemiyle nümerik simülasyonlar yap larak test edilmi ve yap sal hasar

artt kça lineer davran tan uzakla ld bunun da depremin hemen akabinde özellikle

vi

yap sto unun ilk tasnifi esnas nda hasars z binalar ay rmakta etkili olabilece i

gözlenmistir.

Anahtar Kelimeler: Lineer olmayan; Eigensistem Realizasyon Algoritmas ; Titre im

bazl hasar tespiti; Sistem tan mlamas ; Deprem hasar .

vii

To my mother and father

my brothers and sisters

and my wife and daughter

viii

ACKNOWLEDGMENTS

I wish to express my gratitude to my supervisor, Asst. Prof. Dr. Burcu GÜNE for

her continuous support throughout my graduate career and during the completion of

this thesis. She offered me so much advice, patiently supervising me, and always

guiding me in the right direction. I have learned alot from her, without her help I

could not have finished my research successfully.

I also express my appreciation to the committe members, Prof. Dr. Selçuk Soyupak,

Prof. Dr. S. Erol Ulu , Asst. Prof. Dr. Tolga Ak , and Asst. Prof. Dr. Ahmet Yakut,

for thier valuable comments and for the time they have spent applying their expert

knowledge to the examination of this thesis .

I wish to thank my friends for encouraging me during my whole research. Special

thanks go to my friends who have attended the presentation of my thesis.

Finally, I wish to thank my family for the support they provided me through my

entire life and in particular, I must acknowledge my wife, Ghada, without her love,

encouragement and editing assistance, I would not have finished this thesis. I thank

my daughter, Munira, for all the happy time she gives.

ix

TABLE OF CONTENTS

ABSTRACT ............................................................................................................. iii

ÖZ .......................................................................................................................... v

DEDICATION.......................................................................................................... vii

ACKNOWLEDGMENTS ..................................................................................... viii

TABLE OF CONTENTS ......................................................................................... ix

LIST OF TABLES .................................................................................................... xi

LIST OF FIGURES .................................................................................................. xii

LIST OF SYMBOLS ............................................................................................... xiv

CHAPTER

1. INTRODUCTION ............................................................................................. 1

1.1 Vibration Based Damage Detection ............................................................. 1

1.2 Objective and Scope .................................................................................... 6

1.3 Thesis Organization ..................................................................................... 7

2. EIGENSYSTEM REALIZATION ALGORITHM WITH OBSERVERKALMAN FILTER IDENTIFICATION ......................................................... 9

2.1 Introduction ................................................................................................ 9

2.2 State Space Representation .......................................................................... 9

2.3 Eigensystem Realization Algorithm .......................................................... 11

2.3.1 Basic Formulations .......................................................................... 12

x

2.3.2 Extraction of Modal Parameters from System Realization .............. 16

2.4 Observer Kalman Filter Identification ....................................................... 19

2.4.1 Modifications on basic ERA formulations ...................................... 19

2.5 Illustrative Example ................................................................................... 23

3. NON-LINEARITY AS A DAMAGE INDICATOR ..................................... 32

3.1 Hysteretic behavior .................................................................................... 32

3.2 Bouc-Wen Model ....................................................................................... 34

3.3 Damage Indicators ..................................................................................... 37

3.3.1 Maximum Deformation Damage Indices ......................................... 38

3.3.2 Cumulative Damage Indices ............................................................ 39

3.3.3 Combined Indices: Maximum Deformation and

Cumulative Damage ........................................................................... 39

3.4 Proposed Approach .................................................................................... 40

4. STRUCTURAL DAMAGE IDENTIFICATION PROBLEM ........................ 41

4.1 System Description .................................................................................. 41

4.2 Modal Identification ................................................................................. 43

5. CONCLUSIONS AND FUTURE WORK ...................................................... 60

5.1 Conclusions ............................................................................................. 60

5.2 Recommendations for Future Work ......................................................... 62

REFERENCES ....................................................................................................... 63

APPENDIX

MATLAB CODES ................................................................................................... 68

A. 1 A Matlab Code for the Simulation ............................................................... 68

A. 2 A Matlab Code for non-linear modeling (Dash3dx.m) ................................ 73

A. 3 A Matlab Code works with Dash3dx code ................................................. 84

xi

LIST OF TABLES

TABLE

2.1 True Modal Parameters .......................................................................................24

2.2 Identified modal damping ratios and frequencies for cases 1, 2, and 3 .............27

2.2 Identified modal damping ratios and frequencies for cases 4 ............................27

4.1 Earthquake ground motion records .................................................................... 43

4.2 Data set of earthquake ground motions ............................................................. 48

4.3 Identified modal damping ratios and frequencies (with 4 sensors) ................... 50

4.4 Identified modal damping ratios and frequencies (with 2 sensors) ................... 52

4.5 Results of case 1 (4 sensors) for the records at different levels

of the ground motion ............................................................................................. 57

xii

LIST OF FIGURES

FIGURE

2.1 Flowchart for the ERA (Juang, 1994) ................................................................18

2.2 Flow Chart for OKID (Juang, 1994) ............................................................... 22

2.3 Structural model used for system identification .................................................23

2.4 Acceleration Records .........................................................................................25

2.5 Singular values of Hankel matrix for case 1 (8 sensors) .................................. 26

2.6 Singular values of Hankel matrix for case 2 (4 sensors) .................................. 26

2.7 Mode shapes of structural and identified model for case 1 ............................... 28

2.8 Computed vs predicted acceleration responses .................................................30

2.9 Absolute errors in response predictions of the identified models

(8-storey shear building with four sensors) .......................................................31

3.1 The effect of shaping parameters on the hysterestic loop

(a) linearly elastic systems .................................................................................36

(b) softening systems with different degree of hysteresis ..................................36

(c) hardening systems with different degree of energy dissipation ...................36

4.1 Structural model of a 4-story shear building ......................................................41

4.2 Earthquake ground motions ...............................................................................42

4.3 Force-displacement curves at different levels of the ground

motion of the Northridge Earthquake (Whitter Narrows record) ......................45

4.4 Flow chart of the approach (Burcu and Bernal, 2004) ......................................46

xiii

4.5 Ground motions recorded at the Whitter Narrows station during

the Northridge Earthquake (1994) .....................................................................48

4.6 Hankel matrix singular values and identified system with 8 sensors

(Whiter Narrows record, undamaged case) .......................................................49

4.7 Hankel matrix singular values and identified system with 8 sensors

(Cape Campbell earthquake, level 4 of damage) .................................................49

4.8 Computed vs. predicted acceleration response for the Whitter Narrows record

(level 1)

(a) the initial segment of the record ...................................................................55

(b) the entire duration of the record ...................................................................55

4.9 Computed vs. predicted acceleration response for the Whitter Narrows record

(level 3)

(a) the initial segment of the record ...................................................................56

(b) the entire duration of the record ...................................................................56

4.10 Percent error in the predicted response of the first floor versus

damage index .....................................................................................................58

xiv

LIST OF SYMBOLS

α - Ratio of post-yielding to pre-yielding stiffness

Λ - Eigenvalue matrix containing squared natural frequencies

Λ - a diagonal matrix containing the identified eigenvalues

A - Discrete time state matrix

A,δ ,γ ,a - Parameters describing shape and amplitude of hysteresis

CBA ,, - Estimates of the state-space matrices for the discrete-timestructural model

A - Observer state matrix

cA - Continuous-time state matrix

2b - Spatial distribution of the loading

B - Discrete-time input-influence matrix

B - Observer input-influence matrix

cB - Continuous-time input-influence matrix

mB - Input matrix in modal coordinates

rxB - a non-square matrix mapping the displacements to the relativedeformations

C - Output-influence matrix

mC - Output matrix in modal coordinates

Γ - Damping matrix

D - Direct-transmission matrix

xv

dE - Incremental absorbed energy

DMI - Damage Index

DOF - Degree-of-Freedom

dt - Sampling period

i∆ - Maximum interstory displacement

ERA - Eigensystem Realization Algorithm

ERA/DC - Eigensystem Realization Algorithm with Data Correlation

)(tEH - Hysteretic energy

)(tES - Maximum elastic stored energy

φ - Eigenvectors of the system matrix cA

)(tFg - Ground acceleration at time t

FDR - Flexural Damage Ratio

RF - Total restoring force

EF - Nonhysteretic restoring force

HF - Hysteretic restoring force

G - Gain matrix

h - Story height

H - Hankel matrix

iI - Identity matrix of order i

ψ - Mode shapes

k - Time index

0k - Initial stiffness of the inelastic system

Tk - Pre-yielding stiffness

xvi

K - Stiffness matrix

EK - Nonhysteretic stiffness

HK - Hysteretic stiffness

l - Data length

iλ - Identified eigenvalues

m - Number of outputs

M - Mass matrix

MAC - Modal Amplitude Coherence

MIMO - Multi-Input Multi-Output

MSV - Mode Singular Value

Rµ - Ductility ratio

n - Order of the state-space model

N - Number of degree-of-freedom

NCD - Normalized Cumulative Deformation

NHE - Normalized Cumulative Dissipated Energy

OKID - Observer Kalman Filter Identification

0i - Zero-square matrix of order i

p - Integer determining maximum order of system

P - Observability matrix

Q - Controllability matrix

rQ - Yield strength

r - Number of inputs

RMS - Root Mean Squares

s - an integer that determines the size of such a matrix

xvii

S - Singular-value matrix

SVD - Singular Value Decomposition

iδ - Drift ratio

t - Time

u(k) - Inputs vector at time index k

mu - Maximum deformation

yu - Yield deformation

U - Left singular matrix

V - Matrix contains the input vectors for different time steps arranged in an upper-triangular form.

V - Right singular matrix

- Modal angular frequency

w - Displacements vector

w& - Velocities vector

w&& - Accelerations vector

x(k) - State vector at time index k

)(),( 21 txtx - The state variable

x(t) , )(tx& - Displacements and velocities patitions of the state variable

- Damping ratios

y - Output measurement vector

y(k) - Output-measurement vector at time index k

y - Matrix whose columns are the output vectors for the l time steps

Y - System Markov parameters

Y(k) - Markov parameters Matrix

xviii

Y - Observer Markov parameters

z(t) - Vector containing the corresponding hysteretic information

1

CHAPTER 1

INTRODUCTION

1.1 Vibration Based Damage Detection

The interest in the ability to monitor a structure and detect damage at the earliest

possible stage is recognized as an important objective of structural engineering. An

earthquake, an unpredictable and unpreventable event, is regarded as one of the most

destructive natural disasters on earth. Hence, assessing the health of structural

systems after a major earthquake is vital because structural safety has to be reassured

before the structures are reoccupied.

Current methods of detecting post-earthquake damage in civil structures are manual,

visual inspection performed by experienced engineers. However, visual inspections

reveal significant problems sometimes. These inspections are time consuming, labor

intensive, and expensive. Because it is not possible to access all members of

structure, it is very hard to make a final decision if inspectors have not enough

information about that particular structure. Two different types of methodologies are

available to assess the health of structures, localized and global techniques. Localized

techniques are used to identify the health of a structural member using technology

such as X-rays and ultrasound. To utilize these techniques previous knowledge of the

location of damage and direct access to the structural member are required.

Additionally, none of these techniques provide a quantitative assessment of the

damage magnitude. Subjected to these limitations, these experimental methods can

detect damage on or near the surface of the structure. Global techniques, on the other

hand, use the vibration data of the structure to identify damage, its approximate

location and its severity. Global techniques are very attractive to civil engineers

because they can be used without direct access to the structural components and

without a priori knowledge of damage of the structure which as a result reduce the

2

time and cost of assessing damage of the structure.

In the most general terms, damage can be defined as changes introduced into a

system that adversely affects its current or future performance. The effects of damage

on a structure can be classified as linear or nonlinear. A linear damage situation is

defined as the case when the initially linear-elastic structure remains linear-elastic

after damage. The changes in modal properties are the result of changes in geometry

and/or material properties of the structure, but the structural response can still be

modeled using linear equations of motion.

Nonlinear damage is defined as the case when the initially linearly-elastic structure

behaves in a nonlinear manner after the damage has been introduced. One example

of nonlinear damage is the formation of a fatigue crack that subsequently opens and

closes under the normal operating vibration environment.

Another classification system for damage-identification methods, defines four levels

of damage identification, as follows Rytter (1993)

- Level 1: Identification of damage presence in a structure,

- Level 2: Localization of damage,

- Level 3: Quantification of damage severity, and

- Level 4: Prediction of the remaining service life of the structure.

Level 1 techniques determine whether or not damage exists in a structure. Level 2

techniques determine the existence of damage, as well as its’ location. Level 3

techniques identify the existence, location and severity of damage. Level 4

techniques identify the existence, location and severity of damage, as well as

characterizing the remaining life of the structure. Each damage identification method

can also be classified by the type of data and numerical technique used. Doebling et

al. (1996) made an extensive survey and classification of different damage

identification and health monitoring methods.

In civil engineering structures, numerous methods have been applied to assess

structural health using changes in vibration characteristics. The vibration-based

damage identification methods rely on the fact that occurrence of damage or loss of

3

integrity in a structural system leads to changes in the dynamic properties of the

structure (eigenfrequencies, modal damping rates, mode shapes). The changes in

structural dynamic characteristics, if properly identified and classified, can be used as

quantitative measures that provide the means for assessing the state of damage of the

structure. The literature on vibration based damage detection is vast and growing,

and therefore a complete review of the subject is beyond the scope of this study and a

brief review is summarized below.

The concept of tracking natural frequency changes for damage detection was first

reported by Adams et al. (1978). Damage is characterized based on the ratio of

frequency changes for two modes. Cawley and Adams (1979) focus on damage

detection using the information of natural frequency changes. Based on the work of

Cawley and Adams, Stubbs and Osegueda (1990) propose a damage detection

method using the sensitivity of modal frequencies change. Farrar et al. (1994) used

natural frequency shifts to identify damage in a highway bridge. A review of using

frequency changes for damage detection is compiled by Salawu (1997).

Mode shape changes have also been monitored to detect and locate damage. West

(1984) proposes using the modal assurance criteria (MAC), correlating mode shapes

between damaged and undamaged structures, to characterize damage. Srinivasan and

Kot (1992) find that changes in mode shapes, for a shell structure, are a more

sensitive indicator of damage than changes in resonant frequencies. Mayes (1992)

presents a method for model error localization based on mode shape changes known

as structural translational and rotational error checking. Natke (1997) uses changes in

natural frequencies and mode shapes to detect damage in a finite element model of

the cable-stayed steel bridge. Doebling and Farrar (1997) examine changes in the

frequencies and mode shapes of a bridge as a function of damage. Yuen (1985),

Rizos et al. (1990), Osegueda et al. (1992), Kam and Lee (1992), Kim, et al. (1992),

Ko et al. (1994), Salawu and Williams (1994, 1995), Lam et al. (1995), and Salawu

(1995) provide examples of other studies that examine changes in mode shapes,

primarily through MAC and coordinate MAC (or COMAC) values, to identify

damage.

4

Mode shapes curvature is also investigated as a means of identifying and localizing

damage. Pandey et al. (1991) demonstrate that absolute changes in mode shape

curvature can be a good indicator of damage for the FEM beam structures they

consider. Ho and Ewins (1999) proposes a damage index method using the quotient

squared of a structure’s modal curvature in the undamaged state to the structure’s

corresponding modal curvature in its damaged state. Ho and Ewins (2000) state that

higher derivatives of mode shapes are more sensitive to damage and they propose

changes in the mode shape slope squared as a feature.

Damping properties, when compared to frequencies and mode shapes, have not been

used as extensively as frequencies and mode shapes for damage diagnosis. Modena

et al. (1999) show that visually undetectable cracks cause very little change in

resonant frequencies and require higher mode shapes to be detected, while these

same cracks cause larger changes in the damping.

Another class of vibration-based damage identification methods uses the dynamically

measured flexibility matrix to estimate changes in the static behavior of the structure.

Aktan et al. (1994) propose the use of measured flexibility as a “condition index” to

indicate the relative integrity of a bridge. Pandey and Biswas (1994) present a

damage-detection and location method based on changes in the measured flexibility

of the structure. Mayes (1995) uses measured flexibility to locate damage from the

results of a modal test on a bridge. Peterson et al. (1995) propose a method for

decomposing the measured flexibility matrix into elemental stiffness parameters for

an assumed structural connectivity. This decomposition is accomplished by

projecting the flexibility matrix onto an assemblage of the element-level static

structural eigenvectors. Bernal and Gunes (2000) propose a damage locating vector

approach that uses the synthesized flexibility matrix at sensor locations. The

technique is applied to a 4 story steel building for a variety of damage patterns. Reich

and Park (2000) focus on the use of localized flexibility properties for structural

damage detection. Topole (1997) discusses the use of the flexibility of structural

elements to identify damage.

Strain energy is another parameter examined for the purpose of damage detection.

Stubbs et al. (1992) present a method based on the decrease in modal strain energy

5

between two structural DOF, as defined by the curvature of the measured mode

shapes. Zhang, Qiong, and Link (1998) propose a structural damage identification

method based on element modal strain energy, which uses measured mode shapes

and modal frequencies from both damaged and undamaged structures as well as a

finite element model to locate damage. Worden et al. (1999) present another strain

energy study using a damage index approach. Carrasco et al. (1997) discuss using

changes in modal strain energy to locate and quantify damage within a space truss

model.

Vibration based damage detection which provides a global means for detecting

damage has recently received significant attention by researchers. These vibration

methods allow the engineer to use sensing of the structural responses in conjunction

with appropriate data analysis and modeling techniques to monitor the condition of a

structure. After extreme events, such as an earthquake or blast loading, the vibration

data can be utilized for condition screening in the hope of providing reliable

information regarding the integrity of the structure.

The problem most commonly considered in vibration based damage detection is that

where data is recorded at two different times and it is of interest to determine if the

structure suffered damage in the time interval between the two observations. The

behavior of the system during the data collection is typically assumed linear and the

damage, which may result from an extreme event occurring inside the time segment,

is characterized as changes in the parameters of a linear model. Hence, linear

methods are utilized to analyze the two signals, namely before and after the damage.

In civil engineering applications, however, the assumption of linearity is hardly ever

satisfied. The problem for the non-linear behavior of the system becomes a major

difficulty in a lot of cases. In general, any structure contains nonlinearities stemming

from the nonlinear material behavior, geometric nonlinearities, nonlinearities in the

supports or the connections. In many cases this results in strongly non-linear

behavior which cannot be approximated by a linear response. The introduction of

damage (defect) in a structure can be regarded as an additional non-linearity. The

introduction of a new non-linearity as well as its growth changes the vibrational

response of the structure and is expected to influence its non-linear dynamic behavior

6

Using system identification algorithms that are based on the assumption of linearity

cannot account for the non-linear effects of such damage scenarios. The review

articles by Billings (1980) and Imregun (1998) reveal a survey of nonlinear system

identification algorithms. However, because of their specialized nature and limited

applicability, there seems to be some consensus that selection of a particular

algorithm depends on the objectives of the analysis.

An alternative for damage detection associated with response to extreme events is to

assess damage by estimating the degree of non-linearity present within a given

response measurement. To be more specific, if the system is characterized during a

non-damaging event in which the behavior of the system remains linear and a model

is obtained for the healthy state of the structure, response prediction can be obtained

for any other input based on the identified model of the structure (Gunes and Bernal,

2004). The difference between the recorded data and the predicted response under

the premise of linear behavior can then be used as a criterion to estimate the state of

the structure. In obtaining the model of the nominally healthy system, the measured

signals from small events can be utilized. An alternative to this, which we have

employed in this study, is to treat the initial segment of the data recorded during an

earthquake ground motion as the data obtained from a small event. This initial data

segment is then processed using eigensystem realization algorithm with observer

Kalman filter (ERA-OKID) for identifying the system matrices of the healthy state.

Based on the obtained realization, responses at the sensor locations can be predicted

for the entire duration of the ground motion. The residuals of the difference between

the recorded and the predicted values can then be used to extract information on the

damage state of the structure.

1.2 Objective and Scope

This research aims to provide a method for the detection of damage by estimating the

degree of non-linearity present within the given response measurement. The initial

segment of the data recorded during an earthquake ground motion is used to identify

the nominally healthy system. This thesis focuses on the eigensystem realization

7

algorithm with observer Kalman filter (ERA-OKID) for identifying the system

matrices of the healthy state.

The fundamental objective of this study is to investigate the effect of nonlinearities in

the response of the system on the system parameters identified by ERA-OKID

algorithm. In this investigation the hysteretic behavior is based on Bouc-Wen model,

which is mathematically convenient and can provide a good approximation to the

conditions found in practice.

1.3 Thesis Organization

This thesis consists of five chapters and one appendix:

Chapter 1 is an introduction to the work completed and a brief description of the

problems addressed.

Chapter 2 presents an overview of the state space formulation followed by a

description of the observer kalman filter. The methodology presented herein uses the

Eigensystem Realization Algorithm (ERA) to identify the natural frequencies and

mode shapes of the structure. Using these natural frequencies and mode shapes it is

possible to determine the stiffness coefficients of structural members through a least

squares solution of the eigenvalue problem. The eigensystem realization algorithm

(ERA) is adopted because it is quite effective for identification of lightly damped

structures and is applicable to multi-input/multi-output systems.

Chapter 3 presents the fundamentals of non-linearity as a damage index. The first

part of this chapter presents the basics of Bouc-Wen Model that is used for general

random response analysis of hysteretic systems. The next part of the chapter reviews

the literature on possible damage indicators and presents the one employed herein

this study.

The modeling and identification of linear and nonlinear dynamic systems through the

use of measured experimental data is given in Chapter 4. Here, MATLAB programs

are used to perform the system identification. The performance of the method is

examined with simulated data on a 4 degree-of-freedom system; the response of a

8

four-story, steel moment resisting structure is analyzed to show the application of the

adopted approach. Ten real ground motions with moment magnitude between 5.7 and

7.5 are considered. The adopted method is successfully applied to detect damage in

the structure.

Chapter 5 contains some concluding remarks and recommendations for future

research.

Finally, the appendix provides some of the MATLAB codes that were used to

simulate the dynamic response of the structure. These codes correspond to the

discussion of the experimental results in chapter four.

9

CHAPTER 2

EIGENSYSTEM REALIZATION ALGORITHM WITHOBSERVER KALMAN FILTER IDENTIFICATION

2.1 Introduction

This chapter discusses the application of a two-step methodology that consists of,

first, finding a first order minimal state space realization of the system using only

input-output measurements, and then, extracting the modal parameters (i.e. natural

frequencies, damping, and mode shapes) of the underlying second-order system.

Initially, it is assumed that the structure (model) under investigation is subjected to

measured dynamic excitations (inputs), and that concurrently measurements, in the

form of either displacements, velocities, and/or accelerations, are obtained at various

locations on the structure (outputs). These input and output measurements are then

used in the Observer/Kalman filter Identification (OKID) algorithm (Juang et al.,

1993) to identify the so called observer Markov parameters, and the system Markov

parameters that are required to build the Hankel matrix for the realization algorithm

are later retrieved from these observer Markov parameters.

2.2 State Space Representation

The dynamic behavior of an N degree-of-freedom (DOF) linear dynamic system can

be represented by the second-order vector differential equation as

)()()()( 2 tubtwKtwtwM =+Γ+ &&& (2-1)

where w, w& and w&& are the displacement, velocity and acceleration vectors,

respectively, u(t) are generally externally applied forces, t is the time index, and M, Γ

and K, are the mass, damping and stiffness matrices ( NN × ), respectively, while 2b

10

indicates the spatial distribution of the loading.

The 2nd order differential equation (2-1) can be represented as a system of a first-

order differential equations. The state space formulation of the dynamic system

requires n = 2N states (the model order) to equivalently represent the second-order

system of eqn. (2-1). By defining the displacement and velocity as the states of this

2nd order differential equations, the state variable is written as

)()(1 twtx = (2-2)

)()(2 twtx &= (2-3)

Inserting eqns. (2-2) and (2-3) into eqn. (2-1), one gets:

)()()()( 2122 tubtxKtxtxM +−Γ−=& (2-4)

Comparing eqns. (2-2) and (2-3), one can write

)()( 21 txtx =& (2-5)

Multiplying eqn. (2-4) by 1−M , one gets:

)()()()( 21

21

11

2 tubMtxMtxKMtx −−− +Γ−−=& (2-6)

Writing eqns. (2-5) and (2-6), in matrix format we get

)(00

21

2

111

2

1 tubMx

xMKM

Ixx

+

Γ−−

=

−−−&

& (2-7)

or in a compact form as

)()()( tuBtxAtx cc +=& (2-8)

The output vector which is a linear combination of states and inputs can be written as

)()( tuDtxCy += (2-9)

11

where nncA × is the square matrix called the system matrix and n = 2N, rncB × is the

input-state influence matrix, nmC × is the state-output influence matrix, and rmD × is the

matrix which represents any direct connection between the input and output, y is the

m×1 output vector (measurements) such as displacement, velocity or acceleration

and m is the number of outputs, x(t) and )(tx& are the r×1 state vector and its first

derivative with respect to time, respectively, where r is the number of independent

input functions, u(t) are generally externally applied forces.

Since the input and output signals are recorded in a digitized manner, it is convenient

to work in discrete time domain so the discrete counterpart of the continuous state-

space model can be expressed as

)()()1( kuBkxAkx +=+ (2-10)

)()()( kuDkxCky += (2-11)

where the integer k denotes the time-step number, x(k), u(k) and y(k) are the vectors

of states, inputs and outputs respectively, and A, B, C and D and are the system

matrices.

2.3 Eigensystem Realization Algorithm

Eigensystem Realization Algorithm (ERA) is a time domain technique proposed by

Juang and Pappa (1985). It is an extended version of the Ho-Kalman algorithm (Ho

and Kalman, 1965) and has become an accepted and widely used method. This

method was first proposed for modal parameter identification from measured

responses. Some modifications were later considered to improve the ERA method.

Juang et al. (1988) introduced a modification to ERA algorithm, using response data

correlations (ERA/DC) rather than the pulse response values in the formulation of

the Hankel matrix. The ERA/DC modified method was found to reduce measurement

noise bias without model over-specification. However, when over-specification is

permitted and singular value decomposition is used to obtain a minimum order

realization, both old and modified methods give equally good results for the data

12

used. Here the eigensystem realization algorithm (ERA) is adopted because it is

applicable to multi-input/multi-output systems, and especially those that are lightly

damped. This algorithm uses impulse response functions to obtain modal

properties of multi-input multi-output (MIMO) systems.

The ERA method uses a singular-value decomposition to derive the basic

formulation for a minimum-order realization. First, a block Hankel matrix is obtained

by arranging the pulse response data into the blocks of the Hankel matrix. By

examining the singular values of the Hankel matrix, the order of the system is

determined. A minimum-order realization (A, B, and C state-space matrices) is

constructed using a shifted block Hankel matrix. By finding the eigensolution of the

realized state matrix, modal damping rates and frequencies may be obtained. The

method then evaluates coherence and co-linearity accuracy parameters to separate

system modes from noise modes. Based on these accuracy parameters, the system

model is determined and the Hankel matrix based on identified state space matrices

is reconstructed and compared with the measurement data.

2.3.1 Basic Formulations

To present, in a brief form, the fundamental theoretical principles of ERA and

ERA/DC, consider a system with r input and m outputs. The system response, )(ky j

at time step k due to unit impulse ju can be written as

,......2,1,)](...)()([)( 21 == kkykykykY r (2-12)

and the rsms× Hankel matrix is formed as

1))-2(Y(...)Y(1)-Y(::

)Y(...2)Y(1)Y(1)-Y(...1)Y()Y(

)1(

+++

+++++

=−

sksksk

skkkskkk

kH (2-13)

where Y(k) is the pulse response matrix at the kth time step, and s is an integer that

determines the size of such a matrix. By definition, the sub-matrices Y(k)

13

correspond to the system Markov parameters. The first Markov parameter, i.e. D, can

be readily identified by considering that

DY =)0( (2-14)

Having identified the D matrix, the triplet (A, B, C) that will reproduce the data

sequence Y(k), k=1,2,… is considered. If the data permits a realization, then the full

data sequence can be generated from the triplet (A, B, C) via the following equation:

,...2,1,)( 1 == − kBACkY k (2-15)

Substituting eqn. (2-15) into eqn. (2-13) leads to the following representation of the

Hankel matrix:

...0,1,i ,

BCA...BCABCA::

BCA...BCABCABCA...BCABCA

iH

S2iSi1Si

Si2i1i

1Si1ii

S =

=

−++−+

+++

−++

)1(

)( (2-16)

A realization is a set of A, B, C, and D matrices that describe the behavior of the

structure. The minimum realization is the realization with the minimum number of

states required to describe the behavior of the system. Ideally, the modal parameters

(natural frequencies and mode shapes) of a minimum realization model of the

structure will be the same as the modal parameters of the structure (Juang, 1994).

Once the system’s Markov parameters have been determined and the corresponding

Hankel matrix has been built, let the singular value decomposition (SVD) of H’s(0)

be denoted by

[ ] TT

TT VSUV

VSUUVSUH 112

121 00

0)0( =

== (2-17)

where S is a square diagonal matrix with the singular values in the diagonal, and the

rsmsU × and rsrsV × are unitary matrices that contain the left and right eigenvectors of

H(0). The matrices SN, UN and VN are then obtained by eliminating the rows and

14

columns corresponding to small singular values produced by computational modes.

Since SN is a diagonal matrix, eqn. (2-17) can be expressed as

)()()0( 2/12/1 TNNNN VSSUH = (2-18)

Using the discrete-time state space form of the equation and the Markov parameters,

eqn. (2-13) becomes

][ BABABA

CA

CAC

QAPkH rk

p

k .....)1( 11

1

1 −

−

−

==− (2-19)

where A, B, and C are the matrix coefficients of a state space realization of the

system, P and Q are controllability and observability matrices respectively. For k=1

PQH =)0( (2-20)

From eqns. (2-18) and (2-20), both P and Q could be balanced as

TNNNN VSQandSUP 2/12/1 == (2-21)

For k=2

PAQH =)1( (2-22)

Combining eqns. (2-19), (2-21) and (2-22), the estimates of the state-space matrices

for the discrete time dynamic model are found using

2/12/1 )1( −−= NNTNN SVHUSA (2-23)

rT

NN EVSB 21= (2-24)

21SUEC NTm= (2-25)

15

where TmE is [Im 0m … 0m] and rE is [Ir 0r … 0r] and 0i is a null matrix of, Ii is an

identity matrix of order i. This is the basic ERA formulation.

The realized discrete-time model represented by the matrices CBA ,, and D can be

transformed to the continuous-time model. The system frequencies and damping may

then be computed from the eigenvalues of the estimated continuous-time state

matrix. The eigenvectors allow a transformation of the realization to modal space

and hence the determination of the complex (or damped) mode shapes and the initial

modal amplitudes (or modal participation factors).

With the matrix D identified directly (as shown in eqn.(2-14)), all the discrete time

system matrices belongs to a first order model with the smallest state space

dimension. A major benefit of this approach is that there is no requirement for an a

priori knowledge of the order of the system. If the data comes from an nth order

controllable and observable model and is noise free, then the singular value

decomposition of H(0) reveals exactly n non-zero singular values, and therefore the

order of the system can be picked after the decomposition has been done. Even when

the measurements are polluted by noise, the magnitude of the singular values reveal

the true order of the system, since the noise modes usually correspond to very small

singular values. A detailed presentation of the ERA and ERA/DC procedures can be

found in Juang (1994).

The identified discrete-time model in modal coordinates can be expressed as

)()()1( kuBkxkx mmm +Λ=+ (2-26)

)()()( kDukxCky mm += (2-27)

with r inputs and m outputs, where Λ is a diagonal matrix containing the

identified eigenvalues, iλ (i = 1,2, ...,n), of the system, and mB and mC

are the input and output matrices in modal coordinates, respectively. Because

the measurement vector y is real, all complex quantities in eqns. (2-26)

and (2-27) including the eigenvalues occur as complex conjugate pairs.

16

2.3.2 Extraction of Modal Parameters from System Realization

The system generated by the ERA is transformed to the corresponding continuous-

time equivalent one using the relation

)(ln dtAAc = (2-28)

It is then possible to extract the modal frequencies and damping ratios of the

identified dynamic system. The complex eigenvalues are given by;

2, 1 iiiinii i ξωωξ −−=Λ + m (2-29)

where, n is the number of identified modes. Expressing eqn. (2-29) as;

iii iβα +=Λ (2-30)

The modal angular frequency and damping ratios are given by

22iii βαω += (2-31)

and

22ii

ii

βα

αξ

+

−= (2-32)

The corresponding mode shapes can then be computed as

hC −Λ= φψ (2-33)

where h = 0, 1 or 2 for displacement, velocity or acceleration sensing respectively,

while φ is eigenvectors of the system matrix cA .

To distinguish true modes from noise modes, Juang and Pappa (1985) develop two

indicators, (i) the modal amplitude coherence (MAC) and (ii) the mode singular

values of the H’s(0) matrix. The first indicator, Modal Amplitude Coherence (MAC),

17

gives a measure of how well a computed mode shape and frequency reproduce the

measured system response. A value of 1.0 indicates perfect reproduction. The second

indicator, Mode Singular Value (MSV), gives a measure of the contribution of each

mode to the identified pulse response time history. The measure is normalized such

that the strongest responding mode has an MSV value of 1.0.

The computational steps of ERA shown in Figure 2.1 are summarized as follows:

1) Construct a block Hankel matrix H(0) by arranging the Markov parameters

(pulse response samples) into blocks.

2) Decompose H(0) using singular value decomposition (SVD).

3) Determine the order of the system by examining the singular values of

the Hankel matrix H(0).

4) Construct a minimum order realization [ CBA ,, ] using a shifted block

Hankel matrix H(l).

5) Find the eigensolution of the realized state matrix and transform the

realized model to modal coordinates to calculate the system damping and

frequencies.

6) Calculate the modal amplitude coherence and mode singular values to

quantify the system and noise modes.

7) Determine the reduced system model based on the accuracy indicators

computed in step 6, reconstruct Markov parameters kY and compare with the

measured Markov parameters.

18

Figure 2.1: Flowchart for the ERA (Juang, 1994)

Mode shapes

Right singular vectors

Input-outputdata

Pulse response matrix(Markov parameters)

Singular values

State matrix,A

Eigensolution of A

Natural frequencies andmodal damping

Eigenvectors

Shifted Hankelmatrix, H(1)

H(0)

Hankel matrix,H(0)

Output matrix,C

EigenvaluesEigenvectors

Reduced model

Reconstruction and comparison with data

Left singular vectors

Modal amplitudes

Input matrix,B

19

2.4 Observer Kalman Filter Identification

The ERA-OKID is used to identify the Markov parameters of the system, which are

in turn used in the Eigensystem Realization Algorithm (ERA) (Juang and Pappa,

1985) to realize the discrete time first-order system matrices. To reduce the number

of Markov parameters needed to adequately model a system, an observer is

introduced into the OKID technique. The observer Markov parameters will become

identically zero after a finite number of terms. For lightly damped systems, this

means that the system can be described by a reduced number of observer Markov

parameters. Furthermore, an unstable system can be represented using this technique.

2.4.1 Modifications on Basic ERA Formulations

The basic idea of the OKID approach consists of re-writing eqns. (2-10) and (2-11)

by adding and subtracting the term Gy(k) in the state equation, so that a “new”

system can be formulated as

)()()1( kvBkxAkx +=+ (2-34)

)()()( kuDkxCky += (2-35)

where

GCAA += ; ][ GGDBB −+= ; [ ]TTT kykukv )()()( = (2-36)

and G is an mn × “arbitrarily chosen” gain matrix to make the system as stable as

desired. Although the systems in eqns. (2-34) and (2-10) are mathematically

identical, the equations in (2-34) can be viewed as observer equations, and the

Markov parameters of this new system are called the observer's Markov parameters.

If the matrix G is chosen in such a way that A is asymptotically stable, then

0≈BAC h for pk ≥ , and the input/output relations can be written as

lrpmrrpmrmYy ×++++×≈ ))(())(( V (2-37)

20

where

][ )1(...)(...)2()1()0( −= lypyyyyy (2-38 a)

][ BACBACBCDY p 1... −= (2-38 b)

and the matrix V contains both input and output data, i.e.

1)-p-(...(0)...::

2)-(...1)-(p...(1))0(1)-(...(p)...(2))1((0)

V

=

lvv

lvvvvluuuuu

(2-39)

The important thing to note is that a small number of observer Markov parameters

are sufficient to describe the mapping in eqn. (2-37). The so called “observer Markov

parameters” are the block partitions of the matrixY , and they are obtained by finding

the least squares solution to eqn. (2-37) as tyY V= ( tV = pseudo inverse of V).

Through manipulation, system Markov parameters Y can then be recovered from the

observer Markov parameters through partition of Y as:

[ ]PYYYYY ...210= (2-40)

where

DY =0 (2-41 a)

BACY kk

)1( −= (2-41 b)

The observer Markov parameter 0Y is readily identified as the first block partition of

Y and has a smaller dimension than the remaining Markov parameters. Once the

observer Markov parameters are obtained, the system Markov parameters can then

be retrieved from the observer Markov parameters via back substitution as

21

DYYDCGGDBCCBY )2(1

)1(11 )()( −=−+== (2-42)

To obtain the Markov parameter CAB, first consider the product )1(2Y

)2(2

)2(12

)1(2 )()( YYYGDBGCACY −−=++= (2-43)

Hence,

DYYYYCABY )2(21

)2(1

)1(22 −−== (2-44)

and so on.

According to the above derivation, the general relationship between the actual

system Markov parameters and the observer Markov parameters is

00 YYD == (2-45)

∑=

− =−=k

iikikk pkforYYYY

1)(

)2()1( .....,,1 (2-46)

∞+=−= ∑=

− .......,,11

)()2( pkforYYY

p

iikik (2-47)

Once the system’s Markov parameters have been identified, they can be used in the

previous ERA formulation for the identification of the dynamic structural

characteristics as mentioned before. Further details omitted in this presentation can

be found in the works of Juang et al. (1993), Lus¸ et al. (1999), and Lus¸ (2001).

The computational steps of OKID, shown in Figure (2.2) are summarized as follows:

1) Compute the observer Markov parameter. Choose a value of p of eqn. (2-37)

which determine the number of Markov parameter to be identified from the

given set of input and output data. In general, p is required to be sufficiently

larger (at least four or five times) than the effective order of the system for

identification of the Kalman filter gain with accuracy.

22

2) Recover system and observer gain Markov parameters.

3) Realize a state-space model of the system and corresponding observer gain

from the recovered sequence of the system and observer gain Markov

parameters by using ERA or ERA/DC.

4) Find the eigensolution of the realized state matrix and transform

the realized model to modal coordinates for modal parameter identification.

The modal parameters include frequencies, damping, and mode shapes at the

sensor locations.

Figure 2.2: Flowchart for OKID (Juang, 1994)

Input and Output Time Histories

Observer Markov Parameter

Observer Gain MarkovParameters

System MarkovParameters

System Matrices A, B, C, D & Observer GainMatrix

23

2.5. Illustrative Example

In order to illustrate the application of the ERA/OKID identification scheme, a

simple numerical example is presented. The structure selected, shown in Figure 2.3,

is a two-dimensional eight-story shear building, with a floor weight of 345.6 tons and

floor stiffness equal to 340 400 kN/m. The damping ratios and the modal frequencies

are displayed in Table 2.1. The input is an ambient excitation at each floor in the

horizontal direction, while the output measurements are the simulated floor

accelerations in the same direction. The noise is simulated by adding to the

analytically computed acceleration response of all floors white noise having a root-

mean-squared (RMS) equal to 5% of the response measured on the sensor located at

the first floor. Figure 2.4 shows the accelerations records simulated for the structural

model.

Figure 2.3: Structural model used for system identification

24

Table 2.1: True Modal Parameters

Mode Damping ratio Natural frequencyNumber ( %) (Hz)

1 5 0.922 5 2.733 5 4.454 5 6.025 5 7.386 5 8.497 5 9.318 5 9.82

The effect of the number of sensors used in the identification is considered with four

cases:

Case 1. There is a sensor at each floor,

Case 2. There is a sensor at each other floor starting from the second floor,

Case 3. There is a sensor at the fourth and eighth floor, and

Case 4. There is one sensor at (the first floor or the fourth floor or the roof).

The initial step consists of identifying the order of the system model. For all cases,

varying the number and the location of sensors used in the identification, 16

significant singular values corresponding to 8 modes were identified. Figures 2.5 and

2.6 show typical plots of the singular values of the Hankel matrix for case 1 and case

2 respectively.

It is important to mention that when there is no noise in the measurements, the

system can be identified (for all the numbers of sensors investigated) with perfect

accuracy for any p (the number of observer Markov parameters) such that npm ≥×

where m is the number of outputs and n is the order of the system (Juang, 1994).

However, when there is noise present in the measurements, one needs to have a value

for p large enough so that more data points are employed, leading to a better

identification of modal characteristics. The value for p is arbitrarily selected as p =

20 in case 1 to make the maximum system order pm = 160, which is higher than the

25

Figure 2.4 Acceleration Records

26

Figure 2.5: Singular values of Hankel matrix for case 1 (8 sensors)

Figure 2.6: Singular values of Hankel matrix for case 2 (4 sensors)

27

anticipated system order of n = 8. By looking at the singular value plots of the initial

models, the order is then reduced to 16th.

The identified modal frequencies and damping ratios for all cases are given in Tables

2.2 and 2.3 and a sample of the corresponding mode shapes for case 1 (8 sensors) are

shown in Figure 2.7. Examination of the identified frequencies and dampings and

mode shapes shows that the algorithm is very robust in the presence of noise and the

modal frequencies and damping rates are identified accurately for the four different

sensor configurations examined here.

Table 2.2: Identified modal damping ratios and frequencies for cases 1, 2, and 3.

CASE 1: 8 sensors CASE 2: 4 sensors CASE 3: 2 sensorsModeNo. (%) f (Hz) (%) f (Hz) (%) f (Hz)1 4.99 0.92 4.99 0.92 4.99 0.922 5.01 2.73 5.03 2.73 5.12 2.733 5.00 4.45 5.00 4.45 5.01 4.454 5.00 6.02 5.00 6.02 5.00 6.025 5.00 7.38 5.00 7.38 5.01 7.386 5.00 8.49 5.00 8.49 5.00 8.497 5.00 9.31 5.00 9.31 5.00 9.328 5.00 9.82 5.00 9.82 4.99 9.82

Table 2.3: Identified modal damping ratios and frequencies for case 4

CASE 4: 1 sensorat the first floor at the forth floor at the roofMode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz)

1 5.03 0.92 5.05 0.92 4.89 0.922 4.89 2.73 5.00 2.74 5.05 2.733 4.98 4.45 5.02 4.45 5.00 4.454 4.98 6.02 5.00 6.02 5.02 6.025 5.03 7.38 5.02 7.38 5.02 7.386 5.02 8.49 4.99 8.49 5.00 8.507 5.08 9.32 5.01 9.37 5.02 9.308 4.93 9.82 4.96 9.82 4.97 9.82

28

Figure 2.7 Mode shapes of structural and identified model for case 1

29

It should also be mentioned that when there is only one sensor that is placed at the

roof, the modal properties can still be identified accurately as shown in Table (2-3).

The natural frequencies can be identified accurately as long as the sensor is placed at

a point that is observable, and the frequency resolution has to capture the frequencies

of the system.

Once the identification process has been completed and system realization is

obtained, it is essential to test the accuracy of the identified modes in predicting the

response time histories when subjected to different ground motions. In this case, the

structure is excited with a recorded time history from the El-Centro (1940)

earthquake. The response of the system is predicted using the result of the realization

with the identified modes for the case of four sensors. Figure 2.8 shows the

acceleration time histories for the predicted and the actual accelerations of the

second, fourth, sixth, and eighth floor, respectively. Figure 2.9 shows the time

histories of the actual and the predicted relative accelerations on the eighth floor and

confirms that the predicted response of such an identified model is very close to the

actual response (with the maximum relative error less than 10 percent).

30

Figure 2.8 Computed vs predicted acceleration responses

31

Figure 2.9 Absolute errors in response predictions of the identified models (8-storey shear building with four sensors)

32

CHAPTER 3

NON-LINEARITY AS A DAMAGE INDICATOR

When a structure is subjected to an extreme event such as an intense ground shaking

it will deform into its inelastic range. During an earthquake, structures undergo

oscillatory motion with reversal of deformation and the force-deformation plots show

hysteresis loops because of inelastic behavior. The force-deformation relation would

be linear at small deformation but would become nonlinear at large deformation. In

this study, any deviation from linearly-elastic behavior is regarded as an anomaly and

considered as a result of the extreme event and hence as ‘damage’. So the first part of

this chapter discusses the general characteristics of a mathematical model called the

Bouc-Wen model providing an analytical expression for the hysteretic behavior of an

inelastic system. This will be used in the next chapter to simulate the measured

vibration data of an inelastic system. The next part of the chapter reviews the

literature on possible damage indicators and presents the one employed herein this

study.

3.1 Hysteretic Behavior

Over the years considerable attention has been given to the identification of linear

systems. However, most real-life systems inherently show nonlinear dynamic

behavior. Consequently, the use of linear models has its limitations. When

performance requirements are high, the linear model is no longer accurate enough,

and nonlinear models have to be used.

To describe the behavior of hysteretic processes several mathematical models have

been proposed (Baber and Noori 1985; Baber and Noori 1986). As stated earlier,

structures under severe loading usually become inelastic and exhibit nonlinear

hysteretic behavior. When the restoring force is plotted against the structural

33

deformation, inelastic behavior often manifests itself in the form of hysteresis loops.

Hysteresis depicts the hereditary nature of an inelastic system, in which the restoring

force depends not only on instantaneous deformation, but also on the past history of

the deformation. As a result, the hysteretic restoring force cannot be expressed by an

algebraic function of the instantaneous displacement and velocity. The history-

dependence makes the hysteretic systems more difficult to model and analyze than

other nonlinear systems.

Consider a structure idealized by an N degree-of-freedom system under a one-

dimensional earthquake ground motion. The equation of motion for the system can

be expressed as:

{ } )()()()()()( tFtulMtzKtxKtxtxM ggHE =−=++Γ+ &&&&& (3-1)

in which x(t) is a vector containing the displacement of each degree of freedom

relative to the ground, and z(t) is a vector containing the corresponding hysteretic

information for each element. M is the mass matrix, EK is the non-hysteretic

stiffness, HK is the hysteretic stiffness and Γ is the viscous damping matrix. The

ground motion, gF , is found by mapping the horizontal ground acceleration, gu&& , to

the horizontal degrees of freedom through the vector { }l and multiplying by M. As in

the single element case, the elastic ( EF ) and hysteretic components ( HF ) of the

structural restoring force can be separated such that:

)()( tzKtxKFFF HEHER +=+= (3-2)

so that the restoring force is a function of both x(t) and z(t). The equation of motion

for the system can be written in a nonlinear state-space format, as follows:

[ ])(

0

0

)()()(

0/0

00

)()()(

1111 tFMtztxtx

dxdzKMMKM

I

tztxtx

gHE

+

−Γ−−=

−−−− &&&&

(3-3)

where [ ]dxdz / is a non-square matrix function found by

34

[ ] [ ] rxBdrdzdxdz // = (3-4)

and [ ]drdz / is a diagonal matrix function of )(tx& and )(tz with entries ii drdz /

found in Equation (3-7), and rxB is a non-square matrix mapping the displacements

x(t) to the relative deformations r(t). The system can now be solved using any of a

number of numerical algorithms.

3.2 Bouc-Wen Model

Various hysteretic models for the restoring force of an inelastic structure have been

developed in recent years. The model used in this study is the Bouc-Wen (Wen,

1976) an approximate method for general random response analysis of hysteretic

systems. This model, widely used in structural and mechanical engineering, gives an

analytical description of a smooth hysteretic behavior. It represents a smooth

transition between elastic and yield states and includes a number of parameters,

allowing a mathematically tractable state-space representation capable of expressing

several hysteretic properties. The restoring force and the deformation are connected

through a nonlinear differential equation containing unspecified parameters. By

choosing the parameters suitably, it is possible to generate a large variety of different

shapes of the hysteresis loops.

As mentioned before, the restoring force, iRF )( for a single nonlinear element i

decomposed into two parts can be written as

)()1()()()( tzkxxktzKtxKFFF hbaTHEHER αα −+−=+=+= (3-5)

where α is the ratio of post-yielding to pre-yielding stiffness and Tk is the pre-

yielding stiffness. The variable x(t) is the relative deformation, ax and bx are the

absolute displacements at coincident end nodes a and b respectively, and z(t) is the

corresponding variable introduced to describe the hysteretic component. The force-

deformation curve is described by

35

]))sgn((A[ azzxxdrdzxz &&&& γδ +−== (3-6)

where A, δ , γ , and a are shaping parameters (Wen, 1976) and the term in square

brackets, drdz / , describes the hysteretic curve. Since we require drdz / to be unity

at small values of z, then A = 1. The yield displacement aY /1)( −+= γδ ; taking δ and

γ as equal, Equation (3.2) can now be written as:

+−=

a

Yzzxxz ))sgn(1(5.01 &&& (3-7)

The resulting hysteretic behavior described above is a stable force-deformation

curve. The use of constant strain-hardening with the stable hysteretic loop ignores the

presence of cyclic hardening and does not permit modeling of deterioration due to

local instabilities. These effects could be captured through modification of the above

equations.

Eqn. (3-2) can be integrated in close form which one can show that a hysteretic

relationship exists between z and x. The shape and scale of the hysteresis loop are

governed by the combination of the shaping parameters A,γ , and δ while the

smoothness of the force-displacement curve is controlled by a. Therefore by

adjusting these parameters, one can construct a variety of restoring forces, such

hardening or softening, narrow or wide-band systems (Wen, 1976). For example, for

a=1, the z-x curve is of the exponential type; restoring forces under periodic

displacement for A=1.0 and several combinations of the values of γ and δ are

shown in Figure 3.1. Figure 3.1(a) clearly depicts softening systems with different

degree of hysteresis. The case γ =0.5, δ =0.5, can be used as a model for an

elastoplastic system with smooth transition. Figure 3.1(b) describes, on the other

hand, hardening systems with different degree of energy dissipation. It should be

mentioned that when γ =δ =0 the relation between z and x, as shown in Figure

3.1(a), is linear.

36

Figure 3.1: The effect of shaping parameters on the hysterestic loop (a) linearly elastic systems (b) softening systems with different degree of hysteresis. (c) hardening systems with different degree of energy

dissipation

37

3.3 Damage Indicators

In analyzing a structure, performing a damage evaluation in detail at every point of

the structure is impossible. Several methods to determine an indicator of damage at

the global level have been presented in literature. Generally, these methods can be

divided into four categories of structural demand parameters (Luciana R. Barroso,

1999):

1. Strength demands (elastic and inelastic): If strength demands remain below the

yield capacity of the structure, the structural damage will be small. However, if

demands approach or exceed the ultimate strength of the structure, the damage to

structure may also be severe. Once yield is exceeded, strength capacity may become

reduced in future cycles into the inelastic range.

2. Ductility demand: Ductility is the ability of an element to deform inelastically

without total fracture. It is usually expressed in terms of a ratio between the

maximum deformation incurred during loading and the yield deformation. Any

deformation quantity may be used to determine the ductility demand.

3. Energy dissipation: Energy dissipation is the capacity of member to dissipate

energy through hysteretic behavior. An element has a limited capacity to dissipate

energy in this manner prior to failure. As a result, the amount of energy dissipated

serves as an indicator of how much damage has occurred during loading.

4. Stiffness degradation: Damage suffered during loading may result in a loss of

stiffness, and consequently longer natural periods. As determination of the

fundamental period is easily accomplished, this parameter can also be used as a

damage indicator.

An alternative way to classify damage indices is based on the response parameter

that can be used in determining the level of damage that the structure suffers during a

ground motion. The maximum deformation, the cumulative damage, and the

combination of maximum deformation and the cumulative damage can be used as the

response parameter of interest

38

3.3.1 Maximum Deformation Damage Indices

Maximum deformation damage indices are based on the peak value of a specified

deformation, such as element rotation or member displacement. Two of the earliest

and simplest forms of a damage index were the ductility and interstory drift.

Ductility Ratios

Ductility is defined as ability to deform inelastically without total fracture and

substantial loss of strength. In literature, it is commonly expressed as a ductility ratio,

Rµ , as defined below:

y

mR u

u=µ (3-8)

where mu is the maximum deformation experienced and yu is the yield deformation.

Interstory Drift

Interstory drift is defined as the relative interstory displacement of a story. Toussi

and Yao (1983) proposed a damage index defined as the ratio between the maximum

interstory displacement, i∆ , and the story height, h , as given below, and provided

guidelines for interpretation of results. This drift ratio, iδ , has been widely used in a

variety of structural systems as an indicator of the deformation demands on a

structure.

hi

i∆

=δ (3-9)

As with ductility ratios, peak interstory drift measures cannot take into account the

effects of repeated cycling, which can be a significant source of damage to structural

members.

39

3.3.2 Cumulative Damage Indices

Cumulative damage indices also are proposed. To capture the accumulation of

damage sustained during dynamic loading, the energy absorbed by the system during

loading is calculated. Banon and Veneziano, (1982) normalized the cumulative

deformations and proposed the ratio of the sum over all half-cycles of all the

maximum plastic deformations to the deformation at yield as a damage index.

∑=

=m

i y

pi

u

uNCD

1

(3-10)

where NCD is the normalized cumulative deformation, piu is the maximum plastic

deformations, and yu is the deformation at yield. Another index proposed by the

same researcher is the normalized cumulative dissipated energy (NHE) as a damage

index. The NHE is defined as the ratio of the total energy dissipated in inelastic

deformation to the elastic energy that would be stored in a member.

3.3.3 Combined Indices: Maximum Deformation and Cumulative Damage

An alternative to the previously mentioned two classes of damage indices is to have a

combined index of maximum deformation and cumulative damage. Park and Ang

(1985) defined a local damage index which combines the influence of the normalized

maximum deformation and absorbed hysteretic energy. The damage index is

expressed as the following linear combination:

∑+= dEuQu

uD

uruiPA

βmax, (3-11)

where maxu is the maximum deformation, uu is the ultimate deformation under

monotonic loading, rQ is the yield strength, dE is the incremental absorbed energy,

and β is a non-negative strength deteriorating parameter.

40

3.4 Proposed Approach

In this study we have used the hysteretic energy (EH) normalized with respect to the

maximum elastic stored energy (ES) as the damage index, DMI

S

H

EEDMI = (3-12)

The hysteretic energy dissipation, represented by the cumulative area of the force-

displacement hysteresis loops, indicates how much of the input seismic energy is

dissipated through various inelastic mechanisms associated with the member. Since

inelastic mechanisms such as formation of cracks and plastic hinges induce overall

structural degradation of the member, the amount of energy dissipated has been

frequently used in quantifying the damage of the member (Park et.al, 1985).

The input energy imparted to an inelastic system by an earthquake dissipated by

yielding. )(tEH can be expressed as

)(),()(0

tEdtuufutE S

t

RH −

= ∫ && (3-13)

where )(tES is the recoverable strain energy of the linear system and given by

[ ]0

2

2)(

)(ktftE S

S = (3-14)

where 0k is the initial stiffness of the inelastic system

41

CHAPTER 4

STRUCTURAL DAMAGE IDENTIFICATION PROBLEM

This chapter presents the application and results of the adopted approach assessing

damage by estimating the degree of nonlinearity present within a given response

measurement. Numerical simulation of a four-story structure subjected to ten strong

ground motion records in the presence of noise and nonlinearity was carried out for

each ground motion to simulate the measured vibration data. Four levels of damage

for each ground motion record are considered in this study.

4.1 System Description

A numerical example is presented herein to show the application of the adopted

approach. The structure is a two-dimensional four-story shear building, with a floor

mass and floor stiffness as shown in Figure 4.1. The modal parameters of such a

k = 7.5 × 104 kN/mm = 45.6 tons

Mode

No.

Damping

(%)

Frequency

(Hz)1 5 2.242 5 6.453 5 9.894 5 12.13

Figure 4.1: Structural model for numerical study

42

Figure 4.2: Earthquake ground motions

43

system also shown in the figure is used as benchmark for comparing the various

identification attempts. The restoring force relationship for the first floor is assumed

to be of the Bouc-Wen (Wen, 1976) type with parameters that result in a smooth

transition from elastic to plastic behavior. Other floors are assumed to remain

linearly elastic.

4.2 Modal Identification

Ten different earthquake ground motion records described in Table 4.1 and displayed

in Figure 4.2 have been used in this study. Since smaller events are not expected to

cause significant structural damage to engineered structures, priority in selection of

ground motions was given to records from events with moment magnitude as large as

possible. These time histories are applied only in one direction (the horizontal

direction) as the input excitation.

Table 4.1: Earthquake ground motion records

No.Earthquake

NameStation Name Date Mag

1 Anza Alpine - Fire Station 30-Oct-2001 5.1

2 Cape Campbell GNS: Wellington 18-Jan-1977 6.0

Reserve Bank A

3 San Simeon Cambria -Hwy 1 22-Dec-2003 6.5

Caltrans Bridge Grnds

4 Imperial Valley Aeropuerto Mexicali 15-Oct-1979 6.53

5 Superstition Hills Westmorland - Fire Station 24-Nov-1987 6.6

6 Big Bear Silent Valley 28-Jun-1992 6.6

7 Northridge Whitter Narrows 17-Jan-1994 6.7

8 Northridge Alhambra 17-Jan-1994 6.7

9 Marmara Region Duzce 17-Aug-1999 6.7

10 Landers Amboy, 1-Story Bldg 28-Jun-1992 7.5

44

These input ground motions are scaled to represent four different states of the

structural response and hence four different damage states: Level 1 represents the

elastic range with slight non-linearity, level 2 represents the transition stage from

elastic to plastic range before yielding, level 3 represents the yielding stage and

finally at level 4 significant inelastic action takes place and the ductility demand

imposed by the ground motion exceeds the allowable ductility. These four levels are

used to estimate four different damage states: no damage, slight damage, moderate

damage, and severe damage. In addition to the four level states, we also consider the

no damage state in the linearly elastic range when the values of shaping parameter of

the Bouc-Wen model are taken as ,0=α and 0=β . Figure 4.3 shows the force-

deformation curves corresponding to each level of the Northridge Earthquake

(Whitter Narrows record). The ductility demand for each ground motion-calculated

as the ratio of the maximum lateral displacement of the first floor under the premise

of linearly elastic behavior during the ground motion to the yield displacement is also

displayed in the figure. For each case sensor noise is simulated by contaminating the

analytically computed acceleration response of all four floors with white noise

having an RMS equal to 5% of the initial segment of the record.

Figure 4.4 shows a flowchart of the adopted approach for the detection of damage.

For each of the earthquake ground motion, the initial segments representing small

vibration amplitude were used for identifying the linear systems. The specified initial

segment of the recorded inputs and the responses is then processed with the ERA-

OKID approach to obtain the realizations of the system. Based on the obtained

realization, responses at sensor locations can be predicted for the entire duration of

the ground motion. The residual of the difference between the recorded data and the

predicted response under the premise of linear behavior can be used to extract useful

information on the damage state of the structure using

100)(

%2/1

12

2

×

−= ∑

=

N

i recorded

predictedrecorded

uuu

err&&

&&&& (4-1)

45

Figure 4.3: Force-displacement curves at different levels of the ground motion of the Northridge Earthquake (Whitter Narrows record)

46

Figure 4.4 Flowchart of the approach (Gunes and Bernal, 2004)

2/1

2

2)(

−= ∑

m

pm

yyy

ε

u

t10

s1 s2

ym

u

ym

Use mappingsto

predict outputfor S2

ResidualFormulation of

Mappings

+

yp

_

47

Initial examination of the earthquake ground motion data shown in Figure 4.5

suggests that for the Whitter Narrows record (dt=0.02 sec, N=2000 points), the initial

300 sample points, i.e. 6 seconds of recorded data can be used to represent the ‘small

event’ data to identify the system matrices and the modal properties of the healthy

system. Table 4.2 displays the set of earthquake ground motion records used in this

study for the numerical simulations. For each of these records, the initial segment

used for identifying the healthy system is tabulated in the last column of the table.

The first step of realization is determination of the system order. For all damaged and

undamaged states of each of the examined earthquakes except the Cape Campbell

earthquake, 8 significant singular values corresponding to 4 modes were identified.

Figures 4.6 and 4.7 display two sample plots of the Hankel matrix singular values for

the no damage state of the 1979 Imperial Valley earthquake and for the level 4 state

for the Cape Campbell earthquake respectively. The singular value decomposition of

the Hankel matrix for the 1994 Northridge earthquake indicates the order of the

system as 8 (4 modes). On the other hand, the singular value decomposition of the

Hankel matrix for the Cape Campbell earthquake indicates the order of the system as

6 (3 modes).

It should be mentioned that because of the presence of noise in the measurement, the

value for p (the number of observer Markov parameters) is required to be sufficiently

larger (at least four or five times) than the effective order of the system for

identification of the Kalman filter gain with accuracy (Juang, 1994). The value for p

is arbitrarily selected as p=20 when there is sensor at each floor to make the

maximum system order pm=80, which is higher than the anticipated system order of

n=4.

For each record at each level of scaling the ERA-OKID approach is carried out with

the specified initial segments of the recorded inputs and the responses and system

realizations are obtained. Table 4.3 displays the identified values of the natural

frequencies and the damping ratios for each of these cases when there is one sensor

at each floor. Comparison of these results with the exact ones indicates that the

maximum error in the identified frequencies is 7.53% and the maximum error in the

identified damping is 98%.

48

Figure 4.5: Ground motions recorded at the Whitter Narrows station during the Northridge Earthquake (1994)

Table 4.2: Data set of earthquake ground motions

No.Earthquake

NameStation Name

Sampling

dt

No of

Steps

Identification

Steps

1 Anza Alpine - Fire Station 0.01 3699 500

2 Cape CampbellGNS: Wellington

Reserve Bank A0.02 2724 600

3 San SimeonCambria - Hwy 1

Caltrans Bridge0.02 8063 1510

4 Imperial Valley Aeropuerto Mexicali 0.01 1955 280

5 Superstition HillsWestmorland-

Fire Station0.02 2999 180

6 Big Bear Silent Valley 0.02 1999 280

7 Northridge Whitter Narrows 0.02 2000 300

8 Northridge Alhambra 0.02 2999 200

9 Marmara Region Duzce 0.005 5437 1400

10 Landers Amboy 0.02 3999 550

49

Figure 4.6: Hankel matrix singular values and identified system with 8 sensors(Imperial Valley earthquake, no damage case).

Figure 4.7: Hankel matrix singular values and identified system with 8 sensors(Cape Campbell earthquake, level 4 of damage).

50

Table 4.3 Identified modal damping ratios and frequencies (with 4 sensors)

Anza EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.13 2.24 5.18 2.24 5.85 2.23 7.98 2.21 11.10 2.192 5.11 6.45 5.16 6.45 5.43 6.43 6.07 6.37 6.68 6.263 5.26 9.91 5.29 9.91 5.43 9.91 5.71 9.91 5.89 9.924 5.32 12.18 5.36 12.20 5.43 12.20 5.50 12.19 5.23 12.18

Cape Campbell EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.03 2.24 5.28 2.24 6.32 2.22 9.90 2.15 2.57 2.102 4.93 6.46 5.23 6.43 6.33 6.47 5.55 6.62 3.96 6.903 5.11 9.92 5.10 9.91 5.91 9.85 - - 6.85 10.634 5.87 12.26 5.82 12.31 6.11 12.13 - - - -

San Simeon EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.03 2.24 5.00 2.24 5.04 2.24 5.02 2.24 5.03 2.242 5.08 6.45 5.28 6.46 5.32 6.46 5.26 6.46 5.22 6.453 5.34 9.90 5.76 9.92 5.78 9.93 5.70 9.92 5.58 9.914 5.48 12.16 6.03 12.19 6.06 12.20 5.97 12.20 5.76 12.19

Imperial Valley EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 4.72 2.25 4.85 2.25 5.08 2.25 5.42 2.24 6.11 2.232 5.15 6.47 5.13 6.47 5.22 6.47 5.41 6.46 5.71 6.453 5.55 9.92 5.54 9.91 5.57 9.91 5.67 9.89 5.77 9.864 5.39 12.09 5.42 12.09 5.42 12.08 5.50 12.07 5.66 12.04

Superstition Hills EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.44 2.24 5.41 2.24 5.45 2.24 5.57 2.24 5.69 2.242 5.14 6.46 5.11 6.46 5.17 6.46 5.17 6.46 5.22 6.453 4.94 9.89 4.99 9.88 4.97 9.89 4.96 9.89 4.98 9.894 4.71 12.16 4.81 12.16 4.79 12.17 4.74 12.17 4.75 12.18

51

Table 4.3 (cont.)

Big Bear EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 4.98 2.24 5.14 2.24 5.86 2.23 8.20 2.21 11.52 2.182 5.03 6.46 5.04 6.45 5.14 6.43 5.80 6.37 7.29 6.313 5.12 9.89 5.13 9.88 5.12 9.85 4.99 9.78 4.80 9.714 5.21 12.14 5.17 12.14 4.99 12.12 4.54 12.06 4.11 11.98

Northridge Earthquake (Whitter Narrows Station)No Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.04 2.24 5.08 2.24 5.19 2.24 5.61 2.23 6.20 2.232 5.09 6.46 5.14 6.45 5.11 6.45 5.18 6.43 5.27 6.413 5.10 9.87 5.13 9.87 5.13 9.87 5.24 9.86 5.38 9.844 5.23 12.19 5.29 12.20 5.28 12.19 5.32 12.19 5.31 12.20

Northridge Earthquake (Alhambra Station)No Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.10 2.24 5.11 2.24 5.19 2.24 5.38 2.24 5.70 2.232 4.95 6.46 4.98 6.46 4.96 6.46 4.99 6.46 4.96 6.453 5.25 9.88 5.32 9.88 5.18 9.88 5.11 9.88 5.04 9.874 5.88 12.15 6.23 12.17 6.13 12.16 6.32 12.18 6.48 12.21

Marmara Region EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.10 2.24 5.14 2.24 5.38 2.24 6.23 2.20 9.57 2.142 4.94 6.46 4.94 6.45 5.24 6.42 6.02 6.34 5.10 6.383 5.38 9.90 5.26 9.89 5.47 9.87 6.74 9.96 5.47 10.044 8.23 12.32 8.64 12.43 8.88 12.39 9.54 12.28 7.96 12.49

Landers EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 4.99 2.24 5.08 2.24 5.41 2.23 6.73 2.22 9.00 2.192 5.08 6.46 5.06 6.46 4.95 6.44 4.93 6.40 5.17 6.363 5.11 9.90 5.16 9.90 5.16 9.89 5.63 9.88 6.12 9.884 7.81 12.50 7.90 12.51 7.42 12.43 7.62 12.44 7.33 12.39

52

Table 4.4 Identified modal damping ratios and frequencies (with 2 sensors)

Anza EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.35 2.24 5.41 2.24 6.12 2.23 8.33 2.22 11.45 2.192 5.25 6.44 5.31 6.43 5.60 6.41 6.34 6.35 7.04 6.243 6.83 10.02 9.90 10.02 7.27 10.02 8.90 10.13 11.29 10.424 6.35 12.23 6.40 12.22 6.60 12.22 7.72 12.26 10.49 12.34

Cape Campbell EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.02 2.24 5.26 2.24 6.38 2.22 10.45 2.16 - -2 5.07 6.46 5.40 6.44 5.92 6.46 3.28 6.87 - -3 4.86 9.92 5.09 9.95 6.36 9.92 - - - -4 5.87 12.09 6.22 12.07 6.30 11.83 - - - -

San Simeon EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.01 2.24 4.94 2.24 5.02 2.25 4.99 2.24 5.00 2.242 5.14 6.45 5.35 6.46 5.40 6.46 5.35 6.45 5.30 6.453 5.66 9.91 6.24 9.93 6.30 9.94 6.10 9.93 5.99 9.924 5.46 12.15 6.05 12.17 6.02 12.18 5.92 12.18 5.74 12.16

Imperial Valley EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.86 2.25 5.96 2.25 6.21 2.25 6.53 2.24 7.24 2.232 5.21 6.47 5.20 6.47 5.29 6.47 5.48 6.46 5.76 6.453 6.08 9.86 6.03 9.86 6.08 9.85 6.00 9.83 5.73 9.784 5.96 12.14 5.98 12.14 6.00 12.13 6.12 12.12 6.35 12.11

Superstition Hills EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.58 2.25 5.48 2.25 5.58 2.25 5.74 2.25 5.87 2.252 5.18 6.45 5.15 6.46 5.20 6.46 5.21 6.45 5.25 6.453 5.21 9.90 5.23 9.90 5.22 9.91 5.21 9.90 5.23 9.904 5.37 12.14 5.40 12.13 5.41 12.15 5.37 12.16 5.44 12.16

53

Table 4.4 (cont.)

Big Bear EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.00 2.24 5.15 2.24 5.88 2.23 8.19 2.21 11.46 2.182 5.02 6.46 5.02 6.45 5.10 6.43 5.67 6.37 7.12 6.303 5.17 9.86 5.15 9.84 5.09 9.81 4.99 9.71 5.21 9.644 5.19 12.17 5.13 12.17 4.99 12.15 4.62 12.10 4.43 12.05

Northridge Earthquake (Whitter Narrows Station)No Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.19 2.24 5.23 2.24 5.33 2.24 5.75 2.23 6.37 2.232 5.06 6.46 5.11 6.46 5.10 6.45 5.22 6.43 5.36 6.413 5.53 9.87 5.57 9.87 5.57 9.87 5.73 9.85 5.90 9.834 5.44 12.18 5.49 12.18 5.49 12.18 5.62 12.18 5.60 12.16

Northridge Earthquake (Alhambra Station)No Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.09 2.24 5.11 2.24 5.19 2.24 5.38 2.24 5.71 2.232 5.00 6.47 5.03 6.47 5.00 6.47 5.07 6.46 5.15 6.453 5.15 9.90 5.22 9.90 5.09 9.90 4.97 9.89 4.97 9.884 6.12 12.17 6.43 12.19 6.28 12.19 6.35 12.20 6.51 12.22

Marmara Region EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.12 2.24 5.09 2.24 5.20 2.24 5.67 2.23 6.54 2.222 5.23 6.46 5.26 6.45 5.19 6.45 5.04 6.43 4.23 6.413 6.21 9.93 5.87 9.92 6.38 9.90 6.83 9.87 6.58 10.004 7.99 12.42 7.89 12.45 7.77 12.48 6.78 12.38 7.57 12.13

Landers EarthquakeNo Damage Level 1 Level 2 Level 3 Level 4Mode

No. (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f (Hz) (%) f(Hz)1 5.02 2.24 5.12 2.24 5.44 2.23 6.76 2.21 9.02 2.182 5.35 6.45 5.33 6.45 5.22 6.44 5.06 6.38 5.48 6.333 5.28 9.99 5.30 9.99 5.28 9.96 5.47 9.91 5.40 9.954 8.70 12.53 8.74 12.52 8.00 12.41 6.90 12.19 6.67 12.12

54

For comparative purposes, the results for the case where there are sensors at the

second and fourth floors are also computed and the results of identified values of the

natural frequencies and the damping ratios are displayed in Table 4.4. Comparison of

these results with the exact ones indicates that the maximum error in the identified

frequencies is 7.65% and the maximum error in the identified damping is 129%.

As can be seen, for the case of full sensors the accuracy of the identification (at least

in frequency) is excellent. Even with only two sensors, the frequencies values are

identified quite successfully, which highlights the efficiency of the adopted

approach. These results clearly indicate that while the estimates of frequency remains

almost constant, nonlinearity manifests itself with high damping ratio estimates. The

estimation of damping is more problematic than that of system frequencies. This is a

well-known problem in system identification. A large number of data are typically

required to extract damping information accurately. Unfortunately, earthquake

response data is often short in duration, making the identification of this parameter

more difficult.

After the identification is carried out with the initial segments of the response, the

response of the system for the rest of the duration is predicted using the result of the

realization with the identified modes. Figure 4.8 and 4.9 show a sample of the

acceleration time histories for the predicted and the actual accelerations of the roof

when the structure is subjected to Whiter-Narrows record at different levels. The

predicted response is very close and almost identical to the actual response as long as

the response remains linear as shown in Figure 4.7. As the response changes from

linear to nonlinear, the realization is no more capable of predicting the structural

response as shown in Figure 4.8.

Table 4.5 summarizes the results of the simulations for the ten earthquake records

scaled at different levels of peak ground acceleration. The maximum lateral

displacement of the first floor in the case of linearly elastic force-deformation

relationship )( mlu , ductility demand )( dµ , damage index (DMI), and percent error in

the predicted response (%err) are tabulated below. An examination of the table

indicates that the damage index increases with the increase of damage level. It can be

55

Figure 4.8: Computed vs. predicted acceleration response for the Whitter Narrows record (level 1). (a) the initial segment of the record (b) the entire duration of the record

56

Figure 4.9: Computed vs. predicted acceleration response for the Whitter Narrows record (level 3). (a) the initial segment of the record (b) the entire duration of the record

57

Table 4.5: Results of case 1 (4 sensors) for the records at different levels of theground motion

Anza Earthquake Cape Campbell EarthquakeGroundMotionLevel uml (cm) dµ DMI % err uml (cm) dµ DMI % err

1 0.20 0.10 0.01 3.79 0.23 0.10 0.014 8.782 1.25 0.60 1.04 7.77 1.20 0.60 1.64 29.573 4.00 2.00 24.92 15.58 4.00 2.00 59.06 65.834 9.00 4.50 132.07 21.02 9.00 4.50 388.06 69.23

San Simeon Earthquake Imperial Valley EarthquakeGroundMotionLevel uml (cm) dµ DMI % err uml (cm) dµ DMI % err

1 0.25 0.10 0.02 9.06 0.20 0.10 0.01 7.022 1.20 0.60 1.64 38.09 1.10 0.60 1.18 22.373 4.10 2.00 42.64 81.96 4.00 2.00 35.94 52.884 9.00 4.50 207.74 108.42 9.00 4.50 223.10 74.77

Superstition Hills Earthquake Big Bear EarthquakeGroundMotionLevel uml (cm) dµ DMI % err uml (cm) dµ DMI % err

1 0.20 0.10 0.011 10.86 0.23 0.10 0.0162 7.072 1.20 0.60 0.76 33.95 1.20 0.60 1.52 25.823 4.00 2.00 20.39 80.16 4.15 2.10 45.88 56.824 9.00 4.50 138.91 111.51 9.00 4.50 194.33 60.81

Whitter Narrows Station Alhambra StationGroundMotionLevel uml (cm) dµ DMI % err uml (cm) dµ DMI % err

1 0.23 0.10 0.018 10.35 0.27 0.10 0.0149 11.582 1.10 0.60 1.39 39.91 1.25 0.60 1.47 47.483 4.20 2.10 49.44 85.93 4.10 2.10 44.31 96.004 9.00 4.50 186.55 102.92 9.00 4.50 240.69 115.93

Marmara Region Earthquake Landers EarthquakeGroundMotionLevel uml (cm) dµ DMI % err uml (cm) dµ DMI % err

1 0.21 0.10 0.01 9.32 0.25 0.10 0.02 9.672 1.12 0.60 0.89 37.42 0.12 0.60 1.69 35.103 4.10 2.10 31.47 88.27 4.20 2.10 47.45 70.064 9.00 4.50 207.38 99.42 9.00 4.50 256.04 83.77

58

0

50

100

150

200

250

300

350

400

0 20 40 60 80 100 120 140% error

DM

I

Anza Cape Cambell San Simon Imperial Valley Superstition Hills

Big Bear Whitter Narrow s Alhambra Duzce Landers

Figure 4.10: Percent error in the predicted response of the first floor versus damage index

59

seen that as the maximum lateral displacement of the first floor increases the ductility

demand, the damage index, and the relative error increase, indicating a larger

dissipation of input energy through action.

The relation between the prediction error of the acceleration response of the first

floor and the hysteretic energy normalized with respect to the maximum elastic

stored energy are depicted in Figure 4.10. Note that even for the level 1 of damage

state (DMI is around zero) which means there is no damage in the structure, the

cumulative error on the prediction error on the prediction estimates can get as much

as 11.58 % in the case of the 1994 Northridge Earthquake (Alhambra record). This is

mainly due to the presence of noise in the measurements. When the data length is

sufficiently long, the ERA/OKID approach provides excellent estimates of the modal

parameters. Since only the initial segment of the data is used which is a relatively

short duration to identify the healthy system, the error in the predicted response is

within the expected level of accuracy. Furthermore, although for ground motion level

1 one operates with a ductility demand of 10% since transition from the linearly

elastic range to the plastic range is a smooth one in the Bouc-Wen model. Hence, a

very mild nonlinear behavior that the identification procedure is not taking into

account. As the non-linearity increases and inelastic action grows, percent error in

the predicted response grows exponentially with the damage index.

60

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

Currently, vibration-based damage detection is an area of significant research activity.

Several methods have been proposed to assess structural health using changes in

vibration characteristics. In these methods, the structure is assumed to behave linearly

both before and after damage. The research presented in this thesis, however, adopts

an alternative approach to assess the state of the system for post-earthquake damage

detection. The early segment of the recorded data is used to identify the nominally

healthy system. The relative error can be used as a good criterion in assessing the

state of the structure. The main reason for considering this approach is the fact that it

does not rely on the assumption of linearity of the structure, while most of the

existing vibration-based methods do, although most real structures demonstrate

nonlinear behaviour. In the context of damage detection technology this technique

provides an initial screening methodology which can tell whether damage is present

in the structure or not. The amount of the input energy dissipated through hysteretic

action is used as the response parameter for determining the level of damage. This

damage index and the relation between the residual can be used to quantify the level

of damage. The Bouc-Wen model, widely used to have a mathematical description of

hysteretic patterns appearing in structural engineering systems, is instrumented for

modeling the hysteretic behavior of the structural system in the numerical

simulations.

This proposed approach was employed to a finite element model of a 2-dimensional

four-story shear building excited with earthquake ground motions. Ten earthquake

records have been selected such that each ground motion has initial low-vibration

amplitude and a proceeding strong motion segment. The first floor is assumed to

61

display hysteretic behavior of the Bouc-Wen type with parameters that result in a

smooth transition from elastic to plastic behavior. Other floors are assumed to remain

linearly elastic. Various issues relevant to the implementation of this methodology are

investigated. The first issue considered in this study is the accuracy of the

methodology when different sensors were used in calculating the modal parameters.

The results are computed for the case when there is sensor at each floor and also for

the case when there are sensors at the second and fourth floors. Comparison of the

estimate of the modal parameters with the exact ones indicate that while the

estimates of frequency remain almost constant, the nonlinearity manifests itself with

increased damping ratio estimates. The estimation of damping is more problematic

than that of system frequencies and this is a well-known problem in system

identification. In fact, a large number of data are typically required to extract

damping information accurately. Unfortunately, earthquake response data is often

short in duration, and with this methodology one is using the initial segment of the

earthquake which is even short, or which makes the identification of the modal

parameter more difficult. Damage detection was accurately carried out even with

two sensors one on the second floor and one on the roof.

In conclusion, a new technique for detecting damage is proposed and experimentally

validated through numerical simulations. The results of the numerical simulations

have shown that the system realization methodology (ERA-OKID) is robust to noise

and nonlinearities reflect themselves with the artificially high damping estimate

especially for the fundamental vibration mode. The percent error in the predicted

response grows exponentially with the damage index as the non-linearity grows. It is

also shown that the realized model mimics the behavior of the healthy system very

well during the linear portion of the response when subjected to a different ground

excitation. However, as nonlinear action takes place, the realization is no more

capable of predicting the structural response. The discrepancy between this predicted

response on the premise of linear behavior and the measured response data is then

used to predict whether the system has suffered any damage or not.

62

5.2 Recommendations for Future Work

Based on the conclusions presented herein, the following suggestions for future

research are made:

• The model presented in this thesis is based on observations made regarding

the hysteretic behavior of steel buildings. The hysteretic behavior of

reinforced concrete buildings has not been studied. To apply the approach

implemented here to reinforced concrete buildings, future study has to be

performed with a model that accurately represents the hysteretic behavior of

reinforced concrete type of structure.

• The numerical simulations in this thesis assumed only the first floors getting

damage. The methodology can be tested for multiple damage scenarios and as

damage progresses to the other floors.

• In this thesis, the damage index (DMI) is used only to predict whether the

structure has suffered any damage or not. In order to characterize the damage

that is to locate and if possible to quantify the severity, an additional criterion

has to be implemented to the proposed methodology.

• Finally, the approach adopted here providing a means of identifying the

existence of damage although is tested with numerical simulations with

realistic level of noise still awaits validation using measured data.

63

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68

APPENDEX

MATLAB CODES

This appendix contains Matlab codes for simulating the response of hysteretic

structures when subjected to earthquake ground motions. The first code is a sample

which is used to demonstrate the steps related to the approach. The code uses the

ERA-OKID for modal parameter identification. In addition, it uses the other two

codes for nonlinear modeling. The second and third codes are used for a nonlinear

shear building model with element of Bouc-Wen type.

A.1 A Matlab Code for the Simulation

clear;

% time step and number of steps in the analysis

s=[1 2]';

nsteps=2000;

load whitter1 % Northridge EQ Whitter Narroes Record

px=1*(whitter1(1:nsteps+1))';

dt=0.02;

% Integration time step

dti=0.0;

m=[45600 45600 47776 45600 45600 47776 45600 45600 47776 45600 ...

45600 47776];

% scaling factors to convert mass to weights for p-delta computations

pd=[386.4*0]; % applies to all levels;

% radius of gyration for p-delta

rg=[0]; % uses the values of the radius of gyration for the mass

% interstory heights

69

h=[144]; % same in every level

% global coordinates of the centers of mass

ccm=[0 0;0 0;0 0;0 0];

% modal damping ratios

mdr=0.05*ones(1,12);

% elements

aw=0;

bw=0;

fy=7.5*10^7*0.02;

elp1=[0 0 0 1 aw bw 1 7.5*10^7 fy;-120 0 90 1 0 0 1 100 fy;...

120 0 90 1 0 0 1 50 fy];

elp2=[0 0 0 2 0 0 1 7.5*10^7 fy;-120 0 90 2 0 0 3 100 fy;...

120 0 90 2 0 0 3 50 fy];

elp3=[0 0 0 3 0 0 1 7.5*10^7 fy;-120 0 90 3 0 0 3 100 fy;...

120 0 90 3 0 0 3 50 fy];

elp4=[0 0 0 4 0 0 1 7.5*10^7 fy;-120 0 90 4 0 0 3 100 fy;...

120 0 90 4 0 0 3 50 fy];

elp=[elp1;elp2;elp3;elp4];

% Deterioration

deter1=3*ones(12,1);

deter2=1*ones(12,1);

deter3=1*ones(12,1);

deter=[deter1 deter2 deter3];

% load is base excitation

pe=[0;0;0];

% applied loads

p=zeros(12,nsteps+1);

pl1=-m(1:3:12)'*px;

p(1,:) =pl1(1,:);

p(4,:) =pl1(2,:);

p(7,:) =pl1(3,:);

p(10,:)=pl1(4,:);

% displacement and rotation limits to terminate the run

70

maxd=20;

maxr=0.5;

term=[maxd maxr 10000];

% Run

[tt,d,v,ac,f,drift,bu,bum,per,msh,worki,ENERGY]=dash3dx(m,pd,rg,h,...

ccm,mdr,elp,deter,dt,dti,pe,p,nsteps,term);

%Evaluate the floor drifts

drift(:,1)=d(:,1);

drift(:,2)=d(:,4)-d(:,1);

drift(:,3)=d(:,7)-d(:,4);

drift(:,4)=d(:,10)-d(:,7);

[DRIFT,i]=max(max(abs(drift))*100);

max(abs(d(:,1)));

input=p(1:3:10,:);

output=ac(:,1:3:10)';

% Define output noise

randn('state',s);

noiseout=randn(size(output));

noiseout=noiseout*std(output(1,1:300))*0.05;

% Define input noise

randn('state',s);

noisein=randn(size(input));

noisein=noisein*std(input(1,:))*0.0;

% Add noise

inputt=input+noisein;

outputt=output+noiseout;

%Do the identification with the linear portion

nsteps=300;

input2=inputt(:,1:nsteps+1);

input2=input2';

output2=outputt(:,1:nsteps+1);

output2=output2';

[A,B,C,D]=okid(4,4,dt,input2,output2,'lq',20);

71

%Prediction

lamda=eig(Ac);

n=length(pl1);

t=[0:n-1]*dt;

pll=pl1';

Y=dlsim(A1,B1,C,D,pll);

Y=Y';

[w_es,zai_es]=ceig(lamda);

f_es=sort(w_es(1:2:7))/2/pi;

%error

et=(outputt(1,:)-Y(1,:)).^2;

et2=sqrt(sum(et)./sum((outputt(1,:)).^2))*100;

%damage index

DMI=max(worki(:,1))*6;

hysteret=max(worki(:,1))

fre=1./per;

fex=fre(9:12);

error_fre=(f_es-fex)./fex*100;

error_zai=(zai_es-0.05)/0.05*100;

%plot the measured vs predicted acceleration responses using the

% initial segment of the record

figure(2)

plot(t,outputt(1,:))

hold on

plot(t,Y(1,:),'r')

xlabel('Time (sec)')

ylabel('a1')

legend('measured','predicted')

%plot the measured vs predicted acceleration responses using the

% entire duration

72

figure(3)

plot(t(1:300),outputt(1,1:300),':')

hold on

plot(t(1:300),Y(1,1:300),'r')

xlabel('Time (sec)')

ylabel('a1')

legend('measured','predicted')

% plot the restoring force vs drift of the first floor

figure(4)

plot(drift(:,1),f(:,1))

xlabel('Displacement (m)')

ylabel('Restoring Force (N)')

title('Force-displacement curve')

73

A.2 A Matlab Code for non-linear modeling (Dash3dx.m)

% **********************************% * DASH3DX.m *% * *% * *% * *% **********************************

% DESCRIPTION% Dynamic response of 3D shear buildings. An arbitrary number of resisting% elements per level - each with a potentially different restoring force relationship% can be considered. The restoring force relationships may be elastic (either linear,% nonlinear softening or nonlinear hardening) or hysteretic. Deterioration of%strength, stiffness, or both, as a function of the hysteretic energy dissipated, can be%considered in the analysis.

% The current version assumes damping to be classical with damping ratios for each%of the 3*N modes provided by the user,(modes are computed using the initialtangent stiffness of the elements). A more general damping model can be easily%considered by blocking the appropriate section of the program and reading the%damping matrix directly. Translational and torsional secondary effects can be%considered. The formulation allows for arbitrary location of the center of mass in%each level (i.e. - the centers of mass need not be on a common vertical axis).

% An arbitrary 3 component ground acceleration plus arbitrarily defined loads at%each one of the 3*N dof of the model can be considered in a single run.

% INTEGRATION OF THE EQUATIONS OF MOTION% The equations are integrated by casting them in state form and envoking the built%in MATLAB integrator ODE23. Two options are% available, see description of dti in the input list.

% ****************************** USERS GUIDE *****************% The program is a MATLAB function with the input output entries shown

% function [tt,d,v,ac,f,drift,bu,bum,per,msh,worki]=% dash3d(m,pd,rg,h,ccm,mdr,elp,deter,dt,dti,pe,p,nsteps,term)

% The DOF are ordered X, Y, ZETA starting at the first level.% Counterclockwise rotation is positive.% The origin of the global coordinates located arbitrarily.% The X-Y plane is horizontal.% N = number of stories.% NE = number of elements (total in all stories).

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% INPUT DATA% m = diagonal of the mass matrix (3Nx1);% pd = scaling factor applied to the masses to compute weights for P-delta effects -% level#1 is the first element (Nx1).% If a single value is input it is assumed to apply to all masses. Input [0] to% neglect P-delta.% rg = radius of gyration for p-delta calculations (N x 1). If a single (nonzero) value% is input it is assumed to apply to all levels. Input rg=[0] to make the values of rg %the same as the mass radius of % gyration of the stories.% Definition: Consider Rj and hj as the total axial force and the interstory height in% level j. If the diaphragm in level j is rotated a unit about an axis that coincides with% Rj the p-delta torque generated in level j = (Rj/hj)*rgj^2.% h = story heights - level#1 first (Nx1)(used only for p-delta computations). If a% single value is given it is assumed to be the same at all levels. When p-delta is not% considered one can input any (nonzero) value for h.% ccm = coordinates of the centers of mass (N x 2) (X in column #1,Y in column#2);% mdr = modal damping ratios (first mode first) (3Nx1);% elp = element description matrix (NE x9). Each row refers to one element; the% entries in each row are: [X Y Fi LV alw bew nw k0 Zc], where:%% X and Y = coordinates of a point in the line of action of the force in the element.% Fi = angle from the positive global X axis to the line of action of the force in the% element (measure counterclockwise) (in degrees).% LV = level at the upper side of the element.% alw, bew and nw = parameters in Bouc-Wen's model (see next section for a quick% guide on how to specify these parameters).% k0 = initial element stiffness.% Zc = force used to scale the restoring force. Equal to the strength in% softening models and to the force when the stiffness is twice of% the stiffness at the origin in hardening models.%% deter = matrix providing information on the deterioration of strength, stiffness% or both, as a function of the dissipated hysteretic energy. Each row% corresponds to one element (in the order of elp).

% Column #1 - ductf = coefficient used to compute the hysteretic energy at which% the element deterioration levels off.% In particular, the hysteretic energy is normalized by% Zc^2*(ductf)/k0. One can readily show that "ductf" can% be interpreted as an "effective ductility at failure". A% number in the order of two to 3 times the ductility at% failure for monotonic loading is reasonable.%% Column #2 - FCL = fraction of original strength when the normalized% dissipated energy is >=1. For no strength deterioration% input FCL=1.%% Column #3 - FSL = fraction of original stiffness at zero force when the% normalized dissipated energy is >=1. For no stiffness

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% deterioration input FSL=1.%% ** Notes on deter **% 1) The reduction of stiffness is applied only to the loading branch.% 2) The function used to modify the strength or the stiffness has% been taken as% f(wb)=1-(3wb^2-2*wb^3)*(1-F)% where wb is the normalized hysteretic energy and% F = FCL for strength and FSL for stiffness.%% dt = time discretization of the prescribed loading.% dti = integration time step. Input dti=0 to perform a variable time step% solution. In the option dti=0 the time step is adjusted automatically% during the solution - the execution time in this option is typically% significantly longer than when a constant time step is used. If there% is uncertainty about the size of the time step needed to attain% sufficient accuracy use dti=0;% pe = time history of earthquake accelerations (3 x ns1). First two rows% are the X and Y translational components and the third is a% rotational input (any or all can be zero).% p = time history of applied loads (3*N x ns2).% nsteps = the solution is obtained from t= 0 to t = nsteps*dt% (nsteps <= to the largest of ns1-1 or ns2-1).% term = termination criteria (1x3). The entries are [maxd maxr maxw]:% maxd - maximum displacement - program stops if the displacement at any DOF% response reaches this value.% maxr - rotation of a floor slab that terminates the analysis.% maxw - normalized hysteretic energy that terminates the analysis.%% RESTORING FORCE CHARACTERISTICS%% Softening bew = +1 Zc = Strength% Elastic Behavior (alw = 0) Linear bew = 0 Zc = Any value% Hardening bew = -1 Zc = Force when kt = 2ko%% Softening (alw+bew) = + Zc = Strength% Loading Linear (alw+bew) = 0 Zc = Any value% Hardening (alw+bew) = - Zc = Force when kt =2ko% Inelastic Behavior (alw ~=0)% Softening (alw-bew) = +% Unloading Linear (alw-bew) = 0% Hardening (alw-bew) = -%% The values of +1 and -1 are specified for convenience, any positive or negative% values will lead to the same results. nw affects the smoothness of the restoring% curves. A typical choice to represent stable steel elements that undergo inelastic% response is% alw = bew = 0.5 and nw = 3 or 5.%

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% OUTPUT% The dynamic response is ordered in columns with the rows being subsequent time% steps.% tt = time - note that the time interval is dt when dti=0 and dti when dti ~= 0.% d = displacements of the diaphragms at the center of mass.% v = velocities of the diaphragms at the center of mass.% ac = absolute accelerations of the diaphragms at the center of mass (the% accelerations are absolute even when the loading is only base excitation).% f = shear forces in the elements (same order as input).% drift = relative displacement at the various resisting elements (in same order as f).% bu = fundamental buckling eigenvalue (scaling of vertical loads to attain elastic% instability).% bum = fundamental buckling eigenvector.% per = natural periods for initial tangent stiffness (no geometric stiffness).% msh = mass normalized mode shapes for initial tangent stiffness% (no geometric stiffness).% worki = normalized hysteretic energy in each of the elements (see% description of "deter" Column #1 for the normalizing value).% ENERGY = a matrix containing the history of several energy quantities – see% listing.%% ****************************************************************

function [tt,d,v,ac,f,drift,bu,bum,per,msh,worki,ENERGY]=dash3dx... (m,pd,rg,h,ccm,mdr,elp,deter,dt,dti,pe,p,nsteps,term);

% Extract the termination criteriamaxd=term (1);maxr=term (2);maxw=term (3);

% Form the transformation matrix and evaluate the initial stiffnessN=max(elp (:,4));[NE,dum]=size(elp);k0=elp (:, 8);TS=zeros (NE, 3*N);for j=1:NE; lv=elp(j,4); xt=elp(j,1)-ccm(lv,1); yt=elp(j,2)-ccm(lv,2); fi=elp(j,3)*pi/180; dtt=[cos(fi) sin(fi) (xt*sin(fi)-yt*cos(fi))]; if lv~=1; xb=elp(j,1)-ccm((lv-1),1); yb=elp(j,2)-ccm((lv-1),2); db=[cos(fi) sin(fi) (xb*sin(fi)-yb*cos(fi))]; dd=[-db dtt]; n1=(lv-2)*3+1; n2=n1+5;

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else; dd=dtt; n1=1; n2=3; end; TS(j,n1:n2)=dd; end;

% Initial StiffnessK=TS'*diag(k0)*TS;

% Mass matrix. M=diag(m);

% Eigensolution and Damping (damping taken as modal damping - if an arbitrary% [C] is desired block the line that defines C and enter it directly. [aa,bb]=eig(K,M); mn=aa'*M*aa; mn=sqrt(mn); aa=aa*inv(mn); [bb,I]=sort(diag(bb)); aar=[]; for j=1:3*N; aar(:,j)=aa(:,I(j)); end; D=2*sqrt(bb).*mdr'; C=M*aar*diag(D)*aar'*M; C=real(C); per=real(2*pi./sqrt(bb)); msh=real(aar);

% Geometric Stiffness.npd=length(pd);nh=length(h);nrg=length(rg);

if npd==1; pd=pd*ones(1,N);end;

% perform calculations only if gravity load is prescribed if cumsum(pd)~=0;if nh==1; h=h*ones(1,N);end;count=0;mm=[];J=[];for j=1:3*N

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count=count+1; if count==1; mm=[mm m(j)]; count=-2; J=[J m(j+2)]; else end; end; pd1=mm.*pd; R=fliplr(cumsum(fliplr(pd1))); r2=J./mm; if nrg==1; rg=rg*ones(1,N); end; if sum(rg)==0; rg=sqrt(r2); end;

% global cordinates of the total vertical load at a given level. for j=1:N; myy=sum(pd1(j:N).*ccm(j:N,1)'); mxx=sum(pd1(j:N).*ccm(j:N,2)'); xpd(j)=myy/R(j); ypd(j)=mxx/R(j); end;

for j=1:N; if R(j)==0; disp(' zero total load detected at level') j break end; end;

for j=1:N; if h(j)<=0; disp(' the height of level'); j disp(' has been input as zero or negative'); disp(' change to positive'); break; end; end;

% initialize KG=[];

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% assemblefor j=1:N; AB=[]; jj=j+1;

if jj==N+1; R(jj)=0.0; h(jj)=1; xpd(jj)=1; ypd(jj)=1; rg(jj)=0; ccm(jj,:)=[1 1]; else end;

kg11=(R(j)/h(j)+R(jj)/h(jj)); kg12=0; kg13=-(R(j)*(ypd(j)-ccm(j,2))/h(j)+R(jj)*(ypd(jj)-ccm(j,2))/h(jj)); kg14=-R(jj)/h(jj); kg15=0; kg16=R(jj)*(ypd(jj)-ccm(jj,2))/h(jj);

kg21=kg12; kg22=(R(j)/h(j)+R(jj)/h(jj)); kg23=(R(j)*(xpd(j)-ccm(j,1))/h(j)+R(jj)*(xpd(jj)-ccm(j,1))/h(jj)); kg24=0; kg25=-R(jj)/h(jj); kg26=-R(jj)*(xpd(jj)-ccm(jj,1))/h(jj);

kg31=kg13; kg32=kg23; kg33=R(j)/h(j)*((ypd(j)-ccm(j,2))^2+(xpd(j)-ccm(j,1))^2+rg(j)^2); kg33=kg33+R(jj)/h(jj)*((ypd(jj)-ccm(j,2))^2+(xpd(jj)-ccm(j,1))^2+rg(jj)^2); kg34=R(jj)*(ypd(jj)-ccm(j,2))/h(jj); kg35=-R(jj)*(xpd(jj)-ccm(j,1))/h(jj); kg36=(ypd(jj)-ccm(j,2))*(ypd(jj)-ccm(jj,2))+(xpd(jj)-ccm(j,1))*(xpd(jj)-ccm(jj,1)); kg36=-R(jj)/h(jj)*((kg36+rg(jj)^2));

A=[kg11 kg12 kg13;kg21 kg22 kg23;kg31 kg32 kg33]*0.5; B=[kg14 kg15 kg16;kg24 kg25 kg26;kg34 kg35 kg36]; AB=[A B];

if j==1; li=0;else li=(j-1)*3; end;

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if j==N; ui=0;AB=AB(:,1:3);else ui=3*N-li-6; end;comp1=zeros(3,li);comp2=zeros(3,ui);strip=[comp1 AB comp2];KG=[KG;strip];end; KG=KG+KG';

% if gravity is not prescribed else KG=zeros(3*N); bu=' p-delta not considered' bum=bu; end;

% Calculate the fundamental buckling eigenvalueif cumsum(pd)~0; [bum,bu,flag]=eigs(K,KG,1,0); if flag==1; disp(' buckling eigensolution did not meet specified tolerance '); end if bu<=1; disp(' structure is statically unstable - check data'); return %break end;end;

% Extract remaining vectors from elp alw=elp(:,5); bew=elp(:,6); nw=elp(:,7); Zc=elp(:,9);

% Check that the number of steps does not exeed the load duration t0=0.0; [dum,nc1]=size(pe); [dum,nc2]=size(p); nc=max(nc1,nc2); pe=[pe zeros(3,(nc-nc1+1))]; p=[p zeros(dum,(nc-nc2+1))];

if nc<nsteps+1; disp('longest temporal load must be defined for at least nsteps+1')

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return; %break else; end;

% Calculate G. for jj=1:NE; yt=abs(alw(jj)+bew(jj)); if yt==0; G(jj)=1; else G(jj)=k0(jj)/(yt*(Zc(jj)^nw(jj))); end; end;

% Time span and effective loadif dti~=0; ddt=dti;else ddt=dt;endR=[];ts=0:ddt:nsteps*dt; tt=ts; rr=[1 0 0;0 1 0;0 0 1]; for j=1:N; R=[R;rr];end;pp=p-M*R*pe;

% Create the system matrix invm=inv(M); y1=-invm*(-KG); y2=-invm*C; y3=-invm*TS'; nul=zeros(3*N); nul1=zeros(3*N,NE); nul2=zeros(NE); AAA=[nul eye(3*N) nul1 nul1;y1 y2 y3 nul1;nul1' nul1' nul2 nul2;... nul1' nul1' nul2 nul2]; unk=6*N+2*NE;

% Compute the "hysteretic work at failure"workf=(Zc.^2).*deter(:,1)./k0;

% Numerical integration% Select the integration strategyif dti==0; options=odeset('Events','on');

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else options=odeset('Events','on','AbsTol',100,'NormControl','on', 'MaxStep',dti,... 'InitialStep',dti);end

[T,y,te,ye,ie]=ode23('oddash3dx',ts,zeros(1,unk),options,pp,AAA,N, NE,dt,alw,... bew,nw,unk,invm,TS,G,maxd,maxr,maxw,deter,workf,k0);nt=length(T);nsteps=nt-1;XX=isempty(ie);if XX==1; ie=100; end;if ie==1; disp(' Maximum displacement exceeded at time '); tett=0:ddt:nsteps*ddt;elseif ie==2; disp(' Maximum rotation exceeded at time '); te tt=0:ddt:nsteps*ddt;elseif ie==3; disp(' Maximum normalized hysteretic energy exceeded at time '); te tt=0:ddt:nsteps*ddt;else;end;

% Extract the solution; d=y(:,(1:3*N)); v=y(:,(3*N+1):(6*N)); f=y(:,(6*N+1):(6*N+NE));worki=y(:,(6*N+NE+1):unk);

% Normalize the hystertic energy dissipatedfor j=1:NE; worki(:,j)=worki(:,j)/workf(j); end;

% Obtain local drifts (used to plot hysteresis loops the elements);drift=[];for i=1:nsteps+1; drifti=TS*d(i,:)'; drift=[drift drifti]; end; drift=drift';

% Calculate the acceleration vector;

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ac=[];nm=dt/ddt;if nm-round(nm)~=0 disp(' the time interval of the loading divided by the integration time step'); disp(' must be an integer for the computation of accelerations'); breakelse ppa=[]; for j=1:3*N; ppai=interp(p(j,:),nm,1,0.0001); ppa=[ppa;ppai]; end; p=ppa; end;

for i=1:nsteps+1; aci=invm*(p(:,i)-C*v(i,:)'-TS'*f(i,:)'-(-KG)*d(i,:)');ac=[ac aci];end;ac=ac';

% Some energy quantities - the strain energy is correct only for elastic response

for j=1:nsteps; SE(j)=0.5*d(j,:)*(K-KG)*d(j,:)'; DF(j)=v(j,:)*C*v(j,:)'; KE(j)=0.5*v(j,:)*M*v(j,:)'; TF(j)=v(j,:)*M*ac(j,:)';endDE=(cumtrapz(DF))*ddt;Total=(cumtrapz(TF))*ddt;for j=1:nsteps; RIE(j)=Total(j)-KE(j);end;ENERGY=[SE' KE' DE' RIE'];

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A.3 A Matlab Code works with the Dash3dx code

% **************************************% * ODDASH3DX.m *% * *%% **************************************%% ODE file that works with DASH3DX.m

function varargout = oddash3dx(T,X,flag,pp,AA,N,NE,dt,alw,bew,nw,unk,invm, ... TS,G,maxd,maxr,maxw,deter,workf,k0);

switch flagcase ' ' varargout{1}=f(T,X,pp,AA,N,NE,dt,alw,bew,nw,unk,invm,TS,G,maxd, maxr,... maxw,deter,workf,k0); case 'events' [varargout{1:3}]=events(T,X,N,NE,unk,maxd,maxr,maxw,workf);otherwise error(['unknown flag'])% error(['unknown flag'''flag'''.'])end;

function [xd]=f(T,X,pp,AA,N,NE,dt,alw,bew,nw,unk,invm,TS,G,maxd,... maxr,maxw,deter,workf,k0);

ind=round(T/dt+1);x1=X(1:3*N);x2=X((3*N+1):(6*N));x3=X((6*N+1):6*N+NE);x4=X((6*N+NE+1):(unk));

% Normalized hysteretic energy (set a maximum of one to impose limit on% reductions).wb=x4./workf;wb=min(wb,ones(NE,1));

% Define the functions used to account for deteriorationF=3*wb.^2-2*wb.^3;fs=1-F.*(1-deter(:,2));qs=1-F.*(1-deter(:,3));

% Adjust qs to 1 if the element is unloadingvv=sign((TS*x2)./(x3+0.000001));v1=(vv+ones(NE,1))/2;qs=qs.*v1+(ones(NE,1)-vv)/2;

z1=zeros(3*N,1);

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z2=invm*pp(:,ind);z3=-alw.*(abs(TS*x2)).*(abs(x3.^(nw-1)).*x3)-bew.*(TS*x2).*(abs(x3.^nw));z3=z3.*G'.*(1./fs).^nw+k0.*(TS*x2);z3=z3.*qs;z4=x3.*(TS*x2);

xd=AA*X+[z1;z2;z3;z4];

% Subtract the rate of elastically stored energy.ku=k0-(bew-alw).*G'.*(abs(x3).^nw);xd((6*N+NE+1):unk)=xd((6*N+NE+1):unk)-(x3.*xd((6*N+1):(6*N+NE)))./ku;% ********************************************

function[value,isterminal,direction]=events(T,X,N,NE,unk,maxd,maxr,maxw,workf)

x4=X((6*N+NE+1):(unk));wb1=x4./workf;rota=resample(X(3:3*N),1,3,0);value1=[maxd-max(abs(X(1:3*N)))];value2=[maxr-max(abs(rota))];value3=[maxw-max(wb1)];value=[value1;value2;value3];isterminal=[1;1;1];direction=[0;0;0];