D A M A G E INDICES FOR REINFORCED CONCRETE F R A M E S E V A L U A T I O N A N D C O R R E L A T I O N by I S A B E L L E V I L L E M U R E B. Eng., Ecole Polytechnique de Montreal, 1993 A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE \ \ in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Civi l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A July 1995 © Isabelle Villemure, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT The purpose of this study was to assess the effectiveness of damage indicators for predicting damage in reinforced concrete structures. Two damage assessment approaches were investigated. One of these approaches leads to damage indices based on structural properties. The other approach leads to damage indices based on structural dynamic characteristics. Both approaches were used to characterize the damage of five 0.45 scale reinforced concrete bridge bents that were subjected to lateral slow cyclic loading at the Structures Laboratory of the University of British Columbia. These bents were tested as part of a seismic retrofit program undertaken by the Ministry of Transportation and Highways of British Columbia (MOTH), in collaboration with Klohn-Crippen Consultants and the University of British Columbia. The specimens underwent lateral slow cyclic loading which was monotonically increased until failure or very significant damage of the specimen occurred. During the loading process, load-deformation relationships were determined to assess structural properties of the specimens. Vibration measurement tests (ambient and impact testing) were performed at different stages of the loading cycles in order to identify the dynamic properties of the specimens. Damage assessment based on structural properties, such as displacement, stiffness and energy absorption, consisted in the evaluation of three damage indices: displacement ductility, modified stiffness ratio and the modified Park and Ang index. For the five specimens tested in the laboratory, these three indices were evaluated at different stages of damage as the displacement level increased. Damage based on dynamic properties, such as natural frequencies and damping ratios, was assessed using three damage indices: ultimate stiffness degradation, maximum softening and normalized damping ratio. ABSTRACT iii For each damage index considered in this study, the specimens were ranked according to their performance and the results from these evaluations were compared with the physical damage observed at different stages of the load testing. The results of the study showed that the agreement between analytical predictions and experimental observations was, in general, satisfactory. This study also indicated that, while two of the structural damage indices could provide some indication on the failure mode, none of the modal damage indices could give specific information on the failure mode (shear/flexure). Finally, comparison of index values from the two approaches indicated that modal damage indices generally provide a better damage characterization than structural damage indices. TABLE OF CONTENTS page A B S T R A C T ii T A B L E O F C O N T E N T S iv L I S T O F T A B L E S viii L I S T O F F I G U R E S . . ix A C K N O W L E D G M E N T S xi D E D I C A T I O N xii CHAPTER 1 I N T R O D U C T I O N 1 1.1 P R O B L E M O V E R V I E W 1 1.2 O B J E C T I V E S A N D O U T L I N E O F S T U D Y 2 CHAPTER 2 T H E O R E T I C A L B A C K G R O U N D O N D A M A G E A S S E S S M E N T 4 2.1 D A M A G E A S S E S S M E N T B A S E D O N S T R U C T U R A L P R O P E R T I E S 6 2.1.1 G L O B A L D A M A G E INDICES 6 2.1.2 L O C A L D A M A G E I N D I C E S 7 2.1.2.1 N O N - C U M U L A T I V E INDICES 8 2 . 1 . 2 . 2 C U M U L A T I V E INDICES 9 2.1.3 S U M M A R Y O F D A M A G E INDICES B A S E D O N S T R U C T U R A L P R O P E R T I E S . . . . 13 2.2 D A M A G E A S S E S S M E N T B A S E D O N D Y N A M I C P R O P E R T I E S 14 2.2.1 G L O B A L I N D I C E S 15 2 . 2 . 2 L O C A L I N D I C E S 2 3 2.2.3 S U M M A R Y O F D A M A G E INDICES B A S E D O N D Y N A M I C P R O P E R T I E S 2 8 2.3 I N T R O D U C T I O N T O D A M A G E A S S E S S M E N T B A S E D O N N E U R A L N E T W O R K S 3 0 2.3.1 T Y P I C A L I N T E R N A L S T R U C T U R E 3 1 2 . 3 . 2 T Y P E S O F N E U R A L N E T W O R K 3 2 2.3.3 B A C K - P R O P A G A T I O N N E U R A L N E T W O R K S 3 2 2.3.3.1 DESCRIPTION O F T H E N E T W O R K A N D ITS O P E R A T I O N 3 2 2 . 3 . 3 . 2 N E T W O R K D E V E L O P M E N T 3 4 2 . 3 . 3 . 2 . 1 Input and Output Layers 3 5 2 . 3 . 3 . 2 . 2 Hidden Layers 3 5 2 . 3 . 3 . 2 . 3 Training Patterns 3 6 2 . 3 . 3 . 2 . 4 Training Process 3 7 2 . 3 . 3 . 2 . 5 Validation of Network 3 7 2 . 3 . 3 . 3 E X A M P L E S O F APPLICATION T O D A M A G E A S S E S S M E N T 3 8 2 . 4 C O N C L U D I N G R E M A R K S 3 9 iv TABLE OF CONTENTS V CHAPTER 3 E X P E R I M E N T A L P R O C E D U R E 40 3.1 T E S T O B J E C T I V E S 41 3.2 D E S C R I P T I O N O F S P E C I M E N S 41 3.2.1 S P E C I M E N OSB1 42 3.2.2 S P E C I M E N O S B 2 43 3.2.3 S P E C I M E N OSB3 43 3.2.4 S P E C I M E N O S B 4 43 3.2.5 S P E C I M E N O S B 5 44 3.3 L O A D I N G P R O C E D U R E 44 3.3.1 V E R T I C A L L O A D I N G 45 3.3.2 L A T E R A L L O A D I N G 45 3.3.3 S T R U C T U R E I N S T R U M E N T A T I O N 46 3.4 D Y N A M I C T E S T I N G P R O C E D U R E 46 3.4.1 T Y P E O F V I B R A T I O N S 47 3.4.1.1 A M B I E N T VIBRATIONS 47 3.4.1.2 F O R C E D VIBRATIONS 47 3.4.2 S T R U C T U R E I N S T R U M E N T A T I O N 47 3.4.2.1 A C C E L E R O M E T E R S 48 3.4.2.2 I N S T R U M E N T E D H A M M E R 51 3.4.3 D A T A A C Q U I S I T I O N S Y S T E M 52 3.4.3.1 M E A S U R E M E N T H A R D W A R E 53 3.4.3.2 M E A S U R E M E N T S O F T W A R E • 53 3.4.3.2.1 Ambient Vibration Testing Parameters 54 3.4.3.2.2 Forced Vibration Testing Parameters 54 3.4.4 D Y N A M I C T E S T I N G S E Q U E N C E S 56 3.4.4.1 PRELIMINARY A N D F I N A L S E Q U E N C E S 56 3.4.4.2 I N T E R M E D I A T E S E Q U E N C E S 57 3.5 P R E L I M I N A R Y D E S C R I P T I O N O F D A M A G E O B S E R V E D 58 3.6 C O N C L U D I N G R E M A R K S : 61 CHAPTER 4 D A M A G E A S S E S S M E N T B A S E D O N S T R U C T U R A L P R O P E R T I E S 62 4.1 C H O I C E O F D A M A G E I N D I C E S '. 62 4.2 E V A L U A T I O N O F D A M A G E I N D I C E S 63 4.2.1 D I S P L A C E M E N T D U C T I L I T Y 63 4.2.2 M O D I F I E D S T I F F N E S S R A T I O 65 4.2.3 M O D I F I E D P A R K A N D A N G I N D E X 66 4.3 R E S U L T S A N D C O M P A R I S O N S O F INDICES 67 4.3.1 C O M P A R I S O N O F S P E C I M E N B E H A V I O R S 67 4.3.1.1 D U C T I L I T Y D I S P L A C E M E N T 67 4.3.1.2 M O D I F I E D STIFFNESS R A T I O 69 4.3.1.3 M O D I F I E D P A R K & A N G I N D E X 71 4.3.1.4 S U M M A R Y O F SPECIMEN CLASSIFICATION 72 4.3.2 D A M A G E INDICES C O R R E L A T I O N 73 4.3.3 C O M P A R I S O N O F N O R M A L I Z E D INDICES F O R E A C H S P E C I M E N 76 4.4 C O N C L U D I N G R E M A R K S 78 TABLE OF CONTENTS vi C H A P T E R 5 D A M A G E A S S E S S M E N T B A S E D O N M O D A L P R O P E R T I E S 80 5.1 C H O I C E O F M O D A L D A M A G E INDICES 80 5.2 E V A L U A T I O N O F N A T U R A L F R E Q U E N C I E S F R O M E X P E R I M E N T A L D A T A 81 5.2.1 T H E O R E T I C A L B A C K G R O U N D 82 5.2.1.1 T H E F O U R I E R T R A N S F O R M 82 5.2.1.2 INDICATORS O F N A T U R A L FREQUENCIES 83 5.2.1.3 N A T U R A L F R E Q U E N C Y R E L A T E D T O D I F F E R E N T DIRECTIONS O F M O T I O N 86 5.2.2 E V A L U A T I O N O F N A T U R A L F R E Q U E N C Y 86 5.2.2.1 T Y P I C A L T I M E SIGNALS 87 5.2.2.2 INITIAL S T U D Y O F N A T U R A L FREQUENCIES 89 5.2.2.3 F U N D A M E N T A L L O N G I T U D I N A L F R E Q U E N C Y S T U D Y 91 5.2.3 N A T U R A L L O N G I T U D I N A L F R E Q U E N C Y H I S T O R Y 94 5.3 E V A L U A T I O N O F D A M P I N G F R O M E X P E R I M E N T A L D A T A 98 5.3.1 T H E O R E T I C A L B A C K G R O U N D 98 5.3.1.1 V i s c o u s D A M P I N G 99 5.3.1.2 H Y S T E R E T I C D A M P I N G 101 5.3.2 E V A L U A T I O N O F D A M P I N G 103 5.3.2.1 V i s c o u s D A M P I N G 104 5.3.2.2 H Y S T E R E T I C D A M P I N G 105 5.3.3 D A M P I N G H I S T O R Y 105 5.4 E V A L U A T I O N O F D A M A G E I N D I C E S 109 5.4.1 U L T I M A T E S T I F F N E S S D E G R A D A T I O N . . 109 5.4.2 M A X I M U M S O F T E N I N G 110 5.4.3 N O R M A L I Z E D D A M P I N G R A T I O '.. 110 5.5 R E S U L T S A N D C O M P A R I S O N O F D A M A G E INDICES I l l 5.5.1 C O M P A R I S O N O F S P E C I M E N B E H A V I O R S I l l 5.5.1.1 U L T I M A T E STIFFNESS D E G R A D A T I O N 112 5.5.1.2 M A X I M U M SOFTENING 114 5.5.1.3 N O R M A L I Z E D D A M P I N G R A T I O 115 5.5.1.4 S U M M A R Y O F SPECIMEN CLASSIFICATION 117 5.5.2 D A M A G E I N D I C E S C O R R E L A T I O N 118 5.5.3 C O M P A R I S O N O F N O R M A L I Z E D INDICES F O R E A C H S P E C I M E N 121 5.6 C O N C L U D I N G R E M A R K S . • . . . . . . 123 C H A P T E R 6 C O M P A R A T I V E A N A L Y S I S 125 6.1 D A M A G E I N D I C E S C O R R E L A T I O N 125 6.1.1 U L T I M A T E S T I F F N E S S D E G R A D A T I O N 126 6.1.2 M A X I M U M S O F T E N I N G 128 6.1.3 N O R M A L I Z E D D A M P I N G R A T I O 131 6.1.4 S U M M A R Y O F I N D I C E S C O R R E L A T I O N 134 6.2 G E N E R A L C O M P A R I S O N O F D A M A G E A S S E S S M E N T A P P R O A C H E S 135 6.3 C O N C L U D I N G R E M A R K S 137 C H A P T E R 7 C O N C L U S I O N S A N D F U R T H E R S T U D I E S 138 R E F E R E N C E S 140 TABLE OF CONTENTS vii A P P E N D I X A 146 A P P E N D I X B 162 B. 1 T Y P I C A L T E S T I N G P R O C E D U R E 163 B.2 D E T A I L E D T E S T I N G C H A R A C T E R I S T I C S 164 B.2.1 S P E C I M E N OSB1 164 B.2.2 S P E C I M E N O S B 2 166 B.2.3 S P E C I M E N OSB3 167 B.2.4 S P E C I M E N O S B 4 169 B.2.5 S P E C I M E N O S B 5 171 B.3 D E T A I L E D S E N S O R L O C A T I O N 173 B.4 H A M M E R A N D S E N S O R S P E C I F I C A T I O N S 174 B . 5 D E T A I L S O N P R O C E S S I N G O F L O N G I T U D I N A L F R E Q U E N C Y 175 A P P E N D I X C 176 C . l T Y P I C A L S P R E A D S H E E T F O R D I S P L A C E M E N T D U C T I L I T Y - S P E C I M E N OSB1 177 C.2 T Y P I C A L S P R E A D S H E E T F O R M O D I F I E D S T I F F N E S S R A T I O - S P E C I M E N OSB1 . . . .178 C.3 T Y P I C A L S P R E A D S H E E T F O R M O D I F I E D P A R K A N D A N G I N D E X -S P E C I M E N OSB1 179 A P P E N D I X D 181 LIST OF TABLES CHAPTER 2 page Table 2.1 Damage indices based on structural properties 14 Table 2.2 Global damage indices based on dynamic properties 29 Table 2.3 Local damage indices based on dynamic properties 30 CHAPTER 3 Table 3.1 Ambient vibration testing characteristics 54 Table 3.2 Impact vibration testing characteristics 55 Table 3.3 Preliminary and final sequences 57 Table 3.4 Experimental lateral loading characteristics 58 CHAPTER 4 Table 4.1 Ranking of specimen behaviors 72 CHAPTER 5 Table 5.1 Natural frequencies of OSB1 extracted from impact vibrations 89 Table 5.2 Natural frequencies of OSB1 extracted from ambient vibrations 90 Table 5.3 Input/output combinations for longitudinal FRF calculations 94 Table 5.4 Fundamental natural longitudinal frequency history for each of the 5 specimens . . . .94 Table 5.5 Viscous damping ratios 105 Table 5.6 Equivalent hysteretic damping ratios 107 Table 5.7 Ultimate stiffness degradation characteristics 113 Table 5.8 Maximum softening characteristics 115 Table 5.9 Normalized damping ratio characteristics 116 Table 5.10 Ranking of specimen behaviors 117 CHAPTER 6 Table 6.1 Summary of index correlation 135 v i i i LIST OF FIGURES CHAPTER 2 page Figure 2.1 Fundamental period time history 17 Figure 2.2 Back-propagation neural network 32 CHAPTER 3 Figure 3.1 Specimen OSB5 ready for testing 41 Figure 3.2 Experimental model of bent S28 42 Figure 3.3 Experimental setup 44 Figure 3.4 Load cycles in a single sequence 45 Figure 3.5 Typical tri-axial accelerometer setup 49 Figure 3.6 Installation device for accelerometers of OSB3 and OSB4 50 Figure 3.7 Setup for dynamic testing 51 Figure 3.8 Longitudinal impact applied with the instrumented hammer 52 Figure 3.9 Typical recording setup for vibration measurements 53 Figure 3.10 Low sampling rate [200 sps] 55 Figure 3.11 High sampling rate [1000 sps] 55 Figure 3.12 Hysteresis curves 60 CHAPTER 4 Figure 4.1 Typical hysteresis loop and parameter definitions 64 Figure 4.2 Comparison of specimen behaviors: displacement ductility 68 Figure 4.3a Comparison of specimen behaviors: modified stiffness ratio 69 Figure 4.3b Comparison of specimen behaviors: modified Park and Ang index 71 Figure 4.4a Index correlation: modified stiffness ratio vs displacement ductility 74 Figure 4.4b Index correlation: modified Park and Ang index vs displacement ductility 75 Figure 4.4c Index correlation: modified Park and Ang index vs modified stiffness ratio 75 Figure 4.5 Comparison of normalized indices 78 CHAPTER 5 Figure 5.1 Ambient vibration output signal. 88 Figure 5.2 Impact vibration input and output signals 88 Figure 5.3 Fundamental natural mode shapes of OSB1 91 Figure 5.4 Comparison of FRF formulations 93 Figure 5.5 Sensitivity of natural longitudinal frequency to structural damage 96 Figure 5.6 Example of stiffness study for specimen OSB5 96 Figure 5.7 Viscously damped free-vibration response 100 Figure 5.8 Hysteretic damping from hysteresis loop 102 ix LIST OF FIGURES X Figure 5.9 Sensitivity of viscous damping to structural damage 106 Figure 5.10 Sensitivity of hysteretic damping to structural damage 106 Figure 5.11 Modulation in the response time signals 108 Figure 5.12a Comparison of specimen behaviors: ultimate stiffness degradation 112 Figure 5.12b Comparison of specimen behaviors: maximum softening 114 Figure 5.12c Comparison of specimen behaviors: normalized damping ratio 116 Figure 5.13a Index correlation: ultimate stiffness degradation vs maximum softening 119 Figure 5.13b Index correlation: ultimate stiffness degradation vs normalized damping ratio . . 120 Figure 5.13c Index correlation: normalized damping ratio vs maximum softening 121 Figure 5.14 Comparison of normalized damage indices 123 CHAPTER 6 Figure 6.1 Correlation between ultimate stiffness degradation and displacement ductility . . . .126 Figure 6.2 Correlation between ultimate stiffness degradation and modified stiffness ratio . . .127 Figure 6.3 Correlation between ultimate stiffness degradation and modified Park and Ang index 128 Figure 6.4 Correlation between maximum softening and displacement ductility 129 Figure 6.5 Correlation between maximum softening and modified stiffness ratio 130 Figure 6.6 Correlation between maximum softening and modified Park and Ang index 131 Figure 6.7 Correlation between normalized damping ratio and displacement ductility 132 Figure 6.8 Correlation between normalized damping ratio and modified stiffness ratio 133 Figure 6.9 Correlation between normalized damping ratio and modified Park and Ang index .134 APPENDIX A Figure Q-l 17-01 PierS28 148 Figure Q-l 17-11 specimen OSB1 149 Figure Q-l 17-13 specimen OSB2 150 Figure Q-l 17-12 specimen OSB3 151 Figure Q-l 17-14 specimen OSB4 152 Figure Q-l Elevation view of specimen OSB5 153 Figure A . l Typical accelerometer setup on a column 154 Figure A.2 Typical accelerometer setup on the cap beam 155 Figure A.3 Instrumented hammer 155 Figure A.4 Transverse impact applied with the instrumented hammer 156 Figure A.5 Vertical impact applied with the instrumented hammer 157 Figure A.6 Data acquisition system 157 Figure A.7 Overall view of specimen OSB1 at failure 158 Figure A.8 View of specimen OSB2 at failure (north column, east side) 158 Figure A.9 View of specimen OSB2 at failure (north column, west side) 159 Figure A.10 Overall view of specimen OSB3 at failure. . 159 Figure A.11 Overall view of specimen OSB4 at failure 160 Figure A.12 View of specimen OSB5 at failure (north half) 160 Figure A.13 View of specimen OSB5 at failure (south half) 161 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my thesis supervisor, Dr. Carlos E. Ventura, for his inestimable assistance and constant encouragement in the advancement and completion of this thesis. I would also like to gratefully acknowledge my thesis co-supervisor, Dr. Robert G. Sexsmith, for his excellent guidance throughout this research project. I am indebted to my colleague Norman Schuster. During his graduate studies at the University of British Columbia, he upgraded the data acquisition software which was used in this study. Devoted and patient, he offered me helpful guidance and advice throughout the vibration testing program. The financial support of the Natural Sciences and Engineering Research Council of Canada (N.S.E.R.C.) is gratefully acknowledged. This includes a Postgraduate Scholarship as well as Research Grants awarded to both Dr. C.E. Ventura and Dr. R.G. Sexsmith. The research was also made possible by financial support of the Ministry of Transportation and Highways of British Columbia. Special thanks are expressed to Howard Nichol, earthquake laboratory technician, for his helpful assistance and constant technical support. Furthermore, the helpful suggestions of colleagues Mahmoud Rezai and Vincent Latendresse are gratefully acknowledged as well as the contribution of Thomas Horyna in the development of two computer programs used in the analytical part of this study. I would also like to thank all of the other people who provided technical support in the testing of the specimens. These people include Paul Symons, Tony Cigic, Daryl English, Brad Kemp and Mike Baraka. Finally, I would like to thank Dr. R. Foschi who, along with Dr. C. E. Ventura and Dr. R. G. Sexsmith, reviewed this thesis. Mic^effe et Pierre CHAPTER 1 INTRODUCTION 1.1 PROBLEM OVERVIEW During its lifetime, a structure is subjected to loads arising from different sources. Depending on the intended use and occupancy of the structure, serviceability loads produce stresses and deformations on the different structural components, in general, below critical levels. Although unusual, extreme loads, such as earthquakes and hurricanes, may generate stresses and deformations on the structural members that will be so high as to cause a certain level of damage or even failure of members, or the whole structure. Over the last few decades, one of important areas of research for structural engineers has been the characterization and evaluation of structural damage. Considering the complexity involved in the structural degradation processes, quantifying damage often represents a difficult assignment. Different approaches have been developed to provide reliable predictions of the state of a damaged structure. Performed from analytical predictions or from experimental measurements, damage assessment investigates the potential or actual degradation state of a structure. Damage assessment techniques have been applied in different situations such as disaster planning, structural assessment, retrofit and repair operations, maintenance inspection and post-earthquake evaluation. Among the different approaches to characterize damage, damage indices provide useful means to quantify damage of structures or rank their vulnerability relative to each other. Damage indices can be evaluated either based on the response of a structure to a particular loading pattern or based on the dynamic response of a structure. A different type of approach that has been recently developed considers the application of neural networks to the degradation process experienced by a structure. 1 INTRODUCTION 2 Although damage assessment techniques are based on different concepts, they aim to quantify the same effect, structural degradation. However, investigating how different approaches characterize damage presents a topic of significant interest. Damage characterization is understood here to be how physical elements, such as displacement, crack propagation, yielding, stiffness degradation, etc., are taken into account in the rate of approach to failure of a damage index. Moreover, damage indicators of a same approach can quantify differently the structural degradation. Consequently, comparing damage indicators of a same approach is also of significant interest. 1.2 OBJECTIVES AND OUTLINE OF STUDY In order to investigate different damage assessment approaches and different damage indicators of a same approach, a combined experimental and analytical program was developed and implemented during the course of this study. The objectives of this study were: 1. to measure structural and dynamic properties of the specimens tested; 2. to evaluate damage indices based on structural properties and damage indices based on modal properties; 3. to compare damage indices within a same approach; 4. to correlate damage indices obtained from the two different approaches. A third approach, damage assessment based on neural networks, was also investigated. Since no conclusive developments were derived from this approach, results of this approach are not presented as part of this study. This thesis describes how these objectives were implemented on an experimental program developed at the University of British Columbia. Chapter 2 presents a literature review of most commonly used damage indices, based on structural or modal properties, as well as an introduction to damage assessment based on neural networks. Chapter 3 describes the experimental procedure performed, including details on the INTRODUCTION 3 specimens tested and description of the loading and dynamic testing procedures. Chapter 4 investigates damage assessment based on structural properties and describes its application to the specimens tested in the laboratory. Damage assessment based on modal properties, also applied to the experimental specimens, is presented in Chapter 5. A comparative analysis of these two approaches is presented in Chapter 6. Finally, Chapter 7 summarizes results obtained from this study. CHAPTER 2 THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT During its lifetime, the capacity of a structure may be reduced due to different types of structural faults, such as cracking, buckling, unbonding, corrosion losses, loosening of fastened parts and yielding of steel reinforcement. The complexity involved in these degradation processes sometimes limits their complete understanding. As a consequence, quantifying damage often represents a difficult assignment. Nevertheless, quantification of damage remains a useful assessment tool in several situations. Maintenance inspection and post-earthquake evaluation are examples of structure assessment requiring damage quantification. As a result, different approaches have been developed to provide reliable predictions of the state of a damaged structure. Three of these approaches are discussed below. A first approach is based on the response of the structure to a particular loading pattern. Since seismic events present a significant damage threat, this approach usually considers structural degradation caused by earthquakes or cyclic loads. Response to these loadings is usually measured in terms of force applied and corresponding displacements experienced by the structure. Resulting load-deformation curves are commonly called hysteresis curves. These curves can either be predicted analytically with models of variable performance or determined experimentally. Since hysteresis curves obtained from experimental measurements imply partial or complete destruction of the structure under study, they contain information about degradation levels sustained by the structure. Maximum displacements experienced by the structure, stiffness degradation and levels of energy absorption can be determined from these hysteresis curves. Combination of these structural characteristics led to the development of a category of damage indices, that provide quantification of damage levels sustained by the structure. 4 THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 5 Modal analysis also provides information about structural damage. This approach is based on the concept that degradation of structural elements and/or joints alters the dynamic response of the structure. These changes in the vibration response are in turn reflected in the experimentally measured dynamic properties of the structure. The modal properties usually are the fundamental frequencies, damping ratios and mode shapes. They can be evaluated from processing of time histories obtained during vibration tests or recorded during an actual seismic event. They can also be predicted with analytical models of variable performance. Unlike structural damage indices, damage assessment based on experimental dynamic properties may involve either destructive or non-destructive measurement techniques. Combination of these dynamic properties or their relative changes has generated another category of damage indices. The third approach, briefly discussed in this study, is damage assessment based on neural networks. Basically, application of neural networks attempts to overcome the complexity involved in damage assessment. By definition, neural networks are computing environments modelling a complex system behavior. In this case, the system behavior refers to the degradation process experienced by a structure. In order to model a system behavior, the network must initially be trained to a specific condition. For damage evaluation, the system is trained to assess a certain type of structural degradation. Thereafter, based on the training conducted, the network functions as an associative memory capable of diagnosing unknown degradation levels. The following sections describe alternative ways of implementing these three approaches. Section 2.1 introduces damage indices based on structural properties. Damage indices based on dynamic properties are discussed in section 2.2. Finally, an introduction to damage assessment based on neural networks is presented in section 2.3. THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 6 2.1 DAMAGE ASSESSMENT BASED ON STRUCTURAL PROPERTIES When subjected to an earthquake, a structure might suffer excessive deformations, causing structural damage in individual members or parts of the structure. Moreover, the repeated load reversals caused by the earthquake can generate low-cycle fatigue damage, leading to structural deficiencies in the system. The structure deterioration generally originates from a combination of these two effects. Consequently, damage indices based on structural properties usually include a large deformation term and/or a fatigue loading term. These damage indices can be evaluated locally on a particular member or globally on parts of the structure from measurements or predicted response to simple cyclic loadings. Several of the most commonly used global and local damage indices are presented in the following sections. 2.1.1 GLOBAL DAMAGE INDICES Global indices, DG, quantify damage for the complete structure, or for parts of the structure when several of its structural elements are considered. They provide an overall assessment of structure performance based on damage distribution and level of degradation sustained by its individual components. They are typically evaluated by weighting local damage indices of the different members composing the structure. Different types of weighting functions have been formulated to consider the state of heavily damaged individual elements. The most common global index uses the amount of energy absorbed at different locations as a weighting function (Park, Ang and Wen, 1987, Chung et al, 1990, Kunnath et al, 1992). Evaluated for a complete structure of N elements or part of a structure composed of N members, the global damage index, DG, is defined as: THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 7 N Z D L , i Ei DG = L T [21] i=\ where DL, is a local index evaluating damage at location i (see section 2.1.2) and £, is the energy absorbed at that same location. Severely damaged members of a structure can limit its overall stability. This is not reflected in the averaging effect of Equation 2.1. Hence, Bracci et al (1989) developed a global damage index that emphasizes the severity of damage in a structural member. It is expressed as: N DG= T • [2-2] i = 1 High values of parameter b are used when more emphasis on the most severely damaged members is required. This formulation defines the weights w, as the ratio of the gravity load supported by member i to the total gravity load on the structure. These weighting functions reflects the greater dependence of the overall structural stability to the damage occurring at the base of the structure. Tests performed on reinforced concrete frames (Bracci et al, 1989) verified the ability of the indicator to quantify damage. Corresponding index values showed good correlation with observed and measured damage. 2.1.2 LOCAL DAMAGE INDICES Local indices, DL, usually characterize damage of individual members or joints, and are typically based on ductility measurements, energy absorption or a combination of both. Some indices also model the accumulation of degradation induced by the cyclic part of the motion. Others are non-cumulative and THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 8 characterize a fixed damage state of the structure. Generally, local index values range between zero, for an undamaged structure, and one, for a collapsed structure. Commonly used local damage indices are described below. 2.1.2.1 N O N - C U M U L A T I V E INDICES The displacement ductility, p-g, represents the most elementary index to quantify structural damage (Newmark and Rosenblueth, 1974). It is defined as the ratio of the maximum displacement sustained by the structure to its yield displacement: 5 5 - 5 y y where 8^ , is the yield displacement and 8 m represents the maximum displacement of each cycle. The displacement ductility index is based exclusively on peak displacement and it neglects the fatigue contribution of cyclic loading. Nevertheless, it is still used as damage indicator because of its simplicity in evaluation and practical interpretation (Sordo et al, 1989). Banon et al (1981) developed a measure of the local stiffness degradation and called it the flexural damage ratio (FDR). In terms of stiffnesses, it is expressed as: k DL= FDR - T^- [2.4] m where k0 represents the initial tangent stiffness of the structural element considered while km refers to the maximum stiffness of this same member during a complete cycle. Considering a particular cycle, its maximum stiffness km is evaluated for both forward and reverse parts of the cycle and the minimum km value, yielding the maximum index value, is retained. Stiffnesses are, by definition, derived from the ratio of force over displacement. Therefore, they can be evaluated from hysteresis curves of the element studied. Based on experiments of reinforced concrete frames (Banon et al, 1981), FDR indicated THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 9 adequately damage originating from large deformations. The FDR is considered as a better damage indicator than displacement ductility since it takes into account stiffness and strength degradations in the deteriorated member. Roufaiel and Meyer (1987a) used a modified version of this index, considering the increase in flexibility at maximum deformation and at failure state. Expressed in terms of stiffnesses, the modified stiffness ratio, MSR, is defined as: D^MSR=iw^ [2-51 where kf represents the stiffness of the structure at failure. The index retained is the maximum ratio considering both positive and negative cycles. Experimental data obtained from diverse laboratory specimens were used to verify the damage index. Corresponding index calculations indicated good correlation with residual strength and stiffness obtained from experimental specimens. 2.1.2.2 C U M U L A T I V E INDICES The cumulative ductility (Banon et al, 1981) represents a measure of ductility and captures the effect of repeated loading on the structure. Considering M cycles of loading, it is expressed as: M DL= Z 0 * 5 , , - ! ) I 2 ' 6 ' where 1x5^ is the maximum displacement ductility at cycle j, defined by Equation 2.3. Tests of reinforced concrete frames, conducted by Banon et al (1981), showed that this index was closely associated with the hysteretic behavior of the structure. Note that while displacement ductility represents a measure of damage due to excessive deformations, cumulative ductility carry information on fatigue damage, inflicted by the cyclic part of motion. THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 10 The Stephens & Yao index (1987) was developed based on plastic displacement increments, A8p, for a complete cycle. Considering M cycles, it is expressed as: M 7 = 1 A5;-A 5 / _ 1 -br [2.7] in which A 8 p + is defined as the value of the positive plastic decrement and ASy- is the positive plastic decrement in a single-cycle test to failure. Coefficient r represents the ratio of positive to negative plastic displacement increments, A 5 p + / A5p", for each cycle. Parameter b is a calibrated constant based on different type of structure and damage levels and it has a recommended value of 0.77. Based on Stephens and Yao experimental studies of two small structural systems, the index provided useful measures of damage sustained by these structures. However, calibration of parameter b represents a potential limitation of this method. The Wang & Shah index (1987) was developed on the assumption that the rate of accumulation of damage is proportional to the damage already affecting the structure. They proposed the following exponential equation to characterize damage, based on M loading cycles: exp (sa) - 1 Dj = F \ / . [2.8] L exp (s) - 1 M in which, CL = c / -g— 7=1 f Parameters 8mj and 8y are respectively the maximum displacement of cycle j, and the final displacement after the complete loading pattern of M cycles. Parameters c and s are constants with recommended values of 0.1 and 1.0 for well-reinforced concrete member. Small-scale reinforced concrete beam-column joint were used to verify the index ability to quantify structural degradation (Wang and Shah, 1987). Analytical models, based on damage index calculations, showed good correlations with experimental response of the THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 11 specimens subjected to different loading histories. However, parameter s, which appeared to be correlated to the member design properties, such as the amount of joint and beam reinforcement, limits the application of this method in general. The Jeong & Iwan index (1988) quantifies damage under cyclic loading using an expression combining the effects of cycles at different amplitudes. It measures the influence of both duration and ductility of response. Considering P loading cycles of different amplitudes, it is defined as: P nk D, = T — [2.9] k= 1 where nk is the number of cycles at amplitude k and np is the number of cycles to failure at that same amplitude, evaluated as: nf,k H,k = c Constants c and s have recommended values of 6.0 and 416 respectively for reinforced concrete structures. The index provided an adequate qualitative estimation of structural damage. However, calibration of parameters c and s, affected by the design details of the structure under study, is a limitation of the index. The Kratzig & al index (1989) is based on the hysteretic energy absorbed by a member. Considering a positive loading cycle, the first load cycle at a given amplitude represents the primary half cycle. The subsequent cycle at the same or lower amplitude is denoted follower cycle. A similar definition is used for the negative part of a complete loading cycle. Positive and negative energy terms, denoted D+ and D~, are then evaluated as follows for each loading cycle;': THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 12 M M Z E \ J + I E=J n = ^ — M ^ — j=l [2.10] where Epj* is the energy absorbed in a primary half cycle, E* is the energy of the follower cycles and E* is the energy absorbed in a monotonic test to failure. The overall damage index is finally expressed as: DL = DL+ + DL- + DL+DL From various databases, obtained from laboratory experiments or on literature, Kratzig et al successfully verified the ability of the index to detect damage evolution in a structure. This index involves considerable calculation effort compared to other local indices including a fatigue damage term. The Park & Ang index (1985) combined a deformation term and a hysteretic energy absorption term to take into account the peak deformation as well as the damage related to fatigue. The corresponding damage index is expressed as: [2.11] The integral takes into account the accumulation of energy absorbed up to the cycle under study. Parameters 8 r a and 6y are respectively the maximum displacement of the cycle considered and the final displacement after the complete loading pattern of M cycles. Fy represents the yield strength of the structure, p is a strength degradation parameter and is assumed random with a mean of 0.27 and coefficient of variation of 0.6 (Ciampoli et al, 1989). For well-reinforced concrete, its value is 0.1. Limitations of the index could be associated to the calibration of p for the structure under study. This index has been calibrated and validated against a significant amount of observed seismic damage on different structures. It has been used in a number of seismic vulnerability studies and probabilistic studies (Ang, THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 13 1987, Barenberg and Foutch, 1988, Ciampoli et al, 1989, Seidel et al, 1989, Stone and Taylor, 1993). The modified Park & Ang index (Kunnath et al, 1992) slightly transforms Equation 2.11 to consider only the permanent deformation in the first term. The index is expressed as: 2.1.3 S U M M A R Y OF D A M A G E INDICES B A S E D ON S T R U C T U R A L PROPERTIES Global and local damage indices are summarized in Table 2.1. Data required in calculation as well as calibrated parameters required in the formula are presented for each damage index. For more details on these structural damage indices as well as their background, one can consult the comparative study by Williams and Sexsmith (1994). E = E, D = [2.12] THEORETICAL BACKGROUND ON DAMAGE ASSESSMENT 14 Type liquation Number I-'xpurimentiil