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Optimal retrofit strategy determination for bridges using decision analysis Khan, Saqib A. 2001

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OPTIMAL RETROFIT STRATEGY DETERMINATION FOR BRIDGES USING DECISION ANALYSIS by SAQIB A . K H A N B . S c , University of Engineering and Technology, Lahore, Pakistan, 1998  A THESIS SUBMITTED IN PARTIAL 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  THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A August 2001  © Saqib A . Khan, 2001  U B C Special Collections - Thesis Authorisation Form  Page 1 of 1  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my w r i t t e n p e r m i s s i o n .  Department of  tiviL  FMGlMEERlMG  The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date  4 • oi • i o o l  ht1p://www.library.ubc.ca/spcoll/thesauth.html  7/30/01  ABSTRACT W i t h our increased knowledge about seismicity and risks related to earthquakes, there is a greater need for retrofitting deficient structures to ensure the functioning o f a transportation network and to minimize the life and economic loss associated with catastrophic events. Given the scarcity o f funds, the decision maker should try to determine the optimal level o f retrofit for a structure and the priorities among various candidates. A decision analysis methodology is thus proposed and demonstrated through two example bridges in order to determine the most preferable level o f retrofit and the retrofit order for the two candidates. The decision alternative minimizing the total cost o f the structure over its life is the best retrofit strategy while the bridge w i t h the lower costto-benefit ratio should be retrofitted first. Seismic assessment o f the Colquitz river south structure was performed in terms o f three different levels of seismicity. Various earthquake records were scaled to match the site-specific spectra corresponding to each level o f seismicity. These records were then used to drive the non-linear dynamic analysis o f the bridge pier. Overall damage states for the bridge were determined based on damage index values for the pier and expert judgement for other bridge components corresponding to various pre-defined levels o f retrofit. The structural damage was then translated into dollar damage and this information was used in the decision analysis algorithm as consequence costs. A n expected annual cost of future damage was then calculated and converted to present worth for each retrofit level. The estimated retrofit costs and the present value o f future damages were then added to find the total expected cost for each decision alternative. Sensitivity analyses were carried out to examine the variations in decision outcome due  ii  to changes in different input parameters. Also, the effectiveness of decision analysis techniques not employing probability and risk attributes was briefly examined. For the Interurban overpass, it was assumed that the safety level retrofit would be the optimal strategy. Cost-to-benefit ratios for the two structures were then calculated corresponding to the optimal retrofit decision for each, thus determining the order of retrofit.  iii  TABLE OF CONTENTS  ABSTRACT  ii  T A B L E O F CONTENTS  iv  LIST OF FIGURES LIST OF TABLES ACKNOWLEDGEMENTS  viii x xii  CHAPTER I: INTRODUCTION  1  1.1 Background 1.1.1 Bridge retrofit in British Columbia 1.1.2 Retrofit levels for acceptable performance 1.2 Objectives of study 1.3 Scope of study  1 2 3 5 6  CHAPTER 2: GENERAL BRIDGE AND RETROFIT DESCRIPTION  9  2.1 Colquitz river south structure 2.1.1 Geometric and structural description 2.1.2 Structural deficiencies 2.1.2.1 Superstructure 2.1.2.2 Bearings 2.1.2.3 Pier 2.1.2.4 Abutments 2.1.2.5 Piles and foundations 2.1.3 Retrofit scenarios for Colquitz south bridge 2.1.4 Site soil conditions for Colquitz 2.2 Interurban overpass 2.2.1 Geometric and structural description 2.2.2 Structural deficiencies 2.2.2.1 Girder diaphragms 2.2.2.2 Anchor bolts 2.2.2.3 Pier and abutment piles and foundations 2.2.2.4 Columns in Pier No.2 2.2.2.5 Abutment backwall 2.2.3 Retrofit of Interurban overpass 2.2.4 Site soil conditions for Interurban  9 9 11 11 12 12 13 13 14 15 16 16 17 18 18 18 19 19 19 20  iv  2.3 Costs of retrofit 2.3.1 Colquitz river south bridge 2.3.2 Interurban overpass  20 21 22  CHAPTER 3: DECISION ANALYSIS METHODOLOGIES  23  3.1 Introduction 3.2 Basis for ranking competing alternatives 3.3 Determination of expected value 3.4 Decision tree for present study 3.5 Decision making with out probability values  23 25 25 26 29  CHAPTER 4: SEISMIC HAZARD EVALUATION  31  4.1 Methodologies for seismic hazard prediction 4.2 Synopsis of methodology for present study 4.3 Considered earthquake levels 4.3.1 Current design level earthquake 4.3.2 Cascadia subduction event 4.3.3 Future design level earthquake 4.4 Annual occurrence rates and earthquake probabilities 4.4.1 Earthquake probability determination 4.5 Earthquake records selected for present study 4.5.1 Loma Prieta earthquake 4.5.2 El Centra earthquake 4.5.3 Miyagi earthquake 4.6 Ground motion amplifications and site-specific design spectra 4.7 Site-specific record generation 4.7.1 Computer program SYNTH 4.7.2 Modified records CHAPTER 5: ASSESSMENT  NON-LINEAR  ANALYSIS  AND  5.1 General aspects of the Colquitz bent modelling 5.2 Non-retrofitted Vs retrofitted pier 5.3 Moment-curvature and push over analyses for Colquitz pier 5.4 Damage indices 5.4.1 Park and Ang damage index 5.4.2 Residual energy damage index 5.4.3 Categorization and calibration of damage 5.4.4 Discussion 5.5 Non-linear dynamic analysis and damage index determination 5.5.1 Computer program CANNY 5.5.2 Analysis results 5.5.3 Damage index summary  31 32 33 33 34 35 37 37 38 38 39 40 41 46 46 47 DAMAGE  STATE 52 53 55 57 60 61 62 64 65 66 66 68 72  v  5.6 Additional damage indicators for Colquitz  73  5.7 Damage indicators for Interurban overpass for safety retrofit  78  CHAPTER 6: ESTIMATION OF D A M A G E COSTS  81  6.1 Direct damage costs 6.1.1 Relating direct damage costs to damage states 6.1.1.1 Direct damage costs for Colquitz south bridge 6.1.1.2 Direct damage costs for Interurban overpass 6.2 Indirect damage costs CHAPTER 7: DECISION AND SENSITIVITY ANALYSIS  81 82 89 91 92 95  7.1 Net present cost (NPC) criterion 7.2 Decision cost comparisons 7.3 Sensitivity analysis 7.4 Discussion 7.5 Decision making with out probability knowledge 7.6 Summary  95 97 99 102 104 108  CHAPTER 8: BRIDGE RETROFIT PRIORITIZATION PROCEDURES  109  8.1 Existing screening procedures 8.1.1 The ATC/FHWA methodology 8.1.2 The CALTRANS procedure 8.1.3 TheWSDOT approach 8.1.4 The IDOT scheme 8.1.5 Methodology proposed by Basoz and Kiremidgian 8.1.5.1 Vulnerability assessment 8.1.5.1.1 Seismic hazard analysis 8.1.5.1.2 Classification of bridges 8.1.5.1.3 Fragility analysis 8.1.5.2 Importance assessment 8.1.5.3 Overall ranking 8.2 Proposed methodology 8.3 Results  110 110 Ill 113 114 115 116 116 117 118 119 120 121 121  8.4 Discussion  123  CHAPTER  9:  CONCLUSIONS  AND APPLICATION  OF  PROPOSED  METHODOLOGY  125  9.1 Conclusions  125  9.2 Application of proposed methodology  128  o  REFERENCES  130  APPENDIX A:  135  APPENDIX B:  145  APPENDED C:  161  APPENDIX D:  166  vii  LIST OF FIGURES CHAPTER 2 Figure 2.1: Elevation of the Colquitz south structure Figure 2.2: Photograph of the Colquitz pier and superstructure looking east Figure 2.3: A view of the Interurban overpass roadway Figure 2.4: A viewfromunderneath the Interurban overpass showing Pier No.2 and the steel superstructure  10 11 17 17  CHAPTER 3 Figure 3.1: General layout of a decision tree Figure 3.2: Decision analysis tree for bridge retrofit problem  24 29  CHAPTER 4 Figure 4.1: Victoria 10%/50 year robust UHRS (50 percentile values) (Private correspondence, 1999) 34 Figure 4.2: Cascadia subduction earthquake scenario spectrum (84 percentile values) for Victoria (Private correspondence, 1999) 35 Figure 4.3: Victoria 2%/50 year robust UHRS (50* percentile values) (Private correspondence, 1999) 36 Figure 4.4: Loma Prieta earthquake acceleration time history 39 Figure 4.5: El Centro earthquake acceleration time history 40 Figure 4.6: Miyagi subduction earthquake acceleration time history 41 Figure 4.7: Design spectra for bridge sites with soil amplification effects 45 Figure 4.8: Firm ground and site-specific design spectra 45 Figure 4.9: Target and computed spectra for level-1 earthquake 48 Figure 4.10: Target and computed spectra for level-2 earthquake 48 Figure 4.11: Target and computed spectra for level-3 earthquake 48 Figure 4.12: Acceleration time histories for level-1 earthquake 49 Figure 4.13: Acceleration time histories for level-2 earthquake 50 Figure 4.14: Acceleration time histories for level-3 earthquake 51 th  th  CHAPTER 5 Figure 5.1: (a) Colquitz bridge prototype pier (b) SDOF lumped-parameter model.... 54 Figure 5.2: Simplified bi-linear moment-curvature for non-retrofitted Colquitz pier.. 58 Figure 5.3: Simplified bi-linear moment-curvature for retrofitted Colquitz pier 58 Figure 5.4: Push over curve for non-retrofitted Colquitz pier 59 Figure 5.5: Push over curve for retrofitted Colquitz pier 59 Figure 5.6: Residual energy index model elaboration 63 Figure 5.7: CANNY sophisticated bilinear/trilinear hysteresis model (From CANNY-E technical manual, 1996) 67 Figure 5.8: Analysis results for level-1 earthquake 69  Figure 5.9: Analysis results for level-2 earthquake... Figure 5.10: Analysis results for Ievel-3 earthquake  70 71  CHAPTER 6 Figure 6.1: Deterministic mapping for direct damage estimation (From Gunturi and Shan, 1993) Figure 6.2: DDI Vs damage state plots for various bridge components  83 85  CHAPTER 7 Figure 7.1: Hurwicz rule results for different values of a  106  CHAPTER 8 Figure 8.1: Hierarchical order for primary structural attributes (From Basoz and Kiremidgian, 1996) Figure 8.2: A generic fragility curve  117 118  LIST OF TABLES  CHAPTER 2 Table 2.1: Retrofit strategies for Colquitz south structure 15 Table 2.2: Retrofit activity costs for Colquitz south structure 21 Table 2.3: Costs for different retrofit levels as defined for Colquitz south structure... 22 Table 2.4: Break down of safety retrofit costs for Interurban overpass 22 CHAPTER 4 Table 4.1: Annual probabilities of various earthquake levels 38 Table 4.2: Description of various site classes based on soil average shear wave velocity (Private correspondence, 2000) 42 Table 4.3: Values of Fa as a function of site class and T=0.2 sec spectral acceleration (Private correspondence, 2000) 44 Table 4.4: Values of Fv as a function of site class and T=1.0 sec spectral acceleration (Private correspondence, 2000) 44 CHAPTER 5 Table 5.1: Damage classification according to Park and Ang (1985) Table 5.2: Damage classification according to Stone and Taylor (1993) Table 5.3: Proposed damage classification by Hindi and Sexsmith (2001) Table 5.4: Damage index values for the base scenario Table 5.5: Damage index values for the upper bound damage scenario Table 5.6: Damage assessment of Colquitz for retrofit A Table 5.7: Damage assessment of Colquitz for retrofit B Table 5.8: Damage assessment of Colquitz for retrofit C Table 5.9: Damage assessment of Colquitz for retrofit D Table 5.10: Damage assessment of Interurban overpass for retrofit C  64 64 65 72 73 74 75 76 77 79  CHAPTER 6 Table 6.1: Assumptions for bridge component cost assessment Table 6.2: Average replacement costs for various bridge types in 1995 U S dollars (from Caltrans, 1995) Table 6.3: Replacements costs in millions, 2001 C D N dollars Table 6.4: Removal and total costs in millions, 2001 C D N dollars Table 6.5: Direct damage costs for the base scenario (2001 C D N dollars) Table 6.6: Direct damage costs for the upper bound damage scenario (2001 C D N dollars)  86 87 88 88 90 91  x  Table 6.7: Direct damage costs for Interurban overpass safety retrofit (2001 C D N dollars) Table 6.8: Assumed closure days and indirect costs for Colquitz (Base scenario) Table 6.9: Assumed closure days and indirect costs for Colquitz (Upper bound damage scenario) Table 6.10: Assumed closure days and indirect costs for Interurban overpass for safety retrofit (2001 C D N dollars)  92 93 94 94  CHAPTER 7 Table 7.1: N P C s for direct costs only (Base scenario) 97 Table 7.2: N P C s for direct plus indirect costs (Base scenario) , 98 Table 7.3: N P C s for direct costs only (Upper bound damage values) 98 Table 7.4: N P C s for direct plus indirect costs (Upper bound damage values) 98 Table 7.5: N P C s corresponding to different i values and direct costs only T = 100 years, F C R F / S F R F = 1.5 (Base scenario) 100 Table 7.6: N P C s corresponding to different i values and direct plus indirect costs T = 100 years, F C R F / S F R F =1.5 (Base scenario) 100 Table 7.7: N P C s for direct plus indirect costs after increasing indirect costs corresponding to base scenario T = 100 years, F C R F / S F R F = 1.5 101 Table 7.8: N P C s for direct plus indirect costs after decreasing indirect costs corresponding to upper bound damage scenario T = 100 years, F C R F / S F R F = 1.5 101 Table 7.9: N P C s for direct plus indirect costs for i=4%, T = 100 years and F C R F / S F R F =1.4 102 Table 7.10: N P C s for direct plus indirect costs for i=4%, T = 100 years and F C R F / S F R F = 1.33 102 Table 7.11: Consequence matrix for direct plus indirect cost scenario corresponding to the lower bound damage case i = 4%, F C R F / S F R F =1.5 105 Table 7.12: M i n i m i n and M a x i m i n rule results 105 Table 7.13: Regret matrix for the M i n i m a x regret rule 107 Table 7.14: M i n i m a x regret rule results 107  CHAPTER 8 Table 8.1: Priority index calculation for Colquitz and Interurban Table 8.2: Priority index determination for Interurban for different percentages o f estimated consequence cost  122 123  ACKNOWLEDGEMENTS  I am immensely grateful to my supervisor, Dr Robert.G.Sexsmith for providing me with guidance, ideas and advice during this research work. I would like to thank Dr. R.O.Foschi and Dr.D.L.Anderson for their help and guidance. I also wish to express my appreciation to Mr. Peter Brett and Mr. Brock Radloff of BC MoTH for helping me in getting technical information about the bridges selected for this study. Above all, I am heavily indebted to my parents who provided me with financial and moral support and encouragement during this time. The financial support of the Natural Sciences and Engineering Research Council of Canada and the BC MoTH is gratefully acknowledged.  xii  Chapter 1 - Introduction  CHAPTER 1 INTRODUCTION  It is important for a transportation system to remain functional after an earthquake so that rescue activities can take place and long and short-term economic losses are minimized. Bridges, being an essential part of the transportation network, are a risk to the functionality of a transportation network. A great deal of effort is currently being put into retrofitting bridges in North America and other parts of the world so that they can withstand these catastrophic events while maintaining the required degree of functionality. The present study is a continuations of work aimed at obtaining the aforementioned objective.  1.1 B A C K G R O U N D : The province of British Columbia is located in one of the most earthquake prone zones in the world, the risks of getting a devastating earthquake here being greater than in any other part of Canada. This region has had earthquakes of similar magnitude and intensity in the past as the ones that have recently happened in Turkey, Taiwan, Kobe, California and India. Due to a considerable increase in population, such an earthquake could cause much more damage i f it were to happen today. This damage would not only comprise of life loss but would also have long-term economic consequences from a transportation network perspective due to increased traffic congestion and long detours resulting in lost work time, business losses and other miscellaneous indirect factors.  1  Chapter 1 - Introduction  1.1.1 BRIDGE RETROFIT IN BRITISH C O L U M B I A : The British Columbia Ministry of Transportation and Highways (BC MoTH) started a program of bridge upgrading in 1989 to ensure that bridges built before the 1983 A A S H T O requirements would be as safe as possible before the next major earthquake. A brief compendium of the objectives of this program is as follows (BC M o T H , 2000): 1. To minimize the risks of bridge collapse 2. To ensure that important highway routes are preserved for disaster response and economic recovery after earthquakes 3. To reduce damage, life loss and injury during earthquakes A total of 1100 bridges in the high-risk seismic zones of western British Columbia were initially selected for screening purposes while 470 out of those were identified as potential retrofit candidates. The program decisions for the remaining bridges have been deferred at this time but they might be included in the retrofit program in future. The taxonomy of bridges for the purpose of retrofit is as below (BC M o T H , 2000): A. Lifeline Bridges, which are some of the essentially important and major structures in the Lower Mainland such as Oak Street, Mission, Pattullo, Port Mann, Second Narrows, etc and others. B. Disaster Response Route (DRR) Bridges, which are part of a system of routes in the Lower Mainland and on Vancouver Island and hence, most exposed to earthquakes. Disaster Response Routes are such corridors, which must be available for emergency vehicle response following a disastrous earthquake.  2  Chapter 1 - Introduction C. Bridges on Economic Sustainability Routes (ESR's), which are part of corridors in the Lower Mainland, and on Vancouver Island considered essential for maintaining minimum effective transportation for economic survival after a major event. D.  Other Bridges, comprising of the remaining not so critical ones in terms of  emergency response operations and post-earthquake economic sustainability or recovery.  1.1.2 RETROFIT L E V E L S FOR A C C E P T A B L E P E R F O R M A N C E : Several retrofit options and scenarios could be considered for a defined performance objective for a bridge. The underlying factors for selecting a specific level of retrofit for a given structure are its seismic vulnerability, which depends on the site seismicity and structural deficiencies, and the importance of the route on which it is located. The importance attributes consist of traffic access, detour lengths, loss of time and business, expected life loss (if any), etc. The B C M o T H has defined the following retrofitting levels for recommended seismic performance criteria (BC MoTH, 2000): A.  Superstructure Retrofitting, which precludes possible unseating  of bridge  superstructures by tying spans to supporting piers or abutments. Devices such as restrainer cables, shear keys, etc can be used for this purpose. It might also be warranted in certain cases to repair egregious deficiencies in the substructure to ensure the effectiveness of restrainers. Inadequate single column piers with lap splices etc can be cited as an example in this regard. B. Safety retrofitting which is described as the prevention of collapse during the design earthquake that has a return period of 475 years. New bridges are required by current codes to withstand this design level earthquake without collapse and hence it can be  3  Chapter 1 - Introduction assumed that Safety Retrofitting provides a safety level, which is comparable to that of ordinary new bridges designed according to the present codes. Although not guaranteed, it is expected that bridges that have undergone safety retrofit will be usable, at least for emergency vehicles, after a design level earthquake. C. Functional Retrofitting is more of a general term encompassing rigorous and more extensive retrofitting designed to provide a higher level of assurance that bridges will remain in service after an earthquake. Functional Retrofitting can be assumed to provide such performance levels for retrofitted bridges that are comparable to the ones specified by recent codes for important new bridges. As per the program of the B C MoTH, the Lifeline bridges and bridges on or crossing the Disaster Response Routes are being given the highest priority. These are initially being provided with Safety level retrofit. The level of protection for selected Lifeline and Disaster Response Route bridges may be upgraded in future to have a greater assurance that they will remain in service after the design level earthquake. The need for this additional work will however have to be assessed in light of the current performance of bridges in recent earthquakes to ascertain whether the additional expenditure is justified or not. Bridges carrying traffic on Economic Sustainability Routes are being provided with Safety retrofitting. Decisions regarding the less important "Other" bridges are being deferred at this time. It is however expected that a majority of these bridges will require some retrofit and most of these would be retrofitted with the Superstructure retrofit option initially. Selected ones may be provided with a higher level in future. It should be noted that some bridges may already meet the required seismic performance criteria or it may be more cost effective to not go for any kind of retrofit.  4  Chapter 1 - Introduction  Hence the option of no retrofit must always be considered while exploring the effectiveness of various retrofit levels for a specific structure or a set of bridges.  1.2 OBJECTIVES OF STUDY: The primary and foremost objective of this study is to develop and illustrate a risk-based methodology so that given a number of alternatives, the optimal retrofit strategy can be determined for a structure (or a set of structures) based on decision analysis principles employing probability and risk attributes. This entails various steps such as, the identification of required information, how it can be obtained, and how it can be organized in a decision analysis algorithm. Using the decision analysis methodology was thought to be the most appropriate strategy for rational decision-making in the face of inexact or uncertain information. Another main objective of this study is to analyse the sensitivity of data and its influence on the decision outcome since the information about damage states and the corresponding "consequences" is quite subjective and relatively crude. A comparison for ascertaining the effectiveness of decision analysis techniques not using probability and risk attributes in structural retrofit decision-making is also carried out. Finally, a brief review of the screening procedures that are currently in use for ranking potential candidates for retrofit is given in Chapter 8. A secondary objective of devising a two-step methodology for assigning priority indices to bridges based on a costto-benefit ratio approach is also illustrated thereafter.  5  C h a p t e r 1 - Introduction  1.3: SCOPE OF STUDY: Two bridges located on the disaster response route on Vancouver Island were selected for the purpose of this study. The two structures are located quite close to each other and therefore promised to be good candidates for the sake of comparison. The Colquitz river south structure was selected to illustrate the primary objectives of this study in detail i.e. to determine optimal retrofit strategy level for a given structure using decision analysis employing probability and risk attributes, carry out sensitivity analyses, and examine the effectiveness of decision analysis techniques that do not consider probability and risk attributes. The initial assessment of Colquitz was based on the information provided by Choukalos Woodburn Mckenzie Maranda Ltd (Choukalos Woodburn Mckenzie Maranda Ltd, 1994). This is however carried out only for the earthquake with a return period of 475 years. Due to our increased knowledge about the seismicity of B C , the Cascadia subduction scenario and the earthquake with a return period of 2500 years are now considered critically important. Hence all three levels of seismicity were considered for this study. Since Colquitz is a single-pier, singlecolumn structure, the pier is the most vulnerable component in this bridge. The pier was therefore subjected to detailed non-linear analyses to gain insight into the physical damage states of the bridge corresponding to different levels of seismicity. In order to drive the analyses, three earthquake records were selected and scaled to site-specific response spectra corresponding to each level of seismicity. Two damage index definitions were used to predict and compare the nature and extent of damage in the bridge pier. Since the damage index approach can be used readily only for concrete bents presently, damage in other vulnerable bridge components was considered from a demand-to-  6  Chapter 1 - Introduction capacity ratio point of view. The Choukalos Woodburn Mckenzie Maranda Ltd report (Choukalos Woodburn Mckenzie Maranda Ltd, 1994) outlines various deficiencies and gives demand-to-capacity  ratios for different  components  corresponding to  the  earthquake with a return period of 475 years. Using this information and the likely increase in force demands for higher levels of seismicity from the site-specific spectra along with expert judgment, overall scenarios for four different levels of retrofit were considered for the bridge and the physical damage states were translated into dollar damage by deterministic mapping procedures. Indirect damage costs for this structure due to economic loss were also taken into consideration. This dollar damage information was then treated as "consequences" and used in an "expected value" decision analysis algorithm to ascertain the optimal level of retrofit. Sensitivity analyses were then carried out to see i f changes in different input parameters affected the decision outcome. The second structure, namely, the Interurban overpass, was used to have a comparison between the two bridges for retrofit priority purposes. In order to assign a priority index to a bridge, one ought to know the optimal retrofit level for the specific structure. This exercise was not carried out for this bridge and it was assumed that the "Safety Retrofit" (as carried out by the B C MoTH) would be the optimal strategy for this overpass. Damage for the safety level retrofit and the no retrofit scenarios for this structure was assessed on the basis of information provided by Sargent and Vaughan (Sargent and Vaughan, 1998). Priority indices were calculated for both structures to find out the order of retrofit given a hypothetical situation of paucity of funds. This methodology can be extended to set priorities for any number of structures being considered for retrofit. This is of particular importance when funds are scarce, all  7  Chapter 1 - Introduction structures cannot be retrofitted simultaneously and the goal is to make cost effective decisions to gain maximum benefits in terms of money spent and the corresponding levels of safety that can be achieved. Since the two selected structures were retrofitted in different years, the hypothetical decisions examined in his study i.e. determining the optimal retrofit level for Colquitz bridge and the correct order for retrofitting the two candidates are considered for the year 2001.  8  Chapter 2 - General bridge and retrofit description  CHAPTER 2 G E N E R A L BRIDGE A N D RETROFIT DESCRIPTION  2.1 COLQUITZ RIVER SOUTH STRUCTURE: The Colquitz river south bridge is located in Victoria on Vancouver Island, and is a critical structure on the disaster response route. The bridge was constructed in 1977.  2.1.1 GEOMETRIC A N D STRUCTURAL DESCRIPTION: The Colquitz South Structure consists of two spans. It is a single bent bridge with a fixed single column. The two spans are almost equal, being 37.743m and 37.759m respectively, and the total span is 75.502 m from back to back of diaphragms. The central pier is circular in shape with an approximate diameter of 1.68m. The length from the top of the footing to the bottom of the cap beam is 6.42m. The pier supports a cap beam cantilevering on both sides of the column. The cap beam is 2.06m deep at the top of the pier and uniformly tapers off to a depth of approximately l m at the farthest points on both sides of the column. The total length of the cap beam is 9.63m and it supports 5 steel girders, which are continuous over the pier. The 200mm thick bridge deck is then supported over the steel girders. The bridge superstructure has no apparent skew. The  column supporting the above-described system has 40-3 5M bars as  longitudinal reinforcement while 15M spirals at a pitch of 75mm is provided as transverse or shear reinforcement and confinement. There is a 1.38m long lap splice provided at the base of the pier. The pier is founded on battered H-piles. The  9  Chapter 2 - General bridge and retrofit description superstructure is fixed to the pier but the abutments are expansion type as the connection here is through neoprene bearings. The steel girders are composite with the deck in the positive moment regions, being connected through Nelson studs. The abutments are "push-through" type carried by battered, concrete filled pipe piles. The girders are therefore embedded in a concrete diaphragm at their ends, bearing directly against the end fill. Short ballast wall sections project behind the diaphragm at both ends thus separating it from the fill.  Figure 2.1: Elevation of the Colquitz south structure  10  Chapter 2 - General bridge and retrofit description  Figure 2.2: Photograph of the Colquitz pier and superstructure looking east  2 . 1 . 2 S T R U C T U R A L DEFICIENCIES: Although overall problems seemed to be minimal with this structure, there were quite a few deficiencies that needed to be addressed and retrofitted. It should be kept in mind that these defects are pertinent to the design level earthquake with a return period of 475 years. The deficiencies assessed by the Choukalos Woodburn Mckenzie Maranda Ltd about this structure are summarised as follows (Choukalos Woodburn Mckenzie Maranda Ltd, 1994):  2 . 1 . 2 . 1  SUPERSTRUCTURE: The superstructure shows no noticeable deficiencies. The deck is integral with end  concrete diaphragms over the abutments while the superstructure is composite with the deck and has the capability to span in case the pier forms a pin at the base. A small deficiency however does exist in transferring the loads out of the deck and into the  11  C h a p t e r 2 - G e n e r a l bridge a n d retrofit description  diaphragm at the pier. This is due to the fact that the steel is non-composite over the pier and the force transfer from deck to girder is through bond only which is considered to be an unreliable load path.  2.1.2.2 BEARINGS: The "Spencer" type fixed bearings at the pier are thought to exhibit adequate level of performance as the internal high tensile bolts and the anchor bolts have ample capacity to transfer the lateral forces given each bearing picks up its due share of load. There is good reason to believe that the load would be shared uniformly since the deck delivers load to each girder directly and also because the girders are connected transversely with cross frames. The bearings show a good deal of reserve even i f there is an "unzippering" effect that leads to non-uniform load sharing and hence causes the failure of one bearing. Also, the fact that the there might be column base hinging means that loads on the bearing would be further reduced. The elastomeric pads used as bearing at the abutments however show a deficiency since they do not have enough friction capacity to prevent movement transversely. The keeper angles intended to restrain the bridge transversely would therefore be grossly over loaded for the design level earthquake.  2.1.2.3 PIER: The bridge is continuous longitudinally and the longitudinal loads are primarily taken care of, by the passive resistance at the abutments. This is due to the high stiffness of the abutments relative to the pier, which limits the overall displacement in the  12  Chapter 2 - General bridge and retrofit description longitudinal direction, meaning that the pier forces in this direction are not critical. However the column does indicate a tendency of lap splice failure under transverse loading despite the 1.38m of lap provided. This is due to the presence of heavy reinforcement and thus inadequate amounts of transverse steel to confine the splice. Although the pier forms a pin or partial pin/hinge at the base, this would not amount to over all failure since the load is shed through the superstructure to the abutments, which need further assessment. Shear in the column is not thought to be a problem.  2.1.2.4 A B U T M E N T S : Longitudinal loads are carried through passive resistance of the end fill bearing on the diaphragm. The bearing area in this case is equal to the height of the diaphragm and is just adequate given the Caltrans maximum recommended values for the design level earthquake forces. Since the diaphragm actually bears against the ends of the short ballast wall, these portions could potentially fail in bending. The piles carry the most of the transverse forces. The seat lengths at the abutment bearings are inadequate according to ATC6-2. It should however be noted that the predicted movements corresponding to the design level earthquake forces using multi mode spectral analysis are much less than the provided seat length.  2.1.2.5 PILES A N D FOUNDATIONS: The piles beneath the pier foundation have sufficient capacity to carry vertical loads. The dead load of the structure could easily be supported on the central piles even i f  13  Chapter 2 - General bridge and retrofit description the pier pile cap was to fail in tension at the top. There is no indication of pile capacity for the abutments under transverse loading. The pile group just appears to be adequate for the design level earthquake forces based on the nominal capacity value of 40 kips per pile as per Caltrans specifications.  2.1.3 RETROFIT SCENARIOS FOR COLQUITZ SOUTH BRIDGE: The retrofit strategies proposed for considering different retrofit levels for the Colquitz south structure are given in Table 2.1. The proposed safety retrofit is the same as was carried out by the B C M o T H for this structure as part of their retrofit program while other levels are hypothetical in nature suggested merely for the purpose of this study. The pier retrofit was considered as part of the safety retrofit since it can likely retain its axial capacity after a splice failure for a 10% in 50 years earthquake. The abutment backfill and piles are just adequate for the force levels being considered. Their retrofit is considered as a part of the functional retrofit to provide greater assurance for vehicular access after an earthquake.  14  Chapter 2 - General bridge and retrofit description T A B L E 2.1: Retrofit strategies for Colquitz south structure RETROFIT A  RETROFIT B  RETROFIT C  RETROFIT D  (NO RETROFIT)  (SUPERSTRUCTURE  (SAFETY  (FUNCTIONAL  RETROFIT)  RETROFIT)  RETROFIT)  Leave the structure  1 .Add new seat extensions  1. Include all of  1 .Include all of  as it is and accept  at abutments.  retrofit B.  retrofit C.  the future  2.Add shear stud  2. Add new shear  2. Retrofit the  consequences  connectors to girders over  keys at abutments.  abutments.  pier.  3. Add steel jacket 3. Retrofit the to confine column  abutment diaphragms  splice.  and piles.  2.1.4 SITE SOIL CONDITIONS FOR COLQUITZ: The bridge site comprises of different types of soils at varying depths. The top 1.8m contain loose brown and grey sandy gravel and some traces of clay. Mottled brown silty clay and some traces of sand and some gravel to 50 mm diameter are present from a depth of 1.8m to 5.2m. From a depth of 5.2m to 16.5m, there is soft to firm silty clay along with medium plastic sand in seams. There is clayey gravel present from a depth of 16.5m to 18.6m. The average shear wave velocity based on the information given about the top 18.6m of soil is 153.72 m/s.  15  Chapter 2 - General bridge and retrofit description  2 . 2 I N T E R U R B A N OVERPASS: The interurban overpass (Portage Creek Bridge) was constructed in 1983 and crosses the Interurban road and Colquitz river at McKenzie avenue in Victoria on the Vancouver Island. It is also a part of the D R R on the Vancouver Island like the Colquitz river south structure.  2 . 2 . 1 GEOMETRIC A N D STRUCTURAL DESCRIPTION: The Interurban overpass is three spans; double bent steel structure, which is 125.38m long having a central span of 49.85m. The two unequal end spans are 44.95m and 30.58m. Each bent comprises of two concrete piers with a cap beam while the superstructure is a concrete deck having a roadway width of 15.9m along with 1.53m wide sidewalks and aluminium railings. The concrete piers and abutments are founded on steel H piles. The only fixed bearing is at the east abutment with all other ones being expansion type. A l l columns are approximately 1.68m in diameter. The pier no.l and pier no.2 columns have 22-35M and 29-35M bars respectively as longitudinal reinforcement. A l l four columns have 15M spirals at 75mm pitch as transverse reinforcement. The cap beam is 2.14m deep and 19.11m in length with an over-hang of 3.13 m beyond each column. The height of columns from the top of the foundation to the bottom of the cap beam for pier no.l is 7.64m while that for pier no.2 is 4.13m. The superstructure comprises of an assembly of girders, floor beams and stringers.  16  Chapter 2 - General bridge and retrofit description  Figure 2.3: A view of the Interurban overpass roadway  2.2.2 STRUCTURAL DEFICIENCIES: The following deficiencies exist in the as-built structure (Sargent and Vaughan Engineering Limited, 1999):  17  Chapter 2 - General bridge and retrofit description  2.2.2.1 GIRDER DIAPHRAGMS: The necessity to transfer large forces from the deck slab to the piers and abutments means that there would be a failure of the connections between the existing floor beams and the girders, thus leading to their instability and collapse.  2.2.2.2 A N C H O R BOLTS: The forces generated from the superstructure would grossly overload the anchor bolts at the abutments and the piers leading to insufficient transverse restraint. This is a similar problem regarding the longitudinal resistance of anchor bolts at the fixed bearing of the east abutment.  2.2.2.3 PIER A N D A B U T M E N T PILES A N D FOUNDATIONS: It is expected that the pier piles would survive the design seismic event forces in combined bending and axial load although only marginally. Pile pull out at piers is also not a big problem although the assumption of rocking motion of the pier footings does push the limits of pile capacity along with causing potential shear failure of the pile caps. Pile pull out at the east abutment is somewhat of a bigger concern due to the fixed nature of connection between the superstructure and the abutment. Also, the toe projection at the east abutment pile cap footing shows a potential for shear failure i f the piles beneath the backwall retain their anchorage.  18  Chapter 2 - General bridge and retrofit description  2.2.2.4 C O L U M N S IN PIERNO.2: A redistribution of forces after the formation of plastic hinges in columns of pier no.l causes higher shear forces to be resisted by the columns in pier no.2. Since the push over analysis shows a shear failure in pier no.2 columns before forming a plastic hinge, the aforementioned redistribution would cause a sudden shear failure, which is an unacceptable mode of failure.  2.2.2.5 A B U T M E N T B A C K W A L L : The east abutment backwall could fail in bending under passive pressure due to longitudinal loads from the superstructure.  2.2.3 RETROFIT OF I N T E R U R B A N OVERPASS: As mentioned in chapter 1, the Interurban Overpass was only considered for having a comparison of priority indices for the two structures. For the sake of illustration, it was deemed reasonable to assume that the Safety Retrofit level (as carried out by the B C M o T H for this structure) would be the optimal strategy for this bridge. Hence only the Safety Retrofit scenario is described herein for the stated bridge. A summary of the pertinent details is as follows (Sargent and Vaughan Engineering Limited, 1999): (a) Introduce diaphragm braces at the floor beams over the piers and abutments. Reinforce floor beam to exterior girder connections with welds to prevent block tear out at the end of the beam.  19  Chapter 2 - General bridge and retrofit description (b) Anchor structural steel weldments to the pier caps and abutment seats for transverse restraint. Extend abutment seats to accommodate this. Add a longitudinal restrainer to the west abutment, leaving allowance for expansion. (c) Provide FRP wraps to increase the shear capacity without altering the moment capacity in order to prevent the sudden shear failure in columns of pier no. 2. (d) Strengthen the toe projection at the east abutment pile cap in shear by a concrete thickening of the footing, dowelled into the abutment wall.  2.2.4 SITE SOIL CONDITIONS FOR INTERURBAN: This bridge site has a top layer of gravel, which is about 0.5m thick. There is soft to firm brown silty clay from a depth of 0.5m to 5.5m. Soft to firm grey silty clay is present along with traces of medium plastic fine sand from a depth of 5.5m to 15.0m along with some gravel which is 20mm to 50mm. Below this layer, there is dense gravely sand along with some silt and small gravel up to 22.25m depth. The average shear wave velocity based on the given information is 171.61m/s.  2.3 COSTS OF RETROFIT: A break down of the costs for various retrofit measures for the two structures considered for this study is given in Tables 2.2 to 2.4. The costs corresponding to different retrofit levels are given in terms of C D N dollars for both the retrofit year of the structure and 2001 assuming an inflation rate of 4% for the Colquitz bridge.  20  Chapter 2 - General bridge and retrofit description  2.3.1 COLQUITZ RIVER SOUTH BRIDGE: The costs for retrofitting various components in the Colquitz South structure are shown in Table 2.2. A l l estimates except for the last activity are based on actual numbers given in the report by Choukalos Woodburn Mckenzie Maranda Ltd (Choukalos Woodburn Mckenzie Maranda Ltd, 1994). Since an actual retrofit was not designed for the abutment piles and diaphragm, a hypothetical value had to be assumed for this activity. The cost of functional retrofit was assumed as 1.5 times that of safety retrofit.  Table 2.2: Retrofit activity costs for Colquitz south structure Retrofit Activity  COST (1994 C D N $)  Add new shear keys at abutments  15000  Provide new seat extensions at abutments  18000  Add shear stud connectors to girders over pier  4000  Provide steel jacket to confine column splice  15000  Retrofit abutment piles and diaphragm  26000 (assumed)  The total costs for different retrofit scenarios as defined in table 2.1 would therefore be as follows:  21  Chapter 2 - General bridge and retrofit description Table 2.3: Costs for different retrofit levels as defined for Colquitz south structure COST (1994 C D N $)  COST (2001 C D N $)  0  0  Superstructure retrofit (Retrofit B)  22,000  29,000  Safety retrofit (Retrofit C)  52,000  68,500  Functional retrofit (Retrofit D)  68,000  102,250  RETROFIT L E V E L No retrofit (Retrofit A)  2.3.2 INTERURBAN OVERPASS: The costs for safety level retrofit for the Interurban overpass are shown in table 2.4 as follows (Sargent and Vaughan Engineering Limited, 1999):  Table 2.4: Break down of safety retrofit costs for Interurban overpass RETROFIT A C T I V I T Y  COST (1999 C D N $ )  Cost of Mobilization  20,000  Cost of West abutment retrofit  48,106  Cost of Pier N o . l retrofit  49,080  Cost of Pier No.2 retrofit  76,400  Cost of East abutment retrofit  75,910  Total cost of Safety retrofit  264,496  The safety retrofit cost in terms of 2001 C D N $ is thus 295,000 dollars for an inflation rate of 4%.  22  Chapter 3 - Decision analysis methodologies CHAPTER 3  DECISION A N A L Y S I S METHODOLOGIES  3.1 INTRODUCTION: Complex decision-making involves uncertainty due to our inability to predict accurately the factors influencing future events. Decision analysis methodologies are powerful tools that enable the decision maker to take into account all of the various factors such as different courses of action available, outcomes of nature and corresponding consequences of each outcome to arrive at the optimal decision given the miscellaneous constraints and uncertainties. Since it is impossible to take all of the aforementioned factors simultaneously into account when faced with a complex problem, decision analysis entails the decomposition of the problem into a decision tree. It starts with all possible courses of action (ai,a2,.. ..a ), followed by the various possible states of nature ( 9 i , 0 2 , . . . 0 ) and their n  n  probabilities [P(0i)P(02), ....P(G )] and consequences (un,ui2,...ui ,...u )[Figure 3.1]. A 1  n  m  nm  decision tree therefore starts with a decision node at the left representing alternative courses of action followed by chance nodes representing chance events that comprise of the outcome of nature. The occurrence of a chance event can be considered as a random event such as an earthquake over which the decision maker has no control. The main problem is thus decomposed into a number of smaller problems that help decide the optimal course of action through the standard folding back procedure. A detailed treatment of the folding back procedure is given by Benjamin and Cornell (Benjamin and Cornell, 1970), Fabrycky and Thuesen (Fabrycky and Thuesen 1984), and many others.  23  Chapter 3 - Decision analysis methodologies  e.[P(eo] Ull  Figure 3.1: General layout of a decision tree  It is usually convenient to ensure that the events considered in a decision problem should be mutually exclusive and collectively exhaustive i.e. they should have no overlap and should completely fill up the sample space. The sum of their probabilities will therefore be equal to one.  24  Chapter 3 - Decision analysis methodologies  3.2 BASIS FOR R A N K I N G COMPETING A L T E R N A T I V E S : A n expected value for a random variable can be calculated i f its probability mass function or probability density function is known. If we use such probability distributions to describe the profit or cost of an investment as a random variable, the expected values of these parameters can be assumed as reasonable basis for the sake of comparing alternatives. This means that the expected profit or cost of an investment proposal depicts the long-term profit or cost that would be realised i f the investment were repeated a large number of times and the pertinent probability distributions remained unchanged. Since this study focuses on bridge retrofit decision-making and as bridges are structures with considerable life spans, the expected value criterion appears to be a logical basis for comparing alternatives. Also, since the retrofit decision-making has to be done by the government, the expected value criterion seems plausible, as the money to be spent on the retrofit program on a yearly basis is small as compared to the total annual budget. This leads to an effective and logical use of the taxpayer's money on a long-term basis.  3.3 DETERMINATION OF EXPECTED V A L U E : The expected value for a course of action can be determined by solving the decision tree through the use of the "folding back" or "roll back" procedure. The following two rules have to be employed while starting from the tips of the decision tree's branches and working back towards the initial nodes: 1. If the node happens to be the first chance node, calculate the expected value of that node by summing the products of the possible outcomes and their  25  Chapter 3 - Decision analysis methodologies corresponding probabilities. For the following chance nodes, determine the expected value based on the 'folded back' values on the adjacent nodes to the right of the one being considered. 2. If the node is a decision node, select the minimum cost or maximum profit from the adjacent nodes to the right of the node under consideration. The expected costs thus obtained for each alternative have then to be discounted to the present.  3.4 DECISION TREE FOR PRESENT STUDY: The current policy by the B C M o T H is to retrofit deficient bridges for various performance levels corresponding to the current design earthquake i.e. one with a return period of 475 years. The earthquakes with smaller return periods and hence less magnitude (and intensity) are likely to cause only negligible damage to the unretrofitted bridges while retrofitted structures would most likely have no damage from the lower level earthquakes. The decision would thus be sensitive to the design level earthquake and the ones that have considerably longer return periods such as the Cascadia subduction event or the crustal event with a probability of 1 in 2500 years but not so much to the lower level earthquakes. It was therefore decided to take the design earthquake, as the lowest level to be incorporated into the decision analysis, as still lower level earthquakes would not have much of a bearing on the decision. The three different earthquake levels and hence chance events identified for the decision tree for this study (Figure 3.2) are:  26  C h a p t e r 3 - Decision analysis methodologies  (a) Earth quake level-1: Design Level Earthquake abbreviated as EQ1 on the decision tree (b) Earth quake level-2: Cascadia subduction event with a probability of 1 in 2500 years, abbreviated as EQ2 on the decision tree (c) Earth quake level-3: Crustal earthquake with a return period of 1 in 2500 years, abbreviated as EQ3 on the decision tree The probabilities of occurrence of various earthquakes as shown in the decision tree of figure 3.2 are annual probabilities of these events. Also, it has been assumed that the probability of having joint events in a year e.g. two earthquakes of level-1, one earthquake each of level-1 and level-2, etc is negligibly small and hence neglected. Since there is considerable variability in the nature of earthquakes even i f they have similar magnitudes, it was decided to use three different earthquake records for each level of earthquake under consideration. This could lead to considerably different damage states and hence different consequences for each level of earthquake. The states of nature were therefore identified as different damage levels corresponding to each earthquake level. Assigning a probability of 1/3 to each damage level corresponding to the same level of seismicity was based on the "Laplace principle" or the "Principle of Insufficient Reason". It effectively means that in the absence of other information, each damage state is considered equally likely. The three earthquake records used in the present study are Loma Prieta (LP), E l Centro and Miyagi (MI) and would be referred to as record-1, record-2 and record-3 respectively in the subsequent chapters. The consequences associated with experiencing a certain type of earthquake for a given level and a given retrofit strategy are shown at the tip of the decision tree. The  27  Chapter 3 - Decision analysis methodologies letter " C " denotes the consequence index while the first subscript shows whether the damage corresponds to the first, second or third record used. The second subscript is for the level of earthquake and hence 1,2 and 3 show the design level earthquake, the Cascadia subduction event and the crustal event with the return period of 1 in 2500 years. The following alphabets designate the level of retrofit for which the consequences have been calculated. The possible courses of action are clearly the retrofit options available, identified as NRF (for no retrofit), SSRF (for superstructure retrofit), SFRF (for safety retrofit) and F C R F (for functional retrofit) in the decision analysis tree as shown in figure 3.1. As an example, the expected value E (per annum) of NRF would be give by: E (NRF) = S l , R F * P l + S ,NRF*P2 + S , N R F * P 3 N  2  3  where, 51,N R F 52, N R F  -  +C21NRF*l/3 + C 3 i N R F * l / 3  C]2NRF*l/3 + C22NRF*l/3 + C32NRF*l/3  =  53, N R F  CllNRF*l/3  =  Cl3NRF*l/3 + C23NRF*l/3 + C33NRF*l/3  Si, S2 and S3 values for other retrofit scenarios can be similarly determined using the corresponding consequence indices. Expected values per annum for other alternatives can then be determined and discounted to the present for the purpose of comparison.  28  Chapter 3 - Decision analysis methodologies  H3(Pi) B$(P2) XiS)  Q.SSRF  03i Qgj^ IiS)— G i s w •05)— G s w  H?(P3)  H3(Pl)  GZSSSF  NRF  -Q2L  H?(P3)  S5KF  -03L  SM2L- Qtsw  -m—Qs*  H3(P1)  SEW  X^3)—  H?(P2)  COTF  KHt  H3(H)  H?(P2)  -Q3L  GlFOT  : -  (j2FOT  -G2>—Ora* ClKKF -05)  GKRF  -03)— GlKPF  Figure 3.2: Decision analysis event tree for bridge retrofit problem  3.5 DECISION M A K I N G WITHOUT PROBABILITY V A L U E S : A decision maker may be faced with a situation wherein he or she finds it difficult to assign probabilities to future events with a sufficient degree of confidence. When dealing with earthquakes, one may find it hard to have too much confidence in assigning probabilities to events of such nature. For example, the probability of getting a  29  Chapter 3 - Decision analysis methodologies large earthquake after having one of similar nature in the recent past could be considerably smaller. It was therefore thought appropriate to look at the proposed decision problem based on the premise that there is no rational way of assigning probabilities to the associated events. This was however done only to have a comparison between the results of the two broadly different categories of decision analysis. The various methods employed for this comparison are: (a) The Maximin and the Maximax rule, which can be termed as Minimax and Minimin rules since we are dealing with costs for our problem. The first rule is based on an extremely pessimistic view of the outcome of nature while the second one is based on an extremely optimistic view of the outcome of nature. (b) The Hurwicz rule which embraces a relative degree of optimism and pessimism through an index of optimism a, which varies between 0 and 1. (c) The Minimax Regret rule has a conservative underlying philosophy as it seeks to minimize the maximum regret for a given set of actions. The effectiveness of these techniques is examined and discussed in Chapter 7 along with sensitivity analyses.  30  Chapter 4 - Seismic hazard evaluation  CHAPTER 4  SEISMIC HAZARD EVALUATION  Evaluation of the nature, extent and likelihood of seismicity is an essential part of obtaining data to be used for damage state assessment and the decision analysis algorithm. This chapter discusses in detail the various aspects of seismic hazard evaluation and elucidates the methodology adopted for the purpose of this study. The determination of earthquake occurrence rates and probabilities, selection of earthquake records, ground motion amplification aspects and site-specific record generation are described herein.  4.1 METHODOLOGIES FOR SEISMIC H A Z A R D PREDICTION: The two fundamental  procedures  for seismic hazard prediction are the  deterministic and the probabilistic approaches. The deterministic analyses make use of discrete, single-valued events to arrive at scenario-like description of earthquake hazard. The probabilistic methodology however allows the incorporation of multi-value or continuous events and models. Of most importance, the probability of different magnitude (or intensity) earthquakes occurring is included in the analysis (Reiter, 1990). The latter methodology can be used in two ways i.e. by developing the response spectrum for a specific value of peak ground acceleration or by constructing the uniform hazard response spectrum (UHRS) for the site under consideration. A thorough discussion of the  31  Chapter 4 - Seismic hazard evaluation probabilistic seismic analysis is given by the EERI committee on seismic risk (EERI committee on seismic risk, 1989). Since the bridges for this study are located in Victoria, the spectra for Victoria for various earthquake levels (with different mean return periods or probabilities of exceedence) were used as target spectra to assess the expected seismicity at the bridge sites as explained in the following section.  4.2 SYNOPSIS OF M E T H O D O L O G Y FOR PRESENT STUDY: This study employs three different levels of earthquakes for considering the effects of various seismicity levels on the extent of damage. The three levels of seismicity considered correspond to the current code design level earthquake with a return period of 475 years, the Cascadia subduction event scenario, and the future code design earthquake with a return period of 2500 years with the level of seismicity increasing from the first to the last case respectively. Since different earthquake records have different characteristics such as the number of pulses, duration, etc, three natural earthquake records were selected for the purpose of damage state assessment corresponding to each level of seismicity. These records are from the 1940 E l Centro earthquake, the 1978 Miyagi earthquake and the 1989 Loma Prieta earthquake. The soil information for the bridge sites indicated that these are overlain by soft soils and therefore the firm-ground response spectra for Victoria corresponding to each level of seismicity were increased to take the effects of soil amplification into account. Each of the selected natural earthquake records was then modified to match its response spectrum to the site-specific (target) spectra for each  32  Chapter 4 - Seismic hazard evaluation seismicity level obtained in the previous step thus leading to nine modified records. These records were then used to drive the non-linear dynamic analysis for the Colquitz south structure.  4.3 CONSIDERED E A R T H Q U A K E L E V E L S : The suggested methodology relies on an integrated approach whereby the first and third levels of earthquake considered are based on probabilistic UHRS methodology while the second level corresponds to the deterministic Cascadia subduction event response spectrum. A brief background of the various earthquake levels thus considered is as follows:  4.3.1 CURRENT DESIGN L E V E L E A R T H Q U A K E : Since the B C M o T H is currently retrofitting bridges to achieve certain performance objectives corresponding to the current code design level earthquake with a return period of 475 years, it was deemed reasonable to incorporate this as the basic/lowest earthquake level. As discussed in Chapter 3, the still lower level earthquakes were not considered as they would most likely cause negligible damage to the existing infrastructure and hence would not have any significant bearing on the decision outcome. The UHRS spectral values correspond to the Robust 50 percentile with a probability of th  0.0021 p.a. on firm ground for a damping of 5% as shown in figure 4.1 (Private correspondence, 1999).  33  Chapter 4 - Seismic hazard evaluation  Figure 4.1: Victoria 10%/50 year robust UHRS (50 percentile values) th  (Private correspondence, 1999)  4.3.2 C A S C A D I A SUBDUCTION EVENT: The next higher-level earthquake considered is the Cascadia subduction scenario whereby the seismicity is primarily driven by the subduction of the Juan de Fuca plate under the North American plate. There is evidence that large offshore subduction earthquakes have occurred many times in the past several thousand years with moment magnitudes M w in the range of 8-9 on the thrust fault between the subducting Juan de Fuca plate and the North American plate (At Water et al., 1995; Clague et al., 1995). Hence there is a realistic threat of damage from the subduction scenario event, which was therefore included in this study. The scenario earthquake has a magnitude Mw=8.2 on the subduction zone at the depth of 25 kilometres (Private correspondence, 1999). The response spectrum values selected for this study correspond to the 84 percentile values (exceeded 16% of the th  times). The 84th percentile values were considered for this study because the spectrum so obtained lies between the 10%/50 year UHRS and the 2%/50 year UHRS for the period  34  Chapter 4 - Seismic hazard evaluation range of interest. The spectral values for the 84 percentile Cascadia subduction scenario th  correspond to the 2% chance of exceedence in 50 years. The target spectrum for this scenario event is shown in figure 4.2 as follows:  0  1  2  3 T(sec)  4  5  •  Figure 4.2: Cascadia subduction earthquake scenario spectrum (84 percentile values) for Victoria th  (Private correspondence, 1999)  4.3.3 FUTURE DESIGN L E V E L E A R T H Q U A K E : It is now being increasingly recognised that the threat to the Vancouver metropolitan area and Victoria stems from shallow crustal earthquakes and not so much from the subduction earthquakes. There has been a greater consensus about this opinion due to extensive work conducted by structural and geotechnical engineers over the past few years. It has to be recognised that as a society becomes more and more wealthy and at the same time aware of the damage potential of earthquakes, the level of acceptable  35  Chapter 4 - Seismic hazard evaluation risk decreases. The upcoming Canadian code to be introduced in the near future therefore specifies a higher design level earthquake with a return period of 2500 years or a probability of exceedence of 2% in 50 years (0.000404 p.a.). This is being regarded as the maximum credible earthquake level and since future performance requirements are going to be based on this level of seismicity, it was thought appropriate to include this as the highest level for the present study. The UHRS spectral values thus selected correspond to the Robust 50 percentile with a probability of 0.000404 p.a. on firm ground for a th  damping of 5% (Figure 4.3).  Figure 4.3: Victoria 2%/50 year Robust UHRS (50 percentile values) l  (Private correspondence, 1999)  36  Chapter 4 - Seismic hazard evaluation  4.4  ANNUAL  OCCURRENCE  RATES  AND  EARTHQUAKE  PROBABILITIES: When considering individual zones around a site, it is normal practice to use the Gutenberg-Richter relationship to relate the magnitude of earthquakes to their recurrence rate. The relationship in its simplest from is as follows: log N = a - b M Where ' M ' is the magnitude, ' N ' is the average number of earthquakes per year with magnitude more than or equal to M while 'a' and 'b' are constants which are generally obtained by regression on a database of seismicity of the source zone of interest.  4.4.1 E A R T H Q U A K E PROBABILITY DETERMINATION: It is imperative to determine the annual probabilities of encountering the aforementioned levels of earthquakes so that the element of risk can be introduced into the decision process. The occurrence rates for level 1, 2 and 3 earthquakes are 1 in 475 years, 1 in 2500 years and 1 in 2500 years respectively (Private correspondence, 1999). Since we are dealing with events having long return periods and the chance of having more than one per annum is negligible, the annual probabilities of occurrence would be approximately equal to the annual occurrence rates. Table 4.1 gives the annual probabilities of the various earthquake levels. It should be noted that the zero-level earthquake as shown in table 4.1 comprises of all events with a lower level of seismicity as compared to the current design level earthquake. It is therefore only given for the sake of completeness.  37  Chapter 4 - Seismic hazard evaluation  T A B L E 4.1: Annual probabilities of various earthquake levels EQ Level 0 1 2 3  Recurrence interval  Annual occurrence rate  1 in 475 years 1 in 2500 years 1 in 2500 years  0.0021 0.000404 0.000404  Annual Probability (Pi) 0.9971 0.0021 0.0004 0.0004 » 1  4.5 E A R T H Q U A K E RECORDS SELECTED FOR PRESENT STUDY: A brief discussion about the selected earthquakes and the corresponding records is as follows:  4.5.1 L O M A PRIETA E A R T H Q U A K E : The Loma Prieta earthquake occurred at 5:04 pm on the 17 of October 1989 th  causing 62 deaths and over $ 10 Billion US in damage. The magnitude of earthquake on the Richter scale was 7.1 and it was centred about 101 K m (60 miles) south of San Francisco. The acceleration-time history for this earthquake is shown in figure 4.4.  38  Chapter 4 - Seismic hazard evaluation  t(sec)  Figure 4.4: Loma Prieta earthquake acceleration time history (North-South component, Capitola fire station, 2000 points at 0.02 sec)  The record shows strong shaking for about 20 seconds with a number of pulses in excess of 0.4 g in the first 10 seconds. The rumbling can be seen to continue up to about 40 seconds.  4.5.2 E L CENTRO E A R T H Q U A K E : The Imperial Valley E l Centro earthquake occurred on the 18 of May 1940 th  causing 9 deaths and about $ 6 M US in damage. It had strong motion duration of about 25 seconds. The acceleration-time history of the Elcentro earthquake is shown in Figure 4.5.  39  Chapter 4 - Seismic hazard evaluation  0.4  t(sec)  Figure 4.5: El Centro earthquake acceleration time history (North-South component, Japan arch centre, 1134 points at 0.02 sec)  As can be seen form the record, the Elcentro record does not have as high a P G A as that of Loma Prieta. There is only one pulse in excess of 0.3g but there is a large pulse of about 0.2g around the 12-second mark after the initial high acceleration levels seem to be on the decrease.  4.5.3 M I Y A G I E A R T H Q U A K E : A subduction type earthquake record was also used in this study. This is the magnitude 7.4 Miyagi event in the Pacific Ocean off the coast of Miyagi (Japan). This earthquake occurred on the 12 of June 1978 resulting in the death of 27 while causing th  injuries to about 1100 people (Fowler, 1980). The acceleration time history shown for this record in Figure 4.6 depicts a number of pulses quite close to or in excess of 0.2g while considerable intensity of shaking can be seen to last for a longer duration as compared to the first two records.  40  Chapter 4 - Seismic hazard evaluation  0.3  -0.2  :  —  IM  ¥  -0.3 t(sec)  Figure 4.6: Miyagi Subduction earthquake acceleration-time history (North-South component, 2000 points at 0.02 seconds)  4.6  GROUND  MOTION AMPLIFICATIONS A N D SITE-SPECIFIC  DESIGN SPECTRA: The summary logs show that the sites for bridges selected for this study are overlain by soft soils. The average shear wave velocities calculated for the Colquitz river south structure and the Interurban overpass were 153.72 m/sec and 171.16 m/sec respectively. The soil amplification effects therefore had to be taken into account. A simple procedure employing the methodology proposed for the upcoming Canadian code was used. This procedure comprises of classifying a given soil into one of six different classes based on average soil shear wave velocity and multiplying the Sa values at different periods for the firm ground conditions with suitable coefficients depending upon the range of values at periods of 0.2 and 1.0 seconds. The various site classes defined as below in table 4.3 are based on the average properties in top 30m (Private correspondence, 2000).  41  Chapter 4 - Seismic hazard evaluation  Table 4.2: Description of various site classes based on soil average shear wave velocity (Private correspondence, 2000).  Site Class  Soil Profile Name  Soil Shear Wave Average Velocity, Vs (m/sec)  A  Hard Rock  V s > 1500  B  Rock  760 < V s < 1500  C  Very Dense Soil and Soft  360 < Vs < 760  Rock D  Stiff Soil  180<Vs<360  E  Soft Soil  V s < 180 and/or 3m < thickness < 15m  F  Others  Although the above classification is based on the average shear wave velocities for the top 30m of soil, the shear wave velocities used were calculated from the data provided for the top 16.3m for Colquitz bridge and the top 15.05m for the Interurban overpass respectively. It is clear that both sites can be classified as Class E and therefore the design spectra would be the same for both bridge structures. It must however be kept in mind  42  Chapter 4 - Seismic hazard evaluation that explicit analyses and calculations of damage were only carried out for the Colquitz river south structure. The next step after assigning the site soil to a specific class is to determine the new design spectral acceleration values that take into account the amplification due to soft soil. The design spectral values S(T) are determined as follows using linear interpolation for intermediate values of T (Private correspondence, 2000): S (T) = FaSa(o.2) for T < 0.2sec = FvSa (0.5) or FaSa (0.2) which ever is smaller for T = 0.5 sec = FvSa (i.o) for T = 1.0 sec = FvSa (2.0) for T = 2.0 sec = FvSa (2.0)/ 2 for T > 4.0 sec where Fa is an acceleration-based site coefficient, Fv is a velocity-based site coefficient and Sa is the 5% damped spectral response acceleration for a period T. The values of Fa as a function of site class and T=0.2 sec spectral acceleration and Fv as a function of site class and T=1.0 sec spectral acceleration are given in tables 4.4 and 4.5 respectively as follows.  43  Chapter 4 - Seismic hazard evaluation  Table 4.3: Values of Fa as a function of Site class and T = 0.2 sec spectral acceleration (Private correspondence, 2000). Values of Fa Site class  Sa o. )<0.25  Sa (0.2)= 0.5  Sa (o,2) =0.75  Sa .2)=1.0  Sa .2) = 1-25  A  0.7  0.7  0.8  0.8  0.8  B  0.8  0.8  0.9  1.0  1.0  C  1.0  1.0  1.0  1.0  1.0  D  1.3  1.2  1.1  1.1  1.0  E  2.1  1.4  1.1  0.9  0.9  F  ~  (  2  (0  (0  ~  Table 4.4: Values of Fv as a function of site class and T = 1.0 sec spectral acceleration (Private correspondence, 2000)  Values of Fv Site class  Sa .2)<0.1  Sa .2)= 0.2  Sa (0.2) =0.3  S a . = 0.4  A  0.5  0.5  0.5  0.6  0.6  B  0.6  0.7  0.7  0.8  0.8  C  1.0  1.0  1.0  1.0  1.0  D  1.4  1.3  1.2  1.1  1.1  E  2.1  2.0  1.9  1.7  1.7  F  ~  ~  ~  (0  (0  (0  2)  Sa .2)>0.5 (0  44  Chapter 4 - Seismic hazard evaluation  The design spectra corresponding to the various earthquake levels considered in this study hence obtained are shown as below in figure 4.7. Figure 4.8 shows all the six spectra (modified and unmodified) for the sake of comparison.  0  0.5  1  1.5  2  2.5  3  2%/50 year design spectrum — — Cascadia(84th perc) scenario  3.5  4  4.5  5  10%/50 year design spectrum  Figure 4.7: Design spectra for bridge sites with soil amplification effects  •10%/50 years spectrum for firm ground • Cascadia spectrum for firm ground •2%/50 years spectrum for firm ground -10%/50 year spectrum with soil amplification effects -Cascadia spectrum with soil amplification effects -2%/50 year spectrum with soil amplification effects  Figure 4.8: Firm ground and site-specific design spectra 45  Chapter 4 - Seismic hazard evaluation It is evident from Figure 4.8 that the site-specific design spectra are much higher than their firm ground counterparts. As can be seen from tables 4.4 and 4.5, the values of Fa and Fv vary between 2.1 and 0.9 for Soil Class E. Considering 20 and 50 percent of mass contribution from the bridge structure to the pier, the elastic period of the pier for the Colquitz bridge pier lies in the range of 0.8 seconds to 1.0 seconds. Clearly, soil amplification effects for the period range of interest are considerable.  4.7 SITE-SPECIFIC RECORD GENERATION: The final step in the seismic hazard evaluation is to obtain site specific earthquake records which would then help drive the non-linear structural analyses. The selected earthquake accelerograms were therefore modified so that their response spectra matched the obtained design spectra as described in section 4.6. The computer software S Y N T H was used for this purpose. A brief discussion about the program follows.  4.7.1 C O M P U T E R P R O G R A M SYNTH: S Y N T H can be used to generate artificial time histories compatible with a target spectrum. The initial time history is a real accelerogram with a maximum number of values equal to 8192 (Naumoski, 1985). There are three input files where by the user inputs information such as the number of initial and generated acceleration values, the time interval between these values, PGA, etc along with initial accelerogram values at equal time intervals, and periods and pseudo acceleration spectral values for the target spectrum. The output also comprises of three different files where by information such as  46  Chapter 4 - Seismic hazard evaluation periods and spectral values of the computed spectrum are output along with acceleration values of the generated time history.  4.7.2 MODIFIED RECORDS: The site-specific spectra determined earlier were input into S Y N T H as target spectra along with time histories for the various earthquakes. Each earthquake record was changed to match its spectrum with the three different target spectra for various earthquake levels and hence 9 different site-specific records were generated for driving the non-linear dynamic analyses of the Colquitz bridge pier. A reasonable agreement was reached between the target and computed Spectra for the various time histories by using S Y N T H as evident from Figures 4.9 through 4.11. Each of the modified records (or its spectrum) is abbreviated hereinafter by two alphabets depending on the seed record followed by the level of seismicity (1,2 or 3) to which it has been matched. EC1, LP2 and MI3, etc thus show the E l Centra record matched to the earthquake level-1 spectrum, Loma Prieta record matched to earthquake level-2 spectrum and Miyagi record matched to earthquake level-3 spectrum respectively.  47  C h a p t e r 4 - Seismic h a z a r d evaluation  • 10%/50 years EQ target spectrum -EC1 computed spectrum • LP1 computed spectrum MM computed spectrum Period,T(sec)  Figure 4.9: Target and computed spectra for level-1 earthquake  Cascadia EQ target spectrum  1  0.8  EC2 computed spectrum  0.6 Sa(g) 04  LP2 computed spectrum  0.2 0 Period,T(sec)  Ml2 computed spectrum  Figure 4.10: Target and c o m p u t e d spectra f o r level- 2 earthquake  -2%/50yearEQ target spectrum - E C 3 computed spectrum - LP3 computed spectrum MI3 computed spectrum 2  3 Period,T(sec)  Figure 4.11: Target and c o m p u t e d spectra for level-3 earthquake  The various time histories corresponding to the three different earthquake levels are shown in Figures 4.12 through 4.14.  48  Chapter 4 - Seismic hazard evaluation  Figure 4.12: Acceleration-time histories for level-1 earthquake  The P G A ' s for EC1 and LP1 occur at different times in the time histories. In LP1, there is a pulse of about 0.4g around the 14-second mark while there are a number of pulses in access of 0.25g that are absent in EC1. There is more sustained shaking in M i l lasting for a longer duration. This record has a number of pulses of 0.2g to 0.25g around the 15-second mark.  49  Chapter 4 - Seismic hazard evaluation  t(sec) Figure 4.13(c)-MI2  Figure 4.13: Acceleration-time histories for level-2 earthquake  It can be seen from the level-2 records that the P G A for these is less as compared to the level-1 records. However, the expected displacement demand on the structure would be more due to the shape of the target spectrum for this level. Looking carefully at the records reveals that large pulses in LP2 are more scattered during the time history and it has more pulses with amplitudes around 0.3g as compared to EC2. MI2 shows more uniform shaking with only one pulse in excess of 0.3g but a large number of pulses in excess of 0.2g.  50  Chapter 4 - Seismic hazard evaluation  t(sec) Figure 4.14(c)-MI3  Figure 4.14: Acceleration-time histories for level-3 earthquake  It is evident from level-3 records that the P G A for these is considerably higher as compared to records for the first two levels. EC3 has one pulse each with amplitudes around 0.8g and 0.4g. In comparison, LP3 has one pulse around 0.8g, one around 0.7g and two around 0.4g amplitude levels. More yield excursions may therefore be suspected for the bridge pier when subjected to this record. MI3 is similar to M i l and MI2 inasmuch as it has sustained, uniform shaking for a longer duration. It has one pulse of about 0.8g and a number of pulses with approximately 0.4g amplitude. The strong shaking duration is longer for this record as compared to the other two.  51  Chapter 5 - Non-linear analysis and damage state assessment  CHAPTER 5 NON-LINEAR ANALYSIS AND DAMAGE STATE ASSESSMENT  This chapter focuses on quantifying damage for the two bridges under consideration. The Colquitz river south structure is given a more thorough treatment whereby damage in the single column pier is assessed through non-linear analyses for different earthquake levels (using various scaled records) for each retrofit scheme as defined in Chapter 2. Damage in other structural components of this structure for various retrofits is determined on the basis of the information provided by Choukalos Woodburn Mckenzie Maranda Ltd (Choukalos Woodburn Mckenzie Maranda Ltd, 1994) and expert judgment. For the Interurban overpass, no analyses were carried out and structural damage was determined on the basis of expert judgment, information provided in the report produced by Sargent and Vaughan Engineering Ltd for B C M o T H (Sargent and Vaughan, 1999) and the trends shown by the design spectra for different seismicity levels. Safety retrofit was assumed to be the optimal strategy for the Interurban overpass and hence damage has only been assessed for this scenario. This chapter therefore  illustrates the application of non-linear analysis  techniques to assess pier damage with the help of damage indices. Important aspects pertinent to bridge bent modelling, damage index calibration and eliciting the overall damage for various retrofits are discussed in detail in the following sections.  52  Chapter 5 - Non-linear analysis and damage state assessment  5.1 G E N E R A L ASPECTS OF THE COLQUITZ BENT M O D E L L I N G : The Colquitz bridge is a single bent, single pier structure with a fixed column and two expansion type abutments. Since this forms a relatively simple structural configuration, it was deemed suitable to employ the Lumped-Parameter model for this bridge. The bridge bent was modelled as a single degree of freedom structure with seismic mass lumped at the top node [Figure 5.1(b)]. The column length was divided into five flexural and one rigid element. The rigid element was introduced to take into account the stiffness difference between the column and the superstructure. The length of this link was determined to be 3m starting from the lower end of the cap beam. The total length of the column thus modelled was 9.42m from the base of the column to the centre of the lumped mass. The pier has piles driven to the bedrock and was thus treated as fixed at the base. Since the bridge is longitudinally continuous and the abutment stiffness is much higher than that of the pier, the pier forces in the longitudinal direction are not critical (Choukalos Woodburn Mckenzie Maranda Ltd, 1994). The longitudinal modelling of the bridge bent was thus avoided. In order to calculate the value of the lumped mass at the top of the pier, a small linear elastic model of the bridge was produced. Hand calculations showed that for an infinite stiffness of the abutments as compared to the pier, the pier would attract approximately 12.5% of a unit uniform load applied at the super structure level taking the deck stiffness into account. Since the effects of shear deformation and abutment flexibility were neglected in the calculations, the lumped mass value was determined by taking 20% of the super structural mass, half of the column mass and the mass of the  53  Chapter 5 - Non-linear analysis and damage state assessment beam to account for abutment flexibility and shear deformation. This was thought of as a reasonable estimate and was treated as the base case for this study.  0.2m 1.2m  Rigid link=3m  5@ 1.28m  Figure 5.1(b)  Figure 5.1(a)  Figure 5.1: (a) Colquitz Bridge Prototype Pier (b) SDOF Lumped-Parameter Model  54  Chapter 5 - Non-linear analysis and damage state assessment Analyses were also carried out by calculating the lumped mass by considering 50% of the super structural mass for determining an upper bound for the damage assuming a pin at the superstructure level. This would help to have an idea about the pragmatic range of values for sensitivity analyses to be carried out later.  5.2 NON-RETROFITTED V S RETROFITTED PIER: Since the Colquitz bridge has only one single column pier, it is the most critical element in the structural assembly. The pier was thus investigated in detail to have confidence in predicting its performance. The as-built or non-retrofitted column has 4035M bars as longitudinal reinforcement while 15M spirals are provided at a pitch of 75mm. This seems to be a well-confmed column but the presence of a lap splice at the column base renders it susceptible to lap-splice failure. Calculations indicate that despite the 1.37m (54 inch) of lap splicing provided, the spiral reinforcement is not adequate to prevent a splice failure at the base. Lap splices in hinge zones, such as the base of columns should not be used, as they tend to break down under cyclic inelastic action even when very long splice lengths are provided. Another critical issue in this regard is the shortening of effective plastic hinge length due to the doubling of the effective longitudinal reinforcement ratio, thus causing an early onset of ductility failure (Priestly et al; 1997). In order to avoid the aforementioned type of failure, the column base was retrofitted with a 10mm thick steel jacket, 1.5m in height, extending upwards from the column footing. A typical gap of 50mm was provided between the jacket and the footing of the column to avoid the  55  C h a p t e r 5 - N o n - l i n e a r analysis a n d damage state assessment  possibility of the jacket acting as compression reinforcement by bearing against the footing thus avoiding excessive flexural strength of the plastic hinge region. A steel jacket provides passive confinement by inducing lateral confining stress in the concrete. We can consider the jacket as continuous hoop reinforcement and the level of confinement depends on the hoop strength and stiffness of the steel jacket. For assessment purposes, the jacket was assumed to have a yield strength of 248 MPa. The following equation was used to estimate the maximum effective compression strain e  c m  for the retrofitted pier (Priestly et al; 1997): e  c m  = 0.004 + 5.6 tj f e yj  s m  / Df  where tj is the jacket thickness, f j is the jacket yield stress, £ y  stress and f  c c  (5-1)  c c  s m  is the strain at maximum  is confined concrete compression strength.  For assessing the non-retrofitted pier, an extreme fibre compression strain (e ) c  equal to 0.002 was used as degradation begins to happen after reaching a curvature ductility corresponding to this value of extreme fibre compression strain (Priestly at al; 1997). A n appreciable increase in the ultimate strain capacity and hence the maximum lateral displacement of the pier is achieved by using the steel jacket. This makes the pier more ductile and imparts it the capability to meet much higher displacement demands.  56  Chapter 5 - Non-linear analysis and damage state assessment  5.3  MOMENT-CURVATURE  A N D PUSHOVER  ANALYSES FOR  COLQUITZ PIER: The computer program Response 2000 was used to determine the momentcurvature relationship for the non-retrofitted and the retrofitted Colquitz bridge pier. The following issues had to be addressed before the software could be used to obtain the required relationships: Confinement of concrete can cause an appreciable increase in the ultimate compression strain that can be carried. This increase was taken into account for modelling the retrofitted Colquitz pier by initially having an estimate of the enhanced f  c  value of concrete due to confinement from the steel jacket. Equation 5-1 was then used to determine the maximum effective compression strain that could be carried by the confined concrete. The £  c m  values for confined concrete can range from 0.012 to 0.05,  which is a 4-16 fold increase over the traditionally assumed value for unconfined concrete. For the given steel jacket retrofit, the f leading to an e  cm  c c  value was determined to be 50 M P a  value of 0.034.  This approach not only takes into account the increase in the yield and ultimate displacement of the pier but also considers the jump in the yield moment that is caused due to the enhanced concrete stress due to confinement. This is necessary to have realistic results from the non-linear dynamic analyses. Shear was not considered to be a problem as there would be a considerable increase in the shear carrying capacity of the pier due to the jacket retrofit. Simplified bi-linear moment-curvature plots were employed to carry out the non-linear dynamic analyses of the bridge pier. These are shown in figures 5.2 and 5.3 as follows:  57  C h a p t e r 5 - Non-linear analysis a n d damage state assessment  *  15000 -  t  (2.3,9000)  Moment (KN-m)  100005000 -  I  -80  !  I  I  I  i  0 I  I  4  1*  -70 -60 -50 -40 -30 -20 -10  )  i  I  10  i  I  20  30  40  i  I  50  60  Curvature (rad/Km)  -5000 /-  i  I  70  80  ^>  -10006* -15000 -  mt-curvature for non-retrofitted Figure 5.2: Simplified bi-linear momcpier Colquitz  |  15000 -  M o m e n t (KN-m) |  (10,11500) • /  10000  • (70,11500)  5000 O < W J -70 -60 -50 -40 -30 -20 -10 /() -5000 I  -80  I  I  I  I  !  I  '  I  10  I  I  20  30  I  40  i  i  i  i  50  60  70  80  Curvature (rad/Km) •  •  -10000 • -15000 -  Figure 5.3: Simplified bi-linear moment-curvature for retrofitted Colquitz pier  The above moment-curvature plot values were used to perform the pushover analyses (Figures 5.4, 5.5) of the Colquitz pier for the two scenarios. A large increase in  58  Chapter 5 - Non-linear analysis and damage state assessment the ultimate displacement capacity for the pier was observed after retrofit. This is due to the 17-fold increase in the ultimate strain carrying capacity for the retrofitted pier as compared to one taken to define the ultimate condition for the non-retrofitted pier. Since this is a single column pier, the Pushover curve also defines the monotonic forcedisplacement curve for the column to the failure state.  1400  500  600  Figure 5.4: Pushover curve for non-retrofitted Colquitz Pier  1400  600  Figure 5.5: Pushover curve for retrofitted Colquitz Pier  59  Chapter 5 - Non-linear analysis and damage state assessment It is evident form the two pushover curves that the ultimate displacement capacity of the retrofitted pier is about 7 times that of the non-retrofitted pier while there is a 1.25 times increase in the ultimate force taken by the retrofitted pier in comparison to the nonretrofitted pier. Also, the pushover curve for the non-retrofitted pier shows that the ultimate condition happens almost immediately after the yielding of the column thus indicating the propensity of the lap splices to come apart due to lateral expansion of concrete in the tension region.  5.4 D A M A G E INDICES: Damage indices are tools in the form of non-dimensional parameters that help engineers and decision makers with quantification of damage sustained in concrete structures under earthquake loading. This information is employed to have an idea about the consequences related to different levels of earthquake. The damage indices do not rely on the subjectivity of the evaluator and provide a means by which different retrofit or design options can be assessed objectively. Damage indices can quantify damage in individual structural elements, storeys or complete structures and can be obtained from non-linear dynamic analysis. They are calculated on the basis of a non-linear dynamic analysis. A number of damage index definitions have been proposed by different researchers, which mostly employ deformation and energy absorption as measure of the level of damage. A comprehensive review of damage indices is given by Williams and Sexsmith (Williams and Sexsmith, 1994). The two broad categories of damage indices are the non-cumulative and cumulative damage indices. Non-cumulative indices fail to  60  Chapter 5 - Non-linear analysis and damage state assessment take into account the accumulation of damage occurring under cyclic loading. Usage of cumulative damage indices is thus recommended for determining the state of damage due to seismic loading.  5.4.1 P A R K A N D A N G D A M A G E INDEX: The best known and most widely used of all the cumulative damage indices is that of Park and Ang (1985). This damage index is defined as follows: DI ~ 8  m  5  U  where 8  m  and 8  U  +  PE F 8 y  (5-2) U  are the maximum displacement and the ultimate displacement  respectively, Fy is the yield force of the structural component, E is the amount of dissipated hysteretic energy and (3 is a parameter representing the extent of strength degradation after yielding. The first term in the equation 5-2 is a simple pseudo-static displacement measure and is referred to as deformation damage, which takes no account of the cumulative damage. The second term accounts solely for the cumulative damage due to cyclic loading and is termed as the strength damage. The term p in this term can have values ranging from 0.0 to 0.5. Low values of this parameter are used for properly reinforced and well-confined columns while higher values are recommended for poorly detailed structural sections. A more recent and slightly modified form of the Park and Ang index by Stone and Taylor (1993) is based on a formulation wherein the recoverable deformation is removed from the first term, and moment and curvature are used in place of force and displacement:  61  Chapter 5 - Non-linear analysis and damage state assessment  D l = (b - (j) m  (5E  y  (5-3)  +  <(>„ - <)>y  M <|> y  u  where (j) , (j) and (j) represent the maximum, yield and ultimate curvatures, M y is the m  y  u  yield moment while E and fi have the usual meaning.  5.4.2 RESIDUAL E N E R G Y D A M A G E INDEX: A very recently proposed measure of damage is the Residual Energy damage index (Hindi and Sexsmith, 2 0 0 1 ) . This index is based on the predicted hysteretic behaviour of a concrete column and yields a damage index at a point in the time history for the element taking into account parameters such as stiffness degradation, strength deterioration and ultimate displacement reduction. This model primarily considers the amount of work required to fail a reinforced concrete column monotonically after it has been damaged due to cyclic loading. It takes the energy A n under a monotonic loaddisplacement curve up to failure as a reference capacity, and then uses the actual loaddisplacement history up to point n, followed by a monotonic load-displacement curve from the end of last cycle n (zero force point) to failure representing the amount of work required henceforth to fail the column, A (figure 5.6). The damage index is then n  calculated by using the following expression: Dl = A - A 0  A  (5-4)  n  0  Figure 5.6 elaborates on the model definition, graphically showing the meaning of the reference capacity An, and the residual capacity A of the column. n  62  Chapter 5 - Non-linear analysis and damage state assessment  Failure  I 1  /  T I  7 A  A  ° / / /  /  I / / / /  /////  A  (a) Loaded column  (b) Monotonic force-displacement envelope  F  (c) Cyclic loading (n cycles)  (d) Degraded monotonic energy after n cycles  Figure 5.6: Residual energy index model elaboration  63  Chapter 5 - Non-linear analysis and damage state assessment  5.4.3 CATEGORIZATION A N D CALIBRATION OF D A M A G E : The various proposed damage indices are calibrated against observed damage. Each one of these correlates the specific damage index with the observed damage based on visual distress in the structure or to the repairability of the building. The Park and Ang index as described in the previous section employs the former approach while Stone and Taylor (Stone and Taylor, 1993) applied the modified index definition to 82 tests on Caltrans circular bridge columns by using the latter method. The two damage classifications are described as follows in tables 5.1 and 5.2.  Table 5.1: Damage classification according to Park and Ang (1985) D <0.1  No damage or localized minor cracking  0.1 <D<0.25  Minor damage - light cracking throughout  0.25 < D < 0.4  Moderate damage - Severe cracking, localized spalling  0.4 <D< 1.0  Severe damage - crushing of concrete, reinforcement exposed  D > 1.0  Collapse  Table 5.2: Damage classification according to Stone and Taylor (1993) D<0.11  No damage or localized minor cracking  0.11 < D < 0 . 4  Repairable - extensive spalling but inherent stiffness remains  0.4 < D < 0.77  Irrepairable - still standing but failure imminent  D > 0.77  Collapse  64  Chapter 5 - Non-linear analysis and damage state assessment  The residual energy model concludes the following scale from the comparison between the proposed model and the observed damage (Table 5.3).  Table 5.3: Proposed damage classification by Hindi and Sexsmith (2001) D< 0.1  No damage  0.1 < D < 0 . 2  Minor damage - light cracking, very easy to repair  0.2<D<0.4  Moderate damage - Severe cracking, cover spalling, repairable  0.4<D<0.6  Severe  damage  -  extensive  cracking, reinforcement  exposed,  crushing, reinforcement  buckling,  repairable with difficulty 0.6<D<1.0  Severe  damage  -  concrete  irrepairable D = 1.0  5.4.4:  Complete collapse  DISCUSSION: The recently proposed residual energy damage index (Hindi and Sexsmith,  2001) takes into account the parameters that describe the hysteretic behaviour of concrete elements including stiffness degradation, strength deterioration and ultimate displacement reduction due to low cycle fatigue in the longitudinal reinforcement. It has been applied to many tested bridge columns and compared to existing damage indices by the researchers and gives a realistic prediction of damage through out the various loading cycles of a concrete member. Since other damage indices show a lower degree of  65  Chapter 5 - Non-linear analysis and damage state assessment correlation with the observed damage, it was decided to use this index for quantifying damage in the Colquitz bridge pier. The damage index values using the modified Park and Ang index were also calculated but only for the sake of comparison and the numbers obtained from this index were not used for eliciting the damage states.  5.5 NON-LINEAR  DYNAMIC ANALYSES AND DAMAGE  INDEX  DETERMINATION: The computer software C A N N Y was used to perform the non-linear dynamic analyses of the bridge pier to yield deformation-time histories for various earthquake records.  5.5.1 C O M P U T E R P R O G R A M C A N N Y : The computer program C A N N Y can be used to carry out three-dimensional nonlinear analyses of structures (Li, 1996). It can also be used for static or modal analysis. When used for non-linear structures under earthquake excitation, it produces a timehistory response, which can then be used to extract the damage index values. The New mark-Beta method was used for the current analyses with P=0.25 for the integration of the equation of motion. A detailed treatment of the Newmark-Beta method is given by Chopra (1997). To model the material behaviour of the pier of the Colquitz south bridge, the C A N N Y sophisticated bi-linear/tri-linear model was used. This hysteresis model has been designed to represent the stiffness degradation, strength deterioration and pinching behaviour by a series of control parameters (Figure 5.7). It was assumed that the unloading stiffness change would start from the zero-displacement point while moderate  66  Chapter 5 - Non-linear analysis and damage state assessment unloading stiffness degradation was assumed (8=0, 0=3). Since the steel jacket retrofit imparts a good deal of confinement, only a small change in strength was assumed to take place while no softening yielding stiffness or pinching was taken into account.  OF'y •-•-/• E F , * max  Fy  y  •V '•  "  \ OK  o  D  : t-  **4  F a tax  1:7  y  . . . BFy  (a) Unloading Stillness Degradation  '1  F nun  (b) Strength Deterioration  (c) Pinching Behavior  Figure 5.6: CANNY sophisticated bi-Iinear/tri-linear hysteresis model (from CANNY-E technical manual, 1996)  67  Chapter 5 - Non-linear analysis and damage state assessment  5.5.2 A N A L Y S I S RESULTS: The principal objective of carrying out the non-linear dynamic analyses was to obtain the force-displacement history of the retrofitted pier for the various earthquake records corresponding to the different levels of seismicity. The force-displacement histories would then be used to extract damage index values by both definitions discussed earlier. Force versus displacement and displacement versus time plots are shown for the base case (i.e. with 20% of superstructure mass lumped at the top of the column along with mass of the beam and half the mass of the column). Analysis results (Figure 5.7-5.9) show hardly any non-linearity depicted by the force-displacement plots in the retrofitted Colquitz column for the level-1 earthquake records. This would evidently lead to no or negligible damage. Some non-linearity is observed when the pier is subjected to the three intermediate level earthquake records leading to some damage. Since the extent of non-linearity is still quite small, minimal damage is expected in the pier for this level of seismicity. However, subjecting the retrofitted column to the highest earthquake level produces noticeable non-linearity along with a number of yield excursions of the column for this level of seismicity. Analyses carried out for the non-retrofitted structure showed greater inelastic displacement demands as compared to the capacity of the pier even for the basic level earthquake thus indicating failure. It was therefore assumed that the non-retrofitted pier would have failure for all levels of seismicity. Analyses for the other scenario whereby 50% of the superstructure mass was taken into account for determining the seismic mass were also carried out but only for assessing an upper bound of damage for sensitivity analyses. Damage index values corresponding to these analyses are shown in table 5.5.  68  Chapter 5 - Non-linear analysis and damage state assessment  -4590-  400  -4500t(sec)  -d(mm)  (a) Force Vs Displacement and Displacement Vs time plots for E C 1  -450&  t  F(KN) -400.  200  400  -4S00d(mm)  ^.  . t(sec)  (b) Force Vs Displacement and Displacement Vs time plots for L P 1  1500  t  F(KN)  400  400  -4§0& d(mm).  (c) Force Vs Displacement and Displacement Vs time plots for M i l  Figure 5.8: Analysis results for level-1 earthquake  69  Chapter 5 - Non-linear analysis and damage state assessment  -4500-  t  F(KN)  -450C+ • d(mm) •  (a) Force Vs Displacement and Displacement Vs time plots for E C 2  (b) Force Vs Displacement and Displacement Vs time plots for L P 2  -4-500-  t  F(KN) -400  200  -200  400  -1500d(mm).  (c) Force Vs Displacement and Displacement Vs time plots for MI2  Figure 5.9: Analysis results for level-2 earthquake  70  Chapter 5 - Non-linear analysis and damage state assessment  4500-  400 300 200 100 0 d(mm)-ioo © -200 -300 -400 -500  t  -1500-  d(mm)  ^.  t(sec)  (a) Force Vs Displacement and Displacement Vs time plots for EC3  (b) Force Vs Displacement and Displacement Vs time plots for LP3  (c) Force Vs Displacement and Displacement Vs time plots for MI3  Figure 5.10: Analysis results for level-3 earthquake  71  Chapter 5 - Non-linear analysis and damage state assessment  5.5.3 D A M A G E INDEX S U M M A R Y : It was observed that the modified Park and Ang index generally depicted a higher level of damage in the Colquitz pier as compared to the residual energy index for the various earthquake records used in this study. For example, in case of MI3, the residual energy index value is 0.26, which shows moderate and repairable damage while the corresponding Stone and Taylor value is 0.43, which means that the structure has suffered irreparable damage. However, as mentioned earlier the Stone and Taylor values were tabulated for comparison purposes only while the damage state assessment is based on the residual energy damage definition. Tables 5.4 and 5.5 show the various damage index values corresponding to both the base and the upper bound scenarios for the two damage index definitions.  Table 5.4: Damage index values for the base scenario Earthquake Designation  Residual energy Index  Park and Ang Index  EC 1  0.02  0.01  LP 1  0.01  0.01  MI 1  0.08  0.11  EC 2  0.06  0.09  LP 2  0.1  0.15  MI 2  0.1  0.13  EC 3  0.2  0.31  LP 3  0.42  0.56  MI 3  0.26  0.43  72  Chapter 5 - Non-linear analysis and damage state assessment Table 5.5: Damage index values for the upper bound damage scenario Earthquake Designation  Residual energy Index  Park and Ang Index  EC 1  0.14  0.24  LP 1  0.14  0.25  MI 1  0.15  0.19  EC 2  0.22  0.31  LP 2  0.24  0.41  MI 2  0.25  0.32  EC 3  0.26  0.42  LP 3  0.48  0.63  MI 3  0.6  0.74  5.6: ADDITIONAL D A M A G E INDICATORS FOR COLQUITZ: The overall damage assessment for the Colquitz bridge corresponding to the various levels of retrofit as defined in Chapter 2 was based largely on the information provided in the Sargent and Vaughan report (Sargent and Vaughan, 1994) along with the data acquired from the elastic design spectra produced for the bridge site. It can be seen from these spectra that the elastic demands for level-2 and level-3 earthquakes are 1.2 and 1.6 times those of the level-1 earthquake respectively for the elastic period of the pier. A description of extent of expected damage in various bridge elements for different retrofits is thus listed as follows:  73  Chapter 5 - Non-linear analysis and damage state assessment Retrofit A (No retrofit):  Table 5.6: Damage assessment of Colquitz for retrofit A Damage description/General Performance  Retrofit A EQ level-1  EQ level-2  EQ level-3  Seats  O.K. (non-retrofitted)  O.K (non-retrofitted)  O.K (non-retrofitted)  Deck  No reliable force path over  No reliable force path  No reliable force path over  the pier, some damage  over the pier, damage  the pier, higher degree of  expected  expected  damage expected  Splice failure, but will likely  Splice failure, severe  Splice failure, much higher  retain its axial capacity  damage expected  inelastic displacement  Bridge Component  Pier  demands, unlikely to retain its axial capacity Keepers and  Can't transfer shear at the  Failure, as force demand  Failure, as force demand is  Bearings  abutments, heavily  is higher than level -1  higher than levels 1 and 2  overloaded Abutment  Show a tendency of failure  Failure  Failure  Diaphragms  (D/C= 1.33)  Severe (repairable)  Severe irrepairable to total  Moderate damage expected  damage assumed  damage assumed  Abutment  Just adequate  Moderate  Piles  (D/C= 1.02)  (repairable) expected  to  severe  Failure  damage  Irreparable to total damage expected  74  Chapter 5 - Non-linear analysis and damage state assessment Retrofit B (Superstructure retrofit):  Table 5.7: Damage assessment of Colquitz for retrofit B Retrofit B  Damage description/General Performance EQ level-1  EQ level-2  EQ level-3  Seats  O.K (retrofitted)  O.K  O.K  Deck  Reliable force path over the  Bridge Component  pier due to the addition of  Same as for EQ level-1  Same as for EQ level-1  shear stud connectors to  (No damage assumed)  (No damage assumed)  girders Pier  Splice failure, but will likely  Splice failure, severe  Splice failure, much higher  retain its axial capacity  (irrepairable) damage  inelastic displacement  expected  demands, unlikely to retain its axial capacity  Keepers and  Can't transfer shear at the  Failure as force demand is  Failure as force demand is  Bearings  abutments, heavily  higher than level -1  higher than levels 1 and 2  overloaded (failure) Abutment  Show a tendency of failure  Failure  Failure  Diaphragms  (D/C= 1.33)  Severe (repairable) damage  Severe irrepairable to total  moderate damage expected  assumed  damage assumed  Just adequate  Failure  Failure  (D/C= 1.02)  Moderate to severe  Irrepairable to total damage  (repairable) damage  expected  Abutment Piles  expected  75  Chapter 5 - Non-linear analysis and damage state assessment Retrofit C (Safety retrofit):  Table 5.8: Damage assessment of Colquitz for retrofit C Damage description/General Performance  Retrofit C EQ level-1  EQ level-2  EQ level-3  Seats  O.K (retrofitted)  O.K  O.K  Deck  Reliable force path over the pier due to the addition of  Same as for EQ level-1  Same as for EQ level-1  shear stud connectors to  (No damage assumed)  (No damage assumed)  Negligible damage expected  Negligible to minimal  Moderate to severe  (retrofitted with steel jacket)  damage expected  (repairable) damage  Bridge Component  girders Pier  expected Keepers and  Keepers and bearings would  Shear keys assumed to be  Shear keys assumed to have  Bearings plus  be fine, Shear keys added for  fine with hardly any damage  minor to moderate damage  shear keys  transferring transverse shear  for this earthquake level  force Abutment  Show a tendency of failure  Failure  Failure  Diaphragms  (D/C= 1.33)  Severe (repairable) damage  Severe irrepairable to total  moderate damage expected  assumed  damage assumed  Just adequate  Failure  Failure  (D/C= 1.02)  Moderate to severe  Irrepairable to total damage  (repairable) damage  expected  Abutment Piles  expected  76  Chapter 5 - Non-linear analysis and damage state assessment Retrofit D (Functional retrofit):  Table 5.9: Damage assessment of Colquitz for retrofit D Retrofit D  Damage description/General Performance EQ level-1  EQ level-2  EQ level-3  Seats  O.K (retrofitted)  O.K  O.K  Deck  Reliable force path over the pier due to the addition of  Same as for EQ level-1  Same as for EQ level-1  shear stud connectors to  (No damage assumed)  (No damage assumed)  Negligible damage expected  Negligible to minimal  Moderate to severe  (retrofitted with steel jacket)  damage expected  (repairable) damage  Bridge Component  girders Pier  expected Shear keys assumed to have  Keepers and  Keepers and bearings would  Bearings plus  be fine, Shear keys added for  Shear keys assumed to be  minimal to moderate  shear keys  transferring transverse shear  fine with hardly any damage  damage  force Abutment  Considered to be adequate  Diaphragms  after retrofit  Abutment Piles  Considered to be adequate after retrofit  Minor to moderate damage No damage assumed  assumed Minor to moderate damage  No damage assumed  assumed  77  C h a p t e r 5 - Non-linear analysis a n d damage state assessment  A number of assumptions in damage assessment for Colquitz were made due to the lack of data/knowledge about the specific states of damage in the various components of the bridge. It was considered adequate to make these plausible assumptions since a lot of geotechnical input is required to assess abutment diaphragms and piles before and after retrofit.  5.7 D A M A G E INDICATORS FOR I N T E R U R B A N OVERPASS FOR S A F E T Y RETROFIT: As mentioned earlier, the Interurban overpass was selected to show the priority index assignment and since safety retrofit was assumed to be the optimal strategy for this bridge, the structure has only been assessed for the mentioned retrofit level under various degrees of seismicity (Table 5.10).  78  Chapter 5 - Non-linear analysis and damage state assessment  Table 5.10: Damage assessment of Interurban overpass for safety retrofit  Damage description/General performance  Safety Retrofit (C)  EQ level-1  EQ level-2  EQ level-3  O.K  O.K  O.K  O.K  O.K  O.K  (Restrainers at E and W  Some rotation and  Same as earthquake  Same as earthquake  abutments)  pile pull out  level-1  level-1  with RFP)  O.K  No damage assumed  No damage assumed  East abutment pile cap footing  O.K  Bridge Component Floor beam-girder connections (retrofitted) Deck under transverse shear (restrainers added) East abutment piles  Columns at pier 2 (retrofitted  (Toe projection  Moderate damage No damage assumed  expected (D/C for EQ level-1 is 0.95  strengthened)  after retrofit Abutment back wall  Failure in bending  Failure  Irrepairable to total  (D/C= 1.76)  Severe (irrepairable)  damage assumed  Severe (repairable)  damage assumed  damage assumed Pier piles  Marginal survival in  Moderate damage  Severe (repairable)  compression plus  expected  damage assumed  bending and pull out  79  Chapter 5 - Non-linear analysis and damage state assessment  It should be noted that the same degree of damage was assumed for rotation and pull out of piles corresponding to the different degrees of seismicity since the restrainers at the east and west abutment will come into play after the initial rotation. It was however assumed that the restrainers are capable of resisting forces produced by the two higher levels of earthquakes.  80  Chapter 6 - Estimation of damage costs  CHAPTER 6 ESTIMATION OF D A M A G E COSTS  The determination and quantification of damage corresponding to the various retrofits and different levels of seismicity was elaborated in the previous chapter. The next step towards the implementation of the decision analysis algorithm is to relate the damage states thus determined to the dollar damage and use this data as the "Consequence Costs" in the decision tree as laid out in Chapter 3. This chapter deals with issues related to translating physical damage to direct dollar damage and indirect costs incurred as a result of the direct damage.  6.1 DIRECT D A M A G E COSTS: The direct damage costs arise from the physical damage to the structure and its contents, and deaths and injuries should there be a collapse. The latter factor depends on the function of the structure and on the value of life. The costs related to the life loss and injuries were however not taken into account for the purpose of this study, as this is a complex issue to deal with. It is assumed herein that occupancy on a small bridge has a very high probability of being zero at the time of an earthquake, and hence the effect of life loss and injury may be neglected. The direct costs thus considered are the ones associated with repair or replacement of the structures only.  81  Chapter 6 - Estimation of damage costs  6.1.1 R E L A T I N G DIRECT D A M A G E COSTS TO D A M A G E STATES: The  terms Damage Index (DI) and Response Damage Index (RDI) are  sometimes used interchangeably while developing relationships between physical damage and dollar damage. Before relating the damage costs to damage states of different bridge components, it is necessary to establish a relationship between the response damage index or the damage state and the Dollar Damage Index (DDI) for each bridge component. The DDI is defined as the ratio between the cost of repair and total cost of replacement of the particular bridge component. The RDI for the bridge pier for this study is the residual energy damage index as described in Chapter 5. The damage in other components was however assessed in terms of particular damage states. Also, since the residual energy damage index values can also be attributed to certain damage states of the pier, it was decided to express the mapping relationship for all components in terms of the DDI and the corresponding damage states. The damage states considered for this purpose are similar to the ones defined in Table 5.3 i.e. no or negligible damage, minor damage, moderate damage, severe repairable damage, severe irrepairable damage and collapse or total damage. Several researchers have proposed a number of strategies for mapping structural damage to dollar damage in the absence of a great deal of data corresponding to each damage state for various types of bridges. A synopsis of such methods is given by Gunturi and Shah (1993). The method used for this study is Deterministic Mapping, which is the simplest of all approaches. It comprises of developing a relationship between DDI and RDI (or the different damage states) based on expert judgement (see Figure 6.1). It must be emphasised here that different numbers thus obtained are relatively crude and  82  Chapter 6 - Estimation of damage costs approximate. These are however deemed to be appropriate for the illustration of the proposed methodology. Bridge engineers and decision makers who have access to bettercost estimates can certainly use that information to obtain more precise data.  1.6  0  0.2  0.4  RDI (or D A M A G E STATE) Expert 1  0.6  0.8  1.0  • Expert 2  Average  Figure 6.1: Deterministic mapping for direct damage estimation (From Gunturi and Shah, 1993)  Based on judgement, the above relationship for various bridge components is shown in figure 6.2. For the sake of better representation, the damage states are assigned specific numbers as follows:  83  Chapter 6 - Estimation of damage costs •  No or negligible damage - Damage state 1  •  Minimal damage - Damage state 2  •  Moderate damage - Damage state 3  •  Severe, repairable damage - Damage state 4  •  Severe, irreparable damage - Damage state 5  •  Collapse or total damage - Damage state 6  84  Chapter 6 - Estimation of damage costs  SHEAR KEYS,PIER/PILE CAP, PILES AND BEARINGS  Figure 6.2: DDI Vs damage state plots for various bridge components  85  Chapter 6 - Estimation of damage costs Assigning costs to various bridge components is a hard task and the desired amount of data is unavailable at present. To overcome this problem, expert judgement and cost estimates from certain jobs carried out by the B C M o T H were used to elicit these costs as specific percentages of the total cost of the bridge. It is generally seen that the cost of substructure equals roughly the cost of superstructure for small to medium-span bridges. Then assuming equal amounts of money to be spent at each support, the two abutments and the pier(s) in Colquitz South structure and the Interurban overpass were assigned one-sixth and one-eighth of the total cost of the bridge respectively. Table 6.1 shows how various components and their sub-components were assigned cost figures for the purpose of this study.  T A B L E 6.1: Assumptions for bridge component cost assessment COMPONENT  COST GUIDELEINES  Pier(s) and each abutment  1/6-1/8 of the total bridge cost  Pier piles  35% of total pier cost  Pier pile cap  15% of total pier cost  Abutment diaphragm  20% of total abutment cost  Abutment piles  55%) of total abutment cost  Abutment pile cap  15%o of total abutment cost  Cost of fixing abutment pile pull out  50%) of cost of piles  86  Chapter 6 - Estimation of damage costs In order to calculate the cost of various bridge components and hence the direct damage costs, we need the average replacement costs for the two bridges. Table 6.2 gives the average replacement costs for various bridge types as reported by Caltrans (Caltrans, 1995). The reported values are given in 1995 US dollars in units of cost per square foot of bridge deck.  Table 6.2: Average replacement costs for various bridge types in 1995 US dollars (from Caltrans, 1995) T Y P E OF B R I D G E  TOTAL  NO AMOUNT  SQ.FT OF D E C K  A V G . COST PER SQ.FT  OF BRIDGES R C Slab  17  6,466, 77  80,407  80.42  R C Box Girder  10  14,744,702  173,560  85.13  CIP/PS S L A B  5  5,260,219  48,281  108.95  CIP/PS Girder  70  211,691,470  2,315,672  91.42  PC/PS "I" Girder  2  1,862,557  14,474  128.68  PC/PS Slab  2  750,502  6,973  107.63  Steel girder  6  74,064,563  448,965  164.97  Totals  112  314,870,190  3,088,332  101.95  The same numbers were used as the Canadian Dollar estimates for obtaining the replacement costs of the bridges selected for this study. The costs were then determined in terms of 2001 C D N dollars using the following equation: P = P (l+i) r  st  n  (6-1)  87  Chapter 6 - Estimation of damage costs where P is the present worth, P is the past worth, i is the interest rate and n is the r  st  number of years. Since both structures are steel girder bridges, the replacement costs were determined using the Table 6.2 figure of $ 165 per square foot or $1765 per square meter. A n interest rate of 4% was used to calculate the present cost of the bridges as given in Table 6.3.  Table 6.3: Replacement costs in millions, 2001 CDN dollars BRIDGE  D E C K A R E A (Sq m)  R E P L A C E M E N T COST  Colquitz river south structure  910  2.02  Interurban overpass  2500  5.57  Since one would need to remove a structure in case of excessive damage due to an earthquake, we also need the removal costs for a bridge under consideration. According to the Division of Structure at Caltrans (Caltrans, 1992), the average bridge costs about $20 per square foot or $215 per square meter to remove (in 1992 US dollars). Removal costs for. the two bridges in this study were calculated in a similar way by using an inflation rate of 4% and using the US dollar figures as the C D N dollar worth. The removal and total costs for the two structures are given in Table 6.4.  Table 6.4: Removal and total costs in millions, 2001 CDN dollars BRIDGE  DECK AREA  R E M O V A L COST  T O T A L COST  Colquitz  910  0.25  2.27  Interurban  2500  0.68  6.25  88  Chapter 6 - Estimation of damage costs Given the over all damage states for the two bridges as described in Chapter 5, the DDI and damage state relationships for various bridge components, the total costs of various bridge components as percentage of the total bridge costs, we are now in a position to calculate the direct dollar damage costs for the two structures corresponding to each of the pre-defined levels of retrofit.  6.1.1.1 DIRECT D A M A G E COSTS FOR COLQUITZ SOUTH BRIDGE: The direct damage costs for the Colquitz river bridge for the proposed retrofits corresponding to different earthquake levels are given in Tables 6.4 and 6.5 for the base scenario and the upper bound scenario respectively. The differences in the inelastic displacement demands for the pier as obtained from the non-linear analyses and the likelihood of different degrees of damage in various components of the bridge corresponding to the same level of seismicity were taken into account. This sometimes led to different cost numbers for the three records corresponding to the same seismicity level. A higher damage state was assessed for certain vulnerable bridge components for records that showed a higher degree of damage in the pier. The costs for the upper bound were only varied as a function of the damage in the pier. In other words, damage in other components was assumed to remain the same for this case as in the base scenario. Also, the retrofit cost for the damaged pier for non-retrofitted Colquitz pier was taken as $50,000. This comprises of the present cost of steel jacket ($20,000) and the assumed labour charges. As described in chapter 3, the first subscript in the consequence index notation denotes the first, second or the third record (1 record: Loma Prieta, 2 st  E l Centro, 3  rd  nd  record:  record: Miyagi), the second subscript shows the level of earthquake  89  Chapter 6 - Estimation of damage costs (level-1: 10% in 50 years return period, level-2: Cascadia subduction event, level-3: 2%o in 50 years return period) while the following alphabets designate the level of retrofit for which the consequences have been calculated. Hence  C32SFRF  indicates the damage index  corresponding to the Miyagi record scaled to Cascadia subduction earthquake level corresponding to the safety level of retrofit.  Table 6.5: Direct damage costs for the base scenario (2001 CDN dollars) N O RETROFIT (A)  SUPERSTRUCTURER  S A F E T Y RETROFIT (C)  FUNCTIONAL  (NRF)  ETROFIT (B)  (SFRF)  RETROFIT (D) (FCRF)  (SSRF) Consequence  Cost  Consequence  Cost  Consequence  Cost  Consequence  Cost  Index  (M)  Index  (M)  Index  (M)  Index  (M)  CnNRF  0.1250  CuSSRF  0.1197  CuSFRF  0.0337  CiiFCRF  0  C2INRF  0.11  C2ISSRF  0.116  C2ISFRF  0.0337  C21FCRF  0  C31NRF  0.1212  C31SSRF  0.1047  C31SFRF  0.0337  C31FCRF  0  C12NRF  0.9506  C12SSRF  0.9453  C12SFRF  0.5052  C12FCRF  0  C22NRF  0.8582  C22SSRF  0.853  C22SFRF  0.438  C22FCRF  0  C32NRF  0.8212  C32SSRF  0.8159  C32SFRF  0.438  C32FCRF  0.0675  C13NRF  2.27  C13SSRF  2.27  C13SFRF  0.6887  C13FCRF  0.2754  C23NRF  2.27  C23SSRF  2.27  C23SFRF  0.6467  C23FCRF  0.4912  C33NRF  2.27  C33SSRF  2.27  C33SFRF  0.6467  C33FCRF  0.2754  90  Chapter 6 - Estimation of damage costs Table 6.6: Direct damage costs for the upper bound damage scenario (2001 CDN dollars) N O RETROFIT (A)  SUPERSTRUCTURES  S A F E T Y RETROFIT  FUNCTIONAL  (NRF)  TROFIT (B)  (C)  RETROFIT (D)  (SSRF)  (SFRF)  (FCRF)  Consequence  Cost  Consequence  Cost  Consequence  Cost  Consequence  Cost  Index  (M)  Index  (M)  Index  (M)  Index  (M)  CnNRF  0.2434  CliSSRF  0.2381  CuSFRF  0.1011  CnFCRF  0.0675  C21NRF  0.5464  C21SSRF  0.5401  C2ISFRF  0.1011  C2IFCRF  0.0675  C3INRF  0.5464  C31SSRF  0.5401  QlSFRF  0.1011  C31FCRF  0.0675  C12NRF  0.9506  C12SSRF  0.9453  C12SFRF  0.5726  C12FCRF  0.1347  C22NRF  0.9506  C22SSRF  0.9453  C22SFRF  0.5727  C22FCRF  0.1347  C32NRF  2.27  C32SSRF  2.27  C32SFRF  0.5727  C32FCRF  0.1347  C13NRF  2.27  CBSSRF  2.27  C13SFRF  0.6887  C]3FCRF  0.2754  C23NRF  2.27  C23SSRF  2.27  C23SFRF  0.6467  C23FCRF  0.4912  C33NRF  2.27  C33SSRF  2.27  C33SFRF  0.992  C33FCRF  0.7942  6.1.1.2 DIRECT D A M A G E COSTS FOR I N T E R U R B A N OVERPASS: For determining the direct damage costs for the interurban overpass, no variation of damage was assumed corresponding to a given level of seismicity. The direct consequence costs for the safety level retrofit of this structure are shown in Table 6.6.  91  Chapter 6 - Estimation of damage costs Table 6.7: Direct damage costs for Interurban overpass safety retrofit (2001 CDN dollars) EARTHQUAKE LEVEL  DIRECT D A M A G E COST (M)  1  0.2611  2  0.5744  3  0.679  6.2 INDIRECT D A M A G E COSTS: It would be preferable to consider most factors proposed by Basoz and Kiremidgian (Basoz and Kiremidgian, 1996) such as emergency response, lone-term economic impact, interaction with other lifelines, historical significance of the bridge, etc in the evaluation of indirect damage costs. However, for the sake of present study, simple guidelines used by Nishimura (Nishimura, 1997) were employed for eliciting indirect damage costs for the two structures. The approach is to have a reasonable estimate of how much each commuter would be willing to pay in order to use a particular bridge every day. The aforementioned study suggests that it is safe to assume that each motorist would agree to a one-way toll of about 45 cents in terms of 1994 US dollars. This is based on incentive contracts to replace bridges damaged in the Loma Prieta earthquake and comes out to be around 60 cents in terms of 2001 US dollars. For the purpose of this study, a value of 67 cents C D N was deemed to be a fair estimate with the underlying premise that motorists would be willing to pay more for the convenience of using the bridges. Also for major highway bridges, it can be assumed that it would take around 90  92  Chapter 6 - Estimation of damage costs days to remove and replace an irreparable bridge based on the information given by Caltrans (Caltrans, 1994b). The Average Daily Traffic (ADT) figures for Colquitz were assumed on the basis of a rough traffic count conducted during a site visit to the Colquitz Bridge in September 2000. The number of vehicles per minute was counted to be 125. Assuming such rate of traffic for 10 hours in the day, the A D T for Colquitz south bridge was determined to be 50,000 vehicles. The Interurban Overpass is a very short distance to the Colquitz structure and is a 4-lane bridge carrying traffic in both directions. A n average loss of $100,000 per day was assumed for this overpass in the absence of any meaningful data for this structure. Based on the over all damage states of the two bridges, the assumed closure time periods and the resulting indirect losses are summarised in Tables 6.7 to 6.10. Some variations in closure times due to the variation in damage states for Colquitz corresponding to each level of seismicity were taken into account in the indirect cost estimation. However, no variations were assumed for the Interurban overpass.  Table 6.8: Assumed closure days and indirect costs for Colquitz (Base scenario) RETROFIT A  RETROFIT D  RETROFIT C  RETROFIT B  EQ  CLOSURE  COST  CLOSURE  COST  CLOSURE  COST  CLOSURE  COST  LEVEL  PERIOD  (M)  PERIOD  (M)  PERIOD  (M)  PERIOD  (M)  (DAYS)  (DAYS)  (DAYS)  (DAYS)  1  15,15,15  0.75,0.75,0.75  15,15,15  0.75,0.75,0.75  3,3,3  0.15,0.15,0.15  0,0,0  0,0,0  2  30,30,30  1.5,1.5,1.5  30,30,30  1.5,1.5,1.5  7,7,7  0.35,0.35,0.35  0,0,3  0,0,0.15  3  90,90,90  4.5,4.5,4.5  90,90,90  4.5,4.5,4.5  15,15,15  0.75,0.75,0.75  7,7,7  0.35,0.35,0.35  93  Chapter 6 - Estimation of damage costs Table 6.9: Assumed closure days and indirect costs for Colquitz (Upper bound damage scenario) RETROFIT A  RETROFIT B  RETROFIT C  RETROFIT D  EQ  CLOSURE  COST  CLOSURE  COST  CLOSURE  COST  CLOSURE  COST  LEVEL  PERIOD  (M)  PERIOD  (M)  PERIOD  (M)  PERIOD  (M)  (DAYS)  (DAYS)  (DAYS)  (DAYS)  1  15,30,30  0.75,1.5,1.5  15,30,30  0.75,1.5,1.5  7,7,7  0.35,0.35,0.35  3,3,3  0.15,0.15,0.15  2  30,60,90  1.5,3.0,4.5  30,60,90  1.5,3.0,4.5  15,15,15  0.75,0.75,0.75  3,7,7  0.15,0.35,0.35  3  90,90,90  4.5,4.5,4.5  90,90,90  4.5,4.5,4.5  30,30,30  1.5,1.5,1.5  7,15,30  0.35,0.75,1.5  Table 6.10: Assumed closure days and indirect costs for Interurban overpass for safety retrofit (2001 CDN dollars) C L O S U R E PERIOD  COST  (DAYS)  (M)  1  7  0.7  2  21  2.1  3  21  2.1  EQ L E V E L  The overall costs for the two bridges can be determined using the information given in Tables 6.4 through 6.10. Despite the roughness of the above cost estimates, these are satisfactory numbers for use in the decision analysis algorithm as outlined in Chapter 3. In order to have considerable confidence in the decision outcomes, a discussion of sensitivity analyses performed by varying these data with in reasonable limits is given in Chapter 7.  94  Chapter 7 - Decision and sensitivity analysis  CHAPTER 7 DECISION AND SENSITIVITY ANALYSIS After determining the consequence costs, the next step is to organize this information in the decision analysis algorithm to ascertain the optimal level of retrofit for the structures under consideration. The use of decision analysis ensures that the likelihood of the occurrence of various levels of seismicity, the uncertainty in the nature of seismicity and consequences corresponding to various levels of retrofit are taken into consideration appropriately. Since the cost information is relatively crude and based on a number of assumptions and expert judgement, a sensitivity study is the best tool to reveal how the decision is affected by altering various input parameters. This exercise gives valuable insight into issues such as; how much of a consequence cost variation causes a change in the optimal decision, at what value of a certain parameter a decision maker would be indifferent between two alternatives, etc. A comparison of results from the above mentioned approach with results obtained from decision analysis techniques, which do not rely upon probability determination, is also worth carrying out to see i f the latter methodology is of use for decision-making in earthquake retrofit of structures.  7.1 NET PRESENT COST (NPC) CRITERION: For the purpose of making comparisons between different retrofit options, the net present cost criterion was adopted in this study. The net present cost of damage caused due to ground shaking is defined as the sum of the initial cost of retrofit and the  95  Chapter 7 - Decision and sensitivity analysis present value of the annual damage cost expected to accrue each year during the planning period. The N P C can then be mathematically represented as  NPC = C + E [l-(l+ir ]  (7-1)  T  0  i where Co is the initial cost of retrofit, E is the expected annual damage cost which is assumed to stay constant for each year, i is the discount rate, and T is planning period depending upon the effective life of the structure. E for the base scenario considering both direct and indirect costs, i = 4%, T = 100 years and the ratio of functional to safety retrofit cost as 1.5 can be determined as follows. Referring to figure 3.2 and the notation described in Chapter 3, Sl.SFRF  =  (CnsFRF Q2ISFRF + C31SFRF) * 1/3 = 0.2525  S ,SFRF  =  (C12SFRF + C22SFRF + C32SFRF) *  S3.SFRF  =  2  +  (Q3SFRF+ C23SFRF + C33SFRF)  1/3 = 0.9218  * 1/3 = 1.4798  and, E(safety retrofit) = S FRF * Pi +S SFRF * P + S ,SFRF * P ];S  2  2>  3  3  where Pi = 0.0021, P = 0.000404 and P = 0.000404 and hence, 2  3  E(safety retrofit) = 0.2525 * 0.0021 + 0.9218 * 0.000404 + 1.4798 * 0.000404 = 0.0015M The N P C for the safety retrofit scenario is then given by, NPC(safety retrofit) = 0.0685 + 0.0015*[l-(l+0.04) ] / 0.04 = 0.10 M 100  NPC's for other alternatives can be similarly calculated. The option with the least N P C is then the optimal retrofit strategy.  96  Chapter 7 - Decision and sensitivity analysis  7.2 DECISION COST COMPARISONS: For carrying out the basic decision cost comparisons, a real interest rate of 4% not considering inflation was assumed. It is also realized that beyond about 100 years, the time span considered has very little effect (Nishimura, 1997). A value of T equal to 100 years was thus assumed. Tables 7.1 and 7.2 give N P C of each retrofit option for both direct and direct plus indirect cost cases corresponding to the base scenario. Tables 7.3 and 7.4 show the same information for the upper bound damage values. A l l costs are given in 2001 C D N dollars. As described in Chapter 3, the various retrofit options i.e. no retrofit, superstructure retrofit, safety retrofit and functional retrofit are abbreviated as NRF, SSRF, SFRF and FCRF respectively.  Table 7.1: NPCs for direct costs only (Base scenario values) ACTION  NPC (M)  NRF  0.037 (Optimal)  SSRF  0.066  SFRF  0.081  FCRF  0.106  97  Chapter 7 - Decision and sensitivity analysis Table 7.2: NPCs for direct plus indirect costs (Base scenario values) ACTION  NPC (M)  NRF  0.135  SSRF  0.164  SFRF  0.1 (Optimal)  FCRF  0.108  Table 7.3: NPCs for direct costs only (Upper bound damage values) ACTION  NPC (M)  NRF  0.059 (OPTIMAL)  SSRF  0.0877  SFRF  0.87  FCRF  0.122  Table 7.4: NPCs for direct plus indirect costs (Upper bound damage values) ACTION  NPC (M)  NRF  0.2024  SSRF  0.2261  SFRF  0.1273 (OPTIMAL)  FCRF  0.1313  98  Chapter 7 - Decision and sensitivity analysis The results show that considering direct costs only, the no retrofit option comes out to be the optimal strategy for both base and upper bound damage considered for damage. However considering the effect of the indirect costs for these two cases changes the decision outcome as it considerably increases the N P C of the no retrofit option. The optimal decision now is to retrofit the structure to the safety level for each of the lower and upper bound damage cases.  7.3 SENSITIVITY A N A L Y S I S : Due to a number of assumptions and approximations in deriving the consequence costs, it is logical to examine the sensitivity of the decision to critical input parameters. It has already been shown in the previous section that the decision is not sensitive to the amount of superstructure mass assumed at the pier for realistic values of 20 and 50 percent. Since the variations in the nature and extent of seismicity were modelled thoroughly in the decision tree, no further variations were considered in this regard. However a number of other parameters were varied and their effect on decision outcome analysed. This was carried out as follows: •  Taking i as 3%, 5% and 6% for the lower bound damage costs assuming the initial functional retrofit cost to safety retrofit cost ratio as 1.5  •  Varying indirect costs by +10%, +20%, +25% and +50% for lower bound damage and by -10%, -20%, -25% and -50% for upper bound damage respectively (for i = 4%).  •  Keeping i as 4 % and taking the initial functional retrofit to safety retrofit cost ratios as 1.4 and 1.33 for both upper and lower bound damage cases  99  Chapter 7 - Decision and sensitivity analysis The results for various sensitivity analyses are shown in Tables 7.5 through 7.9.  Table 7.5: NPCs corresponding to various i values and direct costs only T=100, FCRF/SFRF = 1.5 (Base scenario) ACTION  NPC (i = 3%)  NPC (i = 5%)  NPC (i = 6%)  NRF  0.048 (Optimal)  0.030 (Optimal)  0.024 (Optimal)  SSRF  0.0766  0.059  0.052  SFRF  0.085  0.079  0.077  FCRF  0.107  0.105  0.104  Table 7.6: NPCs corresponding to various i values for direct plus indirect costs T=100 years, FCRF/SFRF = 1.5 (Base scenario)  ACTION  NPC (i = 3%)  NPC (i = 5%)  NPC (i = 6%)  NRF  0.174  0.109  0.092  SSRF  0.203  0.138  0.120  SFRF  0.109 (Optimal)  0.094 (Optimal)  0.089 (Optimal)  FCRF  0.110  0.107  0.106  100  Chapter 7 - Decision and sensitivity analysis Table 7.7: NPCs for direct plus indirect costs after increasing indirect costs corresponding to base scenario T = 100 years, FCRF/SFRF = 1.5 INDIRECT COST V A R I A T I O N ACTION  +10%  +20%  +25%  +50%  NPC  NPC  NPC  NPC  NRF  0.144  0.154  0.159  0.184  SSRF  0.173  0.183  0.188  0.213  SFRF  0.102 (Optimal)  0.104 (Optimal)  0.104 (Optimal)  0.109 (Optimal)  0.1096  0.1097  0.1104  FCRF  0.1090  Table 7.8: NPCs for direct plus indirect costs after decreasing indirect costs corresponding to upper bound damage scenario T = 100 years, FCRF/SFRF = 1.5 INDIRECT COST V A R I A T I O N -10%  -20%  -25%  -50%  NPC  NPC  NPC  NPC  NRF  0.188  0.174  0.166  0.131  SSRF  0.212  0.198  0.191  0.157  SFRF  0.123  0.119 (Optimal)  0.117 (Optimal)  0.107 (Optimal)  FCRF  0.123  0.121  0.12  0.115  ACTION  101  Chapter 7 - Decision and sensitivity analysis Table 7.9: NPCs for direct plus indirect costs for i = 4 % , T =100 years and FCRF/SFRF = 1.4 NPC FOR L O W E R  NPC FOR UPPER  B O U N D OF D A M A G E  B O U N D OF D A M A G E  SFRF  0.999 (Optimal)  0.1273  FCRF  0.102  0.1249 (Optimal)  ACTION  Table 7.10: NPCs for direct plus indirect costs for i = 4%, T =100 years and FCRF/SFRF = 1.33 NPC FOR L O W E R  NPC FOR UPPER  B O U N D OF D A M A G E  B O U N D OF D A M A G E  SFRF  0.099  0.1273  FCRF  0.097 (Optimal)  0.12 (Optimal)  ACTION  7.4 DISCUSSION: The decision and sensitivity analyses show that when only direct costs are considered, no retrofit option is the optimal strategy for both lower bound and upper bound damage states (for i = 4%). However accounting for the indirect costs in the decision makes the safety retrofit as the optimal course of action for both lower and upper bounds of damage. The decision is relatively insensitive to the variations in the discount rate as N R F is optimal for all direct cost scenarios while SFRF is optimal for all direct  102  C h a p t e r 7 - Decision a n d sensitivity analysis  plus indirect scenarios. However for i = 3%, the NPCs for SFRF and F C R F are close enough that the decision maker might be indifferent between these two strategies. Variations in indirect costs for the lower bound damage scenario do not affect the decision outcome yielding SFRF as the optimal level. For the upper bound damage values, a variation o f - 1 0 % in the indirect costs make the NPCs for SFRF and F C R F equal to each other and hence the decision maker would be indifferent between the two retrofit levels for such consequence costs. However for other values of indirect costs, the SFRF is the optimal strategy. For FCRF/SFRF = 1.4 and i = 4%, SFRF is the optimal level for the lower bound damage case while FCRF is the optimal strategy for the upper bound damage case. However for FCRF/SFRF =1.33 and i = 4%, FCRF is the optimal course of action for both cases. Considering the lower bound damage values and i = 4%, the decision maker would be indifferent between SFRF and FCRF for FCRF/SFRF = 1.37 while for the upper bound case this value is 1.43. The effect of variation in T on the decision was also analysed. It was found that the decision outcome was only changed for a value of T = 25 years. The decision analysis indicated N R F as the optimal retrofit scheme for T = 25 years when both direct and indirect damage costs are taken into account for the lower bound damage case taking i = 4%. For T =50 and 75 years, there was no bearing on the decision which still yielded the SFRF as the most favourable course of action for direct plus indirect costs.  103  Chapter 7 - Decision and sensitivity analysis  7.5 DECISION-MAKING WITHOUT PROBABILITY K N O W L E D G E : In a situation where meaningful data are unavailable to assign probabilities to future events, the problem can be formulated in a structured manner and standard decision rules may be applied. For the current study, such techniques were used to ascertain their usefulness in structural retrofit decision-making. The base scenario for both direct and indirect cost cases with i = 4% was considered for this purpose. The Minimin and Minimax rules were first used. The former is based on an extremely optimistic view of nature while the latter is based on an extremely pessimistic view of nature. A compromise between these two approaches is achieved through the Horwicz rule that allows the decision maker to select an index of optimism, a. Thirdly, the Minimax Regret rule is a conservative approach with the underlying premise that the decision maker wishes to minimise his maximum regret about a decision. The application of all three rules requires the construction of a consequence matrix that exhibits the interaction of decision alternatives and the states of nature. The consequence matrix for the studied case is given in Table 7.10. For the Minimin rule, the minimum value of consequences corresponding to each retrofit level is identified. The retrofit level corresponding to the minimum of all such values would then be the required retrofit strategy. For the Minimax rule, the maximum consequence value for each retrofit level is determined and the retrofit level corresponding to the minimum of all such values is then deemed to be the preferred line of action. The results from both these methods show that the functional retrofit is the most suitable level of retrofit to be carried out (Tables 7.11 and 7.12).  104  Chapter 7 - Decision and sensitivity analysis Table 7.11: Consequence matrix for direct plus indirect cost case corresponding to base scenario, i = 4%, FCRF/SFRF = 1.5 STATES OF N A T U R E ALTERNATIVES  SI  S2  S3  S4  S5  S6  S7  S8  S9  NRF  0.875  0.86  0.871  2.45  2.36  2.32  6.77  6.77  6.77  SSRF  0.899  0.884  0.895  2.474  2.382  2.345  6.799  6.799  6.799  SFRF  0.252  0.252  0.252  0.923  0.856  0.856  1.508  1.465  1.465  FCRF  0.102  0.102  0.102  0.102  0.319  0.102  0.527  0.743  0.727  Table 7.12: Minimin and Maximin rule results ALTERNATIVE  MINIMIN R U L E  MINIMAX RULE  NRF  0.86  6.77  SSRF  0.884  6.799  SFRF  0.252  1.508  FCRF  0.102 (min, optimal)  0.743 (min, optimal)  The Minimin and Maximin rules have extremely pessimistic and optimistic underlying philosophies respectively. In order to achieve a compromise between optimism and pessimism, the decision maker can employ the Hurwicz rule which allows the selection of an index of optimism a such that 0<a<l. Once a is selected, the Hurwicz rule requires the computation of Max j {a [max j P,j] + (1- a) [minj P,j]  105  Chapter 7 - Decision and sensitivity analysis A value of a = 1 corresponds to the Minimin rule while a value of a = 0 corresponds to the Minimax rule. Figure 7.1 shows that the preferred retrofit level for all values of a between 0 and 1 is the FCRF.  Figure 7.1: Hurwicz rule values for different values of a  Finally, the regret matrix is shown in Table 7.12 for the Minimax Regret rule. Table 7.13 then shows that the most favourable retrofit level is the functional retrofit.  106  Chapter 7 - Decision and sensitivity analysis  Table 7.13: Regret matrix for the Minimax Regret rule STATES OF N A T U R E ALTERNATIVES  SI  S2  S3  S4  S5  S6  S7  S8  S9  NRF  0.772  0.757  0.768  2.348  2.041  2.218  6.243  6.027  6.043  SSRF  0.796  0.782  0.793  2.372  2.063  2.433  6.272  6.056  6.072  SFRF  0.150  0.150  0.150  0.821  0.537  0.754  0.981  0.722  0.738  FCRF  0  0  0  0  0  0  0  0  0  Table 7.14: Minimax Regret rule results ALTERNATIVE  M A X I M I N REGRET R U L E  NRF  6.243  SSRF  6.272  SFRF  0.981  FCRF  0 (min, optimal)  107  Chapter 7 - Decision and sensitivity analysis  7.6 S U M M A R Y : A n inspection of the results obtained from the Minimin, Minimax, Hurwicz and Minimax Regret rule reveals that all the aforementioned methods yield the functional retrofit as the most suitable strategy. This is in contrast with most results obtained from the standard decision analysis methodology employing probability and risk attributes. A l l of these methods tend to pick out the course of action that reduces the consequence costs to a minimum. Since the functional retrofit level would generally correspond to the minimum consequence costs in bridge retrofit decisions (unless the cost to retrofit the structure to the functional retrofit level approaches the cost of reconstruction), these methods would yield this strategy, as the most preferred every time. The aforementioned methods evidently have an inherent conservative bias. Although this approach is quite consistent with standard engineering practise, it does not fulfill the objectives of a detailed decision analysis. Such methods are therefore not adequate for structural retrofit decision problems and employing them for long-term policy making would lead to an inefficient use of limited resources.  108  Chapter 8 - Bridge retrofit prioritization procedures  CHAPTER 8 BRIDGE RETROFIT PRIORITIZATION PROCEDURES  Prioritization of bridges for the purpose of retrofit has been an area of great interest and consideration for highway departments and ministries in North America for the past two decades or so. This is due to a large number of bridges in need of retrofit and a corresponding lack of funds thus driving them to consider efficient and cost-effective ways of managing their retrofit programs. A number of procedures have been proposed in this regard by various groups and researchers and are being used by the different bridge governing bodies. These procedures seek to rank bridges based on their seismicity, vulnerability and importance attributes, the ultimate goal being to identify the prime candidates for retrofit in an order that is most cost-effective thus ensuring the maximum benefit returns in the long run. This chapter gives a compendium of the various screening methods being used in North America for prioritization of bridges for retrofit purposes. These methodologies can be described as rapid screening procedures for sifting through large bridge inventories and ranking candidates for retrofit. A two step methodology is then proposed for ranking bridges whereby the decision maker has to set priorities among few candidates that have either already been screened based on certain criteria, or selected for retrofit due to political, social, or economic reasons. The proposed methodology is based on results obtained from decision analysis algorithm for each bridge as illustrated in the previous chapter.  109  Chapter 8 - Bridge retrofit prioritization procedures  8.1 EXISTING SCREENING PROCEDURES: The most commonly used prioritization procedures in the United States are the ones developed by Caltrans (Sheng and Gilbert, 1991) and the Federal Highway Administration (FHWA, 1983). Two other methodologies in common use are the procedures proposed by the Washington State Department of Transportation (WSDOT) [Babei and Hawkins, 1993] and the Illinois Department of Transportation (IDOT) [Cherng and Wen, 1992]. A comprehensive methodology was developed by Stanford University researchers more recently (Basoz and Kiremidgian, 1995; 1996) and is believed to be the most complete and rational way of prioritizing bridges to date. The following sections briefly deal with the various in-use methodologies at the present.  8.1.1 THE A T C / F H W A M E T H O D O L O G Y : This approach is a two-step procedure whereby the preliminary screening process is followed by a more detailed evaluation of seismic capacity of bridge components along with identifying retrofit measures and assessing their effectiveness. The preliminary screening process rates each bridge in the inventory based on the seismicity of the bridge site, vulnerability of the various parts of the structure and the importance of the bridge. The seismicity rating is assessed on the basis of the expected peak ground acceleration at the bridge site. The acceleration coefficient maps (ATC-3, 1978) are utilized in this regard. The vulnerability aspects are addressed thorough the assessment of bearings, columns, piers and footings, abutments and foundations, etc. Finally, the importance attributes are based on the importance classification that takes into account  110  Chapter 8 - Bridge retrofit prioritization procedures miscellaneous factors such as the average daily traffic on or under the bridge, length and width of the bridge, detour length and importance of the bridge in terms of the overall functionality of the transportation system. The individual ratings are then combined to obtain an overall seismic rating for each bridge in the following manner:  Seismic rating = X [(Ratings * weighty] i A n updated manual by A T C / F H W A known as the Seismic Retrofitting Guidelines for Highway Bridges has been published more recently (Buckle et al., 1994) containing a significant number of changes. However, the additive formulation (comprising of the weighted sum of the three main criteria) to obtain the overall seismic rating is still maintained. A drawback of this procedure is that it only deals with conventional steel and concrete construction with girder and box girder format with spans not exceeding 500ft. Also,  this methodology  ignores the natural relationship between  a structure's  vulnerability and the expected nature and extent of ground motion. The procedure does, however, account in an approximate way for basic decision analysis concepts. It aims to provide a seismic rating that is related to consequences.  8.1.2 THE C A L T R A N S PROCEDURE: The Caltrans procedure has the underlying philosophy of first retrofitting such bridges that are at greatest risk and are most vital for the functionality of the transportation network. Their approach is to identify the structures most susceptible to  111  Chapter 8 - Bridge retrofit prioritization procedures collapse during a large earthquake (Sheng and Gilbert, 1991) as the ultimate goal is to ensure that all bridges in California are capable of surviving the maximum credible earthquake. The initial risk analysis algorithm developed by Caltrans (Roberts, 1991) is based on a pre-weight factor and a weight for various attributes corresponding to the three main criteria i.e. seismicity, vulnerability and importance. The risk number is then calculated as the summation of the product of the assigned weight and pre-weight score of each attribute:  n Risk Number = £  [(weighty * (preweightj)]  i=l The various attributes considered in this regard entail bedrock acceleration, soil conditions, number and type of hinges, column design (single or multiple bents), height, skew, length of the bridge, abutment type, year of construction (which relates to the confinement details of columns), traffic exposure (average daily traffic), facilities crossed, route type (major and minor) and the detour length. The risk number obtained varies between 0 and 1 with a number closer to 0 implying a lower risk and a number closer to 1 implying a higher risk. In the revised algorithm being used by the Caltrans currently, a new risk index is defined (Maroney, 1991) which is determined as the weighted product of the risk indices for the three main criteria. The weighting factors for each criterion express their relative importance while the criteria and attribute weighting coefficients are obtained based on expert opinion of Caltrans engineers and managers. The revised prioritization algorithm  112  Chapter 8 - Bridge retrofit prioritization procedures is therefore a two-level approach encompassing the "seismic hazard" and "impact and structural vulnerability" criteria. The seismic hazard is now a function of seismic activity, soil conditions, peak ground acceleration and duration of strong motion shaking. The overall rank of a bridge is thus calculated as follows:  R = A*H (0.61 + 0.4V) where, R = prioritization ranking, A = seismic activity, H = hazard, I = impact and V = vulnerability. Clearly, vulnerability is weighted as two thirds of the importance criterion while seismicity has a relative weighting factor of one. Although the procedure proposed by Caltrans does not consider vulnerability as an explicit function of the nature and extent of site seismicity, it is a considerable improvement over the previous approach since it addresses the seismicity related issues in appropriate detail.  8.1.3 THE WSDOT A P P R O A C H : The WSDOT has various objectives behind the development and evaluation of retrofit procedures such as: (1) To minimize risk of bridge collapse (2) Prioritize projects to minimise risk of life loss (3) To insure continued service of all interstate/essential lifeline bridges (4) To accept moderate damage, and (5) To address both substructure and superstructure seismic retrofit needs for each bridge concurrently.  113  Chapter 8 - Bridge retrofit prioritization procedures For retrofit purposes, bridges are first classified into five main priority groups each having a different set of deficiencies. These structures are then ranked within themselves based on the importance criteria (Babei and Hawkins, 1993). Mathematically, this model can be represented as follows: I =C* V where, I = Priority index (0-100), increasing with priority increase C = Criticality factor depending on various importance attributes of the structures and, V = Factor representing vulnerability of the bridge to seismic failure which increases with an increase in bridge vulnerability.  8.1.4 THE IDOT SCHEME: The IDOT scheme is to rank bridges based on their seismic risk. This methodology comprises of a two-stage approach whereby the first stage provides a preliminary ranked list of all bridges under consideration based on their seismic risk while the second stage encompasses a more detailed evaluation. Seismic risk is given by the product of the probability of failure of a bridge and the consequences of such a failure (Woodward Clyde consultants, 1991). Risk = Probability of failure * consequences of failure A separate equation used for the determination of a "Bridge Score" is as follows: Bridge Score = Bridge Vulnerability Factor * Importance Factor which can also be written as follows: R = I * B VF  114  Chapter 8 - Bridge retrofit prioritization procedures where, R = Bridge score, I = importance, B V F = bridge vulnerability factor = f (SVF*GVF), SVF = structural vulnerability factor, which is a function of peak ground acceleration, soil amplification index and soil liquefaction index and G V F = ground vulnerability factor. The candidates with a relatively high bridge score are then subjected to the second stage analysis. It may be pointed out here that the IDOT methodology does consider the relationship between vulnerability and seismicity of a bridge by calculating a probability of failure.  8.1.5 M E T H O D O L O G Y PROPOSED B Y BASOZ A N D KIREMIDGIAN: The methodology developed by Basoz and Kiremidgian is a more thorough treatment of the bridge prioritization problem based on Vulnerability and Importance attributes of each structure under consideration. This methodology differs from the ones used by C A L T R A N S and F H W A in that it considers the Vulnerability of a structure as a function of the Site Seismicity based on the premise that a bridge subjected to a higher level of seismicity is more vulnerable than a structure with potentially lower level of ground motion. Bridges that are of interest for seismic retrofitting are grouped in a set B = { B i , B2,.. . , B j , . . . , B } , where B j denotes bridge i , and N is the total number of bridges N  being considered in the ranking process. The bridges in set B are ranked in decreasing order, i.e., for R i > R2 >...>Rj>...> R N , and the bridge having the highest ranking R | is identified as the first candidate for seismic retrofitting. The ranking R i is related to the  115  C h a p t e r 8 - B r i d g e retrofit p r i o r i t i z a t i o n procedures  two main criteria of Vulnerability and Importance through the following functional relationship (Basoz and Kiremidgian, 1996): Ri = f(Vi,I0  (8-1)  8.1.5.1 V U L N E R A B I L I T Y ASSESSMENT: The vulnerability assessment for a bridge entails the following aspects: •  Seismic hazard analysis at the bridge site  •  Classification of bridges based on their structural characteristics  •  Fragility analysis  The mathematical expression for Vulnerability evaluation is as follows: V = f(D,A,Cn,m)  (8-2)  where: D = damage state assuming values dr in D = {di,d ,....,d }, where Z is the total number 2  z  of damage states, A = seismic hazard at the bridge site, C = bridge class n defined based on the primary structural attributes Y , n  M = modifier based on the secondary structural attributes Y .  8.1.5.1.1 SEISMIC H A Z A R D A N A L Y S I S : The parameter A is a function of local soil conditions at the bridge site and location of the bridge relative to potential seismic hazard sources and represents either ground shaking or severity of the liquefaction. The results of seismic hazard analysis  116  Chapter 8 - Bridge retrofit prioritization procedures comprise of the probability of exceeding various levels of a site parameter over a future time period presented by a seismic hazard curve (Kiremidgian, 1992).  8.1.5.1.2 CLASSIFICATION OF BRIDGES: The classification of bridges is based on primary structural attributes, Y . Figure 8.1 shows a hierarchical scheme of these elements. The bridge attributes considered in this regard are the material and structural type and miscellaneous properties such as number of spans and span continuity, number of columns and bents, abutment type, etc. The material type Y i refers to the material of the substructure while the structural type represents the superstructure configuration of a bridge.  Figure 8.1: Hierarchical Order for Primary Structural Attributes (from Basoz and Kiremidgian, 1996) 117  Chapter 8 - Bridge retrofit prioritization procedures 8.1.5.1.3  FRAGILITY A N A L Y S I S :  The fragility  analysis comprises  of developing ground  motion-damage  relationships for computing the probability of a bridge being in a particular damage state for a given ground motion level, P (D = d / a,C ). A simple approach for obtaining the r  n  fragility curves is by considering the combination of all possible failure modes based on components. This information can then be used to develop the fragility curves by defining relationships between the component states and the system damage states. The authors have also proposed a set of modifiers m for modifying the fragility curves, based on secondary vulnerability attributes such as skewness, effect of seat width, etc. A modifier for a given attribute may or may not be a function of the ground motion level or the bridge class [Basoz and Kiremidgian, 1996]. A generic fragility curve is shown in Figure 8.2 where 'a' is a specific ground motion parameter.  118  Chapter 8 - Bridge retrofit prioritization procedures Performing all the above-mentioned analyses, the Vulnerability Vj of the bridge Bj can be evaluated through the use of equation (8-2).  8.1.5.2 IMPORTANCE ASSESSMENT: The Importance  criterion of bridge assessment entails the appraisal of  consequences arising from life safety and socio-economic aspects of bridge damage. Mathematically the bridge importance criterion can be expressed as follows: Ii = f(Si,E ,Gi,L ,Q ,H ) i  i  i  i  (8-3)  The various factors in equation (8.2) are explained as below: •  Sj is the life safety factor for bridge Bj that depends upon route carried on and under the bridge and the corresponding A D T along with the damage level of the bridge.  •  Ej is the emergency response factor for bridge Bj depending on factors such as; whether or not bridge Bj belongs to an ensemble of bridges whose failure delays accessibility to a disaster area from available resource locations, travel time based on the travel times between two points before and after the failure of bridge Bj and spatial importance of bridge B i in the given highway network system.  •  Gi is the factor taking into account the long-term economic impact based on the A D T for the route carried on the bridge, traffic capacity of the route carried on the bridge and the origin-destination trip demands for various origin destination pairs.  •  Lj is a factor accounting for the interaction of other lifelines carried on bridge Bj.  •  Qj is the defence route factor evaluated on the basis of whether the bridge carries and/or crosses a defence route.  119  Chapter 8 - Bridge retrofit prioritization procedures •  Hj is factor corresponding to the historical significance of bridge Bj.  The importance assessment is thus carried out by obtaining a decision maker's values and developing utility functions and scaling factors for all importance attributes, determining lifeline network analysis attributes for a given bridge, performing network analysis (connectivity analysis for emergency response, serviceability analysis for long term economic impact) and obtaining the overall utility value for importance assessment employing equation (8-3).  8.1.5.3 O V E R A L L R A N K I N G : The overall ranking of a structure is obtained through the integration of the Vulnerability and Importance criteria. The rank of bridge Bj as a function of these two criteria as defined by equation (8-1) can be expressed as below: U B j = k U i + kjU.i v  v  where U B j is the utility value for bridge B i to be used in obtaining R;, k and kj are the v  scaling constants for vulnerability and importance respectively and U i and V  are the  utility values for vulnerability and importance respectively. Bridges in set B are then ranked by decreasing values of U B j to obtain the rank order Ri > R >.. >RN2  Unlike other approaches, this methodology rationally considers vulnerability as a function of site seismicity and incorporates the various pertinent parameters in a thorough fashion. Similarly, the importance assessment is carried out by considering a number of factors that have not been taken into account in other studies. The two criteria are then combined using relative weighting factors. The methodology fits well with decision analysis principles, since it attempts to quantify damage in structures along with the  120  Chapter 8 - Bridge retrofit prioritization procedures determination of consequences. It is the most comprehensive and rational approach of prioritizing bridges in a large inventory developed so far.  8.2 PROPOSED METHODOLOGY: The approach proposed herein is a systematic two-step procedure that can be summarised as follows:  STEP 1:  Subject the bridges under consideration to detailed decision analysis and  determine the optimal course of action for each structure based on the Net Present Cost (NPC) criterion as illustrated in Chapter 7.  STEP 2: Determining the opportunity loss R; for each structure, which is the difference between the present consequences of doing nothing, CNRF, and the N P C of the optimal course of action as determined in the previous step for each bridge. The logical action now is to determine the benefit-to-cost ratio yj for each bridge by dividing R with the cost of retrofit Coi corresponding to the optimal level of retrofit for each candidate (Sexsmith, 1994). Ranking can now be carried out in the decreasing order of RJ/COJ ratios for each candidate under consideration.  8.3 RESULTS: In order to assign the index y, = Rj/Coi as defined above to Colquitz for the sake of prioritization, the results obtained for this structure corresponding to the lower bound of damage for direct plus indirect costs and an i value of 4% are used. For the Interurban overpass, the optimal retrofit strategy is assumed as the safety retrofit level. For assigning the priority index to this structure, we need to determine the present consequence cost of  121  Chapter 8 - Bridge retrofit prioritization procedures the no retrofit option. Given the deficiencies in the unretrofitted structure, it is reasonable to assume that the overpass would have to be replaced for all earthquake levels. This leads to a total consequence cost of C D N $15.25 million and hence a N P C of $1,085 million. The priority index values are given in Table 8.1 for the two structures.  Table 8.1: Priority index calculations for Colquitz and Interurban BRIDGE  OPTIMAL  CNRF  NPC  Ri  Coi  Yi= Ri/Coi  RETROFIT COLQUITZ  SFRF  0.135  0.1  0.035  0.0685  0.51  INTERURBAN  SFRF  1.085  0.398  0.687  0.295  2.33  The priority index values clearly show that the Interurban overpass should be retrofitted first in order to gain the maximum long-term benefit from the retrofit program. The priority index values for the Interurban overpass were re-calculated by taking the consequence costs corresponding to the 75%, 67%, 60% and 50% of the calculated consequence costs for this structure (Table 8.2). B y keeping all other variables uniform, it is seen that the order of retrofit only changes when the considered percentage of consequence costs is taken to be as small as 50% of the estimated consequence costs for the Interurban overpass.  122  Chapter 8 - Bridge retrofit prioritization procedures Table 8.2: Priority index determination for Interurban for different percentages of estimated consequence cost P E R C E N T A G E OF ESTIMATED C O N S E Q U E N C E COST  Yi = Ri/Coi  100  2.18  75  1.29  66  1.01  60  0.77  50  0.49 (< 0.51)  8 . 4 DISCUSSION: The methodology proposed herein is a level-two procedure whereby bridges that have already been screened for retrofit based on certain criteria, or selected for undergoing up gradation as a policy matter due to political, social and economic factors, can be ranked to make the greatest possible reduction in the opportunity loss, or gain the maximum benefit, given a fixed total of construction costs. The proposed scheme is therefore not a rapid screening procedure like other methods discussed in the previous sections. This is a logical way of assigning priority indices to bridges as it first employs decision analysis to figure out the optimal level of retrofit for each bridge thereby eliminating  irrationalities in the  decision-making process,  and then takes  the  corresponding relative benefit gained for each bridge to decide the order of retrofit of various structures given a scarcity of funds. Other approaches fail to take this aspect into  123  Chapter 8 - Bridge retrofit prioritization procedures  account as they only focus on the vulnerability and importance attributes of bridges. The proposed approach thus adds a very important third dimension to the retrofit prioritization process.  124  Chapter 9 - Conclusions and application of proposed methodology  CHAPTER 9 CONCLUSIONS AND APPLICATION OF PROPOSED METHODOLOGY  9.1 CONCLUSIONS: A methodology was developed and illustrated for determining the best retrofit strategy given a set of alternatives for a structure or a number of structures, and for assigning priority indices to candidates being considered for retrofit. Two bridges that are a part of the D R R on the Vancouver Island were selected for demonstration of the proposed procedure. The processes of assessing the nature and extent of site seismicity, subjecting the bridge pier to non-linear analyses, deriving damage states for the pier and other bridge components, relating the damage states to dollar damage indicators and using consequence costs in the decision analysis algorithm were described. The priority index calculation based on cost-benefit ratio analysis was also illustrated. It can be concluded that the damage index approach is a reasonable methodology for determining damage in concrete piers. The residual energy damage index not only takes into account the stiffness and strength degradation of concrete and the confinement effect of the spiral reinforcement, it also considers the reduction in ultimate displacement due to the low cycle fatigue of the longitudinal reinforcement. Damage states so obtained are clearly more accurate than simply deriving them from demand-to-capacity ratio values determined from linear elastic analysis. A considerable scatter of the damage index was found corresponding to each level of seismicity. This is evident from the various damage index values obtained for  125  Chapter 9 - Conclusions and application of proposed methodology different records scaled to the same response spectrum. The scatter is more pronounced for higher seismicity levels as the structure becomes more and more non-linear. It was found that damage is not only dependent on the P G A of a record but more so on the number of large pulses that a seismograph might present. A higher level of damage in such a case is caused due to the reduction in ultimate displacement of the bridge pier due to the low cycle fatigue effects of longitudinal reinforcement. A great deal of difficulty was encountered in determining damage costs due to lack of documented information. The cost of fixing abutment pile pull out can be cited as an example in this regard. Rough estimates had therefore to be made to obtain cost figures for use in decision analysis. There is some fiizziness in damage states for each seismicity level and hence the damage costs corresponding to each state. Fuzzy logic mapping would therefore be the best way of relating damage states to the dollar damage index. This study however employed the deterministic mapping for this purpose due to its simpler nature. The decision analysis showed that taking only the direct costs into account did not change the outcome of the decision for the base and upper bound scenarios. However, accounting for the indirect costs changed the decision outcome to safety retrofit as the optimal course of action. Leaving out the indirect costs can therefore lead the decision maker into taking a course of action that may not be the best strategy. The discount rate was found to have minimal effect on the decision for this study. Also, it was found that the decision outcome only changed for a very short planning period of 25 years, which yielded the no retrofit as the optimal retrofit strategy. For higher values of T such as 50, 75 and 100 years, the decision to retrofit Colquitz to the safety level was found to be the  126  Chapter 9 - Conclusions and application of proposed methodology most preferable course of action considering both direct and indirect costs. Certain importance aspects such as the emergency response, interaction with other lifelines, etc were not given a thorough treatment in this study. Considering these factors in detail would lead to better indirect cost estimates thus leading to consistent and efficient decision-making. The decision analysis techniques not considering probability and risk attributes are not effective in revealing the optimal course of action for a given choice of alternatives. These techniques tend to lead the decision maker into unreal optimism or pessimism and usually lead to the course of action with the minimum consequence cost, which may not be the best strategy for long-term policy making. Finally, there can be some confusion in the definition of various retrofit levels since there may be an overlap of performance objectives from these i f the bridge is not grossly deficient. Colquitz is an example that can be considered as a borderline case for the safety and functional retrofits for the design level earthquake. For such an event, the safety level retrofit only leaves the abutment diaphragm with a potential deficiency as it shows a tendency of failure due to passive pressure from abutment backfill. This minor defect can be repaired very quickly following a design level earthquake and the bridge made functional. However i f the bridge also had major deficiencies in the abutment and/or pier footings and piles corresponding to the design level earthquake, the functional retrofit would ensure a considerably higher performance level as compared to the safety retrofit. In the present case, the performance levels of the bridge for the two retrofit strategies corresponding to the design level earthquake are not too different. The proposed methodology can however be employed regardless of this obscurity to  127  Chapter 9 - Conclusions and application of proposed methodology determine whether the additional expense to retrofit a structure to a higher level is justified or not.  9.2 APPLICATION OF PROPOSED M E T H O D O L O G Y : Given the nature of the retrofit program by B C MoTH, this study provides a rational tool for determining the optimal level of retrofit for a given structure or set of structures. This could be applied to any given bridge classified under any given type (whether Lifeline, DRR, ESR or Other). The proposed methodology could therefore provide hindsight into whether what has been done (or is being done) by the B C M o T H was/is the optimal strategy in terms of money spent and levels of safety achieved. It would also help the ministry make future decisions such as; whether or not to upgrade Lifeline, DRR, ESR bridges to the Functional level, to upgrade bridges crossing ESR's to a level higher than the Superstructure retrofit, to retrofit Other bridges to the Superstructure or a higher level or to not retrofit them at all, etc. Sensitivity analyses could be carried out to take into account the subjective nature of information in order to determine the influence of change in values of various parameters on the outcome of the decision. Another facet of this study was to illustrate a cost-benefit ratio based method of assigning priority indices to a set of bridge structures so that they can be retrofitted in a preferable order. This approach can be used to determine the retrofit order of a number of bridges after they have been initially screened and the corresponding optimal retrofit strategies have been determined for these structures. This would ensure that the bridge  128  Chapter 9 - Conclusions and application of proposed methodology retrofit program of B C is cost-effective and the limited resources are allocated in a rational and pragmatic manner.  129  References  REFERENCES 1. Adams, J., D.H.Weichert, and Stephen Halchuk. [1999]. "Trial Seismic Hazard Maps of Canada- 1999: 2%/50 Year Values for Selected Canadian Cities", Geological Survey of Canada, Open File 3724. 2. Anderson, D.L., R.G.Sexsmith, D.S.English, D.W.Kennedy, and J.B.Jennnings. [1994]. "Oak Street and Queensborough Bridges- Two Column Bent Tests". 3. ATC-3 [1978]. "Tentative Provisions for the Development of Seismic Regulations for Buildings". Report ATC-3-06. Applied Technology Council, Redwood city, California. 4. ATC-6-2 [1983]. "Seismic Retrofitting Guidelines for Highway Bridges". Applied Technology Council, Redwood city California, (also Report No. F H W A / R D 83/007). 5. Atwater, B.F., A.R.Nelson, JJ.Clague, G.A.Carver, D.K.Yagamuchi, P.T.Bobrowsky, J.Bourgeois, M.E.Darienzo, W.C.Grant, E.Hemphill-Haley, H.M.Kelsey, G.CJacoby, S.P.Nishenko, S.P.Palmer, C.D.Peter, and M.A.Reinheart. [1995]. "Summary of Coastal Geological Evidence for Past Earthquakes at the Cascadia subduction zone". Earthquake spectra, 11: 1-18. 6. Babei, K . , and N.Hawkins. [1993]. "Bridge Seismic Retrofit Planning Program". Report No. WA-RD217.1. Washington State Department of Transportation, Olympia, Washington. 7. Basoz, N . , and A.S.Kiremidgian. [1995]. "Prioritization of Bridges for Seismic Retrofit". Report No.l 14. The John.A.Blume Earthquake Engineering Center, Department of Civil Engineering, Stanford University. 8. Basoz, N . , and A.S.Kiremidgian. [1996]. "Risk Assessment for Highway Transportation Systems". Report No.l 18. The John.A.Blume Earthquake Engineering Center, Department of Civil Engineering, Stanford 9. B C M o T H . [1989]. "Contract documents". Bridge Project No. B-3769, Nechako River Bridge No.0250, Burrard Avenue-Vanderhoof. 10. B C M o T H . [2000]. "Seismic Retrofit Design Criteria". British Columbia Ministry of Transportation and Highways, Victoria, B C , Canada 11. Benjamin, J.R., and C.A.Cornell. [1970]. "Probability, Statistics and Decision for Civil Engineers". McGraw-Hill Book Company.  130  References  12. Buckle, I.G., I.M.Friedland, and J.D.Cooper. [1994]. " A Seismic Retrofitting Manual for Highway Bridges". N C E E R Bulletin 8, No.2: 9-14. 13. C A L T R A N S . [1992]. "Summary of Seismic Work Costs". Internal Literature. Caltrans division of Structures, Sacremento, C A . 14. C A L T R A N S . [1994 (a)]. "Post Earthquake Investigation Report". Internal Report. Caltrans division of Structures, Sacramento, C A . 15. C A L T R A N S . [1994 (b)]. "Northridge Informal B i d Contracts". Literature. Caltrans division of Structures, Sacramento, C A .  Internal  16. C A L T R A N S . [1995]. "Construction Statistics, 1995". Internal Literature. Caltrans division of Structures, Sacramento, C A . 17. Cherng, R., and Y.K.Wen. [1992]. "Reliability Based Cost Effective Retrofit of Highway Transportation Systems against Seismic Hazard". University of Illinois. 18. Chopra, A . K . [1997]. "Dynamics of Structures, Theory and Applications to Earthquake Engineering". Prentice Hall. 19. Choukalos Woodburn Mckenzie Maranda Ltd. [1994]. "Retrofit Strategy Report for Colquitz River Bridges, North and South Nos. 1378 and 2655". Prepared for B C M o T H by Choukalos Woodburn Mckenzie Maranda Ltd. 20. Clague, J.J., P.T.Bobrowsky, and R.D.Hyndman. [1995]. "The Threat of a Great Earthquake in South Western British Columbia". B C Professional Engineer, Volume 46, No.9, November 1995. 21. EERI Committee on Seismic Risk. [1989]. "The Basics of Seismic Risk Analysis". Earthquake Spectra, Volume 5, No.4: 675-702. 22. Fabrycky, W.J., and G.J.Thuesen. [1974]. "Economic Decision Analysis". Prentice-Hall Inc. Englewood Cliffs, N J . 23. F H W A . [1983]. "Seismic Retrofitting Guidelines for Highway Bridges". Report FHWA/RD-83/007. (Also Report No.ATC-6-2, Applied Technology Council, Redwood City, California). 24. Fowler. [1980]. "Investigation of the Miyagi-Ken-Oki, Japan, Earthquake of June 12, 1978". Editor: Bruce.R.Ellingwood. 25. Gunturi, S.K.V., and H.C.Shah. [1993]. "Mapping Structural Damage to Monetary Damage". Structural Engineering in Natural Hazards Mitigation, Proceedings of papers presented at the A S C E Structures Congress 1993, Irvine, C A . Vol.2: 1331-1336.  131  References  26. Hindi, R., and R.G.Sexsmith. [2001]. " A Proposed Damage Model for R C Bridge Columns under Cyclic Loading". Earthquake Spectra, Volume 17, No.2, May 2001:261-290. 27. Idriss, I.M. [1993]. "Procedures for Selecting Earthquake Ground Motions at Rock Sites". NIST G C R 93-625. A Report to the US Department of Commerce, Technology Administration, National Institute of Standards and Technology, Gaithersburg, M D 20899. 28. Klohn-Crippen C B A Consultants. [1999]. "Seismic Safety Retrofit, Seismic Retrofit Strategy Report". Tsawassen Overhead No.2452, Highway 17 over Deltaport Way. 29. Kramer, L.S. [1996]. "Geotechnical Earthquake Engineering". Prentice-Hall. 30. Kwan, J., R.G.Sexsmith, and R.O.Foschi. [1997]. "Decision Analysis for Seismic Retrofit of Historic Buildings". Reprint ICASP, Paris, 1997. 31. L i , K . N . [1996]. " C A N N Y - E : Three Dimensional Non-Linear Dynamic Structural Analysis Computer Program Package". Technical and User's Manuals. 32. Maroney, B . [1991]. "Risk Evaluation Update to Seismic Advisory Board". Memorandum (prepared for the C A L T R A N S Seismic Advisory Board). Division of Structures, California Department of Transportation, Sacramento, California.  33. Naumoski, N . [1985]. " S Y N T H Program - Generation of Artificial time History Compatible with a Target Spectrum". McMaster Earthquake Engineering Software Library, Department of Civil Engineering Mechanics, McMaster University, Hamilton, O N . 34. Nishimura, K.S. [1997]. "Risk Based Decision Model for Bridges". M.A.Sc Thesis. University of British Columbia, Vancouver, B C . 35. Park, Y . J . , and A.H-S Ang. [1985]. "Mechanistic seismic damage model for reinforced concrete". Journal of Structural Engineering, A S C E 111, No.ST4: 722- 739. 36. Penelis, G.G., and A.J.Kappor. [1997]. "Earthquake Resistant Concrete Structures". E & F N SPON, A n Imprint of Chapman and Hill, London, U K . 37. Penzien, J. [1995]. "Seismic Performance of Bridges- Learning from California's experience". Seventh Canadian Conference on Earthquake Engineering, Montreal, 1995: 25-44  132  References 38. Private correspondence, 1999. 38 Private correspondence, 2000. 40. Reiter, L . [1990]. "Earthquake Hazard Analysis, Issues and Insights". Columbia University Press. 41. Roberts, J. [1991]. "Recent advances in Seismic Design and Retrofit of California Bridges". Proceedings of the Third US National Conference on Lifeline Earthquake Engineering, Los Angeles, California: 52-64.  42. Sargent and Vaughan Engineering Ltd. [1999]. "Portage Creek Bridge Retrofit Strategy Report". Bridge No. 2728, Contract No. 642CS006. Prepared for B C M o T H by Sargent and Vaughan Engineering Ltd. July 1999. 43. Sexsmith, R . G . [1994]. "Seismic Risk Management for Existing Structures". Canadian Journal of Civil Engineering, Volume 21, No.2, 1994: 180-185 44. Sexsmith, R.G., R.Welch, and P.Koubsky. [1993]. "Seismic Retrofit of Bridge Bents". Proceedings 1993 C P C A / C S C E Structural Concrete Conference, Toronto, ON. 45. Sheng, L . H . , and A.Gilbert. [1991]. "The Prioritization and Screening Process". California Department of Transportation Seismic Retrofit Program. Proceedings of the Third US National Conference on Lifeline Earthquake Engineering, Los Angeles, California: 1110-1119. 46. Stone, W.C., and A.W.Taylor. [1993]. "Seismic Performance of Circular Bridge Columns Designed in Accordance with A A S H T O / C A L T R A N S Standards". NIST Building Science Series 170. February 1993. 47. Stone, W.C., and G.S.Cheok. [1989]. "Inelastic Behaviour of Full-Scale Bridge Columns Subjected to Cyclic Loading". NIST Building Series 166. January 1989. 48. Taylor, P.R., A.M.Van Selst, W.E.Hodge, and R.G.Sexsmith. [1985]. "Annacis Cable Stayed Bridge- Design for Earthquake". Canadian Journal of civil Engineering, No. 12: 472-482. 49. Williams, M.S., I.Villemure, and R.G.Sexsmith. [1997]. "Evaluation of Seismic Damage Indices for Concrete Elements Loaded in Combined shear and Flexure". A C I Structural Journal, Volume 94, No.3: 50. Williams, M.S., and R.G.Sexsmith. [1995]. "Seismic Damage Indices for Concrete Structures: A State-of-the-Art Review". Earthquake Spectra, Volume 5, No.2, May 1995: 319-349.  133  References  51. Williams, M.S., and R.G.Sexsmith. [1997]. "Seismic-Assessment of Concrete Bridges using Inelastic Damage Analysis". Engineering Structures, Volume 19, No.3: 208-216. 52. Williams, M.S., I.Villemure, and R.G.Sexsmith. [1997]. "Evaluation of Seismic Damage Indices for Concrete Elements Loaded in Combined shear and Flexure". A C I Structural Journal, Volume 94, No.3: 315-322. 53. L i , K . N . [1996]. " C A N N Y - E : Three Dimensional Non-Linear Dynamic Structural Analysis Computer Program Package". Technical and User's Manuals.  134  APPENDIX A  STRUCTURAL DRAWINGS OF COLQUITZ RIVER SOUTH BRIDGE AND INTERURBAN OVERPASS  135  4^ Co  4*. 4^  APPENDIX B  E X A M P L E SYNTH FILES FOR L O M A PRIETA -  SpLP3.dat  94 85 0  04 0  050  8 73  .10 0  99 0  10 5  99 9  110  1 . 00 8 1 . 03 5  115  1 . 0 17  .12 0  1 . 02 6  1 . 044 1 . 071  13 5  1 . 0 53  .14 0  1 . 06 2  1 . 08 0 1 . 080  15 5  1 . 08 0  .16 0  1 . 08 0  175  1 . 08 0  .18 0  1 . 08 0  18 5  1 . 08 0 1 . 08 0  19 0  1  08 0  19 5  1  08 0  .200  1  08 0  2 10  1  07 8  2 20 25 0  1 1  075 06 8  23 0  1  0 73  .240  1  07 0  26 0 29 0  1 06 6 1 .059 1 .056  2 70  1  06 3  .280  1  06 1  3 10  1 .054  .320  1 .051  1 .049 1 .047  3 50  1 .044  .360  1 .042  1 .039 1 .037  390  1 .035  .400  1 .032  1 .030 1 .028  4 30  1 .025  .440  1 .023  470  1 .016  .480  1 . 0 13  .520  .995  .540  .981  .560  1 .008 .967  .58 0  .953  .600  .93 9  .620  .92 5  .64 0 .660  .911 .897  .680  .883  .700  .870  .720  .856  .74 0  .842  .760  .828  .780  . 8 14  .800 .820  .800 .786  .840  .772  .860  .758  .880  .745  .900  .73 1 .68 9 .675  .920  . 7 17  .94 0  .703  1 .000  .661  1.10 0  .628  12 5 13 0 145 150 16 5 17 0  30 0 3 30 34 0 370 38 0 4 10 420 4 50 46 0 49 0 .500  .960 .980  1 .020 1 . 0 18 1 .011  1 .200 1 .300 1 .600  . 5 95 .562 .463  1 .400  .529  1.500  .496  1 .700 2 .000 2 .250  .43 0  1 .800  .397  1.900  . 3 64  .331 .279  2 .500  .270  2.750  .266  SpLP3.dat  3.000 3.500  .216 .219  4.000  .117  147  Thr s L P3 .dat  94 8 72  .10 0  1 . 03 6  .115  1 . 032 .  .12 0  98 3  1 .02 9 1 . 110  .13 5  1 . 04 6  .14 0  1 . 08 7  1 . 108 1 . 12 9 1 . 099  .15 5  1 . 07 9  .16 0  1 . 03 8  .17 5  1 . 08 8  .18 0  1 . 08 6  .19 5  1  10 5  .200  1 . 12 3  033 08 2  .230  1  07 9  .240  1  03 9  .270  1 .147  .280  1  14 0  .310  1 .073  .320  1 .029  .330  08 9 1 .020 1 .056 1 .027  .340 .370  1 .058 1 .045  .350  1 .027  .360  1 .043  .380  1 .051 1 . 0 17  .390  1 .036  .400  1 .026  1 .042 1 .006 1 .003  .430  1 .053  .440  1 .042  .470  .995  .480  1 .026  .520  .975  .540  .97 5  .600  .94 6  .620  .92 0  .640  .97 0 .904  .660  .899  .680  .892  .700  .882  .720  .872  .74 0  .847  .760  .800  .780  . 8 15  .800  . 8 13  .820  .800  .840  .782  .860  .769  .880  .757  .900 .960 .980  .72 9 .700 .67 1  .920  .703  .940  .711  1.000  .654  1.10 0  .627  1.200 1.300  .627 1.400  .53 0  1.500  . 5 17  1.600  .54 6 .445  1.700  .454  1.800  .372  1.900  .360  2.000  .357  2.250  .265  2.500  .284  2.750  . 2 52  .040  85 9  .105  914  .110  1 . Oil 1 . 12 0  .125 .13 0 .14 5 .15 0 .16 5 .17 0  .210  1 . 10 8 1 0 98 1 00 7  .220  1  .250  1 1  .185 .190  .260 .290 .300  .410 .420 .450 .460 .490 .500 .560 .580  1 . 0 53 1 .058 .977  .050  •  ThrsLP3.dat  000 500  2 61 203  4.000  12 7  149  SynthLP3 . dat  2 62  17 8  0 4 6  10 5  18 1  0 17  06 1  02 2  05 5  106  12 1  ,161  118  .260  0 62  .057  13 0  .077  03 2  .066  122  .0 1 1  10 9  .047  265  .304  20 1  .116  03 4  .058  .0 1 7  .113  .19 8  .19 8  .251  .272  .118  .115  .10 0  .060  .250  .002  .12 6  .121  .224  .175  .010  .025  .033  .002  .005  .0 95  .008  .044  .10 3  .0 1 0  .18 7  .011  .200  .10 6  .245  .073  .098  .13 0  .423  .3 16  .10 7  .083  .12 7  .037  .0 6 3  .200  .117  .051  .13 0  .168  .026  .0 53  .0 12  .14 0  .177  .10 6  .033  .010  .0 1 7  -.078  .289  -.317  .117  .19 2  .246  .299  -.158  10 6  .007  .15 1  .023  03 4  .118  .14 6  -.127  0 92  .17 0  .0 65  .071  155  .13 1  .089  - . 044  .083  - . 113  20 9  .082  ,157  .0 7 3  -.222  .085  .10 1  -.084  .0 17  -.038  .117  -.139  -.212  -.308  .279  -.203  -.225  -.038  .063  .006  -.082  -.024  .004  .15 0  .18 9  .13 6  .116  .071  .025  .237  .440  .780  .2 97  .10 0  .102  .124  .12 1  -.179  .247  -.340  -.368  -.015  .025  .207  .425  .206  .209  .225  .199  -.336  .162  .12 8  .308  .250  .032  -.209  .267  -.243  .02 8  -.068  .026 -.090  .085  .19 3  .293  .281  .10 3  .147 .020 - . 134 -.062 .163 .026  150  SynthLP3  0 04  dat  -.023 033  - . 0 52  -.107  - . 144  - . 2 85  -.150 128  - . 143  - • 153  - . 18 6  - . 08 8  -.114 02 1  . 0 83  . 105  - . 0 57  - . 13 8  - . 0 63 006  .045  - 172  .164  .147  113 2 12 340  -.150 027  -.088  -.202  -.126  -.218  -.314 -.268  -.074  .076  .014  -.123  167  - . 07 8 .074  .147  .295  .274  03 0 154 159 099 068 073 02 4 000  044  .171  17 0  . 0 93  175  .248  .281  .210  .100  .072  071  .1 18 2 02  .008  .118  .122  -.013  077  00 1 .001 015  .079  .096  .068  .067  022 .022 063  -.010  -.023  -.080  -.199  .404  - . 4 1 99 154  . 0 15  .03 7  - . 15 1  - . 1 87  .116  .1 02 1 69  .092  .048  .074  .062  .. 1 03 9 03  .164  .146  .101  -.011  -.048  -.172  -.208  .087  -.088  .096  -.126  -.096  .018  .092  .124  .059  .104  .130  .124  .112  .111  .092  .080  .174  .133  .083  .064  .053  .056  .095  .110  -.074  -.053  -.037  -.107  -.142  -.124  -.128  -.027 .010  -.005 -.008  .016 -.02 0  -.164 04 3  .039 .037 .068 .050 .296  .040  .011  .034  .010  .124  .121  .109  .105  .048  .029  .044  .109  .085  .094  .076  .072  .098  .056  .042  .002  .164  - . 116  .158 .002 .. 2 5 0 11 7 .042  -.207 -.010 -- .. 012417 -.020  -.001  151  SynthLP3 . dat  102  .000  101  -.105  0 53  - . 0 64  067  -.053  045  -.083  075  -.032  0 85  - . 0 74  016  .015  0 76  .080  059  -.077  -.081  -.061  -.054  -.020  .010  -.019  .012  .056  .085  .033  -.006  -.047  -.061  -.018  .035  .019  .134  .076  04 0  - . 00 2 7 48  - . 073  - . 042  0 19  06 0 - .006690  -.007  .077  .065  .022  05 0 .100520  .116  .186  .265  -188  .106 03 3  .0 2 1  .0 1 1  - . 0 24  .151 068  .022  -.009  .009  .028  -.016 015  -.018  -.005  -.021  -.010  - . 151 321  - . 325  -. 235  - . 108  -.054 02 5  - . 0 17  .0 13  .0 12  0 15 0 90 036 0 1 6632 0 35 12 0 10 0 0 01 0 08 078 16 0 041 0 68 0 62 0 19 0 63 10 0 0 61 0 17 12 9 203 208  .088  -.096  - . 020  .0 0 9  .0 16  - . 04 5 - . 037  -.007 007  .004  -.087 0 93  -.071  - . 043  - . 02 0  .062 025  -.019  .027  .067  .095  .167 208  .205  .191  .298  .362  .177 13 2  . 110  177 044 •077  -.086 -.062  .151  - . 144  •161  -.164  .15 7  -.036  .031  .044  .066  .084  .088 .050 •098  .002  .0 8 5  .006  . 058  -.019 .0 10  .0 0 8  -.054  -.082  -.108  -.134  -.197  -.246  -.237  -.197  .059  .053  .067  .067  .076  .118  .144  .134  .064  -.131 -.063  -.088  -.041  .050  .072  152  SynthLP3.dat  0 19 046  -.032 -.041  089 040 104 132  -.060 -.031 - . 122 -.123  023 019 190 154  - . 02 9 .014 .184 .153  -.037  -.044  -.061  -.081  -.030  -.061  -.073  -.082  -.082  -.042  -.030  -.022  .041  .078  .130  .171  .185  .187  .148  .086  -.006  -.040  -.056  -.032  .038  .017  -.010  .002  .000  -.006  -.006  .007  -.074  -.079  -.106  -.119  -.002  .019  .040  .036  .040  .048  .037  .014  -.088  -.079  -.064  -.067  .016  .016  .014  .012  .065  .069  .082  .093  .008  .003  -.015  -.059  -.056  -.047  -.029  .003  .101  .114  .102  .110  .037  .026  .019  .015  .006  .019  .023  .018  .014  .016  .026  .032  .045  .046  .030  .001  -.061  -.049  -.030  -.013  -.002  -.014  -.025  -.042  0 5 7 . 0 5 0 046  .027  03 9 029 02 6 030 002 060 111 .087 .035 .011 .011 .057 .077 .012  -.039 -.006 .037 .010 -.032 -.075 - . 097 -.038 .023 .016 - . 029 -.078 - . 052 .005  .015  .012  .030  .059  .087  .077  .047  .015  .088 .072  -.084 -.058  .015  .019  .037  .064  .074  .045  .049 .004 .024  .037 -.016 -.013  .001  -.002  .005  .009  .033  .030  .028  .034  .024  - . 045  .057  -.063  .006 .004  .010 .013  153  SynthLP3.dat  051  - . 0 62  065  -.072  119  - . 116  -.087  -.109  -.119  -.119  - .103  - .103  - .107  - . 0 95  -.010  .006  .018  .033  .157  .159  .156  .119  .041  .040  .041  .038  .020  .016  .007  -.001  -.004  -.014  -.024  -.030  -.049  -.046  -.046  -.047  -.013  .005  .030  .031  0 0 1 .042 005  .017  .036  .051  .053  .076  042 .062 047  .047  .058  .075  .081  .046  .041  .042  .040  .035  .024  -.004  -.005  -.009  -.019  -.101  -.116  -.127  -.126  -.116  -.117  -.095  -.060  .018  .036  .056  .081  .107  .110  .115  .113  .082  .065  .054  .040  114 074 0 3 6 072  110 - . 056 056 022 117 .117  142  153  073  .062 062  05 9  045  03 0  .027 027  0 19 0 03 0 0 2 03 7  017 - . 003 003 001 - . 047  054  047  052  - . 055 049  .037  049  .007  .001 026  .0 03 .046 .045  .0 12  .001  .004  -.003  .03 6  - . 055  .072  -.089  .123  - . 123  .119  -.119  .055 .032  -.047 -.006  .099  .108  .111  .109  .117  .122  .121  .106  .028  .025  059  -.067  -.074  -.082  .021  .013  .003  -.009  -.016  -.025  . 032  - . 038  .044 0 8 3  - .- .051 062  .095 04 6 .123 0 4 5  - .. 108 03 3 -.127 .066  - . 003 -.106  . 0 15 -.102  0 94  . 110  - . 0 19 -.116 . 107  .0 8 1  .0 5 5  0 3 2 -.098 0 5 0  154  SynthLP3  04 9 051 0 17 03 9 042 0 14 021 021 022 038 0 78 105 050 035 025 045 028 0 18 002 0 08 .008 .024 . 0 14 .009 .007 .001 .043 .059 .02 6 .028 .022 .040 .030 .035 .0 12 .016 .018 .03 8 .043 .033 .000 . 0 12 .020 .020 .009 .000  .052 .03 5 -.030 -.045 - . 023 - . 013 -.025 -.005 .03 0 .046 .091 .116 .044 .022 -.032 - . 055 - . 024 -.010 .001 . 0 16 -.015 -.032 -.011 -.006 .005 - . 010 - . 053 - . 055 -.026 - . 023 .033 .03 6 .03 7 .02 8 - . 015 - . 015 -.031 -.038 - . 045 -.026 .006 .021 .019 .022 .004 .002  dat  - . 001  - . 009  -.012  -.053  -.060  -.064  -.058  - . 015  -.014  -.014  - . 019  .007  .011  . 0 10  . 0 13  .056  . 0 63  .069  .072  .10 8  .083  .068  .060  .009  .001  -.007  -.015  -.062  -.069  -.066  - . 044  .016  -.001  . 0 02  . 0 02  .002  .024  .032  .03 3  .03 2  -.035  -.036  -.033  -.023  -.007  -.006  -.001  .006  -.016  -.021  -.027  -.036  - . 044  - . 035  - . 028  -.025  -.015  -.004  .004  . 0 13  .028  . 0 18  .018  .023  . 0 18  .009  .000  -.007  -.014  -.013  -.012  - . Oil  -.035  -.030  -.030  - .0 3 6  -.022  -.018  - . 014  - . 006  .031  .032  .026  .021  .018  . 0 14  . 0 14  .0 12  .007  .018  .025  .027  155  SynthLP3.dat  03 0 028 03 1 02 1 020 02 6 0 10 005 034 020 Oil 004 036 048 040 0 19 04 9 0 55 038 038 02 9 03 3 025 .016 .023 . 0 02 .014 .000 .03 1 .03 8 .0 19 .008 .008 .005 .015 . 0 18 .0 19 .03 1 .011 .010 .002 .004 .018 .02 5 . 0 12 .003  028 027 .027 .016 .024 .026 .008 .009 .026 .016 .006 . 0 04 .042 .057 .03 1 . 0 07 .053 .057 .037 .03 7 .032 .032 .021 . 0 16 .009 .0 11 .008 .009 .033 .044 . 0 13 .001 .005 .005 .015 .022 . 0 25 .03 0 . 0 10 .008 .004 -.001 .022 .03 1 .008 -.002  02 8  .030  .032  .033  010  .003  - . 006  020  -.012  -.001  0 16  .025  .034  .03 9  0 13  .013  .016  . 0 15  ,009  .016  .02 1  .028  .0 64  . 0 64  .062  .053  -.014 .007  .005  -.018  -.029  - . 040  .056  -.053  -.047  -.041  .03 3  -.028  -.025  -.025  .03 0  -.029  -.031  -.030  .019  -.021  -.026  -.029  . 0 17  .019  . 0 17  . 0 16  .013  -.019  - . 025  -.027  .047  - . 044  -.036  -.028  .005  .008  .011  .009  .006  .009  . 0 13  . 0 15  .025  .023  . 018  .018  .027  .024  .021  .015  .006  .001  -.004  -.002  .001  .003  .008  . 0 14  .03 8  .038  .032  .020  .007  -.011  -.015  -.018  156  SynthLP3.dat  0 16 001 0 15 02 6 Oil 004 0 12 0 14 000 005 005 001 005 005 0 19 0 15 .007 .02 0 .02 1 .021 .006 .0 13 .000 .008 .007 .010 .009 .015 .03 7 .042 .023 .006 .010 .011 .001 .005 .009 . 0 04 . 0 10 .021 .033 .025 .024 .0 13 .010 .010  009 006 023 02 8 .008 .002 .0 12 . 015 .003 .007 . 0 04 .001 .006 . 0 04 . 0 18 . 0 12 . 0 13 .023 .022 . 0 17 .009 .013 .005 .006 . 0 10 .006 . 0 14 .02 1 .04 0 .040 . 0 14 .000 . 0 12 .010 .004 .006 .006 .008 -.017 • . 024 -.029 - . 023 -.019 -.006 .011 . 0 10  008  .004  004  .009  026  .023  019  . 0 14  .008  -.010  0 12  -.013  .0 12  -.009  007  -.004  . 0 12  . 0 12  . 0 10  .008  .000  .001  .001  .003  .006  .010  .015  .016  .009  .005  .002  -.006  .021  -.017  . 0 16  -.020  . 0 14  -.010  .007  -.004  .009  -.007  .004  -.003  .001  -.002  .002  .002  .005  .003  .001  -.004  .027  - . 032  .034  -.035  .038  -.035  .032  -.028  .006  .009  . 0 11  .007  .003  .000  -.003  .006  .007  .007  .007  . 0 12  . 0 13  .010  -.000  .010  • . 023  -.022  .026  -.030  -.021  - . 024  -.026  -.026  -.001 .010  .001  .005  . 0 13  .017  .010 .019  157  SynthLP3  022 029 03 5 045 047 04 9 04 5 045 0 18 0 10 0 14 007 0 10 0 12 0 13 .024 .050 .033 .057 .053 .060 . 0 64 .050 .030 .025 .022 .021 .031 .0 12 .0 11 .005 .019 .001 .0 11 .025 .023 .033 .021 .0 02 .002 .002 .006 .024 .02 9 .027 .035  .025 .03 3 .03 8 .051 .048 .051 .046 .046 . 0 12 . 0 14 .006 . 0 15 .009 .009 -.020 -.023 - . 044 -.027 -.058 -.036 -.064 -.063 -.039 -.025 -.022 -.018 -.026 -.031 -.011 -.011 . 0 12 .024 .005 . 0 13 .024 .021 .02 9 . 0 14 -.001 .007 .002 .016 .028 .029 .030 .033  dat  .035  .03 5  .03 5  .03 6  .055  .056  .053  .049  .051  .044  .044  .04 5  .043  .038  .033  .025  . 0 18  .02 1  .026  .025  .025  .028  .019  . 0 12  .008  . 0 04  -.002  -.005  - . 019  - . 021  -.030  -.045  -.027  -.035  - . 044  -.051  -.022  -.022  -.040  -.056  -.061  -.060  - . 061  - . 057  -.021  - . 023  -.025  -.025  -.015  -.014  -.013  -.016  -.026  -.019  - . 014  -.012  -.012  -.012  -.010  -.003  .021  .009  .000  -.003  . 0 14  .016  . 0 18  .022  .022  .026  .03 1  .03 5  . 0 10  .006  .003  -.001  .008  .008  .007  .004  .025  .027  .024  .022  .025  .024  .023  .02 6  .032  .031  .033  .035  158  SynthLP3  038 029 03 0 029 008 0 09 0 05 0 13 0 05 0 12 002 003 0 12 Oil 0 15 0 13 033 03 2 .047 .050 .053 .042 . 0 14 . 0 17 . 0 18 .011 .005 . 0 13 .007 .00 9 .007 . 0 17 .020 .006 .03 2 .03 9 .037 .03 6 .03 9 .03 8 .041 .037 . 0 16 .0 11 .001 .001  .034 .028 .03 0 .028 . 0 07 .009 .008 . 0 14 .009 . 0 14 .000 - . 008 -.012 -.013 -.013 -.014 - . 033 -.035 - . 047 -.055 -.047 -.037 -.013 -.021 -.014 -.011 . 0 10 . 0 14 .009 .004 - . Oil - . 023 -.012 .001 .03 5 .04 0 .036 .03 8 .037 .041 .03 8 .031 . 0 13 .008 .000 .001  dat  .026  .027  .028  .02 9  .024  .021  .015  .010  .008  .006  .004  . 0 03  .010  . 0 04  .001  .003  . 013  .008  .002  .000  -.012  -.013  - . 012  - . Oil  -.015  -.019  -.021  -.020  -.018  -.023  -.030  -.033  -.038  -.044  -.048  - . 048  -.059  -.060  -.059  -.057  - . 034  -.029  -.023  -.018  - . 024  -.025  -.025  - . 024  - . Oil  -.008  -.007  -.002  . 014  .011  . 0 10  .001  .000  - . 001  -.003  -.028  - . 031  -.030  -.027  .008  . 0 14  .021  .027  .03 8  .037  .03 6  .037  .042  .045  .046  .043  .042  .044  .04 5  .044  .026  .024  .022  .019  .003  -.000  -.001  .001  .0 15  . 0 13  .007  . 0 13  . 0 07  159  SynthLP3.dat  Oil 015 010 007 008 003 0 07  .012 .016 .002 -.012 -.005 -.005 . 0 05  .015  .013  .013  . 013  -.012  -.010  -.009  - . 0 09  -.010  -.Oil  -.00 6  . 0 02  160  APPENDIX C  RESPONSE INPUT AND MOMENT-CURVATURE PREDICTION FOR COLQUITZ PIER  £91  P9\  co  c  £1  E E  Em  U.  OS co <£  5 E .co i  o  CN  o o  E §  10  o o o  CD CO  •w  c>u: < > / c-  CO  CU >  CU  Q  9191 o  O CO  m co  XI ai co .  iri II  *!  o  co cu cu  Q o 61 ! CD IE io i® lO  > O'i  5 •a Z  CO  r-fco  o!  <->  o.  § >>  CD CO  <  X  (-1  X  E £  E JE  cd"  of  c  'o to  d +  Q  .InT5  :o  CN  X  E  O  >  o d o d  0) .  :8 /  a 'ro co O  If "I  * " co  XI  +:  E E,  o d  II CO  CN II  £ E •in ;  APPENDIX D  CANNY DATA INPUT AND OUTPUT FILES FOR SAFETY RETROFITTED COLQUITZ PIER FOR E L CENTRO-1  CSFEC1.dat  //analysis  a s s u m p t i o n s and o u t p u t o p t i o n s  t i t l e : s i n g l e column c o l q u i t z s o u t h f o r c e u n i t = KN length unit = m time u n i t = s e c  structure,Victoria,B.C  g r a v i t y a c c e l e r a t i o n g = 9.81 analysis i n y-direction including p-delta effect o u t p u t o f o v e r a l l response o u t p u t a l l o f beam response output o f nodal displacements o u t p u t a l l o f beam, column r e s p o n s e o u t p u t a l l o f s u p p o r t response o u t p u t extreme response //dynamic r e s p o n s e c o n t r o l  data  i n t e g r a t i o n time i n t e r v a l . 0 . 0 2 s t a r t t i m e 0, end time 40.96 Damping c o n s t a n t 0.05 p r o p o r t i o n a l t o mass [M] newmark method,Beta-value 0.25 including Z - t r a n s l a t i o n a l i n e r t i a forces s c a l e f a c t o r 9.81 t o Y-EQ f i l e = c : \ s a q t h e s i s \ s y n t h e c l . d a t master DOFs f o r a n a l y s i s c o n t r o l : Y - t r a n s l a t i o n s , 7-node response l i m i t 0.7 binary  f o r m a t t e d output a t e v e r y  0-steps  //node l o c a t i o n s node 1 t o 6 e v e r y 1, (0 0 0 ) , Zi=1.07 node 7 (0 0 9.42) //node d e g r e e s o f freedom g e n e r a l d e g r e e s o f freedom : 5-components node 1 e l i m i n a t e a l l components //node w e i g h t node 7, w=2450 //element d a t a : column 1 2 BU100 TU100 AU90  CSFEC1.dat  2 3 4 5 6  3 4 5 6 7  BU100 BUIOO BUIOO BUIOO BUIOO  TUIOO TUIOO TUIOO TUIOO TUIOO  AU90 AU90 AU90 AU90 AU90  r ( 0 3)  / / s t i f f n e s s and h y s t e r e s i s p a r a m e t e r s UlOO CA7 3.3e7 0.0346 C(0 0) Y(11500 11500) A ( l 1) B(0.0000001 0.0 000001) P(0 3 0.01 0.01 0 0 0) U90 E L I 2.4647e7 2.22187 / / i n i t i a l load node7, Pz=4160, p o s i t i v e i s c o m p r e s s i o n  168  csfeel.TRF  CANNY-E main program A u t h o r : Kang-Ning L i C o p y r i g h t b y Canny C o n s u l t a n t s P t e L t d ( S i n g a p o r e ) , 1995-97 Report a t Wed Feb 14 11:25:53 2001 s i n g l e column c o l q u i t z s o u t h s t r u c t u r e , V i c t o r i a , B.C 3 - d i m e n s i o n a l dynamic a n a l y s i s U n i t system:KN,m,sec,rad. 1. ELEMENT EXTREME RESPONSE (1) Element F o r c e s a) Column (Mj) [x: (Mb)Q(Mt)] [y: ( M b ) Q ( M t ) ] A x i a l F C l ( l - 2 ) [x: (0.000-0.000) (0.000-0.000)] [y: (11500.000--11425 .831) (10 12 8.033~-10286.299)]4160.000~416 0.000 C2(2-3) [x: (0.000-0.000) (0.000-0.000)] [y: (10286.2 94~-1012 8.03 0) (88 30.228~-8 968.212)]4160.000~4160.000 C3(3-4) [x: (0.000-0.000) (0.000-0.000)] [y: (8968.219--8830.235) (7532 . 441--7650.158)]4160.000-4160.000 C4(4-5) [x: (0.000-0.000) (0.000-0.000)] [y: (7650 . 149--7532 . 455)'(6234 . 635--6332.068)] 4160.000-4160.000 C5(5-6) [x: (0.000-0.000) (0.000-0.000)] [y: (6332.074--6234 . 614) (4936 .83 6--5013.941)]4160.000-4160.000 C6 (6-7) [x: (0.000-0.000) (0.000-0.000)] [y: (5013.946--4936. 711) (3 63 9 .069--3695.846)]4160.000-4160.000 (2) Element D u c t i l i t y a) Column baseR(, ShearD) (, TAxialD) , topR C l ( l - 2 ) bx:0-0.0000-0.0000 by:-0 . 9 9 2 1 0 1 . 2947Y, Ae, tx:0-0.0000-0. 0000 ty:-0.8945C-0.8807C C2(2-3) bx:0-0.0000-0.0000 by:-0.8807C-0.8945C, Ae, tx:0-0.0000-0. 0000 ty:-0.7798C-0.7678C C3(3-4) bx:0-0.0000-0.0000 by:-0.7678C-0.7798C, Ae, t x : 0 - 0.0000-0. 0000 t y : - 0 . 6 6 5 2 0 0 . 6550C C4(4-5) bx:0-0.0000-6.0000 by:-0.6550C-0.6652C, Ae, tx:0-0.0000-0. 0000 ty:-0.5506C-0.5421C C5(5-6) bx:0-0.0000-0.0000 by:-0.5421C-0.5506C, Ae, tx:0-0.0000-0. 0000 t y : - 0 . 4 3 6 0 0 0 . 4 2 9 3 C C6(6-7) bx:0-0.0000-0.0000 by:-0.4293C-0.4360C, Ae, tx:0-0.0000-0. 0000 t y : - 0 . 3 2 1 4 0 0 . 3 1 6 4 C (3) Extreme r e s p o n s e s a t c o n t r o l master DOF N7 TY D(0.28298911 - -0.29356885), A(4.9324 - -4.8565), V(1.1842 -1 . 3128) 169  csfecl.TRF  2. COMPUTATION TIME F i n i s h e d c o m p u t a t i o n s t e p s =2048, w i t h 127 s t e p s s t i f f n e s s change T o t a l CPU time =1.00 s e c Time o f s t i f f n e s s i n i t i a l i z a t i o n =0.00 Time o f b i n a r y f i l e o u t p u t =0.00 Time o f member r e s p o n s e computation =0.00 Time o f m a t r i x LDU d e c o m p o s i t i o n =0.00 Time o f m a t r i x s u b s t i t u t i o n =0.00 Time o f n u m e r i c a l i n t e g r a t i o n =1.0 0 ***  ANALYSIS NORMAL END  ***  170  CSFECla.dat  / / a n a l y s i s assumptions and o u t p u t o p t i o n s t i t l e : s i n g l e column c o l q u i t z s o u t h f o r c e u n i t = KN length unit = m t i m e u n i t = sec  structure,Victoria,B.C  g r a v i t y a c c e l e r a t i o n g = 9.81 analysis i n y-direction i n c l u d i n g p-delta e f f e c t output o f o v e r a l l response o u t p u t o f n o d a l d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n o u t p u t a l l o f column r e s p o n s e //dynamic response c o n t r o l  data  i n t e g r a t i o n time i n t e r v a l 0.02 s t a r t t i m e 0, end time 40.96 Damping c o n s t a n t 0.05 p r o p o r t i o n a l t o mass [M] newmark method,Beta-value 0.25 i n c l u d i n g Z - t r a n s l a t i o n a l i n e r t i a forces s c a l e f a c t o r 9.81 t o Y-EQ f i l e = c : \ s a q t h e s i s \ s y n t h e c l . d a t m a s t e r DOFs f o r a n a l y s i s c o n t r o l : Y - t r a n s l a t i o n s , 7-node r e s p o n s e l i m i t 0.7 b i n a r y f o r m a t t e d o u t p u t a t e v e r y 0-steps //node  locations  node 1 t o 6 e v e r y 1, (0 0 0 ) , Zi=1.07 node 7 (0 0 9.42) //node degrees o f freedom g e n e r a l degrees o f freedom : 5-components node 1 e l i m i n a t e a l l components //node w e i g h t node 7, w=4050 //element 1 2 BU100 2 3 BU100 3 4 BU100 4 5 BU100  d a t a : column TU100 AU90 TU100 AU90 TU100 AU90 TU100 AU90  CSFECla.dat  5 6 BU100 TUIOO AU90 6 7 BUIOO TUIOO AU90  r ( 0 3)  / / s t i f f n e s s and h y s t e r e s i s p a r a m e t e r s U100 CA7 3.3e7 0.0346 C(0 0) Y(11500 11500) A ( l 1) B(0.0000001 0.0 000001) P(0 3 0.01 0.01 0 0 0) U90 E L I 2.4647e7 2.22187 / / i n i t i a l load node7, Pz=4160, p o s i t i v e i s c o m p r e s s i o n  172  csfecla.TRF  CANNY-E main p r o g r a m A u t h o r : Kang-Ning L i C o p y r i g h t by Canny C o n s u l t a n t s P t e L t d (Singapore) , 1995-97 R e p o r t a t Wed J a n 31 14:09:57 2001 s i n g l e column c o l q u i t z south s t r u c t u r e , V i c t o r i a , B . C 3 - d i m e n s i o n a l dynamic a n a l y s i s U n i t system:KN,m,sec,rad. 1. ELEMENT EXTREME RESPONSE (1) Element F o r c e s a) Column (Mj ) [x: (Mb)Q(Mt)] [y: ( M b ) Q ( M t ) ] A x i a l F C l ( l - 2 ) [x: (0.000-0.000) (0.000-0.000)] [y: (11500.001--11500.004) (10 258.143--10348.912)]4160.000-4160.000 C2(2-3) [x: (0.000-0.000) (0.000-0.000)] [y: (10348.919--10258.139) (89 43.654--9022.814)] 4160.000-4160 . 000 C3 (3-4) [x: (0.000-0.000) (0.000-0.000)] [y: (9022.814--8943.669) (7629 .213--7696.72 5 ) ] 4 1 6 0 . 000-4160.00 0 C4 (4.-5) [x: (0.000-0.000) (0.000-0.000)] [y: (7696 . 743--762 9 . 209) (6314 . 744--6370.592)] 4160.000-4160.000 C5 (5-6) [x: (0.000-0.000) (0.000-0.000)] [y : (6370.606--6314.718) (5000 .240--5044.4 84)] 4160 . 000-4160.0 00 C6(6-7) [x: (0.000-0.000) (0.000-0.000)] [y: (5044.443--5000.213) (3685 .73 6--3718.380)]4160. 000-4160.000 (2) Element D u c t i l i t y a) Column b a s e R ( , ShearD)(, A x i a l D ) , topR by: -4 .5675Y- 1 9973Y, Ae, t x : 0- 0 .0000- 0 . 0-0 . 0000-0.0000 bx: CI (1 -2) 0000 t y -0 .8999C-0.8920C C2 (2-3) bx: 0-0.0000-0.0000 by: -0 .8920C- 0 .8999C, Ae, t x : 0- 0 .0000- 0 . 0000 t y -0 .7846C-0.7777C C3 (3 -4) bx: 0-0 . 0000-0.0000 by: -0 .7777C- 0 .7846C, Ae, t x : 0- 0 .0000- 0 . 0000 t y -0 .6693C-0.6634C C4 (4-5) bx: 0-0 . 0000-0.0000 by: -0 .6634C- 0 .6693C, Ae, t x : 0- 0 . 0000-0 . 0000 t y : -0 .5540C-0.5491C C5 (5 -6) bx: 0-0 . 0000-0.0000 by: -0 .5491C- 0 . 5540C, Ae, t x : 0- 0 . 0000-0 . . 0000 t y : -0 4387C-0.4348C C6 (6 -7) bx: 0-0.0000-0.0000 by: -0 .4348C-•0.4386C, Ae, t x : 0- 0 . 0000--0 . 0000 ty:-0.3233C-0.3205C (3) Extreme r e s p o n s e s a t c o n t r o l m a s t e r DOF N7 TY D(0.34885630 - -0.30588371), A(3.0020 - -2.9756), V(1.0427 -1.1148) V  173  csfecla.TRF  2. COMPUTATION TIME F i n i s h e d computation s t e p s =2048, w i t h 73 s t e p s s t i f f n e s s change T o t a l CPU time =1.00 s e c Time o f s t i f f n e s s i n i t i a l i z a t i o n =0.00 Time o f b i n a r y f i l e o u t p u t =0.00 Time o f member response c o m p u t a t i o n =1.00 Time o f m a t r i x LDU d e c o m p o s i t i o n =0.00 Time o f m a t r i x s u b s t i t u t i o n =0.00 Time o f n u m e r i c a l i n t e g r a t i o n =0.00 ***  ANALYSIS NORMAL END  ***  174  

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