Copyright by Jeremiah David Fasl 2013
The Dissertation Committee for Jeremiah David Fasl
certifies that this is the approved version of the following dissertation:
Estimating the Remaining Fatigue Life of Steel Bridges Using Field
Measurements
Committee:
Todd A. Helwig, Supervisor
Sharon L. Wood, Co-Supervisor
Michael D. Engelhardt
Karl H. Frank
Dean P. Neikirk
Estimating the Remaining Fatigue Life of Steel Bridges Using Field
Measurements
by
Jeremiah David Fasl, B.S.C.E.; M.S.E.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
May 2013
Dedication
To Dad, Mom, Travis, Kati, Thomas, Jasmin, Matthew, Anna, Mikayla, and the rest of my family
v
Acknowledgments
I am very grateful to the National Institute of Standards and Technology (NIST)
for funding my research project and PhD education. The research was very rewarding
and offered me with many new and unforgettable opportunities.
I was very fortunate to have a great doctoral committee who continuously
challenged and helped me to develop a better dissertation. My two primary advisors, Dr.
Todd Helwig and Dr. Sharon Wood, were fantastic. I have appreciated Dr. Helwig’s
friendship and mentorship over the past eight years, and was always impressed how he
maintained a healthy work-life balance. He provided significant input into my research
and dissertation while keeping the work environment fun and entertaining. Dr. Wood
challenged me to be a better researcher and was always available to sit down and talk
through research ideas. I cannot express enough thanks for her friendship and help over
the last several years. The other members of my committee, Dr. Karl Frank, Dr. Mike
Engelhardt, and Dr. Dean Neikirk, also deserve many thanks for their feedback, insight,
and encouragement during my time at UT.
I was privileged to work with so many great researchers while at UT. I could not
ask for better research partners than Matt Reichenbach, Vasilis Samaras, and Ali Abu
Yosef. A lot was accomplished on the NIST project due to their enthusiasm and
willingness to complete tasks and make the time enjoyable. We monitored many bridges
over the last few years, and I would not trade away any of the time. I am blessed to call
each of them my friends, and I know each will have very successful futures in
engineering. Beyond these three, I also worked with many other researchers on the NIST
project: Dr. Praveen Pasupathy (ECE), Rich Lindenberg (WJE), David Potter (NI), Eric
Dierks (ME), Travis McEvoy (ME), Jason Weaver (ME), Sumedh Inamdar (ME),
Krystian Zimowski (ME), Dr. Richard Crawford (ME), Dr. Kristin Wood (ME), and
multiple NI employees. Each of you helped to make the research gratifying!
So many smart, talented people come through FSEL on an annual basis and I was
fortunate to meet many of them. I am truly thankful for the friendships that developed
with Alejandro Avendaño and Analissa Icaza, Catherine Hovell, Kerry Kreitman, Jason
vi
and Samantha Stith, Nancy Larson, Jason Varney, Daniel Williams, Jamie Farris, Amy
Barrett, Dr. Jack Breen, Craig Quadrato, Anthony Battistini, and countless others.
Writing up all the fun we had will take too long, so I will just summarize by saying you
kept me busy and amused outside of research with intramural sports, golf outings, movie
nights, game nights, team in training, fun journeys, tailgates, conference outings,
BLDG24 meetings, etc. You made my time at FSEL memorable, so thank you for all
you did!
The technical and support staff at FSEL provided much needed assistance to keep
the research going. Special thanks to Barbara Howard and Jessica Hanten for their help
with trip logistics and supply purchases. I also appreciate the help over the years from
Blake Stasney, Dennis Fillip, Andrew Valentine, Scott Hammond, Eric Schell, and Mike
Wason.
It is difficult to condense my family’s encouragement into only a few words. So,
I will simply state that their love and support have made me who I am today… thank you
for always being there!
vii
Estimating the Remaining Fatigue Life of Steel Bridges Using Field
Measurements
Jeremiah David Fasl, Ph.D.
The University of Texas at Austin, 2013
Supervisors: Todd Helwig
As bridges continue to age and budgets reduce, transportation officials often need
quantitative data to distinguish between bridges that can be kept safely in service and
those that need to be replaced or retrofitted. One of the critical types of structural
deterioration for steel bridges is fatigue-induced fracture, and evaluating the daily fatigue
damage through field measurements is one means of providing quantitative data to
transportation officials.
When analyzing data obtained through field measurements, methods are needed
to properly evaluate fatigue damage. Five techniques for evaluating strain data were
formalized in this dissertation. Simplified rainflow counting, which converts a stress
history into a histogram of stress cycles, is an algorithm standardized by ASTM and the
first step of a fatigue analysis. Two methods, effective stress range and index stress
range, for determining the total amount of fatigue damage during a monitoring period are
presented. The effective stress range is the traditional approach for determining the
amount of damage, whereas the index stress range is a new method that was developed to
facilitate comparisons of fatigue damage between sensors and/or bridges. Two additional
techniques, contribution to damage and cumulative damage, for visualizing the data were
conceived to allow an engineer to characterize the spectrum of stress ranges. Using those
two techniques, an engineer can evaluate whether lower stress cycles (concern due to
electromechanical noise from data acquisition system) and higher stress ranges (concern
and Sharon L. Wood
viii
due to possible spike from data acquisition system) contribute significantly to the
accumulation of damage in the bridge.
Data from field measurements can be used to improve the estimate of the
remaining fatigue life. Deterministic and probabilistic approaches for calculating the
remaining fatigue life were considered, and three methods are presented in this
dissertation. For deterministic approaches, the output of the equations is the year when
the fatigue life has been exceeded for a specific probability of failure, whereas for
probabilistic approaches, the probability of failure for a given year is calculated.
Four different steel bridges were instrumented and analyzed according to the
techniques outlined in this dissertation.
ix
Table of Contents
CHAPTER 1 Introduction....................................................................................................1
1.1 Overview of Research ...................................................................................... 1
1.2 Project Description .......................................................................................... 3
1.3 Implications of Research ................................................................................. 6
1.4 Organization of Dissertation ............................................................................ 7
CHAPTER 2 Background & Literature Review ..................................................................9
2.1 Definition of Fatigue ........................................................................................ 9
2.2 Characterizing Fatigue Resistance ................................................................. 11
2.2.1 Stress-Based Approach ...................................................................... 11
2.2.1.1 Palmgren-Miner’s Rule .............................................................. 16
2.2.1.2 Cycle-Counting Methods ........................................................... 20
2.2.2 Linear-Elastic Fracture Mechanics .................................................... 23
2.3 Literature Review .......................................................................................... 27
2.3.1 Development of Fatigue Resistance Under Variable-
Amplitude Loading ............................................................................ 28
2.3.1.1 NCHRP Project 12-12 ................................................................ 28
2.3.1.2 NCHRP Project 12-15(4) ........................................................... 31
2.3.1.3 Swenson & Frank ....................................................................... 33
2.3.1.4 ASCE Committee on Fatigue and Fracture Reliabilty ............... 33
2.3.1.5 NCHRP Project 12-15(5) – Fatigue Database ............................ 34
2.3.1.6 NCHRP Project 12-28(03) ......................................................... 36
2.3.1.7 NCHRP Project 12-15(5) – Variable-Amplitude Fatigue
Tests ........................................................................................... 40
2.3.1.8 Manual for Evaluation in AASHTO (2011) ............................... 41
2.3.1.9 Regression Values for Current AASHTO S-N Curves .............. 42
2.3.1.10 Summary .................................................................................... 44
2.3.2 Field Monitoring of Bridges for Fatigue Evaluation ......................... 45
x
2.3.2.1 Summary .................................................................................... 49
2.3.3 Structural Reliability Fatigue Analysis .............................................. 50
2.3.3.1 General Reliability Concepts ..................................................... 51
2.3.3.2 Fatigue Reliability Analysis ....................................................... 53
2.3.3.3 Fatigue Limit State Function ...................................................... 55
2.3.3.4 Summary .................................................................................... 57
CHAPTER 3 Techniques for Fatigue Analysis .................................................................58
3.1 Simplified Rainflow Analysis ........................................................................ 60
3.2 Amount of Fatigue Damage ........................................................................... 62
3.2.1 Effective Stress Range ....................................................................... 65
3.2.2 Index Stress Range ............................................................................ 68
3.2.3 Comparison of Effective Stress Range and Index Stress Range ....... 71
3.3 Characterization of Fatigue Damage ............................................................. 76
3.3.1 Contribution to Damage .................................................................... 77
3.3.2 Cumulative Damage .......................................................................... 79
3.4 Example ......................................................................................................... 80
3.5 Summary ........................................................................................................ 85
CHAPTER 4 Field Monitoring of Four Highway Bridges ................................................86
4.1 Bridge A ......................................................................................................... 87
4.1.1 Geometry ........................................................................................... 87
4.1.2 Condition of Bridge and Characterization of Fatigue Details ........... 90
4.1.3 Retrofit ............................................................................................... 92
4.1.4 Instrumentation .................................................................................. 95
4.1.5 Calculated Fatigue Response of Bridge A ......................................... 98
4.2 Bridge B ....................................................................................................... 102
4.2.1 Geometry ......................................................................................... 102
4.2.2 Instrumentation and Characterization of Fatigue Details ................ 104
4.2.3 Calculated Fatigue Response of Bridge B ....................................... 105
xi
4.3 Bridge C ....................................................................................................... 108
4.3.1 Geometry ......................................................................................... 109
4.3.2 Instrumentation and Characterization of Fatigue Details ................ 110
4.3.3 Calculated Fatigue Response of Bridge C ....................................... 111
4.4 Bridge D ....................................................................................................... 116
4.4.1 Geometry ......................................................................................... 117
4.4.2 Instrumentation and Characterization of Fatigue Details ................ 118
4.4.3 Calculated Fatigue Response of Bridge D ....................................... 119
4.5 Data Acquisition System & Sensors ............................................................ 122
4.5.1 Wired System - CompactRIO .......................................................... 122
4.5.1.1 Programming ............................................................................ 123
4.5.2 Wireless System - Wireless Sensor Networks (WSN) .................... 123
4.5.2.1 Programming ............................................................................ 125
4.5.3 Sensors ............................................................................................. 127
4.6 Summary ...................................................................................................... 128
CHAPTER 5 Interpretation of Measured Fatigue Response of Four Bridges ............... 129
5.1 Duration of Monitoring ................................................................................ 130
5.1.1 Bridge A ........................................................................................... 131
5.1.2 Bridge B ........................................................................................... 134
5.1.3 Bridge C ........................................................................................... 135
5.1.4 Bridge D ........................................................................................... 136
5.2 Measured Fatigue Response During Monitoring Period ............................. 136
5.2.1 Bridge A (Before construction of the retrofit) ................................. 139
5.2.2 Bridge A (After the construction of the retrofit) ............................. 141
5.2.3 Bridge B ........................................................................................... 145
5.2.4 Bridge C ........................................................................................... 148
5.2.5 Bridge D ........................................................................................... 151
5.2.6 Summary of Daily Fatigue Damage at Bridges ............................... 154
xii
5.3 Variations in Measured Fatigue Damage .................................................... 155
5.3.1 Variation with Time of Day ............................................................. 156
5.3.1.1 Bridge A (Before the construction of the retrofit) .................... 157
5.3.1.2 Bridge A (After the construction of the retrofit) ...................... 158
5.3.1.3 Bridge B ................................................................................... 159
5.3.1.4 Bridge C ................................................................................... 160
5.3.2 Variation with Day of the Week ...................................................... 161
5.3.2.1 Bridge A (Before the construction of the retrofit) .................... 163
5.3.2.2 Bridge A (After the construction of the retrofit) ...................... 164
5.3.2.3 Bridge B ................................................................................... 165
5.3.3 Weekly ............................................................................................. 166
5.3.4 Summary of Variation in Measured Fatigue Response ................... 168
5.4 Tracking Progress of Construction at Bridge A .......................................... 168
5.4.1 Rainflow Results .............................................................................. 169
5.4.2 Behavior of Bridge During Construction ........................................ 170
5.5 Summary ...................................................................................................... 174
CHAPTER 6 Comparison of the Calculated and Measured Responses of Bridge A ..... 176
6.1 Bending-Stress Histories ............................................................................. 176
6.2 Distribution Factor ....................................................................................... 182
6.3 Effect of Speed ............................................................................................ 183
6.4 Benefit of Lane Changes ............................................................................. 186
6.5 Effective Stress Range ................................................................................. 188
6.6 Summary ...................................................................................................... 191
CHAPTER 7 Calculation of Remaining Fatigue Life .................................................... 193
7.1 General Considerations ................................................................................ 194
7.1.1 Annual traffic growth ...................................................................... 196
7.1.2 Infinite fatigue life ........................................................................... 199
7.2 Deterministic Approach For Calculating the Remaining Fatigue Life ........ 200
xiii
7.3 AASHTO Approach For Calculating the Remaining Fatigue Life ............. 203
7.4 Probabilistic Approach For Calculating the Remaining Fatigue Life ......... 206
7.5 Calculating the Remaining Fatigue Life at Bridges Monitored During
This Investigation ........................................................................................ 211
7.5.1 Bridge A ........................................................................................... 212
7.5.1.1 Deterministic Method ............................................................... 212
7.5.1.2 AASHTO Method .................................................................... 213
7.5.1.3 Probabilistic Method ................................................................ 215
7.5.1.4 Comparison .............................................................................. 216
7.5.2 Bridge B ........................................................................................... 218
7.6 Summary ...................................................................................................... 218
CHAPTER 8 Conclusions............................................................................................... 219
8.1 Summary ...................................................................................................... 219
8.2 Conclusions .................................................................................................. 221
8.2.1 Techniques for Fatigue Analysis ..................................................... 221
8.2.2 Field Monitoring .............................................................................. 223
8.2.3 Remaining Fatigue Life ................................................................... 225
8.3 Recommendations ........................................................................................ 226
APPENDICES .................................................................................................................230
APPENDIX A Strain data from Bridge A (4/1/2011 to 5/1/2012) ..................... 230
APPENDIX B Strain data from Bridge A (3/1/2012 to 8/1/2012) ..................... 270
APPENDIX C Strain data from Bridge B ........................................................... 281
APPENDIX D Strain data from Bridge C .......................................................... 290
APPENDIX E Strain data from Bridge D ........................................................... 323
References .....................................................................................................................333
xiv
List of Tables
Table 2-1: Fatigue constant ( ) and constant-amplitude fatigue limit (CAFL) for each fatigue detail category (AASHTO LRFD Specifications 2010) ...................15
Table 2-2: Regression analysis coefficients for AASHTO curves in 1974 (Keating and Fisher 1986). ..........................................................................................35
Table 2-3: Proposed lower-bound fatigue constant (Keating and Fisher 1986). ...............36
Table 2-4: Mean and design stress ranges and coefficient of variation (COV) of fatigue data. ...................................................................................................43
Table 2-5: Derived values from regression model. ............................................................44
Table 2-6: Limiting stress range. .......................................................................................48
Table 2-7: Equivalent probabilities of failure and reliability indexes. ..............................53
Table 4-1: Summary of strain gages installed along the longitudinal girders at Bridge A during phase one of the instrumentation. ..................................................96
Table 4-2: Summary of strain gages installed along the longitudinal girders at Bridge A during phase two of the instrumentation. ..................................................97
Table 4-3: Summary of strain gages installed along the longitudinal girders at Bridge A during phase three of the instrumentation. ................................................97
Table 4-4: Live load distribution factors for Bridge A ......................................................98
Table 4-5: Calculated stress range at locations of strain gages used to monitor Bridge A due to the AASHTO fatigue truck. .........................................................102
Table 4-6: Live load distribution factors for Bridge B. ...................................................105
Table 4-7: Calculated stress range at locations in the west girder of Bridge B due to the AASHTO fatigue truck. ........................................................................108
Table 4-8: Summary of strain gages installed at Bridge C ..............................................111
Table 4-9: Live load distribution factors for Bridge C based on lever rule. ....................112
Table 4-10: Live load distribution factors for Bridge C based on AASHTO. .................114
xv
Table 4-11: Calculated stress range at locations of Bridge C due to AASHTO fatigue truck. ...........................................................................................................116
Table 4-12: Summary of strain gages installed at Bridge D ............................................118
Table 4-13: Live load distribution factors for Bridge D based on lever rule. ..................119
Table 4-14: Live load distribution factors for Bridge D based on AASHTO. .................120
Table 4-15: Calculated stress range at locations of Bridge D due to AASHTO fatigue truck. ...........................................................................................................122
Table 5-1: Distribution of full days of rainflow data before retrofit at Bridge A. ...........134
Table 5-2: Distribution of full days of rainflow data after construction of the retrofit at Bridge A. .....................................................................................................134
Table 5-3: Distribution of full days of rainflow data at Bridge B. ..................................135
Table 5-4: Summary of average daily fatigue damage before the construction of the retrofit at Bridge A. .....................................................................................140
Table 5-5: Summary of average daily fatigue damage after the construction of the retrofit at Bridge A. .....................................................................................142
Table 5-6: Occurrence of abnormally large stress cycles for gages attached to the west longitudinal girder after the construction of the retrofit. ............................144
Table 5-7: Summary of average daily fatigue damage at Bridge B. ................................146
Table 5-8: Summary of average daily fatigue damage at Bridge C. ................................150
Table 5-9: Summary of average daily fatigue damage at Bridge D. ...............................152
Table 5-10: Summary of average daily fatigue damage at four bridges. .........................155
Table 5-11: Summary of daily fatigue damage before the construction of the retrofit at Bridge A. .................................................................................................164
Table 5-12: Summary of daily fatigue damage after construction of the retrofit at Bridge A. .....................................................................................................165
Table 5-13: Summary of daily fatigue damage after construction of the retrofit at Bridge A. .....................................................................................................166
xvi
Table 6-1: Equivalent bending stress ranges ( 1 ) in longitudinal girders for trucks crossing the bridge at 30 mph. ....................................................................182
Table 6-2: Live load distribution factors for Bridge A based on measurements. ............183
Table 6-3: Summary of effective stress range ( 4,000 ) before the construction of the retrofit at Bridge A. ...............................................................................191
Table 7-1: Fatigue constant ( ), mean fatigue constant ( ), and constant-amplitude fatigue limit (CAFL) for each fatigue detail category (AASHTO LRFD Specifications 2010). ...................................................................................196
Table 7-2: Stress-range estimate partial load factors (AASHTO Manual for Bridge Evaluation 2011). ........................................................................................204
Table 7-3: Resistance factor ( ) for evaluation, minimum, and mean fatigue life (AASHTO Manual for Bridge Evaluation 2011). .......................................205
Table 7-4: Resistance factor ( ) for evaluation, minimum, and mean fatigue life (Bowman, et al. 2012). ................................................................................206
Table 7-5: Lognormal parameters for random variable ( ) used in this dissertation. .....208
Table 7-6: Variation in fatigue constant ( ) from AASHTO Manual for Bridge Evaluation (2011) and derived parameters for lognormal distribution. ......209
Table 7-7: Fatigue damage in year 37 and extrapolated damage in year 1 for annual growth rates of 2%, 4%, and 6%. ................................................................213
Table 7-8: Calculated fatigue life in years for the west and east girders using the deterministic approach. ...............................................................................213
Table 7-9: Calculated fatigue life in years for different considerations using the AASHTO approach for the west and east girders. ......................................215
Table 7-10: Mean and standard deviation in year 37 for the west and east girders. ........215
Table 7-11: Calculated fatigue life in years for AASHTO, Deterministic, and Probabilistic approaches for the west and east girders. ..............................217
Table A-1: Summary of fatigue damage prior to the construction of the retrofit. ...........230
Table A-2: Summary of fatigue damage after the construction of the retrofit. ...............231
Table C-1: Summary of fatigue damage at Bridge B. .....................................................281
xvii
Table D-1: Summary of fatigue damage for gages attached to the bottom flange at Bridge C. .....................................................................................................290
Table E-1: Summary of daily fatigue damage at Bridge D. ............................................323
xviii
List of Figures
Figure 1-1: Age distribution of bridges in the United States (National Bridge Inventory 2008). ..............................................................................................1
Figure 1-2: Schematic of data flow for envisioned bridge application. ...............................4
Figure 2-1: Stress concentrations due to geometry of tension coupons. ............................10
Figure 2-2: Definition of a stress cycle ..............................................................................12
Figure 2-3: Example of data from a representative fatigue test. ........................................13
Figure 2-4: Regression lines from a representative fatigue test. ........................................14
Figure 2-5: AASHTO S-N curves for design of steel bridges. ..........................................16
Figure 2-6: Example of damage accumulation index. .......................................................17
Figure 2-7: Revised example of damage accumulation index. ..........................................19
Figure 2-8: Stress history for example of simplified rainflow counting. ...........................22
Figure 2-9: Step-by-step procedure for simplified rainflow counting. ..............................22
Figure 2-10: Change in crack-growth rate, / , with range of stress intensity factor. ............................................................................................................25
Figure 2-11: Change in number of cycles due to change in (a) critical crack length, (b) initial crack length, (c) stress range, and (d) type of detail. ..........................27
Figure 2-12: Probability of occurrence ..............................................................................29
Figure 2-13: Example truck passage ..................................................................................29
Figure 2-14: Comparison of cover-plate beam results with AASHTO allowable stress for Category E (from Schilling, et al. (1978)) ..............................................30
Figure 2-15: The variable-amplitude loading cases considered in NCHRP Project 12-15(4) (Fisher, Mertz, and Zhong 1983). .......................................................32
Figure 2-16: Dimensions and distribution of weight of fatigue truck. ..............................37
Figure 2-17: Probability density curves for (a) and and (b) . .............................52
xix
Figure 3-1: Example stress history at the west girder from truck event and the results of the simplified rainflow analysis. ...............................................................60
Figure 3-2: Example histogram of stress ranges for 14 days of data for the east girder at Bridge A. Rainflow bins were 0.15-ksi wide and cycles less than 0.06 ksi were truncated from the histogram..........................................................64
Figure 3-3: Different levels of the damage accumulation index on a S-N graph. .............66
Figure 3-4: Graphical representation of the method for determining the effective stress range. ...................................................................................................67
Figure 3-5: Varying sets of effective stress ranges and measured cycles for different gage locations................................................................................................67
Figure 3-6: Graphical representation of the index stress range method. ...........................69
Figure 3-7: Graphical representation of determining an index stress range from an effective stress range. ....................................................................................70
Figure 3-8: Change in effective stress range and number of cycles at index stress range as the lower cycles are truncated from the calculation for Category E detail. .........................................................................................................72
Figure 3-9: Change in number of cycles at index stress range as the lower cycles are truncated from the calculation for Category E detail. ...................................73
Figure 3-10: Change in effective stress range and number of cycles at index stress range as the lower cycles are truncated from the calculation for Category D detail. .........................................................................................................74
Figure 3-11: Change in number of cycles at index stress range as the lower cycles are truncated from the calculation for Category D detail. ..................................74
Figure 3-12: Change in effective stress range and number of cycles with day of the week for Bridge A. ........................................................................................75
Figure 3-13: Change in number of cycles at index stress range with day of the week for Bridge A. .................................................................................................76
Figure 3-14: Contribution of each bin to the total fatigue damage during the 14-day monitoring period for Bridge A. ...................................................................78
Figure 3-15: Contribution of each bin to , 4.5 during the 14-day monitoring period for Bridge A. ......................................................................................79
xx
Figure 3-16: Average cumulative damage during the 14-day monitoring period for Bridge A. .......................................................................................................80
Figure 3-17: Example histogram of stress ranges for 4 days of data for Bridge D. Rainflow bins were 0.15-ksi wide and cycles less than 0.06 ksi were truncated from the histogram. .......................................................................81
Figure 3-18: Change in effective stress range and number of cycles at index stress range as the lower cycles are truncated from the calculation for Bridge D. ...................................................................................................................82
Figure 3-19: Change in number of cycles at index stress range as the lower cycles are truncated from the calculation for Bridge D. ................................................82
Figure 3-20: Contribution of each bin to , 12 during the 4-day monitoring period for Bridge D. ......................................................................................83
Figure 3-21: Average cumulative damage during the 4-day monitoring period for Bridge D. .......................................................................................................84
Figure 3-22: Stress history with single-point spike at Bridge D. .......................................84
Figure 3-23: Lightning strike (middle photo) during single-point spike at Bridge D. ......85
Figure 4-1: Bridge A. .........................................................................................................87
Figure 4-2: Elevation of girder at Bridge A with spacing of floor beams shown. .............88
Figure 4-3: Location of cover plates on the longitudinal girder at Bridge A to increase the section modulus.......................................................................................89
Figure 4-4: Cross section and location of lanes after widening (looking north). ..............89
Figure 4-5: Close-up of connection between the longitudinal girder and the floor beams and cantilever brackets. .....................................................................90
Figure 4-6: Representation of crack between longitudinal girder and floor beam. ...........91
Figure 4-7: Location of cracks at welded connections between top flanges of longitudinal girder and floor beams identified during fracture-critical inspections of Bridge A. ...............................................................................91
Figure 4-8: Longitudinal girder cross section of Bridge A at (a) floor beam 33 and (b) floor beam 34. ...............................................................................................92
Figure 4-9: Retrofit locations at Bridge A. ........................................................................93
xxi
Figure 4-10: Removal of concrete bridge deck above connections retrofitted. .................94
Figure 4-11: Developed crack at connection between the floor beam and longitudinal girder. ............................................................................................................94
Figure 4-12: Photo of completed retrofit at connection between the longitudinal girder and floor beam at Bridge A. ..........................................................................95
Figure 4-13: Location of strain gages. Floor beams are not shown for clarity. ................96
Figure 4-14: Strain gage located on the top flange of the longitudinal girder of Bridge A approximately 2 ft away from the connection of the floor beam. .............97
Figure 4-15: Crack propagation gage installed at tip of active crack at floor beam 34 of Bridge A. ..................................................................................................98
Figure 4-16: Variation of moment of inertia along the length of Bridge A used for the line-girder analysis. .......................................................................................99
Figure 4-17: Moment envelope from a line analysis of an AASHTO fatigue truck crossing a single girder at Bridge A. ...........................................................100
Figure 4-18: Calculated stress ranges for each girder due to an AASHTO fatigue truck crossing Bridge A in the (a) left lane and (b) right lane. ...................101
Figure 4-19: Bridge B. .....................................................................................................102
Figure 4-20: Plan view of Bridge B (internal diaphragms are not shown). .....................103
Figure 4-21: Bridge cross section and location of lane at Bridge B (looking north). ......104
Figure 4-22: Gage locations at Bridge B (internal diaphragms are not shown). .............104
Figure 4-23: Variation of moment of inertia for Bridge B used for the line analysis. .....106
Figure 4-24: Moment envelope from a line analysis of an AASHTO fatigue truck crossing a single girder at Bridge B. ...........................................................107
Figure 4-25: Calculated stress ranges in the (a) top flange and (b) bottom flange for each girder due to an AASHTO fatigue truck crossing Bridge B. ..............108
Figure 4-26: Bridge C. .....................................................................................................109
Figure 4-27: Plan view of Bridge C (cross frames are not shown). .................................109
Figure 4-28: Bridge cross section and location of lane at Bridge C (looking east). ........110
xxii
Figure 4-29: Strain gages installed within 1″ of the exterior edge of the flange. ............110
Figure 4-30: Gage locations at Bridge C. ........................................................................111
Figure 4-31: Moment envelope from a line analysis of an AASHTO fatigue truck crossing a single girder at Bridge C. ...........................................................114
Figure 4-32: Calculated stress ranges for each girder due to an AASHTO fatigue truck for crossing Bridge C in the (a) left lane, (b) right lane, and (c) exit lane. .............................................................................................................115
Figure 4-33: Bridge D. .....................................................................................................116
Figure 4-34: Plan view of Bridge D. ................................................................................117
Figure 4-35: Cross section of Bridge D (looking west). ..................................................117
Figure 4-36: Locations of gages at I-girder Bridge D. .....................................................118
Figure 4-37: Moment envelope from a line analysis of an AASHTO fatigue truck crossing a single girder at Bridge D. ...........................................................120
Figure 4-38: Calculated stress ranges for each girder due to an AASHTO fatigue truck for crossing Bridge D in the (a) left lane and (b) left-center lane. .....121
Figure 4-39: CompactRIO data acquisition system with NI-9237 modules (wired). ......123
Figure 4-40: Wireless strain node data acquisition system (WSN-3214). .......................124
Figure 4-41: Typical network configuration for bridges. ................................................124
Figure 5-1: Solar panels installed at (a) Bridge A and (b) Bridge B. ..............................131
Figure 5-2: Crack growth at floor beam 34 in east longitudinal girder during 28-day period. .........................................................................................................132
Figure 5-3: Contribution of each bin to , 12 at Bridge B for gages connected to the CompactRIO and WSN data acquisition systems. ............................138
Figure 5-4: Histogram of stress ranges before construction of the retrofit (average of 71 days) for gages attached to the top flange of the (a) west girder and (b) east girder. .............................................................................................139
Figure 5-5: Contribution of each bin to , 4.5 before the construction of the retrofit for gages attached to the longitudinal girder near floor beam 34 (Bridge A). ..................................................................................................140
xxiii
Figure 5-6: The cumulative fatigue damage before the construction of the retrofit for gage attached to the (a) west girder and (b) east girder. .............................141
Figure 5-7: Histogram of stress ranges after the construction of the retrofit (average of 94 days) for gages attached to the top flange of the (a) west girder and (b) east girder. .............................................................................................142
Figure 5-8: Contribution of each bin to , 4.5 during the monitoring period after the construction of the retrofit for gages on the west and east girders. ........................................................................................................143
Figure 5-9: The cumulative fatigue damage during the monitoring period after the construction of the retrofit for gages on the (a) west and (b) east girders. .144
Figure 5-10: The cumulative fatigue damage with cycles above 15 ksi truncated for gage on the west girder. ..............................................................................145
Figure 5-11: Histogram of stress ranges at Bridge B for gages in (a) Span 1 (average of 68 days) and (b) Span 3 (average of 31 days). .......................................146
Figure 5-12: Contribution of each bin to , 12 at Bridge B for gages in Span 1 and Span 3. ..................................................................................................147
Figure 5-13: The cumulative fatigue damage at Bridge B for gages in (a) Span 1 and (b) Span 3. ...................................................................................................148
Figure 5-14: The cumulative fatigue damage at Bridge B if stress ranges below 0.4 ksi were truncated for gages in (a) Span 1 and (b) Span 3. ........................148
Figure 5-15: Histogram of stress ranges at Bridge C for (a) girder 3 (average of 19 days) and (b) girder 5 (average of 10 days). ...............................................149
Figure 5-16: Contribution of each bin to , 12 at Bridge C for gages at girder 3 and girder 5. ................................................................................................150
Figure 5-17: The cumulative fatigue damage at Bridge C for gages at (a) girder 3 and (b) girder 5. .................................................................................................151
Figure 5-18: Histogram of stress ranges at Bridge D (average of 4 days) for gage locations along (a) girder 2 and (b) girder 3. ..............................................152
Figure 5-19: Contribution of each bin to , 12 at Bridge D for gages attached to girders 2 and 3.........................................................................................153
xxiv
Figure 5-20: The cumulative fatigue damage at Bridge D for gages attached to (a) girder 2 and (b) girder 3. .............................................................................153
Figure 5-21: Variation in 4.5 with time of day before the construction of the retrofit at Bridge A. .....................................................................................157
Figure 5-22: Contribution of each bin to 4.5 during the specific 6-hour periods before the construction of the retrofit for the east girder at Bridge A. .....................................................................................................158
Figure 5-23: Variation in 4.5 with time of day after construction of the retrofit at Bridge A. .................................................................................................158
Figure 5-24: Contribution of each bin to 4.5 during the specific 6-hour periods after construction of the retrofit for the east girder at Bridge A. ...159
Figure 5-25: Variation in 12 with time of day at Bridge B (cycles less than 0.4 ksi were truncated). ...............................................................................160
Figure 5-26: Contribution of each bin to 12 during the specific 6-hour periods for Span 1 at Bridge B. ...............................................................................160
Figure 5-27: Variation in 12 with time of day at Bridge C (cycles less than 0.4 ksi were truncated for girder 3 only). ....................................................161
Figure 5-28: Contribution of each bin to 12 during the specific 6-hour periods for girder 5 at Bridge C. ..............................................................................161
Figure 5-29: Average variation of fatigue damage ( , 4.5 ) for the west and east girders before the construction of the retrofit at Bridge A. .........................163
Figure 5-30: Variation of fatigue damage ( , 4.5 ) for the west and east girders after construction of the retrofit at Bridge A. .............................................164
Figure 5-31: Variation of fatigue damage ( , 4.5 ) for Span 1 and Span 3 at Bridge B (cycles less than 0.4 ksi were truncated). ....................................165
Figure 5-32: Variation in average daily damage ( , 4.5 due to a week of continuous monitoring for gage E-35n-BE at Bridge A. ............................167
Figure 5-33: Variation in average daily damage ( , 4.5 ) due to a week of continuous monitoring for gage location E-34s-TW at Bridge A. ..............167
xxv
Figure 5-34: Histogram of stress ranges during the construction of the retrofit (average of 57 days) for gages on the bottom flanges of the (a) west and (b) east girders. ............................................................................................169
Figure 5-35: Histogram of stress ranges during the construction of the retrofit (average of 57 days) for gages on the top flanges of the (a) west and (b) east girders. .................................................................................................170
Figure 5-36: Daily fatigue damage ( , 4.5 ) during the construction process for the bottom flanges in the west and east girders at Bridge A. ......................171
Figure 5-37: Daily fatigue damage ( , 4.5 ) during the construction process for the top flanges in the west and east girders at Bridge A. ............................172
Figure 5-38: Contribution to fatigue damage for the top and bottom flanges in the east girder of Bridge A (a) before and (b) after construction of the retrofit. .....173
Figure 5-39: Contribution to fatigue damage for the top flange of the east girder and retrofit plate at Bridge A. ............................................................................173
Figure 5-40: Daily fatigue damage ( , 4.5 ) during the construction process for the east longitudinal girder near floor beam 27 at Bridge A. ......................174
Figure 6-1: Truck weight and spacing of axles. ...............................................................177
Figure 6-2: Proposed new center lane for Bridge A. .......................................................178
Figure 6-3: Isolation of flexural bending and flange warping stresses. ...........................178
Figure 6-4: Flexural stress history in top flange of east and west girders at floor beam 34 due to test truck crossing the bridge at 10 mph in the (a) left lane and (b) right lane. ...............................................................................................179
Figure 6-5: Flexural stress history in top flange of east and west girders at floor beam 34 due to test truck crossing the bridge at 63 mph in the (a) left lane and (b) right lane. ...............................................................................................180
Figure 6-6: Histogram of flexural stress ranges in the top flange of the east girder near floor beam 34 due to the test truck crossing the bridge at 30 mph in the (a) left lane and (b) right lane. ...............................................................180
Figure 6-7: Influence of speed on at gages located near floor beam 34. ...................185
Figure 6-8: Influence of speed on at gages located near floor beam 35. ...................185
xxvi
Figure 6-9: Proportion of load carried by the (a) west girder and (b) east girder for trucks in the left and right lanes. .................................................................186
Figure 6-10: Possible distributions of truck traffic. .........................................................188
Figure 6-11: Graphical representation of the method for determining an equivalent stress range for an assumed number of cycles ( ). ...................................189
Figure 7-1: (a) Annual traffic volume and (b) accumulated traffic volume for different growth models. ............................................................................................197
Figure 7-2: (a) Annual traffic volume and (b) accumulated traffic volume for constant, annual growth rates of 2%, 4%, and 6%. .....................................199
Figure 7-3: Probability of failure in the (a) west and (b) east girders. .............................216
Figure 8-1: Increase in fatigue damage ( , 4.5 ) due to change in width of the rainflow bin. ................................................................................................229
Figure A-1: Response of bridge at gage E-35n-TE. ........................................................232
Figure A-2: Response of bridge at gage E-35n-TW. .......................................................234
Figure A-3: Response of bridge at gage E-35s-TE. .........................................................236
Figure A-4: Response of bridge at gage E-35s-TW. ........................................................238
Figure A-5: Response of bridge at gage E-35n-BE. ........................................................240
Figure A-6: Response of bridge at gage E-35n-BW. .......................................................242
Figure A-7: Response of bridge at gage E-34s-TE. .........................................................244
Figure A-8: Response of bridge at gage E-34s-TW. ........................................................246
Figure A-9: Response of bridge at gage W-35s-TE. ........................................................248
Figure A-10: Response of bridge at gage W-35s-TW. ....................................................250
Figure A-11: Response of bridge at gage W-35s-BE. .....................................................252
Figure A-12: Response of bridge at gage W-35s-BW. ....................................................254
Figure A-13: Response of bridge at gage W-34s-TE. ......................................................256
Figure A-14: Response of bridge at gage W-34s-TW. ....................................................258
xxvii
Figure A-15: Response of bridge at gage E-27s-TE. .......................................................260
Figure A-16: Response of bridge at gage E-27s-TW. ......................................................261
Figure A-17: Response of bridge at gage E-27s-BE. .......................................................262
Figure A-18: Response of bridge at gage E-27s-BW. .....................................................263
Figure A-19: Response of bridge at gage E-34-BE. ........................................................264
Figure A-20: Response of bridge at gage E-34-BW. .......................................................265
Figure A-21: Response of bridge at gage E-34s-TE-P. ...................................................266
Figure A-22: Response of bridge at gage E-34s-TW-P. ..................................................267
Figure A-23: Response of bridge at gage E-33n-TE. ......................................................268
Figure A-24: Response of bridge at gage E-33n-TW. .....................................................269
Figure B-1: Response of bridge at gages E-35n-TE and E-35n-TW. ..............................271
Figure B-2: Response of bridge at gages E-35s-TE and E-35s-TW. ...............................272
Figure B-3: Response of bridge at gages E-35n-BE and E-35n-BW. ..............................273
Figure B-4: Response of bridge at gages E-34s-TE and E-34s-TW. ...............................274
Figure B-5: Response of bridge at gages E-34-BE and E-34-BW. ..................................275
Figure B-6: Response of bridge at gages E-34s-TE-P and E-34s-TW-P. ........................276
Figure B-7: Response of bridge at gages E-33s-TE and E-33s-TW. ...............................277
Figure B-8: Response of bridge at gages W-35s-TE and W-35s-TW. ............................278
Figure B-9: Response of bridge at gages W-35s-BE and W-35s-BW. ............................279
Figure B-10: Response of bridge at gages W-34s-TE and W-34s-TW. ..........................280
Figure C-1: Response of bridge at gage W-1-BE. ...........................................................282
Figure C-2: Response of bridge at gage W-1-BW. ..........................................................283
Figure C-3: Response of bridge at gage W-2-BE. ...........................................................284
Figure C-4: Response of bridge at gage W-2-BW. ..........................................................285
xxviii
Figure C-5: Response of bridge at gage W-3-BE. ...........................................................286
Figure C-6: Response of bridge at gage W-3-BW. ..........................................................287
Figure C-7: Response of bridge at gage W-4-BE. ...........................................................288
Figure C-8: Response of bridge at gage W-4-BW. ..........................................................289
Figure D-1: Response of bridge at gage L1-1-TN. ..........................................................291
Figure D-2: Response of bridge at gage L1-1-TS. ...........................................................292
Figure D-3: Response of bridge at gage L1-1-BN. ..........................................................293
Figure D-4: Response of bridge at gage L1-1-BS. ..........................................................294
Figure D-5: Response of bridge at gage L1-3-TN. ..........................................................295
Figure D-6: Response of bridge at gage L1-3-TS. ...........................................................296
Figure D-7: Response of bridge at gage L1-3-BN. ..........................................................297
Figure D-8: Response of bridge at gage L1-3-BS. ..........................................................298
Figure D-9: Response of bridge at gage L1-5-BN. ..........................................................299
Figure D-10: Response of bridge at gage L1-5-BS. ........................................................300
Figure D-11: Response of bridge at gage L2-1-TN. ........................................................301
Figure D-12: Response of bridge at gage L2-1-TS. .........................................................302
Figure D-13: Response of bridge at gage L2-1-BN. ........................................................303
Figure D-14: Response of bridge at gage L2-1-BS. ........................................................304
Figure D-15: Response of bridge at gage L2-3-TN. ........................................................305
Figure D-16: Response of bridge at gage L2-3-TS. .........................................................306
Figure D-17: Response of bridge at gage L2-3-BN. ........................................................307
Figure D-18: Response of bridge at gage L2-3-BS. ........................................................308
Figure D-19: Response of bridge at gage L2-5-BN. ........................................................309
Figure D-20: Response of bridge at gage L2-5-BS. ........................................................310
xxix
Figure D-21: Response of bridge at gage L3-1-TN. ........................................................311
Figure D-22: Response of bridge at gage L3-1-TS. .........................................................312
Figure D-23: Response of bridge at gage L3-1-BN. ........................................................313
Figure D-24: Response of bridge at gage L3-1-BS. ........................................................314
Figure D-25: Response of bridge at gage L3-3-TN. ........................................................315
Figure D-26: Response of bridge at gage L3-3-TS. .........................................................316
Figure D-27: Response of bridge at gage L3-3-BN. ........................................................317
Figure D-28: Response of bridge at gage L3-3-BS. ........................................................318
Figure D-29: Response of bridge at gage L3-5-TN. ........................................................319
Figure D-30: Response of bridge at gage L3-5-TS. .........................................................320
Figure D-31: Response of bridge at gage L3-5-BN. ........................................................321
Figure D-32: Response of bridge at gage L3-5-BS. ........................................................322
Figure E-1: Response of bridge at gage L1-2-TN. ..........................................................324
Figure E-2: Response of bridge at gage L1-2-BN. ..........................................................325
Figure E-3: Response of bridge at gage L1-3-TS. ...........................................................326
Figure E-4: Response of bridge at gage L1-3-BS. ...........................................................327
Figure E-5: Response of bridge at gage L2-3-TS. ...........................................................328
Figure E-6: Response of bridge at gage L2-3-BS. ...........................................................329
Figure E-7: Response of bridge at gage L3-2-BN. ..........................................................330
Figure E-8: Response of bridge at gage L3-3-TS. ...........................................................331
Figure E-9: Response of bridge at gage L3-3-BS. ...........................................................332
1
CHAPTER 1
Introduction
1.1 OVERVIEW OF RESEARCH
Transportation officials face the difficult task of maintaining the nation’s
inventory of bridges under the pressure of reduced budgets and an aging infrastructure.
Highway bridges are critical elements of the transportation network, providing the means
to transport people and goods across the nation. As more bridges reach or exceed their
intended design lives, transportation officials must make difficult decisions on which
bridges can be safely kept in service versus those that need to be replaced or retrofitted.
Figure 1-1: Age distribution of bridges in the United States (National Bridge Inventory
2008).
The US bridge inventory comprises over 600,000 bridges, and a significant
portion is beyond 50 years in age (Figure 1-1). In the current economic environment,
transportation officials do not have the resources to replace a bridge simply because it has
reached a certain age. Instead, transportation officials rely on qualitative data from
routine visual inspections to assess the overall condition and maximize the service life of
each bridge in their inventory. If deterioration is identified, actions can be taken to
0
10,000
20,000
30,000
40,000
50,000
60,000
Nu
mb
er o
f b
rid
ges
Current age of bridge (yr)
≈ 600,000 bridges
2
mitigate further damage. However, visual inspections alone cannot detect all forms of
deterioration (Moore, et al. 2000).
Structural deterioration and safety problems are generally identified in bridges in
the US through visual inspections. The National Bridge Inspection Program (USDOT
2006) requires all bridges with spans greater than 20 ft to be inspected at least once every
two years. For most bridges, the inspection is cursory and identifies safety problems.
However, more detailed, hands-on inspections are required for fracture-critical bridges.
Bridges that have non-redundant structural systems are designated fracture critical
because the loss of a single structural member has the potential of causing wide-spread
damage or collapse of the bridge. One of the primary modes of deterioration for steel
bridges is the growth of fatigue cracks, which may lead to brittle fracture of structural
components. Hands-on inspections of fracture-critical bridges provide transportation
officials with data relative to the growth and locations of cracks. Depending on the
inspection results, transportation officials can perform maintenance to arrest the crack
growth, increase the frequency of inspections, or, if the cracks are not growing, do
nothing.
Although fixed-interval inspections ensure that each bridge will be reviewed for
damage, one of the drawbacks with this protocol is the lack of recognition of damage
accumulation. Current procedures generally do not provide a distinction among bridges
with different truck-traffic volumes. As such, a bridge along a major interstate requires
the same inspection frequency as a rural bridge along a country road, even though the rate
of damage accumulation is likely quite different. The age of the bridge is also currently
not considered to have an impact, despite the fact that bridges approaching their design
lives are at a higher risk for deterioration as compared to newer bridges. Thus, if a bridge
was put into service tomorrow, it would require the same inspection frequency as one that
has been in service for 50 years. Finally, another limitation is the possibility that the
damage rate can escalate between inspections.
Recent legislation in the US Congress highlights the need for quantitative data to
set priorities amongst bridges. MAP-21, the Moving Ahead for Progress in the 21st
3
Century Act, was signed into law on July 6, 2012 (Federal Highway Administration
2012). MAP-21 allows federal money to be used for new construction or rehabilitation
of highway bridges based on performance-based or risk-based criteria.
By providing appropriate technology and methodology, inspection practices can
be enhanced through real-time monitoring of bridges. Real-time monitoring systems are
advantageous because they can detect deterioration, as long as the sensors are installed in
the correct location, between inspection visits and notify transportation officials of
problems. For instance, strain gage data can be used as a measure of fatigue damage and
the remaining fatigue life can be estimated. The remaining fatigue life can then be used
by transportation officials to prioritize bridges for inspection, retrofit, and/or replacement
using quantitative data. It is important to emphasize that instrumentation does not
generally identify damage, but instead provides information on the likely progression of
damage to help focus inspections to the locations with the most damage.
Within the context described above, the National Institute of Standards and
Technology (NIST) sponsored a research project entitled, “Development of Rapid,
Reliable and Economic Methods for Inspection and Monitoring of Highway Bridges.”
The primary goal of the project was to provide transportation officials with the
technology and methodology to collect quantitative information on bridge performance to
complement the qualitative data obtained during hands-on-inspections.
1.2 PROJECT DESCRIPTION
The research project was awarded by NIST in 2008 through the Technology
Innovation Program (TIP). The project involves a multi-disciplinary team and is a joint
venture between the University of Texas at Austin (UT), National Instruments (NI), and
Wiss, Janney, Elstner Associates (WJE). UT is the lead for the project and involves
students and faculty from the Departments of Civil, Architectural and Environmental
Engineering; Electrical and Computer Engineering; and Mechanical Engineering.
A low-power, wireless sensor network (WSN) capable of long-term deployment
was developed to monitor steel bridge systems. Early in the project, the project team
4
chose to use a wireless data acquisition system because of the lower hardware and
installation costs as compared to wired systems. The wireless system is based on IEEE
802.15.4, which is a low-power, low-bandwidth network. Fracture-critical bridges were
the primary target of the research; as such, monitoring fatigue damage was the major
focal point of the research and this dissertation.
A description of the wireless system is presented in Fasl, et al. (2012a) and briefly
discussed in Section 4.5.2. The components of the system are sensors, WSN nodes,
gateway, and remote access. The gateway is the key component of the system that
establishes and manages the wireless network and also allows remote access to the
network through a cellular modem. All monitoring programs for the instrumentation
were written in the NI LabVIEW (Laboratory Virtual Instrumentation Engineering
Workbench) format. A variety of sensors may be used on the bridge depending on the
desired data to be collected. However, strain gages were the primary source of data used
in this investigation.
Figure 1-2: Schematic of data flow for envisioned bridge application.
In terms of data flow (Figure 1-2), the sensors are connected to wireless nodes,
which can perform simple LabVIEW programs in real time using the microprocessor on
the node. Data are processed at the node to save power (transmitting analysis results is
5
more power effective than transmitting data histories (Lynch, et al. 2004)) and to limit
network bandwidth. Each node can be configured as either an end node or a router node.
End nodes are connected to the sensors and can transmit the data to either the gateway or
to another node that is configured as a router node. Router nodes are also potentially
connected to sensors, but as the name implies can also receive and transmit data from
other nodes. The combination of the end and router nodes forms the WSN system that
collects and transmits data back to the gateway. Additional, more-complicated analyses
can be performed at the gateway. Data can be stored on the gateway or if a modem is
connected to the gateway, the data can be transferred to a cloud server. On the cloud
server, the engineer or transportation official can access the data from anywhere. The
cloud server can also be configured to send alerts to transportation officials, if problems
arise. Thus, each piece of hardware (node, gateway, and/or cloud) can be used for data
analysis while maintaining the data of interest for the transportation official in order to
maximize the efficiency of the monitoring system. The transportation official also has
the ability to reconfigure the nodes and gateway, as needed for a particular application.
The monitoring system was envisioned to function for 10 years with minimal
maintenance. Though wireless systems have many advantages, there are some
limitations that must be overcome for a 10-year life, namely power and interference
issues. The interference issues were researched and the tests are summarized in Fasl, et
al. (2012b). Because the goal is to power the sensors free from the power grid, energy-
harvesting techniques were studied to obtain a targeted battery life of ten years.
Several different energy-harvesting techniques were considered for the various
components of the monitoring system (Weaver 2011). Solar panels have been used in
many situations and offer high power densities, such that the panels are appropriate for
nodes and gateways (Inamdar 2012). Wind turbines have been studied by the research
team and found to be capable of powering the wireless nodes (McEvoy 2011; Zimowski
2012). In addition, vibrational energy from vehicular-induced excitations was found to
be sufficient to power the wireless nodes for certain bridge types (Dierks 2011;
Reichenbach 2012). Through a combination of these energy-harvesting sources, it is
6
possible to power the monitoring system and reach the ten-year service life. At this stage
of the project, solar panels are the only commercial option for energy harvesting.
Beyond powering the monitoring system, the sensors (primarily strain gages or
crack propagation gages) need to be properly chosen and installed to survive at least ten
years. The harsh environmental conditions, severe temperature ranges and humidity
fluctuations, make monitoring bridges challenging. A wide variety of strain sensors have
been tested in different environmental conditions to identify robust gages and installation
techniques. The preliminary results are published in Samaras, et al. (2012).
1.3 IMPLICATIONS OF RESEARCH
The research presented in this dissertation represents a small portion of the
research project as a whole. This dissertation formalizes a methodology for analyzing
strain data for a fatigue analysis. Techniques were developed to normalize the data, such
that gages at different locations within the bridge or even on different bridges could be
easily compared. With normalized data, the variation in fatigue damage can be easily
evaluated on an hourly, daily, and weekly basis. In addition, the influence of a retrofit at
reducing the fatigue damage can be quickly assessed. Though the methods were
developed to be a part of a wireless data acquisition system for fracture-critical bridges,
they are also appropriate for wired data acquisition systems and redundant bridges.
A primary concern from using field-monitored data is whether smaller cycles
should be truncated or considered in a fatigue analysis. There is currently not
international agreement on whether stress cycles below the constant-amplitude fatigue
limit (CAFL) for a particular fatigue detail influence the fatigue life. Therefore,
techniques were developed so an engineer can characterize the entire spectrum of stress
ranges and make a decision, based on engineering judgment, whether to include or
discard the cycles from an analysis.
Compared to the results from strain data, the design approach for fatigue that
utilizes structural analysis in the AASHTO Load and Resistance Factor Design (LRFD)
Bridge Design Specifications (2010) may be overly conservative because each bridge has
7
a unique geometry. For instance, the girder distribution and dynamic impact factors that
were determined from strain gages are very different than the ones provided by
AASHTO. As such, the remaining fatigue life that is calculated from structural analysis
may be incorrect. Approaches for calculating the remaining fatigue life using field-
measured data and the assumptions associated with each method are discussed. Using the
remaining fatigue life, transportation officials can set priorities for maintenance or
replacement among an inventory of bridges.
1.4 ORGANIZATION OF DISSERTATION
This dissertation consists of eight chapters, with a focus on the development of
methodologies for assessing fatigue in steel highway bridges. Background material and a
literature review are presented in Chapter 2. Specifically, fatigue is defined and the
methods of characterizing the fatigue resistance are discussed. The literature review
focuses on three areas: response and behavior of steel structures to variable-amplitude
loading, uncertainties from monitoring bridges in the field, and application of
probabilistic approaches to estimating remaining fatigue life.
Five techniques for analyzing strain data for a fatigue analysis are discussed in
Chapter 3. Simplified rainflow counting, which converts a stress history into a histogram
of stress cycles, is an algorithm standardized by ASTM and the first step of a fatigue
analysis. Two methods, effective stress range and index stress range, for determining the
total amount of fatigue damage during a monitoring period are presented. In the
AASHTO Manual for Bridge Evaluation (2011), the effective stress range is the
traditional approach for determining the amount of damage. The index stress range is a
new method that was developed to facilitate comparisons of fatigue damage between
sensors and/or bridges. Visualization of data is important, especially in wireless systems
where strain data are processed in real time and not archived due to the bandwidth
restrictions of the wireless network. Thus, two techniques, contribution to damage and
cumulative damage, were conceived to allow an engineer to characterize the spectrum of
stress ranges. Using those two methods, an engineer can evaluate whether lower stress
8
cycles (concern due to electromechanical noise from the data acquisition system) and
higher stress ranges (concern due to possible spike from the data acquisition system)
contribute significantly to the accumulation of damage in the bridge.
Chapter 4 summarizes the four different bridges that were instrumented and
analyzed during this project. Two of the bridges were classified as fracture critical,
whereas the other two were considered to be redundant. The instrumentation locations
and data acquisition systems are described.
Representative strain data from the four bridges are presented in Chapter 5. The
results from all of the strain gages are presented in Appendices A-E. Likely fatigue
damage was evaluated using the techniques discussed in Chapter 3. By using the index
stress range, the hourly, daily, and weekly variations in the fatigue damage are
characterized.
The results of a live load test at one of the bridges are presented in Chapter 6.
The girder distribution factor and dynamic impact factor were determined from
measurements and compared to the design values obtained from the AASHTO LRFD
Specifications (2010). In addition, the distribution of measured stress ranges was
compared to the assumed distribution of trucks crossing the bridge based on the
AASHTO fatigue truck.
Three methods for calculating the remaining fatigue life for steel bridges are
presented in Chapter 7. The fatigue life of a particular bridge depends on the inherent
variability of the bridge details, accumulation of fatigue cycles (past, current, and future),
and the method of calculation. The uncertainties from each source are described.
A summary of the research conducted is provided in Chapter 8, which also
includes the conclusions and recommendations of this study.
9
CHAPTER 2
Background & Literature Review
Summaries of the background information relevant to fatigue and the evaluation
of fatigue based on field monitoring of steel bridges are provided in this chapter. In
particular, a definition of fatigue, methods for characterizing fatigue resistance, and a
literature review in three areas (resistance under variable-amplitude loading, field
monitoring of bridges, and structural reliability) are presented.
2.1 DEFINITION OF FATIGUE
Fatigue is the process of crack growth in steel structures due to the cyclic
variations in the applied stress. In high-cycle fatigue, the magnitudes of the stress are
often well below the yield stress of the material; however, stress concentrations due to the
presence of defects or microcracks in the material and/or residual stresses lead to crack
growth (Barsom and Rolfe 1999). In these situations, the number of cycles necessary for
the crack to grow to the critical size, such that brittle fracture occurs, range from several
hundred thousand to several million.
The initial cracks in the material often initiate from small flaws resulting from the
manufacturing and/or fabrication processes. The cracks propagate under the applied
stress cycles, and lead to fracture once the fracture toughness of the material has been
exceeded. In rolled shapes, the flaws arise from surface and edge imperfections,
irregularities in mill scale, inclusions, and from the mechanical notches due to handling,
straightening, cutting, and shearing (Fisher, Kulak, and Smith 1998). When bolts or
rivets are used, additional flaws can be introduced from drilling or punching the holes.
Many more fatigue issues develop from the welding process, such that welded details are
in general more severe to fatigue damage than bolted details.
Two problems are created by the welding process: (1) residual stresses from the
shrinking of the weld and differential heating and/or cooling and (2) the potential
10
introduction of initial flaws. An applied stress less than the yield strength of the material
can cause inelastic behavior locally due to residual stresses. Continued application of the
applied loads causes repeated plastic deformation to occur locally and eventual growth of
the crack. Flaws are generated from the welding process when the weld has partial
penetration, lack of fusion, micro flaws at the weld toe, and start and stop locations
(Fisher, Kulak, and Smith 1998). While flaws in the welds cannot be totally eliminated,
proper welding procedures and inspections can help minimize the frequency and size of
the flaws.
The geometric shape of a member or detail also affects the fatigue behavior.
Changes in the geometry often result in stress concentrations that reduce the fatigue
resistance of a specimen (Barsom and Rolfe 1999). The magnitude of the stress
concentration is related to the severity of the change in geometry. For instance, three
shapes are shown in Figure 2-1. The specimen with no change in geometry (Figure
2-1(a)) will have the best fatigue behavior. The specimen on the right (Figure 2-1(c))
will have the most severe stress concentration due to the highly localized change in
geometry. Compared with the other two details, the middle detail (Figure 2-1(b)) will
have an intermediate stress concentration. The change in geometry in Case B will
produce a stress concentration that will lead to lower fatigue resistance compared with
case A; however, the fatigue behavior for Case B would be better than Case C because
the change is geometry is more gradual.
Figure 2-1: Stress concentrations due to geometry of tension coupons.
More severe Most severeBase(a) (b) (c)
Weld, typ.
11
In welded structures, such as steel bridges, the primary variables used to evaluate
the fatigue performance are the magnitude of the stress range and the type of detail
(Fisher, et al. 1970). The minimum, mean, and maximum stresses do not significantly
affect fatigue behavior in welded structures due to high residual stresses (Schilling, et al.
1978). Those parameters can be very important in certain fatigue applications (i.e.
airplanes); however, they are not a major concern in welded steel bridges. Fatigue is
mainly a concern in structures subjected to tensile stress cycles because the tension tends
to open and grow the crack. Although compressive stresses will tend to close the crack,
crack growth can still occur under compression if residual stresses tend to keep the crack
tip open and subjected to tension. The type of detail is a primary variable due to the
stress concentrations that form from geometric changes. In steel bridge design, the
typical arrays of details are classified into various design categories, each having a
different fatigue performance. If a new detail is desired, the fatigue resistance can be
determined both experimentally and numerically. As is discussed in Section 2.2.2, the
initial crack size and fracture toughness also affect the fatigue behavior.
2.2 CHARACTERIZING FATIGUE RESISTANCE
There are two prevalent methods for characterizing the fatigue resistance of a
specimen. The resistance can be determined experimentally following a stress-based
approach (S-N curves) or numerically from a fracture-mechanics model. Both methods
result in considerable uncertainty in the fatigue resistance, partly because fatigue is a
complex process that involves localized damage accumulation. In addition, the actual
nature of the applied loads and the extent of imperfections are not known.
2.2.1 Stress-Based Approach
The classical experimental method to characterize fatigue resistance is to test a
given detail in tension at a constant-amplitude stress range and count the number of
cycles to failure. Nominally identical details are tested at different constant-amplitude
12
stress ranges until failure. The stress range is defined as the difference between the
highest and lowest values in a stress history (Figure 2-2).
Figure 2-2: Definition of a stress cycle
After testing the detail at various stress ranges, the data can be plotted on a graph
of number of cycles to failure ( ) for various cyclic stress ranges ( ). The data are
typically plotted on a log-log plot, as demonstrated in Figure 2-3. It was determined
empirically that stress range and type of detail are the primary variables affecting fatigue
resistance. It is important to note that the stress range is typically the nominal stress
range. The flow of stresses at discontinuities and/or welds creates locations of stress
concentrations that lead to higher magnitudes than calculated from engineering
mechanics (bending stress, axial stress, etc.). Because the stress concentrations vary
with the detail, the nominal stress (stress calculated away from the discontinuities and/or
welds) is typically used to characterize the S-N curves.
0.0
2.0
4.0
6.0
8.0
10.0
0 1 2 3 4
Stre
ss (
ksi)
Time (sec)
One cycle
13
Figure 2-3: Example of data from a representative fatigue test.
By considering a single type of detail, the fatigue resistance can be described by
Equation 2-1. The constants and are determined empirically from the fatigue data.
Equation 2-1
where
numberofcyclesuntilfailureat
empiricalconstantforspecificdetailfromfatiguedataksi‐B
nominalconstant‐amplitudestressrange ksi
slopeoftheS‐Ncurve
To determine the fatigue resistance, regression techniques are used to define the
line that goes through the mean of the data. For most metals, the slope of the S-N curve
( ) will vary between -2 to -4 (-3 used by most specifications). The code value is
determined by moving the mean failure line approximately two standard deviations to the
left (Figure 2-4). The code value in the AASHTO Load and Resistance Factor Design
(LRFD) Bridge Design Specifications (2010) typically corresponds to a 95% confidence
interval of a 95% probability of survival (probability of failure of approximately 5%)
(Keating and Fisher 1986).
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Mean
Data points
Constant-amplitude fatigue limit
“Run-out” test
14
Figure 2-4: Regression lines from a representative fatigue test.
As shown in Figure 2-3, there is a stress range at which the detail is assumed to
have an infinite fatigue life for a given constant-amplitude stress range. This horizontal
line corresponds to the constant-amplitude fatigue limit (CAFL). The determination of
the CAFL historically corresponded to two million cycles, at which point the specimen
would be declared a “run-out” test. The two-million cycle limit was based upon
equipment limitations during the period of testing (1970s-1980s) (Munse 1964). With
modern testing facilities, specimens can be and are often tested to much higher cycle
counts.
The equation currently used in the AASHTO LRFD Specifications (2010) is
shown in Equation 2-2. As expected, the form follows Equation 2-1, with the exception
that all detail categories are assumed to have a value of equal to -3. Because Equation
2-2 is used for design, the fatigue constant ( ) for each fatigue category reported in Table
2-1 are lower-bound values (approximately 5% probability of failure).
Equation 2-2
where
numberofcyclesuntilfailureat
fatigueconstantfordetailcategory,definedbyAASHTO
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Code value
Constant-amplitude fatigue limit
5% probability of failure
95% probability of survival
Number of cycles to failure
Mean
15
LRFDSpecifications 2010 ksi3
constant‐amplitudestressrange ksi
Table 2-1: Fatigue constant ( ) and constant-amplitude fatigue limit (CAFL) for each
fatigue detail category (AASHTO LRFD Specifications 2010)
Category Fatigue constant, (ksi3) CAFL (ksi) A 250×108 24.0 B 120×108 16.0 B′ 61×108 12.0 C 44×108 10.0 C′ 44×108 12.0 D 22×108 7.0 E 11×108 4.5 E′ 3.9×108 2.6
The AASHTO fatigue categories for design are best understood by graphing the
expressions in the form of S-N curves as shown in Figure 2-5. The AASHTO LRFD
Specifications (2010) provide examples of typical types of details that fit into each of the
eight fatigue categories. For instance, Category A corresponds to the fatigue resistance
of the base metal (i.e. flat plate with no weld attachments) and Category B corresponds to
continuous longitudinal fillet welds. The specimen from Figure 2-1(a) would be
considered a Category A detail. Because the CAFL are high and the fatigue constants are
so large, these types of details never control the fatigue resistance of a bridge. Instead,
the design of details with discontinuities or attachments with fillet or groove welds
parallel and perpendicular to the applied stress are more likely to be controlled by fatigue
considerations. In those situations, the fatigue category varies between C and Eʹ. The
specimen Figure 2-1(c) could be classified as a Category E detail, whereas the
classification of specimen from Figure 2-1(b) would depend on the radius of the
transition.
16
Figure 2-5: AASHTO S-N curves for design of steel bridges.
2.2.1.1 Palmgren-Miner’s Rule
Though fatigue tests have historically been performed using a constant-amplitude
stress range, real structures are subjected to varying-amplitude stress ranges. For bridges,
the amplitudes vary with the weight and length of the crossing vehicles. For offshore
structures, the amplitudes depend on the wave height and frequency. Though one event
may generate a 5-ksi cycle and the next event produces a 3-ksi cycle, each will contribute
to the fatigue damage.
A cumulative damage theory is needed to relate the varying-amplitude cycles to
the constant-amplitude fatigue data. Palmgren-Miner’s rule is the most commonly-used
cumulative damage theory because it is simple and agrees well with historic fatigue data
(see Section 2.3.1). The rule follows a linear-damage hypothesis and is expressed by
Equation 2-3 and Equation 2-4 (Miner 1945).
,
Equation 2-3
Equation 2-4
1
10
100
1E+05 1E+06 1E+07 1E+08
Stre
ss r
ange
, Sr
(ksi
)
Number of cycles to failure, Nf
ABB′
C & C′D
EE′
C′ C
17
where contributionofcycles toPalmgren‐Miner’sdamage
accumulationindex
Palmgren‐Miner’sdamageaccumulationindex numberofcyclesmeasuredat ,
, numberofcyclesuntilfailureat , numberofdifferentstressranges
In Palmgren-Miner’s rule, each cycle causes damage; however, the damage
induced is proportional to the number of cycles corresponding to failure for the actual
stress range. The total damage in an element subjected to multiple stress ranges can be
determined by simply summing the damage that accumulates at each stress range. For
example, considering a detail with an AASHTO Category D rating that is tested for
300,000 cycles at 8 ksi and 300,000 cycles at 16 ksi, the damage at each stress range can
be evaluated and summed. As shown in Figure 2-6, although the number of cycles are
the same for the two stress ranges, the 16-ksi cycles contribute eight times as much
damage as compared to the 8-ksi cycles. The reason for the difference is because , is
inversely proportional with the cube of the stress range.
Figure 2-6: Example of damage accumulation index.
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
4,000,000
4,500,000
8 16
Nu
mb
er
of
cycl
es
Stress range (ksi)
4.3x106
cycles at 8 ksi
0.54x106
cycles at 16 ksi
0.3x106
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
8 16
Dam
age
accu
mu
lati
on
ind
ex
Stress range (ksi)
0.07
0.56
18
The two stress ranges considered in Figure 2-6 can be converted to a single,
equivalent stress range using the concept of an effective stress range. While summing the
damage for the two stress ranges discussed above is relatively simple, in actual bridges
the stress ranges have a wide variation such that combining the damage from cycles at
many stress ranges is not practical. An effective stress range relates the damage caused
by many variable-amplitude stress ranges to a constant-amplitude stress range. Because
the distribution of stress ranges is condensed into the effective stress range, it is a more
practical design solution.
If Equation 2-2 is combined with Equation 2-4, the damage from a spectrum of
stress ranges (left side) must equal the damage from a single, effective stress range (right
side) as seen in Equation 2-5. The effective stress range ( can then be determined by
Equation 2-6. As is discussed in Section 2.3.1, the effective stress range can be used
directly with the AASHTO S-N curves.
, Equation 2-5
,
/
,
/
Equation 2-6
where
effectivestressrange ksi
totalnumberofcyclesmeasured
ratioof to
For the example in Figure 2-6, the effective stress range is 13.2 ksi for the
600,000 stress cycles. Due to the cubing term on the stress range, the effective stress
range is weighted more towards the 16-ksi stress range. If a different distribution of
cycles were used to produce the same damage accumulation index, the effective stress
range will change for the given number of cycles. For example, as shown in Figure 2-7,
if there were only 100,000 cycles at 16 ksi and 1,900,000 cycles at 8 ksi, the damage
19
index (0.63) is the same as the previous example (Figure 2-6); however the effective
stress range is 8.84 ksi for 2,000,000 cycles. Therefore, the effective stress range varies
depending on the number of measured cycles, even if the damage level is the same.
The two examples outlined above produce the same amount of damage; however,
the effective stress ranges are quite different. As such, the effective stress range by itself
is not a reliable indication of damage; two effective stress ranges cannot be compared
with one another to determine the amount of damage. Rather, the effective stress range
must be considered along with the number of cycles to make the comparison.
Figure 2-7: Revised example of damage accumulation index.
According to Palmgren-Miner’s rule, failure is assumed to occur when 1.0.
Due to the scatter in measured fatigue data, failure can occur at a damage accumulation
index that varies considerably from 1.0. The value of the damage accumulation index
corresponding to failure would be 1.0 if the actual number of cycles to failure for each
stress range were known for the connection beforehand (Equation 2-4). Instead, the
number of cycles corresponding to failure used in these calculations is based on the
lower-bound and/or average values from tests of typical connection details within the
fatigue category. For a test with two different stress ranges, the damage can be reflected
by Equation 2-7 (Tanaka and Akita 1975).
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
3,500,000
4,000,000
4,500,000
8 16
Nu
mb
er
of
cycl
es
Stress range (ksi)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
8 16
Dam
age
accu
mu
lati
on
ind
ex
Stress range (ksi)
4.3x106
cycles at 8 ksi
0.54x106
cycles at 16 ksi
0.1x106
1.9x106
0.44
0.19
20
, ,
Equation 2-7
where
, averagenumberofcyclesuntilfailurefor ,
numberofcyclesmeasuredat ,
Equation 2-7 can be rewritten slightly as shown in Equation 2-8.
,
,
, ,
,
,∆
, , Equation 2-8
where
,
,
Palmgren‐Miner’scriticaldamageindex,ratioofnumberofcyclesuntilfailureofspecimentoaveragenumberofcyclesuntilfailurefor ,
The uncertainty from using Palmgren-Miner’s rule can be encapsulated in the
critical damage index ( ). If the mean S-N curves are used to determine , , the value
of is expected to have a mean of 1.0. After reviewing data from over a thousand tests
of welded connections, a lognormal distribution with a median of 1.0 and coefficient of
variation of 65% was recommended for the critical damage index (ASCE 1982).
However, some of the data included in the review were from small-scale tests, which had
higher variability than full-scale tests. For fatigue in offshore structures, the critical
damage index is modeled by a lognormal distribution with a median of 1.0 and
coefficient of variation of 30%, which is based on a survey of fatigue tests of structural
sections (Wirsching 1984). For most tests of offshore fatigue details, the median is 1.0
and the coefficient of variation ranges between 30-60% (Wirsching and Chen 1988).
2.2.1.2 Cycle-Counting Methods
To use Palmgren-Miner’s rule, the stress spectrum must be determined. If field
measurements are utilized, a cycle-counting method is needed to transform the stress
21
history into a histogram of stress amplitudes. Four types of counting methods for fatigue
analysis are included in ASTM E1049 (2011): (1) level-crossing counting, (2) peak
counting, (3) simple-range counting, and (4) rainflow counting or related methods. The
various methods will produce different stress spectrums for a given stress history; thus,
each method should be validated with Palmgren-Miner’s rule and fatigue data (S-N
curves).
A simplified rainflow method is considered to be well-suited for fatigue analyses
(Dowling 1972), and differs slightly from the rainflow method in ASTM E1049 (2011).
In the ASTM rainflow method, half cycles are counted; whereas, the simplified rainflow
method resolves everything in a stress history into full cycles. The method is outlined by
Downing and Socie (1982), and the analysis can be performed in real time or offline. If
the simplified rainflow method is used to determine amplitudes of cycles and the cycles
are transformed into an effective stress range using Palmgren-Miner’s rule, the results
have matched historic S-N curves (Swenson and Frank 1984).
The simplified rainflow method can be applied to a strain/stress history, such as
the one in Figure 2-8. The first step of the method is to identify all of the relative peaks
and valleys (the black dots). Next, the history has to be re-ordered to find the maximum
stress (point A). All data points that occur prior to the maximum are moved to the end of
the stress history.
The peaks and valleys shown in Figure 2-8 were re-ordered as shown in Figure
2-9(a). A full cycle is counted when a hysteresis has been formed by three data points.
For example, consider the three data points A, B, and C in Figure 2-9(a). The stress
difference between data points A and B is designated as stress range “Y” while the stress
difference between data points B and C is designated as stress range “X.” If Y is equal to
or less than X, the range is counted and the data points associated with the counted range
(Y) are discarded. If Y is greater than X, the next data point is considered (i.e., data point
D). New data points are considered until all ranges have been counted (i.e. there are no
longer three data points).
22
Figure 2-8: Stress history for example of simplified rainflow counting.
Figure 2-9: Step-by-step procedure for simplified rainflow counting.
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
0 1 2 3 4
Stre
ss (
ksi)
Time (sec)
0.75
0.5
0.25
0.5
11
A
B
C
DF
G
H
I
JE
Summary:(1) 0.25-ksi cycle(2) 0.5-ksi cycles(1) 0.75-ksi cycle(1) 11-ksi cycle
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Stre
ss (
ksi)
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Stre
ss (
ksi)
A
B
C
D
E
F
G
H
I
J
A
Count CD = 0.5 Count EF = 0.5
Count GH = 0.75
Count IJ = 0.25
Count AB = 11
(a) (b)
(c) (d)
A
B
G
H
I
J
AA
B
I
J
A
A
B
E
F
G
H
I
J
A
23
The rules for the method can be followed in Figure 2-9. In part (a), the first time
that Y is less than X is when data points C-E are considered; Y (CD) would be counted
with a stress range of 0.5 ksi and those points discarded. The process continues in part
(b) when Y (EF) is less than X (FG) for another stress range of 0.5 ksi. Cycle GH is the
next stress range counted (part (c)) at 0.75 ksi. Finally, in part (d), cycles IJ (0.25 ksi)
and AB (11 ksi) are counted and all cycles have been counted for the example stress
history.
2.2.2 Linear-Elastic Fracture Mechanics
Fatigue resistance can also be characterized by fracture-mechanics principles.
Fracture-mechanics methods are not used as often as a stress-range model (S-N curves);
however, fracture mechanics provides more insight into the fatigue problem because the
propagation of cracks until fracture can be calculated. In the stress-based approach, only
the number of cycles until failure is determined. In a fracture-mechanics approach, the
initial flaw size and fracture toughness are included in the analysis of fatigue resistance.
The linear-elastic fracture mechanics (LEFM) method is the most common approach to
engineering problems. This approach assumes small displacements, that the material is
isotropic and linear elastic, and that a small plastic zone exists at the crack tip. The
method is mainly an analytical approach; however, some empirical tests are needed to
determine correction factors.
The most common fracture-mechanics model relates the fracture toughness of a
material to the applied stress ( ) and crack length ( ). The stress intensity factor ( ) is
used as a measure of the fracture toughness of the material. The value varies with
temperature and is difficult to determine experimentally. However, the Charpy V-notch
test can be used to indicate fracture toughness. It only provides a qualitative approach
because stress and crack length values are not assessed directly (Fisher, Kulak, and Smith
1998). The stress intensity factor can be calculated using Equation 2-9.
√ Equation 2-9
24
where
stressintensityfactor ksi√in.
correctionfactorfornon‐uniformlocalstressfields
correctionfactorforplateandcrackgeometry
appliedstress ksi
cracklength in.
The correction factors in Equation 2-9 can be determined empirically or through
numerical methods. The results are summarized in fracture mechanics handbooks, such
as Barsom and Rolfe (1999).
Equation 2-9 can be modified for specimens subjected to fatigue loading, as seen
in Equation 2-10.
∆ ∆ √ Equation 2-10
where
∆ stressintensityfactor range ksi√in.
∆ appliedstressrange ksi
Equation 2-10 can be related empirically to the crack-growth rate ( / ) which
is obtained from the slope of the curve of crack-growth measurements. If that is done,
the results can be plotted on a log-log plot (Figure 2-10). From the tests, there is a region
where cracks do not grow or the cracks grow at a very slow rate (Region I). The region
corresponds to the threshold of the stress intensity factor range ( ). In Region III,
crack growth accelerates very rapidly until fracture occurs. As such, the region intersects
with the fracture toughness of the material ( ). Region II corresponds to the propagation
of the crack. Paris, Gomez, and Anderson (1961) proposed an equation to relate the
crack-growth rate to the stress intensity factor range (Equation 2-11). The equation is
commonly referred to as Paris’ law.
25
Figure 2-10: Change in crack-growth rate, / , with range of stress intensity
factor.
∆ Equation 2-11
where
materialconstantfromregressionanalysisoftestdata
upperboundisapproximately3.6 10 √ .forbridgesteelsandweldedconnections Fisher,etal.1993
materialconstantfromregressionanalysisoftestdatatypically3.0forsteelstructures
If Equation 2-11 is rearranged and integrated, the number of cycles until failure
can be determined. If Equation 2-10 is also substituted, the result is Equation 2-12.
1 1
∆1 1
∆ √ Equation 2-12
where
numberofcyclestofailure
criticalcracklength in.
initialcracklength in.
log
da
/dN
log ΔK
log
da
/dN
log ΔKΔKth KI
Reg
ion
I
ΔKth KI
Idealized
Actual
Fracture toughness
Reg
ion
II
Reg
ion
III
(b)(a)
26
With Equation 2-12, the parameters affecting the number of cycles until failure
can be evaluated. The impact of changing the critical crack length, initial crack length,
stress range, and type of detail are shown in Figure 2-11. As shown in Figure 2-11(a),
increasing the critical crack length, or improving the fracture toughness of a material, has
a slight effect on fatigue strength. Doubling the critical crack length does not necessarily
double the number of cycles until failure because the crack-growth rate increases rapidly
as the material approaches its fracture toughness. A bigger benefit can be realized by
reducing the initial crack length, or reducing the initial flaw size of a material, as shown
in Figure 2-11(b). Such a small change in the initial flaw size produces a great increase
in number of cycles until failure. The large scatter in the fatigue data in the S-N curves
could partly be attributed to differences in the initial flaw size. Though tests performed at
the same time would be expected to have similar flaw sizes, differences in groups of tests
could be attributed to the initial flaw size. As expected, reducing the stress range will
increase the number of cycles until failure (Figure 2-11(c)). Finally, changing the fatigue
category would lead to different values for and in Equation 2-9, which could
improve the fatigue life (Figure 2-11(d)). Any or all of these methods can be used to
impact the fatigue life.
The LEFM approach is not often used due to the limitations of the method.
Mainly, controlling and/or determining the initial crack size is difficult, and as shown in
Figure 2-11(b), it has a significant impact on fatigue life. In addition, estimating the
stress intensity factor range for complex geometric shapes (finding and ) makes the
method impractical for design.
27
Figure 2-11: Change in number of cycles due to change in (a) critical crack length, (b)
initial crack length, (c) stress range, and (d) type of detail.
2.3 LITERATURE REVIEW
A literature review in three areas is presented below. The first area summarizes
the research on the response and behavior of steel structures subjected to variable-
amplitude loading (Section 2.3.1). The uncertainties that encompass monitoring bridges
in the field for fatigue evaluation are discussed in Section 2.3.2. Finally, the concepts of
structural reliability and the application of probabilistic approaches to estimating
remaining fatigue life are presented in Section 2.3.3.
Cra
ck le
ngt
h
Number of cycles
Cra
ck le
ngt
h
Number of cycles
Cra
ck le
ngt
h
Number of cycles
Cra
ck le
ngt
h
Number of cycles
ac,1ac,2 Change in critical
crack length
Change in number of cycles
Nf,2 Nf,1
ai
ac
Change in initial crack length
Change in number of cycles
Nf
ai,1ai,2
(b)(a)
(c)
ac
aiNf,1 Nf,2
Change in number of cycles
Higher stress range
Lower stress range
Nf,1Nf,2
ac
ai
Base detail category
Better detail category
Change in number of cycles
(d)
28
2.3.1 Development of Fatigue Resistance Under Variable-Amplitude Loading
Prior to 1970, the fatigue guidelines in the AASHTO LRFD Specifications were
based on small-scale specimens that were tested at constant-amplitude loading.
However, bridges are subjected to variable-amplitude cycles from vehicles of varying
weights crossing the bridge. Thus, in the 1970-1980s, National Cooperative Highway
Research Program (NCHRP) sponsored a series of projects to determine the fatigue
resistance of various details used in bridges under variable-amplitude loading. The
fatigue behavior under variable-amplitude loading is summarized below. Except where
noted, all fatigue tests were performed on beam elements with a constant-moment region.
2.3.1.1 NCHRP Project 12-12
NCHRP Project 12-12 investigated the effects of variable-amplitude loading on
fatigue life under simulated traffic loading (Schilling, et al. 1978). The researchers found
that the variable-amplitude data could be related to the constant-amplitude data using an
effective stress range concept. It was also observed that the primary variables affecting
variable-amplitude loading were the same as the variables affecting constant-amplitude
loading.
Previous variable-amplitude fatigue tests utilized block loading. The block
loading consisted of organizing the sequence of loads in a fixed sequence that was often
very different from random traffic. In this study, the specimens were loaded following a
Rayleigh probability-density function, which is defined by two variables: (1) modal stress
range ( ) and (2) dispersion ratio ( / ) (Figure 2-12).
The researchers reviewed 51 sets of data from 37,000 truck passages to determine
the impact of a truck passage. Most truck passages at bridges produce a major stress
cycle with smaller vibrational stresses, as shown in Figure 2-13. From the data analysis,
the researchers determined that the vibrational stresses were small enough to be
neglected. As such, only major cycles were applied to the test specimens.
29
Figure 2-12: Probability of occurrence
Figure 2-13: Example truck passage
Over 300 specimens were tested during the study to understand the fatigue
behavior under variable-amplitude loading for two types of steels, A514 and A36. To aid
in planning subsequent beam tests and to study secondary test parameters, 84 small-scale,
plate specimens with a simulated cover plate were tested first in an axial setup.
Fabricated beams were then tested, with and without welded cover plates. The flange
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 2 4 6 8 10
Pro
bab
ility
of
occ
urr
ence
Stress range (ksi)
= 0.25
= 0.5
= 1.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
0 1 2 3 4
Stre
ss (
ksi)
Time (sec)
Main stress cycle
Vibrational stress cycles
30
plates (6 ¾-in. width) were welded to web plates (25-in. deep) to fabricate the I-shaped
beams. A total of 156 beams with partial-length cover plates were tested to obtain the
lower bound of the fatigue strength under variable-amplitude loading (Figure 2-15).
Sixty-three beams without cover plates were tested to obtain the upper bound of the
fatigue strength.
Figure 2-14: Comparison of cover-plate beam results with AASHTO allowable stress
for Category E (from Schilling, et al. (1978))
To relate the variable-amplitude fatigue data to the constant-amplitude fatigue
data, an effective stress range concept was used (Equation 2-13).
,
/
Equation 2-13
where
proportionofcyclesatstressrange ,
fitofeffectivestressrangedata
31
Values of 2 (root mean square, RMS) and 3 (Palmgren-Miner’s rule)
were considered to fit the variable-amplitude data to the constant-amplitude data. If
3, the effective stress range from Equation 2-13 matches the stress range from
Palmgren-Miner’s rule because three corresponds to the assumed slope of the S-N curve.
The researchers found that the RMS fit the data slightly better than Palmgren-Miner’s
rule. However, Palmgren-Miner’s rule was recommended because it is more
conservative (maximum difference is approximately 11%) than RMS and the engineering
community have accepted the rule.
The primary variables affecting the fatigue resistance were ascertained from the
tests to be the stress range and type of detail. The conclusion matches the primary
variables discovered from historic, constant-amplitude fatigue tests (Schilling, et al.
1978). The minimum stress and type of steel had similar effects as reported in NCHRP
Report 12-7 (Fisher, et al. 1970; Fisher, et al. 1974) for constant-amplitude loadings and
were considered secondary parameters from that report.
By transforming the variable-amplitude data into an effective stress range, the
researchers found that the differences in the constant-amplitude and variable-amplitude
data were not statistically significant. As such, the fatigue strength of specimens subject
to variable-amplitude loading could be determined from the constant-amplitude tests.
2.3.1.2 NCHRP Project 12-15(4)
NCHRP Project 12-15(4) considered the effect on fatigue if the majority of the
cycles were less than the constant-amplitude fatigue limit (CAFL) (Fisher, Mertz, and
Zhong 1983). The results from the tests performed in this project established the
conditions under which cycles below the CAFL contribute to fatigue damage. The
frequency of occurrence above CAFL and the magnitude of the peak stress range were
both investigated.
Eight, full-size specimens (W18x50 rolled beams) were used to test gusset-plate
attachments and cover plates in fatigue. The beams were subjected to a variable-
32
amplitude stress range using a Rayleigh distribution, with stress widths ( , ,
,) of
3.75, 4.0, 4.75, and 4.75 (the stress width of NCHRP Project 12-12 (Schilling, et al.
1978) was approximately 4.67). The stress range of the variable-amplitude input
exceeded the CAFL from 0.0 to 10.16 percent of the time. As seen in Figure 2-15, three
cases were considered: (1) the effective stress range ( ) and the maximum stress range
( , ) were above CAFL, (2) the maximum stress range was above CAFL and the
effective stress range was below CAFL, and (3) the maximum stress range and effective
stress range were below CAFL.
Figure 2-15: The variable-amplitude loading cases considered in NCHRP Project 12-
15(4) (Fisher, Mertz, and Zhong 1983).
The tests revealed that as long as a portion of the applied stress was above the
CAFL (Cases 1 and 2), fatigue cracking occurred at both the web attachments and the
cover plates. If none of the cycles were above the CAFL (Case 3), fatigue cracking did
not occur. It was found that the fatigue life could be calculated using a straight-line
extension of the S-N curve for the detail, which suggests that all of the stress cycles
contribute to the fatigue damage. The scatter in the test data from using the straight-line
extension is similar to the scatter from constant-amplitude tests.
Stre
ss r
ange
(lo
g)
Number of cycles (log)
Extension of S-N curve
Constant-amplitude fatigue limit (CAFL)
Constant-amplitude S-N curve
Case 2
Case 3
Case 1
33
2.3.1.3 Swenson & Frank
A study at the University of Texas at Austin examined the influence of small
stress cycles upon the fatigue life of welded components (Swenson and Frank 1984).
After reviewing truck histories, the researchers determined that the minor cycles may
increase the fatigue damage by as much as 15% in medium-span girder bridges. The
truck histories were also used to create a variable-amplitude loading scheme for
laboratory tests, which found that all cycles in a stress spectrum are damaging. The
specimens were two plates welded together to form a tee section. The flange of the tee
section was bolted to a reaction wall and load was applied at the tip of the web (cantilever
loading).
The researchers considered four different methods for counting varying-amplitude
cycles: peak counting, range counting, simplified rainflow counting, and reservoir
counting. Simplified rainflow counting and reservoir counting produced the same stress
spectrum and were observed to be the most appropriate methods for counting cycles.
When the simplified rainflow algorithm was used to count varying-amplitude cycles and
Palmgren-Miner’s rule was applied to determine an effective stress range, the results of
the fatigue tests matched historic constant-amplitude fatigue tests.
2.3.1.4 ASCE Committee on Fatigue and Fracture Reliabilty
The American Society of Civil Engineering (ASCE) Committee on Fatigue and
Fracture Reliability published a state-of-the-art review of fatigue and fracture reliability
in 1982. One of the articles focused on variable-amplitude loading for fatigue specimens.
To relate variable-amplitude loading to constant-amplitude tests, the cycles must be
correctly identified by a cycle-counting method and then evaluated using a cumulative
damage theory.
As discussed in Section 2.2.1.1, the most commonly-used cumulative damage
theory is Palmgren-Miner’s rule. The damage theories were most useful before the
development of fracture-mechanic models provided better mathematical models of crack-
growth data; however, the theories remain useful today due to their simplicity.
34
Palmgren-Miner’s rule is a linear damage hypothesis that is simple in application: the
rule does not agree with all experimental data. For instance, Palmgren-Miner’s rule is not
very accurate if the load sequence or mean stress are important variables in the fatigue
behavior. Load sequencing and mean stress are typically not important in situations
where there is significant inelastic behavior at all strain levels. As mentioned before,
welding creates significant residual stress, which causes inelastic behavior at low stress
levels.
The variability in using Palmgren-Miner’s rule was evaluated by reviewing data
from eleven researchers and over a thousand laboratory tests. Based on that data, a
lognormal distribution with a median of 1.0 and coefficient of variation of 65% was
suggested to characterize the variability in using the rule. However, some of the data
included in the review were from small-scale tests, which had higher variability than full-
scale tests. Despite the limitations of Palmgren-Miner’s rule, it continues to be used
because of its simplicity and because other damage theories, whether linear or nonlinear,
have not proven to provide consistently better results.
The committee recommended use of the simplified rainflow method because it
can lead to a better prediction of fatigue life (Dowling 1972). The simplified rainflow
method is considered superior to other methods because it identifies stress ranges
associated with closed-loop hysteresis, which is compatible with constant-amplitude
fatigue data.
2.3.1.5 NCHRP Project 12-15(5) – Fatigue Database
NCHRP projects from 1966 to 1972 formed the basis of the 1974 AASHTO
fatigue specification (Keating and Fisher 1986). Over 800 constant-amplitude, full-sized
fatigue tests were used to define the S-N curves. Due to the large number of tests
performed since then, part of the objective of NCHRP Project 12-15(5) (Keating and
Fisher 1986) was to review and expand the current fatigue database.
Past NCHRP research had shown that the fatigue data (stress range versus number
of cycles) can be represented on a log-log graph. The fatigue data can be assumed to
35
follow the form of Equation 2-14. The results of a regression analysis performed on the
1974 fatigue data are summarized in Table 2-2 (each fatigue category had a different
slope ( )).
log log log Equation 2-14
where
log thelog ‐axisinterceptofthelogS‐Ncurve
slopeoftheS‐Ncurve
Table 2-2: Regression analysis coefficients for AASHTO curves in 1974 (Keating and
Fisher 1986).
Category Slope,
Intercept
(mean), Standard
deviation, Intercept (lower)
A 3.178 11.121 0.221 10.688 B 3.372 10.870 0.147 10.582 C 3.25 10.038 0.063 9.915 D 3.071 9.664 0.108 9.453 E 3.095 9.292 0.101 9.094 E′ 3.00 -- -- 8.61
Data from several sources including the United States, Japan, Germany, Canada,
and The Office of Research and Experiments of the International Institute of Technology
(ORE), were evaluated and added to the fatigue database, expanding the database to
nearly 2,000 tests. Based on the updated data, the slope ( ) of all of the fatigue
categories became -3.0 and another category was added to the list of S-N curves. The
authors determined that the distribution of initial flaws, residual stress fields, and
specimen size played a major role on which data sets were grouped together. The initial
size of the flaw was of particular importance, which matches the results of the fracture-
mechanics approach to fatigue evaluation (Figure 2-11).
The researchers proposed a design equation that is used in the modern AASHTO
fatigue code (Equation 2-2). The proposed values of the lower-bound fatigue constant
36
( ) for each fatigue category are listed in Table 2-3 and are approximately the same
values used in the current AASHTO LRFD Specifications (2010) (Table 2-1).
Table 2-3: Proposed lower-bound fatigue constant (Keating and Fisher 1986).
Category Fatigue constant, (ksi3) A 2.500×1010
B 1.191×1010 B′ 6.109×109 C 4.446×109 D 2.183×109 E 1.072×109 E′ 3.908×108
2.3.1.6 NCHRP Project 12-28(03)
The objective of NCHRP Project 12-28(03) was to develop more accurate
procedures for evaluating existing bridges and designing new bridges for fatigue (Moses,
Schilling, and Raju 1987). The method for estimating the remaining fatigue life was
developed considering realistic bridge conditions and a probabilistic model for analysis.
The method developed by the researchers was published by AASHTO as the Guide
Specifications for Fatigue Evaluation of Existing Steel Bridges (1990).
The procedure provided engineers with realistic values to model fatigue
conditions and estimate the remaining fatigue life for bridges. The authors evaluated
bridge data and past research to recommend values for the weight and axle spacing of the
fatigue truck (Figure 2-16), presence of multiple trucks on a bridge ( , dynamic impact
( , moment range ( , lateral distribution ( , member section ( , reliability
factors, cycles per truck passage ( ), and lifetime average truck volume. The fatigue
truck is based on weigh-in-motion data from 30 sites nationwide and 27,000 observed
trucks. The fatigue truck is expected to produce a similar amount of fatigue damage on
average as the typical assortment of trucks at a site. If the site-specific distribution of
weights of trucks is known, the weight of the fatigue truck ( ) can be modified using
Equation 2-15. Otherwise, a gross weight of 54 kips can be assumed. Based on the
37
weigh-in-motion data, the maximum weight of a single truck is expected to be twice as
much as the fatigue truck. Though, that truck would rarely cross a given bridge.
Figure 2-16: Dimensions and distribution of weight of fatigue truck.
/
Equation 2-15
where
weightofthefatiguetruck kips
frequencyofgrossweightswithininterval
grossweightoftruckswithininterval kips
The design stress range for fatigue can be calculated using Equation 2-16. The
values recommended by the authors for , , , and are not as conservative as
might be used to check the flexural capacity because fatigue is an averaging process; one
vehicle will not cause fracture, but thousands to millions will. The authors also suggest
how to modify the section modulus ( ) to account for unintended composite action.
Equation 2-16
where
momentrangecausedbypassageofthefatiguetruck
sectionmodulus
lateralgirderdistribution
impactfactor
30′-0″ 6′-0″14′-0″
6 kips 24 kips 24 kips W = 54 kips
38
multiplepresencefactor
cyclespertruckpassage
A fatigue reliability model was created to provide a consistent safety margin
(Equation 2-18). If 0, failure is assumed to occur.
Equation 2-17
where
ageofbridgeatwhichfailureoccurs randomvariable
ageofbridge deterministic
The age of the bridge at which failure occurs ( ) depends on a number of
variables. The uncertainty of each variable can be modeled and normalized to determine
the reliability factor ( ) of the suggested equation for the remaining fatigue life. A
lognormal distribution was used for each random variable in Equation 2-18.
P Z 0 P365
0 Equation 2-18
where
numberofstresscycles deterministic
ageofbridge deterministic
reliabilityfactor tobedetermined
Palmgren‐Miner’srule 1.0,COV 15%
normalizedmomentrange 0.97,COV 3%
normalizedgrossweightoftheequivalentfatiguetruck1.0,COV 10%
multiplepresenceoftrucksonthebridge 1.03,COV 0.6%
impactfactor 1.0,COV 11%
girderlateraldistributionfactor 1.0,COV 13%
normalizedeffectivesectionmodulus 1.0,COV 10%
equivalentnumberofstresscyclesduetosingletruckpassage1.0,COV 5%
39
lifetimeaveragedailytrucktraffic 2500,COV 10%
normalizedfatigue‐strengthcurve 1.297,COV 15.3%
The means and coefficients of variation (COV) are presented for each variable.
The uncertainty in all of the variables, except for Palmgren-Miner’s rule ( , was
determined from field data. For Palmgren-Miner’s rule ( , the COV value of 15% was
estimated because there was insufficient data on the validity of Palmgren-Miner’s rule for
welded steel bridges from typical highway loadings. However, work by Fisher, Mertz,
and Zhong (1983) does validate the assumed COV. One of the most important benefits
of a reliability analysis is that it shows the interrelationship of the various conservative
assumptions that are made at each step in the design or evaluation procedure.
The reliability factors ( ) were determined from the reliability model to be 1.35
and 1.75 for redundant and non-redundant members, respectively. Using these reliability
factors and Equation 2-19, the remaining fatigue life can be estimated with a more
consistent level of reliability for the variety of bridges types. Either the safe life (2.3%
and 0.1% probability of failure for redundant and non-redundant members, respectively)
or mean life (50% probability of failure) can be calculated, depending on the parameters
used.
10
Equation 2-19
where
remainingfatiguelifeinyears
detailconstant ksi3
estimatedlifetimeaveragedailytruckvolume
stresscyclespertruckpassage
stressrange ksi
combinedreliabilityfactor
base reliability factor (same as )
40
correction if stresses are measured (0.85)
correction if site-specific truck traffic weighs less than standard fatigue truck
correction for a more accurate calculation of the lateral distribution factor
presentageofbridgeinyears
1.0forcalculatingsafelifeand2.0forcalculatingmeanlife
Due to the scatter of the fatigue data, there can be a large difference between the
calculated mean and safe lives. In an example in the report, the calculated mean life was
138 years and safe life was 14 years. A non-redundant detail was considered in the
example, which is partly why there was such a large difference between the mean (50%
probability of failure) and safe life (0.1% probability of failure). Another potential
source of error was the use of design values for the gross weight of the fatigue truck,
average daily truck traffic (ADTT), and impact factor. The uncertainty from those
variables can be minimized if strain gages were used to determine the distribution of
fatigue cycles.
2.3.1.7 NCHRP Project 12-15(5) – Variable-Amplitude Fatigue Tests
Additional tests were conducted in NCHRP Project 12-15(5) (Fisher, et al. 1993)
to supplement and verify the preliminary observations of NCHRP Project 12-15(4)
(Fisher, Mertz, and Zhong 1983). The fatigue behavior of specimens in which the
majority of the stress ranges are below the CAFL was tested.
To accomplish the objective, eight full-size welded girders (34-in. deep web plate
with 12-in. wide flange plates) with Category Eʹ web attachments and cover-plated
flanges as well as Category C transverse stiffeners and diaphragm connection plates were
subjected to long-life, variable-amplitude loading. It was found that fatigue cracking
developed at the Category Eʹ details when the CAFL was exceeded by more than 0.05%
of the stress cycles. For the Category C details, fatigue cracks did not develop unless the
peak stress range exceeded the CAFL by more than 33%. The results supported the
41
extension of the exponential S-N relationship even when the effective stress range is
below the constant-amplitude fatigue limit.
A fracture-mechanics model was developed to analyze how smaller stress cycles
contributed to fatigue damage. The analysis indicated that the contribution is dependent
on the crack-growth threshold. If the lower stress cycles are assumed to contribute to the
damage, the estimate of fatigue life is typically conservative.
For the cover-plate details (Category Eʹ) that were tested, it was recommended
that all cycles above 50% of the CAFL should be considered to cause fatigue damage.
2.3.1.8 Manual for Evaluation in AASHTO (2011)
In 2011, AASHTO released the second edition of the Manual for Bridge
Evaluation (AASHTO 2011) for evaluating fatigue in steel bridges. The method is
similar to the procedure described in NCHRP Project 12-28(03) (Moses, Schilling, and
Raju 1987); however, the updated method utilizes new formulas and values for several
parameters from the AASHTO LRFD Specifications (2010). The updated values more
accurately reflect bridge conditions. In addition, the evaluation manual utilizes updated
values for the resistance and reliability/load factors. Equation 2-20 can be used to
estimate the fatigue life. It is similar to Equation 2-19, with some of the parameters
redefined.
365
Equation 2-20
where
fatiguelifeinyears
resistancefactor
detailconstant ksi3
stresscyclespertruckpassage
averagenumberoftrucksperdayinasinglelaneaveragedovertheentirefatiguelife
stressrange ksi
42
partialloadfactor
The guide previously used (resistance parameter) and (reliability factor) in
Equation 2-19, whereas the evaluation manual used (resistance factor) and (partial
load factor) in Equation 2-20.
According to Chotickai and Bowman (2006), the updated method utilizes lower
load factors and higher resistance factors, which produces a higher probability of failure
for a calculated fatigue life (i.e. longer calculated fatigue life). The evaluation manual
does not distinguish between redundant and non-redundant details and provides three
levels of safety for the fatigue life: minimum life (design fatigue life, two standard
deviations away from the mean fatigue resistance), evaluation life (one standard deviation
away from the mean fatigue resistance), and mean life. The values for the resistance
factor ( ) and partial load factor ( ) vary with detail and desired level of safety.
2.3.1.9 Regression Values for Current AASHTO S-N Curves
The regression values for the current AASHTO S-N curves were determined by
Keating and Fisher (1986), yet the values were not published in that NCHRP report.
Instead, the values were published in a report that is no longer available (Keating, Halley
and Fisher 1986), and were later presented in Moses, Schilling, and Raju (1987). The
regression values are in terms of the mean ( ) and design ( ) stress ranges at two
million cycles for a slope ( ) of -3.0 (Table 2-4).
By rearranging Equation 2-14, the intercept of the regression equation can be
determined (Equation 2-21). By substituting the mean stress range, the mean intercept
for the fatigue constant can be calculated (Equation 2-22).
log log 2 10 3 log Equation 2-21
log log 2 10 3 log Equation 2-22
where
meanvalueofthefatigueconstant intheAASHTOLRFDSpecifications
43
designstressrangeat2x106 cyclesforeachfatiguecategory
meanstressrangeat2x106 cyclesforeachfatiguecategory
Table 2-4: Mean and design stress ranges and coefficient of variation (COV) of fatigue
data.
Category Fatigue
constant, (ksi3)
at 2x106 cycles (ksi)
at 2x106 cycles (ksi)
COV
A 250×108 33.0 23.2 21.7% B 120×108 22.8 18.1 14.1% B′ 61×108 18.0 14.5 13.2% C 44×108 16.7 13.0 15.3% C′ 44×108 16.7 13.0 15.3% D 22×108 13.0 10.3 14.2% E 11×108 9.5 8.1 9.7% E′ 3.9×108 7.2 5.8 13.2%
The standard deviation for each fatigue category can be calculated from the
coefficient of variation ( ) (Equation 2-23).
ln 1 Equation 2-23
where
standarddeviationoflog
coefficientofvariationforeachfatiguecategoryat2x106cycles
Using the regression values from Table 2-4, the regression values for log can
be calculated and are reported in Table 2-5.
44
Table 2-5: Derived values from regression model.
Category Mean of fatigue
constant, (ksi3)
Intercept (mean),
Intercept (lower),
Standard deviation of
, A 719×108 10.86 10.40 0.21 B 237×108 10.37 10.08 0.14 B′ 117×108 10.07 9.79 0.13 C 93×108 9.97 9.64 0.15 C′ 93×108 9.97 9.64 0.15 D 44×108 9.64 9.33 0.14 E 17×108 9.23 9.03 0.10 E′ 7.5×108 8.87 8.59 0.13
2.3.1.10 Summary
The following conclusions can be inferred from the literature on the development
of fatigue resistance under variable-amplitude loading.
Variable-amplitude loading tests can be related to constant-amplitude data
through the concept of an effective stress range (Schilling, et al. 1978).
Based on updated data, the slope ( ) of all of the fatigue categories was
-3.0 (Keating and Fisher 1986).
Palmgren-Miner’s rule is most often used to determine the effective stress
range due to its simplicity and has been verified by variable-amplitude
tests (ASCE 1982; Swenson and Frank 1984).
If the effective stress range is less than the constant-amplitude fatigue
limit (CAFL), fatigue failure can occur if the maximum stress range is
greater than the CAFL. However, if the maximum stress range is less than
the CAFL, fatigue failure will not occur. A straight-line extension of the
S-N curves can be used for effective stress ranges less than the CAFL
(Fisher, Mertz, and Zhong 1983; Fisher, et al. 1993).
The minor stress cycles due to the vibration of the bridge from the passage
of a truck can cause fatigue damage. As such, the minor stress cycles
45
should be counted and considered in a fatigue evaluation (Swenson and
Frank 1984).
Because the rainflow algorithm identifies stress cycles from closed-loop
hysteresis, it is the best method for determining the amplitude of cycles in
a varying-amplitude stress history (ASCE 1982; Dowling 1972). When
the rainflow algorithm is used to determine amplitudes and Palmgren-
Miner’s rule is used to calculate an effective stress range, there is good
agreement with historic constant-amplitude fatigue tests (Swenson and
Frank 1984).
There is uncertainty in using Palmgren-Miner’s rule, which must be
considered when evaluating the fatigue life of a bridge or designing details
for fatigue (ASCE 1982).
There is still uncertainty on which cycles should be considered when
evaluating fatigue damage. Some state that all cycles are damaging
(Swenson and Frank 1984), whereas others recommend that stress ranges
that are less than 50% of the CAFL can be neglected (Fisher, et al. 1993).
2.3.2 Field Monitoring of Bridges for Fatigue Evaluation
This section contains some of the approaches used by engineers and researchers to
calculate the fatigue damage when field monitoring a bridge. Some of the approaches are
not appropriate, as they ignored fatigue cycles or produced unconservative results.
Recommended techniques for a fatigue analysis of a bridge with field-monitored data are
presented in Chapter 3.
The fatigue life of a bridge may be evaluated through structural analysis and the
application of equations in the AASHTO Manual for Bridge Evaluation (2011). The
process requires an engineer to make a number of assumptions to calculate the effective
stress range. Namely, uncertainty can arise from choosing a representative truck that
causes an equivalent amount of damage as the vehicles that cross the bridge, the number
of truck passages, how load is distributed between longitudinal girders, if composite
46
action exists, and the increase in stresses due to dynamic effects. Because the geometry
and traffic patterns at each bridge are different, the uncertainty from these assumptions
may be reduced by field monitoring to determine the actual distribution of stress ranges.
Field studies are often performed on bridges that already have problems
(determine why cracks have formed) or to justify continued use of a bridge that is past its
design life. The data are useful to bridge owners as they make plans for replacement,
retrofit, or continued use of the bridge inventory, as well as bridge inspectors to help
focus efforts on critical members. In Zhou (2006), the Cleveland Central Viaduct was
monitored after 40 years of service and determined to have an infinite fatigue life due to
the low-measured stress ranges. It has often been found that the stress ranges measured
in the field are lower than the stress ranges calculated from conservative design values
(Zhou 2006).
Despite the advantages of directly measuring the spectrum of stress ranges, there
is still some uncertainty when monitoring bridges. The uncertainty can be classified into
three areas: (1) location of sensor, (2) representation (quality of data) of traffic, and (3)
interpretation of data. Some engineering judgment is required to determine the optimum
locations for the gages. If a gage is installed at a location with low stress ranges, the
bridge could be incorrectly evaluated. Zhou (2006) recommended that bridges be
analyzed before monitoring to determine the locations that are subjected to the highest
levels of stress. Inspection reports can also be used to identify locations where cracks
have been observed.
The data obtained from field monitoring is useful only if the collected spectrum of
stress ranges is representative of actual traffic. If only a few days of data are captured,
the amount of data could be overestimated or underestimated, especially if the data
includes a holiday (Fasl, et al. 2012c). Because damage will vary with the day of the
week (more damage during the weekdays as compared to the weekend), it is necessary to
capture at least a week’s worth of data to determine the average damage per day. There
can also be fluctuations in damage on a weekly basis; as such, two to four weeks are
often needed to make long-term damage projections (Connor and Fisher 2006).
47
Obtaining data beyond a month will help define the maximum stress range; however,
those large stress ranges occur with relatively low frequency and do not significantly
influence projections of damage (Connor and Fisher 2006).
The other main cause of uncertainty deals with interpretation of the data, mainly
which cycles should be considered in the fatigue analysis. A concern of any field
instrumentation involving sensors is error in the measurements due to electromechanical
noise and/or analog resolution of the data acquisition system. The electromechanical
noise cycles are fictitious stress fluctuations that do not induce fatigue damage, and
should be truncated from a fatigue analysis. Most data acquisition systems can limit
electromechanical noise to less than 10 microstrain (Connor and Fisher 2006), with many
systems limiting noise between 2-5 microstrain.
In addition to ignoring cycles from electrical noise, many researchers truncate
data at small stress ranges because those cycles are not expected to cause much damage.
Truncating cycles that are less than 25-50% of the CAFL for each fatigue category are
typically recommended (Connor and Fisher 2006; Zhou 2006). Where ignoring electrical
noise is required (not real damage), care must be taken to ensure that damaging cycles are
not truncated, even if they correspond to small stress ranges.
Part of the reason engineers truncate lower stress ranges is because the calculation
of the effective stress range (Equation 2-6) is very sensitive to the number of fatigue
cycles in each bin (Section 2.2.1.1). In most monitoring programs, a majority of the
cycles are in the first few bins due to electromechanical noise and actual cycles from light
vehicles (cars and two-axle trucks). As a result, the calculation of the effective stress
range produces a very small value if all the data were considered.
Truncating the lower cycles produced a more realistic value of the effective stress
range. A more realistic value was important when performing a fatigue evaluation using
the AASHTO Guide for Fatigue Evaluation (1990) because whether a detail had infinite
life was not dependent on the maximum-measured stress range. Instead, it was based on
whether the calculated or measured effective stress range was less than a limiting stress
48
range; a bridge connection had infinite life if Equation 2-24 was true. The combined
reliability factor ( ) was discussed in Section 2.3.1.6.
Equation 2-24
where
limitingstressrange ksi – seeTable 2-6
Table 2-6: Limiting stress range.
Category (ksi) A 8.8 B 5.9 B′ 4.4 C 3.7 C′ 4.4 D 2.6 E 1.6 E′ 0.9
Without truncating the lower stress ranges, Equation 2-24 might be satisfied for
many bridge details that actually had a finite fatigue life. In the most recent AASHTO
Manual for Bridge Evaluation (2011), the infinite fatigue life is based on the maximum-
measured stress range, rather than Equation 2-24. Using the maximum-measured stress
range as the metric for infinite fatigue life is consistent with the long-life fatigue tests
performed by Fisher, et al. (1983).
When truncating cycles, Connor and Fisher (2006) suggested that vehicles less
than 20 kips could be neglected from a fatigue analysis because those vehicles are not
included in the load spectrum used for the AASHTO LRFD Specifications (2010). This
implies that vehicles less than 20 kips are assumed to cause only a small amount of
damage. At the Patroon Island Bridge in New York, a 20-kip vehicle was used to
determine the level of truncation for each gage, ranging from 5-33 microstrain (Alampalli
and Lund 2006). At the I-95 Bridge over James River in Richmond, Virginia, a slightly
different method was employed. There, the authors truncated lower stress ranges until
the number of cycles in the stress-range histogram matched the known ADTT at the
49
bridge (each truck caused only one stress cycle) (Zhou 2006). The effective stress range
was then calculated based on the truncated histogram.
The practice of truncating load-induced stress ranges, even though they are small,
conflicts with the conclusion that all cycles can contribute to the accumulation of fatigue
damage (Swenson and Frank 1984). In most bridges, truncating lower stress ranges may
not change the calculated amount of damage because the number of cycles is also
reduced. However, in some bridges, truncating lower stress ranges can reduce the
calculated amount of fatigue damage, which will produce unconservative estimates of the
remaining fatigue life.
The last complication of data interpretation is due to the calculation of a negative
remaining fatigue life. Leander, Anderson, and Karoumi (2010) evaluated the fatigue life
of a railway bridge in Stockholm based on the Swedish Regulations for Steel Structures,
and estimated that the design fatigue life had been exceeded. When the authors evaluated
the bridge, they found that the approximately 40% of the design fatigue life was
consumed in a year (failure expected in just over two years). However, the bridge has
been in service for over 50 years with no apparent damage. The authors attributed the
negative remaining fatigue life to conservative safety margins in the code, errors due to
Palmgren-Miner’s rule, and improper classification of the detail (by changing the fatigue
constant for the detail by 100%, the damage reduced from 0.4 to 0.018 each year). This
example underlines the importance of understanding the procedure for how the fatigue
life is calculated (probability of failure for method) and dealing with uncertainty properly
(from Palmgren-Miner’s rule and other sources). To express the uncertainty in fatigue
lives, the AASHTO Manual for Bridge Evaluation (2011) provides load and resistance
factors for three levels of safety: minimum (two standard deviations from the mean),
evaluation (one standard deviation from the mean), and mean life.
2.3.2.1 Summary
The following points can be concluded from the literature on monitoring bridges
for fatigue evaluation.
50
Capturing data in the field is beneficial to bridge owners, helping them
make decisions in terms of replacement, retrofit, or continued use of the
bridge inventory. Often the stress ranges measured in the field are much
less than those predicted from code equations because the equations
cannot accurately reflect the unique geometry and traffic conditions of
each bridge (Zhou 2006).
Two to four weeks of data are needed to make long-term damage
projections (Connor and Fisher 2006).
There is a lot of confusion on how to interpret data obtained in the field.
Electromechanical noise from data acquisition systems should be ignored
from the analysis because they do not represent actual fatigue cycles.
Many engineers truncate the lower stress cycles because they can skew
(lower) the calculation of the effective stress range. As much as 25-50% of
the CAFL has been recommended (Connor and Fisher 2006). However,
by truncating the lower stress ranges, some actual fatigue damage may be
neglected from the analysis (Swenson and Frank 1984).
The procedure for calculating the remaining fatigue life should be
understood or the results from the analysis could be misinterpreted,
especially if the remaining fatigue life is negative (Leander, Andersson,
and Karoumi 2010).
2.3.3 Structural Reliability Fatigue Analysis
If the strength and loading are known exactly, establishing fatigue failure for
critical bridge connections would be very simple. However, due to uncertainty in
material strength, section dimensions, loading, and many other sources, estimating failure
is uncertain. Structural reliability is a convenient method to characterize the probability
of failure considering the uncertainty from various sources. For specimens designed for
flexure and shear, modern codes use structural reliability to determine the appropriate
safety factors on the load and resistance to provide a consistent level of reliability. The
51
reliability can be defined as the probability that failure will not occur. Reliability
concepts can be used to develop a probabilistic approach to fatigue analysis. By using
such an approach, the uncertainty inherent in fatigue is treated in a rational manner
(Chung 2004).
2.3.3.1 General Reliability Concepts
If a specimen is assumed to have a resistance ( ) and loading ( ), each with a
given mean ( and ) and uncertainty ( and ), failure is a possibility at the overlap
between the curves (Figure 2-17(a)). In this situation, failure occurs when the loading is
greater than the resistance. A possible opportunity for failure is when the resistance is
below its mean and the loading is above its mean. The probability of failure can be
formalized by defining a limit state function ( ). If the limit state is defined in
Equation 2-25, failure will occur when 0.
0 Equation 2-25
where
uncertaintymodelforthelimitstatefunctionof and withamean andstandarddeviation
uncertaintymodelfortheresistance withamean andstandarddeviation
uncertaintymodelfortheload withamean andstandarddeviation
If and are assumed to be independent, normally‐distributed random
variables,themean anduncertainty canbedeterminedforthelimitstate
function( ) using Equation 2-26 and Equation 2-27.
Equation 2-26
Equation 2-27
52
The limit state function will also be normally distributed and the probability that
failure occurs can be determined from Equation 2-28. The probability of failure can be
shown graphically in Figure 2-17(b).
0 Φ0
Φ Equation 2-28
where
Φ cumulativedistributionfunctionofastandardnormalrandomvariable
Figure 2-17: Probability density curves for (a) and and (b) .
A reliability index ( ) is often used and is related to the probability of failure ( )
using Equation 2-29 and Equation 2-30.
Φ 1 Φ Equation 2-29
Φ Equation 2-30
The reliability index and the probability of failure are inversely related to each
other; as the probability of failure decreases, the reliability index increases. A set of
Pro
bab
ility
den
sity
fu
nct
ion
Region where failure could occur
,,
(a) (b)
Pro
bab
ility
den
sity
fu
nct
ion
53
and values are summarized in Table 2-7. Typical values for probabilities of failure in
civil engineering applications range between 10-3 to 10-5, or a reliability index between
3.09 to 4.26.
Table 2-7: Equivalent probabilities of failure and reliability indexes.
Probability of failure, Reliability index, 10-1 1.2810-2 2.33 10-3 3.09 10-4 3.72 10-5 4.26 10-6 4.75
2.3.3.2 Fatigue Reliability Analysis
Due to the considerable uncertainty from fatigue, a probabilistic approach is
appropriate for estimating the remaining fatigue life. The method presented by NCHRP
Project 12-28(03) (Moses, Schilling, and Raju 1987) used a probabilistic approach to
determine the average reliability factors for different bridges and probabilities of failure.
The approach is appropriate for certain bridge conditions. Although the NCHRP
approach provides a more consistent level of reliability for the typical assortment of
bridges, a more general probabilistic approach can be used to account for uncertainty in
all variables.
The study presented by Zhao, Halder, and Breen (1994) was one of the first to
apply a general reliability approach to bridges. The authors suggested a technique to
update the reliability index using Bayesian techniques based on the results of non-
destructive inspections (NDI). Depending on whether a crack was detected (crack length
was determined or crack length was not determined because too small) or not detected
(crack was too small for NDI method), the reliability index could be updated. By
updating the reliability index, bridge owners could make decisions on the inspection
interval or whether to repair, replace, or do nothing at a bridge.
54
Recently, Chung (2004), Orcesi, Frangopol, and Kim (2010), and Bocchini and
Frangopol (2011) have utilized reliability concepts to determine the optimal schedule for
inspections. The process involves setting a target level of reliability ( ) and
optimizing the inspection interval in terms of the cost of the inspections, repairs, and
failure. The result of these analyses produces a schedule that is non-regular, rather than
the regular schedule (every two years) specified by the current inspection requirements
(USDOT 2006)).
The reliability techniques have also been used by researchers to estimate the
remaining fatigue life. Szerszen, Nowak, and Laman (1999) developed a model that
utilized weigh-in-motion stations and S-N curves developed from test data to estimate the
reliability index for a bridge in Michigan. The data were used to determine the locations
where the fatigue damage was the highest. The authors found that the strains that were
measured were rather low; as such, the reliability index and remaining fatigue life were
both large.
Kim, Lee, and Mha (2001) applied a probabilistic method to a railway bridge to
estimate the remaining fatigue life. The probabilistic model accounted for the weight of
each train and when two trains might cross over the bridge at the same time. The
remaining fatigue life determined from the probabilistic method was compared with a
simple, deterministic approach. Because the deterministic approach is for a single
probability of failure, the methods could only be compared at that single data point.
Comparing the methods, the deterministic approach was found to have 10-20% longer
estimate for the fatigue life.
Orcesi, Frangopol, and Kim (2010) estimated that the reliability index will reach 0
(corresponds to a probability of failure of 50%) at the I-39 Northbound B-35-75 Bridge
over the Wisconsin River in 2024. The authors developed probability density functions
(PDFs) of the stress-range histogram that were validated from field data to make the
estimate of the fatigue life.
55
Kwon and Frangopol (2010) developed a reliability model that accounted for
measurement errors from misalignment of the strain gages and/or data acquisition error.
The measurement error was modeled using a lognormal distribution with a mean of 1.0
and coefficient of variation of 0.03. Two bridges were considered in the paper: Neville
Island Bridge and Birmingham Bridge. At the Neville Island Bridge, no traffic growth
was considered. Using the AASHTO equations (5% probability of failure), a positive
remaining fatigue life was calculated (4 years and 29 years) for the two locations
considered; however, one of those locations had active cracks. Using a general
probabilistic approach that is discussed in the paper, the calculated life was positive (2
years) at one location and negative (-6 years) at the other. Thus, the general method
estimated that the bridge could have active cracks for a probability of failure of 5%.
In the study on the Birmingham Bridge, the bridge was expected to have an
infinite fatigue life based on the monitored data and no traffic growth. If traffic growth
was considered, then the bridge had a finite fatigue life. The stress ranges were modeled
using PDF developed from the measured data; but the authors truncated the lower stress
ranges (up to 33% of the CAFL) because the value for the effective stress range was
skewed.
Building on the study by Kwon and Frangopol (2010), Liu, Frangopol, and Kwon
(2010) created a model for using the reliability equations with field data. The monitored
data were used to validate an FEM model and the model was used to determine critical
locations. Thus, the authors used measurements from a few locations to validate a model
that could be used to evaluate the entire bridge. The authors assumed traffic growth rates
of 2% to 5% for the fatigue analysis.
2.3.3.3 Fatigue Limit State Function
In general, there are two limit state functions used by researchers (Equation 2-31
and Equation 2-32). However, if an effective stress range ( ) is used with Equation 2-2
and Equation 2-8 to determine and , both limit state functions simplify to
the same fatigue limit state function (Equation 2-33). Equation 2-33 can be rearranged so
56
that it involves only products (Equation 2-34). By modeling the equation with products,
the limit state function ( ) can be modeled by a lognormal distribution function.
0 Equation 2-31
where
uncertaintymodelforthelimitstatefunctionof and
numberofcyclesuntilfailureforagivenstressrangeresistance
numberofcyclesatbridgeatgivenstressrangesincebridgewasputinservice loading
0 Equation 2-32
where
uncertaintymodelforthelimitstatefunctionofΔand
Δ Palmgren‐Miner’scriticaldamageindex resistance
Palmgren‐Miner’sdamageaccumulationindexsincebridgewasputinservice loading
0 Equation 2-33
where
uncertaintymodelforthefatiguelimitstatefunction
ln ln 1 0 Equation 2-34
If all of the terms of Equation 2-34 are modeled with random variables, there will
be a mean ( ) and standard deviation ( ) associated with each variable. If all of the
random variables have lognormal distributions, the reliability index for the fatigue limit
state function can be determined using Equation 2-35.
3
3
Equation 2-35
57
where
parameterforthelognormaldistributionforrandomvariablethatdependsonthemean andstandarddeviationoftherandomvariable
ln2
parameterforthelognormaldistributionforrandomvariablethatdependsonthemean andstandarddeviation oftherandomvariable
ln 1
These fatigue limit state functions have been used by many researchers to
estimate the remaining fatigue life (Szerszen, Nowak, and Laman 1999; Kwon and
Frangopol 2010; Orcesi, Frangopol, and Kim 2010; Liu, Frangopol, and Kwon 2010).
2.3.3.4 Summary
The following concepts can be deduced from the literature on applying structural
reliability concepts to fatigue evaluation.
Reliability equations can be used to evaluate fatigue damage by
accounting for uncertainty from resistance and load sources in a rational
method (Chung 2004).
A general limit state function can be used with measured data to estimate
the remaining fatigue life (Szerszen, Nowak, and Laman 1999; Kwon and
Frangopol 2010; Orcesi, Frangopol, and Kim 2010; Liu, Frangopol, and
Kwon 2010).
The probabilistic approach to estimating remaining fatigue life can be a
better approach than typical deterministic approaches (Kim, Lee, and Mha
2001; Kwon and Frangopol 2010).
Truncating the spectrum of stress ranges from data is also a concern when
applying the data to reliability equations (Kwon and Frangopol 2010).
58
CHAPTER 3
Techniques for Fatigue Analysis
When analyzing data obtained from field measurements, methods are needed to
properly evaluate fatigue damage. The traditional method of converting the distribution
of stress cycles used for a rainflow analysis to an effective stress range using Palmgren-
Miner’s rule has some shortcomings. Namely, the calculation of the effective stress
range ( ) can be skewed if there is a large percentage of the cycles in the first few bins.
As such, even if two different stress spectra generate the same amount of fatigue damage,
the spectra will likely produce different values for the effective stress range.
A new method, denoted as the index stress range, was developed as part of this
research investigation. The method is targeted for evaluating and comparing fatigue
damage at multiple locations in a structural system based on rainflow data from strain
gages. The index stress range is advantageous to the effective stress range because the
extent of fatigue damage induced at the location of each strain gage is normalized.
Rather than calculating the effective stress range for a given number of cycles, the
number of equivalent cycles at an index stress range ( ) is determined for a set amount
of induced fatigue damage.
Two techniques for visualizing the influence of the stress spectrum to the fatigue
damage are also presented. The contribution to damage and cumulative damage can be
calculated for measured strain histories. Both techniques are used to characterize the
effect of cycles within specific stress ranges of the induced fatigue damage. Many
engineers truncate the cycles in the lower stress ranges of the stress spectrum because
these cycles are either attributed to electromechanical noise within the data acquisition
equipment, or the amplitude of the cycles are so low that they are assumed not to
contribute to the fatigue damage. The electromechanical noise cycles do not correspond
to stress cycles experienced by the bridge, and therefore, do not induce damage. These
cycles should clearly be truncated from a fatigue analysis. However, the low-amplitude
59
load-induced cycles should not necessarily be truncated. Truncating the cycles
corresponding to the lower stress ranges increases the calculated value of the effective
stress range for a given strain history.
At the other end of the stress spectrum, the larger stress cycles may be load
induced or caused by electromechanical spikes from lightning or radios. If a cycle was
caused by a spike in the data acquisition system, the cycle should not be included in the
fatigue analysis. Determining the cause of the large-amplitude stress cycles has
historically been very difficult; however, several approaches were used in this
investigation The consequences of truncating cycles corresponding to lower and higher
stress ranges can be assessed by plotting the contribution to damage and cumulate
damage for the stress spectra.
Data from a few strain gages at two bridges were used to demonstrate the
importance of the methods discussed in this chapter. Data from the remaining gages and
bridges are summarized in Chapter 5 and the appendices. The specific details of the
bridges are not important for the examples discussed in this chapter; however, the
structural characteristics of the bridges are described in detail in Chapter 4. Bridge A is a
statically-determinant, three-span bridge that features two longitudinal girders. Strain
gages were concentrated near the north end of the bridge, near the connections to floor
beams 34 and 35. Two gages were considered in this chapter for Bridge A: W-34s-TE
and E-34s-TE. W-34s-TE corresponds to a gage installed on the west longitudinal girder
of Bridge A near floor beam 34 on the east side of the top flange, whereas E-34s-TE
corresponds to the east longitudinal girder. Throughout this discussion, data from W-
34s-TE will be identified as the west girder and data from E-34s-TE will be referred to as
the east girder.
Bridge D is a continuous, three-span bridge with seven girders across the bridge
width. Only one gage was considered for Bridge D in this chapter and it is located on the
bottom flange of girder 3 in the middle of the center span (L1-3-BN).
60
3.1 SIMPLIFIED RAINFLOW ANALYSIS
The first step to evaluate the fatigue damage using data from field measurements
in a bridge is to determine the distribution of stress cycles. As discussed in Chapter 2, the
simplified rainflow analysis is typically used because it counts cycles due to closed-loop
hystereses (Dowling 1972).
Following the algorithm outlined by Downing and Socie (1982), the distribution
of stress cycles for a single truck event can be determined. The stress history shown in
Figure 3-1 corresponds to gage W-34s-TE for a four-axle vehicle crossing in the right
lane of Bridge A. The primary cycle has an amplitude of 3.4 ksi; however, there are
seven other smaller stress-range cycles, due to bridge vibrations, with amplitudes
between 0.4 and 1.2 ksi. By considering all eight cycles and using Palmgren-Miner’s
rule (Section 2.2.1.1), the fatigue damage was increased by 11% as compared to
considering only the primary cycle (3.4 ksi).
Figure 3-1: Example stress history at the west girder from truck event and the results
of the simplified rainflow analysis.
Rather than record each specific stress range (as in Figure 3-1), the stress ranges
are typically accumulated in rainflow bins. The rainflow bins correspond to a specific
‐2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 1 2 3 4 5
Stre
ss (
ksi)
Time (sec)
Summary:(2) 0.4-ksi cycles(1) 0.5-ksi cycle(2) 0.8-ksi cycles(1) 1.1-ksi cycle(1) 1.2-ksi cycle(1) 3.4-ksi cycle
11% increase in damage due to secondary cycles
61
range and make it easier to summarize the spectrum of stress ranges measured at a bridge.
An example histogram is presented in the next section.
Following the procedure outlined by Downing and Socie (1982), the simplified
rainflow method was implemented in LabVIEW for use in the research outlined in this
dissertation. LabVIEW is a graphical programming language that was developed by
National Instruments (NI). It can be used with many different computer platforms;
however, LabVIEW is ideally compatible with NI hardware.
The program can be executed offline on a Windows-based computer or in real-
time on NI hardware. The user inputs the data (strain/stress history), the size of the bins
for the histogram, the number of total bins, the minimum amplitude to count, and the
length of the analysis. The program will reduce the strain/stress history into peaks and
valleys, save the peaks and valleys into memory, and determine the amplitude of each
cycle and put it into the appropriate bin of the rainflow histogram.
If a cycle is less than the minimum specified amplitude, the corresponding peak
and valley are deleted and not counted. As such, the minimum amplitude should be set at
the limit for the expected electromechanical noise from the data acquisition system
(typically between 1-10 microstrain).
The size of the bins times the number of total bins is the total spectrum of stress
ranges for the bridge that can be identified. The required spectrum could vary for each
gage location and bridge that is monitored. As such, care should be taken to ensure that
the total spectrum is wide enough to accommodate the largest expected stress range. If a
stress range is larger than the total spectrum, it is put into the last bin; but, the actual
value for that stress range is not known. As an example, if 10 bins were specified, each
0.5-ksi wide, the total spectrum would be 5 ksi. Any cycle above 5 ksi that is captured
would be put into an 11th bin. However, the values of the cycles above 5 ksi would not
be known because the total spectrum was not large enough.
The duration of the analysis is an important consideration. If the duration is too
long, a potentially large, fictitious cycle due to the drift of the strain gage with
temperature may be counted. If the duration is too short, then the count could be biased
62
due to partial cycles being counted as full cycles. A length of 30 minutes was chosen for
all of the analyses in this dissertation. In 30 minutes, the drift of the gage due to
temperature is generally minimal and the time is long enough to provide a realistic count
(Fasl, et al. 2010).
3.2 AMOUNT OF FATIGUE DAMAGE
The primary goal of any monitoring project is to determine the amount of fatigue
damage that is induced during the monitoring period. The damage is then typically used
to estimate the remaining fatigue life by projecting damage in years prior to the
monitoring period and in the future. AASHTO LRFD Specifications (2010) relates the
number of cycles to failure for a given stress range using Equation 3-1; the equation is
often plotted and referred to as S-N curves. Thus, the fatigue damage is described by the
number of cycles times the stress range cubed, and the fatigue resistance is .
Equation 3-1
where
designnumberofcyclesuntilfailureat
fatigueconstantfordetailcategory,definedbyAASHTOLRFDSpecifications 2010
constant‐amplitudestressrange
Because bridges experience variable-amplitude stress ranges, it is useful to
transform the distribution of stress ranges into a single, constant-amplitude stress range
that induces an equivalent amount of damage. In addition, because the S-N curves are
expressed in terms of the number of constant-amplitude cycles, determining the
equivalent stress range is useful for both design and analysis. As outlined in Chapter 2,
the most common way to relate the variable-amplitude stress ranges to a single stress
range is to calculate the effective stress range using Palmgren-Miner’s rule. It is difficult
to compare the amount of fatigue damage for different gage locations using the effective
stress range because the number of measured cycles must be considered along with the
63
effective stress range. As part of this research study, a new method for assessing and
comparing fatigue damage among different locations along a given bridge or within an
inventory of bridges was developed. Palmgren-Miner’s rule is also utilized with the
index stress range. However, because the index stress range is the same for each
instrumented location, the metric of damage becomes only the equivalent number of
cycles at the index stress range. Therefore, the response at different locations can be
compared directly using only the equivalent number of cycles.
Palmgren-Miner’s rule is the most popular cumulative damage theory utilized in
the USA for dealing with fatigue-related damage because the simple method has been
shown to agree well with test results (ASCE 1982). It is used by AASHTO to relate
variable-amplitude loading to the fatigue results of constant-amplitude loading.
Nonlinear, cumulative damage theories have been proposed by some researchers (Li,
Chan, and Ko 2001). However, the equations are more complicated to use and the results
are not consistently better than a linear damage theory.
There are two shortcomings of Palmgren-Miner’s rule. First, the method does not
consider the sequence of loading. In some cases, the sequence has been shown to impact
the fatigue behavior. For instance, if an overload stress range occurs with low frequency
after many low stress ranges, it has been found to increase the fatigue strength of the
specimen (ASCE 1982). Yet, Palmgren-Miner’s rule does not account for such
interaction. Palmgren-Miner’s rule also does not account for any influence that the
average stress may have on the fatigue life of the specimen. However, for welded
structures with high residual stresses, the average stress does not have a large influence
on the fatigue life (Fisher, Kulak, and Smith 1998). For structures subject to random
loading and relatively few overloaded stress ranges, Palmgren-Miner’s rule can
adequately describe the damage.
A concern of applying Palmgren-Miner’s rule is how to deal with cycles with an
amplitude less than the constant-amplitude fatigue limit (CAFL). Assuming that all
cycles contribute to the damage is equivalent to assuming that the value for the fatigue
threshold stress intensity factor is zero. Some researchers have proposed truncating all
64
cycles less than 25-50% of the CAFL (Connor and Fisher 2006), whereas European
practice weights the lower stress ranges differently (slope constant, , of 5, rather than 3)
(ECCS Technical Committee 6 1985). The European practice has been shown to be
unconservative for some welded longitudinal attachments (Fisher, Kulak, and Smith
1998). In addition, Swenson & Frank (1984) have shown that all cycles can contribute to
the accumulation of fatigue damage, even cycles with amplitudes less than the CAFL.
Though, there is not agreement among the international community on how to deal with
the lower stress ranges, an engineer may assess the influence of the stress ranges using
the methods discussed in this chapter.
Figure 3-2: Example histogram of stress ranges for 14 days of data for the east girder
at Bridge A. Rainflow bins were 0.15-ksi wide and cycles less than 0.06 ksi were
truncated from the histogram.
A histogram of stress ranges for fourteen days of data for the east girder at Bridge
A is presented in Figure 3-2. The first seven days occurred in April 2011 and the second
seven days in July 2011. The strain gages were attached to a connection that is classified
as AASHTO Category E. The stress history was evaluated in 30-minute periods and
cycles less than 0.06 ksi were considered to be caused by electromechanical noise within
the data acquisition system and truncated. The rainflow bins were 0.15-ksi wide. A total
of 800,000 cycles were measured during the two weeks (daily average is 57,100 cycles),
with a maximum stress range of 20.8 ksi. The histogram was used to compare the two
0
1
10
100
1,000
10,000
100,000
1,000,000
0 5 10 15 20 25 30
Nu
mb
er
of
cycl
es
Stress range (ksi)
Fourteen days of data (one week in April and one week in July)
800,000 cycles20.8 ksi
-
65
methods for evaluating the amount of fatigue damage: (1) the effective stress range and
(2) the index stress range. The following two subsections document the results from the
two methods.
3.2.1 Effective Stress Range
The effective stress range is an efficient method to relate the cycles from a
spectrum of stress ranges to a single, equivalent stress range. The method was discussed
briefly in Section 2.2.1.1 and utilizes Palmgren-Miner’s rule as the cumulative damage
theory. Rather than listing each stress range that was determined from a rainflow
analysis, the ranges are typically sorted into bins. All of the cycles in a particular bin are
assumed to have the same stress range as the average stress of the bin. As such, the
damage accumulation index ( ) and effective stress range ( ) for a rainflow histogram
can be calculated using Equation 3-2 and Equation 3-3, respectively.
,
∑ Equation 3-2
where
measurednumberofcyclesinbincorrespondingto recordedduringmonitoringperiod
, designnumberofcyclesuntilfailureat
averagestressrangeforbin ksi
fatigueconstantfordetailcategory,definedbyAASHTOLRFDDesignSpecifications 2010 ksi3
∑ /
Equation 3-3
where
effective stress range for cycles (ksi)
total number of stress cycles measured during monitoring period
66
If the damage accumulation index is graphed on a log-log S-N plot, different
levels of plot parallel to the descending branch of the design S-N curve for the given
fatigue category (Figure 3-3). The lines plot parallel because the damage accumulation
index can also be defined as the ratio of the number of cycles experienced to the design
fatigue life. As demonstrated in the figure, the damage accumulation index is larger as it
gets closer to the design S-N curve for the fatigue category.
Figure 3-3: Different levels of the damage accumulation index on a S-N graph.
In principle, the effective stress range is calculated for a given number of
measured cycles ( ) and damage accumulation index ( ). This is shown graphically in
Figure 3-4. If the number of cycles change, but the damage stays constant, the effective
stress range can change as discussed in Section 2.2.1.1. As such, calculating the effective
stress range does not provide a relative measure of damage at a gage location because
each set of data could produce a different set of effective stress ranges (Figure 3-5). With
effective stress ranges that vary depending on the monitoring period and number of
cycles captured in the first few bins of a rainflow histogram (Section 3.2.3), comparing
multiple gage locations is very difficult.
The effective stress range for the example histogram that was presented for
Bridge A in Figure 3-2 was calculated to be 2.24 ksi for the 800,000 cycles recorded
during the 14-day period. The effects of truncating low-amplitude stress cycles and
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Parallel lines correspond to constant levels of the damage accumulation index )
Design S-N curve for fatigue category
Increasing
67
variation in traffic volume based on the day of the week on the calculated effective stress
range are discussed in Section 3.2.3. A new method, index stress range, was developed
to overcome the limitations of using the effective stress range, and allow relative
comparison between gage locations.
Figure 3-4: Graphical representation of the method for determining the effective stress
range.
Figure 3-5: Varying sets of effective stress ranges and measured cycles for different
gage locations.
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Effective stress range calculation: (1) Determine (2) Determine (3) Calculate
(1)
(3)
Design category
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Design category
Gage locations
68
3.2.2 Index Stress Range
Relative damage accumulation is a useful tool for bridge owners because it can
identify problems within a single bridge or across a bridge inventory. If the locations of
highest damage accumulation are known, inspection efforts can be focused on such
locations. Fatigue damage is characterized by the number of cycles at a specific stress
range; thus, either the number of the cycles or the stress range has to be the same to
compare two locations directly.
An index stress range was developed as a method to evaluate the relative extent of
fatigue accumulation for different locations along the same bridge. It can also be used to
compare damage among different bridges. With the index stress range, all measured
data, whether in the form of an effective stress range or a distribution of stress ranges, are
normalized to the same stress range, which is selected by the engineer. By normalizing
the data to an index stress range, the number of cycles at the selected stress range
becomes the direct metric of relative fatigue damage: twice as many cycles at the same
index stress range causes twice the fatigue damage.
The index stress range can be derived using two approaches: (1) with an effective
stress range or (2) with a histogram of stress ranges. If the effective stress range obtained
from measured data has already been calculated, it can be indexed to an arbitrary stress
range as long as the damage accumulation index ( ) stays the same. The damage index
can be defined for an effective stress range using Equation 3-4.
Equation 3-4
Because damage is characterized by the number of cycles at a specific stress
range and the AASHTO detail category constant does not change with the stress range,
the number of cycles at the index stress range is modified (Equation 3-5) to keep the
damage index the same as at the effective stress range. By keeping the damage index the
same, the remaining fatigue life will be the same whether the index stress range (with
69
cycles at index stress range) or the effective stress range (with cycles at effective stress
range) is used for the calculation.
Equation 3-5
where
number of equivalent cycles at
indexstressrange
Because the number of cycles and corresponding stress range are needed to
calculate the amount of fatigue damage, the index stress range can be referenced using
the notation of Equation 3-5. By referencing the stress range, the damage accumulation
index can be calculated easily or used to calculate the fatigue life (see Chapter 7). For
instance, if an index level of 4.5 ksi is chosen, the notation for the number of cycles can
be expressed as 4.5 . A line over the top and a subscript can be used to
represent the daily average: , 4.5 . Other subscripts can be used for monthly ( )
or yearly ( ) averages.
Figure 3-6: Graphical representation of the index stress range method.
By using an index stress range, the number of cycles is determined for an
arbitrary stress range and damage index (Figure 3-6). In contrast, the effective stress
range is determined from the number of measured cycles and damage index (Figure 3-4).
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Design category Index stress range calculation: (1) Choose (2) Determine (3) Calculate
(3)
(1)
70
The index stress range can be set to any desired magnitude; however, a convenient value
is the threshold stress of the given AASHTO detail category.
The application of Equation 3-5 is shown graphically in Figure 3-7 for the
example histogram. As determined from the previous section, the effective stress range
for the example histogram was 2.24 ksi for 800,000 cycles (57,100 daily cycles, , ),
which can be used to determine the damage index ( ). If an index stress range of 4.5 ksi
is used, which is the CAFL for detail Category E (AASHTO LRFD Specifications 2010),
an average of 7,075 cycles per day ( , 4.5 ) would be required to produce an
equivalent amount of damage.
Figure 3-7: Graphical representation of determining an index stress range from an
effective stress range.
If Equation 3-3 is substituted into Equation 3-5, an alternative method to
calculating is presented (Equation 3-6). With this alternative method, the
effective stress range does not need to be calculated.
Equation 3-6
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Design category
71
Equation 3-6 is sufficient to normalize the data if all the connections that are
being monitored correspond to the same fatigue category. If data are collected from
connections with different fatigue categories, Equation 3-6 can be modified as shown in
Equation 3-7. The / term of Equation 3-7 accounts for the difference in number
of cycles to failure for the different AASHTO fatigue categories.
:
Equation 3-7
where
: number of equivalent cycles at and indexed to AASHTO Category
AASHTOfatigueconstantforcategoryofindex
AASHTOfatigueconstantforcategoryofdetail
As an example, consider two details: a welded connector plate corresponding to
an AASHTO Category E detail and a riveted connection corresponding to an AASHTO
Category D detail. If Category E is used for the index category, then the index stress
range becomes 4.5 ksi and the / term is 1.0 for the Category E details. For the
riveted connections, the / term becomes / 0.5 because the
fatigue constant for Category D is twice as large as the fatigue constant for Category E.
After scaling for the fatigue category, the relative damage accumulation for different
fatigue categories can be compared on the same figure by looking at the number of cycles
at the index stress range. The notation for the number of cycles at the index stress range
for the given example would be: 4.5 : .
3.2.3 Comparison of Effective Stress Range and Index Stress Range
The relative merits of each method, effective stress range and index stress range,
can be compared by evaluating the effects of truncation and daily traffic variations on the
calculation of damage. If the lower stress ranges do not impact the accumulation of
damage, the effect of truncating those lower rainflow bins should also be minimal. As
72
much as 25-50% of the threshold stress range is recommended to be truncated (Connor
and Fisher 2006).
The fourteen days of data from Bridge A were used to compare both methods. If
the bins with the lower stress ranges are truncated from the histogram in Figure 3-2, the
impact of level of truncation on the effective stress range and number of cycles at an
index stress range can be evaluated as shown in Figure 3-8. The effective stress range
changes quite drastically by truncating the lower bins; the effective stress range changes
by more than 100% if the stress ranges below 1.0 ksi are neglected from the calculation.
The reason for the change is the large reduction in total number of cycles (i.e. there are
nearly 500,000 cycles in the first bin of the histogram in Figure 3-2). While there is an
increase in the effective stress range, there is also a reduction in number of cycles that
could offset the apparent change in damage. Thus, the effective stress range by itself, as
obtained from applying Palmgren-Miner’s rule, is not an effective indication of damage
because it depends heavily on the number of cycles in the lower bins.
Figure 3-8: Change in effective stress range and number of cycles at index stress
range as the lower cycles are truncated from the calculation for Category E detail.
By transforming the effective stress range to an index stress range of 4.5 ksi using
Equation 3-5, the impact on damage accumulation at this bridge from truncating lower
stress ranges is negligible. As seen in Figure 3-9, the percent decrease in , 4.5 is
-20%
20%
60%
100%
140%
180%
-20%
20%
60%
100%
140%
180%
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Incr
eas
e in
S re
Truncated stress range (ksi)
Category E DetailCAFL: 4.5 ksi
1.125 ksi (25% of threshold stress range)
2.25 ksi (50% of threshold stress range)
Change in Sre
Change in D
ecr
eas
e in
73
less than 2% if cycles below 2.25 ksi are truncated. This analysis shows that the lower
bins do not significantly contribute to the overall fatigue damage at this particular bridge,
which was not immediately obvious when using the effective stress range as a metric.
Figure 3-9: Change in number of cycles at index stress range as the lower cycles are
truncated from the calculation for Category E detail.
It would have been acceptable to neglect 50% of the threshold stress range for a
Category E detail at this particular bridge. However, if the rivets controlled for this detail
(Category D) and stress ranges below 3.5 ksi were truncated (50% of 7-ksi threshold
stress range), the decrease in , 7 would approach 5% and start decreasing quickly
from there as higher stress ranges are truncated (Figure 3-10 and Figure 3-11). It may not
be appropriate to truncate up to 50% of the threshold stress range for all bridges, but it
did not significantly impact this bridge. The index stress range allows for a quick
evaluation on the influence of truncating the lower stress bins.
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Incr
ease
inS r
e
Truncated stress range (ksi)
Category E DetailCAFL: 4.5 ksi
1.125 ksi (25% of threshold stress range)
2.25 ksi (50% of threshold stress range)
Change in SreChange in
Dec
reas
e in
74
Figure 3-10: Change in effective stress range and number of cycles at index stress
range as the lower cycles are truncated from the calculation for Category D detail.
Figure 3-11: Change in number of cycles at index stress range as the lower cycles are
truncated from the calculation for Category D detail.
The histogram of strain data for Bridge A (Figure 3-2) were split into individual
days. The corresponding daily damage levels were determined using each method. For
the method using the effective stress range, both the stress range and number of cycles
must be plotted (Figure 3-12). The effective stress ranges for the days in April were
higher than the days in July. However, the number of measured cycles were sometimes
-20%
20%
60%
100%
140%
180%
220%
-20%
20%
60%
100%
140%
180%
220%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Incr
ease
inS r
e
Truncated stress range (ksi)
Category D DetailCAFL: 7.0 ksi
1.75 ksi (25% of threshold stress range)
3.50 ksi (50% of threshold stress range)
Change in Sre
Change in
De
cre
ase
in
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Incr
eas
e in
S re
Truncated stress range (ksi)
Category D DetailCAFL: 7.0 ksi
1.75 ksi (25% of threshold stress range)
3.50 ksi (50% of threshold stress range)
Change in Sre Change in
De
cre
ase
in
75
higher in April and sometimes higher in July. In such a case, it becomes difficult to
characterize the relative damage. The number of measured cycles and the effective stress
ranges by themselves are not good indicators.
Because the effective stress range is not the same for each day, comparing the
relative damage between days is slightly convoluted. To evaluate the relative damage on
Friday (2.34 ksi and 65,500 measured cycles) and Sunday (2.10 ksi and 39,100 measured
cycles), the damage accumulation index could be calculated. Comparing the calculated
damage accumulation indexes, it can be determined that 2.3 times as much damage was
accumulated on Friday as compared to Sunday, which was not apparent from comparing
the measured cycles and effective stress ranges.
Figure 3-12: Change in effective stress range and number of cycles with day of the
week for Bridge A.
In contrast, if the index stress range were used, the method is able to characterize
the relative damage by comparing 4.5 directly (Figure 3-13). For instance, by
dividing the damage ( 4.5 ) on Friday (9,200 cycles) by the damage on Sunday
(4,000 cycles), 2.3 times as much damage occurred on Friday as compared to Sunday.
0
20,000
40,000
60,000
80,000
Mon Tues Wed Thur Fri Sat Sun Dailyaverage
Nd
,m
Day of the week
1.8
2.0
2.2
2.4
S re
Week in JulyWeek in April
76
In summary, the index stress range is a better indicator of damage because the
damage is normalized. The method allows for easy comparison between different gage
locations, days, and/or bridges because each set of data can be indexed to the same
arbitrary stress range. Therefore, the number of cycles at the index stress range ( )
is the only value needed to compare two gage locations. Additionally, using the index
stress range makes it easier to evaluate the variation in fatigue damage (Section 5.3) and
characterize the benefit of a retrofit by tracking the daily changes in the fatigue damage
(Section 5.4). In constant, comparing two gage locations using the effective stress range
is convoluted because the damage accumulation index has to be calculated. As such, the
effective stress range and measured number of cycles have to be multiplied following
Palmgren-Miner’s rule for comparison of two gage locations.
Figure 3-13: Change in number of cycles at index stress range with day of the week
for Bridge A.
3.3 CHARACTERIZATION OF FATIGUE DAMAGE
Section 3.2 focused on identifying the amount of fatigue damage for a given
distribution of stress cycles. This section focuses on characterizing the fatigue damage
by identifying which stress ranges are contributing the most to the damage. Because
fatigue damage is proportional to the stress range cubed, the larger stress cycles (greater
than 15 ksi) in Figure 3-2 are alarming. Though each cycle can cause a lot of damage,
the larger cycles do not occur very often. As such, they may not significantly affect the
0
2,000
4,000
6,000
8,000
10,000
Mon Tues Wed Thur Fri Sat Sun DailyaverageDay of the week
Week in July
Week in April
77
accumulation of fatigue damage. The contribution of a stress range to damage and
cumulative damage with stress range can be used to characterize the fatigue damage.
3.3.1 Contribution to Damage
The contribution of a particular bin or stress range to the fatigue damage ( )
across a histogram of stress ranges can be evaluated using Equation 3-8. The total fatigue
damage for a monitoring period is the sum of all stress ranges. The damage accumulation
index ( ) can be calculated if the total fatigue damage is divided by the fatigue constant
( ) for the particular detail.
Equation 3-8
where
fatigue damage from bin
If Equation 3-8 is applied to the example histogram for Bridge A (Figure 3-2), the
contribution of each bin can calculated (Figure 3-14). The average fatigue damage for
the data is 644,700 ksi3/day. The largest amount of damage occurred between stress
ranges of 2 ksi and 10 ksi. The limits (2 ksi and 10 ksi) were determined based on the
technique described in the next section. Despite the large number of cycles in the first
few bins, the lower stress ranges did not significantly contribute to the accumulation of
fatigue. This matches the result of truncating the lower stress ranges in Figure 3-8. The
larger cycles also did not contribute significantly to the fatigue damage as seen by the
small peaks in Figure 3-14. The contribution of stress cycles greater than 15 ksi is
insignificant because there are so few occurrences of the larger stress ranges relative to
the mid-level stress ranges.
78
Figure 3-14: Contribution of each bin to the total fatigue damage during the 14-day
monitoring period for Bridge A.
The contribution of stress ranges to fatigue damage can also be determined using
the index stress range method. In Equation 3-6, an alternate method for calculating the
number of equivalent cycles at an index stress range was presented. If the sum was not
present in that equation, the contribution of each stress range to can be calculated
using Equation 3-9.
Equation 3-9
where
number of equivalent cycles at due to number of cycles in bin ( ) with an average bin stress range of
The contribution of damage can be calculated using an index stress range for the
example histogram for Bridge A. As seen in Figure 3-15, the contribution of each stress
range to , 4.5 has the same shape as the contribution to damage in Figure 3-14.
The only difference is how the damage is represented: directly as in Figure 3-14 or as an
equivalent number of cycles at an index stress range as in Figure 3-15. Thus, either
approach can be used to determine the contribution of each stress range to the damage. If
the number of equivalent cycles in each bin is summed across the distribution of stress
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
0 5 10 15 20 25
Ave
rage
co
ntr
ibu
tio
n t
o
fati
gue
dam
age
(ksi
3)
Stress range (ksi)
Average fatigue damage: 644,700 ksi3/day
79
ranges, the total average number or cycles at the index stress range (4.5 ksi) is 7,075
cycles, which matches what was previously calculated for the histogram for Bridge A.
Figure 3-15: Contribution of each bin to , . during the 14-day monitoring
period for Bridge A.
3.3.2 Cumulative Damage
The contribution to damage can be formalized by calculating how much damage
is accumulated by the measured stress ranges. Equation 3-10 can be used to calculate the
cumulative stress range for a given bin. The method of index stress range could also be
used to determine the cumulative damage, and it would produce an equation similar to
Equation 3-10.
, ∑
Equation 3-10
where
, cumulative fatigue damage from bins 0 to
sum of fatigue damage from all bins
The cumulative damage was calculated for the example histogram for Bridge A
(Figure 3-2). The cumulative damage graph was split into regions of lower 2.5%
0
50
100
150
200
250
300
350
400
0 5 10 15 20 25Stress range (ksi)
Co
ntr
ibu
tio
n t
o
Sum of : 7,075 cycles
80
damage, middle 95% damage, and upper 2.5% damage (Figure 3-16). The graph shows
that the majority of the damage was induced by stress cycles between 2 ksi and 10 ksi.
The cumulative damage plot formalizes the conclusions from the contribution to
damage plot: neither the lower or larger stress ranges contributed significantly to the
accumulation of fatigue damage at this bridge. Many researchers truncate lower bins
because the large number of cycles in the lower bins might affect the calculation of the
effective stress range and the cycles are typically a result of electromechanical noise,
rather than real fatigue cycles. For this bridge, the lower bins can be truncated with very
little impact on the analysis because cycles below 1 ksi contributed less than 2.5% to the
total damage. The damage at the site is dominated by typical truck traffic, whereas
cycles greater than 10 ksi only contributed approximately 2.5% of the damage.
Figure 3-16: Average cumulative damage during the 14-day monitoring period for
Bridge A.
3.4 EXAMPLE
The techniques discussed in this chapter were applied to the example histogram of
stress ranges for Bridge A. An example histogram for a different bridge (Bridge D) is
considered in this section to exemplify further the benefit of the techniques. The
histogram for nearly four days of data of Bridge D is presented in Figure 3-17. Over 1.2
million cycles were measured in the time period, with most of the cycles in the first few
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25
Cu
mu
lati
ve d
amag
e
Stress range (ksi)
Maximum2.5 ksi
10.5 ksi
95% of damage
2.5% of damage
20.8 ksi
81
bins. A minimum amplitude of 0.06 ksi was chosen for this bridge to delete the majority
of cycles caused by electromechanical noise. A few cycles with larger stress ranges
(greater than 10 ksi) were identified, with a maximum stress range of 29.1 ksi. The
effective stress range ( ) for the histogram was determined to be 0.52 ksi, with an
average daily count of 325,000 cycles. The detail at Bridge D corresponds to a Category
C′ detail. Using an index stress range of 12 ksi, , 12 was 26.3 cycles per day.
Figure 3-17: Example histogram of stress ranges for 4 days of data for Bridge D.
Rainflow bins were 0.15-ksi wide and cycles less than 0.06 ksi were truncated from the
histogram.
The impact of the lower stress ranges can be evaluated through truncation or by
considering the contribution to damage. In Figure 3-18, the truncation of the lower stress
ranges were evaluated. If all of the stress ranges below 1.0 ksi are truncated, the effective
stress range increased by approximately 350%. In contrast, , 12 decreased by
only 5% if stress ranges below 1.0 ksi were truncated (Figure 3-19). Therefore, despite
the change in effective stress range, cycles less than 1.0 ksi are not contributing
significantly to the estimate of fatigue damage. Because the majority of the cycles
caused by electromechanical noise of the data acquisition equipment were truncated by
the minimum amplitude set for the rainflow program, the cycles less than 1.0 ksi did not
need to be truncated.
0
1
10
100
1,000
10,000
100,000
1,000,000
0 5 10 15 20 25 30
Nu
mb
er
of
cycl
es
Stress range (ksi)
-
Four days of data1,286,000 cycles0.52 ksi
29.1 ksi
82
Figure 3-18: Change in effective stress range and number of cycles at index stress
range as the lower cycles are truncated from the calculation for Bridge D.
Figure 3-19: Change in number of cycles at index stress range as the lower cycles are
truncated from the calculation for Bridge D.
If cycles greater than 1.0 ksi were truncated, the estimate of fatigue damage
started to decrease quickly. Truncating stress ranges below 3.0 ksi, which corresponds to
25% of the CAFL for a Category C′, the estimate of fatigue damage was reduced by 50%.
The large reduction in fatigue damage is due to some of the load-induced cycles being
-60%
20%
100%
180%
260%
340%
420%
500%
-60%
20%
100%
180%
260%
340%
420%
500%
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Incr
ease
inS r
e
Truncated stress range (ksi)
Category C′ DetailCAFL: 12 ksi
3.0 ksi (25% of threshold stress range)
Change in Sre
Change in
De
cre
ase
in
-60%
-40%
-20%
0%
20%
-60%
-40%
-20%
0%
20%
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Incr
ease
inS r
e
Truncated stress range (ksi)
Category C′ DetailCAFL: 12 ksi
3.0 ksi (25% of threshold stress range)
Change in SreChange in
Dec
reas
e in
83
deleted from the analysis. Thus, truncating stress ranges based on the CAFL is not
always justified, and should be evaluated prior to truncating some load-induced cycles.
The contribution to damage was evaluated and is shown in Figure 3-20. As seen
on the left side of the graph, the lower stress ranges (less than 1 ksi) do not impact the
damage and matches the truncation analysis in Figure 3-19. A significant amount of
damage occurs between 1-4 ksi. However, as seen, the upper stress ranges do contribute
to the amount of fatigue damage.
Figure 3-20: Contribution of each bin to , during the 4-day monitoring
period for Bridge D.
The cumulative damage for the example histogram for Bridge D is shown in
Figure 3-21. As seen, the middle 95% of damage takes up the majority of the distribution
of stress ranges due to the effect of the larger stress ranges. Because the large cycles
significantly affect the amount of damage during the monitoring period, the raw stress
history was reviewed to determine if the cycles were load induced or due to spikes in the
data acquisition system.
With the simplified rainflow analysis, the raw data are not typically saved because
of limitations in data storage of dynamic data on data acquisition systems. To validate
the larger stress cycles, an event capture program was added that saved data in
conjunction with the rainflow program on the data acquisition system. The event capture
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 5 10 15 20 25 30
Stress range (ksi)
Sum of : 26.3 cycles
Co
ntr
ibu
tio
n t
o
84
program saved the raw history for events (seconds) when the stress level exceeded a
given threshold so that the large cycles could be validated when reviewing the data.
After review, the large cycles were attributed to single-point spikes in the data acquisition
system (Figure 3-22).
Figure 3-21: Average cumulative damage during the 4-day monitoring period for
Bridge D.
Figure 3-22: Stress history with single-point spike at Bridge D.
A video camera was installed at this bridge to record the types of vehicles that
caused the larger cycles. The still images associated with the spike in Figure 3-22 are
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25 30
Cu
mu
lati
ve d
amag
e
Stress range (ksi)
Maximum0.4 ksi 95% of damage
29.1 ksi
-20.0
-15.0
-10.0
-5.0
0.0
5.0
10.0
0.00 0.25 0.50 0.75 1.00
Stre
ss (
ksi)
Time (sec)
Single-point spike in the data
85
shown in Figure 3-23. As can be seen, no vehicles were traveling across the bridge at the
time of the spike. Instead, as seen in the middle photo, there is a large flash from
lightning in the area around the bridge. The spike in the stress history is likely due to the
lightning strike, and the cycle should be truncated from the fatigue analysis.
Figure 3-23: Lightning strike (middle photo) during single-point spike at Bridge D.
3.5 SUMMARY
Using the procedures described in this chapter, the amount of fatigue damage can
be determined and characterized. The index stress range provides a method for assessing
the relative damage accumulation between gage locations and/or bridges. Because the
damage is normalized using the method, the number of cycles at the index stress range is
a better indicator of damage. The method can also be used to calculate the contribution
of damage from each stress range. Because both lower (electromechanical noise) and
higher (possibility of a spike in the data acquisition system) stress ranges are questioned
during a monitoring program, the impact of all stress ranges can be evaluated. If the
cycles contribute more to the damage than expected (as the larger cycles for Bridge D),
those cycles can be analyzed further. Engineers can use judgment on how to handle low-
amplitude cycles in the absence of international agreement. However, in most cases (as
seen for both Bridge A and Bridge D), the lower cycles (less than 1.0 ksi) do not
significantly contribute to fatigue damage and will be normalized by the index stress
range.
86
CHAPTER 4
Field Monitoring of Four Highway Bridges
Four steel bridges were instrumented to characterize fatigue behavior and estimate
the remaining fatigue life. Fatigue-induced fracture is a primary design concern for steel
structures, especially if the bridge involves fracture-critical members. Because fracture-
critical bridges have non-redundant structural systems, the loss of a single structural
member has the potential of causing wide-spread damage or collapse. Both box girder
and I-girder bridges were monitored, and two were considered to be fracture critical. The
geometry and instrumentation at the bridges are explained in this chapter. In addition, the
data acquisition systems and sensors are summarized.
Bridge A was monitored the longest of the four bridges because the bridge was a
prime candidate for the technologies and methodologies developed as part of the NIST-
sponsored research project (Section 1.1). When Bridge A was initially instrumented, the
wireless data acquisition system (Section 4.5.2) had not yet been developed; as such, a
wired data acquisition system was used (Section 4.5.1). Bridge A was monitored for over
a year, and considers periods prior to construction of a retrofit, construction of a retrofit,
and after the construction of a retrofit. The data from select gages are presented in
Chapter 5, and all of the gages are summarized in Appendices A and B.
Bridges B, C, and D were not monitored to the same extent as Bridge A. The
main focus of monitoring Bridges B, C, and D was to validate the operation of the
wireless data acquisition equipment (Section 4.5.2) in a variety of bridge environments.
Of the three bridges, Bridge B was monitored the longest, at a length between one and
two months (depending on the gage location). Bridge D was monitored the least amount
of time (less than one week), as it was the first bridge to utilize the wireless system.
Bridge C was monitored for approximately one month. Fatigue data from a couple strain
gages are presented in Chapter 5, while the remaining gages are summarized in
Appendices C (Bridge B), D (Bridge C), and E (Bridge D).
87
4.1 BRIDGE A
Bridge A is a fracture-critical bridge along a major transportation corridor with
significant truck traffic and a maximum speed limit of 70 mph (Figure 4-1). The bridge
is considered to be fracture critical because the twin-girder system is non-redundant and
the failure of a flange from a longitudinal girder would be expected to cause collapse of
the entire bridge. The annual daily truck traffic was reported to be 4,000 in 2005.
Figure 4-1: Bridge A.
The bridge was constructed over 75 years ago and has exceeded its intended
design life. For nearly 40 years, one lane was dedicated to traffic in each direction. At
that point, the bridge was widened and traffic changed to only the north direction. Lane
widths increased from 12 ft to 14 ft. A separate bridge was added next to the fracture-
critical bridge to allow for a third lane of traffic. There is a gap between the two bridges
that allows the bridges to act independently. The right-lane structure is not discussed
further in this dissertation because the primary purpose of the structure is for traffic
entering and exiting the highway and does not influence the behavior of the fracture-
critical bridge.
4.1.1 Geometry
Bridge A is a three-span, plate-girder bridge that features two longitudinal girders.
The total length of the bridge is 272 ft with 73.5-ft end spans and a 125-ft center span.
88
The longitudinal girders in the end spans are continuous over the interior supports and
extend 30.6 ft into the center span (Figure 4-2). The center section is suspended between
the cantilevers by hangers, which makes the bridge statically determinate. The transverse
floor beams are 2.25-ft deep and typically spaced at 7.5-ft intervals.
Figure 4-2: Elevation of girder at Bridge A with spacing of floor beams shown.
The built-up, longitudinal, plate girders are riveted sections with a transverse
center-to-center spacing of 23 ft. The girders are haunched, with a depth of 8 ft over the
interior supports, 5 ft at the middle of the suspended span, and 5.54 ft at the end supports.
The built-up section comprises web plates riveted to double angles that act as flanges.
Cover plates are utilized to increase the moment-carrying capacity at the interior
supports and middle of the suspended span. Rivets are used to connect the cover plates to
the longitudinal girder. The locations of the cover plates are identified in Figure 4-3 and
are symmetric about the middle of the bridge. Over the interior supports, three sets of
cover plates are used along the top and bottom flanges, with thicknesses varying between
3/8″ to 3/4″. Along the suspended span, only two layers of cover plates are needed for
the top and bottom flanges.
A cross section of the widened bridge is shown in Figure 4-4. The widening was
accomplished with the addition of transverse, cantilever brackets that allowed the lane
widths to increase from 12 ft to 14 ft. The top flanges of the cantilever brackets and floor
beams were welded to the top flange of the longitudinal girders to provide moment
capacity at the overhang as depicted in Figure 4-5. The current locations of the lanes are
shown in Figure 4-4. Because vehicles have travelled northbound only across the bridge
Hanger
North spanCenter spanSouth span
73′-6″ 73′-6″125′-0″
End support, typ. Interior support, typ.
104′-1″ 63′-10″
I I II I II I I III I IIIIIIII IIIII I III I I III III I14 spa @ 7′-6″ 14 spa @ 7′-6″1′-0″ 8 spa @ 7′-6″ 1′-0″
Direction of traffic
89
since the deck was widened, over the entire life of the bridge, the majority of the trucks
are expected to have crossed the bridge in the right lane, which is supported by the east
girder.
Figure 4-3: Location of cover plates on the longitudinal girder at Bridge A to increase
the section modulus.
Figure 4-4: Cross section and location of lanes after widening (looking north).
Though the deck was replaced during the bridge widening, it was not designed for
full composite action with the longitudinal girders. Shear studs were not installed along
the longitudinal girders; however, a single line of shear studs was installed along the
cantilever bracket on each side of the bridge (Figure 4-5). The shear studs were spaced
every 2.5 ft along the length of the bridge. The line of shear studs created some
North spanCenter spanSouth span
I I II I II I I III I IIIIIIII IIIII I III I I III III I
5′-0″8′-0″5′-6½″
48′-0″
34′-0″
33′-6″
25′-0″9′-6″ 7′-9″
18′-9″25′-0″
Length of cover plates are symmetric about center span
Thickness of cover plate varies between ⅜″ and ¾″
4′-0″
14′-0″ (right lane)14′-0″ (left lane)
6′-0″3′-0″ 6′-0″ 8′-0″6′-1½″
West girder East girder
23′-0″ (girder spacing)
90
composite action between the cantilever brackets and the concrete bridge deck.
However, the composite behavior between the longitudinal girder and concrete bridge
deck was minimal as evidenced by the similar magnitudes of the strain-gage readings
attached to the top and bottom flanges during the live load test discussed in Section 6.1.
Figure 4-5: Close-up of connection between the longitudinal girder and the floor
beams and cantilever brackets.
4.1.2 Condition of Bridge and Characterization of Fatigue Details
The structural connections of interest at this bridge from a fatigue perspective are
the transverse stiffeners, riveted connections, and the welded connection between the
longitudinal girder and floor beams ((B) and (C) in Figure 4-5). Using the AASHTO
LRFD Specifications (2010), transverse stiffeners are characterized as Category C′ and
the riveted connection as Category D. As identified in Figure 4-5, a weld connects the
longitudinal girder to the floor beams and cantilever brackets. Depending on the
transition of the connection, that detail can be classified as Category B through Category
E. Because there is no transition between the floor beam and longitudinal girder
connection in the bridge, the detail is considered to be Category E.
(A) Cantilever bracket added to support increased deck from bridge widening
(C) Top flange of bracket is welded to top flange of longitudinal girder
(B) Weld added during bridge widening at top flange of floor beams to top flange of longitudinal girder
(D) Bottom flange of bracket is slotted and welded to the stiffener angle
(B) (C)
(A)(D)
Stiffener angle
Cantilever bracket
Floor beam
(E)
(E) Strain gage attached to the top flange of the longitudinal girder
(E)
(F)
(F) Shear stud along cantilever bracket
91
Past fracture-critical inspection reports at this bridge identified cracks at the welds
between the top flange of the longitudinal girder and floor beams (Figure 4-6). The
affected locations in the north span are shown in Figure 4-7. Cracks formed in the east
longitudinal girder at connections to floor beams 34, 35, 36, 37, and 38. In the west
longitudinal girder, cracks only formed at connections to floor beams 37 and 38. The
south span exhibited a similar cracking pattern: more cracks formed in the east girder
than the west girder and the cracks formed in the floor beams nearest the end of the
bridge.
Figure 4-6: Representation of crack between longitudinal girder and floor beam.
Figure 4-7: Location of cracks at welded connections between top flanges of
longitudinal girder and floor beams identified during fracture-critical inspections of
Bridge A.
Crack locations, typ.
North spanPartial center span
FB 35FB 34
North
East girder
West girder
Plan
Section
I I I I I II I I II I I II I I II I II I II
92
The cracks formed at connections between the longitudinal girders and floor
beams where cover plates were not present. Though increasing the girder depth in the
haunched sections results in an increase in the moment of inertia of the longitudinal
girder, the presence of cover plates causes a greater increase in the moment of inertia
between two floor beams. In the north span, cover plates start between floor beams 33
and 34, and the connection to floor beam 34 is the first location with cracks (Figure 4-7).
A crack formed at the connection of floor beam 34 although the applied moment from a
vehicle crossing the bridge was smaller at floor beam 34 as compared to floor beam 33.
The height of the web was approximately the same at these two floor beams; therefore
the presence of the cover plates is the likely explanation for the improved performance at
floor beam 33. The moment of inertia at floor beam 33 is approximately 1.5 times
greater than it is at floor beam 34 (Figure 4-8). The increased moment of inertia at floor
beam 33 reduces the stress range and increases the fatigue life for that particular
connection as compared to floor beam 34.
Figure 4-8: Longitudinal girder cross section of Bridge A at (a) floor beam 33 and
(b) floor beam 34.
4.1.3 Retrofit
Over the past ten years, the cracks at the locations identified in Figure 4-7 had
been growing. Instrumentation installed as part of this research revealed that the rate of
crack growth was very high and stress ranges from truck traffic were relatively high in
some locations of the bridge. Based upon the results from the field data, the bridge
6′-7½″ 6′-4½″in.4
in.4
FB 33 FB 34
Additional PL 18″x1/2″
(a) (b)
93
owner decided to retrofit the bridge to extend its service life so that sufficient time was
available to design and fabricate a replacement bridge.
Each girder was retrofitted at ten locations (Figure 4-9). The retrofit locations
corresponded to locations with no cover plates and where cracks had been identified.
The bridge owner chose to increase the section modulus by adding cover plates along the
top flange. With the increased section modulus, the stress range at critical locations was
expected to decrease.
Figure 4-9: Retrofit locations at Bridge A.
The retrofit consisted of the removal of a 3-ft by 3-ft region of the concrete bridge
deck above the connections between the longitudinal girders and the floor beams at all
locations where cover plates were not present (Figure 4-10). The removal of the concrete
deck provided a clear view of the crack profiles on the section. At several locations, the
crack extended across the entire width of one of the angles that formed the top flange of
the built-up longitudinal girder (Figure 4-11). The crack started at the weld between the
floor beam and longitudinal girder and grew until it reached the web of the longitudinal
girder. Because two angles formed the flanges, there was not continuity between the
flanges. As such, a crack could not extend across the entire width of the top flange.
North spanCenter spanSouth span
I I II I II I I III I IIIIIIII IIIII I III I I III III I
Retrofit locationsFB 34-FB 38FB 2-FB 6
94
Figure 4-10: Removal of concrete bridge deck above connections retrofitted.
Figure 4-11: Developed crack at connection between the floor beam and longitudinal
girder.
The retrofit used at the bridge is shown in Figure 4-12. The deck was removed at
the floor-beam intersections identified in Figure 4-9. Cover plates (3/4″ thick) were
added to the top and bottom face of the top flange and were connected by 1"-diameter,
high-strength bolts. Another plate was also added perpendicular to the cover plates (3/4″
thick) and connected to the floor beams for redundancy. By adding the cover plates, an
additional load path was created while providing redundancy in sections with cracks.
95
Figure 4-12: Photo of completed retrofit at connection between the longitudinal girder
and floor beam at Bridge A.
4.1.4 Instrumentation
A total of 36 strain gages were installed in three phases along the length of Bridge
A to monitor the live-load response. In phase one, sixteen gages were installed on the
east and west longitudinal girders near the floor beams identified in Figure 4-13. An
additional twelve gages were installed along the top flanges of floor beams 34 and 35.
The gages installed on the floor beams were used to understand the behavior of the floor
beams in transferring load between the longitudinal girders. Based on the response of the
strain gages, the forces in the floor beams were minor, and did not affect the fatigue
performance of the connections between the longitudinal girders and floor beams. As
such, many of the gages were removed from the data acquisition equipment in future
phases of the instrumentation, and are not discussed further in this dissertation. The
locations of the sixteen gages installed on the longitudinal girders during phase one of the
instrumentation are summarized in Table 4-1.
96
Figure 4-13: Location of strain gages. Floor beams are not shown for clarity.
Table 4-1: Summary of strain gages installed along the longitudinal girders at
Bridge A during phase one of the instrumentation.
Girder Floor beam
Position relative to FB
Number of gages
Description
East 27s 2′-0″ south 1 East side of the top flange East 27s 2′-0″ south 1 East side of the bottom flange East 34s 2′-0″ south 2 Each side of the top flange East 35n 2′-0″ north 2 Each side of the top flange East 35n 2′-0″ north 2 Each side of the bottom flange East 35s 2′-0″ south 2 Each side of the top flange West 34s 2′-0″ south 2 Each side of the top flange West 35s 2′-0″ south 2 Each side of the top flange West 35s 2′-0″ south 2 Each side of the bottom flange
In the next two phases of instrumentation, additional gages were installed along
the longitudinal girders. Phase two corresponded to a period before the live load test
(Chapter 6), during which two gages were installed on the longitudinal girder near the
connection at floor beam 27 (Table 4-2). In the final phase of instrumentation, gages
were added to the longitudinal girders and retrofit plates to track the performance of the
east longitudinal girder after the construction of the retrofit. The four strain gages
installed along the east longitudinal girder at floor beam 27 were removed from the data
acquisition system to free up channels and because floor beam 27 was far enough away
from the retrofit to be unaffected. A summary of the additional gages installed at the
bridge during phase three are listed in Table 4-3.
Hanger
North spanCenter spanSouth span
37′-6″
30′-0″
FB 35
FB 27
FB 33
44′-0″
90′-0″FB 34
97
Table 4-2: Summary of strain gages installed along the longitudinal girders at
Bridge A during phase two of the instrumentation.
Girder Floor beam
Position relative to FB
Number of gages
Description
East 27s 2′-0″ south 1 West side of the top flange East 27s 2′-0″ south 1 West side of the bottom flange
Table 4-3: Summary of strain gages installed along the longitudinal girders at
Bridge A during phase three of the instrumentation.
Girder Floor beam
Position relative to FB
Number of gages
Description
East 33n 2′-0″ north 2 Each side of the top flange East 34s 6″ south 2 Each side of the retrofit cover plate East 34 Centerline 2 Each side of the bottom flange
A CompactRIO was used for the data acquisition system (Section 4.5.1). To
measure the nominal response of the girder, the majority of the gages were installed 2 ft
away from the centerline of the floor beams (Figure 4-14). All longitudinal gages were
installed within 1″ of the exterior edge of each side of the flange. Because some of the
cracks at Bridge A were growing, crack propagation gages were also installed along with
the strain gages. Crack propagation gages were installed at cracks near floor beams 34
and 38 (Figure 4-15).
Figure 4-14: Strain gage located on the top flange of the longitudinal girder of
Bridge A approximately 2 ft away from the connection of the floor beam.
98
Figure 4-15: Crack propagation gage installed at tip of active crack at floor beam 34
of Bridge A.
The nomenclature used to identify the strain gages is Girder - Floor beam Position
- Flange. Thus, E-35n-TW is a gage installed approximately 2 ft north of floor beam 35
on the west side of the top flange of the east longitudinal girder.
4.1.5 Calculated Fatigue Response of Bridge A
The east longitudinal girder was modeled as a line girder using SAP2000 (v 15) to
calculate the fatigue response of the bridge. A line analysis was performed using the
standard AASHTO fatigue truck (Figure 2-16) and the live load distribution factors
defined in the AASHTO LRFD Specifications (2010). Following the procedures in the
LRFD Specifications, the live load distribution factors (LLDF) were determined using the
lever rule for the lanes identified in Figure 4-4. The factors are listed in Table 4-4.
Table 4-4: Live load distribution factors for Bridge A
Girder Location of vehicle
Left lane Right lane West 0.74 0.13 East 0.26 0.87
The moment of inertia of the longitudinal girder varies along the length of the
bridge due to the variable depth of the girder and addition of cover plates. Though there
are shear studs along the cantilever brackets, there are not enough shear studs to create
full composite action between the concrete deck and steel girders. As such, a non-
99
composite section was modeled. To simplify modeling the changes in the moment of
inertia, the bridge was divided into segments ranging from 2 ft to 10 ft in length. Each
segment had a moment of inertia that was based on the average depth along the length of
the segment considered. The variation in moment of inertia used for the line-girder
analysis is shown in Figure 4-16. The larger jumps represent changes due to cover plates.
Although the actual variation of the moment of inertia is smoother between the jumps in
Figure 4-16 because the depth of the section varies continuously along the length, the
model provides a reasonable representation of the stiffness of the longitudinal girder.
The calculated moment envelope of an individual girder due to the passage of the
full axle load of an AASHTO fatigue truck is shown in Figure 4-17. The end spans
experience both positive and negative moments, whereas only negative moments develop
in the cantilevers and only positive moments develop within the suspended span. The
largest moment occurs over the supports and the maximum range in moments occurs
within the end spans. The calculated moment range for a particular location is the
difference between the maximum and minimum moments. For example, for a strain-gage
location in the north span, the maximum moment occurs when the truck is in the north
span while the minimum moment occurs when the truck is in the center span. The
moment range is the difference between those two moments.
Figure 4-16: Variation of moment of inertia along the length of Bridge A used for the
line-girder analysis.
0
50,000
100,000
150,000
200,000
250,000
Mo
men
t o
f In
erti
a (i
n.4
)
Distance from center line of south support (ft)73.5 198.5167.9104.10.0 272.0
100
Figure 4-17: Moment envelope from a line analysis of an AASHTO fatigue truck
crossing a single girder at Bridge A.
Because the moment of inertia varies along the length, the maximum stress range
does not necessarily occur at the location of the maximum variation in the live-load
moment. The flexural stress range, Δ , can be calculated using Equation 4-1. The
distance to the neutral axis is the same for the top and bottom flanges because a non-
composite section was assumed for the analysis and the section is doubly symmetric.
The increase in static loads due to dynamic effects (IM) was assumed to be 15% for the
bridge per AASHTO LRFD Specifications (2010).
Δ Δ
Equation 4-1
where
Δ momentrangeinthelongitudinalgirder kip‐in.
distancefromtheneutralaxistotheextremefiberofthecrosssection in.
momentofinertiaofthecrosssection in.4
dynamicimpactfactor 1.15
liveloaddistributionfactor
-1,200
-800
-400
0
400
800
Mo
men
t (k
ip-f
t)
Distance from center line of south support (ft)73.5 198.5167.9104.10.0 272.0
1360 kip-ft
+
Sign convention
-
101
The variations in stress range for an AASHTO fatigue truck in the left lane and
right lane are presented in Figure 4-18. Due to the greater live load distribution factor,
the stress range is larger when the truck is in the right lane. The maximum calculated
stress range from an AASHTO fatigue truck is 16 ksi in the east girder and 14 ksi in the
west girder. The maximum stress range corresponds to a location without cover plates
and is near floor beam 34 in the north span. All vehicles cause fatigue damage to both
longitudinal girders.
Figure 4-18: Calculated stress ranges for each girder due to an AASHTO fatigue truck
crossing Bridge A in the (a) left lane and (b) right lane.
Using the calculated variation in stress range along the length of the bridge, the
stress range from an AASHTO fatigue truck can be estimated at the locations of the strain
gages. Table 4-5 summarizes the stress ranges for five connections between the
longitudinal girders and floor beams. Because the fatigue truck represents an average
truck crossing the bridge, the largest expected stress range from a single truck is double
the values in Figure 4-18 (AASHTO Manual for Bridge Evaluation 2011).
Distance from center line of south support (ft)
0
4
8
12
16
Stre
ss r
ange
(ks
i)
Distance from center line of south support (ft)
73.5 198.5167.9104.10.0 272.0 73.5 198.5167.9104.10.0 272.0
AASHTO fatigue truck in left lane AASHTO fatigue truck in right lane
West girder
East girder
East girder
West girder
(a) (b)
102
Table 4-5: Calculated stress range at locations of strain gages used to monitor
Bridge A due to the AASHTO fatigue truck.
Girder Floor beam Location of AASHTO fatigue truck
Left lane Right lane East 27 1.0 ksi 3.0 ksi East 34 4.8 ksi 16.0 ksi East 35 4.7 ksi 15.5 ksi West 34 13.6 ksi 2.4 ksi West 35 13.3 ksi 2.4 ksi
4.2 BRIDGE B
Bridge B is a trapezoidal box girder bridge that was constructed over 10 years ago
and is part of a horizontal direct connector between two highways (Figure 4-19). The
bridge is considered to be fracture critical because it is a twin-girder system. The direct
connector features a single lane for traffic, an angle change of 90 degrees, elevation
change of 40 ft, and a maximum allowed speed limit of 55 miles per hour.
Figure 4-19: Bridge B.
4.2.1 Geometry
Bridge B comprises four spans and connects two highways. The box girders are
spaced 15.7 ft apart with span lengths of 210 ft, 230 ft, 230 ft, and 210 ft (Figure 4-20).
The first span and a portion of the second span (~100 ft) are straight before the remaining
103
sections of the bridge feature a horizontal radius of curvature of approximately 458 ft
(center of the bridge). Steel-plate diaphragms are used to control distortion at each
intermediate support. Within each box, the plate diaphragms have an access port (1′-4″ x
2′-8″) to allow inspectors to move along the length of the girder.
The typical cross section for the bridge is presented in Figure 4-21. The width of
the bridge is 30 ft (including the rails), with a lane width of 14 ft and shoulders of 6 to 8
ft on each side of the bridge. The web plates are 6.5-ft deep (vertically) and the bottom
flanges are 59-in. wide. For the majority of the bridge, there is a 24-in.-wide flange at the
top of each web. Though over the third interior support, 28-in.-wide flanges are used to
increase the section modulus. Shear studs are attached to each flange to create composite
action between the concrete deck and steel box girders.
Figure 4-20: Plan view of Bridge B (internal diaphragms are not shown).
The plate thickness of the top and bottom flanges varies along the length of the
bridge. For the majority of the bridge, the thickness of the flanges is between ¾″-1″.
However, at the interior supports, thicker plates (1¾″-2¾″) are used to increase the
section modulus.
210′-0″ (span 1)
Steel plate diaphragm
North
West girder
East girder
Direction of traffic
104
Figure 4-21: Bridge cross section and location of lane at Bridge B (looking north).
4.2.2 Instrumentation and Characterization of Fatigue Details
Strain gages were installed on the bottom flange in the middle of each of the
spans (Figure 4-22). Both the CompactRIO (Section 4.5.1) and WSN (Section 4.5.2) data
acquisition systems were used at this bridge. At each location, two gages were attached
to the bottom flange, symmetrically about the mid-width of the flange. Each gage was
located 8 in. from the middle of the bottom flange.
Figure 4-22: Gage locations at Bridge B (internal diaphragms are not shown).
6′-6″
4′-7″ 4′-7″
Access port in steel plate diaphragm
11′-1″
Steel trapezoidal box girder
30′-0″
14′-0″(roadway)
8′-0″ (shoulder)
6′-0″ (shoulder)1′-0″ 1′-0″
4′-10½″4′-10½″
115′-0″ 105′-0″
Gage locations
North
105
The nomenclature used to identify the gages is Girder - Span - Flange. So, W-1-
BE is a gage installed at the middle of span 1 of the west longitudinal girder on the east
side of the bottom flange.
There are three structural connections of interest at this bridge from a fatigue
perspective: transverse stiffeners, flange-flange welds (cover plates), and web-flange
welds. According to the AASHTO LRFD Specifications (2010), the transverse stiffeners
are characterized as Category C′ and the welds (flange-flange and web-flange) as
Category B. The transition to a thicker flange near the interior piers (where a thicker
flange is needed to increase the section modulus) is gradual, such that the flange
transition is classified as Category B. As such, the transverse stiffener is the critical
fatigue detail.
4.2.3 Calculated Fatigue Response of Bridge B
The response of the bridge from the standard AASHTO fatigue truck (Figure 2-
16) was calculated using a line-girder analysis in SAP2000 (v 15). The west longitudinal
girder was modeled with the horizontal curve to obtain the flexural and torsional
moments from the fatigue truck crossing the bridge. Following the procedures in the
AASHTO LRFD Specifications (2010), the lever rule was used to determine the live load
distribution factors (Table 4-6) for the lane identified in Figure 4-21.
Table 4-6: Live load distribution factors for Bridge B.
Girder Live load
distribution factor
West 0.564 East 0.436
The moment of inertia varies along the length of the bridge as a function of
changes in the thicknesses of the flanges. Full composite action was assumed across the
entire length of the bridge. Although the concrete deck is primarily in tension over the
interior supports (negative moment), the service loads are sufficiently low that cracking
of the concrete deck is not expected. Therefore, the composite section was assumed to
106
contribute towards the stiffness along the entire length of Bridge B. The variation in
moment of inertia used for the line-girder analysis is shown in Figure 4-23.
Figure 4-23: Variation of moment of inertia for Bridge B used for the line analysis.
The calculated moment envelope due to the passage of an AASHTO fatigue truck
is shown in Figure 4-24. Due to the horizontal curve, the truck induces both flexural and
torsional moments. At the interior supports, the largest flexural moment is negative
(tension in the top flange), whereas in the middle of the spans, the largest flexural
moments are positive (tension in the bottom flange). For a given gage location, the
flexural moment is positive when the truck is in the span with the gage, and negative
when the truck is in an adjacent span. The greatest range in flexural moment (2,710 kip-
ft) occurs in the third span, near the middle of the span. The torsional moment only
occurs in the curved portion of the bridge. The greatest range in the torsional moment
occurs near the last interior pier (between the third and fourth spans).
The stress range induced by trucks represents a combination of flexural (Δ )
and torsional (Δ ) stresses. The combined effect can be determined by summing the
flexural impact (Equation 4-1) and torsional impact (Equation 4-2). A dynamic impact
factor (IM) of 1.15 was assumed for this bridge. Equation 4-2 is only for shapes that have
closed geometries, such as the box girder.
Δ Δ2
Equation 4-2
where
0
200,000
400,000
600,000
800,000
Mo
men
t o
f in
teri
a (i
n.4
)
Distance from center line of south support (ft)880210 6704400
107
Δ momentrangeinthelongitudinalgirderduetotorsionkip‐in.
thicknessofboundingmaterial in.
areaenclosedbyshape in.2
Figure 4-24: Moment envelope from a line analysis of an AASHTO fatigue truck
crossing a single girder at Bridge B.
The variation in stress range for an AASHTO fatigue truck in the top and bottom
flanges are presented in Figure 4-25. Due to a slightly larger live load distribution factor,
the stress range is highest in the west girder. There is a difference in the stress ranges for
the top and bottom flanges because the centroid for the composite section is generally
closer to the top flange than the bottom flange. As such, the bottom flange experiences
larger stress ranges than the top flange (4 ksi vs. 1 ksi). Due to the effectiveness of the
box girder at resisting torsion, the torsional stress range contributes a maximum of 0.3 ksi
to the total stress range along the length of the bridge.
The stress range from an AASHTO fatigue truck can be estimated at the locations
of the strain gages. For the center of each span, the stress ranges are summarized in
Table 4-7. Because the fatigue truck represents the average assortment of trucks that
might cross the bridge, the largest expected stress range from a single truck is twice the
values in Figure 4-25.
-1,800
-1,200
-600
0
600
1,200
1,800
2,400
Mo
men
t (k
ip-f
t)
Distance from center line of south support (ft)
880210 6704400
2,710 kip-ft
Moment due to flexure
Moment due to torsion
+
Sign convention
-
108
Figure 4-25: Calculated stress ranges in the (a) top flange and (b) bottom flange for
each girder due to an AASHTO fatigue truck crossing Bridge B.
Table 4-7: Calculated stress range at locations in the west girder of Bridge B due to the
AASHTO fatigue truck.
Span Stress range at bottom flange due to the
AASHTO fatigue truck (ksi)
1 2.9 2 3.1 3 3.8 4 2.7
4.3 BRIDGE C
Bridge C was constructed over 50 years ago and is part of a state highway (Figure
4-26). The bridge is not considered to be fracture critical because there are six girders
across the width. The bridge is for one-way traffic, with two lanes for traffic along the
highway and a lane for vehicles exiting the highway.
Distance from center line of south support (ft)
0
1
2
3
4
5St
ress
ran
ge (
ksi)
Distance from center line of south support (ft)
880 210 6704400 880
Stress range in bottom flange
East girder
West girder
(a) (b)
210 6704400
Stress range in top flange
West girder
East girder
109
Figure 4-26: Bridge C.
4.3.1 Geometry
Bridge C has a length of 210 ft, featuring spans lengths of 35 ft, 47.5 ft, 35 ft, 57.5
ft, and 35 ft. The five-span continuous bridge is non-composite and has skewed supports
(20°40′) due to traffic beneath the bridge. The span distances shown in Figure 4-27 are
measured along the centerline of the bridge. Traffic travels in only one direction (west to
east) at the three-lane bridge (two lanes are highway and one lane is for an exit ramp).
Figure 4-27: Plan view of Bridge C (cross frames are not shown).
The typical cross section for the bridge is shown in Figure 4-28. The total width
of the bridge is 49.5 ft, with three lanes that are 12-ft wide and sidewalks on each side of
the bridge. The longitudinal girders are rolled sections (W27x102) that are spaced 8′-4″
35′-0″Span 1
47′-6″Span 2
35′-0″Span 3
57′-6″Span 4
35′-0″Span 5
Girder 6
Girder 5
Girder 4
Girder 1Girder 2
Girder 3
NorthDirection of traffic
110
apart. Cross frames are spaced 16′-3″ on center to provide lateral stability. Cover plates
were added near the interior supports to increase the section modulus.
Figure 4-28: Bridge cross section and location of lane at Bridge C (looking east).
4.3.2 Instrumentation and Characterization of Fatigue Details
A total of thirty-two strain gages were installed on three girders at three locations
along the length of the bridge (Figure 4-30). The WSN (Section 4.5.2) data acquisition
system was used at this bridge. Gages were installed within 1″ of the exterior edge of
each side of the flange (Figure 4-29).
Figure 4-29: Strain gages installed within 1″ of the exterior edge of the flange.
The structural connections of interest at this bridge from a fatigue perspective are
the flange-flange welds (cover plates) and transverse stiffeners. The flange-flange welds
and transverse stiffeners are respectively classified as Category B and Category Cʹ by the
AASHTO LRFD Specifications (2010). As such, the transverse stiffener is the critical
fatigue detail for where the strain gages are attached to the longitudinal girders. Because
5 spa. @ 8′-4″3′-4″ 4′-6″
3′-6″
1′-0″
6′-0″
1′-0″
12′-0″(left lane)
Girder 1
North South
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
12′-0″(right lane)
12′-0″(exit lane)
2′-0″W27x102, typ.
111
of the details used for the ends of the cover plates, those details are classified as Category
E′. However, no strain gages were installed near the ends of the cover plates.
Figure 4-30: Gage locations at Bridge C.
The nomenclature used to identify the gages is Location - Girder - Flange. So,
L1-3-BN is a gage installed at location 1 along girder 3 on the north side of the bottom
flange. The gage locations are summarized in Table 4-8.
Table 4-8: Summary of strain gages installed at Bridge C
Location Girder Number of gages
Description
1 1 4 Each side of the top and bottom flanges 1 3 4 Each side of the top and bottom flanges 1 5 2 Each side of the top flange 2 1 4 Each side of the top and bottom flanges 2 3 4 Each side of the top and bottom flanges 2 5 2 Each side of the top flange 3 1 4 Each side of the top and bottom flanges 3 3 4 Each side of the top and bottom flanges 3 5 4 Each side of the top and bottom flanges
4.3.3 Calculated Fatigue Response of Bridge C
One of the longitudinal girders was modeled as a line girder using SAP2000 (v
15) to calculate the response of the standard AASHTO fatigue truck (Figure 2-16). From
the moment envelope, the stress range from the fatigue truck can be calculated using
Equation 4-1. The increase in static loads due to dynamic effects ( ) was assumed to be
15% for the bridge per the AASHTO LRFD Specifications (2010). For bridges with
25′-0″Loc. 1
Girder 6
Girder 5
Girder 4
Girder 1Girder 2
Girder 3
17′-6″Loc. 2
20′-0″Loc. 3North
112
more than three girders, the live load distribution factor can be calculated using either the
lever rule or the empirical equations in the AASHTO LRFD Specifications (2010). The
live load distribution factors are summarized in Table 4-9 for cases in which the lever
rule is used for the trucks positioned in the center of the lanes identified in Figure 4-28.
When the lever rule is used, the method assumes that the entire load is supported by three
girders.
Table 4-9: Live load distribution factors for Bridge C based on lever rule.
Girder Location of vehicle
Left lane Right lane Exit lane 1 0.1 - - 2 0.64 - - 3 0.26 0.4 - 4 - 0.6 0.16 5 - 0 0.64 6 - - 0.2
AASHTO Table 4.6.2.2.2b-1 in the LRFD Specifications can be used to calculate
the live load distribution factor ( ) for interior girders for bridges with multiple
girders. If this method is used, the distribution between all of the girders is not known,
instead only the maximum distribution factor for a single girder is calculated. The
distribution factor can be used for all of the interior girders because the load is assumed
to be transferred efficiently amongst multiple girders. Because traffic lanes can be
moved with time and vehicles are not always in the center of the lane, the distribution
factor does not depend on the location of the lane. The appropriate equation for steel
girders with a concrete bridge deck is presented below (Equation 4-3). The equation in
the AASHTO LRFD Specifications assumes a multiple presence factor of 1.2. In fatigue,
the presence of a single truck is desired, which is why the factor was removed below.
The parameters of the equation are within the range of applicability listed in the
AASHTO LRFD Specifications (2010).
113
11.2
0.0614
. .
12
.
Equation 4-3
where
distancebetweenlongitudinalgirders ft
length of span (ft)
thicknessofconcretedeck in.
(in.4)
modulusofelasticityofsteelbeam ksi
modulusofelasticityofconcretedeck ksi
momentofinertiaofsteelbeam non‐composite in.4
areaofsteelbeam non‐composite in.2
eccentricitybetweencentroidofbeamandconcretedeck in.
When the Guide Specification for Fatigue Evaluation of Existing Steel Structures
(AASHTO 1990) was developed, a different method for calculating the distribution factor
( ) was used for bridges with multiple girders (Equation 4-4).
3
Equation 4-4
empiricalfactorthatdependsonspanlength
The calculated distribution factors using both methods are summarized in Table
4-10. The distribution factors depend on the span length; thus, the values are different for
positive and negative moments. The values for each method ( and ) are
comparable for each location. Although, the difference between the lever rule
(maximum: 0.64) and (maximum: 0.471) is considerable. Between spans,
varies between 0.396 and 0.471. Because is used in the current specification, only
those values, along with the lever rule, were used in subsequent calculations in this
dissertation.
114
Table 4-10: Live load distribution factors for Bridge C based on AASHTO.
Span Distribution factor for interior girder
1 (positive moment) 0.471 0.463
1-2 (negative moment) 0.452 0.437
2 (positive moment) 0.423 0.430
2-3 (negative moment) 0.423 0.430
3 (positive moment) 0.423 0.430
3-4 (negative moment) 0.442 0.431
4 (positive moment) 0.396-0.403 0.419
4-5 (negative moment) 0.442 0.431
5 (positive moment) 0.471 0.463
The calculated moment envelope due to the passage of an AASHTO fatigue truck
is shown in Figure 4-31. Different sections in SAP2000 were used to represent the
different moment of inertias in the bridge due to the presence of cover plates. All
sections were symmetric and were modeled with a non-composite deck. All of the spans
experience both positive and negative moments. For a given gage location, the flexural
moment is positive when the truck is in the span with the gage, and negative when the
truck is in the adjacent span. The largest range in flexural moment (275 kip-ft) occurs in
the fourth span, near the middle of the span.
Figure 4-31: Moment envelope from a line analysis of an AASHTO fatigue truck
crossing a single girder at Bridge C.
-300
-200
-100
0
100
200
300
Mo
men
t (k
ip-f
t)
Distance from center line of west support (ft)
35 175117.582.50 210
275 kip-ft
+
Sign convention
-
115
If the distribution factors from the lever rule are used, the variation in stress range
for an AASHTO fatigue truck in the left, right, and exit lanes can be determined (Figure
4-32). The maximum calculated stress range from an AASHTO fatigue truck using the
lever rule is 9.2 ksi and occurs in the last span. If the distribution factor from the
AASHTO LRFD Specifications is used (Table 4-10), then the maximum stress range is
less (6.8 ksi).
Figure 4-32: Calculated stress ranges for each girder due to an AASHTO fatigue truck
for crossing Bridge C in the (a) left lane, (b) right lane, and (c) exit lane.
Using the calculated variation in stress range along the length of the bridge, the
stress range from an AASHTO fatigue truck can be estimated at the locations of the strain
gages. Table 4-11 summarizes the stress ranges for the three locations. The stress ranges
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of west support (ft)
Distance from center line of west support (ft)
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of west support (ft)
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of west support (ft)
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of west support (ft)
Distance from center line of west support (ft)
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of west support (ft)
0 35 82.5 117.5 175 210
0 35 82.5 117.5 175 210
AASHTO fatigue truck in left lane and distribution factors from lever rule
0 35 82.5 117.5 175 210
(a) (b)
(c)
G2
G3 G1
G4
G3
G6
G5
G4
AASHTO fatigue truck in right lane and distribution factors from lever rule
AASHTO fatigue truck in exit lane and distribution factors from lever rule
116
listed are for distribution factors using the lever rule and the empirical equations. Girders
1 and 3 are not centered underneath the traffic lane, whereas Girder 5 is centered
underneath the traffic. As such, Girder 5 has a much higher expected stress range using
the lever rule as compared to Girders 1 and 3. The maximum stress range from a single
truck could be twice the magnitude of the values listed in Table 4-11 (AASHTO Manual
for Bridge Evaluation 2011).
Table 4-11: Calculated stress range at locations of Bridge C due to AASHTO fatigue
truck.
Location Maximum stress range due to AASHTO fatigue truck (ksi)
Girder 1† Girder 3† Girder 5† Interior* 1 1.0 (left) 2.7 (left), 4.6 (right) 7.4 (exit) 4.9 2 1.1 (left) 2.8 (left), 4.3 (right) 6.9 (exit) 4.6 3 1.4 (left) 3.7 (left), 5.7 (right) 9.2 (exit) 6.8
†Lever rule was used and fatigue truck is in the center of the lane listed in () *AASHTO LRFD (2010) was used to calculate the live load distribution factor.
4.4 BRIDGE D
Bridge D was constructed nearly 60 years ago and provides highway access
across a river (Figure 4-33). The bridge is not considered to be fracture critical because
there are seven girders across the bridge width. The bridge is for one-way traffic and
features four lanes for traffic along the highway.
Figure 4-33: Bridge D.
117
4.4.1 Geometry
Bridge D is a three-span, continuous, plate-girder bridge. The bridge is
symmetric about its centerline, with spans of 90 ft, 130 ft, and 90 ft (Figure 4-34).
Traffic travels along one direction (east to west) at the bridge. The thickness of the
flanges changes along the length to increase the section modulus over the interior
supports and in the middle of each span.
Figure 4-34: Plan view of Bridge D.
The typical cross section for the bridge is shown in Figure 4-35. The total width
of the bridge is 56.8 ft, with four, 12-ft traffic lanes. Each side of the bridge features
shoulders and sidewalks that are 2′-0″ and 2′-5″ wide, respectively. The plate girders are
64″ deep and spaced 8′-4″ apart. The flanges have a constant width of 16″ and a
thickness that varies at the interior pier and middle span for increased moment capacity.
The bridge is non-composite with cross frames that are spaced 22′-6″.
Figure 4-35: Cross section of Bridge D (looking west).
90′-0″ Span 1
130′-0″ Span 2
90′-0″ Span 3
Girder 1Girder 2
Girder 3Girder 4
Girder 5Girder 6
Direction of traffic
Girder 7
North
12′-0″(left lane)
2′-5″
South
2′-0″
6 spa. @ 8′-4″ 3′-5″
Girder 1
North3′-5″
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
2′-5″12′-0″(center-left lane)
12′-0″(center-right lane)
12′-0″(right lane)
2′-0″
118
4.4.2 Instrumentation and Characterization of Fatigue Details
A total of twenty-eight strain gages were installed on four girders at three
locations along the length of the bridge (Figure 4-36). Both the CompactRIO (Section
4.5.1) and WSN (Section 4.5.2) data acquisition systems were used at this bridge. Gages
were installed 4″ from the exterior edge of the flange. For Girder 3, gages were installed
4″ from each side of the flange.
Figure 4-36: Locations of gages at I-girder Bridge D.
The nomenclature used to identify the gages is Location - Girder - Flange. So,
L1-3-BN is a gage installed at location 1 along girder 3 on the north side of the bottom
flange. The gage locations are summarized in Table 4-12.
Table 4-12: Summary of strain gages installed at Bridge D
Location Girder Number of gages
Description
1 2 2 One gage on the top and bottom flanges 1 3 4 Each side of the top and bottom flanges 1 4 2 One gage on the top and bottom flanges 1 5 2 One gage on the top and bottom flanges 2 3 4 Each side of the top and bottom flanges 2 4 2 One gage on the top and bottom flanges 2 5 2 One gage on the top and bottom flanges 3 2 2 One gage on the top and bottom flanges 3 3 4 Each side of the top and bottom flanges 3 4 2 One gage on the top and bottom flanges 3 5 2 One gage on the top and bottom flanges
Based on where strain gages were installed on the longitudinal girders, the
structural connections of interest at this bridge from a fatigue perspective are the
54′-0″ Loc. 3
65′-0″ Loc. 1
20′-0″ Loc. 2
Girder 1Girder 2
Girder 3Girder 4
Girder 5Girder 6
Girder 7
North
119
transverse stiffeners, flange-flange welds (cover plates), and web-flange welds.
According to the AASHTO LRFD Specifications (2010), the transverse stiffeners are
characterized as Category C′ and the welds (flange-flange and web-flange) as Category
B. As such, the transverse stiffener is the critical fatigue detail for where the strain gages
are attached to the longitudinal girders. Because of the details used for the ends of the
cover plates, those details are classified as Category E′. However, no strain gages were
installed near the ends of the cover plates.
4.4.3 Calculated Fatigue Response of Bridge D
A line analysis of one of the longitudinal girders was performed using SAP2000
(v 15) to calculate the moment envelope from a standard AASHTO fatigue truck (Figure
2-16). The stress range can be calculated using Equation 4-1 and a dynamic impact
factor ( ) of 15%. The live load distribution factor can be computed from either the
lever rule or the AASHTO empirical equations (Equation 4-3 and Equation 4-4). The
center of each line identified in Figure 4-35 was used to calculate the distribution factors
using the lever rule (Table 4-13). The distribution factors from the empirical equations
are summarized in Table 4-14.
Table 4-13: Live load distribution factors for Bridge D based on lever rule.
Girder Location of vehicle
Left lane Center-left
lane Center-right
lane Right lane
1 0.26 - - - 2 0. 64 - - - 3 0.1 0.04 - - 4 - 0.64 0.32 - 5 - 0.32 0.64 0.1 6 - - 0.04 0.64 7 - - - 0.26
120
Table 4-14: Live load distribution factors for Bridge D based on AASHTO.
Span Distribution factor for interior girder
1 (positive moment) 0.429 0.379
1-2 (negative moment) 0.427 0.368 2 (positive moment) 0.400 0.362
2-3 (negative moment) 0.427 0.368 3 (positive moment) 0.429 0.379
The calculated moment envelope due to the passage of an AASHTO fatigue truck is
shown in Figure 4-37. Different sections in SAP2000 were used to represent the different
moment of inertias in the bridge due to the changes in values of the flange thickness.
Though, all sections were symmetric and were modeled with a non-composite deck. All
of the spans experience both positive and negative moments. For a given gage location,
the flexural moment is positive when the truck is in the span with the gage, and negative
when the truck is in the adjacent span. The largest range in flexural moment (1,000 kip-
ft) occurs in the center span, near the middle of the span.
Figure 4-37: Moment envelope from a line analysis of an AASHTO fatigue truck
crossing a single girder at Bridge D.
If the distribution factors from the lever rule are used, the variation in stress range
for an AASHTO fatigue truck in each lane can be computed (Figure 4-38). The
-900
-600
-300
0
300
600
900
Mo
men
t (k
ip-f
t)
Distance from center line of east support (ft)90 220
1,000 kip-ft
0 310
+
Sign convention
-
121
maximum calculated stress range from an AASHTO fatigue truck using the lever rule is
8.3 ksi and occurs in the end spans. If the distribution factor from AASHTO is used
(Table 4-14), then the maximum stress range is less (5.6 ksi).
Figure 4-38: Calculated stress ranges for each girder due to an AASHTO fatigue truck
for crossing Bridge D in the (a) left lane and (b) left-center lane.
Using the calculated variation in stress range along the length of the bridge
(Figure 4-38), the stress range from an AASHTO fatigue truck can be estimated at the
locations of the strain gages. The stress ranges for the three gage locations and different
distribution factors (lever rule and empirical equations) are summarized in Table 4-15.
Location 3 corresponds to the highest stress range for the gage locations. The maximum
stress range from a single truck could be twice the magnitude of the values listed in
Figure 4-38 (AASHTO Manual for Bridge Evaluation 2011).
Distance from center line of east support (ft)
0
2
4
6
8
10
Stre
ss r
ange
(ks
i)
Distance from center line of east support (ft)
AASHTO fatigue truck in left lane and distribution factors from lever rule
(a) (b)
G2
G3 G1
AASHTO fatigue truck in center-left lane and distribution factors from lever rule
G2
G3
G4
The shape of the curve for center-right lane will be similar to (b) and the curve for the right lane will be similar to (a)
122
Table 4-15: Calculated stress range at locations of Bridge D due to AASHTO fatigue
truck.
Location Maximum stress range due to AASHTO fatigue truck (ksi)
Girder 2† Girder 3† Girder 4† Girder 5† Interior*
1 6.6 (left) 6.6 (left-center)
3.3 (left-center)
5.5 (right-center) 4.1
2 4.0 (left) 4.0 (left-center)
2.0 (left-center)
4.0 (right-center) 2.5
3 8.0 (left) 8.0 (left-center)
4.0 (left-center)
8.0 (right-center) 5.4
†Lever rule was used and fatigue truck is in the center of the lane listed in () *AASHTO LRFD Specifications (2010) was used to calculate the live load distribution factor.
4.5 DATA ACQUISITION SYSTEM & SENSORS
Two types of data acquisition systems from National Instruments (NI) were used
to obtain dynamic strain data at the four bridge sites. The wired (CompactRIO) and
wireless (WSN) systems were used to determine the fatigue cycles from vehicles crossing
the bridge using the simplified rainflow algorithm, and the resulting distribution of stress
ranges were used to calculate the remaining fatigue life. Strain gages were the primary
type of sensor used at the bridge sites. Strains were converted to stresses using Hooke’s
Law and a modulus of elasticity for steel of 29,000 ksi.
4.5.1 Wired System - CompactRIO
The NI CompactRIO is a wired system that is capable of high-resolution,
configurable data acquisition and control. For the bridge sites, NI-9237 modules were
combined with the CompactRIO to acquire high-speed quarter-, half-, and full-bridge
measurements. Each module supports four channels and each channel has a separate, 24-
bit analog-to-digital converter (ADC). The module utilizes field-programmable gate
arrays (FPGA) to produce hardware-timed measurements that can be synchronized across
modules. Using the internal master timebase, the module supports sampling rates
123
between 1,613 Hz and 50,000 Hz. A 2.5-V excitation was used along with 350-Ω
quarter-bridge completion resistors.
Figure 4-39: CompactRIO data acquisition system with NI-9237 modules (wired).
4.5.1.1 Programming
With the onboard processor (533-800 MHz) and nonvolatile memory (2-4 GB),
each CompactRIO was programmed for data acquisition and analysis. Each program was
created in the LabVIEW Real-Time (RT) environment. To handle the fast data
acquisition, the FPGA captured hardware-timed data at 2,000 Hz and transferred the data
to the RT environment. In the RT environment, data were downsampled to 50 Hz and
processed using a custom simplified rainflow program. Due to the size of the processor
and memory, other programs can run simultaneously on the CompactRIO. As such, the
raw strain history was also saved at some bridges, while only events greater than a
specified threshold were saved at other bridges.
4.5.2 Wireless System - Wireless Sensor Networks (WSN)
The other data acquisition system employed at the bridge sites was the Wireless
Sensor Network (WSN) platform from NI (Figure 4-40). With such a wireless system,
sensors are connected to nodes that acquire data and send the data wirelessly to a
gateway/computer where it can be stored temporarily or permanently. The gateway
creates the wireless network that enables the nodes to transmit data. The WSN platform
utilizes a narrow-band protocol (IEEE 802.15.4) to transmit data. The protocol is well-
suited for bridge deployments because it is lower power than other broader-bandwidth
protocols (IEEE 802.11). In addition, narrow-bandwidth protocols performed better
124
(longer distances and more consistent communication links) in steel-bridge environments
due to the presence of multi-path effects (Fasl, et al. 2012b).
Figure 4-40: Wireless strain node data acquisition system (WSN-3214).
A specialized gateway, which is connected to a computer or a controller, is
required to create the wireless network. Nodes connect to the gateway and send data
across the network as either mesh routers (always awake) or end nodes (only awake when
scanning and sending data). Because mesh routers are always on, they are not considered
to be low-power devices. However, the mesh routers offer a powerful capability to the
network. Mesh routers increase the redundancy of the network and increase the area that
can be monitored because they can transfer data from end nodes or other routers back to
the gateway (Figure 4-41).
Figure 4-41: Typical network configuration for bridges.
The wireless strain node, WSN-3214, is capable of capturing dynamic data from
four analog channels. A variety of sensors can be used, including quarter-bridge (built-in
350-Ω and 1,000-Ω completion resisters), half-bridge, and full-bridge inputs with a 2.0-V
Gateway
Router
End Nodes
125
bridge excitation. The embedded, Texas Instrument MSP430 microprocessor has the
ability to run LabVIEW WSN graphical programs, thereby enabling onboard data
processing at the node. Using LabVIEW WSN, the strain node can be easily customized
for a particular application. The node also features built-in shunt calibration and remote
sensing to reduce lead-wire errors and 2-MB onboard memory buffers for data storage.
Four internal AA batteries or an external DC power source, such as an energy harvester,
can be used to power the node.
The strain node minimizes electrical noise while maintaining low-power
consumption through a robust ADC design. Because low noise often means more power
and low power often means more noise, a configurable ADC was chosen for the strain
node. Successive-approximate (SAR) converters and delta-sigma converters are the two
most commonly-used ADC in data acquisition applications. SAR converters work well
in multiplexed systems, whereas delta-sigma converters provide high resolution through
oversampling and digitally filtering the analog signal. To deliver the optimum
combination of resolution, low-noise performance, speed, and low-power operation, the
node utilizes a multiplexed, oversampled SAR ADC with digital filtering performed by
an onboard FPGA (fully programmable gate array). The FPGA also enables fast
sampling (1 to 2,000 Hz) without overwhelming the microprocessor. The result is
measurements that are hardware-timed with 20-bit ADC resolution. To balance the
tradeoffs of noise filtering, power consumption, and speed, the user can choose the
measurement time (aperture time) or one of the 50-Hz/60-Hz filtering options.
4.5.2.1 Programming
With a wireless system, data transfer is a primary concern, especially in low-
bandwidth systems. Often, the most power-hungry component in a wireless node is the
radio (Lynch, et al. 2004). By performing a custom analysis with the microprocessor of
each node and only sending the analysis results, the amount of data transmitted can be
reduced. As such, power is saved while not overwhelming the limited bandwidth of the
IEEE 802.15.4 protocol.
126
The impact to the bandwidth of the wireless network from performing an analysis
on the node as compared to the gateway can be imagined by considering the example of a
rainflow analysis. The node might be configured to acquire data at 50 samples per
second to capture the dynamic effects of trucks crossing the bridge. In a 30-minute
period, over 90,000 samples per channel (360,000 samples per node) would need to be
sent over the air if the gateway were to perform the analysis. Assuming no loss of data
and maximum efficiency of the network, approximately ten nodes would cause the
bandwidth of the IEEE 802.15.4 network to be exceeded. As the efficiency of the
network decreased and loss of data occurred, the number of nodes allowed on the
network would also decrease. In contrast, by embedding the program on the node and
performing the analysis in real-time, the number of data points sent over the air could be
reduced to the results of the analysis (a couple hundred samples per channel in a 30-
minute period). Because the engineer or bridge owner can use the results directly, no
information is lost by sending only the results of the analysis, and there would still be
some bandwidth to transmit triggered strain histories (only data above a threshold).
A custom program was developed to allow each strain node to be configured over
the air and operate in a variety of modes of operation. By creating string messages that
are sent by the gateway over the air to the node, a user can configure the scan period,
sampling rate, number of samples, aperture time, channel configuration (quarter bridge,
half bridge, and full bridge), gage factor, and parameters related to the rainflow analysis.
A sub-program was added to the node that can parse the string into configuration
parameters. Thus, the node can be configured for a slow sampling rate for non-critical
times and switched to a faster scan rate when dynamic data are desired. The custom
program has five distinct modes of operation for the node, depending on the application
need: no data, streaming data (data sent back each scan period), triggered data, rainflow
analysis (for counting the number and amplitude of fatigue cycles), and rainflow analysis
plus triggered data. The rainflow algorithm is based on the method described in
Downing and Socie (1982).
127
When multiple nodes are connected to a single gateway, configuring all of the
channels for the rainflow parameters and scan properties can be challenging. A
LabVIEW program was created to automate the process of updating all of the nodes. The
user must first create a configuration file. The configuration file has to be formatted in a
specific manner; thus, a LabVIEW VI was developed to assist with creation of the
configuration file (nodes connected to the gateway are auto-detected to make
configuration easier). The configuration file must then be uploaded to the gateway
through a program that detects all of the nodes, sends the configuration file to each
individual node as string messages, receives confirmation that each node has been
configured, and then starts the desired program on the node (no data, streaming, trigger
data, rainflow, or rainflow and trigger data). The gateway also uses the configuration file
to determine programmatically which node channels to access in the LabVIEW Real-
Time environment so that it can be saved.
4.5.3 Sensors
Two types of sensors were used at the four bridges: strain gages and crack
propagation gages. For Bridges A-C, the CEA-06-250UN-350/P2 strain gage from
Vishay Micromeasurement was used. The general-purpose, foil strain gage is
temperature compensated for mild carbon steel and has three lead wires. The gages are
0.250-in. long with a resistance of 350 Ω. For the wired data acquisition systems, the
gage wires were spliced to a thicker, insulated wire to minimize possible electrical
interference. For the wireless systems, no additional wire was needed as the wireless
node could be placed next to the strain gages. At Bridge D, sealed, weldable strain gages
were used. The weldable gages were from Hitech Products, Inc. Crack propagation
gages from Vishay Micromeasurement (TK-09-CPA02-005/DP) were installed at Bridge
A due to the presence of active cracks.
The sensors were installed to the steel and moisture-protected following
manufacturer recommendations. All of the foil gages (strain and crack propagation) were
applied to the steel girders using the CN adhesive from Texas Measurement. For
128
moisture protection, a two-part scheme was used. A crystalline wax and then a silicone
were applied to the region surrounding and including the gage. The weldable gages were
installed to the steel by adding spot welds around the surface of the gage. Rubber tape
and foil tape were applied over the gage to protect it from moisture. Samaras (2013)
provides a detail discussion on the application and protection techniques for strain gages.
4.6 SUMMARY
Four steel bridges were investigated to characterize the fatigue response of critical
connections due to vehicles crossing each bridge. The instrumentation locations were
identified in this chapter, as were the critical details corresponding to the locations of the
strain gages. At each bridge, a fatigue analysis was conducted, which consisted of a line-
girder analysis and calculating the moment envelope due to the AASHTO fatigue truck,
to determine the locations with the largest stress ranges. The results of the structural
analyses will be used in Chapter 7 to estimate the remaining fatigue life.
The data acquisition equipment used at each bridge are also summarized in
Section 4.5. Both wired (CompactRIO) and wireless systems were utilized at the bridges,
along with two types of sensors (strain gages and crack propagation gages).
129
CHAPTER 5
Interpretation of Measured Fatigue Response
of Four Bridges
Strain gages were installed at four bridges to determine the current distribution of
stress ranges from daily traffic and calculate the accumulation of fatigue damage during
the monitoring period. Descriptions of the four bridges are presented in Chapter 4, as are
the locations of the strain gages. Dynamic strain histories were obtained at an effective
scan rate of 50 Hz at all of the bridges and processed using the simplified rainflow
algorithm (Section 3.1). In general, only the results of the rainflow analysis were
archived, expect during a live load test that occurred at Bridge A, in which strain histories
induced by a test vehicle were recorded. The results of that load test are discussed in
Chapter 6.
The average extent of fatigue damage induced by vehicles crossing the bridge was
characterized using the index stress range (Section 3.2.2). In some situations, the
effective stress range was also calculated (Section 3.2.1); however, only the index stress
range was used to make comparisons between gages and/or bridges discussed in this
chapter. The spectra of stress ranges were verified using the visualization techniques
described in Chapter 3 (contribution to damage (Section 3.3.1) and cumulative damage
(Section 3.3.2)). The amount of fatigue damage determined from the field investigations
were used to calculate the remaining fatigue life of the bridges as presented in Chapter 7.
Data from only a few gages for each bridge are presented in this chapter, but summary
data from all gages are presented in the appendices.
As discussed in Section 3.1, the strain histories were processed by the simplified
rainflow algorithm in 30-minute segments. The bins were summed in various ways to
evaluate the variation in fatigue damage as a function of time of day and day of the week.
With the CompactRIO system (Section 4.5.1) at Bridge A, the 30-min periods were
synchronized on the half hour (1:00 AM, 1:30 AM, etc.) and thus were easy to combine.
130
In contrast, the 30-min periods recorded using the WSN systems (Section 4.5.2) (Bridges
B, C, and D) began when the wireless node started sampling data and were not oriented
on the half hour. As such, to group those acquisition periods, any period that started
between 12:15 AM and 12:45 AM was oriented at 12:30 AM and so forth.
The monitoring periods for each bridge are defined in Section 5.1. Based on
measured data from those monitoring periods, the fatigue damage was evaluated for each
bridge in Section 5.2. The variation in fatigue damage at Bridges A, B, and C are
summarized in Section 5.3.
The construction of the retrofit at Bridge A was tracked using the index stress
range concept (Section 5.4). Using the index stress range, the different phases of the
construction could easily be identified and the benefit of the retrofit quantified.
5.1 DURATION OF MONITORING
Bridges A and B were monitored the longest because both bridges were
considered to be fracture critical and those bridge types were the intended target of the
research project. Whereas the measured fatigue response was tracked for over a year at
Bridge A and multiple months at Bridge B, Bridges C and D were only monitored for
time periods ranging from a few days to a few weeks.
Powering the data acquisition system at each bridge was a concern and impacted
how much data were collected. Because longer monitoring periods were envisioned at
Bridges A and B, solar panels were installed to power the equipment. The goals of
monitoring Bridges C and D were to validate the wireless data acquisition equipment
(Section 4.5.2). As such, both bridges were monitored for a shorter period, and batteries
were used to power the WSN data acquisition systems.
The first iteration of the solar panel (85 W) at Bridge A was too small to power
the system during the winter months and numerous interruptions in the data acquisition
occurred. A larger solar panel (130 W) was installed at the site in August 2011 (Figure
5-1(a)), along with a maximum power point tracking (MPPT) charge controller, which
greatly improved performance. Since August 2011, the system was continuously online
131
with minimal downtown. There were a few instances during prolonged periods of rain
when data were not captured due to inadequate solar exposure. For instance, rainy
conditions prevailed in seventeen of the thirty-one days of December 2011, which caused
loss of data for most of the month.
An 85-W panel was connected to the concrete barrier of Bridge B using post-
installed anchors (Figure 5-1(b)). The panel was not expected to be a long-term solution
for the bridge, but was large enough for data acquisition during spring and summer
months.
Figure 5-1: Solar panels installed at (a) Bridge A and (b) Bridge B.
5.1.1 Bridge A
Monitoring at Bridge A began in March 2011 and continued until August 2012.
Within the first few months, the following key observations were made: (1) some large
stress ranges (greater than 15 ksi) were measured by many of the gages installed along
the top flange of the east longitudinal girder and (2) a few of the cracks identified during
the hands-on inspections were growing. The large stress ranges were alarming from a
fatigue perspective because the fatigue life depends on the cube of the stress range.
A crack propagation gage was installed at the tip of a crack in the welded
connection between the top flange of the east girder and the top flange of floor beam 34.
This crack was identified during the hands-on inspection conducted by the bridge owner
132
in fall 2010. The gage features twenty wires distributed over a width of 0.4″. The wires
of the gage fracture due to growth of the crack, which changes the resistance of the gage.
By measuring the resistance, the crack length can be monitored. Due to the step-wise
nature of the gage, the change in resistance is fairly small (0.2-0.5 Ω) when only a few
wires have broken. However, after the majority of the wires have broken, the change in
resistance due to a wire break is greater than 10 Ω. With the CompactRIO data
acquisition system and measuring the resistance every 30 min, the drift of the gage due to
temperature was less than the change due to a wire break.
The response of the crack propagation gage near floor beam 34 is presented in
Figure 5-2 for a 28-day period in spring 2011. Eleven wires of the crack propagation
gage broke during that period, with a wire breaking approximately every three days. As
such, the crack growth was approximately 0.007″ per day during that period.
Figure 5-2: Crack growth at floor beam 34 in east longitudinal girder during 28-day
period.
Because the crack continued to grow in the flange of the east longitudinal girder
at the location of the welded connection to floor beam 34 over the next 1.5 months at
nearly the same rate, the fatigue life can be assumed to have been exceeded in 2011.
Based on continuous growth of cracks in critical regions of Bridge A, the bridge owner
decided to construct a replacement bridge and remove this structure from service.
However, because it will take 2-3 years to replace the bridge, a retrofit plan was
0
3
6
9
12
15
18
05/02 05/07 05/12 05/17 05/22 05/27 06/01
Res
ista
nce
(o
hm
s)
Time
11 wire breaks | Crack growth of 11x0.02″ = 0.22″ (28 days)
Assumed
Res
ista
nce
(Ω
)
133
developed to maintain the load-carrying capacity of the bridge until the replacement was
designed and constructed.
The chosen retrofit is discussed in Section 4.1.3. The fatigue response of Bridge
A was drastically different before and after the construction of the retrofit. As is
discussed in Section 5.4, the retrofit significantly reduced the fatigue damage induced in
the top flanges of the longitudinal girders at floor beams 34 and 35. For gage locations
on the bottom flange, the rate of change in the fatigue damage decreased, but not nearly
as much as the top flanges. Because the accumulation of fatigue damage in the top flange
was nearly zero and much larger in the bottom flange, the behavior is representative of
composite action between the concrete deck and steel longitudinal girder.
Due to the changes in fatigue behavior of the bridge, the monitoring periods were
broken into different phases. The first phase occurred before the construction of the
retrofit and lasted from March 2011 to August 2011. Due to power issues, a total of 71
full days of data were obtained during this phase. The fatigue damage was characterized
based on the data recorded during those 71 days and used to calculate the remaining
fatigue life in Chapter 7. Only full days (24 hours in a single day) were considered in the
analysis. Partial days were neglected from the analysis because the traffic patterns varied
significantly with the time of day (Section 5.3.1). The distribution of the full days among
the days of the week is summarized in Table 5-1. As is subsequently discussed, the
amount of fatigue damage depended on the day of the week. Therefore, if the estimate of
fatigue damage is based on a non-uniform distribution of days, there is the possibility that
the estimate will be skewed. The data before the construction of the retrofit (Table 5-1)
were used to evaluate the skew in the estimate.
Construction of the retrofit took place from August 2011 to November 2011. The
characterization of fatigue damage was not as interesting during the construction of the
retrofit because a new detail was assumed once the construction was finished. After the
retrofit was finished, the bridge was monitored from November 2011 to August 2012, in
which over 180 days of data were collected (Table 5-2). For this phase, the distribution
134
of days was fairly uniform. The fatigue damage for that period was determined to
evaluate the benefit of the retrofit.
The response of two gages will be discussed in this chapter while the responses of
all gages are summarized in Appendix A & Appendix B.
Table 5-1: Distribution of full days of rainflow data before retrofit at Bridge A.
Day of the week Number of full days Monday 9 Tuesday 8
Wednesday 12 Thursday 10
Friday 10 Saturday 13 Sunday 9
Total 71
Table 5-2: Distribution of full days of rainflow data after construction of the retrofit at
Bridge A.
Day of the week Number of full days Monday 25 Tuesday 28
Wednesday 27 Thursday 24
Friday 27 Saturday 28 Sunday 27
Total 186
5.1.2 Bridge B
The fatigue behavior of Bridge B was tracked for nearly four months. Both
CompactRIO and WSN data acquisition systems were utilized at this bridge. One of the
strain gages from the CompactRIO system will be discussed in Section 5.2, and two
strain gages connected to the WSN system will be discussed in Section 5.2.3. Initially, in
135
April 2012, only two wireless nodes were installed at the bridge, one node in the first
span and one node in the second span. Two months later, June 2012, several WSN
routers were added to the bridge to expand the network to feature a node in each of the
spans. A router in each span was required to wirelessly transmit data from gages located
in Span 4 to the gateway in Span 1. The wireless system proved beneficial at this bridge
because thousands of feet of cable would have been needed if a CompactRIO would have
been used for all of the gage locations. The responses of two gages are discussed in this
chapter while the responses of all gages are summarized in Appendix C.
When evaluating the fatigue damage during the monitoring period, only full days
were considered. The distribution of those days is summarized in Table 5-3. As seen
from the table, the days were fairly evenly distributed.
Table 5-3: Distribution of full days of rainflow data at Bridge B.
Day of the week Number of full days Span 1 Span 3
Monday 10 5 Tuesday 9 5
Wednesday 9 4 Thursday 9 4
Friday 11 4 Saturday 10 5 Sunday 10 4
Total 68 31
5.1.3 Bridge C
Bridge C was monitored for approximately one month. Because batteries were
used at the site to power the WSN data acquisition system, data were not collected
continuously. As such, long-term estimates of the remaining fatigue life were not made
from the data at this bridge because so few full days of data were collected. Because nine
wireless nodes were needed for the instrumentation, one of the nodes was configured as a
136
router while the other eight could be end nodes. The results of all of the WSN gages are
summarized in Appendix D.
5.1.4 Bridge D
Bridge D was monitored for less than one week. Long-term estimates of the
remaining fatigue life were not made from the data at this bridge because so few full days
of data were collected. However, the techniques described in Chapter 3 can be applied to
the data set. Both CompactRIO and WSN data acquisition systems were utilized at this
bridge. Data from the CompactRIO system were discussed in Section 3.4, whereas data
from two of the gages on the WSN system are discussed in this chapter. The responses of
all of the WSN gages are summarized in Appendix E.
5.2 MEASURED FATIGUE RESPONSE DURING MONITORING PERIOD
Two strain gages from each bridge are presented in this section to calculate the
measured fatigue responses during the monitoring periods. For Bridge A, data are
presented for the monitoring periods before and after the retrofit. The index stress range
was used to make comparisons between the bridges. For Bridge A, an index stress range
of 4.5 ksi was used because the critical fatigue detail for this bridge was Category E, and
4.5 ksi corresponds to the CAFL for this detail category. Transverse stiffeners (Category
C′) were the critical fatigue details for the other three bridges, which corresponds to a
CAFL of 12 ksi. Therefore an index stress range of 12 ksi was used at Bridges B, C, and
D.
As discussed in Chapter 3, electromechanical noise produces fictitious stress
fluctuations that do not induce fatigue damage and should be truncated from a fatigue
analysis. The electromechanical noise of each data acquisition system was determined in
the laboratory. For the CompactRIO, it was approximately 1 microstain and 2-3
microstrain for the WSN system. As such, the minimum amplitude of the rainflow
analysis was set at 2 microstrain for the CompactRIO and 5 microstrain for the WSN
system; cycles less than the minimum amplitude were identified, but not included in the
137
rainflow data. As a reference, a 1 microstrain strain change corresponds to a stress
change of approximately 0.03 ksi.
The electromechanical noise at a particular bridge can be higher than these typical
values because the concentration of steel and power lines at each site can affect the noise.
To validate the levels of electromechanical noise at some of the bridges, strain gages
were attached to plates that were not connected to the bridge. The plates were able to
expand and contract freely; thus, any cycles that were identified by the rainflow
algorithm for the gages attached to these plates were considered to be caused by
electromechanical noise and/or temperature drift. The WSN nodes with external batteries
at Bridges B and C were susceptible to higher levels of electromechanical noise
(approximately 15-20 microstrain) because the nodes were not grounded properly.
The electromechanical noise was determined at Bridge B by evaluating the cycles
in the rainflow bins for the plates that were not attached to the bridge. Two gages were
installed next to each other on the bottom flange of the longitudinal girder in Span 1 at
Bridge B, with one gage connected to the CompactRIO data acquisition system and one
gage connected to the WSN system (Figure 5-3). The contribution of each rainflow bin
to the fatigue damage ( , 12 ) was similar for both systems for stress ranges above
0.4 ksi. However, the two systems diverge at stress ranges below 0.4 ksi, with the
rainflow bins from the WSN system contributing significantly more than the rainflow
bins from the CompactRIO system. The response of a strain gage attached to one of the
plates that was not attached to the bridge is also presented in Figure 5-3. As can be seen,
stress ranges below 0.4 ksi contribute significantly to the damage, at a similar scale as the
longitudinal-girder gage connected to the WSN system. Because stress cycles were
measured in a plate with no load applied (free to expand and contract), the cycles are due
to fluctuations in the data acquisition system from electromechanical noise. Therefore,
cycles less than 0.4 ksi were truncated from the WSN nodes at Bridge B.
At Bridge C, both internal-AA and external batteries were used to power the
WSN nodes. For the nodes with internal batteries, the contribution of the lower bins was
138
minimal, whereas the contribution of the lower bins was significant for the nodes with
external batteries (Section 5.2.4). One of the nodes powered by internal-AA batteries
was used for the gages attached to the plate. With that node, no cycles were identified by
the rainflow algorithm. Therefore, the minimum amplitude was set high enough for the
WSN nodes powered by internal-AA batteries. For the node powered by the external
battery, cycles less than 0.4 ksi should be truncated from the rainflow analysis due to
electromechanical noise.
Figure 5-3: Contribution of each bin to , at Bridge B for gages connected
to the CompactRIO and WSN data acquisition systems.
Based in part on the experiences of the research team at Bridges B and C, the
WSN-3214 user guide (National Instruments 2012) provides a method for properly
grounding the external battery and strain gages to the node to minimize the
electromechanical noise.
For each bridge, the rainflow bins had a width of 5 microstrain (0.145 ksi). Based
on the results of the structural analysis discussed in Chapter 4, Bridge A had the largest
expected stress ranges from the crossing of an AASHTO fatigue truck. As such, at
Bridge A, at total of 200 rainflow bins were used, which allowed a maximum stress range
of 29 ksi to be identified and sorted into one of the bins. At the other three bridges, only
100 rainflow bins were used, which corresponds to a maximum stress range of 14.5 ksi.
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10Stress range (ksi)
Co
ntr
ibu
tio
n t
o
WSN system (longitudinal girder)
cRIO system (longitudinal girder)
WSN system (plate not attached to girder)
139
5.2.1 Bridge A (Before construction of the retrofit)
The total accumulation of fatigue damage estimated from field monitoring can be
determined from the histogram of stress ranges using either the effective stress range or
index stress range. Two gages are discussed in this section, one attached to the west
girder and one attached to the east girder. As a reminder of the nomenclature outlined in
Chapter 4, Gage W-34s-TE was installed on the east side of the top flange of the west
longitudinal girder near the connection to floor beam 34, whereas gage E-34s-TW was
installed on the west side of the top flange of the east longitudinal girder near the
connection to floor beam 34. In this section, W-34s-TE and E-34s-TW are referred to as
the gages on the east and west girders, respectively.
Figure 5-4: Histogram of stress ranges before construction of the retrofit (average of
71 days) for gages attached to the top flange of the (a) west girder and (b) east girder.
The histogram of average number of cycles in a day recorded by each gage is
presented in Figure 5-4. The maximum measured stress ranges were 17.2 ksi and 24.9
ksi for the gages installed on the west and east girders, respectively. Some strain
histories were captured at Bridge A before the construction of the retrofit and verified
that the larger stress ranges were induced from trucks crossing the bridge. The fatigue
damage was calculated using both the effective stress range ( , and ) and the index
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er d
ay
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er d
ay
Stress range (ksi)(a) (b)
38,000
17.2 ksi
41,400
24.9 ksi
140
stress range ( , 4.5 ). Based on the index stress range (Table 5-4), the east girder
accumulated nearly five times as much fatigue damage as the west girder.
Table 5-4: Summary of average daily fatigue damage before the construction of the
retrofit at Bridge A.
Gage Effective stress range Index stress range
, , . W-34s-TE (West girder) 38,000 1.44 ksi 1,250 E-34s-TW (East girder) 41,400 2.37 ksi 6,070
The influence of each rainflow bin to the fatigue damage of Bridge A was
evaluated in Chapter 3 using a limited dataset (only fourteen days). The evaluation of the
entire dataset (71 days) is presented in Figure 5-5. As with the limited dataset, more
damage occurred for stress ranges between 6.5-7.5 ksi for the gages on the west and east
girders than it did for ranges greater than 15 ksi. The lower bins (electromechanical
noise) did not have a significant impact on damage accumulation. The cumulate damage
plots for the gages on the west and east girders are shown in Figure 5-6, which indicates
that the majority of the damage occurred between a range of 2-10 ksi.
Figure 5-5: Contribution of each bin to , . before the construction of the
retrofit for gages attached to the longitudinal girder near floor beam 34 (Bridge A).
0
100
200
300
400
0 5 10 15 20 25 30Stress range (ksi)
Co
ntr
ibu
tio
n t
o East girder
West girder
1,250 cycles (W-34s-TE)
6,070 cycles (E-34s-TW)
141
Figure 5-6: The cumulative fatigue damage before the construction of the retrofit for
gage attached to the (a) west girder and (b) east girder.
5.2.2 Bridge A (After the construction of the retrofit)
The same two gages discussed for before the construction of the retrofit are
discussed for the period after the cracked connections had been retrofitted. The gages are
located on the top flanges of the west and east longitudinal girders near the connections
to floor beam 34. For the remainder of this section, W-34s-TE is referred to as the gage
on the west girder, whereas E-34s-TW is referred to as the gage on the east girder.
The average daily cycles of stress ranges from November 2011 to April 2012 are
presented in Figure 5-7. The effective stress range and index stress range were calculated
for each distribution of stress ranges (Table 5-5). The maximum stress ranges were 29.1
ksi and 10.2 ksi for the gages on the west and east girders, respectively. As will be
discussed, the larger stress ranges in the west girder were not believed to be load induced,
but were due to spikes in the data acquisition system. However, the raw stress histories
from the large events at Bridge A were not saved and the responses of the gages could
not be verified. Validating these larger stress cycles highlights the need for an additional
program (besides the rainflow algorithm) on the data acquisition systems that can save
the stress histories from large events (seconds of data). As such, a triggering program
was developed for the data acquisition systems at the other three bridges.
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
1.2 ksi
9.4 ksi 17.2 ksi 24.9 ksi
2.7 ksi
10.8 ksi
95% of damage
2.5% of damage
95% of damage
2.5% of damage
142
Figure 5-7: Histogram of stress ranges after the construction of the retrofit (average of
94 days) for gages attached to the top flange of the (a) west girder and (b) east girder.
Table 5-5: Summary of average daily fatigue damage after the construction of the
retrofit at Bridge A.
Gage Effective stress range Index stress range
, , . W-34s-TE (West girder) 121,000 0.48 ksi 150 E-34s-TW (East girder) 83,700 1.02 ksi 980
The contribution of each rainflow bin to the fatigue damage was evaluated (Figure
5-8). The damage in the east girder was larger than in the west girder. For the east
girder, the most damage occurred for stress ranges between 2-4 ksi, which is much
smaller than the response (6.5-7.5 ksi) at that location before the retrofit of the connection
detail (Figure 5-5). The measured stress ranges above 5 ksi did not have a significant
impact on the damage accumulation at that particular gage location, which reinforces the
idea that some composite action was gained in the retrofit. For the west girder, the most
damage occurred for stress ranges between 2-3 ksi. However, the larger stress ranges
(greater than 25 ksi) did have an influence on the total amount of damage.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)(a) (b)
121,000
29.1 ksi
83,700
10.2 ksi
143
Figure 5-8: Contribution of each bin to , . during the monitoring period
after the construction of the retrofit for gages on the west and east girders.
Considering the cumulative damage graphs for both gages (Figure 5-9), the
majority of the damage occurred over a rather large range (0.2-29 ksi) for the west girder,
whereas the range of damage was more reasonable (1.1-4.7 ksi) for the east girder. The
lower bins did not contribute much to the damage accumulation for east girder because
truncating cycles less than 1 ksi contributed less than 2.5% to the total damage. For the
west girder, the lower bins contributed more to the fatigue damage. The lower 2.5% area
was at 0.2 ksi in part because the total damage was so small in the west girder.
The large cycles affected the two gage locations differently. For the east girder,
the larger cycles had a minimal effect on the damage. In fact, cycles greater than 5 ksi
contributed less than 2.5% of the damage. For the west girder, the larger stress cycles
caused significant steps in the plot of the average daily cumulative damage. Because the
maximum stress ranges for the majority of the gages were much less than 15 ksi for gages
along the top flange of retrofit locations, the larger stress ranges are suspect. In addition,
many of the large cycles corresponded to days of rain and lightning, which was observed
to cause spikes in the wired data acquisition system installed on Bridge D (Section 3.4).
Six strain gages were attached to the west longitudinal girder for the monitoring
period after the construction of the retrofit. If the large cycles measured during that
period were due to large trucks crossing the bridge in the left lane, all of the strain gages
0
20
40
60
80
100
0 5 10 15 20 25 30Stress range (ksi)
Co
ntr
ibu
tio
n t
o East girder
West girder
150 cycles (W-34s-TE)
980 cycles (E-34s-TW)
144
should have recorded a large stress range. However, not all of the gages featured
abnormally large cycles (Table 5-6). For instance, for the gage on the west girder (W-
34s-TE), two stress cycles with amplitudes of 29.1 ksi were recorded on 4/2. In contrast,
the maximum measured stress range on the other side of the top flange at the same
position along the west longitudinal girder (W-34s-TW) was only 3.3 ksi. As such, the
abnormally large cycles measured during those dates are likely due to spikes in the data
acquisition system and should be truncated from the fatigue analysis.
Figure 5-9: The cumulative fatigue damage during the monitoring period after the
construction of the retrofit for gages on the (a) west and (b) east girders.
Table 5-6: Occurrence of abnormally large stress cycles for gages attached to the west
longitudinal girder after the construction of the retrofit.
Gage Date of abnormally large stress cycle
1/15 1/15 3/17 3/25 4/2 4/2 4/6
W-34s-TE (West girder)
* * * * * * *
W-34s-TW W-35s-TE * * W-35s-TW * * * W-35s-BE * * * * * * W-35s-BW * Abnormally large stress cycle occurred. Actual stress range depends on the gage location.
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
0.2 ksi
29 ksi
10.2 ksi
1.1 ksi
4.7 ksi
97.5% of damage
95% of damage
2.5% of damage
145
If the seven cycles above 15 ksi for were truncated from the histogram for the
gage on the west girder (W-34s-TE), the plot of cumulative damage assumes the expected
shape (Figure 5-10). With those larger cycles deleted, the majority of the damage (95%)
occurred between 0.2-4 ksi. For cycles greater than 4 ksi, the graph was much smoother
and does not have the large steps as seen with the unmodified histogram. Because of this
behavior, the larger cycles were likely due to spikes in the data acquisition system. Some
of the other stress ranges (between 10-15 ksi) might also be due to spikes in the data
acquisition system. However, those cycles (between 10-15 ksi) did not contribute
significantly to the accumulation of damage and can be included in the analysis without
much error. By truncating stress ranges above 15 ksi, the fatigue damage ( , 4.5 )
reduces from 150 cycles per day to 130 cycles per day.
Figure 5-10: The cumulative fatigue damage with cycles above 15 ksi truncated for
gage on the west girder.
5.2.3 Bridge B
The fatigue damage was evaluated at two gages installed along the bottom flange
of the west box girder. One of the gages (W-1-BE) was installed on the east side of the
bottom flange in the middle of span 1, whereas the other gage (W-3-BE) was installed on
the east side of the flange in the middle of span 3. In this section, W-1-BE and W-3-BE
are referred to as the gages in Span 1 and Span 3, respectively. The average daily cycles
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
4.0 ksi
95% of damage
2.5% of damage
0.2 ksi
14.1 ksi
No cycles truncated (unmodified histogram)
Cycles above 15 ksi truncated
146
of stress ranges for the two gages at Bridge B are presented in Figure 5-11. The
maximum stress ranges were 6.2 ksi for Span 1 and 8.0 ksi for Span 3. If no bins are
truncated from the analysis, the fatigue damage at the index stress range ( , 12 )
would be 3.70 and 5.44 cycles per days Span 1 and Span 3, respectively. However, as
discussed in the introduction for Section 5.2, the electromechanical noise was higher for
the WSN nodes at Bridge B, and all cycles less than 0.4 ksi should be truncated.
Truncating those cycles reduces the amount of damage per day (Table 5-7).
Figure 5-11: Histogram of stress ranges at Bridge B for gages in (a) Span 1 (average
of 68 days) and (b) Span 3 (average of 31 days).
Table 5-7: Summary of average daily fatigue damage at Bridge B.
Gage Effective stress range Index stress range
, , W-1-BE (Span 1) 3,400* 0.96 ksi* 1.71* W-3-BE (Span 3) 4,700* 1.09 ksi* 3.47*
* Cycles less than 0.4 ksi were truncated.
The effect of lower and higher bins can be considered by evaluating the
contribution of each bin to the fatigue damage (Figure 5-12). As expected from the
higher electromechanical noise, the lower bins contribute significantly to the amount of
damage: at a stress range of 0.25 ksi, the damage ( , 12 ) peaks around 1.4-1.9
cycles/day (off the graph in Figure 5-12) for both gage locations. Whereas the minimum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er
of
cycl
es
pe
r d
ay
Stress range (ksi)(a)
538,0000.23 ksi
6.2 ksi
275,0000.33 ksi
8.0 ksi
(b)
147
amplitude of the rainflow analysis at Bridge A was set greater than the level of
electromechanical noise, the minimum amplitude was set too low at Bridge B and that is
why the lower bins contributed so much to the damage.
Ignoring the first few bins, the largest amount of damage occurred due to stress
cycles between 2-3 ksi and 2.5-3.5 ksi for Span 1 and Span 3, respectively. Stress ranges
higher than 4 ksi did not significantly contribute to the accumulation of damage.
Figure 5-12: Contribution of each bin to , at Bridge B for gages in Span 1
and Span 3.
The cumulative damage for both gage locations without any truncation is shown
in Figure 5-13. As seen, there was a large step in the damage at the lower stress ranges,
followed by a gradual increase in damage. The section with the gradual increase looks
similar to the cumulative damage plots for Bridge A. The impact of the higher bins was
minimal as cycles greater than 4 ksi contribute approximately 2.5% of the total damage.
As discussed in Section 5.2, the lower stress ranges in the WSN system were due
to electromechanical noise and not induced by vehicles crossing the bridge. If cycles less
than 0.4 ksi are truncated, the cumulative damage can be re-plotted (Figure 5-14). As
seen, if the first few bins were truncated, there was not as large of an initial step in
damage and the plots were more reasonable. The line for 97.5% of damage was shifted
slightly to the right as a result; however, the higher bins still did not cause a significant
amount of damage.
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
Co
ntr
ibu
tio
n t
o
Span 1
Span 3
3.7 cycles (W-1-BE)
5.4 cycles (W-3-BE)
148
Figure 5-13: The cumulative fatigue damage at Bridge B for gages in (a) Span 1 and
(b) Span 3.
Figure 5-14: The cumulative fatigue damage at Bridge B if stress ranges below 0.4 ksi
were truncated for gages in (a) Span 1 and (b) Span 3.
5.2.4 Bridge C
For Bridge C, the two example strain gages presented in this section are located in
the middle span, at location 2. Both gages are installed along the south face of the
bottom flange of girder 3 (L2-3-BS) and girder 5 (L2-5-BS). These gages are referred to
in this section as girder 3 and girder 5. The gage on girder 3 was connected to a WSN
node that was powered by an external battery. Therefore, there was no loss of data for
0 2 4 6 8 10Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
6.2 ksi 8.0 ksi
97.5% of damage
2.5% of damage
3.0 ksi
97.5% of damage
2.5% of damage
4.0 ksi
0 2 4 6 8 10Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
6.2 ksi 8.0 ksi
97.5% of damage
2.5% of damage
3.6 ksi
97.5% of damage
2.5% of damage
4.3 ksi
0.4 ksi 0.4 ksi
149
that node and more full days (19 days) were obtained from that node than nodes powered
by internal-AA batteries (10 days). Though, as discussed in the introduction for Section
5.2, the electromechanical noise was higher for WSN nodes powered by an external
battery (cycles less than 0.4 ksi should be truncated).
The rainflow results from the bridge monitoring for girder 3 and girder 5 are
presented in Figure 5-15. The fatigue damage for both gages is summarized in Table 5-8.
The maximum stress ranges were 5.0 ksi and 7.3 ksi for girder 3 and girder 5,
respectively. If no bins were truncated from the analysis, the fatigue damage at the index
stress range ( , 12 ) would be 4.67 cycles per day for gage location L2-3-BS and
5.51 cycles per day for L2-5-BS. If cycles less than 0.4 ksi are truncated from girder 3
(node powered by an external battery), the damage ( , 12 ) reduces to 3.08 cycles
per day.
Figure 5-15: Histogram of stress ranges at Bridge C for (a) girder 3 (average of 19
days) and (b) girder 5 (average of 10 days).
The effects of lower and higher bins were evaluated and the results are presented
in Figure 5-16. For the gage location connected to the WSN router node (girder 3), the
lower bins had a considerable impact to the amount of damage. In contrast, the lower
bins did not have a significant influence on the amount of damage at the other gage
location (girder 5). The larger stress ranges did not contribute significantly to the total
damage for either of the gage locations.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)(a)
206,0000.34 ksi
5.0 ksi
16,5000.83 ksi
7.3 ksi
(b)
150
Table 5-8: Summary of average daily fatigue damage at Bridge C.
Gage Effective stress range Index stress range
, ,
L2-3-BS (Girder 3) 7,300* 0.90 ksi* 3.08* L2-5-BS (Girder 5) 16,500 0.83 ksi 5.51
* Cycles less than 0.4 ksi were truncated.
Unlike Bridges A and B, the fatigue response of Bridge C was not dominated by a
narrow band of stress ranges. Instead, the entire spectrum of stress ranges contributed to
the fatigue response of Bridge C. Therefore, no specific stress range (or set of trucks)
dominated the response at Bridge C.
Figure 5-16: Contribution of each bin to , at Bridge C for gages at girder 3
and girder 5.
The cumulative damage for both gage locations without any truncation is shown
in Figure 5-17. For girder 3, stress ranges less than 1 ksi contributed significantly to the
fatigue damage. In contrast, the accumulation of damage at girder 5 was much more
gradual (Figure 5-17(b)). The effect of the higher bins was insignificant as stress ranges
greater than 3.7 ksi for girder 3 and 5.4 ksi for girder 5 contributed only 2.5% of the
overall damage.
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
Co
ntr
ibu
tio
n t
o
Girder 3
Girder 5
4.7 cycles (L2-3-BS)
5.5 cycles (L2-5-BS)
151
Figure 5-17: The cumulative fatigue damage at Bridge C for gages at (a) girder 3 and
(b) girder 5.
5.2.5 Bridge D
Due to the short installation period, only four full days of data were captured at
Bridge D. The two strain gages discussed in this section are located in the middle of span
2 (Location 1) on the bottom flange of girders 2 and 3. For the remainder of this section,
L1-2-BS and L1-3-BN are referred to as the gages on girder 2 and girder 3, respectively.
The rainflow results from the monitoring period are presented in Figure 5-18 for
girders 2 and 3. As can be seen in Table 5-9, the fatigue damage for both girders is
similar; however, the maximum stress range was 7.2 ksi for girder 2, whereas it was 4.9
ksi for girder 3.
Another strain gage was installed near the gage on girder 2 and was connected to
a CompactRIO. The histogram data from the CompactRIO are presented in Chapter 3
(Figure 3-17). Whereas many problems were identified with cycles in the higher bins in
the CompactRIO system, the larger cycles were much smaller with the WSN system.
The larger cycles for the CompactRIO were attributed to spikes in the data acquisition
system from lightning strikes. Because the WSN system was better grounded than the
CompactRIO and had shorter cables for the gages, the WSN system was not affected by
the lightning spikes.
0 2 4 6 8 10Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
0.1 ksi
5.0 ksi 7.3 ksi
95% of damage 2.5% of
damage
0.4 ksi
3.7 ksi
95% of damage
2.5% of damage 5.4 ksi
152
Figure 5-18: Histogram of stress ranges at Bridge D (average of 4 days) for gage
locations along (a) girder 2 and (b) girder 3.
Table 5-9: Summary of average daily fatigue damage at Bridge D.
Gage Effective stress range Index stress range
, , L1-2-BS (Girder 2) 150,000 0.60 ksi 18.8 L1-3-BN (Girder 3) 125,000 0.63 ksi 17.7
The effect of electromechanical noise (lower bins) and larger cycles (potential
spikes) were evaluated and the results are presented in Figure 5-19. As can be seen,
electromechanical noise did not contribute much to the damage accumulation at this
bridge. For girder 3, there was a large peak in the data around 2.5 ksi, whereas there
were two peaks (1.8 ksi and 3.2 ksi) in the data for location girder 2. The larger cycles
did not contribute much to the damage accumulation either.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)(a) (b)
150,000
7.2 ksi
125,000
4.9 ksi
153
Figure 5-19: Contribution of each bin to , at Bridge D for gages attached
to girders 2 and 3.
The cumulative damage for both gage locations is shown in Figure 5-20. For both
gages, electromechanical noise was not a problem, as cycles less than 0.4 ksi contributed
approximately only 2.5% of the total damage. The majority of the damage occurred
between 0.4 to 4 ksi. The higher bins (greater than 4 ksi) contributed approximately only
2.5% of the overall damage.
Figure 5-20: The cumulative fatigue damage at Bridge D for gages attached to (a)
girder 2 and (b) girder 3.
0
1
2
3
0 2 4 6 8 10Stress range (ksi)
Co
ntr
ibu
tio
n t
o Girder 3
Girder 2
18.8 cycles (L1-2-BN)
17.7 cycles (L1-3-BS)
0 2 4 6 8 10Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)(a) (b)
0.4 ksi
7.2 ksi 4.9 ksi
95% of damage
2.5% of damage
0.4 ksi
4.6 ksi
95% of damage
2.5% of damage 3.7 ksi
154
5.2.6 Summary of Daily Fatigue Damage at Bridges
Because the index stress range was used to evaluate the extent of fatigue damage
induced during the monitoring periods, the amount of relative damage between the
bridges can be compared easily (Table 5-10). For Bridge A, two gages (E-34s-TW and
W-34s-TW) were considered for data sets acquired before and after the construction of
the retrofit. The fatigue damage at these locations were converted to an index stress
range of 4.5 ksi (fatigue Category E). Two gages were also considered at Bridge B (W-1-
BE and W-3-BE), Bridge C (L2-3-BS and L2-5-BS), and Bridge D (L1-2-BN and L1-3-
BS), yet an index stress range of 12 ksi was used for these bridges because the critical
fatigue details corresponding to the locations of the strain gages were classified as
Category C′.
The average daily fatigue damage for the four bridges is summarized in Table
5-10. Because the welded connections used for all bridges were not classified in the
same category, Equation 3-7 was used to normalize all the data and convert the fatigue
damage to an index stress range of 4.5 ksi and fatigue category E ( , 4.5 : ).
Similarly, the data was converted to an index stress range of 12 ksi and fatigue category
C′ ( , 12 : ′ ). Either can be used to compare the relative accumulation of fatigue
damage at the four bridges.
The most damage occurred at Bridge A before the construction of the retrofit,
whereas the least damage occurred at Bridge B. Even after the construction of the
retrofit, Bridge A experienced more damage than the other bridges. Bridge D is
calculated to be the next most critical bridge; however, the average daily fatigue damage
is less than 10% of the average fatigue damage induced in Bridge A after the retrofit.
155
Table 5-10: Summary of average daily fatigue damage at four bridges.
Bridge Gage
location
Fatigue damage
, . , , . : , : ′
A (pre-retrofit)
E-34s-TW 6,070 - 6,070 1,280
A (pre-retrofit)
W-34s-TE 1,250 - 1,250 264
A (post-retrofit)
E-34s-TW 940 - 940 198
A (post-retrofit)
W-34s-TE 130 - 130 27.4
B+ W-1-BE - 1.71* 8* 1.71*
B+ W-3-BE - 3.47* 16* 3.47*
C+ L2-3-BS - 3.08* 15* 3.08*
C L3-5-BS - 4.86 23 4.86
D L1-2-BN - 18.8 89 18.8
D L1-3-BS - 17.7 84 17.7
* Cycles less than 0.4 ksi were truncated because of electromechanical noise. + WSN system was powered by an external battery.
5.3 VARIATIONS IN MEASURED FATIGUE DAMAGE
Because the rainflow analyses were performed in 30-min periods, the induced
fatigue damage can be characterized as a function of time of day and day of the week.
These short-term variations in fatigue damage can influence the required duration for
monitoring a bridge. If the short-term variations are modest, less than a day of data may
be all that is needed to characterize the long-term accumulation of fatigue damage.
However, if the damage varies with both time of day and day of the week, a longer
monitoring period may be needed to obtain reliable estimates of the accumulation rate of
fatigue damage. The variation in fatigue damage was determined in terms of the index
stress range.
156
The same gages discussed in Section 5.2 are used in this section. For Bridge A
(before and after the construction of the retrofit), W-34s-TE is referred to as the gage on
the west girder while E-34s-TW is the gage on the east girder. For Bridge B, W-1-BE
and W-3-BE are referred to as the gages in Span 1 and Span 3, respectively. For Bridge
C, L2-3-BS and L2-5-BS are referred to as the gages attached to Girder 3 and Girder 5,
respectively. Bridge D is not discussed in this section as not enough data were captured
at that bridge to establish the short-term trends.
5.3.1 Variation with Time of Day
To determine the variation in fatigue damage with time of day, all data associated
with a specific, 30-min time slot, regardless of day, were averaged. In Section 5.2, only
data from full days (24 hours of continuous data acquisition in a single day) were
analyzed. If data from a particular day were not continuous, due to starting/stopping the
program on the data acquisition system or loss of power, the data were not considered
when calculating the daily fatigue damage (Section 5.2). However, for this section, all
data were considered, even data from days for which the data acquisition system was
only active for a portion of the day. As such, the daily fatigue cycles discussed in this
section do not necessarily match those discussed in Section 5.2.
Besides characterizing the fatigue damage in terms of the time of day, the
rainflow data were grouped into four, six-hour time periods to evaluate variations in the
distribution of vehicles crossing the bridge. After grouping the data into the four time
periods, the contribution of each bin to the fatigue damage was calculated to determine if
differences in damage for a specific time period were caused by the number of vehicles
and/or the weight of the vehicles crossing the bridge. The four time periods were 12:00
AM to 6:00 AM (0:00-6:00), 6:00 AM to 12:00 PM (6:00-12:00), 12:00 PM to 6:00 PM
(12:00-18:00), and 6:00 PM to 12:00 AM (18:00-0:00).
157
5.3.1.1 Bridge A (Before the construction of the retrofit)
The average number of fatigue cycles at the index stress ( 4.5 ) is plotted as
a function of time of day in Figure 5-21 for the west and east girders at Bridge A before
the construction of the retrofit. As can be seen, the damage in the east girder was always
greater than that in the west girder throughout the day. The largest number of cycles
occurred in the 6-hr period between 12:00-18:00, which corresponds to when a majority
of large trucks are being driven. The least amount of damage occurred between 0:00-
6:00, when truck traffic is expected to be relatively low.
By plotting the contribution of each bin to the fatigue damage for the four, six-
hour periods (Figure 5-22), a difference in the distribution of stress ranges crossing the
bridge can be noted. The peak in damage for 12:00-18:00 occurred at approximately 7.5
ksi, whereas it occurred at 7.0 ksi for 0:00-6:00. Because the magnitudes of each
rainflow bin are larger and the curve is shifted further to the right, a greater number of
trucks crossed the bridge during the 12:00-18:00 period and those trucks were typically
heavier than the trucks that crossed during the 0:00-6:00 period. The shape of the 6:00-
12:00 curve was similar to the 12:00-18:00 curve, whereas the 18:00-0:00 curve was
similar to the 0:00-6:00 curve.
Figure 5-21: Variation in . with time of day before the construction of the
retrofit at Bridge A.
0
50
100
150
200
250
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 0:00Time of Day
East girder
West girder
158
Figure 5-22: Contribution of each bin to . during the specific 6-hour
periods before the construction of the retrofit for the east girder at Bridge A.
5.3.1.2 Bridge A (After the construction of the retrofit)
After the construction of the retrofit, the average variation in fatigue damage with
time of day was similar to the variation before the retrofit; however, the amplitudes of the
damage were smaller due to the benefit of the retrofit. The variation with time of day is
presented in Figure 5-23 for the west and east girders. As it did before the retrofit (Figure
5-22), the most damage occurred between 12:00-18:00. Similarly, before the retrofit, the
trucks that crossed from 12:00-18:00 were heavier and in greater number than the trucks
that crossed during the 0:00-6:00 (Figure 5-24).
Figure 5-23: Variation in . with time of day after construction of the
retrofit at Bridge A.
0
40
80
120
160
0 5 10 15 20 25Stress Range (ksi)
Average
0:00-6:00
6:00-12:00
12:00-18:00
18:00-24:00
Co
ntr
ibu
tio
n t
o
Between 12:00-18:00
Between 18:00-0:00
Average
Between 0:00-6:00
Between 6:00-12:00
Number of cycles (for 6-hour period):0:00-6:00: 7106:00-12:00: 1,29012:00-18:00: 2,30018:00-0:00: 1,580Average: 1,470
0
10
20
30
40
50
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 0:00Time of Day
West girder
East girder
159
Figure 5-24: Contribution of each bin to . during the specific 6-hour
periods after construction of the retrofit for the east girder at Bridge A.
5.3.1.3 Bridge B
The variation in the fatigue damage ( 12 ) with time of day was not as
smooth for Bridge B (Figure 5-25) as it was for Bridge A. The curves are not as smooth
at Bridge B in part from not having as many data sets. Nonetheless, throughout the day,
the damage in Span 3 was always greater than Span 1. The most damage occurred
between 8:00 AM and 4:00 PM for this bridge, which corresponds to typical business
hours. For a given six-hour period, the most damage occurred between 6:00-12:00,
whereas the least amount of damage occurred between 18:00-0:00. Thus, the time
periods for the most and least damage were slightly different for Bridge B than for Bridge
A.
The contribution of each bin to the four, six-hour periods is presented in Figure
5-26. For this bridge, the peak in fatigue damage occurred at approximately the same
stress range (2.55 ksi) for all six-hour periods, which indicates that the same distribution
of vehicles crossed the bridge on average. Therefore, more damage occurred from 6:00-
12:00 due to a greater number of cycles as compared to the other time periods. For the
average number of cycles reported in Figure 5-26, cycles less than 0.4 ksi were truncated
because those cycles were attributed to electromechanical noise (introduction to Section
5.2).
0
10
20
30
40
0 5 10 15Stress Range (ksi)
Average
0:00-6:00
6:00-12:00
12:00-18:00
18:00-24:00
Co
ntr
ibu
tio
n t
o
Between 12:00-18:00
Between 18:00-0:00Average
Between 0:00-6:00
Between 6:00-12:00
Number of cycles (for 6-hour period):0:00-6:00: 1006:00-12:00: 23012:00-18:00: 43018:00-0:00: 200Average: 240
160
Figure 5-25: Variation in with time of day at Bridge B (cycles less than 0.4
ksi were truncated).
Figure 5-26: Contribution of each bin to during the specific 6-hour periods
for Span 1 at Bridge B.
5.3.1.4 Bridge C
As with the variation with time of day observed for Bridge B, the variation in
fatigue damage at Bridge C was not very smooth (Figure 5-27). The damage at girder 5
was slightly greater than girder 3 during the middle of the day. Similar to Bridge B, the
most damage occurred between 8:00 AM and 4:00 PM for this bridge. For a given six-
hour period, the most damage occurred between 12:00-18:00, whereas the least amount
of damage occurred between 0:00-6:00. Due to the high electromechanical noise at
girder 3, cycles less than 0.4 ksi were truncated for that location only in Figure 5-27.
0.00
0.04
0.08
0.12
0.16
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 0:00Time of Day
Span 3
Span 1
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress Range (ksi)
Average
0:00-6:00
6:00-12:00
12:00-18:00
18:00-24:00
Co
ntr
ibu
tio
n t
o
Between 6:00-12:00
Between 12:00-18:00
Average
Between 0:00-6:00 Between
18:00-0:00
Number of cycles (for 6-hour period):0:00-6:00: 0.30*6:00-12:00: 0.66*12:00-18:00: 0.51*18:00-0:00: 0.23*Average: 0.42**Cycles less than 0.4 ksiwere truncated from sum
161
Figure 5-27: Variation in with time of day at Bridge C (cycles less than 0.4
ksi were truncated for girder 3 only).
The contribution of each bin for four, six-hour periods is presented in Figure 5-28.
For this bridge, the peak in fatigue damage occurred between 3-4 ksi during the middle of
the day (6:00-12:00 and 12:00-18:00). In contrast, the peaks occurred between 2-3 ksi
for night periods (0:00-6:00 and 18:00-0:00). As such, the data indicated a higher
number of heavier trucks during the daylight hours as compared to night.
Figure 5-28: Contribution of each bin to during the specific 6-hour periods
for girder 5 at Bridge C.
5.3.2 Variation with Day of the Week
The variation in fatigue damage as a function of the day of the week was
evaluated by considering only data from full days (24 hours in a single day). For each
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 0:00Time of Day
Girder 5
Girder 3
0.00
0.05
0.10
0.15
0.20
0 2 4 6 8 10Stress Range (ksi)
Average
0:00-6:00
6:00-12:00
12:00-18:00
18:00-24:00
Co
ntr
ibu
tio
n t
o
Between 12:00-18:00
Between 6:00-12:00
Average
Between 0:00-6:00
Between 18:00-0:00
Number of cycles (for 6-hour period):0:00-6:00: 0.206:00-12:00: 1.7212:00-18:00: 2.0718:00-0:00: 0.34Average: 1.09
162
day of the week, all of the data were averaged. Only bridges that had at least a month of
data (Bridges A and B) were considered in this section.
The variation in fatigue damage with day of the week is especially important if
the data collected for a particular bridge has a non-uniform distribution of days. To
calculate the daily fatigue damage, the average of all of the data might be used if there are
enough days of data. Though, if there is an uneven distribution of days of the week, the
data may need to be scaled for the correct distribution of days. For instance, for the data
considered at Bridge A before the retrofit (Table 5-1), there were five additional
Saturdays of data as compared to data from Tuesdays. If the difference in fatigue damage
between those two days is significant, the estimate of the daily fatigue damage could be
skewed if the data are simply averaged together.
To evaluate how much the estimate might be skewed, the mean fatigue damage
considering distribution of days (, ,
) can be calculated. The mean fatigue damage
for a particular day can be calculated using Equation 5-1. The distribution of days of
fatigue data for each day of the week ( ) are summarized in Section 5.1 (Table 5-1,
Table 5-2, and Table 5-3). The mean daily fatigue damage (, ,
) can be calculated
using Equation 5-2.
, , , , Equation 5-1
, , , ,
7 Equation 5-2
where
, , meanfatiguedamageatanindexstressrange , forday
, , numberofequivalentcyclesat , fordataset
numberofdaysoffatiguedataforday
dayoftheweek 1to7
counterfornumberofdatasetsforday 1to
163
, , meanfatiguedamageconsideringdistributionofdaysatan
indexstressrange ,
5.3.2.1 Bridge A (Before the construction of the retrofit)
For the period before the retrofit at Bridge A (Table 5-1), the average variation in
fatigue damage ( , 4.5 ) with day of the week for the west and east girders is
presented in Figure 5-29. As can be seen, more damage occurred on weekdays than
weekends due to the larger volume of trucks that travel on the weekdays compared to
weekends. On average, the maximum amount of damage occurred on Tuesday at this
bridge. If the damage for Monday through Friday is compared to the damage on Sunday,
the damage for the weekday is 2.3 to 2.8 times greater than the damage on Sunday in the
west girder and 1.9 to 2.2 times greater in the east girder.
Figure 5-29: Average variation of fatigue damage ( , . ) for the west and east
girders before the construction of the retrofit at Bridge A.
The average daily fatigue damage before the construction of the retrofit is
summarized for the west and east girders (Table 5-11). The data indicate that there is not
much difference between accounting for the distribution of days (, . ) and
average of all full days of data ( , 4.5 ). The small difference is likely due to the
6,172
7,314 7,207 7,005 6,793
4,852
3,280
1,313 1,578 1,454 1,406 1,4331,006
572
0
2,000
4,000
6,000
8,000
10,000
Mon Tue Wed Thu Fri Sat Sun
East Girder
West Girder
(9 days) (8 days) (12 days)Days of data: (10 days) (10 days) (13 days) (9 days)
164
large amount of measured data (71 days for most gages). In addition, of the 71 days, the
distribution of the weekdays (5 out of 7) and weekend days (2 out of 7) is nearly correct.
Table 5-11: Summary of daily fatigue damage before the construction of the retrofit at
Bridge A.
Gage Mean fatigue damage based
on distribution of days (
, . )
Average of days monitored
( , . )
W-34s-TE (West girder) 1,250 1,250 E-34s-TW (East girder) 6,090 6,070
5.3.2.2 Bridge A (After the construction of the retrofit)
The variation in fatigue damage as a function of day of the week was similar after
the construction of the retrofit at Bridge A (Figure 5-30) as it was before the retrofit. As
previously, more damage occurred on weekdays than weekends. The most damage
occurred on Tuesday for the east girder, whereas it occurred on Wednesday for the west
girder. If the damage for Monday through Friday was compared to the damage on
Sunday, the damage for the weekday was 1.4 to 1.9 times greater for the west girder and
1.7 to 2.0 times greater for the east girder than the damage on Sunday.
Figure 5-30: Variation of fatigue damage ( , . ) for the west and east girders
after construction of the retrofit at Bridge A.
947
1,102 1,070 1,066 1,035
800
547
126 142 160 146 140 125 82
0
500
1,000
1,500
Mon Tue Wed Thu Fri Sat Sun
East Girder
West Girder
(25 days) (28 days) (27 days)Days of data: (24 days) (27 days) (28 days) (27 days)
165
The distribution of days monitored after the construction of the retrofit was fairly
uniform (Table 5-2); therefore, the estimate of the daily fatigue damage was similar
whether the distribution of days were used or the average of all the days monitored
(Table 5-12).
Table 5-12: Summary of daily fatigue damage after construction of the retrofit at
Bridge A.
Gage Mean fatigue damage based on distribution of days (
, . )
Average of days monitored
( , . )
W-34s-TE (West girder) 130 130 E-34s-TW (East girder) 938 940
5.3.2.3 Bridge B
The average variation of damage with the day of the week was determined for
Span 1 and Span 3 (Figure 5-31). As with Bridge A, the fatigue damage from a weekday
was greater than a weekend at Bridge B. For Span 1, there was no variation in fatigue
damage between Monday and Friday. In contrast, there was some variation in the fatigue
damage in Span 3, with the most damage on average occurring on Monday.
Figure 5-31: Variation of fatigue damage ( , ) for Span 1 and Span 3 at
Bridge B (cycles less than 0.4 ksi were truncated).
2.0 2.0 2.1 2.0 2.0
1.30.7
4.4 4.23.6
4.0 3.9
2.6
1.5
0
1
2
3
4
5
6
Mon Tue Wed Thu Fri Sat Sun
Span 1
Span 3
166
Because a fairly uniform distribution of data was collected at Bridge B (Table
5-3), there was not expected to be much skew between considering the distribution of
days and simply calculating the average. As seen in Table 5-13, there is not much
difference between the two methods of calculating the daily fatigue damage.
Table 5-13: Summary of daily fatigue damage after construction of the retrofit at
Bridge A.
Gage Mean fatigue damage based
on distribution of days (
,)
Average of days monitored
( , )
W-1-BE (Span 1) 1.72* 1.71* W-3-BE (Span 3) 3.44* 3.47*
* Cycles less than 0.4 ksi were truncated
5.3.3 Weekly
One set of researchers recommended that a bridge should be monitored
continuously for two to four weeks to obtain an accurate representation of the damage
(Connor and Fisher 2006). To evaluate that recommendation, the weekly variation in
fatigue damage was determined at Bridge A for the period after the construction of the
retrofit. The gage was attached to the bottom flange of the east longitudinal girder near
floor beam 35 (E-35n-BE) (Figure 5-32). The damage from seven continuous days of
monitoring was averaged to obtain and estimate the average amount of daily damage
( , 4.5 ).
The average daily damage for all of the data is 6,000 cycles per day. For each
individual week, the damage was within 10% (600 cycles) of the average except for
weeks that included a federal holiday. Thus, one continuous week of data provided an
adequate representation of data for this gage location as long as the week did not contain
a holiday. However, because estimating the remaining fatigue life involves forecasting
the damage, 10% error in the damage may have a significant impact on the calculated
fatigue life.
167
Figure 5-32: Variation in average daily damage ( , . ) due to a week of
continuous monitoring for gage E-35n-BE at Bridge A.
The variation in weekly damage for a gage attached to the top flange of the east
longitudinal girder near floor beam 34 (E-34s-TW) is presented in Figure 5-33. For this
location, the average of all the data was 945 cycles per day. There was more variation in
the weekly damage at this location, such that an individual week of data could cause as
much as 20% difference in damage (excluding weeks with a federal holiday).
Figure 5-33: Variation in average daily damage ( , . ) due to a week of
continuous monitoring for gage location E-34s-TW at Bridge A.
The data in Figure 5-33 suggest that there could be a variation in damage from
month to month. For instance, the damage ( , 4.5 ) from 3/1/2012 to 5/31/2012
had a daily average of 1,040 cycles, whereas the damage was reduced from 6/1/2012 to
7/30/2012 with a daily average of 810 cycles.
5000
5500
6000
6500
7000
Date
Each data point is the average of seven continuous days of monitoring
Average Week had a holiday
Average ±10%
500
1000
1500
Date
Week had a holiday
Each data point is the average of seven continuous days of monitoring
Average
±20%
168
5.3.4 Summary of Variation in Measured Fatigue Response
Due to the weekly, daily, and hourly variation in damage, a continuous week of
data should be collected as the minimum monitoring period. Monitoring for only part of
the day or week will not be representative of the true damage at the bridge. If calculating
the remaining fatigue life is desired, four weeks of data is recommended to make long-
term projections. Because damage may vary month to month, a permanent monitoring
system could be installed at a bridge and data could be collected for one to two weeks
every month for the entire year.
5.4 TRACKING PROGRESS OF CONSTRUCTION AT BRIDGE A
The construction of the retrofit at Bridge A took place from 8/29/2011 to
11/18/2011. Because the index stress range was used to convert the distribution of stress
ranges into normalized fatigue damage, the progress of repairing cracked connections
could be tracked. The results from four gage locations are discussed in this section. The
results for the other gage locations can be found in Appendix A. Of the gages used to
measure the fatigue of the bridge during the initial acquisition period, some gages were
destroyed during construction and were replaced after completion of the construction.
For most of the gages, 57 full days of data were obtained during the retrofit. In the north
span, only the regions around floor beams 34 to 38 of both longitudinal girders were
retrofitted (Section 4.1.3).
Six strain gages will be discussed in this section. Gage W-35s-BW was installed
on the west side of the bottom flange of the west longitudinal girder near the connection
to floor beam 35, whereas gage E-35n-BW was installed on west side of the bottom
flange of the east longitudinal girder near floor beam 35. W-35s-BW will be referred to
as the bottom flange on the west girder and E-35n-BW will be referred to as the bottom
flange on the east girder. For gages installed along the longitudinal girders near floor
beam 34, W-34s-TE will be referred to as the top flange of the west girder and E-34s-TW
will be referred to as the top flange of the east girder. Finally, two gages were installed
on the east longitudinal girder near floor beam 27. E-27-TW will be referred to as the top
169
flange at floor beam 27 and E-27-BW will be referred to as the bottom flange at floor
beam 27.
5.4.1 Rainflow Results
The average daily cycles of stress ranges for the construction period of the retrofit
for the gages attached to the bottom flanges are presented in Figure 5-34. Similar to the
period before the construction of the retrofit, the effect of the lower (electromechanical
noise) and higher (data acquisition spikes) bins were minimal and did not need to be
truncated. The maximum stress ranges were 23.6 ksi and 16.7 ksi for the bottom flanges
of the west and east girders, respectively. The histograms of stress ranges for gages
attached to the top flanges are presented in Figure 5-35. The maximum stress ranges
were 18.5 ksi and 11.2 ksi for the bottom flanges of the west and east girders,
respectively.
Figure 5-34: Histogram of stress ranges during the construction of the retrofit
(average of 57 days) for gages on the bottom flanges of the (a) west and (b) east
girders.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)(a) (b)
141,000
23.6 ksi
143,000
16.7 ksi
170
Figure 5-35: Histogram of stress ranges during the construction of the retrofit
(average of 57 days) for gages on the top flanges of the (a) west and (b) east girders.
5.4.2 Behavior of Bridge During Construction
The characterization of fatigue damage is not as interesting during the
construction of the retrofit because a new detail will be assumed once the construction is
finished. However, by plotting the fatigue damage on a daily basis using the index stress
range, changes in the traffic pattern due to the construction of the retrofit can be
identified. This is an added benefit of using the index stress range to represent the daily
fatigue damage. If the effective stress range were plotted on a daily basis, the changes in
the traffic pattern would not be as readily identified.
To construct the retrofit, one lane was closed at a time so that the concrete deck
could be removed and the cover plates installed at the appropriate floor beam locations
(Figure 4-9). The contractor started with the east girder, which corresponds to the right
lane. Construction along the east girder took place from August 29 until November 6.
Part of the right lane was reopened at that time and the contractor moved to the west
girder and closed the left lane (traffic could cross the bridge along the second bridge at
the site—for exit ramp). The retrofit construction for the west girder went much quicker,
and was completed on November 18.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)(b)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)(a)
48,400
18.5 ksi
55,100
11.2 ksi
171
The daily fatigue damage ( , 4.5 ) for the gages on the bottom flanges are
shown in Figure 5-36. Each phase of the construction period can be identified in the
graph. Before the retrofit, the fatigue damage in the bottom flange of the east girder was
larger than bottom flange of the west girder because more trucks would be expected to
pass in the right lane (east girder) when both lanes are opened. With the right lane
closed, the damage accumulation in the east girder decreased to almost zero, whereas the
damage accumulation at the gage location on the west girder increased substantially.
Because all trucks had to cross the bridge in the left lane when the right lane was closed,
the damage accumulation in the west girder increased almost to the same level as the east
girder before construction. When part of the right lane was reopened and the left lane
was closed, the accumulation of fatigue damage in the west girder decreased to almost
zero. After construction was finished on both girders, the accumulation of damage
measured in the bottom flange of the east girder was larger than the west girder.
However, the values in the bottom flanges were slightly smaller than the values before
the retrofit.
Figure 5-36: Daily fatigue damage ( , . ) during the construction process for
the bottom flanges in the west and east girders at Bridge A.
The daily fatigue damage ( , 4.5 ) for the top flanges near floor beam 34
are shown in Figure 5-37. The accumulation of damage in the top flanges was similar to
0
4,000
8,000
12,000
16,000
Date
E-35n-BWW-35s-BW
B
A. Prior to retrofitB. Right (east) lane closed
A
C. Left (west) lane closedD. Retrofit completed
C D
172
the damage in the bottom flange. When the right lane was closed, the accumulation of
damage in the top flange of the east girder was nearly zero, whereas the accumulation of
damage in the top flange of the west girder increased substantially. When part of the
right lane was reopened and the left lane was closed, the accumulation of fatigue damage
in the top flange of the west girder decreased to almost zero. After the bridge was
reopened, the accumulation of damage in the top flange of the east girder was again
larger than the west girder. Nonetheless, the retrofit greatly reduced the growth rate of
fatigue damage in the top flanges.
Figure 5-37: Daily fatigue damage ( , . ) during the construction process for
the top flanges in the west and east girders at Bridge A.
Because the fatigue damage in the top flanges is nearly zero and much larger in
the bottom flanges, the behavior is similar to composite action between the concrete deck
and steel longitudinal girder. Before the retrofit, the contribution of the top and bottom
flanges was similar in the east girder (Figure 5-38(a)). After the retrofit, the contribution
of the bottom flange is much larger than the top flange (Figure 5-38(b)). The composite
action is believed to have been created by the bolts from the retrofit plates extending into
the concrete deck (Figure 4-12). The composite action was unintentional as the retrofit
detail was not specifically designed for composite action. As such, the detail should be
monitored that the stress reduction in the top flanges continues to occur over time.
0
2,000
4,000
6,000
8,000
Date
E-34s-TWW-34s-TW
B
A. Prior to retrofitB. Right (east) lane closed
A
C. Left (west) lane closedD. Retrofit completed
C D
173
Figure 5-38: Contribution to fatigue damage for the top and bottom flanges in the east
girder of Bridge A (a) before and (b) after construction of the retrofit.
Strain gages were added to the retrofit plates at floor beam 34 after the
construction process. The gages indicated that the retrofit plates were not carrying much
of the load, such that the fatigue damage from the retrofit plate barely shows up in Figure
5-39. The low number of fatigue cycles ( , 4.5 ) in the top flanges and the retrofit
plates after construction emphasizes the benefit of the partial composite action between
the concrete deck and longitudinal girder.
Figure 5-39: Contribution to fatigue damage for the top flange of the east girder and
retrofit plate at Bridge A.
0
100
200
300
400
500
0 10 20 30Stress range (ksi)
0
100
200
300
400
500
0 10 20 30Stress range (ksi)
Bottom flange
Co
ntr
ibu
tio
n t
o
Top flange
Co
ntr
ibu
tio
n t
o
(a) (b)
Top flange
Bottom flange
0
20
40
60
80
100
0 1 2 3 4 5 6Stress range (ksi)
Before construction
Co
ntr
ibu
tio
n t
o After
construction
Retrofit plate
174
Floor beam 34 was the last location that was retrofitted on the north span. At
floor beam 27, the gages were monitored for the period before construction, during
construction, and for a couple days after construction (Figure 5-40). The response of the
gages near floor beam 27 was similar to the gages near floor beam 35 during construction
of the retrofit. However, while there was a significant reduction in the fatigue damage in
the top flanges and slight reduction in the bottom flanges at the retrofit locations, the
fatigue damage after construction at the locations near floor beam 27 were similar to the
fatigue damage before construction. Therefore, the retrofit did not generally provide a
measureable benefit at locations away from the retrofit locations.
Figure 5-40: Daily fatigue damage ( , . ) during the construction process for
the east longitudinal girder near floor beam 27 at Bridge A.
5.5 SUMMARY
Strain data from four bridges were analyzed using the techniques presented in
Chapter 3. The details of the bridges were discussed in Chapter 4. Using the
visualization techniques (contribution to damage (Section 3.3.1) and cumulative damage
(Section 3.3.2)), the spectra of stress ranges for each bridge were verified (Section 5.2).
The lower and higher bins did not significantly influence the total fatigue damage if the
cycles were load induced. If cycles corresponding to electromechanical noise were not
0
200
400
600
800
1,000
Date
E-27s-TWE-27s-BW
B
A. Prior to retrofitB. Right (east) lane closed
A
C. Left (west) lane closedD. Retrofit completed
C D
175
truncated from the rainflow histograms (Bridges B and C), then the contribution to
damage of the lower bins was very large and caused steep steps in how each rainflow bin
contributed to the cumulative damage. Similarly, if the higher bins were due to spikes in
the data acquisition system, considering those bins would lead to erroneous indications of
damage that would lead to overestimating the fatigue damage for the monitoring period
(Figure 5-10). Cycles corresponding to electromechanical noise or spikes in the data
acquisition system should be truncated from the rainflow histogram because they do not
induce fatigue damage.
Using the index stress range, the average daily fatigue damage was computed for
each bridge to allow comparison of the relative accumulation of fatigue damage (Section
5.2.6). If the details from each bridge correspond to different fatigue categories, all of the
data should be normalized to a specific detail category using Equation 3-7. By
normalizing to the same index stress range and fatigue category, the relative
accumulation can be compared directly. Bridge A was the most critical of the four
bridges based on relative accumulation of fatigue damage.
The variation in fatigue damage as a function of time of day and day of the week
was evaluated for the bridges (Section 5.3). The fatigue damage varied both hourly and
daily, such that the minimum monitoring period should be a continuous week of data.
However, due to weekly variations in the fatigue damage, a minimum of four weeks of
data are needed to make long-term projections of fatigue damage for the purpose of
calculating the remaining fatigue life.
An added value of using the index stress range to characterize fatigue damage was
exemplified by tracking the progress of the construction of the retrofit at Bridge A
(Section 5.4). Each phase of construction could be identified using the index stress
range. Additionally, the benefit of the retrofit was immediately apparent from the
reductions in the daily fatigue damage.
176
CHAPTER 6
Comparison of the Calculated and Measured
Responses of Bridge A
The calculation of the remaining fatigue life of a connection or component of a
steel bridge relies on an engineer evaluating the fatigue damage at a specific point in
time. The fatigue damage can be estimated by field monitoring or combining structural
analysis with standard traffic counts and design values from the AASHTO LRFD
Specifications (2010). If a structural-analysis approach is chosen, an engineer must
assume a representative fatigue truck, live load distribution among the girders, and
dynamic impact factor to calculate the effective stress range. The suggested design
values from AASHTO may not provide an accurate representation of the service-load
conditions for a particular bridge, leading to unreliable estimates of the fatigue damage.
The distribution of stress ranges under service loads is measured directly when a bridge is
monitored, which reduces the uncertainty associated with the calculated fatigue damage.
A live load test was conducted at Bridge A to compare the assumed design values
utilized in a structural-analysis approach with values obtained from field monitoring.
Specifically, a load test was conducted to determine the effect of vehicle speed and lane
position on the bridge behavior. Based on the field data, a measure of the girder
distribution factor and dynamic impact factor were obtained. The distribution of the
measured stress ranges was compared to the assumed distribution of trucks crossing the
bridge based on the AASHTO fatigue truck. Details of Bridge A are discussed in
Chapter 4. This load test occurred prior to the construction of the retrofit.
6.1 BENDING-STRESS HISTORIES
The load test was performed in July 2011 by driving a single truck across the
bridge to determine the effects of vehicle speed and lane position. A rolling road block
was used to isolate the bridge from other traffic, while minimizing delays to the traveling
177
public. The rolling road block consisted of a test truck entering the freeway followed by
two trailing vehicles. The trailing vehicles traveled at a reduced speed in adjacent lanes
to create a gap (0.25 to 0.5 miles) between the test truck and normal traffic. With gaps
behind and in front of the test truck, it was the only vehicle crossing the bridge during the
test.
The test truck crossed the bridge in various lane positions to determine the
stresses imposed in each girder for different transverse positions. The axle spacings and
weights of the test truck (total weight of approximately 65 kips) are listed in Figure 6-1.
The bridge is currently striped for two traffic lanes (Figure 4-4). The truck was driven in
the right and left lanes at 10 mph, 30 mph, and 63 mph. In addition, the truck was driven
over a proposed new center lane at 10 mph and 30 mph (Figure 6-2). The center lane was
a potential retrofit to combine the two lanes into a single lane to reduce the stress ranges
on the twin-girder structure until a replacement bridge could be designed and constructed.
When a vehicle crosses the bridge in the right lane, most of the load is transferred to the
east girder. Likewise, when a vehicle crosses the bridge in the left lane, most of the load
is transferred to the west girder. The proposed center lane provided a more uniform load
distribution between the girders.
Figure 6-1: Truck weight and spacing of axles.
Strain gages were installed on both sides of the top and bottom flanges to isolate
the stresses from flexural bending and warping torsion (lateral bending of the flanges).
The stresses from flexural bending and warping torsion can be approximated by using
6′-10″ 6′-0″17-9″
19.6kips
18.8 kips
W = 64.9 kips18.8 kips
7.7kips
4′-6″
178
superposition as demonstrated in Figure 6-3. The bending stresses are obtained from the
average of the two readings while the warping stresses are obtained from the difference
between the two readings.
Figure 6-2: Proposed new center lane for Bridge A.
Figure 6-3: Isolation of flexural bending and flange warping stresses.
The bending stress histories of the east and west longitudinal girders at floor beam
34 due to the test truck crossing the bridge at 10 mph are shown in Figure 6-4. For the
purposes of the live load test, this speed was assumed to be representative of a static test.
The static assumption is reasonable because the data show a single, large-cycle stress
range with few secondary stress cycles associated with the dynamic excitation of the
bridge. As expected, when the truck was in the left lane (Figure 6-4(a)), the west girder
experienced a larger stress range (8.0 ksi) than the east girder (4.3 ksi). Conversely, the
4′-0″6′-1½″
West girder East girder
(proposed center lane)
6′-0″8′-6″ 8′-6″
Strain gage
179
stress range in the east girder (9.1 ksi) was larger than the west girder (2.5 ksi) when the
truck was the right lane (Figure 6-4(b)).
Figure 6-4: Flexural stress history in top flange of east and west girders at floor beam
34 due to test truck crossing the bridge at 10 mph in the (a) left lane and (b) right lane.
When the truck crossed the bridge at higher speeds, there was a noticeable change
in the stress history (Figure 6-5). At 63 mph, in addition to the single, large stress cycle,
there were many secondary cycles due to vibration of the bridge. The secondary cycles
were due to dynamic effects from the bridge vibrating as the truck crossed the bridge and
excitation of the truck suspension. At the higher speeds, the flexural stress range was
larger than the measured values for the slower speeds (8.8 ksi (west girder) and 5.3 ksi
(east girder) for a truck crossing in the left lane and 11.1 ksi (east girder) and 5.1 ksi
(west girder) for a truck crossing in the right lane).
The secondary cycles observed in Figure 6-5 increase the fatigue damage as
compared to considering only the primary stress cycle. To calculate the additional
damage, the spectrum of stress ranges from each truck crossing the bridge must be
identified using the rainflow algorithm. For example, the flexural stress ranges identified
in the top flange of the east longitudinal girder near floor beam 34 due to a truck crossing
the bridge in the left and right lanes at 30 mph are summarized in Figure 6-6. As
discussed in Chapter 3, the fatigue damage from the truck crossing the bridge can be
0 4 8 12 16Time (sec)
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
0 4 8 12 16
Flex
ura
l str
ess
(ksi
)
Time (sec)
East girder
West girder
Truck in right lane (10 mph)Truck in left lane (10 mph)
(a) (b)
East girder
West girder
180
represented by the index stress range or the effective stress range. Alternatively, the
truck crossing the bridge can be related to a single cycle at an equivalent stress range
( 1 ), which is calculated to produce an equal amount of fatigue damage as the
spectrum of stress ranges.
Figure 6-5: Flexural stress history in top flange of east and west girders at floor beam
34 due to test truck crossing the bridge at 63 mph in the (a) left lane and (b) right lane.
Figure 6-6: Histogram of flexural stress ranges in the top flange of the east girder
near floor beam 34 due to the test truck crossing the bridge at 30 mph in the (a) left
lane and (b) right lane.
0 2 4 6 8Time (sec)
East girder
West girder
Truck in right lane (63 mph)
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
0 2 4 6 8
Flex
ura
l str
ess
(ksi
)
Time (sec)
Truck in left lane (63 mph)
(a) (b)
East girder
West girder
(a) (b)
0
1
2
3
4
5
0 2 4 6 8 10
Nu
mb
er o
f cy
cles
Stress range (ksi)
0
1
2
3
4
5
0 2 4 6 8 10
Nu
mb
er o
f cy
cles
Stress range (ksi)
181
The damage accumulation index ( ) discussed in Section 2.2.1.1 can be used to
derive the equivalent stress range for a single truck event ( 1 ). The damage from the
spectrum of stress ranges due to the truck crossing the bridge (left side) must equal the
damage from an equivalent stress range for a single cycle (right side) as seen in Equation
6-1. The equivalent stress range for a single cycle can be calculated using Equation 6-2.
∑ 1
1 Equation 6-1
1 /
Equation 6-2
where
1 effective stress range for a single truck event
numberofcyclesinbincorrespondingto
averagestressrangeforbin
fatigueconstantfordetailcategory,definedbyAASHTOLRFDSpecifications 2010 ksi3
The equivalent stress range for a single truck event ( 1 ) was used to relate the
fatigue damage from the test truck crossing the bridge at a vehicle speed of 30 mph
(Table 6-1). As a reference, the largest stress range determined by the rainflow algorithm
is listed in parenthesis in the table. There is a slight difference between 1 and the
largest, measured stress range because the secondary, vibrational stress ranges increase
the apparent damage. At floor beam 35, the bending stresses measured for the top and
bottom flanges exhibited similar amplitudes, which indicate a lack of composite action
between the steel girders and the concrete bridge deck.
For trucks crossing the bridge in the proposed center lane, the load distribution
between the girders was nearly equal. For instance, for a test truck crossing the bridge at
30 mph, equivalent stress ranges of 6.78 ksi in the west girder and 7.19 ksi in the east
girder were recorded in the top flanges near the connection at floor beam 34 (Table 6-1).
These values are approximately 70% of the maxima when the vehicle was in the left lane
(west girder) and the right lane (east girder).
182
Table 6-1: Equivalent bending stress ranges ( ) in longitudinal girders for trucks
crossing the bridge at 30 mph.
Girder Floor beam
Flange for truck location (ksi)
Left lane Right lane Center lane East 27s Top 1.65 (1.52) 3.29 (3.26) 2.44 (2.39) East 27s Bottom 1.95 (1.81) 3.59 (3.55) 2.73 (2.68) East 34s Top 5.49 (5.44) 9.94 (9.93) 7.19 (7.18) East 35n Top 3.32 (3.26) 7.03 (7.03) 4.87 (4.86) East 35n Bottom 5.09 (5.00) 9.37 (9.35) 6.92 (6.89) East 35s Top 4.61 (4.57) 8.49 (8.48) 6.32 (6.31) West 34s Top 9.24 (9.21) 3.00 (2.97) 6.78 (6.74) West 35s Top 7.93 (7.91) 2.56 (2.54) 5.90 (5.87) West 35s Bottom 9.11 (9.06) 3.50 (3.41) 6.79 (6.74)
Value in () is the largest stress range determined by the rainflow method.
6.2 DISTRIBUTION FACTOR
At longitudinal locations where strain gages are installed on both girders, the live
load distribution factor (LLDF) can be determined from the measured data. In Chapter 4,
the LLDF was estimated for each girder using the lever rule (Table 4-4). By comparing
the LLDF derived from measurements with those obtained from the lever rule, the
conservatism of the code equations can be evaluated.
The LLDF for each girder can be calculated using Equation 6-3 and Equation 6-4.
The equations are applicable only if the stress range corresponds to static loads.
Therefore only the results for the test truck crossing the bridge at 10 mph were used to
evaluate the LLDF for the transverse truck positions. Dynamic effects are considered
using the impact factor ( ), which is discussed in Section 6.3.
Δ
Δ Δ Equation 6-3
Δ
Δ Δ Equation 6-4
where
Δ stressrangeinwestlongitudinalgirderduetostatictruckload
183
Δ stressrangeineastlongitudinalgirderduetostatictruckload
The calculated LLDFs using the lever rule were 0.74 in the west girder for a truck
in the left lane and 0.87 in the east girder for a truck in the right lane (Table 4-4). As can
be seen in Table 6-2, the average LLDF for each girder (0.67 in the west girder and 0.77
in the east girder) calculated from strain measurements is smaller than the value
calculated using the lever rule. The measurements indicate that the secondary girder
(west girder for vehicle in the right lane and east girder for a vehicle in the left lane)
participated more in carrying the load than calculated using the lever rule. If the LLDF
from field measurements were used in an analysis with an AASHTO fatigue truck
(Section 4.1.5), the calculated stress range would be smaller, which would increase the
calculated fatigue life.
Table 6-2: Live load distribution factors for Bridge A based on measurements.
Girder Floor beam Flange Location of vehicle
Left lane Right lane
East
35n Top 0.29 0.76 35n Bottom 0.35 0.76 34s Top 0.35 0.78
Average 0.33 0.77
West
35s Top 0.71 0.24 35s Bottom 0.65 0.24 34s Top 0.65 0.22
Average 0.67 0.23
6.3 EFFECT OF SPEED
Due to the differences between Figure 6-4 and Figure 6-5, the speed of the truck
crossing the bridge was observed to influence the stress range. The AASHTO LRFD
Specifications (2010) addresses the increase in forces from dynamic effects through an
impact factor ( ) for dynamic load allowance. For many bridges, the recommended
increase for fatigue is 15%. The factor can be determined through measurements using
184
Equation 6-5. The stress ranges used in Equation 6-5 were calculated considering the
spectra of cycles determined from a rainflow analysis (Equation 6-2).
ΔΔ
1 Equation 6-5
where
Δ stressrangeinlongitudinalgirderduetotruckcrossingbridgeatspeed
Δ stressrangeinlongitudinalgirderduetostatictruckload
The influence of speed on the factor is presented in Figure 6-7 for gages
installed near floor beam 34. The factor is 0.0 at 10 mph because the live load
response was assumed to be static. At higher speeds, the factor increased due to
dynamic effects. The increase in was calculated for only the primary, load-carrying
girder. Therefore, the data for the east girder correspond to a truck crossing the bridge in
the right lane, whereas data for the west girder correspond to a truck crossing the bridge
in the left lane.
A truck traveling at 63 mph in the right lane corresponds to an average factor
of 0.18 near floor beam 34. There is some scatter ( factor between 0.15 to 0.20 at 63
mph) for the different gages installed near floor beam 34. The average factor is
slightly larger than the 0.15 suggested by the AASHTO LRFD Specifications (2010).
The factor exhibited more scatter in the west girder than the east girder. In fact, in the
west girder, the average factor was larger when the truck traveled at 30 mph (0.14) in
the left lane as compared to 63 mph (0.10).
The results are similar for the gages located near floor beam 35 (Figure 6-8). The
average IM factor was greater in the east girder (0.19) than the west girder (0.12), and the
IM factor gradually increased in the east girder at lower speeds and quickly increased at
higher speeds. Thus, for the east girder, one strategy for reducing the fatigue damage
would be to lower the speed limit for vehicles crossing the bridge.
185
Figure 6-7: Influence of speed on at gages located near floor beam 34.
Figure 6-8: Influence of speed on at gages located near floor beam 35.
The proportion of load that each girder carried can be calculated for all speeds
(Equation 6-3 and Equation 6-4). Because the factor was not the same for both
girders at highway speeds, the calculated result was the combined effect of and
factor. As seen in Figure 6-9, the primary girder (west girder when a vehicle is in the left
lane and east girder when a vehicle is in the right lane) carried proportionally less at
higher speeds. As such, the bridge became more efficient at higher speeds as the
secondary girder carried a higher proportion of the load.
0 20 40 60 80Speed of truck (mph)
0.00
0.05
0.10
0.15
0.20
0.25
0 20 40 60 80Speed of truck (mph)
IMfa
cto
rWest girder
(a) (b)
East girder
AverageIndividual gage
AverageIndividual gage
0 20 40 60 80Speed of truck (mph)
0.00
0.05
0.10
0.15
0.20
0.25
0 20 40 60 80Speed of truck (mph)
IMfa
cto
r
West girder
(a) (b)
East girder
AverageIndividual gage
AverageIndividual gage
186
Figure 6-9: Proportion of load carried by the (a) west girder and (b) east girder for
trucks in the left and right lanes.
6.4 BENEFIT OF LANE CHANGES
When the bridge owner was evaluating various methods to keep Bridge A in
service until it could be replaced, one option considered was to change the transverse
locations of the lanes on the bridge. Because the bridge only has two girders, the load
distribution factors are high. The maximum stress range for a given girder can be
reduced by decreasing the number of lanes from two to one. Alternatively, the lanes can
be shifted to load one girder more than the other. Neither option erases previously-
induced fatigue damage, but rather future damage will accumulate more gradually as a
result of the lower stress range. As discussed in previous chapters, the bridge owner
eventually chose to retrofit the bridge and the original lane positions were retained.
Three options were considered when evaluating the shifted lane options: (1) a
single center lane (Figure 6-2), (2) a single left lane (Figure 4-4), and (3) no changes in
lanes. Because the east girder (right lane) exhibited more visual fatigue damage than the
west girder, shifting lanes to create a single right lane was not a practical option.
By moving traffic from the right or left lane to a new center lane, the distribution
factor became approximately equal for each girder (Table 6-1). Therefore, the fatigue
damage in the more severely-damaged east girder reduced by approximately two-thirds if
0 20 40 60 80Speed of Truck (mph)
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80
Pro
po
rtio
n o
f lo
ad
Speed of Truck (mph)
West girder
(a) (b)
East girder
Truck in right lane
Truck in right lane
Truck in left lane
Truck in left lane
187
a vehicle traveled in a new center lane as compared to the right lane. A similar reduction
occurred in the west girder if a vehicle travels in the center lane as compared to the left
lane. However, because all traffic must be moved from the two lanes to a single lane, the
total number of vehicles crossing in the single center lane will exceed the number of
vehicles crossing the bridge in either lane.
The benefit of moving from two lanes to a single lane can be examined by
evaluating the change in fatigue damage. For instance, 500 test trucks can be assumed to
cross the bridge on a daily basis, with 100 in the left lane and 400 in the right lane. As
discussed in Section 6.1, the stress ranges in each girder due to a truck crossing in the
center lane is approximately 70% of the maxima when the vehicle was in the left lane
(west girder) and the right lane (east girder). Because fatigue damage is related to the
cube of the stress range, a truck crossing the bridge in the center lane causes a third of the
damage in each girder on average (two-thirds reduction in fatigue damage). Thus, if a
new center lane was chosen by the bridge owner, the fatigue damage would increase in
the west girder by approximately (500/100x0.33) = 1.7, whereas the damage would
decrease in the east girder by approximately (500/400x0.33) = 0.4. If a single left lane
were chosen, the fatigue damage would increase in the west girder by approximately
(500/100) = 5 and significantly decrease in the east girder. Some damage would still
occur in the east girder with the single left lane because the east girder would carry some
of the load.
Due to the existing damage (continued crack growth) in the east girder,
accumulating more damage in the west girder as compared to the east girder for a short
period of time would be beneficial. Both single-lane options (single center lane or single
left lane) increase the damage in the west girder; however, a single left lane would
decrease the damage to the east girder more than a single center lane. Restriping the
bridge to a single lane would have a significant impact on traffic, which was a negative
aspect of this option due to high traffic volume in the vicinity. Therefore, despite the
benefits to the stress range of restriping the bridge, the bridge owner decided to retrofit
the cracked sections.
188
6.5 EFFECTIVE STRESS RANGE
As discussed in Chapter 4, the calculated effective stress range using structural
analysis depends on the LLDF, factor, and assumed distribution of trucks crossing the
bridge. For analysis, many engineers use the AASHTO fatigue truck, which has an
assumed distribution of trucks crossing a bridge (Figure 6-10) that is based on weigh-in-
motion data from the 1970 FHWA Loadometer Survey (Fisher 1977). If the actual
distribution of trucks is heavier or lighter (Figure 6-10), then the fatigue damage will not
be properly represented using the AASHTO fatigue truck and structural analysis.
Figure 6-10: Possible distributions of truck traffic.
The assumed distribution of trucks crossing the bridge can be compared to the
measured distribution of trucks by evaluating the effective stress range calculated from
both structural analysis and measured stress ranges. If the effective stress range from
measured stress ranges is less than the value calculated from structural analysis, a lighter
distribution of trucks is crossing the bridge. Likewise, if the effective stress range based
on measurements is greater than the value calculated from structural analysis, a heavier
distribution of trucks is crossing the bridge.
The effective stress range ( )) that is determined from structural analysis is
used with the average daily truck traffic (ADTT) crossing the bridge (Chapter 4). In
contrast, the effective stress range ( ) from measured data (Chapter 5) is based on the
0
10
20
30
40
20 30 40 50 60 70 80 90 100
Freq
uen
cy o
f tr
uck
s (%
)
Truck weight (kips)
Lighter distribution of truck traffic
Heavier distribution of truck traffic
Assumed distribution of AASHTO fatigue truck
189
average daily number of measured cycles ( . Therefore, because the corresponding
number of cycles are different (ADTT versus ), the effective stress range calculated
from structural analysis and measured stress ranges cannot be compared directly.
As discussed in Chapter 3, the fatigue damage is defined as the number of cycles
times the cube of the stress range. Thus, the fatigue damage calculated from a given
histogram of measured stress ranges can be related to an infinite number of combinations
of arbitrary stress ranges and corresponding number of cycles. The concept of the index
stress range (Section 3.2.2) utilized that idea to normalize all strain gages to the same
arbitrary stress range so that the metric of comparison was the number of cycles at the
index stress range ( ). Similarly, the fatigue damage can be related to a specific
number of cycles ( ) and the corresponding stress range could be calculated ( ),
which is shown graphically in Figure 6-11.
Figure 6-11: Graphical representation of the method for determining an equivalent
stress range for an assumed number of cycles ( ).
The equivalent stress range for an assumed number of cycles can be
derived using the damage accumulation index ( ) discussed in Section 2.2.1.1. The
damage from the spectrum of stress ranges due to field monitoring (left side) must equal
the damage from an equivalent stress range for cycles (right side) as seen in Equation
6-6. The equivalent stress range can be calculated using Equation 6-7.
Stre
ss r
ange
(lo
g)
Number of cycles to failure (log)
Design category Equivalent stress range for cycles: (1) Choose (2) Determine (3) Calculate
(1)
(3)
190
∑ Equation 6-6
/
Equation 6-7
where
effective stress range due to
number of daily truck events
number of rainflow bins in histogram of stress ranges
If the value for ADTT is substituted for , the equivalent stress range
( ) from measurements can be compared to the effective stress range
determined from structural analysis ( )). By normalizing the measured data to the
same number of cycles from structural analysis (ADTT), the assumed distribution of
trucks crossing the bridge can be compared to the measured distribution of trucks. Using
Equation 6-7, all of the measured cycles during a monitoring period are conservatively
assumed to occur from trucks corresponding to the ADTT, as opposed to some occurring
from light-weight vehicles.
The average number of trucks crossing Bridge A according to a traffic survey in
2005 was 4,000 per day. The effective stress range for 4,000 trucks is presented in Table
6-3 for gages located on the east and west longitudinal girders. The equivalent stress
ranges ( 4,000 ) from measurements are much less than the effective stress ranges
calculated from structural analysis. Therefore, the measured distribution of trucks
crossing the bridge is lighter than the assumed distribution that corresponds to the
AASHTO fatigue truck.
191
Table 6-3: Summary of effective stress range ( , ) before the construction of
the retrofit at Bridge A.
Girder Floor beam Flange ,
(ksi)
from structural analysis (ksi)
Truck in left lane
Truck in right lane
East
35n Top 3.95 4.7 15.5 35n Bottom 5.78 4.7 15.5 35s Top 4.70 4.7 15.5 34s Top 5.15 4.8 16.0
West 35s Top 2.57 13.3 2.4 35s Bottom 3.45 13.3 2.4 34s Top 2.80 13.6 2.4
6.6 SUMMARY
The AASHTO Manual for Bridge Evaluation (2011) provides a method for
calculating the remaining fatigue life. Following the method, the effective stress range
can be determined from structural analysis and the recommended design values from the
AASHTO LRFD Specifications (2010) or field monitoring. If a structural-analysis
approach is utilized, an engineer must assume a representative fatigue truck, girder
distribution factor (LLDF), and dynamic impact (IM) factor. The LLDF and IM factor
were evaluted from a load test at the bridge. As compared to the values in the AASHTO
LRFD Specifications (2010), the measured LLDF in the east longitudinal girder is slightly
smaller while the measured IM factor is slightly larger. The two effects are offsetting,
such that there would not be a significant difference in the effective stress range if the
stress range calculated from structural analysis was used with the measured LLDF and
IM. In the west longitudinal girder, both the LLDF and IM factor determined from
measurements were slightly smaller than the design values.
The major difference in the effective stress ranges determined from structural
analysis and field monitoring was the assumed distribution of trucks crossing the bridge.
Using structural analysis, the AASHTO fatigue truck was used, which assumes a
192
distribution of heavier trucks than actually crossed the bridge. Using an effective stress
range based on the monitoring, a more realistic estimate of the remaining fatigue life can
be obtained.
193
CHAPTER 7
Calculation of Remaining Fatigue Life
Three methods for calculating the remaining fatigue life of steel bridges are
presented in this chapter. The methods are applicable for load-induced fatigue, crack
growth caused by in-plane stresses, and can be used by transportation officials, along
with qualitative data from hands-on visual inspections, to determine if a bridge should be
replaced or can remain in service. The methods in this chapter are not appropriate for
distortion-induced fatigue, which typically occurs from out-of-plane forces and/or
deformations. For a detailed discussion of distortion-induced fatigue, see AASHTO
Manual for Bridge Evaluation (2011).
The fatigue life of a particular bridge depends on the inherent variability of the
fatigue response of the connection details, the accumulation of fatigue cycles (past,
current, and future), and the method of calculation. The uncertainty from the fatigue
response and annual accumulation of stress cycles should be considered when
interpreting the results of the remaining fatigue life. For instance, calculating a negative
fatigue life (bridge age is older than calculated fatigue life) does not necessarily
correspond to failure of the connection. Instead, the negative fatigue life corresponds to a
specific probability of failure of the connection. Because variability affects the
interpretation of the calculated fatigue life, the influence of each source of variability is
discussed in this chapter.
The fatigue life was calculated for the bridges discussed in Chapter 5. The fatigue
damage was determined from both structural analyses (Chapter 4) and field monitoring
(Chapter 5). Three different methods of evaluating the fatigue life of an existing
structure were considered: (1) deterministic, (2) AASHTO, and (3) probabilistic. For the
deterministic and AASHTO methods, the output of the equations is the bridge age when
the fatigue life will be exceeded. The calculated fatigue life from these approaches
194
corresponds to a specific probability of failure. In the probabilistic method, the
probability of failure in a given year is estimated.
For all of the methods, fatigue failure corresponds to fracture of the critical bridge
detail. As was discussed in Section 2.2.2, fatigue cracks grow due to cyclic loading. The
rate of crack growth increases rapidly as the length of the crack approaches the critical
length (Figure 2-10). Fracture occurs when the critical crack length has been reached.
The rapid propagation of the crack to fracture may lead to the loss of the load-carrying
capacity of a connection. If the bridge is fracture critical, the loss of a single connection
may lead to the collapse of the bridge.
7.1 GENERAL CONSIDERATIONS
Estimating the remaining fatigue life is one means of providing quantitative data
to transportation officials for setting priorities among older bridges. The fatigue life can
be estimated using either deterministic or probabilistic approaches. Deterministic
approaches are frequently used because the results are relatively easy to calculate and
understand. However, deterministic approaches may not be the most appropriate method
because uncertainty is not included in the analyses. Metal fatigue is not a simple process;
rather it involves localized damage as the metal is subjected to cyclic loading, which
causes variability in the fatigue life. This inherent variability must be considered when
evaluating an existing bridge. Because the probabilistic approach provides a rational
method for including uncertainty, it may be a better method of calculation.
The current design fatigue relationship given by the AASHTO LRFD
Specifications (2010) is based on experimental tests of typical steel connections subjected
to a constant-amplitude stress range (S-N curves). For a given stress range ( ) and detail
category, the number of cycles to failure can be calculated using Equation 7-1. The
number of cycles to failure corresponds to a design value (95% confidence interval of a
95% probability of survival), as discussed in Section 2.2.1.
Equation 7-1
195
where
number of cycles corresponding to failure of connection at stress range
fatigueconstantfordetailcategory,definedbyAASHTOLRFDSpecifications 2010
constant‐amplitudestressrange
The mean number of cycles until failure ( ) can also be calculated if the mean
value of the fatigue constant ( ) is used rather than the design value (Equation 7-2).
Equation 7-2
where
mean number of cycles until failure of connection at stress range
meanfatigueconstantfordetailcategory,definedbyAASHTOLRFDSpecifications 2010 andAASHTOManualforBridgeEvaluation 2011
The design and mean values of the fatigue constant for each AASHTO fatigue
category were presented in Chapter 2, but are summarized again in Table 7-1. The two
values can produce significantly different values for the number of cycles to failure. For
instance, the design numbers of cycles to failure ( ) at 5 ksi for a Category E detail is
8.8 million, whereas the mean number of cycles to failure ( ) is 13.6 million. Thus, the
mean number of cycles to failure is more than 1.5 times the design value for a detail
characterized as Category E.
196
Table 7-1: Fatigue constant ( ), mean fatigue constant ( ), and constant-amplitude
fatigue limit (CAFL) for each fatigue detail category (AASHTO LRFD Specifications
2010).
AASHTO Fatigue
Category
Fatigue constant (ksi3) CAFL (ksi)
(code value) (mean value) A 250×108 719×108 24.0 B 120×108 237×108 16.0 Bʹ 61×108 117×108 12.0 C 44×108 93×108 10.0 Cʹ 44×108 93×108 12.0 D 22×108 44×108 7.0 E 11×108 17×108 4.5 Eʹ 3.9×108 7.5×108 2.6
7.1.1 Annual traffic growth
Though the fatigue damage induced during a particular period of time can be
obtained directly from field measurements, it is the accumulated damage over the service
life of the bridge that influences the fatigue life. In general, traffic volume is irregular; it
can increase or decrease in a given year, day, or hour due to weather, accidents, and many
other sources, which makes modeling actual traffic patterns nearly impossible. The goal
then is to use a traffic model that on average is representative of the actual traffic
patterns. A given traffic model can be validated if multiple years of traffic data (Average
Annual Daily Traffic (AADT) from counting strips) or measured strain data are available.
If the traffic model cannot be validated, a couple traffic models can be assumed and the
fatigue life can be bounded.
A variety of traffic models can be used to characterize the accumulation of fatigue
damage at a particular bridge (Figure 7-1). Five models for traffic volume are shown in
Figure 7-1: annual growth at a constant rate, annual growth with a gradual limit, no
growth, slow growth with a gradual limit, and quick growth with a gradual limit. As can
be seen in Figure 7-1(b), the models will produce significantly different levels of
197
accumulated traffic volume, depending on the year. Any of these models may be
appropriate for a particular bridge, depending on the traffic conditions.
For this particular example, the traffic volume in year 20 was assumed to be the
same for all models in Figure 7-1(a). That estimate of traffic volume can be thought to
have come from field measurements. Though the traffic volume in year 20 is known, the
estimated traffic volume in future and past years depends on the growth model chosen.
The particular model chosen affects the accumulation of traffic volume (Figure 7-1(b)).
For instance, the accumulated traffic volume in year 50 for 4% annual growth is 40%
larger than the model with no growth. However, between years 0-30, the model with no
growth has a higher accumulated traffic volume.
Figure 7-1: (a) Annual traffic volume and (b) accumulated traffic volume for different
growth models.
Depending on which method is used for calculating the fatigue life, certain
growth models are more useful than others. For instance, with the deterministic
approach, the models that feature annual growth or no growth offer mathematical
advantages because a closed-form solution of the fatigue life can be derived. In contrast,
the general solutions for the models with gradual limits require iterative solutions. Any
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 10 20 30 40 50
Traf
fic
volu
me
Year
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
Acc
um
ula
ted
tra
ffic
vo
lum
e
Year
(a)
(d)
(b)
(a) (b)
(e)
(c)
a) Annual growth at constant rateb) Annual growth with gradual limit
after year 30
c) No growthd) Slow growth with gradual limite) Quick growth with gradual limit
(a)
(d)
(b)
(e)
(c)
198
traffic model can be used with the probabilistic approach because the accumulated
damage must be calculated each year to determine the probability of failure.
Because a closed-form solution is possible for constant annual growth and the
model is representative of a typical traffic pattern, the model is discussed further.
Geometrical traffic growth at an annual rate between 2% and 6% is typically considered
to be more realistic than zero growth (constant traffic volume over the design life of the
bridge). That range of growth rates is recommended by the AASHTO Manual for Bridge
Evaluation (2011). Nonetheless, due to limitations on traffic volume, traffic growth will
likely not increase indefinitely. Thus, some restraint should be exercised when
considering long periods of time (greater than 30 years). In those cases, historical data
(counting strips) may provide a more realistic model to use.
For constant annual traffic growth, the particular annual rate of growth chosen
also affects the accumulated amount of traffic volume at a bridge. If monitoring data
were collected in year 20, the unit amount of traffic volume is known for that year, but no
other year. If annual, traffic growth rates between 2% to 6% are assumed, the difference
in the accumulated traffic volume can be projected (Figure 7-2). For any given year, the
annual traffic volume is greater for a growth rate of 2% prior to year 20; but after year 20,
the annual traffic volume is greater at a 6% growth rate. If the traffic volume is
integrated each year, the accumulated damage at 2% growth rate is greater than 6%
growth rate beyond year 20 (Figure 7-2(b)). It is not until year 33 that the accumulated
traffic volume for all growth rates is approximately equal. After year 33, the
accumulated traffic volume for a 6% growth rate is greater than a 2% growth rate.
Therefore, depending on when the field-measured data are collected and the
amount of damage, a 2% growth rate can produce the smallest value for the fatigue life.
If the growth rate is not known, a range of reasonable growth rates can be assumed to
establish the bounds of the estimated fatigue life.
199
Figure 7-2: (a) Annual traffic volume and (b) accumulated traffic volume for
constant, annual growth rates of 2%, 4%, and 6%.
7.1.2 Infinite fatigue life
From the constant-amplitude fatigue tests, there is a stress range at which crack
growth will not occur. The constant-amplitude fatigue limit (CAFL) is listed in Table 7-1
for each detail category and is the stress range that corresponds to an infinite fatigue life.
If a detail is subjected to varying-amplitude stress ranges, all of the stress ranges must be
below the CAFL to achieve infinite life (as discussed in Section 2.3.1). Thus, the detail is
considered to have infinite life if Equation 7-3 is true.
, Equation 7-3
where
, maximum stress range
constant‐amplitudefatiguelimit
AASHTO Manual for Bridge Evaluation (2011) uses a similar approach for
checking whether a detail has a finite or infinite fatigue life. As can be seen in Equation
7-4, must be less than the CAFL for infinite life. A multiplier of two is used
when calculating because the design effective stress range ( ) is based
0
1
2
3
4
5
6
0 10 20 30 40 50
Traf
fic
volu
me
Year
0
20
40
60
80
100
0 10 20 30 40 50
Acc
um
ula
ted
tra
ffic
vo
lum
e
Year
6% growth
2% growth
Monitoring data collected
4% growth
Monitoring data collected
2% growth
6% growth
4% growth
Accumulated damage is equal
(a) (b)
200
on the AASHTO fatigue truck, which has a total weight of 54 kips. Based on truck
survey data, the maximum truck weight could be twice as much.
Equation 7-4
where
maximum-expected stress range using the Manual for Bridge Evaluation (AASHTO 2011)
2
stress-range estimate partial load factor (Table 7-2)
measuredstressrange;orcalculatedstressrangeduetoAASHTOfatiguetruck
In the commentary to the Manual for Bridge Evaluation (AASHTO 2011), the
engineer is permitted to reconsider the multiplier of two if field data are used to evaluate
a bridge. If the types of vehicles crossing the bridge weigh significantly less than 54 kips
(as determined by a weigh-in-motion station), the multiplier may need to be increased.
Whereas, if the trucks are greater than 54 kips, a smaller multiplier can be used. With
field-measured stress ranges, the length of the monitoring time should be considered to
estimate the weight of the vehicles. If a very short time period (hours to days) is used,
the maximum truck weight may not cross the bridge during the monitoring period. Thus,
a multiplier may be needed. If a longer monitoring period (weeks to months) is used,
trucks at or near the maximum vehicle weight are more likely to have crossed the bridge
and a multiplier of 1.0 may be appropriate.
7.2 DETERMINISTIC APPROACH FOR CALCULATING THE REMAINING FATIGUE LIFE
The remaining fatigue life can be estimated using field results in a deterministic
manner. In the deterministic approach, the fatigue damage (number of cycles at an
effective or index stress range) for a given monitoring period is used to estimate the
damage in the past and forecast future damage by assuming an annual growth in the
traffic volume (Section 7.1.1). In this approach, increases in truck weight are not
201
explicitly considered because legal load limits tend to increase in discrete increments.
Cycles are accumulated over the life of the bridge until the number of cycles to failure is
exhausted. Uncertainty in the fatigue damage that is measured during the monitoring
period is not included in the analysis; however, uncertainty in the fatigue data can be
considered. For the fatigue damage, either the design number of cycles to failure ( ) or
the mean number of cycles to failure ( ) can be used to calculate the fatigue life. In the
following discussion, the fatigue life is assumed to be .
The fatigue life can be derived using the equations below. The fatigue life ( )
can be expressed in terms of years of service (Equation 7-5).
Equation 7-5
where
number of loading cycles in year
fatiguelifeinyears
If the traffic volume is assumed to grow at a constant rate ( ) each year, the traffic
volume in year can be calculated from the traffic volume in the first year (Equation
7-6).
1 Equation 7-6
where
number of loading cycles during first year of service
annualrateofincreaseintrafficvolume
By substituting Equation 7-6 into Equation 7-5, the number of cycles to failure
can be expressed as shown in Equation 7-7.
1 Equation 7-7
202
If the annual number of loading cycles for the current age of the bridge ( ) is
known, either through rainflow counting or from counting strips, the number of cycles in
the first year of service ( ) can be approximated by Equation 7-8.
1
Equation 7-8
where
number of loading cycles in year
currentageofthebridgeinyears
Equation 7-7 is a geometric series, which can be rearranged to produce Equation
7-9.
1 1
Equation 7-9
From Equation 7-9, the fatigue life in years ( ) can be calculated using Equation
7-10.
log 1
log 1
Equation 7-10
Expanding all of the terms in Equation 7-10, the fatigue life depends on the stress
range ( ); the current traffic volume ( ); the current age of the bridge ( ); the
assumed annual rate of increase in traffic volume ( ); and the AASHTO detail category
constant for the bridge ( ), as seen in Equation 7-11.
log 1 1
log 1
Equation 7-11
If the mean value for the fatigue constant ( ) is used, rather than the design value
( ), the mean fatigue life ( ) can be calculated using Equation 7-12.
203
log 1 1
log 1
Equation 7-12
If Equation 7-11 is used, the design value of the fatigue constant ( ) corresponds
to a 5th percentile value (approximately 95% of the connection details will sustain more
cycles than the design value before failure). In contrast, the mean values correspond to a
probability of failure of approximately 50%. Because the annual growth rate ( ) and
stress range ( ) must be approximately by the engineer, the probabilities of failure are
approximate. The stress range ( ) and current traffic volume ( ) used in this analysis
should correspond to the damage in year . As such, either the index stress range ( )
and number of cycles at the index stress range ( , ) in year can be used or the
effective stress range ( ) and corresponding number of measured cycles ( , ).
7.3 AASHTO APPROACH FOR CALCULATING THE REMAINING FATIGUE LIFE
The Manual for Bridge Evaluation (AASHTO 2011) provides an approach for
estimating the remaining fatigue life at three calculation levels: minimum, evaluation,
and mean. In this discussion, this approach will be called the “AASHTO approach.”
AASHTO uses load factors that are calibrated to produce certain probabilities of failure.
As the estimate of the stress range gets closer to the actual stress range, the probabilities
of failure will approach 2% (minimum), 16% (evaluation), and 50% (mean). The
minimum level is two standard deviations below the mean of the fatigue data, whereas
the evaluation level is one standard deviation below the mean.
To follow the AASHTO approach, the stress range is calculated per Equation
7-13.
Equation 7-13
where
stress-range estimate partial load factor (Table 7-2)
measuredeffectivestressrangefollowingPalmgren‐Miner’s
204
rule;orcalculatedstressrangeduetoAASHTOfatiguetruck
Two sources of uncertainty are considered in the load factors by the Manual for
Bridge Evaluation (AASHTO 2011): uncertainty in the analysis ( ) and uncertainty in
the assumed effective truck weight ( ). The partial load factor ( ) can be calculated
using Equation 7-14.
Equation 7-14
where
analysis partial load factor
truck-weight partial load factor
Based on recommendations in the Manual for Bridge Evaluation (AASHTO
2011), the stress-range estimate partial load factors ( ) are summarized in Table 7-2.
Table 7-2: Stress-range estimate partial load factors (AASHTO Manual for Bridge
Evaluation 2011).
Fatigue-life evaluation methods
Stress range by simplified analysis, and truck weight per LRFD Design Article 3.6.1.4
1.0
Stress range by simplified analysis, and truck weight estimated through weigh-in-motion study
0.95
Stress range by refined analysis, and truck weight per LRFD Design Article 3.6.1.4
0.95
Stress range by refined analysis, and truck weight by weigh-in-motion study
0.90
Stress range by field-measured strains 0.85
Mean fatigue life (all methods) 1.00
The fatigue life in years ( ) can be calculated using Equation 7-15. The
resistance factor ( ) depends on the fatigue life that is desired (Table 7-3).
365
Equation 7-15
205
where
resistance factor
numberofstress‐rangecyclespertruckpassage
averagenumberoftrucksperdayinasinglelaneaveragedoverthefatiguelife
The average number of trucks per day in a single lane averaged over the fatigue
life ( ) must be calculated to use Equation 7-15. The AASHTO approach
provides a graph in the commentary that can be used to estimate the average lifetime
traffic volume based on growth rates of 2%, 4%, 6%, and 8%. The number of stress-
range cycles ( ) can be estimated through LRFD Design Table 6.6.1.2.5-2 (AASHTO
LRFD Specifications 2010), influence lines, or field measurements.
Table 7-3: Resistance factor ( ) for evaluation, minimum, and mean fatigue life
(AASHTO Manual for Bridge Evaluation 2011).
AASHTO Fatigue
Category
Minimum
life Evaluation
life Mean
life A 1.0 1.7 2.8 B 1.0 1.4 2.0 Bʹ 1.0 1.5 2.4 C 1.0 1.2 1.3 Cʹ 1.0 1.2 1.3 D 1.0 1.3 1.6 E 1.0 1.3 1.6 Eʹ 1.0 1.6 2.5
The provisions in the AASHTO approach were evaluated by the researchers from
NCHRP Project 12-81 (Bowman, et al. 2012). A slightly different equation was
suggested (Equation 7-16). As can be seen, Equation 7-16 follows the same form as
Equation 7-11 (deterministic approach), except with load factors specified by AASHTO.
206
log
1
3651
log 1
Equation 7-16
where
estimated annual traffic-volume growth rate in percentage
presentageofconnectioninyears
averagenumberoftrucksperdayinyear
In NCHRP Project 12-81, the resistance factors were re-evaluated and updated.
There are also four levels to evaluate the fatigue life: minimum, evaluation 1, evaluation
2, and mean, which correspond to the respective probabilities of failure of 5%, 16%,
33%, and 50%.
Table 7-4: Resistance factor ( ) for evaluation, minimum, and mean fatigue life
(Bowman, et al. 2012).
AASHTO Fatigue
Category
Minimum
life Evaluation 1
life Evaluation 2
life Mean
life A 1.0 1.5 2.2 2.9 B 1.0 1.3 1.7 2.0 Bʹ 1.0 1.3 1.6 1.9 C 1.0 1.3 1.7 2.1 Cʹ 1.0 1.3 1.7 2.1 D 1.0 1.3 1.7 2.0 E 1.0 1.2 1.4 1.6 Eʹ 1.0 1.3 1.6 1.9
7.4 PROBABILISTIC APPROACH FOR CALCULATING THE REMAINING FATIGUE LIFE
Due to the uncertainty in calculating the fatigue life, reliability equations may be a
more appropriate calculation method. To solve for the remaining fatigue life using a
probabilistic approach, a fatigue limit state function ( ) must be considered. As
discussed in Section 2.3.3.3, the fatigue limit state function simplifies to Equation 7-17.
207
Because the function involves products, the limit state function can be modeled by a
lognormal distribution function (Equation 7-18).
0 Equation 7-17
ln ln 1 0 Equation 7-18
where
uncertaintymodelforthefatiguelimitstatefunction
Δ Palmgren‐Miner’scriticaldamageindex resistance
numberofaccumulatedcyclesatbridgeinyear correspondingtoastressrange loading
Lognormal distributions can be used for each random variable ( , , , and
) in Equation 7-17. As discussed in previous chapters, the fatigue damage is
characterized by both the stress range ( ) and number of stress cycles ( ). As such,
the two random variables can be combined into a single variable ( ) using Equation
7-19. Equation 7-19 can be substituted into Equation 7-17 and lognormal distributions
can be used to define the three variables ( , , and )).
Equation 7-19
where
total accumulated fatigue damage at time
A random variable is defined for to account for the uncertainty from using
Palmgren-Miner’s rule. As discussed in Section 2.2.1.1, Δ can be modeled by a
lognormal distribution with a median value ( ) of 1.0 and coefficient of variation ( ) of
30% (Wirsching 1984). For most tests, the median is 1.0 and the coefficient of variation
ranges between 30-60% (Wirsching and Chen 1988). When developing a procedure for
estimating the remaining fatigue life in bridges, Moses, Schilling, and Raju (1987) used a
value of of 15%. For this dissertation, = 1.0 and = 0.3 was assumed. The
lognormal parameters can be calculated using Equation 7-20 and Equation 7-21 (Table
7-5).
208
ln Equation 7-20
ln 1 Equation 7-21
where
median value of
coefficientofvariationof
Table 7-5: Lognormal parameters for random variable ( ) used in this dissertation.
Parameter Value 0.0 0.163
The variation in the fatigue constant ( ) can be determined from the test results in
the AASHTO fatigue database. After studying the data from 374 tests, Fisher et al.
(1970) concluded that log can be assumed to follow a normal distribution. As
such, log ) can also be assumed to follow a normal distribution, while follows a
lognormal distribution. To model using a lognormal distribution, the variation in
log ) must be transformed using Equation 7-22 and Equation 7-23. The derived
parameters of the lognormal distribution for each fatigue category are summarized in
Table 7-6.
ln 10 log Equation 7-22
ln 10 log Equation 7-23
where
mean parameter of lognormal function for fatigue constant ( )
deviation parameter of lognormal function for fatigue constant ( )
mean of fatigue constant ( )
normalizedstandard deviationoffatigueconstant
While random variables are used to describe and , the cumulative damage can
be estimated from measurements. Most measurement periods used to monitor highway
209
bridges tend to vary between two to eight weeks during year after the bridge was first
put in service. To calculate the cumulative fatigue damage, the previous amount of
damage has to be estimated. Assuming the average traffic volume grows geometrically at
a constant rate each year, the cumulative fatigue damage can be estimated by Equation
7-24. If a different traffic model is desired (i.e. one that limits the growth), it can be
substituted into the right side of Equation 7-24 because the total damage has to be
calculated each year.
1 1
Equation 7-24
where
mean damage in first year of service
yearevaluated
Table 7-6: Variation in fatigue constant ( ) from AASHTO Manual for Bridge
Evaluation (2011) and derived parameters for lognormal distribution.
AASHTO Fatigue
Category
Fatigue constant
Parameters for lognormal distribution
(code value) (mean value)
A 250×108 719×108 1.64 25.0 0.49 B 120×108 237×108 1.38 23.9 0.32 Bʹ 61×108 117×108 1.35 23.2 0.30 C 44×108 93×108 1.42 23.0 0.35 Cʹ 44×108 93×108 1.42 23.0 0.35 D 22×108 44×108 1.38 22.2 0.33 E 11×108 17×108 1.25 21.3 0.22 Eʹ 3.9×108 7.5×108 1.35 20.4 0.30
As seen from Equation 7-24, the cumulative damage depends on the mean
damage in the first year of service. The mean fatigue damage from the current year ( )
of service, obtained from measurements, can be used to approximate the damage from the
first year by using Equation 7-25.
210
1 Equation 7-25
where
mean damage in year
currentyear
Because and are lognormally distributed and the limit state function
(Equation 7-18) involves products, the reliability factor ( ) can be determined using
Equation 7-26.
Equation 7-26
where
parameterforthelognormaldistributionfor
ln2
parameterforthelognormaldistributionfor
ln 1
The probability of failure can be calculated using Equation 7-27.
Φ Equation 7-27
If there is no variability in the fatigue damage, the reliability index ( ) can be
simplified (Equation 7-28).
ln
Equation 7-28
If the index stress range and number of cycles at the index stress range are used,
rather than the fatigue damage, the reliability index can be calculated using Equation
7-29. Because the value used for the index stress range is selected by the engineer, the
211
number exhibits no variability. As such, all of the variability in fatigue damage would be
in the cumulative number of cycles for the bridge at the index stress range ( ).
Similarly, the mean number of cycles in the first year of service ( ) can be
calculated using Equation 7-30.
3 ln
Equation 7-29
1 Equation 7-30
where
parameterforthelognormaldistributionfor
ln2
parameterforthelognormaldistributionfor
ln 1
7.5 CALCULATING THE REMAINING FATIGUE LIFE AT BRIDGES MONITORED DURING
THIS INVESTIGATION
The remaining fatigue life of the four bridges monitored as part of the bridge can
be estimated using the equations listed in the previous sections. The amount of damage
was determined from both structural analysis and field monitoring. The fatigue lives
from both approaches are compared to evaluate the effectiveness of each source.
Because at least four weeks of data were not captured at Bridges C and D, the remaining
fatigue lives were not calculated for connection details at those bridges. In addition,
those two bridges have redundant structural systems and are not as critical as fracture-
critical bridges.
212
7.5.1 Bridge A
The effective stress range from structural analysis (Chapter 4) and the maximum-
measured stress range from field monitoring (Chapter 5) were larger than the CAFL.
Therefore, Bridge A is expected to have a finite fatigue life. The field-measured data
were used to calculate the fatigue life using all three methods. In contrast, only the
AASHTO approach was used to estimate the fatigue life using the results of structural
analysis. Two strain gages at Bridge A will be discussed in this section to exemplify the
methods for calculating the remaining fatigue life. Gage W-34s-TE was installed on the
east side of the top flange of the west longitudinal girder near the connection to floor
beam 34, whereas gage E-34s-TW was installed on the west side of the top flange of the
east longitudinal girder near the connection to floor beam 34. In this section, W-34s-TE
and E-34s-TW are referred to as the gages on the east and west girders, respectively
7.5.1.1 Deterministic Method
The fatigue life can be estimated from the deterministic method from the fatigue
damage for a given year and the annual rate of growth of traffic volume. If the annual
traffic growth rate is not known, the range of typical growth rates (2%-6%) can be
assumed to establish the bounds of the fatigue life. As more data are collected, the actual
growth rate could be extracted from traffic count data and used to improve the estimate.
To evaluate the remaining fatigue life, the damage in year 1 was estimated using
Equation 7-8. As seen in Table 7-7, there is considerable difference in the amount of
damage calculated in the first year of service for assumed rates of traffic growth between
2% and 6%. For both gage locations, the extrapolated fatigue damage in the first year for
an assumed annual growth rate of 2% is four times larger than the damage for an annual
growth rate of 6%.
213
Table 7-7: Fatigue damage in year 37 and extrapolated damage in year 1 for annual
growth rates of 2%, 4%, and 6%.
Annual growth
West girder East girder
, . in year 1
, . in year 37
, . in year 1
, . in year 37
2% 610 1,250 2,990 6,070 4% 300 1,250 1,480 6,070 6% 150 1,250 750 6,070
The fatigue lives were calculated for the west and east girders (Table 7-8). The
age of the bridge connection was 37 years at the time of the calculation. As seen in Table
7-8, the design and mean fatigue lives have both been exceeded in the east girder, which
corresponds to a negative fatigue life. The mean fatigue life in the east girder has been
far exceeded if the 2% annual growth rate is correct. If the 6% growth rate is more
representative of traffic growth, the mean fatigue life has still been exceeded, but not by
as many years. .
Table 7-8: Calculated fatigue life in years for the west and east girders using the
deterministic approach.
West girder East girder
2% 4% 6% 2% 4% 6%
5% (design) 37 43 45 10 16 22 50% (mean) 50 52 52 15 22 27
Based on the calculations, the west girder has a longer fatigue life than the east
girder. The design fatigue life of the west girder has only been exceeded for 2% annual
growth. For the other two growth rates, the fatigue life has not been exceeded for the
design and mean calculation levels.
7.5.1.2 AASHTO Method
Following the AASHTO method, the fatigue life can be calculated if the lifetime
average number of daily trucks crossing the bridge on a daily basis is known for the
214
calculated effective stress range. The effective stress range can be calculated by either
structural analysis or from field measurements. To provide a consistent level of
reliability for the different calculation levels, AASHTO recommends different partial
load factors depending on the method used to determine the effective stress range (Table
7-2).
The structural analysis of Bridge A was discussed in Section 4.1.5. The effective
stress range calculated for an AASHTO fatigue truck (54 kips) was 16 ksi in the east
girder. In 2005, the ADTT was determined to be 4,000. If 80% of the trucks are
assumed to cross the bridge in the right lane (suggested value in the AASHTO LRFD
Specifications (2010)), the fatigue life would have been exceeded in less than one year
for all calculation levels.
The fatigue life was severely underestimated using the results from the structural
analysis due to the assumptions of the model and because no knowledge of the weight of
typical truck traffic at Bridge A was known. As was discussed in Section 6.5, the stress
range for 4,000 cycles per day corresponds to an effective stress range of 5.15 ksi at floor
beam 34. Thus, the effective stress range from field monitoring was much less than the
effective stress range calculated from structural analysis using an AASHTO fatigue truck
at this bridge.
The fatigue damage obtained from field measurements can also be used with the
AASHTO method. Because the proposed method for calculating annual growth from
NCHRP project 12-81 provides an equivalent method to AASHTO Manual for Bridge
Evaluation (2011), Equation 7-16 was used to calculate the fatigue life.
A partial load factor of 0.85 is recommended for minimum, evaluation 1, and
evaluation 2 calculation levels when field measurements are used to calculate the fatigue
life. However, for the mean calculation level, the recommended value for the partial load
factor is 1.0. Because there is not much difference between the resistance factors ( ) for
the four levels of a Category E detail, the calculated fatigue lives at the minimum,
evaluation 1, and evaluation 2 levels are greater than the mean fatigue lives due to the
215
difference in the partial load factor (Table 7-9). This is unexpected result because the
mean fatigue life has a higher probability of failure than the other three calculation levels.
Table 7-9: Calculated fatigue life in years for different considerations using the
AASHTO approach for the west and east girders.
West girder East girder
2% 4% 6% 2% 4% 6%
5% (design) 51 53 53 16 23 29 16% (evaluation 1) 57 57 56 18 26 31 33% (evaluation 2) 63 61 59 21 28 34
50% (mean) 51 53 53 15 23 29
7.5.1.3 Probabilistic Method
The probability of failure for a given year can be calculated using the probabilistic
approach. The approach allows a bridge owner to evaluate the risk of keeping a bridge in
service for a given duration and probability of failure. The fatigue life from the
probability of failure can be compared to other methods by determining the year when a
given level (i.e. 5%, 50%, etc.) is crossed. Unlike the other two methods, the
probabilistic method considers uncertainty in the fatigue damage. For Bridge A, the
uncertainty in the fatigue damage in year 37 is summarized in Table 7-10.
Table 7-10: Mean and standard deviation in year 37 for the west and east girders.
, . in year 37 , . in year 37
West girder 1,250 113 East girder 6,090 526
The probabilistic approach was applied to the west and east girders (Figure 7-3).
Comparing the two locations, the east girder has a higher probability of failure than the
west girder for any given year. This is expected since more damage occurred in the east
girder than the west girder during the monitoring period.
216
Because the annual rate of growth is not known for the bridge, a range of growth
rates (2%-6%) were considered. If the 2% model of annual growth is correct, the
probability of failure for the current age of the bridge is nearly 100% for the east girder.
If 6% growth is correct, the probability of failure is 95%.
Figure 7-3: Probability of failure in the (a) west and (b) east girders.
For the west girder, the probability of failure is between 0.5% and 11% in year 37.
As was discussed with Figure 7-2, 2% growth will have a higher probability of failure for
years prior to the current age of the bridge and slightly into the future. Though, in year
55 and beyond that point, 6% growth has a greater probability of failure as the
accumulated cycles for 6% growth exceed 2% growth.
7.5.1.4 Comparison
The calculated fatigue lives for different probability levels are considered in Table
7-11 for the three methods discussed (deterministic, AASHTO, and probabilistic).
Whereas the probability of failure for any given year can be determined using the
probabilistic approach, only certain probability levels can be determined using the
AASHTO (5%, 16%, 33%, and 50%) and deterministic approaches (5% and 50%).
0 10 20 30 40 50 60 70
Age of Bridge
0.01
0.10
1.00
0 10 20 30 40 50 60 70
Pro
bab
ilit
y o
f fa
ilu
re
Age of Bridge
6% growth
4% growth2% growth
Current age
= 5%
Mean life
Current age
Mean life
2% growth
6% growth
4% growth
(a) (b)
= 5%
217
Table 7-11: Calculated fatigue life in years for AASHTO, Deterministic, and
Probabilistic approaches for the west and east girders.
West girder East girder
Deterministic AASHTO Probabilistic Deterministic AASHTO Probabilistic
2% annual growth 5% 37 51 33 10 16 9 16% - 57 39 - 18 11 33% - 63 45 - 20 13 50% 50 51 50 15 15 15
4% annual growth 5% 43 53 39 16 23 14 16% - 57 44 - 26 17 33% - 61 49 - 28 20 50% 52 52 52 22 23 22
6% annual growth 5% 45 53 42 22 29 20 16% - 56 46 - 31 23 33% - 59 50 - 34 26 50% 52 53 53 27 27 28
The deterministic and probabilistic approaches produced comparative fatigue
lives for the two probability levels for all growth rates. The fatigue lives in the east
girder were closer between the deterministic and probabilistic approaches because the
uncertainty in the current level of fatigue damage did not have as large of an influence on
the calculation of fatigue life. In contrast, the uncertainty had more of an influence on the
fatigue life for the west girder, which is why the fatigue life for the probabilistic method
had a lower fatigue life than the deterministic approach, in general.
The AASHTO approach was only comparative to the deterministic and
probabilistic approaches at a probability of failure of 50%. For the other three levels
(5%, 16%, and 33%), the fatigue life for the AASHTO method was much higher than the
probabilistic method. This can be attributed in part to the use of the partial load factor of
0.85.
218
7.5.2 Bridge B
From the measurement data, all of the stress ranges were below the CAFL. Thus,
the bridge is considered to have an infinite fatigue life. For the structural analysis
considered in Section 4.2.3, the largest effective stress range that was calculated was 4
ksi. Therefore, if the structural analysis data were used, the bridge would still be
considered to have an infinite life.
7.6 SUMMARY
The fatigue life of a particular bridge depends on the inherent variability of the
fatigue response of the bridge details, accumulation of fatigue cycles, and method of
calculation. Three methods of calculating the remaining fatigue life of a connection were
considered in this chapter. The output of the deterministic and AASHTO approaches is
the bridge age when the fatigue life is exceeded (for a specific probability of failure). For
the probabilistic method, the probability of failure for a given year is estimated.
The remaining fatigue lives of Bridges A and B were calculated using the three
methods discussed in this chapter. Based on the measured maxima at each bridge, Bridge
A was expected to have a finite fatigue life and Bridge B was expected to have an infinite
fatigue life. For Bridge A, the calculated fatigue lives based on the results of the
structural analysis was overly conservative and not representative of the actual behavior
of the connection. In contrast, the calculated fatigue lives corresponding to the measured
data were more representative than using structural analysis.
At Bridge A, the deterministic and probabilistic approaches produced comparative
fatigue lives at the two calculation levels (design and mean fatigue lives). The AASHTO
approach was only comparable to the deterministic and probabilistic approaches at the
mean fatigue life. At the other three calculation levels (design, evaluation 1, and
evaluation 2), the calculated fatigue life for the AASHTO approach was greater due to
the partial load factor of 0.85, which is allowed for field measurements.
219
CHAPTER 8
Conclusions
This chapter summarizes the conclusions of the research that was conducted as a
part of the NIST-funded project entitled, “Development of Rapid, Reliable and Economic
Methods for Inspection and Monitoring of Highway Bridges.” The research presented in
this dissertation represents a small portion of the research project as a whole. The major
conclusions from this dissertation will be discussed, as well as recommendations for field
monitoring.
8.1 SUMMARY
Highway bridges provide vital links in the transportation network, supplying
routes for daily commutes and the infrastructure needed to supply goods and services. As
such, transportation officials seek to maintain the bridge inventory under the pressure of
reduced budgets and an aging infrastructure. One of the critical types of structural
deterioration for steel bridges is fatigue-induced fracture.
If fracture is allowed to occur, localized damage can propagate to member failure
or even bridge collapse. As more bridges near or exceed their intended design lives,
transportation officials must make decisions on which bridges can be safely kept in
service and which need to be replaced or retrofitted. The primary method used to identify
structural deterioration is hands-on visual inspections. The inspections provide
transportation officials with qualitative data related to the number of cracks and rate of
crack growth, but quantitative data are often needed to distinguish among the bridges in
an inventory.
Recent legislation in the US Congress highlights the need for quantitative data to
identify the most vulnerable bridges. MAP-21, the Moving Ahead for Program in the 21st
Century Act, was signed into law on July 6, 2012 (Federal Highway Administration
220
2012). MAP-21 allows federal money to be used for new construction or rehabilitation
of highway bridges based on performance-based or risk-based criteria.
Calculating the remaining fatigue life is one means of providing quantitative data
to transportation officials. To estimate the remaining fatigue life of a bridge, the fatigue
damage must be characterized using either the results of structural analysis or measured
strains. Field monitoring provides a direct measure of the spectrum of stress ranges at the
location of a strain gage, whereas values for the representative fatigue truck, live load
distribution factors among girders, and dynamic impact factor must be assumed to
calculate the stress range using structural analysis.
With field-monitored data, the variable-amplitude stress ranges induced by trucks
of varying weights must be analyzed with a cycle-counting algorithm to produce a
histogram of stress ranges. The distribution of stress ranges is often converted to an
equivalent, constant stress range that can be used with the fatigue database (AASHTO S-
N curves). Historically, this conversion has been accomplished by calculating an
effective stress range using Palmgren-Miner’s linear damage rule. However, a new
method, the index stress range, of normalizing the damage was developed so that
different gage locations could be compared directly using the number of cycles as the
metric of damage accumulation (Chapter 3). Further techniques for visualizing the
contribution to damage and cumulative damage were presented to evaluate the influence
of a given stress range to the total amount of damage.
Once the accumulated fatigue damage has been estimated using the measured
data, the remaining fatigue life can be calculated. The fatigue life of a particular bridge
depends on the inherent variability of the fatigue response of the connection details, the
rate of accumulation of fatigue cycles (past, current, and future), and the method of
calculation. Deterministic approaches do not directly include uncertainty in the analysis,
yet using those approaches, the remaining fatigue life is easy to calculate and understand.
In contrast, probabilistic approaches provide a method for including many types of
uncertainty in the analysis, but the results are not as easily understood.
221
8.2 CONCLUSIONS
The focus of this dissertation was to develop methodologies that enable
transportation officials and/or engineers to collect quantitative information on fatigue
behavior of steel bridges by monitoring the service-load response. As such, techniques
for analyzing fatigue damage were developed and were applied to four steel bridges. The
inherent variability in fatigue damage was also evaluated on hourly, daily, and monthly
bases. The remaining fatigue lives of the bridges were calculated using both
deterministic and probabilistic approaches.
The conclusions from this dissertation are divided into three areas: techniques for
fatigue analysis, field monitoring, and calculating the remaining fatigue life. Conclusions
from each area are summarized in the following sections.
8.2.1 Techniques for Fatigue Analysis
Five techniques for analyzing fatigue damage from measured strain data were
discussed in Chapter 3. The first step in analyzing the fatigue damage is to use a cycle-
counting algorithm, such as the simplified rainflow algorithm, to transform a stress
history into a spectrum of stress ranges. Next, the total accumulation of fatigue damage
during a monitoring period can be expressed using the traditional approach—the effective
stress range—or a new approach—the index stress range. Finally, visualization
techniques, contribution to damage and cumulative damage, can be used to evaluate
whether lower or higher stress ranges are contributing significantly to the accumulation
of damage during the monitoring period.
1. The simplified rainflow method is well suited for fatigue analyses. As discussed
in the literature review, the simplified rainflow method is considered superior to
other cycle-counting methods because it identifies stress ranges associated with
closed-loop hystereses. By counting stress ranges in that manner, the histogram
of stress ranges is compatible with the empirical data from constant-amplitude
fatigue tests (Swenson and Frank 1984).
222
2. An index stress range is a better indicator of damage than an effective stress
range. Fatigue damage is characterized by the number of loading cycles applied
and the stress range. Because Palmgren-Miner’s rule was used to calculate the
fatigue damage, the total fatigue damage calculated using the effective stress
range is equivalent to that calculated using the index stress range. However, the
calculation of the effective stress range is sensitive to the number of cycles
measured by the data acquisition system. Thus, the effective stress range can
change drastically if the low-amplitude cycles are truncated from the calculation.
Because the data are normalized using the index stress range, the cycles
associated with the index stress range are not sensitive to truncating the small-
amplitude stress ranges. Therefore, the index stress range is a better indicator of
fatigue damage because it does not rely on the number of cycles measured.
3. The index stress range can be used to easily compare the measured response at
different locations of a bridge or of different bridges. By normalizing the data to
an index stress range, the number of cycles at the index stress range becomes the
direct metric of relative damage: twice as many cycles at the same index stress
range causes twice the damage. In contrast, the effective stress range and the
number of fatigue cycles at two locations will likely be different. Therefore, to
compare those locations using the effective stress range, the number of measured
cycles and cube of the effective stress range will have to be considered together to
compare the relative damage accumulation.
4. The contribution of each bin in a rainflow histogram to the fatigue damage and
cumulative damage can be calculated. As was noted in the literature review,
many researchers truncate the lower bins of the measured strain response under
the assumption that the lower stress ranges do not contribute to the fatigue
damage. There is not international agreement on how cycles below the constant-
amplitude fatigue life (CAFL) contribute to the fatigue damage. In the US, all
cycles are weighted equally (Palmgren-Miner’s rule). In Europe, stress cycles
less than the CAFL are weighed differently (different slope) than cycles above the
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CAFL. Though studying how to handle cycles below the CAFL was not
explicitly addressed in this dissertation, the contribution to damage and
cumulative damage techniques allow an engineer to assess the influence of all
stress ranges, below and above the CAFL.
8.2.2 Field Monitoring
Strain data from four bridges were obtained and evaluated with respect to the
accumulation of fatigue damage. Two of the bridges featured twin girders and were
considered to be fracture critical, whereas the other two bridges had redundant structural
systems. The following are conclusions relative to the field monitoring of the four
bridges.
1. Fatigue damage obtained from field-monitored data is more representative than
damage calculated from structural analysis and standard design assumptions.
AASHTO LRFD Specifications (2010) recommends values to calculate the
fatigue damage using structural analysis, such as a representative fatigue truck
(Figure 2-16), dynamic impact factor (10% to 30%), and live load distribution
factors. However, those values may not be appropriate for all bridges and could
lead to significant error in the calculated remaining fatigue life. The use of
stresses calculated from measured strains provide representative data regarding
the number and amplitude of fatigue cycles during the measurement period,
thereby improving the estimate of the fatigue damage. For Bridge A, the
distribution factor, dynamic impact factor, and distribution of stress ranges that
were measured using strain gages were different than the values obtained
following AASHTO LRFD Specifications (2010). As such, the calculated
remaining fatigue life based on structural analysis was very conservative as
compared to the calculated remaining fatigue life based on strain gages.
2. Field-monitored data should be captured in weekly intervals due to hourly and
daily variations in fatigue damage. With each bridge, the traffic patterns varied
depending on the hour of the day and day of the week. On average, most fatigue
224
damage was induced between 8:00 AM and 6:00 PM, which corresponds to a
typical work day. For a given six-hour period, the maximum damage depended
on the bridge, but was either from 6:00 AM to 12:00 PM or 12:00 PM to 6:00
PM. On a daily basis, the maximum amount of damage occurred on weekdays,
rather than weekends. Due to such wide variation, the minimum amount of data
collected to calculate the fatigue damage should be a continuous week.
Monitoring for only part of the day or week will not be representative of the
average daily damage at the bridge.
3. One continuous week of data is typically inadequate for long-term projections
of fatigue damage. Traffic patterns are irregular and can increase or decrease due
to weather, accidents, and many other factors. As such, obtaining only a
continuous week of data is inadequate for long-term projections of fatigue
damage. For Bridge A, the amount of induced fatigue damage also varied month
to month. The estimate in the average amount of fatigue damage can be improved
by monitoring over longer periods of time. Four weeks is considered to be the
minimum monitoring period to make long-term projections.
4. Most of the fatigue damage is caused by a narrow range of stress ranges, which
corresponds to typical truck traffic. From the cumulative-damage plots for each
bridge, 95% of the fatigue damage was typically induced by less than 50% of the
range of stress cycles. Those cycles correspond to vehicles that cause a mid-level
amount of stress a moderate number of times per day (typical truck traffic).
Though there are many more cycles at the lower stress ranges, those cycles did
not cause a significant amount of damage due to the low value of stress. If the
cycles corresponding to electromechanical noise from the data acquisition system
were not truncated from the rainflow histograms (Bridges B and C), then the
contribution to damage at the lower stress ranges was very large. However, those
cycles should be truncated because the electromechanical noise does not
correspond to load-induced fatigue cycles and causes no fatigue damage. The
225
largest stress ranges from heavier vehicles also did not cause a significant amount
of damage because they occurred with low frequency.
5. The benefit of retrofitting Bridge A could be tracked effectively using an index
stress range. Each phase in the construction of the retrofit (right lane closed, left
lane closed, both lanes reopened) could be identified from the daily fatigue
damage because the data was normalized using an index stress range. In addition,
the impact of the retrofit could be quantified before and after the construction.
The significant reduction in the fatigue damage in the top flange and moderate
reduction in the bottom flange is consistent with composite action between the
concrete deck and longitudinal girders. The retrofit was not specifically designed
to create composite action and could degrade with time. The gages at the detail
could continue to be monitored to detect if the composite action degrades.
6. Crack propagation gages can be used to track crack growth on steel bridges.
By measuring the resistance of the gage every 30 minutes, the drift of the gage
due to temperature was less than the change in resistance due to a wire break.
Thus, the gage can be used to track crack growth between bridge inspections and
notify transportation officials if problems arise.
8.2.3 Remaining Fatigue Life
After the fatigue damage had been determined and characterized, the remaining
fatigue lives of the bridges were calculated using both deterministic and probabilistic
approaches. The following are conclusions relative to the calculation of the remaining
fatigue life that was discussed in Chapter 7.
1. There is much uncertainty in estimating the fatigue life of a bridge. Traffic
patterns can change drastically from hour to hour, day to day, and year to year.
Weather, traffic accidents, and many other sources affect the accumulation of
traffic and fatigue damage at a bridge. Where possible, historic traffic counts or
strain data should be utilized to validate the growth model chosen for a particular
bridge.
226
2. Deterministic and probabilistic methods can provide similar remaining fatigue
lives for some probabilities of failure. If the uncertainty from determining the
fatigue damage is small (as for Bridge A), deterministic and probabilistic methods
can provide comparable results. Though, using a deterministic approach, the
calculated fatigue life corresponds to a limited number of probabilities of failure
(5%, 16%, 33%, and 50%, depending on the method).
3. For the AASHTO approach (AASHTO Manual for Bridge Evaluation 2011),
reducing the stress range by a factor of 0.85 when field-monitored data are used
is not justified. With the new equation proposed by NCHRP Project 12-81, the
only difference between the deterministic method discussed in Section 7.2 and the
AASHTO approach is the stress-range estimate partial load factor ( ).
AASHTO allows a load factor of 0.85 when field-monitored data are used to
calculate the effective stress range. By reducing the stress range by 0.85, the
fatigue life increases, which is not justified with the other factors considered. In
addition, for certain fatigue categories, such as Category E, the mean fatigue life
will produce a smaller value than the other fatigue lives (minimum, evaluation 1,
and evaluation 2).
4. The probabilistic method provides the framework for calculating the risk of
keeping a bridge in service. Risk involves making decisions with the possibility
of loss. With the probability of failure calculated for each year, the owner can
weigh the potential cost from collapse or failure with the cost from replacing the
bridge. The loss can take into account user costs from the bridge being closed,
cost of bridge replacement, and cost of additional inspections. Knowing all of
these costs, the owner can determine the optimal point of replacing the bridge
and/or increasing the rate of inspections to minimize the loss to the public.
8.3 RECOMMENDATIONS
A variety of steps are needed to characterize the fatigue damage at a bridge and
estimate the remaining fatigue life through field monitoring. First, critical fatigue
227
connection details must be identified through a combination of engineering judgment and
analysis. Some poor fatigue details may not be immediately apparent and could be
discovered from inspection reports. Creating a structural-analysis model of the bridge
can be useful in determining locations with the highest stress range, as those locations
may not always correspond to the location with the highest moment (as in Bridge A).
Once the critical locations have been identified, durable strain gages should be
installed and coupled with a high-resolution data acquisition system. Wired or wireless
data acquisition systems may be used. Though, wireless systems offer many benefits,
such as lower costs, quicker installations, and less susceptibility to high-amplitude spikes
due to lightning strikes, the wireless systems also require more power than a comparable
number of channels on a wired system and are subject to loss of data from wireless
interference. Transportation officials and/or the engineer must weigh those benefits when
choosing a data acquisition system. Nonetheless, the following features are
recommended for the data acquisition system: electromechanical noise is limited to 1-5
microstrain at faster scan rates (50-100 scans per second); the sensors can be calibrated
using shunt calibration; and high-precision completion resistors should be utilized to
minimize drift from temperature fluctuations if quarter-bridge sensors are used.
If a long-term projection of the fatigue damage is desired, four weeks of data are
recommended. As mentioned previously, due to the hourly and daily variation in
damage, a continuous week of data is the minimum for determining the average daily
fatigue damage. If the data acquisition system is a permanent or semi-permanent
installation, multiple months of data can be captured continuously. Though, if power is a
concern, as it is with wireless data acquisition systems, two continuous weeks of data
could be captured in a particular month and two more continuous weeks of data could be
captured in another month. The data should not correspond to a week with a national
holiday, as that will lower the estimate of the fatigue damage.
The dynamic data should be analyzed using the techniques described in Chapter
3. The spectrum of stress ranges from typical traffic can be identified using the
simplified rainflow method. Care should be taken to choose parameters (size of bins and
228
number of bins) for the rainflow method such that wide-enough spectra of stress ranges
are selected. The calculated stress range from an AASHTO fatigue truck crossing the
bridge can be used as an estimate of the required spectrum of the stress ranges: the
spectrum should be at least twice as large as the calculated stress range. Though, the
histogram of stress ranges should be investigated after a few days of monitoring to ensure
that the spectrum is large enough.
The width of each rainflow bin is important. A width of 5 microstrain was used
for each bin in this dissertation, and it provided enough fidelity that the contribution to
damage and cumulative damage plots were smooth. Larger bin widths can be used to
limit the amount of data that needs to be transferred; but the plots may become more
jagged. In addition, the estimate of the fatigue damage might become less accurate
because all of the cycles in a bin are assumed to have the same stress range (Figure 8-1).
If a very wide bin width is used (i.e. 30 microstrain) and the majority of the cycles are
near the bottom of the bin (i.e. 0-5 microstrain), assuming all of the cycles were 15
microstrain would over-estimate the damage. At Bridge C, enlarging the width of the bin
size to 15 microstrain increased the fatigue damage by approximately 1%. Enlarging the
width of the bin size to 30 microstrain increased the fatigue damage by 10%. At Bridge
A, before the construction of the retrofit, increasing the width of the bins did not
significantly influence the fatigue damage. However, after the construction of the retrofit
at Bridge A, the fatigue damage increased by nearly 20% if a bin width of 30 microstrain
was used as compared to 5 microstrain. As such, a bin width between 5-15 microstrain is
recommended.
The time period of a rainflow analysis can also influence the results. If the length
is too long, a large, fictitious cycle due to the drift of the strain gage with temperature
could be counted. If the length is too short, then the count could be biased due to partial
cycles being counted as full cycles. A length of 30 min was chosen for all of the analyses
in this dissertation. In 30 min, the drift of the gage due to temperature will be minimal
and the time is long enough to provide a correct count.
229
Figure 8-1: Increase in fatigue damage ( , . ) due to change in width of the
rainflow bin.
Once the distribution of stress ranges has been determined, the amount of damage
can be calculated using an index stress range. In addition, the effect of each rainflow bin
can be evaluated using the contribution to damage and cumulative damage techniques to
validate that the lower (typically truncated due to electromechanical noise of the data
acquisition system) and higher cycles (often questioned whether the larger cycles are load
induced or due to spikes in the data acquisition system) do not cause significant damage.
The remaining fatigue life can then be calculated using one of the approaches discussed
in Chapter 7.
Because calculating the remaining fatigue life relies heavily on projecting past,
current, and future traffic volumes, a permanent or semi-permanent data acquisition
system should be considered, especially at bridges nearing their intended design life.
With fatigue data from multiple years, the amount of traffic growth can be estimated. In
the absence of fatigue data from strain gages, traffic-count data from transportation
officials can be used to estimate traffic growth.
0%
5%
10%
15%
20%
5 10 15 20 25 30Bin size (microstrain)
Bridge A (after retrofit)
Bridge A (before retrofit)
Bridge C
Incr
ease
in
230
APPENDIX A
Strain data from Bridge A (4/1/2011 to 5/1/2012)
The data from 4/1/2011 to 5/1/2012 for all strain gages at Bridge A are presented
in this Appendix. A summary of the fatigue damage prior to and after the construction of
the retrofit are presented in Table A-1 and Table A-2, respectively.
Table A-1: Summary of fatigue damage prior to the construction of the retrofit.
Gir
der
FB
Pos
itio
n
Fla
nge
Not
e
. , ksi , ksi , Number of
days monitored
, .
Gages installed on the east longitudinal girder
E 27 s TE 11.2 0.93 39,700 71 350
E 27 s TW 9.4 0.86 64,900 23 450
E 27 s BE 9.6 0.79 137,000 71 750
E 27 s BW 7.6 0.74 114,000 23 500
E 34 s TE 23.7 2.26 56,300 71 7,080
E 34 s TW 24.9 2.37 41,400 71 6,070
E 35 n TE 27.6 2.89 39,800 71 10,400
E 35 n TW 11.5 0.83 40,000 71 250
E 35 s TE 23.9 2.69 40,500 71 8,620
E 35 s TW 20.7 1.80 35,000 71 2,260
E 35 n BE 26.3 1.81 133,000 71 8,600
E 35 n BW 27.3 1.85 127,000 71 8,900
Gages installed on the west longitudinal girder
W 34 s TE 17.2 1.44 38,000 71 1,250
W 34 s TW 14.3 1.24 41,300 71 860
W 35 s TE 12.8 1.20 34,700 71 650
W 35 s TW 13.4 1.27 40,100 71 900
W 35 s BE 14.0 0.98 128,000 71 1,340
W 35 s BW 16.9 1.12 144,000 71 2,660
231
Table A-2: Summary of fatigue damage after the construction of the retrofit. G
ird
er
FB
Pos
itio
n
Fla
nge
Not
e
. , ksi , ksi , Number of
days monitored
, .
Gages installed on the east longitudinal girder
E 27 s TE 5.7 0.80 39,394 2 220
E 27 s TW 5.7 0.81 61,679 2 370
E 27 s BE 6.3 0.73 124,748 2 540
E 27 s BW 5.7 0.70 100,334 2 380
E 33 n TE 6.2 0.72 49,772 96 210
E 33 n TW 16.3 0.57 54,300 94 110
E 34 s TE P 2.4 0.32 16,507 96 6
E 34 s TW P 10.9* 0.29 18,884 96 5
E 34 s TE 9.9 1.00 97,518 96 1,060
E 34 s TW 10.2 1.02 83,740 98 980
E 34 BE 19.1 1.52 170,894 83 6,640
E 34 BW 19.6 1.54 164,297 96 6,520
E 35 n TE 5.0 0.61 60,050 98 150
E 35 n TW 6.0 0.58 70,432 96 150
E 35 s TE 3.6 0.42 46,217 98 40
E 35 s TW 3.3 0.45 43,021 96 40
E 35 n BE 18.6 1.45 170,960 98 5,760
E 35 n BW 19.2 1.47 165,975 98 5,810
Gages installed on the west longitudinal girder
W 34 s TE 29.1* 0.48 120,770 94 150
W 34 s TW 10.7 0.36 117,802 98 60
W 35 s TE 11.8* 0.20 46,809 96 4
W 35 s TW 8.9* 0.15 37,880 98 2
W 35 s BE 29.1* 0.82 175,458 98 1,050
W 35 s BW 15.6 0.96 192,092 98 1,860 *Value may be due to spikes in the data acquisition system. See details of gage.
232
Figure A-1: Response of bridge at gage E-35n-TE.
End support, typ.
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
200
400
600
0 10 20 30Stress range (ksi)
North spanCenter spanSouth span
Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(b) Histogram of stress ranges prior to construction (average of 71 days)
(c) Histogram of stress ranges after construction (average of 98 days)
(a) Location of gage (installed 2′-0″ north of FB 35)
After construction
Prior to construction
Maximum
(d) Average daily contribution to effective fatigue damage
(e) Average daily cumulative damage
Co
ntr
ibu
tio
n t
o
233
Figure A-1 (cont’d): Response of bridge at gage E-35n-TE.
0
4,000
8,000
12,000
16,000
Date
0
100
200
300
400
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(f) Daily fatigue damage ( ) for total monitoring period
(g) Daily fatigue damage ( ) for monitoring period prior to construction
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
234
Figure A-2: Response of bridge at gage E-35n-TW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 96 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ north of FB 35)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
235
Figure A-2 (cont’d): Response of bridge at gage E-35n-TW.
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
0
100
200
300
400
Date
0
100
200
300
400
500
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
236
Figure A-3: Response of bridge at gage E-35s-TE.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
100
200
300
400
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
237
Figure A-3 (cont’d): Response of bridge at gage E-35s-TE.
0
20
40
60
80
100
Date
0
4,000
8,000
12,000
16,000
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
238
Figure A-4: Response of bridge at gage E-35s-TW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
100
200
300
400
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 96 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
239
Figure A-4 (cont’d): Response of bridge at gage E-35s-TW.
0
4,000
8,000
12,000
16,000
Date
0
1,000
2,000
3,000
4,000
Date
0
20
40
60
80
100
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
240
Figure A-5: Response of bridge at gage E-35n-BE.
North spanCenter spanSouth span
0
200
400
600
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ north of FB 35)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
241
Figure A-5(cont’d): Response of bridge at gage E-35n-BE.
0
2,000
4,000
6,000
8,000
10,000
Date
0
4,000
8,000
12,000
16,000
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
242
Figure A-6: Response of bridge at gage E-35n-BW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
200
400
600
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ north of FB 35)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
243
Figure A-6 (cont’d): Response of bridge at gage E-35n-BW.
0
4,000
8,000
12,000
16,000
Date
0
4,000
8,000
12,000
16,000
Date
0
2,000
4,000
6,000
8,000
10,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
244
Figure A-7: Response of bridge at gage E-34s-TE.
North spanCenter spanSouth span
0
100
200
300
400
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
End support, typ. Interior support, typ.
37′-6″
FB 34Hanger
West girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 96 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 34)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
245
Figure A-7 (cont’d): Response of bridge at gage E-34s-TE.
0
500
1,000
1,500
2,000
Date
0
3,000
6,000
9,000
12,000
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
246
Figure A-8: Response of bridge at gage E-34s-TW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
100
200
300
400
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
37′-6″
FB 34Hanger
West girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 34)
After construction
Prior to construction
Maximum
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
247
Figure A-8 (cont’d): Response of bridge at gage E-34s-TW.
0
4,000
8,000
12,000
16,000
Date
0
2,000
4,000
6,000
8,000
10,000
Date
0
500
1,000
1,500
2,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
248
Figure A-9: Response of bridge at gage W-35s-TE.
North spanCenter spanSouth span
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 96 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
249
Figure A-9 (cont’d): Response of bridge at gage W-35s-TE.
0
10
20
30
40
50
Date
0
500
1,000
1,500
2,000
Date
0
4,000
8,000
12,000
16,000
g
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
250
Figure A-10: Response of bridge at gage W-35s-TW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
251
Figure A-10 (cont’d): Response of bridge at gage W-35s-TW.
0
4,000
8,000
12,000
16,000
Date
0
500
1,000
1,500
2,000
2,500
Date
0
10
20
30
40
50
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
252
Figure A-11: Response of bridge at gage W-35s-BE.
North spanCenter spanSouth span
0
20
40
60
80
100
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
253
Figure A-11 (cont’d): Response of bridge at gage W-35s-BE.
0
500
1,000
1,500
2,000
2,500
3,000
Date
0
500
1,000
1,500
2,000
2,500
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
254
Figure A-12: Response of bridge at gage W-35s-BW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
20
40
60
80
100
0 10 20 30Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 35)
After construction
Prior to construction
Maximum
West girder
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
255
Figure A-12 (cont’d): Response of bridge at gage W-35s-BW.
0
4,000
8,000
12,000
16,000
Date
0
1,000
2,000
3,000
4,000
5,000
Date
0
1,000
2,000
3,000
4,000
5,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
256
Figure A-13: Response of bridge at gage W-34s-TE.
North spanCenter spanSouth span
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
HangerWest girder East girder
After construction
Prior to construction
Gage
(c) Histogram of stress ranges after construction (average of 94 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 34)
After construction
Prior to construction Maximum
37′-6″
FB 34
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
257
Figure A-13 (cont’d): Response of bridge at gage W-34s-TE.
0
200
400
600
800
1,000
Date
0
500
1,000
1,500
2,000
2,500
Date
0
4,000
8,000
12,000
16,000
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
258
Figure A-14: Response of bridge at gage W-34s-TW.
North spanCenter spanSouth span
0
0.2
0.4
0.6
0.8
1
0 10 20 30
Ave
rage
dai
ly
cum
ula
tive
dam
age
Stress range (ksi)
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
End support, typ. Interior support, typ.
HangerWest girder East girder
After construction
Prior to construction
(c) Histogram of stress ranges after construction (average of 98 days)
(b) Histogram of stress ranges prior to construction (average of 71 days)
(a) Location of gage (installed 2′-0″ south of FB 34)
After construction
Prior to construction Maximum
37′-6″
FB 34
Gage
(e) Average daily cumulative damage(d) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
259
Figure A-14 (cont’d): Response of bridge at gage W-34s-TW.
0
4,000
8,000
12,000
16,000
Date
0
500
1,000
1,500
2,000
Date
0
50
100
150
200
250
Date
Prior to constructionDuring
constructionAfter construction
Prior to construction
After construction
(g) Daily fatigue damage ( ) for monitoring period prior to construction (different vertical scale than (f))
(f) Daily fatigue damage ( ) for total monitoring period
(h) Daily fatigue damage ( ) for monitoring period after construction (different vertical scale than (f))
260
Figure A-15: Response of bridge at gage E-27s-TE.
North spanCenter spanSouth span
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
0
200
400
600
800
g
Date
Prior to constructionDuring
constructionAfter construction
End support, typ. Interior support, typ.
FB 27Hanger
West girder East girder
Gage
(d) Daily fatigue damage ( ) for total monitoring period
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Prior to construction
(b) Histogram of stress ranges prior to construction (average of 71 days)
(c) Average daily contribution to effective fatigue damage
90′-0″
(a) Location of gage (installed 2′-0″ south of FB 27)
Co
ntr
ibu
tio
n t
o
261
Figure A-16: Response of bridge at gage E-27s-TW.
North spanCenter spanSouth span
0
200
400
600
800
Date
End support, typ. Interior support, typ.
FB 27Hanger
West girder East girder
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
Prior to construction
(b) Histogram of stress ranges prior to construction (average of 23 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed 2′-0″ south of FB 27)
90′-0″
Prior to constructionDuring
constructionAfter construction
(d) Daily fatigue damage ( ) for total monitoring period
Co
ntr
ibu
tio
n t
o
262
Figure A-17: Response of bridge at gage E-27s-BE.
North spanCenter spanSouth span
0
20
40
60
80
100
0 10 20 30Stress range (ksi)
0
300
600
900
1,200
1,500
Date
End support, typ. Interior support, typ.
FB 27Hanger
West girder East girder
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Prior to construction
(b) Histogram of stress ranges prior to construction (average of 71 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed 2′-0″ south of FB 27)
90′-0″
Prior to constructionDuring
constructionAfter construction
(d) Daily fatigue damage ( ) for total monitoring period
Co
ntr
ibu
tio
n t
o
263
Figure A-18: Response of bridge at gage E-27s-BW.
North spanCenter spanSouth span
0
300
600
900
1,200
1,500
Date
0
20
40
60
80
100
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
FB 27Hanger
West girder East girder
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Prior to construction
(b) Histogram of stress ranges prior to construction (average of 23 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed 2′-0″ south of FB 27)
90′-0″
Prior to constructionDuring
constructionAfter construction
(d) Daily fatigue damage ( ) for total monitoring period
Co
ntr
ibu
tio
n t
o
264
Figure A-19: Response of bridge at gage E-34-BE.
North spanCenter spanSouth span
0
100
200
300
400
500
0 10 20 30Stress range (ksi)
0
2,000
4,000
6,000
8,000
10,000
Date
After construction
End support, typ. Interior support, typ.
HangerWest girder
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 83 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
37′-6″
FB 34
East girder
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
265
Figure A-20: Response of bridge at gage E-34-BW.
North spanCenter spanSouth span
0
100
200
300
400
500
0 10 20 30Stress range (ksi)
0
2,000
4,000
6,000
8,000
10,000
Date
End support, typ. Interior support, typ.
HangerWest girder East girder
Gage
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 96 days)
(a) Location of gage
(c) Average daily contribution to effective fatigue damage
37′-6″
FB 34
After construction
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
266
Figure A-21: Response of bridge at gage E-34s-TE-P.
North spanCenter spanSouth span
0
10
20
30
40
50
g
Date
0
2
4
6
8
10
0 10 20 30Stress range (ksi)
After construction
End support, typ. Interior support, typ.
HangerWest girder
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 96 days)
(a) Location of gage (installed 6″ south of FB 34 on cover plate from retrofit)
(c) Average daily contribution to effective fatigue damage
Gage
37′-6″
FB 34
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
267
Figure A-22: Response of bridge at gage E-34s-TW-P.
North spanCenter spanSouth span
0
10
20
30
40
50
g
Date
0
2
4
6
8
10
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
HangerWest girder
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 96 days)
(a) Location of gage (installed 6″ south of FB 34 on cover plate from retrofit)
(c) Average daily contribution to effective fatigue damage
Gage
37′-6″
FB 34
After construction
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
268
Figure A-23: Response of bridge at gage E-33n-TE.
North spanCenter spanSouth span
0
100
200
300
400
Date
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
HangerWest girder
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 96 days)
(a) Location of gage (installed 2′-0″ north of FB 33)
(c) Average daily contribution to effective fatigue damage
Gage
45′-0″
FB 33
After construction
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
269
Figure A-24: Response of bridge at gage E-33n-TW.
North spanCenter spanSouth span
0
100
200
300
400
Date
0
10
20
30
40
50
0 10 20 30Stress range (ksi)
End support, typ. Interior support, typ.
HangerWest girder
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 10 20 30
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
After construction
(b) Histogram of stress ranges after construction (average of 94 days)
(a) Location of gage (installed 2′-0″ north of FB 33)
(c) Average daily contribution to effective fatigue damage
Gage
45′-0″
FB 33
After construction
(d) Daily fatigue damage ( ) for monitoring period after construction
Co
ntr
ibu
tio
n t
o
270
APPENDIX B
Strain data from Bridge A (3/1/2012 to 8/1/2012)
The daily variation in fatigue damage ( , 4.5 ) from 3/1/2012 to 8/1/2012
for all strain gages at Bridge A are presented in this Appendix.
271
Figure B-1: Response of bridge at gages E-35n-TE and E-35n-TW.
End support, typ.
North spanCenter spanSouth span
Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
Gages
(a) Location of gages (installed 2′-0″ north of FB 35)
0
100
200
300
400
Date(b) Daily fatigue damage ( ) for monitoring period (E-35n-TE)
(c) Daily fatigue damage ( ) for monitoring period (E-35n-TW)
0
100
200
300
400
Date
272
Figure B-2: Response of bridge at gages E-35s-TE and E-35s-TW.
0
20
40
60
80
100
Date
0
20
40
60
80
100
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
(a) Location of gages (installed 2′-0″ south of FB 35)
Gages
(b) Daily fatigue damage ( ) for monitoring period (E-35s-TE)
(c) Daily fatigue damage ( ) for monitoring period (E-35s-TW)
273
Figure B-3: Response of bridge at gages E-35n-BE and E-35n-BW.
0
2,000
4,000
6,000
8,000
10,000
Date
0
2,000
4,000
6,000
8,000
10,000
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
(a) Location of gages (installed 2′-0″ north of FB 35)
Gages
(b) Daily fatigue damage ( ) for monitoring period (E-35n-BE)
(c) Daily fatigue damage ( ) for monitoring period (E-35n-BW)
274
Figure B-4: Response of bridge at gages E-34s-TE and E-34s-TW.
0
500
1,000
1,500
2,000
Date
0
500
1,000
1,500
2,000
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
37′-6″
FB 34Hanger
West girder East girder
(a) Location of gages (installed 2′-0″ south of FB 34)
Gages
(b) Daily fatigue damage ( ) for monitoring period (E-34s-TE)
(c) Daily fatigue damage ( ) for monitoring period (E-34s-TW)
275
Figure B-5: Response of bridge at gages E-34-BE and E-34-BW.
0
2,000
4,000
6,000
8,000
10,000
Date
0
2,000
4,000
6,000
8,000
10,000
Date
East girder
North spanCenter spanSouth span
End support, typ. Interior support, typ.
HangerWest girder
Gages
(a) Location of gages
37′-6″
FB 34
(b) Daily fatigue damage ( ) for monitoring period (E-34-BE)
(c) Daily fatigue damage ( ) for monitoring period (E-34-BW)
276
Figure B-6: Response of bridge at gages E-34s-TE-P and E-34s-TW-P.
0
10
20
30
40
50
Date
0
10
20
30
40
50
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
HangerWest girder
(a) Location of gages (installed 6″ south of FB 34 on cover plate from retrofit)
Gages
37′-6″
FB 34
(b) Daily fatigue damage ( ) for monitoring period (E-34s-TE-P)
(c) Daily fatigue damage ( ) for monitoring period (E-34s-TW-P)
277
Figure B-7: Response of bridge at gages E-33s-TE and E-33s-TW.
0
100
200
300
400
Date
0
100
200
300
400
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
HangerWest girder
(a) Location of gages (installed 2′-0″ north of FB 33)
Gages
45′-0″
FB 33
(b) Daily fatigue damage ( ) for monitoring period (E-33n-TE)
(c) Daily fatigue damage ( ) for monitoring period (E-33n-TW)
278
Figure B-8: Response of bridge at gages W-35s-TE and W-35s-TW.
0
10
20
30
40
50
Date
0
10
20
30
40
50
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
Gages
(a) Location of gages (installed 2′-0″ south of FB 35)
(b) Daily fatigue damage ( ) for monitoring period (W-35s-TE)
(c) Daily fatigue damage ( ) for monitoring period (W-35s-TW)
279
Figure B-9: Response of bridge at gages W-35s-BE and W-35s-BW.
0
500
1,000
1,500
2,000
2,500
3,000
Date
0
1,000
2,000
3,000
4,000
5,000
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
30′-0″
FB 35Hanger
West girder East girder
Gages
(a) Location of gages (installed 2′-0″ south of FB 35)
(b) Daily fatigue damage ( ) for monitoring period (W-35s-BE)
(c) Daily fatigue damage ( ) for monitoring period (W-35s-BW)
280
Figure B-10: Response of bridge at gages W-34s-TE and W-34s-TW.
0
200
400
600
800
1,000
Date
0
50
100
150
200
250
Date
North spanCenter spanSouth span
End support, typ. Interior support, typ.
HangerWest girder East girder
Gages
(a) Location of gages (installed 2′-0″ south of FB 34)
37′-6″
FB 34
(b) Daily fatigue damage ( ) for monitoring period (W-34s-TE)
(c) Daily fatigue damage ( ) for monitoring period (W-35s-TW)
281
APPENDIX C
Strain data from Bridge B
The average stress spectrum, contribution to fatigue damage ( , 12 ), and
daily variation in fatigue damage ( , 12 ) from 4/1/2012 to 8/1/2012 for all strain
gages at Bridge B are presented in this Appendix. A summary of the daily fatigue
damage for all gages is presented in Table C-1.
Table C-1: Summary of fatigue damage at Bridge B.
Gir
der
Sp
an
Fla
nge
. , ksi , ksi , Number of
days monitored
,
Gages installed on the east longitudinal girder
W 1 BE 6.16 0.23 538,000 68 1.71*
W 1 BW 7.18 0.24 551,000 68 2.04*
W 2 BE 8.19 0.26 527,000 62 3.14*
W 2 BW 6.01 0.22 530,000 62 1.21*
W 3 BE 8.05 0.32 275,000 31 3.47*
W 3 BW 5.44 0.26 305,000 31 0.79*
W 4 BE 7.32 0.42 55,500 30 1.88*
W 4 BW 4.28 0.28 61,300 30 0.40*
* Cycles less than 0.4 ksi were truncated.
282
Figure C-1: Response of bridge at gage W-1-BE.
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
0
1
2
3
4
5
6
Date(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4
ksi were truncated)
(b) Histogram of stress ranges (average of 68 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on east side of bottom flange)
Co
ntr
ibu
tio
n t
o
105′-0″
Gage location
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
283
Figure C-2: Response of bridge at gage W-1-BW.
0
1
2
3
4
5
6
Date
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 68 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on west side of bottom flange)
105′-0″
Gage location
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Co
ntr
ibu
tio
n t
o
284
Figure C-3: Response of bridge at gage W-2-BE.
0
1
2
3
4
5
6
Date
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 62 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on east side of bottom flange)
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
115′-0″
Gage location
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Co
ntr
ibu
tio
n t
o
285
Figure C-4: Response of bridge at gage W-2-BW.
0
1
2
3
4
5
6
Date
0.00
0.05
0.10
0.15
0.20
0.25
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 62 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on west side of bottom flange)
Gage location
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
115′-0″
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Co
ntr
ibu
tio
n t
o
286
Figure C-5: Response of bridge at gage W-3-BE.
0
1
2
3
4
5
6
Date
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 31 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on east side of bottom flange)
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Gage location
Co
ntr
ibu
tio
n t
o
287
Figure C-6: Response of bridge at gage W-3-BW.
0
1
2
3
4
5
6
Date
0.00
0.05
0.10
0.15
0.20
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 31 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on west side of bottom flange)
Gage location
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Co
ntr
ibu
tio
n t
o
288
Figure C-7: Response of bridge at gage W-4-BE.
0
1
2
3
4
5
6
Date
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 30 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on east side of bottom flange)
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Gage location
Co
ntr
ibu
tio
n t
o
289
Figure C-8: Response of bridge at gage W-4-BW.
0
1
2
3
4
5
6
Date
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 30 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage (installed on west side of bottom flange)
Gage location
North
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
(d) Daily fatigue damage ( ) for total monitoring period (cycles less than 0.4 ksi were truncated)
Co
ntr
ibu
tio
n t
o
290
APPENDIX D
Strain data from Bridge C
The average stress spectrum and contribution to fatigue damage ( , 12 ) for
all strain gages at Bridge C are presented in this Appendix. In addition, a summary of the
daily fatigue damage for gages attached to the bottom flange are presented in Table D-1.
Table D-1: Summary of fatigue damage for gages attached to the bottom flange at
Bridge C.
Loc
atio
n
Gir
der
Fla
nge
. , ksi , ksi , Number of
days monitored
,
L1 1 BN 3.84 0.43 13,500 10 0.60
L1 1 BS 3.84 0.40 14,400 10 0.51
L1 3 BN 5.00 0.52 22,000 9 3.14*
L1 3 BS 5.29 0.57 23,000 9 1.21*
L1 5 BN 6.45 0.77 11,100 10 2.98
L1 5 BS 6.60 0.79 13,200 10 3.79
L2 1 BN 3.84 0.44 16,500 10 0.79
L2 1 BS 3.84 0.44 17,000 8 0.81
L2 3 BN 5.00 0.33 215,000 19 2.76*
L2 3 BS 5.00 0.34 206,000 19 3.08*
L2 5 BN 7.47 0.78 13,500 10 3.72
L2 5 BS 7.32 0.79 14,500 10 4.16
L3 1 BN 3.12 0.41 14,600 6 0.58
L3 1 BS 3.26 0.45 13,700 6 0.71
L3 3 BN 5.44 0.65 24,300 10 3.81
L3 3 BS 4.71 0.64 24,000 9 3.65
L3 5 BN 7.32 0.88 12,200 7 4.75
L3 5 BS 8.34 0.85 13,500 7 4.86
* Cycles less than 0.4 ksi were truncated.
291
Figure D-1: Response of bridge at gage L1-1-TN.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Co
ntr
ibu
tio
n t
o
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
292
Figure D-2: Response of bridge at gage L1-1-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
293
Figure D-3: Response of bridge at gage L1-1-BN.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
Co
ntr
ibu
tio
n t
o
294
Figure D-4: Response of bridge at gage L1-1-BS.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
295
Figure D-5: Response of bridge at gage L1-3-TN.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
296
Figure D-6: Response of bridge at gage L1-3-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 8 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
297
Figure D-7: Response of bridge at gage L1-3-BN.
0.00
0.04
0.08
0.12
0.16
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
Co
ntr
ibu
tio
n t
o
298
Figure D-8: Response of bridge at gage L1-3-BS.
0.00
0.04
0.08
0.12
0.16
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
299
Figure D-9: Response of bridge at gage L1-5-BN.
0.00
0.04
0.08
0.12
0.16
0.20
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
Co
ntr
ibu
tio
n t
o
300
Figure D-10: Response of bridge at gage L1-5-BS.
0.00
0.04
0.08
0.12
0.16
0.20
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
25′-0″Loc. 1
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
Co
ntr
ibu
tio
n t
o
301
Figure D-11: Response of bridge at gage L2-1-TN.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 8 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
302
Figure D-12: Response of bridge at gage L2-1-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 8 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
303
Figure D-13: Response of bridge at gage L2-1-BN.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
304
Figure D-14: Response of bridge at gage L2-1-BS.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 8 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
305
Figure D-15: Response of bridge at gage L2-3-TN.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 18 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
306
Figure D-16: Response of bridge at gage L2-3-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 18 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
307
Figure D-17: Response of bridge at gage L2-3-BN.
0.00
0.04
0.08
0.12
0.16
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 19 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
308
Figure D-18: Response of bridge at gage L2-3-BS.
0.00
0.04
0.08
0.12
0.16
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 19 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
309
Figure D-19: Response of bridge at gage L2-5-BN.
0.00
0.04
0.08
0.12
0.16
0.20
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
310
Figure D-20: Response of bridge at gage L2-5-BS.
0.00
0.04
0.08
0.12
0.16
0.20
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
17′-6″Loc. 2
Co
ntr
ibu
tio
n t
o
311
Figure D-21: Response of bridge at gage L3-1-TN.
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 6 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
312
Figure D-22: Response of bridge at gage L3-1-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 6 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
313
Figure D-23: Response of bridge at gage L3-1-BN.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 6 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
314
Figure D-24: Response of bridge at gage L3-1-BS.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 6 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
315
Figure D-25: Response of bridge at gage L3-3-TN.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 8 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
316
Figure D-26: Response of bridge at gage L3-3-TS.
0.00
0.01
0.02
0.03
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Co
ntr
ibu
tio
n t
o
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cle
s p
er
day
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
317
Figure D-27: Response of bridge at gage L3-3-BN.
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 10 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
318
Figure D-28: Response of bridge at gage L3-3-BS.
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 9 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
319
Figure D-29: Response of bridge at gage L3-5-TN.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 7 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
320
Figure D-30: Response of bridge at gage L3-5-TS.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 7 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
321
Figure D-31: Response of bridge at gage L3-5-BN.
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0 2 4 6 8 10Stress range (ksi)
(b) Histogram of stress ranges (average of 7 days)
(c) Average daily contribution to effective fatigue damage
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
322
Figure D-32: Response of bridge at gage L3-5-BS.
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0 2 4 6 8 10Stress range (ksi)
(a) Location of gage
Girder 6
Girder 1
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
South
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
pe
r d
ay
Stress range (ksi)
Gage
(b) Histogram of stress ranges (average of 7 days)
(c) Average daily contribution to effective fatigue damage
20′-0″Loc. 3
Co
ntr
ibu
tio
n t
o
323
APPENDIX E
Strain data from Bridge D
The average stress spectrum and contribution to fatigue damage ( , 12 ) for
all strain gages at Bridge D are presented in this Appendix. The average daily fatigue
damage at Bridge D is summarized in Table E-1 for all of the gages installed on the
bottom flange.
Table E-1: Summary of daily fatigue damage at Bridge D.
Loc
atio
n
Gir
der
Fla
nge
. , ksi , ksi , Number of
days monitored
,
L1 2 BN 7.18 0.60 150,000 4 18.8
L1 3 BS 4.86 0.63 125,000 4 17.7
L2 3 BS 2.68 0.36 95,000 4 2.52
L3 2 BN 5.73 0.59 133,000 4 15.6
L3 3 BS 4.71 0.58 129,000 4 14.4
324
Figure E-1: Response of bridge at gage L1-2-TN.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6 Girder 765′-0″ Loc. 1
Girder 1
Girder 7
Co
ntr
ibu
tio
n t
o
325
Figure E-2: Response of bridge at gage L1-2-BN.
0.0
0.4
0.8
1.2
1.6
2.0
0 2 4 6 8 10Stress range (ksi)
Girder 2
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 3 Girder 4 Girder 5 Girder 6 Girder 765′-0″ Loc. 1
Girder 1
Girder 7
Co
ntr
ibu
tio
n t
o
326
Figure E-3: Response of bridge at gage L1-3-TS.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 2 4 6 8 10Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6 Girder 765′-0″ Loc. 1
Girder 1
Girder 7
Co
ntr
ibu
tio
n t
o
327
Figure E-4: Response of bridge at gage L1-3-BS.
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
0 2 4 6 8 10Stress range (ksi)
Girder 2
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 3 Girder 4 Girder 5 Girder 6 Girder 765′-0″ Loc. 1
Girder 1
Girder 7
Co
ntr
ibu
tio
n t
o
328
Figure E-5: Response of bridge at gage L2-3-TS.
0.00
0.02
0.04
0.06
0.08
0.10
0 2 4 6 8 10Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
Girder 1
Girder 7
20′-0″ Loc. 2
Co
ntr
ibu
tio
n t
o
329
Figure E-6: Response of bridge at gage L2-3-BS.
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10Stress range (ksi)
Girder 2
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
Girder 1
Girder 7
20′-0″ Loc. 2
Co
ntr
ibu
tio
n t
o
330
Figure E-7: Response of bridge at gage L3-2-BN.
0.0
0.4
0.8
1.2
1.6
2.0
0 2 4 6 8 10Stress range (ksi)
Girder 2
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
Girder 1
Girder 7
54′-0″ Loc. 3
Co
ntr
ibu
tio
n t
o
331
Figure E-8: Response of bridge at gage L3-3-TS.
0.00
0.04
0.08
0.12
0.16
0.20
0 2 4 6 8 10Stress range (ksi)
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 2 Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
Girder 1
Girder 7
54′-0″ Loc. 3
Co
ntr
ibu
tio
n t
o
332
Figure E-9: Response of bridge at gage L3-3-BS.
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
0 2 4 6 8 10Stress range (ksi)
Girder 2
0.001
0.01
0.1
1
10
100
1000
10000
100000
0 2 4 6 8 10
Ave
rage
nu
mb
er o
f cy
cles
per
day
Stress range (ksi)
(a) Location of gage
Gage
(b) Histogram of stress ranges (average of 4 days)
(c) Average daily contribution to effective fatigue damage
South
Girder 1
North
Girder 3 Girder 4 Girder 5 Girder 6 Girder 7
Girder 1
Girder 7
54′-0″ Loc. 3
Co
ntr
ibu
tio
n t
o
333
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