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This is to certify that the
dissertation entitled

OPTIMAL DESIGN AND PERFORMANCE OF
LONGITUDINALLY SPLICED
PRECAST/PRESTRESSED CONCRETE BRIDGES

presented by

Pimpida Surakomol

has been accepted towards fulﬁllment
of the requirements for the

PhD. degree in Department of Civil and
Environmental Engineering

 

 

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// Major Profess sSignature
gar/rs

Date

MSU is an Afﬁrmative Action/Equal Opportunity Institution

 

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OPTIMAL DESIGN AND PERFORMANCE OF LONGITUDINALLY
SPLICED PRECAST/PRESTRESSED CONCRETE BRIDGES

VOLUME I
By

Pimpida Surakomol

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Civil and Environmental Engineering

2005

ABSTRACT

OPTIMAL DESIGN AND PERFORMANCE OF LONGITUDINALLY SPLICE
PRECAST/PRESTRESSED CONCRETE BRIDGES
By
Pimpida Surakomol

Design of spliced precast/prestressed concrete girder bridges through continuous
post-tensioning has gained renewed interest; however, information, as well as design and
analysis tools, to assess the feasibility of these systems during the preliminary design
phase are limited. In this study, nonlinear structural optimization analyses were
integrated with sequential time-dependent design and analysis procedures, to evaluate
the performance of single- and two-span continuous spliced precast/prestressed girder
bridges was studied and aids for their preliminary design. Common construction
methods were investigated to asses their effect on system efﬁciency and achievable span
lengths. Component and system optimization procedures were coupled and used as a tool
to develop optimal girder shapes for negative, or pier, segments on two-span continuous
systems. Optimal design solutions for pre- and post-tensioned strands and their locations
along the span, and maximum span lengths, were obtained and presented in the form of
design charts. The solutions satisﬁed a common objective of minimum construction cost
or a set multicriteria objective functions that yielded compromise solutions for different
single objective functions. Studies provided a better understanding of the issues and
parameters in this bridge design type and the developed design aids can be a great asset

to expedite their design potentially resulting in its wider use by transportation agencies.

Acknowledgements

I would like to express my sincere appreciation and special thanks to my advisor,
Professor Rigoberto Burguefio. I am grateful for his guidance and support through the
course of this study. He provided inspiration and helped me greatly in this dissertation.

Without his bright idea and his strong encouragement, this work would not be possible.

In addition, I would like to thank Professor Ronald S. Harichandran, Professor Parviz

Soroushian, and Professor Mohammad Najaﬁ for being in my dissertation committees

and for helpﬁrl ideas.

Furthermore, I also would like to thank Professor William Saul for always guiding me
through these years. I also would like to thank my supervisor, Dr. Khaled Soubra, for

always understanding and supporting me for both work and study.

Also, I would like to give my appreciation to my parents Kosol and Pimphan, my sister
Rose, and my brother Eddy and Oak for their encouragement, support, love and care.
Finally, I would like to thank Kriengkrai for giving me the motivation to ﬁnish this

dissertation. My life would not be at this point without any support from these people.

iii

TABLE OF CONTENTS

 

ACKNOWLEDGEMENTS ....................................................................... iii
LIST OF TABLES .................................................................................. ix
LIST OF FIGURES .............................................................................. xii
LIST OF SYMBOLS ............................................................................ xvii
1 Introduction ..... - -- ..... - - - - 1
1.1 Overview ............................................................................................................ 1
1.2 Precast Prestressed Concrete Bridge Construction ............................................ 3
1.3 Extending the Span Ranges of Precast Bridges ................................................. 4
1.4 Spliced Precast Prestressed Girder Bridges ....................................................... 7
1.5 Structural Optimization of Civil Infrastructure ................................................ 11
1.6 Objective and scope of dissertation ................................................................. 14
2 Design and Analysis of Spliced Precast/Prestressed Girder Bridges ................ 23
2. 1 Introduction ...................................................................................................... 23
2.2 Spliced Precast Prestressed Girder Bridge Systems ........................................ 24
2.3 Girder Cross-Sections ...................................................................................... 26
2.3.] Standard sections ..................................................................................... 26
2.3.2 Non-standard sections .............................................................................. 27
2.4 Splice Location ................................................................................................ 29
2.5 Structural Loads ............................................................................................... 30
2.5.1 Non-composite loads ............................................................................... 30
2.5.2 Composite loads ....................................................................................... 31
2.5.3 Post-tensioning loads ............................................................................... 32
2.6 Flexural Criteria ............................................................................................... 40
2.6.1 Service limit state ..................................................................................... 40
2.6.2 Strength limit state ................................................................................... 42
2.7 Shear Criteria ................................................................................................... 45
2.7.1 Vertical shear at splice location ............................................................... 45
2.7.2 Vertical shear at end blocks ..................................................................... 46
2.8 Construction Sequence Methods ...................................................................... 48
2.8.1 Single-span system with single-stage post-tensioning construction method
49

2.8.2 Single-span system multi-stage post-tensioning construction method 49

2.8.3 Two-span continuous system with single-stage post-tensioning
construction method ................................................................................................. 50

2.8.4 Two- span continuous system with multi-stage post-tensioning
construction method ................................................................................................. 55
2.9 Stress Demands at Critical Construction Stages .............................................. 60
2.9.1 Single-stage post-tensioning construction methods ................................. 61
2.9.2 Multi-stage post-tensioned construction methods ................................... 61

iv

2.10.2 CEB-FIP 90 .............................................................................................. 67

2.11 Simpliﬁed Estimate of Prestress Loss Components ........................................ 75
2.11.1 Relaxation prestress losses at release ....................................................... 76
2.11.2 Shrinkage prestress losses ........................................................................ 76
2.11.3 Creep prestress losses .............................................................................. 77
2.11.4 Elastic shortening prestress losses ........................................................... 80
2.11.5 Relaxation prestress losses at ﬁnal condition .......................................... 81
2.11.6 Additional elastic shortening prestress losses due to post-tensioning ..... 82
2.11.7 Friction prestress losses ........................................................................... 82
2.11.8 Anchorage set prestress losses (N CHRP 517 report) .............................. 83

2.12 Total Prestress Losses at Critical Construction Stages .................................... 83

2.13 Life-Cycle Cost of Spliced Prestressed Concrete Bridges ............................... 86
2.13.1 Life-cycle cost criteria ............................................................................. 87
2.13.2 Effect of chloride ingress on concrete structures ..................................... 88
2.13.3 Reinforcement corrosion initiation period ............................................... 89
2.13.4 Effect of concrete cover on service life of concrete components ............ 91

2.14 Summary .......................................................................................................... 92
Design Optimization of Spliced Precast/Prestressed Girder Bridges ............... 93

3.1 Introduction ...................................................................................................... 93

3.2 Structural Optimization .................................................................................... 93
3.2.1 Problem statement .................................................................................... 96
3.2.2 Data and information ............................................................................... 96
3.2.3 Design variables ....................................................................................... 97
3.2.4 Objective functions 98
3.2.5 Constraints ............................................................................................... 98

3.3 Optimization approach ..................................................................................... 99
3.3.1 Computer aided design ............................................................................ 99
3.3.2 Finite element analyses .......................................................................... 100
3.3.3 Sensitivities analyses ............................................................................. 100
3.3.4 Mathematical algorithm ......................................................................... 101

3.4 Optimization Algorithm Types ...................................................................... 102
3.4.1 Gradient based algorithms ..................................................................... 102
3.4.2 Direct search algorithms ........................................................................ 109
3.4.3 Branch and bound combinatorial search algorithms .............................. 110
3.4.4 Comparison of optimal results ............................................................... 111

3.5 Multicriteria Optimization ............................................................................. 114
3.5.1 Mathematical formulation ...................................................................... 114
3.5.2 s-constraint approach ............................................................................. 116
3.5.3 Compromised programming .................................................................. 118

3.6 Structural Optimization of Spliced Precast/Prestressed Girder bridges ........ 119
3.6.1 Component optimization ........................................................................ 120
3.6.2 Conﬁguration optimization .................................................................... 121
3.6.3 Single objective optimization of spliced girder bridges ........................ 123
a) Minimum superstructure construction cost .................................................... 123
b) Minimum post-tensioning requirements ........................................................ 124

c) Minimum pre-tensioning requirements .......................................................... 124

 

(1) Minimum girder concrete volume ................................................................. 124
3.7 Multicriteria Optimization of Spliced Girder Bridges ................................... 124
3 .8 Implementation .............................................................................................. 1 26
3.9 Summary ........................................................................................................ 129

4 Optimum System Designs of Single-span Spliced Precast/Prestressed

Bridges ............ 130
4. 1 Introduction .................................................................................................... 1 30
4.2 Problem Statement ......................................................................................... 131
4.3 Design Parameters ......................................................................................... 132
4.4 Analysis of Spliced Precast/Prestressed Girder Bridges ................................ 135
4.5 Optimum Design of Single-Span Spliced Girder Bridges ............................. 136

4.5.1 Single-stage post-tensioning after pouring deck (Method 1) ................. 136

4.5.2 Single-stage post-tensioning before pouring deck (Method 2) .............. 153

4.5.3 Multi-staged post-tensioning (Method 3) .............................................. 159
4.6 Multicriteria Optimum Design of Single-Span Spliced Girder Bridges ........ 167

4.6.1 Multicriteria objective ﬁmctions ............................................................ 168

4.6.2 Comparison results of single-objective optimizations ........................... 169

4.6.3 Pareto optima results and discussion ..................................................... 170
4.7 Parametric Studies ......................................................................................... 178

4.7.1 Assumption and procedure for parametric studies ................................. 178

4.7.2 Material parametric studies for PCI-BT 96 beams ................................ 179

4.7.3 Parametric studies on girder types ......................................................... 189
4.8 Life-cycle Analysis and Optimum Design of Single-span SGBs .................. 191

4.8.1 Effect of concrete clear cover on chloride concentration ..................... 191

4.8.2 Effect of concrete cover on initial cost .................................................. 195

4.8.3 Effect of concrete clear cover on life-cycle cost of single-span spliced

girder bridges ......................................................................................................... 199

4.8.4 Comparison of life-cycle cost with initial strength-based design cost .. 200

4.8.5 Discussion of life-cycle-cost of spliced girder bridges .......................... 201
4.9 Effects of Erection Time Delay on Prestress Losses ..................................... 203
4.10 Summary ........................................................................................................ 205

5 Optimum System Design of Two-Span Continuous Spliced Girder Bridges . 207
5.1 Introduction .................................................................................................... 207
5.2 Problem Statement ......................................................................................... 208
5.3 Design Parameters ......................................................................................... 210
5.4 Options to Negative Pier Segments in Continuous SGBs ............................. 210

5.4.1 Pier segment option 1: existing standard positive sections with sofﬁt .. 212

5.4.2 Pier segment option 2: new optimal non-prismatic section with variable

web 21 2

5.4.3 Pier segment option 3: new optimal non-prismatic section with constant

web height .............................................................................................................. 214

5.4.4 Pier segment option 4: new optimal prismatic section for both positive

and negative Segments ........................................................................................... 216

vi

 

5.4.5 Design variables for the development of pier segment options ............. 216

 

5.5 Analysis Continuous-Span Spliced Girder Bridges ....................................... 219
5.6 Optimum Design of Two-Span SGB by Single-Stage Post-Tensioning after
Deck Casting .............................................................................................................. 220
5.6.1 Design variables ..................................................................................... 220
5.6.2 Objective functions ................................................................................ 223
5.6.3 Design constraints .................................................................................. 225
5.7 Optimum Design of Two-Span SGB by Multi-Staged Post-Tensioned Systems
23 1
5.7.1 Design variables ..................................................................................... 231
5.7.2 Objective function .................................................................................. 232
5.7.3 Design Constraints ................................................................................. 232
5.8 Results and Discussion .................................................................................. 240
5.8.1 Prestress requirements for single- and multi-stage post-tensioning
construction methods ............................................................................................. 240
5.8.2 Service moments for single- and multi-stage post-tensioning construction
249
5.8.3 Performance of pier segment option 1: standard beam with sofﬁt ........ 252
5.8.4 Dimensions and performance of optimized pier segments- options 2, 3
and 4 255
5.9 Multi-Criteria Optimum Design of Two-Span Continuous SGB .................. 265
5.9.1 Multi-criteria objective functions .......................................................... 265
5.9.2 Results and comparison to single-objective optimizations .................... 266
5.9.3 Pareto optima results .............................................................................. 268
5.10 Summary ........................................................................................................ 273
6 Optimally-Derived Design Aids and Verification Example 275
6. 1 Introduction .................................................................................................... 275
6.2 Preliminary Design Example 1 ...................................................................... 276
6.2.1 Problem statement ............................................................................................. 276
6.2.2 Preliminary section properties .......................................................................... 278
6.2.3 Preliminary material properties ......................................................................... 279
6.2.4 General information .......................................................................................... 280
6.2.5 Use of design charts .......................................................................................... 281
6.2.6 Service limit state .............................................................................................. 286
6.2.7 Strength limit state ............................................................................................ 300
a) Moment capacity at the midspan of segment 2 .............................................. 300
b) Shear resistance of the interface at spliced locations ..................................... 304
c) Vertical shear at H/2 ...................................................................................... 305
6.2.8 Design evaluation .............................................................................................. 306
6.3 Preliminary Design Example 2 ...................................................................... 310
6.3.1 Problem statement ............................................................................................. 310
6.3.2 Preliminary material properties ......................................................................... 31 1
6.3.3 Calculated section properties of optimized negative section ............................ 313
6.3.4 General information .......................................................................................... 314

6.3.5 Use of design charts 314

vii

6.3.6 Service limit states ............................................................................................ 316

 

6.3.7 Desigr evaluation .............................................................................................. 332
7 Conclusion and Future Research Needs iv
7.1 Overview ........................................................................................................... iv
7.2 Conclusions ........................................................................................................ v
7.3 Future Research ................................................................................................ xi

viii

LIST OF TABLES
Table 2-1 Allowable concrete stress limits for critical construction stages .................... 41

Table 2-2 Load patterns and moment diagrams of single-stage post—tensioning
construction method ......................................................................................................... 53

Table 2-3 Load patterns and moment diagrams of multi-stage post-tensioning
construction method ......................................................................................................... 57

Table 2-4 Construction sequence for single-staged post-tensioning in spliced girder

bridges .............................................................................................................................. 61
Table 2-5 Construction sequence for multi-staged post-tensioning in spliced girder
bridges .............................................................................................................................. 62
Table 2-6 Prestress losses during single-stage post-tensioning after deck casting ......... 84
Table 2-7 Total prestress losses during multi-stagedpost-tensioning ............................ 85
Table 3-1 Comparison of optimization versus algorithm performance ......................... 113
Table 4-1 Assumed material properties ......................................................................... 134
Table 4-2 Components of objective function ................................................................. 141
Table 4-3 Assumed unit cost of materials ...................................................................... 142

Table 4-4 Construction sequence for single-staged post-tensioning applied after pouring

deck in spliced girder bridges ........................................................................................ 143
Table 4—5 Allowable concrete stress limits for single-stagedpost-tensioning applied after
pouring deck at each construction stage ....................................................................... 144
Table 4-6 Maximum span length of SGBs and conventional box—beams ....................... 150

Table 4-7 Maximum span length of SGBs and conventional I- and bulb-tee beams 151

Table 48 Construction sequence for single-staged post-tensioning applied before

pouring deck in spliced girder bridges .......................................................................... 154
Table 4—9 Allowable concrete stress limits for single-staged post—tensioning applied
before pouring deck at each construction stage ............................................................ 155
Table 4-10 Construction sequence for multi-stagedpost—tensioning ............................ 161

ix

 

Table 4-11 Allowable concrete stress limits for multi-staged post—tensioning at each

construction stage .......................................................................................................... 162
Table 4-12 Optimization results for each objective function ......................................... 170
Table 4-13 Pareto set of nprii as objective and npo as constraint (case 1) ................... 173
Table 4-14 Pareto set of npo as objective and npri; as constraint (case 2) ................... 173

Table 4-15 Pareto set of min construction cost as objective and npru as constraint(case

3) .................................................................................................................................... 174
Table 416 Pareto set of min npr,-,~ as objective and min construction cost as constraint
(case 4) ........................................................................................................................... 175
Table 4-17 Design variables of best solutions of each pareto set ................................. 178

Table 4-18 Effect of girder concrete strength on spanning increase of PCI—BT 96 ...... 181
Table 4-19 Effect of girder concrete weight on spanning increase of PCI-B T 96 ........ 182

Table 4-20 Effect of lightweight concrete deck on spanning increase of PCI-BT 96

bridge ............................................................................................................................. 183
Table 4-21 Span increase of lightweight concrete girder PCI-BT 96 and lightweight deck
........................................................................................................................................ 1 83
Table 4—22 Non-composite section properties of 1.8 m (6ft) high girders .................... 189
Table 4-23 Service life time versus concrete cover in cold and warm areas ................ 194
Table 4-24 General bridge data information used in life-cycle-cost analyses .............. 196
Table 4-25 Studies on eﬂect of concrete cover on superstructure cost ......................... 198
Table 4-26 Comparison of prestress losses at critical construction stage .................... 203
Table 5-1 Assumed material properties ......................................................................... 211
Table 5-2 Components of objective function ................................................................. 223
Table 5-3 Concrete volume of new optimal beams ........................................................ 224

Table 5-4 Construction sequence for single—stagedpost-tensioning after deck casting 226

Table 5-5 Construction sequence for multi-stage post-tensioning ................................ 233

 

Table 5-6 Dimensions of optimized prismatic beams (option 4) for two-span continuous
spliced girder bridges — SI units .................................................................................... 257

Table 5-7 Dimensions of optimized non-prismatic beams with constant web height and
varied ﬂange (option 3) for two-span continuous spliced girder bridges — US Units... 259

Table 5-8 Dimensions of optimized non—prismatic beams with constant ﬂange and varied

web (option 2) for two-span continuous spliced girder bridges — SI units .................... 262
Table 5-9 Optimization results for individual objective functions ................................. 267
Table 5-10 Pareto sets of nprn- as an objective and npo as a constraint ....................... 270
Table 5-11 Pareto sets of npo as an objective and npr,-,- as a constraint ....................... 270
Table 5-12 Best solutions to pareto sets ........................................................................ 273

Table 6—1 Comparison of design requirements and capacities from NCHRP 517 and
optimization chart results .............................................................................................. 307

Table 6-2 Comparison of negative pier segment between our method and Lounis method
........................................................................................................................................ 333

xi

 

LIST OF FIGURES

Figure 1-1 Components of spliced precast/prestressed concrete girder bridge Systems .. 8

Figure 1-2 Summary of research approaches .................................................................. 14
Figure 1-3 Typical existing 2-span highway bridge ........................................................ 15
Figure 1-4 Single-span spliced girder bridge .................................................................. 16
Figure 1-5 Typical existing 4—span highway bridge ........................................................ 16
Figure 1-6 T wo-span continuous spliced girder bridge ................................................... 17
Figure 2—1 Three-segment single-span spliced precast/prestressed girder bridge .......... 24

Figure 2-2 T wo-span spliced precast/prestressed bridge with prismatic pier segment... 25

Figure 2—3 T wo—span spliced precast/prestressed bridge with nonprismatic pier segment
.......................................................................................................................................... 25

Figure 2-4 Post-tensioned proﬁle of single-span using prismatic section ....................... 33

Figure 2-5 Equivalent loads and ﬁxed end moments of single-span using prismatic
section .............................................................................................................................. 33

Figure 2-6 Post-tensioned profile of two-continuous span using prismatic section ........ 35
Figure 2-7 Equivalent loads and ﬁxed end moments of two-continuous span using ....... 36

Figure 2-8 Post-tensioned proﬁle of two—continuous span using a non-prismatic section
.......................................................................................................................................... 38

Figure 2-9 Equivalent loads and ﬁxed end moments of two-continuous span using a 39

Figure 2-10 Component forces and centroid distances on composite girder/deck section

.......................................................................................................................................... 68
Figure 3-1 Steps of constrained steepest descent gradient method ............................... 104
Figure 3-2 Steepest descent function direction .............................................................. 105
Figure 3-3 Graphical representation of a two-objective optimization problem ............ 116

Figure 3-4 Negative Segment Alternatives: (a) existing standard girder with attached
soﬂit (b) new optimal standard prismatic girder (c) new optimal standard non-prismatic

xii

girder with variable web depth (d) new optimal standard non-prismatic girder with

variable soﬂit ................................................................................................................. 123
Figure 3-5 Flow chart of computer program for optimization of spliced
precast/prestressed girder bridge systems. .................................................................... 128
Figure 4—1 Bridge cross-section of prototype bridge ..................................................... 132
Figure 4-2 T hree-segment single-span spliced precast/prestressed bridge ................... 132
Figure 4—3. Variables for the design optimization of spliced girder bridges ................. 138
Figure 4-4. Pareto set of pre-tensioned strand and post-tensioned strands .................. 176

Figure 4-5. Pareto set of pre-tensioned strand and minimum superstructure construction

cost ................................................................................................................................. 1 76
Figure 4-6 Eﬂect of girder concrete compressive strength on span length ................... 185
Figure 4-7 Effect of girder unit weight on span length .................................................. 186
Figure 4-8 Effect of deck unit weight on span length ..................................................... 187
Figure 4-9 Effect of girder and deck unit weight on span length ................................... 188

Figure 4-10 Effect of concrete cover on the service life of structures (Warm Area) ..... 193
Figure 4-11 Effect of concrete cover on the service life of structures (Cold Area) ....... 194
Figure 4—12 Effect of concrete clear cover on initial construction cost (Cold Area) 197
Figure 4-13 Maintenance and repair cost versus concrete cover (Cold Area) ............. 199
Figure 4-14 Life-cycle-cost versus concrete clear cover (Cold Area) ........................... 200

Figure 4-15 Comparison of initial, maintenance and life-cycle-cost versus concrete
clear cover ..................................................................................................................... 201

Figure 5-1 Two-span spliced precast/prestressed bridge with prismatic pier segment 209

Figure 5-2 Two-span spliced precast/prestressed bridge with non-prismatic pier segment
........................................................................................................................................ 209

Figure 5-3 Pier segment option 1: existing standard girder attached with soﬂit .......... 212

xiii

Figure 5-4 Pier segment option 2: new optimal non-prismatic girder with variable web
depth ............................................................................................................................... 213

Figure 5-5 Pier segment option 3: new optimal non-prismatic girder with variable
bottom flange thickness .................................................................................................. 215

Figure 5-6 Pier segment option 4: new optimal prismatic section for positive and
negative segments .......................................................................................................... 216

Figure 5-7 Design variable for existing standard girder with soffit (pier segment option
I) .................................................................................................................................... 217

Figure 5-8 Design variable for new non-prismatic section with constant web depth (pier
segment option 2) ........................................................................................................... 218

Figure 5-9 Design variables for new non-prismatic section with varied web depth (pier
segment option 3) ........................................................................................................... 218

Figure 5-10 Design variables for new prismatic section for both positive and negative
segments (pier segment option 4) .................................................................................. 219

Figure 5-11 Variables for the design optimization of two-span continuous SGB built with
single-staged post-tensioning after deck casting ........................................................... 223

Figure 5-12 Post-tensioning requirement variables for the design optimization of two-
staged post-tensioning ................................................................................................... 232

Figure 5-13 Pre-tensioning requirements on positive segments for single- and multi-
staged post-tensioning construction for S = 1.8 m (6ft) ................................................ 241

Figure 5-14 Pre-tensioning requirements on the negative-segment of single- and multi-
staged post-tensioning construction for S = 1.8 m (6 ft) ............................................... 242

Figure 5-15 Pre-tensioning requirements on positive segments of single- and multi—
staged post-tensioning construction for S = 2.4 m (8ft) ............................................... 243

Figure 5-16 Pre-tensioning requirements on negative segment of single- and multi-
Staged Post-Tensioning Construction for S = 2.4 m (8 ft) ............................................ 244

Figure 5-17 Pre-tensioning requirements on positive segments of single- and multi-
staged post-tensioning construction for S = 3. 0 m (1 0 ft) ............................................. 244

Figure 5-18 Pre-tensioning requirements on negative segment of single- and multi-
staged post-tensioning construction for S = 3.0 m (10ft) ............................................. 245

xiv

Figure 5-19 Post-tensioning requirement of single- and multi-staged post-tensioning

construction for S = 1.8 m (6 ft) .................................................................................... 246
Figure 5—20 Post-tensioning requirement of single- and multi-staged post-tensioning
construction for S = 2.4 m (8 ft) .................................................................................... 247
Figure 5-21 Post-tensioning requirement of single- and multi-staged post-tensioning
construction for S = 3.0 m (1 0 ft) .................................................................................. 247
Figure 5-22 Service moment demands on two-span continuous SGB built with single-
staged post-tensioning with L = 68.3 m (224 ft) and S = 1.8 m (6 ft) ........................... 249
Figure 5-23 Service moment of multi-staged post-tensioning with L = 68.3 m (224 ft)
andS= 1.8m (6ft) ........................................................................................................ 252
Figure 5-24 Maximum span length of ASSHTO Type IV standard beam with sofﬁt at
beam spacing 1.8 m (6 ft) ............................................................................................... 254
Figure 5-25 Maximum span length of ASSHTO Type IV standard beam with soﬂit at
beam spacing 2.1 m (7ft) ............................................................................................... 254
Figure 5-26 Maximum span length of ASSHTO Type I V standard beam with soﬂit at
beam spacing 2.4 m (8 ft) ............................................................................................... 255
Figure 5-27 Total beam height of new optimal negative segments for 1.8m (6ft) beam
spacing ........................................................................................................................... 264
Figure 5—28 Pareto set of pre-tensioning and post—tensioning strands .......................... 272
Figure 6-1 Three-segment single-span spliced precast/prestressed girder bridge ........ 276

Figure 6-2 System and girder cross sections for single-span spliced girder bridge study
........................................................................................................................................ 277

Figure 6-3 Variables for the design optimization of single-span spliced girder bridges
........................................................................................................................................ 282

Figure 6-4 Number of pre-tensioning strands per row in end-segment for PCI BT—96
beam ............................................................................................................................... 284

Figure 6-5 Number of pre-tensioning strands per rows in mid-segment for PCI-BT 96
beam ............................................................................................................................... 284
Figure 6-6 Optimized total number of post-tensioning strands for PCI BT—96 beam for

different splice locations ................................................................................................ 285

XV

 

Figure 6-7 Post-tensioning profile at end and mid-segments for PCI BT—96 beam ...... 285

Figure 6-8 Pre- and post-tensioning strands requirement comparison between NCHRP
and optimized aids results .............................................................................................. 308

Figure 6-9 Normalized stress results comparison between NCHRP and optimized aids
designs ............................................................................................................................ 309

Figure 6-10 T wo-span spliced precast/prestressed bridge with non-prismatic pier
segment .......................................................................................................................... 310

Figure 6-11 Design variables for new non-prismatic section with constant web(option 3)
........................................................................................................................................ 315

Figure 6-12 Variables for the design optimization of two-span continuous spliced girder
bridge ............................................................................................................................. 315

xvi

 

a]

BI

850

830

as]:

as:

3t

“Ida

02da

LIST OF SYMBOLS

The angle between the effective post-tensioning strands and the horizontal
reference of lowest post-tensioned proﬁle.

The coefﬁcient of horizontal distance from splicing location to the center
of pier.

The coefﬁcient of eccentricity at the end of girder.

The coefﬁcient of eccentricity at the mid-span of girder.

The time-dependent change in strain at location of prestressing steel
(tension positive).

The value of strain corresponding to the computed stress.

The different in the strain between the concrete and prestressed
reinforcement.

The shrinkage strain for deck.
The shrinkage strain for girder.

The shrinkage strain for steam-cured concrete.
The shrinkage strain at time t.

The total concrete strain at time t.

The bottom girder compression stress constraint

The Bottom girder compressive stresses

 

03da

04da

abln

0b2n

0b3n

Ub4n

00
0t
(00
(00
(0: t

, o

¢Vn

APRET

APS

The bottom girder stress constraint

The compression stress limit under full service load

The girder bottom stress along the negative segment

The girder bottom stress along the negative segment after ﬁrst staged
post-tensioning

The girder bottom stress along the negative segnent after deck and splice
placement

The girder bottom stress along the negative segment after second staged
post-tensioning

The concrete stress at time t = O.

The concrete stress at time t.

The creep coefﬁcient for deck.

The creep coefﬁcient for girder.

The creep coefﬁcient at time t due to load applied at time ta

The nominal shear strength provided by concrete, prestressing and
reinforcement.

The girder goss area along the negative segnent

The pre-tensioned strands.
The cross-sectional area of prestressing steel.

The prestressing steel.

xviii

Apsn

APT

bl

b2

CB

C CD

CPS

CPT

C73

The total pre-tensioned area in the negative segnent

The post-tensioned strands area in the section.

The area of web reinforcement within a distance S.

The total area of all shear reinforcement including post-tensioning strands

and mild reinforcement crossing the shear plane.
The width of a web of a ﬂanged member.

The bottom ﬂange width

The top ﬂange width

The effective bridge width.

The beam ﬂange width.

The permissible pressure at the splice contact area = 0.

The beam cost
The concrete deck cost

The compressive force in the girder ﬂange.

The chloride concentration at the surface

The pre-tensioning cost
The post-tensioning cost

The compression force in the deck.

The temporary support cost

The distance from extreme compressive ﬁber to centroid of prestressing

force.

xix

ED
EC

50

EPS

eob

F 1(1)

ﬂxr')

Fl(x)min

f1(X)
f2(x)

The effective depth.

The diffusion coefﬁcient

The web depth

The eccentricity of prestressing steel relative to centroid of composite
section (down positive).

The modulus of elasticity for deck.

The modulus of elasticity for girder.

The modulus of elasticity at time to_

The elastic modulus of prestressing steel.

The modulus of elasticity of prestressing steel.

The vertical distance at the center pier of beam.

The net eccentricity of pre-tensioning strands along the negative segnent
measured from the neutral axis of the section

The error function, in practice x is equal to the depth of concrete over the
steel reinforcement.

The total post-tensioning forces in the section.

The economics multicriteria optimization

The minimization of total structural costs

The Economics Multi-criteria

The minimum superstructure cost

The minimum number of pre-tensioned strands

XX

f3(X)
fc

k I
fcd
fcd'

fcdp

fcgp

fcgpl

£3ng

for"

fd

fpeﬂnal

The minimum number of post-tensioned strands

The girder compressive strength.

The ﬁnal high-strength concrete compressive strength

The deck compressive strength.

The concrete compressive strength of deck at the ﬁnal age

The change in concrete stress at center of gavity of prestressing steel due
to permanent loads.

The sum of concrete stress at center of gavity of prestressing tendon due
to prestressing force at transfer and the self-weight of member at
maximum moment.

The sum of concrete stress at center of gavity of prestressing tendon due
to post-tensioning force and removal of temporary support.

The sum of concrete stress at center of gavity of post-tensioning tendon
due to pre-tensioned strands and girder self-weight.

The concrete compressive strength at release

The stress due to unfactored dead load at extreme ﬁber section.

The compressive stress in concrete at centroid of cross section resisting
externally applied loads.

The compressive stress in concrete due to effective prestress forces at
extremely ﬁber of section.

The pre-tensioned stress at the ﬁnal stage

xxi

fpel n
fpe2n
fpe3n

fpe4n

fpof

fpoﬂ

fpo/z

fpo/r

fpoﬂY

The effective prestress in the pre-tensioned strands after pre-tensionin g

The effective prestress in the pre-tensioned strands after placing the deck

The effective prestress in the pre-tensioned strands after post-tensioning

The effective prestress in the pre-tensioned strands after applying the live
load

The effective prestress in the post-tensioned strands aﬁer post-tensioning

The effective pre-stress in the ﬁrst post-tensioned strands after ﬁrst post-
tensioning

The effective prestress in the second post-tensioned strands after second
post-tensioning

The effective prestress in the post-tensioned strands after applying the live
load

The effective prestress in the first post-tensioned strands after second
post-tensioning

The prestressing steel.

The speciﬁed tensile strength of prestressing steel.

The yield strength of prestressing steel.

The yield strength of reinforcement.

The relative humidity.

The slab thickness.

xxii

ID
10

kc

The girder web height.

The moment of inertia about the centroid of the cross-section.
The moment of intertia of deck about their own centroids.
The moment of intertia of girder about their own centroids.
The factor for the effect of volume/surface of component.

The factor for effect of concrete strength.

The size factor which is based on PCI equation (1977).
The total span length of each span

The end segment length

The total expected maintenance costs.

The construction load moment along the negative segnent

The moment causing ﬂexural cracking at the section due to externally
applied loads.

The deck moment along the negative segnent

The girder moment along the negative segnent

The haunch moment along the negative segnent

The live load moment due to truck and lane loadings

The maximum factored moment at section due to externally applied load.
The nominal design moment.

The primary moment due to ﬁrst staged post-tensioning

xxiii

M post2

M postn

MsecI
MsecZ

M566!"

npo

npo]

"p02
"Pr 1 i

”PrZi

"Pril

npr i2

nprt;

The primary moment due to second staged post-tensioning
The primary moment due to post-tensioning

The removal moment due to temporary support removal

The secondary moment due to ﬁrst staged post-tensioning
The secondary moment due to second staged post-tensioning
The secondary moment due to post-tensioning

The modular ratio of deck concrete to girder concrete.

The number of identical prestressing tendons.

The total number of post-tensioning strands on beam

The total number of post-tensioned strands in the girder for single-stage
post-tensioning construction or a ﬁrst stage post-tensioning operation for
two-staged post-tensioning construction

The total number of second staged post-tensioned strands
The number of bottom pre-tensioning strands on end segments at i’h row

The number of bottom pre-tensioning strands on mid-span segnents at i“
row

The number of bottom pre-tensioned strands on end-segnents at row ith

The number of top pre-tensioned strands on the mid-span segnents at row

-1
1h

The number of top pre-tensioning strands on end-segnents

 

nprtI
nprt2

nprt2

pra

RF

Sbgn

Sc

t1, t2

t3, t4

tw

The number of top pre-tensioned strands on the end-segnents
The number of top pro-tensioning strands on mid-segnent
The number of bottom pre-tensioned strands on the mid-segment

The permanent net compressive force normal to the shear plane.

The splice location
The removal cost.
The anchor loss.
The beam spacing.

The bottom sectional modulus

The bottom section modulus along the negative segnent

The ﬁrst area moment of the transformed deck section about the centroid
of the composite section.

The maximum sofﬁt thickness

The top sectional modulus

The maturity of concrete (days).
The depth of compression.

The bottom ﬂange thicknesses
The top ﬂange thicknesses

The age of concrete when load is initially applied (days).

The web thickness

XXV

Vc The shear strength provided by concrete.

Vci The shear strength.

Vcw The shear strength.

Vi The factored shear force at section due to externally applied loads.
V" The nominal shear resistance of the interface.

VP The shear force provided by post-tensioning.

Vs The web reinforcement shear.

Va The factored shear force at the section considered.

x The distance from the concrete surface to the steel surface

The stress at the anchorage.

xA

ya The vertical distance from the bottom extreme ﬁber of the girder to the
net centroid of the equivalent post-tensioned duct at end segnents

Yb The vertical distance from the N-A to the net centroid of post-tensioned
duct at the middle of end segnent

Yb b The vertical distance from the bottom extreme ﬁber to the net centroid of
the post-tensioned duct at the mid-segnent

Yc The vertical distance ﬁ'om the neutral axis to the net centroid of the
equivalent post-tensioning duct at the splice location

y d The vertical distance from the neutral axis to the net centroid of the

equivalent post-tensioning duct at the negative pier location

xxvi

The distance from centroidal axis of goss section.

xxvii

1 Introduction

1.1 Overview

A spliced precast prestressed girder bridge is deﬁned as a type of superstructure in
which separate precast concrete beam-type elements are longitudinally joined, typically
using post-tensioning, to a continuous girder. Among the reasons to use spliced girders
are the reduction of substructure units due to increased span lengths, reduction of girder
units due to increased girder spacing, and functionality and aesthetic improvements by
reducing superstructure depth.

Spliced girder bridges have a proven track record, with more than 250 spliced girder
bridges having been constructed in the US, some of them dating back as early as 1952
(Castrodale and White 2004). In spite of their past and continued use, the application of
this technique is not widespread. A sigriﬁcant reason for limited utilization of spliced
girders is the ambiguity in their design and analysis, rooted in the consideration of
various issues with which the desigrer of conventional precast prestressed concrete
girders is typically not familiar. In addition, the information available in the literature
regarding the design, analysis and construction of spliced girder bridges is limited, as the
experience, information, and methods used on these projects have tended to be job-
speciﬁc, and the knowledge gained has not been made widely available for use on
similar projects (Castrodale and White 2004). A recently completed NCHRP research
progam on extending the span of bridges using precast prestressed concrete girders

(Castrodale and White 2004) will certainly improve the state of knowledge on the design

of spliced girder bridges, as the effort has successfully led to recommendations on LRFD
desigr procedures, standard details, and design examples.

While the design requirements for spliced girder bridges are not sigiiﬁcantly
different from conventional prestressed concrete desigi, the analysis procedure must
take additional considerations. Among the most relevant are staged construction,
multiple stressing stages, and combined pro-tensioning and post-tensioning. Thus, the
design of spliced girder bridges involves geater complexity than is required for
conventional precast/prestressed concrete girder designs. The desigi is generally
executed using a computer progam or a series of spreadsheets. Unfortunately, design
guides, aides, and examples of spliced girder bridges to help desigiers are not readily
available or address only limited portions of the design (Castrodale and White 2004).

The design of spliced girder bridges depends on several parameters that sigiiﬁcantly
inﬂuence performance and cost. The most relevant are time dependent effects, splicing
locations, construction sequences, girder segnent geometries, number of beams, and
number or proﬁles of pre-tensioned and post-tensioned reinforcement. Normally most
design variables are determined based on the desigier’s judgnent and on a trial and
error processes. Consequently there is no guarantee of obtaining the most efficient
and/or economical design, which requires more time and effort to explore, and which
typical projects cannot afford.

Since desigr of spliced girder bridges involves a number of desigi considerations,
the use of mathematical optimization methods can provide a systematic approach to
arrive at appropriate desigr solutions. Computational optimization techniques (Arora

2004) can thus be used as a tool to develop the conﬁguration and sectional optimization

of spliced girder bridges and serve as a guide to produce design aids. Design aids can
depict the relations between girder shape, girder spacing, pre-tensioning and post-
tensioning requirements, splicing locations, pier segnent geometries, initial and ﬁnal
concrete compressive strengths, and tendon proﬁles. Bridge engineers would ﬁnd
beneﬁts in design charts and tables, which are based on optimal solutions that will help
expedite the desigi process. As a result the system can be more widely used by state

highway agencies and bridge consulting ﬁrms.

1.2 Precast Prestressed Concrete Bridge Construction

Precast Prestressed concrete bridges have become the preferred type of bridge
construction; approximately 46% of today’s bridges are of this type, primarily for
reasons of economy, savings in life-cycle costs and their fast construction (Tanase et a1.
2002). In spite of this appeal, precast bridge construction has been mostly limited to
simply supported spans primarily due to the ease of construction.

This approach, however, is not in aligiment with current design trends toward
continuous systems to allow for geater span lengths, eliminate deck joints, and reduce
intermediate piers for safety and cost. The use of precast prestressed concrete bridge for
longer spans, for instance, geater than approximately 46 meters (150 feet), has been
limited due primarily to transportation constraints. Consequently, for medium span
bridges, steel plate girders are typically the favored solution, although they have the
drawback of higher material, construction, and maintenance costs. In order to overcome
these drawbacks and to have an alternative to compete with steel superstructures,
methods to achieve continuity with precast prestressed girders have become of geat

interest to increase the spanning capabilities of precast concrete bridges.

Several methods have been proposed to achieve the continuity of precast
prestressed girders (Anderson 1973, Tadros et a1. 1993 and Tadros et a1. 1995). The most
common are the use of mild steel reinforcement in the deck, the splicing of prestressing
strands, and the use of longitudinal posttensioning (Anderson 1973). Of these methods,
splicing of girder segnents through post-tensioning, commonly referred to as
longitudinal girder splicing, appears to have the geatest potential for extending span

ranges for precast prestressed concrete girder bridges (Castrodale and White 2004).

1.3 Extending the Span Ranges of Precast Bridges

Design options for extending the span lengths of conventional precast/prestressed
concrete girder bridges can be divided into goups which include material related
methods; desigr enhancement methods; the post-tensioning method; and spliced girder
construction.

Many options related to the enhancement of material properties have been used for
extending span length. Using lightweight concrete by reducing the density of girder and
deck concrete has been widely used to reduce the weight of concrete bridges for desigi
purposes as well as to minimize transportation limits. Bridge decks constructed with
lightweight concrete have been found to have equal or better durability than normal
weight concrete deck (Castrodale and White 2004). However when using lightweight
concrete girders, the increased prestress losses and the reduced modulus of elasticity
must be considered. The use of high strength concrete can also be used to extend span
length and the workability of concrete. It may not directly affect the increase of span
length, but it does increase the allowable stress limits. This option has an increased cost

compared to the use of standard strength concrete. The increase in strand size can also

 

help increase the span length of precast/prestressed concrete girder bridges. The 42%
increase in strand area from 12.7 mm (0.5 in.) diameter to 15.24 mm (0.6 in.) diameter
strand can sigiiﬁcantly increase the achievable maximum span length for bridge girders,
especially when use in conjunction with high strength concrete girder (Castrodale and
White 2004).

Enhancement of design parameters or procedures can also increase the span length
of bridges. Modiﬁed standard girder sections are used very often since it is less
expensive than proposed new beam sections, however, the designer should confer with
the local producer before proposing modiﬁcation to standard forms. Modiﬁed girder
sections will increase the girder sectional properties. If the side forms for the standard
girders can be adjusted, all horizontal dimension changes can be easily achieved.
Besides increasing girder properties, the increase in top and bottom ﬂange depth allows
an additional row of pre-tension strands or post-tensioning to be placed in the ﬂanges.
These modiﬁcations are considered as economical methods since existing forms are
utilized to modify sections with minimum effort and cost. This girder modiﬁcation will
help increase the section capacity dramatically and should be considered as a ﬁrst
alternative to extend the span length before proposing a new girder section.

In recent years, several new girder cross-sections have been developed and used
since they have been shown to be more efﬁcient than standard girders for project
speciﬁc needs. However, this option should be used only if many girders and many
similar projects are anticipated to use the new girder types. Otherwise forrnwork costs
are relative high and the producer may be reluctant to bid the job competitively.

Furthermore, design of continuous live loads on the girder and use of temporary supports

 

to shore the girders during deck placement can also help gain additional span lengths for
prestressed concrete bridges. Moreover, the LRF D speciﬁcations (AASHTO LRFD
2003) allow designers to use reﬁned methods to compute live load distributions for
girder designs, which can lead to reduced live load demands and as a result an increase
in girder span length.

Utilizing post-tensioning in precast prestressed concrete girder bridges is the most
efﬁcient way to increase the span length beyond conventional simple spans. However,
the design and construction of post-tensioned girders is more complicated and the design
and construction cost is higher. Post-tensioning can be done for a single girder
component or to longitudinally splice several girders. Post-tensioning is typically applied
to ﬁnal girder cross-section (i.e., girder plus deck) where the composite section is
already achieved; therefore it is more efﬁcient than conventional prestressed concrete

girders where pre-tensioning is applied to the noncomposite girder section.
The combination of post-tensioning and pre-tensioning for the splicing of precast
girders can lead to spans that are much longer than typical simply-supported prestressed
Concrete girders. With the use of combined pre-tensioning and post-tensioning, girder
Sections have become longer, heavier, and deeper. Consequently, their length and weight
may exceed the allowable limits for shipment and/or erection. In order to overcome
t1”lese drawbacks and to have an alternative to compete with steel superstructures,
I11€=thods to achieve continuity by making smaller precast prestressed girders continuous
t1‘ll‘ough post-tensioning have become of geat interest to increase the spanning

capabilities of this bridge system. Thus, this method allows for longer spans and has

I‘ecently gained the most attention.

 

1.4 Spliced Precast Prestressed Girder Bridges

Longitudinally spliced precast prestressed girder bridges (SGB) have been built
since 1960. However, they are not commonly used because of limited experience,
limited information since most projects are job-speciﬁc (Castrodale and White 2004),
the added complexity in their desigi and analysis, and the relatively fewer available
desigr guidelines.

Nowadays, spliced girder bridges are primarily used only for speciﬁc projects
where the custom beam sections were used to substitute long-span structural steel
bridges due to the construction schedule, maintenance, and fabrication cost issues of
concern during the design process.

The solution of longitudinally splicing girders can be applied for single- and
multi-span bridges. Standard sections used for simply supported bridges are used for
section lengths under positive moment. These segments are thus typically called positive

or ﬁeld segnents. Pier or negative segnents normally consist of custom-designed
sections for segnents over intermediate supports used for continuous bridge solutions.
“Drop-in” segnents are usually used in long continuous spans located between the pier
Segment and ﬁeld segnent (Figure 1-1). Multi-span bridges typically consist of “pier”

and “ﬁeld” segnents, while single-span systems consist of ﬁeld segments only.

Splicing Joi mm-.. /——— Deck

\3’\ /

\"—’Posti ve/Fi el d segnent \ PDIOPU n segnent

 

 

 

L—-Ne<_:jotive/Pier segnent

 

 

 

 

 

 

Figure 1-1 Components of spliced precast/prestressed concrete girder bridge Systems

All standard precast/prestressed concrete girders currently available in the market
are developed for use in simple span bridges. In general, continuous systems lead to
higher negative moment demands than simple spans. While peak moments can be
reduced by varying the pier spacing and relative span lengths (Ronald 2001), current
practice used for custom pier segnents is to increase the ﬂexural stiffness and strength in

this region by gadually thickening the bottom ﬂange, producing a curved haunch with a
constant sized bottom ﬂange, or using a girder with a variable web depth.

The sigiiﬁcant form modiﬁcation of custom—designed pier segnents limits the

Wide implementation of this concept due to the high cost of girder fabrication. In order
to make spliced girder bridges more economical in comparison to steel plate girders, the
Use of standard precast/prestressed sections for typical pier segnents needs to be
deVeloped and optimized to reduce cost. Thus, the development of standardized
halmched girders for pier segnents has been identiﬁed as a research need by the
C()ncrete Bridges Committee of the National Cooperative Highway Research Progam -

NCHRP 517 (Castrodale and White 2004).

The desigi of spliced girder bridges was geatly propelled forward by the work
of Tadros et al. (1992) who studied the analysis and design of two-span spliced girder
bridges using prismatic I-beams sections. Later Tadros et al. (1995) studied methods to
create continuity in precast girder bridges by using high strength threaded-rod splicing
and pretensioned strand splicing. Holombo et al. (2000) studied the performance of

precast prestressed spliced girder bridges under seismic loads and provided guidelines to
detail reinforcement at piers, bentcap, and joints; as well as discussing design aspects
relating their seismic analysis. Roland (2001) evaluated key aspects on the desigi and
construction of post-tensioned bulb-tee girder bridges and provided speciﬁc
recommendations for future projects. Among the information provided by Roland are
material properties for design, creep and shrinkage parameters, allowable stress and
ultimate strength design considerations, etc. In addition, several recent spliced girder
bridge projects have been published (Caroland et al. 1992, Fitzgeral and Stelmack 1996,
J allsseri and Spaans 1994) providing key aspects of their design, analysis and
Construction to convey knowledge on this bridge type from the experience. The state-of-
1ﬁle-art knowledge on spliced girder desigr and analysis has been recently documented
t111‘0ugh an NCHRP study in (Castrodale and White 2004) in NCHRP Report 517. This
reI:>ort summarizes current approaches to increasing the spanning capabilities of
prestressed concrete girders and identiﬁes longitudinal splicing through post-tensioning
as the most effective method. In addition, the work identiﬁed load-and-resistance-factor-

desigr procedures for spliced girder bridges, including standard details and desigr

ex amples.

The analysis and design tools currently available for spliced girder bridges are
limited. Most desigr work on these systems is made through custom spreadsheets or use
of the CONSPLICE software (LEAP 2004). The design of spliced girder bridges
depends on several design variables (e. g., time dependent effects, prestress locations and
requirements, beam spacing, etc.) that sigriﬁcantly inﬂuence performance and cost.
Normally, most desigr variables are determined based on the designer’s judgnent

through a trial and error process. This approach requires considerable time and effort to
explore solutions and establish an understanding of the system parameters; something
that typical projects in a preliminary desigi phase cannot afford.

In spite of the design knowledge and analysis tools available for spliced girder
bridges, no desigr aids that can expedite the design process are available.
Implementation of mathematical optimization algorithms can eliminate trial-and-error
design procedures and thus lead to the desigi of more efﬁcient systems. Optimization
techniques have been used effectively to develop desigr aids charts and desi gr tables for
determining section dimensions and prestress requirements. Bridge engineers would ﬁnd
beneﬁts in desigr charts and tables that are based on optimal solutions, to expedite the
desigr process. As a result spliced girder bridges can be more widely used by state
hi gl'iway agencies and bridge consulting ﬁrms.

Technical research committees have also identiﬁed the need to develop standard

Segrnents that can be used efﬁciently in the negative moment area of spliced post-
teIlsioned bridges in order to cost-effectively increase the span range for

I)I‘E.‘.cast/prestressed girder bridges. Thus, the current challenge for the industrialization of

10

post-tensioned SGB is the development of analysis, desigi, and construction procedures

that make use of standard precast girder shapes for pier and ﬁeld segnents.

1.5 Structural Optimization of Civil Infrastructure

Mathematical optimization algorithms can be implemented in order to make a
system more efﬁcient and to eliminate the trial and error design procedures. These
methods can result in signiﬁcant cost savings, performance improvements and ease of

design by providing guidance and insight into optimal solutions.

Bridge systems may be associated with three levels of optimization: (1)

components, (2) transverse and longitudinal conﬁgurations, and (3) structural systems.
Component optimization design variables consist of cross-section dimensions of the
girders, slab, and diaphragns, and details for the prestressed and nonprestressed
reinforcements. Conﬁguration optimization desigi variables deal with the number of
girders (or girder spacing), number of spans, position of intermediate supports, and
Slllpbport conditions (simple span, continuous span, or frame). System optimization
consists in exploring different bridge superstructure solutions such as slab on precast

girders, cell box girder, steel girder, etc.

In general, component optimization deals with the study of optimal dimensioning
and section sizing of beams. Component optimization for prestressed concrete girders
has been broadly studied through several efforts sponsored by state highway agencies for

deVeloping standard positive sections in addition to the AASHTO standard sections.

The use of structural optimization for spliced girder bridge design is limited.

Qomponent optimization studies for SGB have been done for prismatic I-Beam sections

11

for medium span lengths as follows. Lounis et al. (1997) investigated the feasibility of

using existing I-beam sections (e.g., AASHTO-PCI standard girders, AASHTO-PCI

Bulb-Tee girders, Florida DOT standard bulb-tee girders, Nebraska university standard

girders, and Canadian prestressed concrete institute standard girders) for both pre-

tensioned and post-tensioned construction. The results showed that no single standard

section set can be considered to be optimal for all structural systems, therefore, a new set
of I-beam sections were proposed based on the optimal analyses for use in both positive
and negative segnents in two equal continuous span arrangements.

In the studies by Lounis et al. (1997), only one construction sequence was

investigated which is a single-staged post-tensioned applied after deck pouring. In
addition, the splice location was ﬁxed. These conditions may not yield optimal system
r eSults. For example, the positive and negative segnent lengths were ﬁxed, which
cOnsequently restricts other possible variables such as the number and layout for the pre-
tetlsioning and post-tensioning reinforcement. Also, the girder sections were considered
to be prismatic along the span without any modiﬁcation to the pier segnent. Finally,
only one post-tensioning stage was investigated, which may not be an efﬁcient
construction process.

Conﬁguration optimization includes the effect of number of spans, pier locations,
I‘\11'nber of girders, pre-tensioning and post-tensioning requirements, and beam spacing.
In addition, conﬁguration optimization can also include the substructure. Since the
I1I-llnber of supports is commonly reduced for SGB, the total cost of the system

Q0l'rstruction will also be reduced.

12

The only studies to date on conﬁguration optimization on spliced girder bridges have
been done for simple spans using I-beam sections. This type of study has been done only
for 1.8 meter (72”) deep I-beams, which are the most widely used precast sections
throughout the country. These precast sections included the Ohio modiﬁed 1.8 meter
(72”), the New England BT1800, the AASHTO Type VI, The PCI BT 72”, and the
Caltrans ET 1850. The purpose of this study was to obtain the maximum span length of

three equal segnents spliced into a simple span bridge system (Collett and Saliba 2002).
In these studies, prestressed I-beams were analyzed using a custom spreadsheet and
using the progam CONSPLICE PT (LEAP 2004) to analyze the post-tensioning
process. This study was useful in providing quantiﬁable limits on the performance
aChieved by post-tensioning and splicing for standard 1.8 meter (72”) beams used in
States. However, the results from this studied were based on trial and error on a custom
Spreadsheet, therefore no guarantee of obtaining the optimal solutions. In addition, the
l'I'Ia.:rimum achievable span length, which is the purpose of this study, was determined by
ﬁxing the splice locations which may not yield the optimal maximum achievable span
length.

In spite of the analysis and design optimization work noted above, at this time there
are no available design aids to determine the pre- and post- tensioning requirements for
Spliced girder bridges or ways to determine the effective conﬁguration of girders and
S3’stem. The use of optimization techniques can thus be used as a tool for the
development of these desigr aids. Design aids can simplify the analysis and design of

Spliced girder bridges by providing guidance on variable sensitivity and reduce effort in

the design process. The availability of desigi aids for spliced girder bridges can be a

13

geat asset to bridge engineers to expedite the desigi process of this bridge system as an

alternative during the preliminary design phase.

1.6 Objective and scope of dissertation

The research objective pursued in this study was to understand performance of
spliced precast/prestressed concrete girder bridges (SGB) to improve their design by
means of single and multi-criteria optimization through the use of standard or new

optimal pier segments with optimal construction sequence including the time-dependent
egff'ects.

The research approaches used in this study can be summarized into four main parts

as represented in the following ﬂowchart.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Research
Approach
I
‘I- 09559" and 2- Optimization and 3. Veriﬁcation 4. Design Aids
Analysis of SGB Sensitivity Studies

 

 

 

 

 

J, I

Construction I l
S equence Methods Component System
T ime-Dependent I l
Effects Single-Objective Multi-Objective

Figure 1-2 Summary of research approaches

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The detail scope of and approach to the conducted research was as follows:
1) Investigation of design and analysis of optimal construction procedures to maximize

effrciency in single- and multi-span structures.

14

Design and analysis of spliced girder bridge were investigated to study on
different construction methods to obtain optimal solutions. The three most commonly
used construction methods for SGB were considered in the studies. These construction
methods were single-stage post-tensioning alter deck pouring, single-stage post-
tensioning before deck pouring, and multi-staged post-tensioning. The main difference
between these construction methods was the sequence of post-tensioning required to
achieve system continuity.
The optimal construction was also used to determine the maximum achievable
span length of single- and multi-span spliced girder bridge to investigate the viability of
replacing typical existing multi-span highway bridges with spliced girder bridges. Our
goal is to replace existing two-span highway bridges (Figure 1-3) with the single-span

Spliced girder bridges (Figure 1-4) to eliminate the intermediate pier.

Span length between 32.3 m (106 it) — 54.25 m (178 ﬁ) 4|

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1-3 Typical existing 2-span highway bridge

15

 

Span length between 37.5 m (123 ﬁ) — 64.0 m (210 it) _|

Post-tensioned
Strands

Pre-tensioned Strands

 

 

Figure 1-4 Single-span spliced girder bridge

The elimination of substructure unit could reduce the construction cost
dramatically especially for the multi-span bridges since their intermediate piers can be
removed. Consequently, our objective is to study the viability of replacing existing
typical existing three to four-span highway bridges (Figure 1-5) with the two-span

continuous spliced girder bridges (Figure 1-6).

Span length between 54.9 m (180 ﬁ) — 85.34 m (280 ft) _.

   
 

Pre-tensioned
Strands

  

Figure 1-5 Typical existing 4-span highway bridge

 

I Span length between 60.9 m (200 it) — 109.7 m (360 ﬁ)

 

 

 

 

 

 

 

Figure 1-6 T wo-span continuous spliced girder bridge

2) Investigation of time-dependent effects on spliced precast/prestressed girder bridge

systems.
The objective was to investigate and compare time-dependent prestress losses on
Spliced girder bridges using the currently recommended code models. The simpliﬁed
method from the AASHTO-LRFD speciﬁcations (AASHTO 2003) and the age-adjusted
Concrete modulus approach of the CEB-FIP 90 recommendations (CEB-FIP 1990) were
Chosen and compared in this study.
In order to determine the best method to describe the actual behavior of time-
dependent losses in SGB, time-dependent effects were integated in the ﬁnite element
analyses. Time-dependent losses were considered and checked at each critical

construction stage.

3) Implementation of optimization approach and sensitivity approach for spliced

precast/prestressed girder bridges.

The objective is to use structural optimization to evaluate the goodness of a

design, to study the sensitivity of different variables, to understand the system better, to

obtain the maximum achievable span length, and to study efﬁciency of different girders
for SGB.

Component optimization approach is used to determine the optimal dimensioning or
sizing of girders and conﬁguration/system optimization is used to determine the best or
more efﬁcient splice location, girder spacing, pro-tensioning and post-tensioning
requirements and proﬁles.

4) Development of an optimal standard negative section over intermediate piers.
The main goal of this task was to investigate the best method to obtain a standard
negative segnent for system cost efﬁciency. New standard negative sections used for
multi-span bridges were developed based on an integated component and conﬁguration
Optimization approach that allowed obtaining the most efﬁcient section. Section
Optimization processes were performed to obtain the dimensions of negative sections and
were coupled with the conﬁguration optimization in order to obtain the best geometry
Solution for a given span length with the most efﬁcient beam at the pier location.
Four methods to increase the section depth over the pier section were considered,
Ilarnely (1) the use of existing standard positive sections with an additional concrete
block (sofﬁt) attached to the underside of the beam; (2) new optimal non-prismatic
negative section with variable web; (3) new optimal non-prismatic negative section with
Variable sofﬁt and constant web; and (4) new optimal prismatic section used over
positive and negative moment regions.
5) Development of a single- and multi-objective optimization approach for spliced

precast/prestressed girder bridges.

l8

The bridge geometry, beam section dimensions, and material properties with
different construction sequences were selected as parameters for component and
conﬁguration optimization of SGB. Different objective functions, such as minimum
construction cost, minimum post-tensioning, minimum concrete volume, and minimum
pre-tensioning were considered and compared through a single-objective optimization
formulation to obtain the best optimal solutions. Multicriteria optimization methods
combined all the single-objective functions as a main objective goup such as economy,
safety, and geometric requirements and results were based on multi-objective

optimization approaches.

Time-dependent effects were part of the optimization process. Therefore, the
Optimized solution obtained for the SGB system included both short and long-term
effects. Construction sequences are not considered as a design variable in the
Optimization. Rather, the results from the multi-objective optimization processes for
each construction sequence were compared. Thus, each construction method was
analyzed separately. This follows from the fact that realistic and/or practical construction

procedures will depend on the contractor’s preference.
6) Investigation of the life-cycle-cost of spliced precast prestressed concrete bridge
Structures affected by chloride ingession.

The aim here was to investigate the durability and life-cycle cost of spliced girder
bridges under chloride attack, and describe the importance of considering durability at
the design stage. Parametric studies to increase the concrete clear cover of the precast
prestressed concrete girder were used to evaluate the service life of the structure.

Strength requirements and durability were simultaneously considered during the design

19

 

process. Thus a compromise between the strength and serviceability requirements versus

the life-cycle-cost of spliced girder bridges was achieved.

7) Veriﬁcation of optimal results obtained from the optimization studies
The objective was to verify obtained optimal results from the optimization
process compared with the results of a documented design in NCHRP 517 Report and
published journal papers. This result veriﬁcation is used to proof that optimal solutions
are acceptable and can be used to develop design aids for the preliminary design of
spliced girder bridges.
8) Development of design aids for the preliminary design of spliced girder bridges.
The objective was to use optimal results from the above-mentioned analyses to
create simpliﬁed design aids for the preliminary design of single-span and two-span
Continuous spliced precast/prestressed girder bridges. The desigr aids are in the form of

t able and gaphs that indicate the best parameter values for minimum construction cost.

The organization and outline of the dissertation is as follows:

Chapter 1 provides an introduction to precast/prestressed concrete bridge
construction and spliced precast/prestressed concrete girder bridges (SGB). Current
application of structural optimization in precast/prestressed concrete and SGB are brieﬂy
reviewed. Outstanding research needs and the motivation behind this work are
discussed, and followed by the research scope and approach.

Chapter 2 presents the desigr and analysis approaches for spliced precast/prestressed

concrete bridges. The major desigi criteria used for the analysis and construction
sequence of SGB are brieﬂy presented. The approaches and models currently

recommended for time-dependent prestress losses as well as the identiﬁcation of critical

20

 

construction stages are reviewed. The available standard section and non—standard
section are evaluated for use in SGB construction. Finally, the analysis of durability in
combination with the life-cycle cost of spliced prestressed girder bridge structures
attacked by chloride is presented.

Chapter 3 reviews the optimization approaches used for single- and multi-objective

optimization of SGB. An overview of the mathematical algorithms used for solving
constrained optimization problems is presented. The weighted objective approach and a-
constraint method for solving multi-objective optimization problems are reviewed. The
concept of generating the pareto optimum set as well as compromise progarnming to
select the best solution in multi-objective problems are presented. Descriptions of single-
and multi-objective functions are discussed in general. Lastly, the implementation of the
Optimization process for SGB in this study is brieﬂy discussed.

Chapter 4 describes the optimal system design of single-span SGB. The optimization
problem formulation for different construction sequence methods of SGB for single- and
thulti-objective optimization are demonstrated to obtain optimal solutions for maximum

achievable span lengths. Parametric studies on material unit weights of the concrete deck
and concrete girder are investigated to determine their inﬂuence in increasing span
length. The durability of SGB due to the chloride ingess is evaluated in terms of
concrete cover and then used to determine the service life and life-cycle-cost of the
bridge system.

Chapter 5 focuses on the investigation of optimal system designs for two-span

continuous SGB. Component optimization formulation for new pier segnents are

implemented and evaluated to obtain a new pier/negative segnent for overall cost

21

efﬁciency. Optimal pier segnents are used in single- and multi-objective optimization
problems to obtain optimal solutions for maximum achievable span lengths. Sensitivity
analyses on material unit weight of the concrete deck and concrete girder were
investigated to determine the best method to increase span length.

Chapter 6 presents the development of aids for the preliminary design of spliced
girder bridges consisting of documented design examples and design charts. Desigr aids
were developed by using the nonlinear structural optimization analyses implemented in a
custom automated progam written in Matlab (Mathworks 2004). Both single- and multi-

objective optimization formulations were used to obtain the optimized solutions. Desigr

and construction limit state requirements were used as constraints in the optimization

processes.

Chapter 7 summarizes and concludes the current research efforts for the single- and
multi-objective optimization for spliced girder bridges and provides recommendations

for future research.

22

2 Design and Analysis of Spliced Precast/Prestressed

Girder Bridges

2.1 Introduction

Desigi of longitudinally spliced precast/prestressed concrete girder bridges through
continuous post-tensioning, or spliced girder bridges (SGB), has gained renewed interest
since their spanning range can exceed 48.77 m (160 ﬂ) and are more efﬁcient and cost-
effective than steel plate-girder designs. Additionally, using SGB can reduce total

project cost due to substructure reduction, fewer number of girders, and increased safety
by elimination of shoulder piers and minimization of superstructure depth.

Spliced girder bridges have been constructed in the US since 1952 (Castrodale and

White 2004), however their use are very limited and not common. A signiﬁcant reason
for their limited use is the ambiguity in their design and analysis, additional issues with
which the designer of conventional precast/prestressed concrete girders is typically not
familiar. In addition, the information available in the literature regarding their desigi,
analysis and construction is limited; as the experience, information, and methods used on
these projects have tended to be job-speciﬁc and the knowledge gained has not been
made widely available for use on similar projects (Castrodale and White 2004).

This chapter presents an overview on the design and analysis approaches for spliced
precast/prestressed concrete bridges. The major design criteria used in structural
analyses and construction sequence methods of SGB are brieﬂy presented. Issues related

to equivalent post—tensioning loading, time-dependent losses and construction sequence

23

methods are reviewed. Lastly, durability issues in combination with the life-cycle cost of

spliced prestressed girder bridge structures attacked by chloride are presented.

2.2 Spliced Precast Prestressed Girder Bridge Systems

Girder splicing is a method in which more than one pre-tensioned precast
concrete segnent is longitudinally spliced to produce span lengths geater than the girder
segnents. Each segnent is connected by full-length post-tensioning. The series of
girders are temporarily simply supported between the piers and the abutrnents and

connection, or splicing, is achieved in the ﬁeld.

The splicing solution can be applied for single-span and multi-span bridges.
Typical standard sections used for simply supported bridges are used for section lengths
under positive moment. These segnents are thus called “positive” or “ﬁeld” segnents. A
single-span spliced-girder bridge (SSSGB) system is normally composed of three
conventional positive girder segnents as shown in Figure 2—1, where the end girder

segnents may have different lengths depending on the splicing location.

 

 

CL Abutment
CL Abutment Totol Spon Length
End Spon L MidSDOD J End Spon
Splicing JoinxL l /— Deck
I II I II

 

 

 

 

\i::— CL Splice
8 CL Temp Support

Figure 2-1 T hree-segment single-span spliced precast/prestressed girder bridge

24

Multi-span continuous spliced girder bridges (CSGB) consists of positive/ﬁeld

segments and “negative” or “pier” segnents. Negative segnents can be either prismatic

segnent as shown in Figure 2-2 or nonprismatic segnent as shown in Figure 2-3. A two-

span CSGB system is normally consisted of custom-designed sections used for

continuous bridge solutions for segnents over intermediate supports. The nonprismatic

segnent has a deeper section over the pier location and is tapered down to match the

positive standard section at the splice location.

 

 

 

 

CL Abutment Totol Spon Length (L) CL Abutment
: End Spon (L1) :I: Midspon (L2) :1: End Spon (L1) 1
Splice joint3|\ CL lIDier /——||Deck
I I H l H Prestressed Beans
L Positive 5.9ng1: or \—Prismotic

Negotive segment

(Field segment)
or Pier segment

 

 

 

 

 

 

 

 

[J

E

Figure 2-2 T wo-span spliced precast/prestressed bridge with prismatic pier segment

 

 

CL AbUTmnI TOIOI SDOn Lengih (L) CL AbUImenT
A End Spon (L1) LL MidSDOH “-2) J4 End Spon (L1) _
CL Pier l

Splice joi mi

/— Deck
I

 

r

(Fl el d segment)

III)

LPositive segnent or

  

“Prestressed Beams
Nonprismotic

Negotive segnent

or Pier segnent

 

 

 

 

 

 

 

Figure 2-3 T wo-span spliced precast/prestressed bridge with nonprismatic pier segment

25

2.3 Girder Cross-Sections

There are several types of standard and newly developed girder cross-sections used
for spliced precast/prestressed girder bridges. Some of them are chosen and set by local
practice or project-speciﬁc requirements. Normally the deeper standards girder sections,
i.e., AASHTO type VI, PCI-BT-72, and NEBT 1800, are used in spliced systems since
they can span longer distances than shallow sections. Some states, i.e., Florida,
Washington, and Nebraska, have developed their own sections for job-speciﬁc projects

in order to accommodate their span length needs.

2.3.1 Standard sections

Currently, only standard precast/prestressed concrete girders are used under
positive moments in simple spans. While these sections can be utilized in spliced girder
construction, some researchers have developed new standard sections to be used more
efﬁciently in both positive and negative regions in continuous SGBs. However, there is
no evidence that these new standard sections are being commonly made.

Available standard sections in the market used for SGB include AASHTO I-
girders and PCI-Bulb Tee girders. Normally, standard sections used in SGB should not
be too shallow in order to beneﬁt from the eccentricity effects of post-tensioning
tendons. In addition, the dimension of standard sections may need to be modiﬁed to
accommodate post-tensioning ducts (i.e., widening of the web). In spite of the
combination of pre-tensioning and post-tensioning, the sizes of ﬂanges are not

necessarily large. Generally, the bottom ﬂange of the spliced prestressed section is used

26

to accommodate the pre-tensioning strands for positive segnents, while the top ﬂange is
used to accommodate pre-tensioning strands for negative segnents.

Several standard sections have been speciﬁcally developed by some states for
their particular design and construction requirements. These sections include the
modiﬁed Washington WSZ400 girder, the modiﬁed Florida bulb-tee PCI-BT 78 and
PCI-BT 96 girders, and trapezoidal box beams. These new standard sections were
developed to increase the span length for job-speciﬁc projects.

I-girder and Bulb-Tee girders are used in SGB more often than box beams due to
their lighter weights. Box beams have two webs, thus their sections are very heavy.
Consequently, their span lengths are much shorter than I- girders or PC] Bulb-Tee girders

for the same girder depth.

2.3.2 Non-standard sections

For multi-span spliced girder bridges, additional moment and shear resistances at
interior supports is required. Segnents over the interior supports are called negative or
pier segnents. These segnents can be custom-desigred with deep sections using
haunches or can be of constant depth girders to carry the high negative bending moments
and high shear demands.

Use of constant depth girder sections for the entire bridge span length is more
economical. However, when the bottom ﬂange compressive stresses, or web shear
stresses at interior supports are not very high, modiﬁed standard girder sections can be
used. Modiﬁcations of standard girders include thickening the web at locations of high

moment and shear, using the beneﬁt of end-block sections, or adding an additional slab

27

to the bottom ﬂange at interior support locations. These modiﬁcations still maintain a
constant section depth.

When desigi requirements are very high the modiﬁcations of the cross-section
usually demand having a deeper section (haunched) over the interior supports
(Castrodale and White 2004). Increase in girder depth at interior supports provides
additional moment and shear capacity that help satisfy the design requirements.
However, these increased depth haunched pier segnents might become very heavy for
highway transportation.

The construction cost of a haunched section is higher than that of a standard
girder. Typically, haunched sections have a linear taper due to ease of construction.
Curved, or parabolic, haunches can also be used but they require more intricate
formwork, which can dramatically increase construction cost.

Currently, there are several available options for achieving haunched sections for

I- and Bulb-Tee girders. These options are listed in the following.

a) Haunched section with constant web

Existing standard girder forms can be used with slight modiﬁcations on the slope
of the existing bottom ﬂange. The haunched section is accomplished by varying the
bottom sofﬁt ﬂange from the standard bottom ﬂange thickness at the splice location to
its deepest section over the interior support, or pier.

This method allows the fabricator to utilize existing forms as part of the
haunched section, thus fabrication is less complicated than other haunching approaches.
However, the increased height of a haunched bottom ﬂange adds signiﬁcant weight to

the section and the deep ﬂange thickness may not be attractive.

28

b) Haunched section with constant bottom ﬂange

This approach requires using special fonnwork and the girders are normally
custom fabricated. Web height is varied along the haunched section while the bottom
ﬂange remains of constant depth along the length of the segnent. Even though the
bottom ﬂange has a constant depth, it must be large enough to resist the large negative
bending moments over interior supports.

The most attractive feature of this option is the reduced weight of the haunched
segnent. This can lead to savings in transportation costs. The girder segnent weight is

much lighter since only the web height is varied.

2.4 Splice Location

The location of splices generally depend on site characteristic, maximum limits for
girder shipping weight and segnent length, fabrication, erection, and/or structural issues.
When the girders are transported through several states they have to satisfy the shipping
weight and length limits of all states on the route. Some states limit the weight of the
girder not to exceed 587 kN (132 kips), in addition to weight limits of 89 kN (20 kips)
for the axles of trucks (Castrodale and White 2004). In addition to limits on
transportation shipping weight and length, available lifting equipment limits should also
be considered.

Splice locations should not be located near mid-span, but rather towards the end of
the span. Splice locations can yield different maximum achievable span lengths for the
same beam type with different splice locations. Thus, the splice location should be

careﬁrlly considered in the desigi phase.

29

2.5 Structural Loads

The analysis and design of post-tensioned spliced girder bridges is very similar to the
analysis and design of segmental bridges. However, it is different from conventional
precast/prestressed concrete girder bridges primarily in terrms of its staged construction.
A time-dependent analysis of the spliced girder system is required to properly
accumulate the actions and reactions at each stage of erection. Additional issues that are
required differently from conventional prestressed concrete desigr are multiple stressing
stages, combined pro-tensioning and post-tensioning, and time-dependent losses. Thus,
the design of spliced girder bridges involves geater complexity than for conventional
precast/prestressed concrete girder designs. Their design is thus generally executed using
computer progarns, such as the commercial progam CONSPLICE PT (LEAP 2005), or
a series of custom spreadsheets developed by design ofﬁces.

The loads in the spliced precast/prestressed girders are a consequence of the
construction stages. Each load case is considered individually and accumulated, and

loads on the structure are thus classiﬁed as non-composite loads and composite loads.

2.5.1 Non-composite loads

Non-composite dead loads include the weights of girders, deck, optional stay-in-
place deck forms, and construction loads. Beam spacing has a direct inﬂuence on the
deck, haunch, and construction loads. All distributed loads are assumed to be applied
over the effective tributary area for each girder. Girder and haunch loads are taken as

beam line loads. The weight density of normal-weight concrete for girders and deck are

assumed to be 2400 kgm3 (150 lb/ﬁ3). For sensitivity analyses, light-weight concrete

30

was considered, with weight densities of 1762 kg/m3 (110) to 2243 kg/m3 (140 lb/ft3).

Uniform 50 mm (2-in.) girder/deck matching haunches are assumed over the top ﬂange

width for construction tolerance and variation in road camber. Corrugated metal stay-in-
place deck forming is assumed to be 0.77 kN/m2 (16 lb/ftz). Construction design loads

were thus resulting from temporary loading during placement of the concrete deck and

taken as 0.96 kN/m2(201b/ft2).

2.5.2 Composite loads

Composite dead loads included the weight of the girder splice joint, the weight of
intermediate diaphragns, the removal of temporary supports, the weight of barriers, the
weight of future wearing surface, and vehicular live loads. Composite dead loads were
assumed to be equally distributed to all girders in the superstructure cross-section.

Loads from splice joints and temporary diaphragms are not immediately transferred
to the superstructure during construction since they are supported by temporary supports.
After post-tensioning is applied and the temporary supports are removed, these loads are
applied to the composite beam. These “support loads” are based on the assumption that
the bridge is lifted off the temporary supports alter post-tensioning.

For a single-span SGB, the three continuous segments are thus analyzed as a simple
single-span system with two point loads applied at the temporary support locations.
Temporary support reactions include all non-composite dead loads and the splice joint
weight. These are calculated as half of the weight from the end and middle girder

segments.

31

For a multi-span continuous SGB, the spliced segments are analyzed as a continuous
system with point loads applied at the temporary support locations. Reactions at the
temporary supports depend on the construction sequence. Secondary reactions due to the
continuous system will be induced at the intermediate pier. Temporary support reactions
include all non-composite dead loads and the splice joint weight. For a two-spam
continuous system, these are calculated as half of the weight from the end together with
a quarter of the weight of the middle girder segnent.

Desigi live loads are based on the ASSHTO-LRFD bridge desigi speciﬁcations
(AASHTO 2003). Design live loads are selected from either the design HL-93 truck
loads in combination with the design lane load or the design tandem loads in
combination with a design lane load (ASSHTO 2003). The live load used in the design is
based on the maximum load combination effect. The dynamic allowance impact factor
(IM) is only applied to the design truck as speciﬁed in the AASHTO speciﬁcations. Live
load demands are based on the load distribution factors for either exterior or interior

beams as given in the AASHTO-LRFD speciﬁcations.

2.5.3 Post-tensioning loads

Normally, post-tensioned tendon proﬁles are arranged in the fonm of parabolic
curves, therefore their eccentricity (distance measured from the girder neutral axis to the
post-tensioning cable) changes along the span length. To simplify the calculations of the
loads introduced by these curved cables, the forces caused by post-tensioning and their
eccentricity proﬁle along the length can be represented by equivalent loads.

Equivalent loads for prismatic sections are commonly available and deﬁned in

multiple references. However, much less information is available on equivalent loads for

32

nonprismatic sections. In the following sections the equivalent loads of post-tensioning
loads on prismatic and nonprismatic sections are outlined for documentation purposes.

a) Single-span girder using a prismatic section

The post-tensioning proﬁle of a single-span girder is deﬁned by a single quadratic
curve (see Figure 2-4) and the resulting load from the post-tensioning cable can be
modeled by the equivalent loads shown in Figure 2-5. Equivalent loads include
horizontal forces, vertical forces, and moments at the external supports; and uniformly
distributed transverse forces along the tendon proﬁle. The uniformly distributed

transverse forces are generated by the curvature in the tendon proﬁle.

 

F. _______________________ X ..................................... -1:

Figure 2-4 Post-tensioned proﬁle of single-span using prismatic section

 

 

 

F5“ FSinO

 

 

 

 

Feasefrélrllllllllll\moso

Fe II: L JFG

Figure 2-5 Equivalent loads and ﬁxed end moments of single-span using prismatic
section

Uniform loads w; represent transverse forces that occur along the span as shown

in Figure 2-5. The distance measured from the neutral axis to the lowest point of tendon

33

proﬁle is deﬁned as d. The angle between the horizontal reference line and the post-
tensioning strand location at x = 0 is deﬁned as 9. The magnitude of the equivalent
uniform load due to the post-tensioning cable curvature is given by:

w, = 8F_(d2‘_e), (2-1)
L

b) Two-span continuous girder system using a prismatic section

The two-span continuous spliced girder system dealt with in this dissertation is
composed of two continuous spans and is assumed to be symmetry about the center of
the pier. Therefore only one half of the system is dealt with herein. The symmetric
proﬁle of a post-tensioned tendon along three prismatic continuous girders in the two-
span system is assumed to be made of three different quadratic curves as shown in
Figure 2-6. The ﬁrst curve (Curve 1) is measured from the end span to a distance aL.
The second curve (Curve 2) is deﬁned between the location at al., measured from the
endspan, and the location at a 1L, measured from the center of the pier. The third curve
(Curve 3) is deﬁned along the segment a distance aIL from the pier to pier location
itself. Assuming quadratic parabolic curves for all parts, the eccentricities of post-
tensioning strands are used to deﬁne the locations of the quadratic curves. The
eccentricity at X = 0 is deﬁned as e(X = 0) = ﬂeog. The eccentricity at X = aL is deﬁned
as e(X = aL) = ﬂjeog. The eccentricity at X = L is deﬁned as e(X = L) = egg. The
terrminology used for the post-tensioning parabolic curve proﬁles is illustrated in Figure

2-6.

34

N-A ﬂ

 

 

 

 

 

 

39031
0 -4
..__.p X
«L (1-a-a1)L 01L
i. .i- ,
Curve 1 Curve 2 Curve 3

 

 

L

Figure 2-6 Post-tensioned proﬁle of two-continuous span using prismatic section

The equivalent distributed loads and ﬁxed-end moments of the post-tensioning
proﬁle of a two-span continuous system using a prismatic beam is illustrated in Figure
2-7. Three uniform transverse loads (W1, W2 and W3) represent the pressure from the
quadratic tendon proﬁle on the symmetric portion of the two-span girder system. The
angle between the horizontal reference line and the post-tensioning strands at X = O is
deﬁned as 6'. This 0 angle is used to determine the vertical end force (FSin6 ) and

horizontal end force (FCos0 ) due to the post-tensioning forces at the end of the spans,

while the end moments (mm and F 3803) are determined by multiplying the horizontal

end force with the eccentricity at each of the end spans.

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CL Pier
“'3
FSinO wz
W1
1 (I A A A A A
F005 0 I/J': I I I I " " ‘ ":11 F0059
F3903 aL (‘l-a-a,)L a1L mm
L

 

 

 

Figure 2-7 Equivalent loads and ﬁxed end moments of two-continuous span using
prismatic section

The magnitude of the equivalent distributed loads and moments can be shown to

be given by the expression in Equation (2-2) through Equation (2-5) below:

 

2F -
W1: (,3 161)€OB,

 

 

2 (2-2)
(aL)
2Fe —
w2 : 2()B[ :62 2:61 2 ]’ (2_3)
L (1—2a-2a1+a +2aa1+a1)
= 2Fe03<ﬂ2 - 1) (24)
2 2 ’
a1 L
mBA =—FeOB[—,Ba2 +(l+a+a1)2 —a1(1-a)—,Bl(3—3a1 +
£112 + 2a — aa1)]0'25 , (2-5)

where F = total post-tensioning force; ,0 = coefﬁcient of eccentricity at the end of the

positive segment; [31 = coefﬁcient of eccentricity at the mid-span of the positive
segment; e03 = vertical eccentricity at the center of the pier segment; a = coefﬁcient of

horizontal distance from spam end to the mid-span of the positive segment; a, =

coefﬁcient of horizontal distance from splicing location to the center of the pier.

36

The sign convention for the eccentricity values assumes that they are positive
below the neutral axis. The coefﬁcients of ,6 and 5, are between 0 and -1 while egg is a
negative value. Consequently, the uniform loads w, and W} are oriented upward and W3 is
oriented downward (see Figure 2-7). Finally the value of | £1 I must be geater than | 3 |
to satisfy the requirements for a quadratic curve requirement.

c) Two-continuous span using nonprismatic section

Pier segments are increased in height for larger negative moment capacity,
consequently a continuous system is typically composed of prismatic positive segments
and non-prismatic negative segments. The proﬁle for the post—tensioning tendon along a
non-prismatic continuous girder is also assumed to be made of three different quadratic
curves on one half of a symmetric structure as shown in Figure 2-8. At the haunched
section the neutral axis is assumed to shift in proportion to the height/depth of the
haunched section. The tendon proﬁle is deﬁned by three continuous quadratic curves as
shown in Figure 2-8. The eccentricity at X = 0 is deﬁned as e(X = 0) = e]. The
eccentricity at X = aL is deﬁned as e(X = aL) = eg. The eccentricity at X = (l-alL) is
deﬁned as e(X = (l-a,L)) = e3, The eccentricity at X = L is deﬁned as e(X = L) = e4,
Finally, the angle between the horizontal reference and the haunch neutral axis is deﬁned

as w.

37

CL Pier

 

 

 

 

 

Prismatic

 

 

 

 

._, :ection section

GL 1 (1-a-a1)L I a1L
I Curve 1 T Curve 2 T Curve 3 r
V L 7

Figure 2-8 Post—tensioned proﬁle of two—continuous span using a non-prismatic section

The equivalent transverse loads and ﬁxed-end moments for the post-tensioning
of a two-span continuous girder using a non-prismatic pier segment is illustrated in
Figure 2-9. The distance measured from the lowest point of the haunched neutral axis of
the pier segnent to the prismatic neutral axis is represented by d in Figure 2-8. The angle
between the horizontal reference lime and the post-tensioning strands at the beam end is

deﬁned as 6 (see Figure 2-8). This angle 0 is used to determine the vertical end force

(FSinB), horizontal end force (FCos0) and segment end moments (mm and F5803),

similar to the end forces in the prismatic section. Three uniform transverse loads (w 1, w;,
and W3) represent the pressure from the post-tensioning strand onto the beam as shown
in Figure 2-9. The values for the uniform loads are given by:

2F(e—e)
w]: 1 2,

(aL)2 (2'6)

38

“22:35 63-82

 

,_ 2 1, (2-7)
L (l—2a—2a1 +a +2aa1+a

2F e — e
w =.___(: 2 3), (2-8)
a1 L
where e, = eccentricity at the end of the positive segment; e; = eccentricity at the mid-
span of the positive segment; e3 = eccentricity at the splice location; and e4 = eccentricity
at the center of the pier segment measured from the centroid of the prismatic section.
It should be noted that that the post-tensioning strand in the non—prismatic segnent

leads to two additional concentrated forces (2FSim/I and FSimp) at the ends of this

element as shown in Figure 2-9.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2FSinip
. w
FSInO “’1 1112414
F003, (viilli v. $1 Peas,
FSinip mm
Fe,

 

 

 

 

Figure 2-9 Equivalent loads and ﬁxed end moments of two-continuous span using a
non-prismatic section

The sign convention for the eccentricity values is positive below the N-A line.
Therefore, the uniform loads w, and w; are oriented upward and W3 is oriented
downward. Notations for all other variables are the same as described in Section

2.5.3(b).

39

2.6 Flexural Criteria

2.6.1 Service limit state

Service limit states include checking concrete stresses in the girder, deck and
splice elements. These stresses are calculated from the applied moments on individual
girder segments, the moment applied to the entire spliced girders, and the forces caused
by prestress loads.

Flexural limit states for both tension and compression stresses are provided in the
AASHTO-LRFD speciﬁcations (AASHTO 2003) for conventional prestressed concrete
and segmental construction. Limit stresses on joints are deﬁned according to segmental
construction while limit stresses at all other locations follow conventional prestressed
concrete construction.

There are two load combinations considered for full service loads after ﬁnal
construction; these are the Service Limit State I (SLS-I) and Service Limit State 111
(SLS—III) conditions. The difference between these limit states is the live load factor,
which is equal to 1.0 for SLS-I and 0.8 for SLS-III. Service limit state I is used to check
the concrete compression stress limit while service limit state III is used to check the
concrete tension stress limit under full service loads. Full service loads include the girder
self-weight, deck load, non-composite dead load, temporary pier removal, construction
dead load, and live load. Top deck stresses, and top and bottom girder stresses at these
limit states are checked against allowable concrete stress as deﬁned in Table 2-1

(AASHTO 2003).

40

Table 2-1 Allowable concrete stress limits for critical construction stages

 

 

Critical Construction Stress Type Girder concrete DECK
Stages concrete
Pre-Tensioning Strand Compression 06]};
Release
Tension -200 psi
Deck/Splice Concrete Compression 0-6fc
Tension -200 p81
Post-Tensioning/Barriers Compression 0.6]; 06de
Tension -O.0914\/fC(MPa) -0.0914\1de (MPa)
-024fo (ksi) -O.24\/f’cd (ksi)
Final Condition — after Compressionl 06¢wa 060ch
losses
Compression 2 0-45fc 045de
Compression 3 0,4fc 04de
Tension 1 -0.0724\/fc (MPa)
-0.19\/fc (ksi)

 

Two additional ﬂexural limit states for spliced girder bridges were recommended in
the recent NCHRP 517 Report (Castrodale and White 2004) that deal with compression
stress limits for partial service load at the ﬁnal construction stage. These two limit states
differ in the load combination to be considered. The ﬁrst one, referred to as Compression
2 in the NCHRP 517 Report, considers all the loads from the SLS-I except the live load
with all other loads using a unit load factor. The second additional limit state, named
Compression 3, considers all the SSL-I loads but with a unit load factor for the live load
and a load factor of 0.5 for all other loads. The limit stresses for the Compression l and
Compression 2 checks (see Table 2-1) are those recommended by the NCHRP 517

Report.

41

2.6.2 Strength limit state

The design of spliced girder bridges is typically controlled by the service limit
state. However, the strength limit state still needs to be checked. Spliced girders should
be checked at critical sections for their ﬂexural and shear capacities. The strength limit
state is checked at ﬁnal conditions.

Assessment of strength limit state is based on the AASHTO-LRFD
recommendations in Article 5.73 (AASHTO 2003) using the simpliﬁed strain-
compatibility analysis approach provided in the PCI Bridge Design Manual (PC1 2002).
Girders for spliced systems tend to have large amounts of post-tensioning strands.
Consequently, the compression zone might extend into the web of the girder. Current
AASHTO-LRFD speciﬁcations (AASHTO 2003) do not consider the different strength
of concrete in the deck and in the girder properly. Therefore, a simpliﬁed strain-
compatibility analysis is best suitable to evaluate the ﬂexural capacity of spliced girder
bridges (Castrodale and White 2004).

Strain compatibility analysis is an iterative approach that can be used to determine

the ﬂexural capacity of a cross-section at its normal strength limit state. First, the depth of

the ﬂexural compression zone, c, is assumed. Decompression strain spa is the difference
in strain between the concrete and prestressed reinforcement and can be deﬁned as ecu =
pr/Ep. The prestressing strain is computed from the curvature 0' = ecu/c plus the
decompression strain. The value of 13,0 should be less than the value of the jacking force

fpj. The ultimate compressive strain, ecu, in the concrete at the top of deck is assumed

equal to 0.003 (sec ASSHTO-LRFD Article 5.7.2.1). The total depth of concrete

42

equivalent stress block is computed as a = [fl/c, where ,0, is the factor based on the
concrete compressive strength. The stress in the prestressing steel is computed using the
modiﬁed Ramberg-Osgood equation:

l—A
fpszEpgpfiA‘l‘ lisfpus (2'9)

 

[1+(BappC1C
where Ep is the elastic modulus of the prestressing steel; epfis the strain corresponding to

the computed stress; and parameters A, B, and C can be varied to deﬁne the shape of the
stress-strain curve. For example, A = 0.0334, B = 116.4 and C = 12 when the stress is
equal to 1675 MPa (243 ksi) at 1% strain and equal to 1862 MPa (270 ksi) at 3.5 %

strain.

The location of effective depth, dp, of the total prestressed reinforcement, which

includes the pre-tensioned and post-tensioned strands, is estimated by

 

dpPTAPT + dpPRETAPRET
APT + APRET

d:

p (2-10)

The total compression force in the top ﬂange, web and deck are compared with
the tension force. The depth of the compression zone, c, is adjusted until the compression
and tension forces are equal, i.e., until section force equilibrium is satisﬁed.

When the concrete stress block extends into the web, all of the ﬂange and part of

the top web are in compression. The tension force in the section is then calculated by:

T : (APT + APRETlfps, (2‘11)

43

where APRET is the area of pro-tensioned strands; A pr is the area of post-tensioned strands;

and fps is the prestressing steel stress. The compression force in the deck, C,, is
calculated as:

C, = 0.85fc'dbehf, (2-12)
where f ’cd is the deck compressive strength; be is the effective bridge width; and hfis the
slab thickness. The compressive force, Cf, in the girder ﬂange is deﬁned as:

of =0.85f;bfhw, (2-13)
where f 'C is the girder concrete compressive strength; bf is the beam ﬂange width; and

hw is the girder web height. The nominal design moment, Mn, for the section is then

deﬁnedas:
M -085f bh d mil—[— 085 b h d —h —k 214
n“ cdef(p 2)+-fcfw(p f 2- (‘)

There is no reduction factor ((15: 1) for the factored moment, see AASHTO-
LRFD Article 5.5.4.2.] (AASHTO 2003). The maximum reinforcement limit needs to be
checked to satisfy C/de < 0.42 according to AASHTO-LRFD Article 5.7.3.3.1
(AASHTO 2003). Normally the minimum reinforcement requirement is satisﬁed since
there is a large amount of reinforcement provided in the section. Finally, according to the
NCHRP 517 report (Castrodale and White 2004), the fatigue limit state does not need to

be considered since the member is designed to satisfy the limiting tensile stress at the

Service 111 limit state according to AASHTO-LRFD Article 5.5.3.1 (AASHTO 2003).

2.7 Shear Criteria

The shear forces on the system include those from the girder, deck, haunch, future
wearing surface, removal of temporary supports and live load including impact factor.
Factored shear loads are determined from service loads multiplied by the appropriate load
factors. Load factors for composite live load and composite dead load are 1.75 and 1.5,
respectively. The load factor for all other dead loads is 1.25. There are several critical
shear sections needed to be checked, which include the end block and the splice
locations. The considerations and relevant formulation for these shear checks are

presented in the next senctions.

2.7.1 Vertical shear at splice location

Shear at the spliced locations must be considered since the sections at the faces of
the splice are weak. The shear capacity of the splice is determined by using the interface
shear method outlined in AASHTO-LRFD Article 5.8.4 (AASHTO 2003). The nominal

shear resistance of the interface is computed as:

V" = cAa, + ,u[Avffy + PC], (2-15)
where Am, is the area of concrete in the shear transfer zone, which is deﬁned as the web
width multiplied by the difference between the total section depth and the section depth

in tension; Avf is the area of all shear reinforcement including post-tensioning strands

and mild reinforcement crossing the shear plane; it is the friction coefﬁcient = 0.6 x A ,

where ,1 =1 for normal weight concrete; c is the permissible pressure at the splice contact

45

area = 0.075; PC is the permanent net compressive force normal to the shear plane and is

deﬁned as fpéA p5; and fy is the yield strength of reinforcement.

The nominal shear resistance must be greater than the factored shear demand, or
¢Vn 2 Vu , where (15 =- 0.9. Since spliced girder bridges contain large amounts of post-
tensioning strands in their section, the shear resistance of the splice is normally adequate.
Although other reinforcement in the section is typically neglected, the post-tensioning
strands usually provide more than the needed resistance. Thus, most of the time, a shear
key is not required to provide additional shear resistance at the splice location (Castrodale
and White 2004). In addition, the inclination of the post-tensioning tendon force crossing

the splice helps increase the shear capacity of the section as deﬁned by the shear

resistance component Vp (AASHTO 2003).

2.7.2 Vertical shear at end blocks

The shear capacity at the end blocks for the spliced girder bridge is calculated in
the same way as for conventional prestressed concrete bridges. The shear capacity is thus

deﬁned as (AASHTO 2003):
¢Vn = ¢(VC +VS+Vp). (2-16)
The component Vp = F sina is the shear force provided by the post-tensioning with a

being the angle between the effective post-tensioning strands and the horizontal reference
of the lowest post-tensioning strand proﬁle, and F being the total post-tensioning force in

the section.

46

The component VC is the shear strength provided by concrete and shall be taken as

the lesser of the values of Vci or VCW. The shear strength, Vci is computed by:

. . V—M
Vci=0.6 fcbd+Vd+'——C’L, (247)

max

but need not be less than 1.7 fc'b'd (psi) and d need not be less than 0.8/i. The distance

d is that from the extreme compressive ﬁber to the centroid of the prestressing force in
inches. The variable b' is the width of a web of a ﬂanged member in inches. V,- is the
factored shear force at the section due to externally applied loads occurring

simultaneously with Mmax. Mmax is the maximum factored moment at the section due to

externally applied loads. V,- and Mmax are computed from the load combination causing
maximum moment at the section. The moment causing ﬂexural cracking at the section

due to externally applied loads, M6,, is computed by:
I .
II/Icr=7(6 fc+fpe_fd)a (2-18)
1

where I is the moment of inertia about the centroid of the cross-section, Y, is the distance
from centroidal axis of the gross sections, fpe is the compressive stress in the concrete due
to effective prestress forces at the extreme ﬁbers, fd is the stress due to unfactored dead
load at the extreme ﬁber; and f ’c is the compressive strength of concrete at 28 days.

The shear strength, V0,, is computed by:

VCW = (3.5 f; + 0.3 fpc )b'd + VP, (2-19)

47

where ﬁx. is the compressive stress in concrete at'the centroid of cross section resisting

externally applied loads The shear strength provided by web reinforcement is taken as:

_ Avfsyd

S—

(2-20)

S

where A, is the area of web reinforcement within a distance 3 . The total nominal shear
strength can then be determined by adding the components of post-tensioning

contribution (VP) with the shear web reinforcement capacity (VS), and the shear capacity

from the concrete (Va or VCW).

2.8 Construction Sequence Methods

Construction process is an important element in the design of spliced girder bridges
since the application of post-tensioning to the structure at different stages can yield
different stress requirements and thus result in different component and system
dimensions and/ or arrangements.

The two most common construction methods currently used for spliced girder
bridges are single-stage post-tensioning and multi-stage post-tensioning. The main
difference between these two construction methods is the sequence of post-tensioning
application required to achieve system continuity.

In traditional spliced girder bridges, prestressing forces are applied once at the
beginning stage of the construction to balance the self-weight of the girder. The amount
of prestressing forces are usually determined to make sure that the total concrete top and
bottom ﬁbers stresses on the girders should be less than the allowable values. When the

deck load is applied prior to continuously post-tensioning the girders, the prestressing

48

forces should be ample enough to balance this additional deck load. Post-tensioning
forces can be applied in single or multiple stages depending on the sequencing of the

deck casting.

2.8.1 Single-span system with single-stage post-tensioning construction
method

In this method, the precast/prestressed beams are ﬁrst installed on temporary
supports at the site. Spliced joints and diaphragms are then poured and cured. Either a
cast in place deck or a precast deck can be added once the splice and diaphragms have
developed enough strength to resist partial loads. Full longitudinal post-tensioning is
applied to connect all segments. The temporary supports are removed once the deck
develops enough strength to achieve composite action with the beams. Wearing surfaces
and barriers are then added prior to trafﬁc opening.

Two separate analyses were used to evaluate stress demands from this
construction process. The ﬁrst analysis considered non-composite girder sections hinged
at closure joints. The second analysis considered the composite girder with ﬁrst-stage
post-tensioning and live load demands. The results from these analyses were
superimposed to approximate ultimate structural behavior. Stresses were calculated for

each stage of erection and accumulated.

2.8.2 Single-span system multi-stage post-tensioning construction
method

For this construction process the precast/prestressed beams are again assumed to
be installed on temporary supports at the site. Spliced joints and diaphragms are then

poured and cured. Partial longitudinal post-tensioning is applied to make the girders

49

continuous prior to the addition of a composite deck. A second stage of post-tensioning
is applied aﬁer the deck is cast to provide residual compression in the deck for
serviceability and deﬂection control. Temporary supports are removed once the deck
develops enough strength to become composite with the beams. Superimposed dead
loads are applied prior to the application of live loads.

Three separate analyses were used to determine demands from this construction
process. The ﬁrst analysis considered the non-composite section hinged at closure joints.
The second analysis considered the non-composite girder in a continuous system with
equivalent externally applied loads to represent the ﬁrst post-tensioning stage. The third
analysis considered the composite girders with second stage post-tensioning equivalent
forces and live load demands. Analysis results were superposed aﬁer each stage to

obtain the total stress state at each stage.

2.8.3 Two-span continuous system with single-stage post-tensioning
construction method

Single-stage post-tensioning has been the favorable construction method for
spliced prestressed girder bridge since it is faster and less complex (Lounis et a1 1997).
Only one stage of post-tensioning forces are applied aﬁer pouring the deck. This permits
construction to be faster since there is no need to apply additional post-tensioning forces.
However, this construction method does not consider for the case of future deck
replacement. Since all the girders are subjected to the entire post-tensioning forces in a
single event after casting the deck, they will likely be overstressed if the deck is

removed.

50

 

This construction method is done by applying a single post-tensioning operation
after the deck has been cast. The prestressed beams are ﬁrst placed on temporary
supports at the site. Spliced joints and diaphragms are then poured and cured. Either a
cast in place deck or a precast deck can be added once the splice and diaphragms have
developed enough strength to resist partial loads.

Temporary supports are used to hold the girders during placement of the deck
concrete and before all segments are longitudinally connected. After the girders are
erected, the support towers are adjusted to touch the bottom of each girder without
applying any signiﬁcant force to them. Temporary supports will resist the partial deck
weight until full longitudinal post-tensioning is applied to connect all segments.

Once the deck concrete reaches enough strength to achieve composite action with
the beams the temporary supports are removed. The force that was carried by the
temporary supports (i.e., reactions) are then resisted by the continuous full-span
structure. The total stresses caused by dead loads together with the support loads are less
than in systems without temporary supports. Therefore, the structure requirements are
expected to be less. Wearing surfaces and barriers are then added prior to trafﬁc
opening.

To evaluate stress demands resulting from this construction process, two separate
analyses were performed. The ﬁrst analysis considered the non-composite girder
sections hinged at closure joints. The second analysis considered the composite girder
with ﬁrst-stage post-tensioning and live load demands. The results from these analyses

were then superimposed to obtain the total structural response. Load patterns and

51

moment diagrams for the single-stage post-tensioning construction precess are illustrated
in Table 2-2.

Positive segments were analyzed as simple spans while negative segments were
analyzed as cantilevers from the pier, even though in reality they will most likely be
supported from a temporary support close to the pier center line. This assumption was
made in order to obtain the maximum negative moment over the pier. Temporary
supports were assumed to be unconnected to the beam, thus serving as compression only
supports. Girder pre-tensioning forces were selected between the range of zero allowable
tension and allowable compression at the bottom ﬁber of the girder section.

Additional deck and joint loads are assumed to be superposed to the system aﬁer
the beams are erected. Stresses in the deck were not considered at this stage since the
slab has not developed enough strength. Therefore, top and bottom stresses in the precast
girders are based on noncomposite section properties.

Post-tensioned splicing forces will induce secondary moments at the pier location. Once
the temporary supports are removed, the released reactions will create additional
demands on the system. Top and bottom stresses on the precast girders and the cast-in-
place deck were determined using composite section properties. Different live load
patterns need to be explored to obtain the maximum positive and/or negative moments at
the desired locations. Truck and lane loads also need to be compared against the effects

caused by tandem and lane loads to obtain the maximum live load demands.

52

Table 2-2 Load patterns and moment diagrams of single-stage post-tensioning
construction method

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Days Event/Loads Load patterns and moment schematics
1 Transfer Pre- Temporarg pl er 1
t6ﬂ8101‘l/ Girder Ref L1 me A Support ngm
SelfWeight & | Ll l I 0. 5t L2) !
Pre-tensronmg I l l a
loads :
Wbeem Wbeom j
r l 1 l l T L l l l l J J
l l l l
AS 0.6 9
Cl rder _i
Deed Load \L/ i
l
Pre-tenSIOning !
I + i
30 Set Girders on SAME AS ABOVE
Falsework
5° 59‘“ Sphce’ SAME AS ABOVE
1aphragms
80 Pour CIP deck/
£1613? welght Of TZrLrjipogirtg Pier 1
Ref Line A pp Sgnm
| Ll l I a. 5( LZLE
l
l i
|
Wdeck Wdeck :
l l l l l l l 1 J r l l h
L 1 F 1
A M 43
Deck I
Deed Load T\\\\\\\\\\~.:;’/////////, :
I

 

53

 

Table 2-2 Load patterns and moment diagrams of single—stage post-tensioning
construction method (Cont...)

 

 

 

 

 

 

 

 

 

 

 

 

  

90 Post- Spllce Pler 1
tensioning Rel Llne A JOI‘r‘It ngm
force L L I

p i "P3
o wpl wp2
Post- ten51on1ng
(a 4
Fae.
Prlmorg Moment /_—\
\
Secondarg Moment
+
Support Removal /
+
Moment. \/

110 Place Splice Pier 1

barrier
Jornt S

(Superimp Ref rte A qmm
osed dead (a, 5( L2)
load)

Superi mposed

Dead Load

Moment

 

54

 

Table 2-2 Load patterns and moment diagrams of single-stage post-tensioning
construction method (Cont...)

 

120

Open to

full
trafﬁc
Live load
per girder

 

 

 

Ref Line A

 

 

 

 

Pier 1

ngm
I
L L A:
i
22.500 lbs g
Wlane = 800 lb/ft E
lllllll I i

l I

 

 

A

Max M- due

 

to live 105dT\\\\\\\\\‘;:ﬂ”ﬂa’,//’///

 

2.8.4 Two— span continuous system with multi-stage post-tensioning
construction method

Multi-stage post-tensioning construction can be used to reduce girder demands
resulting in reduced girder depths. By applying the post-tensioning forces at different
stages, or at least at two stages, the stresses in the girder are redistributed and kept within
allowable limits. Using this incremental technique, the resulting girder sections for this

construction process are expected to be shallower than those obtained from single-stage

construction.

When future maintenance or replacement of the bridge deck is anticipated in a
given region, this has to be considered during the design phase. Especially for the case
when entire deck replacement is expected to occur during the bridge life-time. In this
case, a multi-staged post-tensioned construction method is suitable since it eliminates, or

alleviates, overstress conditions in the girder after the deck is removed.

55

 

For this method, the precast/prestressed beams are installed on temporary
supports at the site. Spliced joints and diaphragms are then poured and cured. Partial
longitudinal post-tensioning is applied to make the girders continuous prior to the
addition of a composite deck. The second stage of post-tensioning is applied after the
deck is cast to provide residual compression in the deck for serviceability and deﬂection
control. Temporary supports are removed once the deck develops enough strength to
become composite with the beams. Superimposed dead loads are applied prior to the
application of live loads.

Typically, the lower post-tensioning tendons are stressed in the ﬁrst stage while
the higher tendons are stressed in the subsequent stages. This tendon stressing scheme
can help improve the pre-compression stress state in the deck and allow second stage
tendons to be easily accessed.

The stress demands developed through this construction were determined from
three separate analyses. The ﬁrst analysis considered the non-composite section hinged
at closure joints. The second analysis considered the non-composite girder in a
continuous system with equivalent externally applied loads to represent the ﬁrst post-
tensioning stage. The third analysis considered the composite girders with the second
post-tensioning stage equivalent forces and live load demands. Analysis results were
superposed after each stage to obtain the total stress state at a given stage. Load Patterns
and moment diagrams for the multi-stage post-tensioning construction process are

illustrated in Table 2-3.

56

Table 2-3 Load patterns and moment diagrams of multi-stage post-tensioning
construction method

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Days Events/Loads Load patterns and moment schematics
1 Transfer Pre- Temporwg Pier
tension/Girder Ref L1 08 A support 59mm
Self Weight & L Ll j a. 5( L2) !
Pre-tensioning L 1 i
loads ,
u
Wbeam WbeamJ
l I I I I I L L l I I J J
[ J [ ‘1
A AA 4
I
I
Dead load /-:
of Girder \i/ '
l
u
Pe / _ H :
Preten51oning) J
+ I
30 Set Girders on SAME AS ABOVE
Falsework
50 Pour Splice, SAME AS ABOVE
diaphragms

 

 

 

57

Table 2-3 Load patterns and moment diagrams of multi-stage post-tensioning
construction method (Cont...)

 

 

80 First stage Re, me A 2;; I
post- L L I
tensioned a;
force I !

Wpost-tensxoned

 

 

(A ‘

Primarg Moment 4///////////”————_‘T\\\\\\\\\\
\

Secondarg Moment
\

Support Removal /////
Moment TTTTTTTTTTTT“‘~—.~“““~‘;:“J///I

 

 

 

 

 

 

 

 

90 Pour CIP Spllce Pier 1
deck/ Ref Line A JON“. 5 mm
Fluid ‘ Ll L a.5(L2)
weight of
deck

Wdeck
I l l l l I I l I l l
A

Dead load of 4”////::/
DeCk \i/

 

 

 

 

 

 

58

Table 2-3 Load patterns and moment diagrams of multi-stage post-tensioning
construction method (Cont. . .)

 

120 second Sp] 1 ca

 

Pl er 1
stage Ref Line A Jox nt. 59 mm
POStj L L
tenswned I
force

Wpost- tensr oned

M

(is ‘

PH mow Moment, A
\

Secondary Moment
\

 

 

 

 

 

 

 

 

130 Place 891‘” Pler‘l
barrier Ref Line A JOIHt St mm
and ' U i a. 5( L2)
superimp I
osed dead
load W51dl

l I l l s l l l l I l
l l
A

 

Superimposed /
Dead Load +
Moment \_/

 

 

 

 

 

 

59

Table 2-3 Load patterns and moment diagrams of multi-stage post-tensioning
construction method (Cont...)

 

 

140 Open to R f A P1 er I
full trafﬁc 8 L‘ ”e Swim
Live load L L i
per girder ~ =i
22v 508 1 b5

 

Wlane = 800 lb/ft
I I I I I I I I

 

 

uh‘b—I-—-—.—.—

 

I
[
5}

.-._9

Max M- due

to live loadv

 

..——i———_—

 

 

 

 

 

The disadvantages of applying post-tensioning in multiple stages are that the
post-tensioning anchorage access must be maintained and some portion of the deck
cannot be cast until all the tendons are installed, stressed and grouted. On bridges that
have girders post-tensioned at the abutments, the backﬁll and casting of diaphragms will

have to be delayed until all post-tensioning stages are done.

2.9 Stress Demands at Critical Construction Stages

Stress constraints for the service limit state include girder and deck ﬂexural stress
limits. Three locations across the composite girder-deck section are checked along the
span length of beam on each critical stage to guarantee that all stresses satisﬁed the
constraints, i.e., they should be within the prescribed limits. These locations are the top

and bottom ﬂexural ﬁbers of the girder and the top of the deck.

60

2.9.1 Single-stage post-tensioning construction methods

There are 7 critical stages for a single-stage post-tensioning construction method.
Stresses at each critical construction stage, as listed in Table 2-4, need to be checked
against the allowable compression and tension stresses. The construction stages
summarized in Table 2-4 have been associated with an assumed time schedule. This time
schedule was proposed based on typical highway bridge projects with precast concrete.
It is understand that this schedule may vary dependent on the nature of the projects state
procedures, or other unforeseen events. These issues are addressed later in the

implementation and discussion of results in Chapter 4 and 5.

Table 2-4 Construction sequence for single-stagedpost-tensioning in spliced girder

 

 

bridges
Stage Time Action Critical Tensioning
(days) Pre-T Post-T

1 - Stress Pre-tensioning strands
2 0 Cast girder segments
3 1 Release pre-tensioning strands X
4 50 Erect girder segments
5 60 Place deck and splice concrete X
6 75 Stress post-tensioning strands X X
7 100 Add barriers X
8 140 Apply live load
9 15000 Add FWS
10 27500 After all prestress losses X X

 

2.9.2 Multi-stage post-tensioned construction methods

There are 10 critical stages for a multi-stage post-tensioned construction method.
Stresses at each critical construction stage, as listed in Table 2-5, need to be checked

against the allowable compression and tension stresses.

61

Table 2-5 Construction sequence for multi-staged post-tensioning in spliced girder

 

 

bridges
Time . Critical Tensionin
Stage (days) Action Pre-T Post-Tg
1 - Stress Pre-tensioning strands
2 0 Cast girder segments
3 1 Release pre-tensioning strands X
4 50 Erect girder segments
5 80 Stress 1St post-tensioning strands X X
6 90 Place deck and splice concrete X X
7 120 Stress 2nd post-tensioning strands X X
8 130 Add barriers X
9 140 Apply live load
10 15000 Add F WS
11 27500 After all prestress losses X X

 

2.10 Time-dependent Prestress Losses

Time dependent effects can signiﬁcantly affect the accuracy of calculated
stresses and deﬂections, particularly in systems employing multi-stage construction,
such as segmental bridges and spliced girder bridges. Unconservative estimates of these
effects can underestimate tendon requirements and member stress resultants.

Normally, SGB rely on large amounts of post-tensioned, pre-tensioned, and
passive reinforcement, consequently the effect of reinforcement on the creep behavior of
SGB components may play an important role and should not be ignored in the analysis,
particularly of negative sections.

Time-dependent prestress loss sources in pre-tensioned and post-tensioned
concrete structures include steel relaxation, concrete shrinkage and concrete creep. Time
dependent effects affect the accuracy of calculated stresses and deﬂections and can have

a signiﬁcant inﬂuence on composite prestressed members (Ronald 2001). The ultimate

62

creep and shrinkage coefﬁcients used in an analysis typically determine the number of
tendons employed to satisfy stress limit states (Ronald 2001).

There are several creep and shrinkage models currently used in bridge design
practice. These methods are those recommended by the American Concrete Institute
Committee 209 (ACI 2002), the American Association of State Highway and
Transportation Ofﬁcials (AASHTO 2003) and the code from the Comite Euro-
lntemational du Beton and the Federation lntemationale de la Precontrainte (CEB-FIP

1990).

2.10.1 Creep and shrinkage prestress losses - AASHTO IACI

The methods of determining creep and shrinkage herein are taken from Collins
and Mitchell (1991). These methods are based on recommendation of the AASHTO-
LRF D code (AASHTO 2003), the ACI-209 recommendations (ACI 1992) and additional
published data by Rusch et al. (1983), Bazant and Wittman (1982), and Ghali and Favre

(1986)

a) Creep coefﬁcient
The creep coefﬁcient ¢(t,t,-) is deﬁned in AASHTO-LRFD Section 5.4.2.3.2

(AASHTO 2003) as follows:

 

(16
_ t-t-
¢(t,t,-)=35.kckf(1.58__1:1_),. 0.118 ( z)

, (2'21)
0 ' lO+(t—t,-)O’6

where:

k]: factor for the effect of concrete strength

63

kf= 1 , [MPa] or kf= 1 , [ksi]. (2-22)

I; 11:.
O.67+(62) 0.67+( 9)

 

 

kc = factor for the effect of volume/surface of component, which is based on PCI

equation (1977):

t

' 0 36(V/S) 4546.52)
_ 26e' +t l.8+1.77e -
kc —1 t 11 2587 1, (2 23)

45+t

 

 

 

where H = relative humidity; t = maturity of concrete (days); ti: age of concrete when

load is initially applied (days); f c = speciﬁed compressive strength at 28 days (ksi); S =
surface area that is exposed to atmospheric drying; and V = concrete volume that is

exposed to atmospheric drying. If the information of humidity (H) is not available, it

may be taken from AASHTO-LRFD Fig. 5.4.2.3.3-1 (AASHTO 2003).

b) Shrinkage strain

The strain due to shrinkage is taken as deﬁned in AASHTO-LRFD Section 5.4.2.3.3
(AASHTO 2003) as follows:
i) Shrinkage strain for moist cured concrete:

1‘

t+35)0.51x10—3; (2-24)

 

53h = -kskh(

ii) Shrinkage strain for steam-cured concrete

t

H 55)0.56 x10_3; (2-25)

 

53h : ’kskh(

where t = drying time (days); and k S : size factor, and is given by:

64

t

 

V V
0.36<——> 1064 _ 94 .—
S + (S)

 

 

 

26e t
k = . 2-26
5 1 t 11 923 1 ( >
45+t
ForH<80%:
140—H
k = , 2-27
h ( 70 ) ( )
and forH>80%:
3(100—H)
k =———. 2-28
h 70 ( )

c) Relaxation losses

To determine prestressing losses due to steel relaxation tensile reinforcement and

strands are assumed to be lumped at a single location with an effective distance to the

passive reinforcement (ds) and effective distance to the prestressing reinforcement (dp).

The average stress in the prestressing steel is then:

c
fps = fpu(l-—k—), (2—29)
dP
where k is the factor of yielding and ultimate strength and is deﬁned as follows:
fpy

pl!

 

k = 2(1.04 — ). (2-30)

The neutral axis depth for the prestressed section can then deﬁned by considering section

equilibrium. For a T-section the neutral axis depth is given by:

65

c _ A p, fpu + Asfy — Asfy — 0.85 ,8] fC (b — bug/if

 

 

(2-31)
. fpu
0.85fcﬂ1bw+ kApS
dP
While for a rectangular-section, it is given by
A + A . — A' '
C: psfpu sfy sfy (2_32)

 

. f ’
0.85 fcﬂlb + MP3 dp“

 

p

where Aps = are of prestressing steel; Jim; = speciﬁed tensile strength of prestressing

steel; and fpy = yield strength of prestressing steel.

The AASHTO-LRFD (AASHTO 2003) guidelines for prestress losses, which
were adopted from the ACI-209 recommendations (AC1 1992), were developed based on
a single loss value at the centroid of the prestressing force, and this single loss value is
assumed to be uniformly distributed across the girder section.

An excessively conservative value of creep or shrinkage can make the allowable
stress design approach nearly impossible without reducing span length or using deeper
members. On the other hand, unconservative values of creep or shrinkage will yield
unconservative estimates of tendon requirements and may lead to serviceability
problems. Rational analyses that offer ﬂexibility in material and geometric variables
may yield accurate results. When mix-speciﬁc data are not available, estimates of
shrinkage and creep may be made using either the AASHTO speciﬁcations, or the CEB-

FIB 90 or ACI -209 recommendations.

66

2.10.2 CEB-FIP 90

The time-dependent strain in concrete can be expressed as the sum of elastic and
creep strains due to initial stresses, elastic and creep strains due to changes in stress and
the shrinkage strain (Wollmann et. al 2003). The total concrete strain at time t is

expressed as follows (Wollmann et. a1 2003):

0' 0' -— 0'
at = 5911+ (01.x, ) +’—E—"<1 + mt, )+ 65,1, (2-33)
0

where 8, = total concrete strain at time t; 00 = concrete stress at time t = 0; 0’, = concrete
stress at time t; (Pate: creep coefﬁcient at time t due to load applied at time to; 85, =
shrinkage strain at time t; ,u = aging coefﬁcient = 0.7; and E0: modulus of elasticity at

time to = 0.
The ﬁrst term in Equation (2-33) represents the elastic and creep strain due to a

stress 00 applied at time to The second term represents the elastic and creep strain due to

stress changes from time to to time t. The last term represents the shrinkage strain at time

The aging coefﬁcient, ,u, is a function of load history and of the time—dependent
concrete properties. The coefﬁcient accounts for the reduced creep of concrete loaded at

a greater age. Generally, the upper bound for p is 1.0, where the load history is a single

step stress increment A0 = a, - 00 applied at time t = to with no further changes (Bazant

1972 and Dilger 1982). The aging coefﬁcient is deﬁned as (Bazant 1972 and Dilger

1982):

67

t

E0 1 do'(r)
1+ t, )d — .
(0', —0'0)(ot,t0 IE(r) dr [ M T] T (Onto

 

 

# = (2-34)

to

Bazant (Bazant 1972) has shown that when the change in stress is increased by
creep and shrinkage, a value of 0.7 and 0.9 is appropriate to use for the loading at age 10
days and 100 days, respectively. Generally, creep and shrinkage properties are subject to
large scatter and cannot be determined with great accuracy. Due to the variability of the
creep properties, high accuracy for the aging coefﬁcient is not required since normally
the aging coefﬁcient is multiplied by the creep coefﬁcient. The assumption of a constant
aging coefﬁcient makes creep and shrinkage calculation simpler and can be reduced to
the solution of linear equation (W ollmann et. al 2003).

The component forces due to section forces acting on the composite section and

distances of deck and girder are illustrated in Figure 2-10.

C.G. Deck

I /—Ao' 10" E0 M3 AMD

 

 

 

 

 

 

 

 

 

 

a
D
a /— C.G.Composrte section M0
v '/ c1313 d M0 Tg'NOAMo
OG (— . . ll" 81" 5 0
' N
11 -I/ No A o
/—DG'IG’EG
/

 

 

K \

Figure 2-10 Component forces and centroid distances on composite girder/deck section

68

The modular ratio, n, of deck concrete to girder concrete is deﬁned as:

n = 50., (235)
EC

where E D and EG are the modulus of elasticity for deck and girder, respectively.

The area of transformed composite section, Ac, includes the girder area AG and

the transformed deck area A D, and can be deﬁned as:
Ac = AG + nAD. (2-36)
The distance aD between centroid of deck and centroid of composite section is

determined by:

aD =:—Ga. (237)
C

The distance a0 between the centroid of girder and the centroid of the composite
section is given by:

71/1 D
AC

 

a0 = a , (2-3 8)

where a is the distance the between centroid of the deck and the centroid of the girder.
The ﬁrst area moment, Sc, of the transformed deck section about the centroid of
the composite section is deﬁned as:

SC = nADaD, (2-39)
where a0 is the distance between centroid of the deck and the centroid of the composite
section.

The moment of intertia of the transformed composite section 1C is then deﬁned

as:

69

IC = [G + nID + aSc, (2-40)
where ID and [G is the moment of intertia of the deck and girder about their own

centroids, respectively.

The component axial forces (ND and NG ) and component moment forces (MD
and MG) due to the moment (M) and axial force (N) acting on the composite section can
be found by integrating the resultant stress distribution over the respective portions of
the cross section. The expressions for the component forces are as follows (Wollmann et.
al 2003).

The axial load in the deck due to axial and moment forces on the composite

section is determined by:

ND=5A—DN—M§—C-. (241)

AC 1C
The axial load the girder due to axial and moment forces of composite section is
determined by:

A 5
N6 =—GN+M—C (242)

AC 1 C '
The moment in deck due to a moment force on the composite section is
determined by:

MD = M ”ID . (243)

1C

 

The moment in the girder due to a moment force on the composite section is

given by:

MG = 1Q- (244)

70

With time, strains and curvature will change due to creep and shrinkage, which
causes the build-up of a self-equilibrating state of stress over the cross-section. This
time-dependent change can be described by eight unknown quantities: -AMD, AND, AMg,
ANg, KB, 80, Kg, and 80. Since there are eight unknown quantities, eight equations are
needed. Two equations can be obtained from compatibility requirements by assuming
plane sections remain plane and perfect bond between deck and girder.

Curvature of deck, K0, and curvature of girder, Kg, due to long term effects are

assumed to be equal and deﬁned as:

KG =K’D = K. (2-45)
The time-dependent strain for deck, ED, is determined from the time-dependent

strain on the girder, 80, and the curvature, K, at the distance between centroid of deck
and centroid of girder.

5D = 86 - Ka. (2-46)

Two more equations (equation 2-47 and equation 2-48) can be obtained from

equilibrium requirements on axial forces and moments. Axial equilibrium in the

composite section can be deﬁned as follows:
AND +ANG =0, (2-47)
where AN D is the time-dependent change of axial loads in the deck; and ANG is the

time-dependent change of axial loads in the girder.

Moment equilibrium in the composite section can be expressed as follows:

AMD+AMG+ANGO=0, (2-48)

71

where AM D is the time-dependent change of moments in the deck, AM G is the time-

dependent change of moments in girder, and a is the distance between the centroid of the
deck and the centroid of the girder.

Even though stresses will change with time, there is no resultant force on the
composite section due to these changes because they are self-equilibrating. The two
equilibrium conditions expressed in Equations 2-47 and 2-48 are true only for a statically
determinate system where long-term deformations do not cause a change in section
forces. For an indeterminate structure, Equations 2-47 and 2-48 need to be modiﬁed to
include terms for the time-dependent redundant forces.

The remaining four equations (2-49 through 2-52) are given by the constitutive
relations describing the time-dependent strains only. The initial strains are not included
in these equations and they are expressed as given by Wollmann et al. (2003) below:

The time-dependent strain in the deck, 80, is determined by:

0
____ ND ¢D+AND
EDAD EDAD

 

 

5D (HM/Johan), (2—49)

where 851) is the shrinkage strain for the deck; N?) is the axial load in the deck at the

beginning of composite action; (oDis the creep coefﬁcient for the deck; E D is the

modulus of elasticity of the deck; and A D is the area of deck.

The time-dependent strain in the girder, 80, is deﬁned as:

 

 

0
N AN
G G
e = + 1+ +5 , 2-50
G EGAG (PG EGAG( #(OG) 56 ( )

72

where 850 is the shrinkage strain for the girder; N g is the axial load in girder at the

beginning of composite action; (PG is the creep coefﬁcient for the girder; E0 is the

modulus of elasticity of the girder; and AG is the area of the girder.

The curvature of the deck due to long-term effects, [(0, is given by:

 

D AMD
+ 1+ , 2-51
E 1 (PD E I ( W19) ( )

 

where M g is the moment in the deck at the beginning of composite action; I D is the

moment of the inertia of the deck about its own centroid; and ,u is the aging coefﬁcient.

The curvature of the girder due to long-term effects, K0, is given by:

 

 

0
M AM
G G
K‘ = + 1+ , 2-52
G EGIG¢G EG1G( MPG) ( )

where M g is the moment in the girder at the beginning of composite action and I G is

the moment of inertia of the girder about its own centroid.
The input data required for the solution of the system of four equations (Equation

2-50 to Equation 2-53) are the cross-section and material properties (E D, E0, AD, A0, ID,
15 and a), the creep producing stress resultants (N 0 ,N g,M 3M 3), and the concrete
creep and shrinkage parameters (¢D,£SD,(DG,£SG, p). Solving this system of equations

with a spreadsheet or numerical analysis programs is simple.
The effect of prestress and prestress losses can be easily implemented in the

system of equations (Equation 2-49 to Equation 2-51). The initial prestress force is

considered with the creep-producing stress resultants M g,M 3N g and N g. Prestress

losses are initially estimated by estimating the time-dependent change in strain 8p at the

73

location of the tendons or strands. The forces acting on the composite section due to
prestress losses are then given by Equation (2-53) through Equation (2-54) (Wollmann

et. a1 2003).
The change of axial load due to prestress losses, AN P , is deﬁned as:
ANP = APS(AfR — Epsé‘p), (253)
where A p5 = cross-sectional area of the prestressing steel; E p5 = modulus of elasticity of

the prestressing steel; 5p: time-dependent change in strain at the location of the

prestressing steel (tension positive); and AfR = prestress loss due to strand relaxation.

The change in moment due to prestress losses, AM P , is deﬁned as:

AMP =ANPe, (2-54)
where e = ecCentricity of the prestressing steel relative to the centroid of the composite

section (down positive).
The force AN P and AM P can be decomposed into component forces
AN SAM SAN (I: , andAMg in the deck and girder, respectively. The component forces

due to prestress losses are self-equilibrating with the change in tendon force. Time-
dependent strains and curvatures including the effects of prestress losses are given by
(W ollmann et al. 2003) and are outlined next.

The time-dependent strain in the deck, 80, including prestress losses is

determined by:

0 P
N AN +AN

D D D
= + 1+ +5 . 2-55
EDAD ¢D EDAD ( MOD) SD ( )

 

 

5D

74

The time-dependent strain in the girder, 80, including prestress losses is deﬁned

N2; ANG +AN§ 1 2 56
= + + +5 . -
EGAG (PG EGAG ( M00) 50 ( )

 

 

50

The curvature of the deck due to long-term effects, KD, including prestress losses

is determined by:

0 P
M AM +AM
D D D
K = + 1+ . 2-57
D EDIDCDD EDID ( #(PD) ( )

 

 

The curvature on the girder due to long-term effects, K6, including prestress

losses is given by:

O P
M AM +AM

G G G
= + 1+ . 2-58
EG 1G (PG EGIG ( MPG) ( )

 

 

KG

2.11 Simpliﬁed Estimate of Prestress Loss Components

The estimate prestress losses according to the “reﬁned” simpliﬁed method of the
AASHTO-LRFD code (LRFD 2003) and the recommendations from the NCHRP 517
report (Castrodale and White 2004) are presented in this section. The principles of the
reﬁned method of the AASHTO-LRFD Speciﬁcations only provide total losses and has
no provisions for computing losses at stages prior to the ﬁnal conditions. The
recommendations from the NCHRP 517 report account for the interaction of pre-

tensioning and post-tensioning in prestress loss computation.

75

2.11.1 Relaxation prestress losses at release

The prestress losses of pre-tensioned strands due to the relaxation at the release

stage can be deﬁned as follows (AASHTO-LRFD Article 5.9.5.4.4b-2):

 

1 24 f -
AfPR1= “if“, ”1 f1” —0.551fpj. (2-59)
py

where Af pm is the relaxation prestress loss at release; f pj is the stress in the strand
during the jacking operation; f py is the yield strength of the strand; and t is the time at

which prestress is released.

2.11.2 Shrinkage prestress losses

The prestress losses in this section are the estimation of time-dependent shrinkage
losses at the ﬁnal condition according to AASHTO-LRF D. The intermediate shrinkage
prestress losses can be determined by multiplying the ﬁnal shrinkage losses with a time
factor as shown in equation (2-62).

Shrinkage prestress losses at the ﬁnal stage of pre-tensioned strands can be

determined as follows (AASHTO-LRFD Article 5.9.5.4.2):

AfpSR = (17 — 0.15H) (ksi), (2-60)
where AfPSR is the shrinkage prestress loss and H is the humidity. Shrinkage prestress
losses at the ﬁnal stage of post-tensioning can be deﬁned as follows:

AfpSR = (13.5—0.123H) (ksi). (2-61)

The time factor of shrinkage losses to determine prestress losses at intermediate

stages is based on AASHTO-LRFD Equation 5.4.2.3.3-1 as follows:

76

t
35.0+z'

 

Mm: em)

2.11.3 Creep prestress losses

NCHRP project 12-57 (Castrodale and White 2004) recommended that the creep
losses equation from AASHTO-LRF D Article 5.9.5.4.3 (AASHTO 2003) be modiﬁed to
include the effects of post-tensioned strands. The creep loss formulas outlined in this
section are for the losses at the ﬁnal stage. Intermediate creep prestress losses can be
determined by multiplying the ﬁnal creep losses with a time factor as it will be shown
later.

Creep prestress losses at the ﬁnal stage of pre-tensioned strands can be determined as

by the following expression:

AfPCR Z lzfcgp _ 7Afcdp + 7fcgp19 (2'63)
where AprR is the creep loss from pre-tensioned strands; fcgp is the summation of
concrete stresses at center of gravity of the prestressing strands due to the prestressing
force at transfer and the member self-weight at maximum moment; fcdp is the change
in concrete stresses at center of gravity of the prestressing steel due to the permanent
loads; fcgpl is the summation of concrete stresses at center of gravity of the

prestressing tendon due to the post-tensioning force and the removal of temporary
support.

The summation of concrete stresses at the center of gravity of the prestressing
tendon due to prestressing force at transfer and the member self-weight at maximum

moment is deﬁned as follows:

77

2 M e
P P GDL
= + — , 2-64
fcgp A I I ( )

 

where A is the gross area of the girder; I is the gross girder moment of inertia; F p is the

total prestress force due to pre-tensioning; Mom is the moment due to girder selfweight;
and e is the eccentricity of the pre-tensioning strands measured from the neutral axis of
noncomposite section to the centroid of the pre-tensioning strands.

The change in concrete stresses at the center of gravity of the prestressing steel

due to permanent loads is given by:

M + M M e

I [C
where MDDL is the moment due to the deck selfweight; MNCDL is the moment due to the

girder/deck‘haunch weight; MCDL is the moment due to construction loads; ec is the

eccentricity of the pre-tensioning strands measured from the neutral axis of composite

section to centroid of the pre-tensioning strands; and IC is the composite girder/deck

moment of inertia.
The summation of concrete stresses at center of gravity of the pre-tensioning
strands due to post-tensioning forces and removal of temporary supports is deﬁned as

follows:

FPT + FPTePTec _ M REMec , (2_66)
AC IC 1C

 

fcgpl =

where F pr is the total prestress force due to post-tensioning; AC is the composite area of

the section; epr is the eccentricity of the post-tensioning strand measured from the

78

neutral axis of the composite section to the centroid of the post-tensioning strands; and
MREM is the moment due to removal of temporary supports.

Creep prestress losses at the ﬁnal stage of post-tensioned strands can be determined
by:

AfPCR =12fcgp _ 7Afcdp + 7fcgp2 , (2’67)
where AprR is the creep loss for post-tensioning strands; fcgp is the summation of

concrete stresses at center of gravity of the post-tensioning tendons due to post—

tensioning forces and removal of temporary supports; fcdp is the change in concrete

stress at center of gravity of the post-tensioning strands due to permanent loads; and

fcgpz is the summation of concrete stresses at the center of gravity of the post-tensioning

tendon due to pre-tensioned strands and girder self-weight.

The summation of concrete stresses at the center of gravity of the post-tensioning
tendon due to post-tensioning forces and the removal of temporary supports is
determined as follows:

2
f _FPT +FPT9PT _MREM6PT
Cgp _ A 1 1
C C C

 

9

(2-68)

where F pr is the total prestress force due to post-tensioning; Ac is the composite area of
the section; epr is the eccentricity of the post-tensioning strand measured from the
neutral axis of the composite section to the centroid of the post-tensioned strands; and

MREM is the moment due to the removal of temporary supports.

79

The change in concrete stress at center of gravity of the post-tensioning strands

due to permanent loads is deﬁned by:

 

M +M M e
fcdp :( DDL 1 NCDL )ePT + CI;L PT . (2-69)
c

The summation of concrete stresses at center of gravity of the post-tensioning

tendon due to pre-tensioning strands and girder self-weight is determined by:

 

F F M , e
PRE + PREePTe_ self PT , (2-70)

fcgp2 = A I I

where F pm is the total prestress force due to pre-tensioning; and MseIf'lS the moment due

to the girder self-weight.
The time factor to determine the prestress losses due to creep at intermediate
stages is based on AASHTO-LRFD Equation 5.4.2.3.2-1 as follows:

t0'6

—. (2-71)
10.0 + :06

AkCRt =

2.11.4 Elastic shortening prestress losses

Instantaneous elastic shortening prestress losses at the initial stage of pre-

tensioning strands can be determined as follows (AASHTO Article 5.9.5.2.3):

E
AfPEs = ﬁfcgpa (2-72)
Cl

where AprS is the elastic shortening losses on the pre-tensioning strands; fcgp is the

summation of concrete stress at the center of gravity of the prestressing tendon as

80

described in Equation (2-63); E p is the elastic modulus of the pre-tensioning strands; and

EC,- is the elastic modulus of girder concrete at the initial stage.

Elastic shortening prestress losses at the initial stage of post-tensioning strands is

deﬁned by:

N—l Ep
A = — —— , 2-73
fPES ( 2N Eci fcgp ( )

where AprS is the elastic shortening losses on the pre-tensioning strands; N is the
number of post-tensioning ducts; fcgp is the summation of concrete stresses at the center

of gravity of the post-tensioning tendon due to post-tensioning forces and removal of

temporary supports, as described in Equation (2-68).

2.11.5 Relaxation prestress losses at final condition

Prestress losses due to steel relaxation at the ﬁnal stage of pre-tensioning strands
is based on AASHTO-LRFD section 5.9.5.4 and can be determined for stress-relieved

strands as follows:

AfPRz = 20 - 0-4AfPEs - 0-2(AfPSR + AfPCR ), (2-74)

where AprZ is the ﬁnal relaxation loss at the ﬁnal stage for the pre-tensioning strands;
AprS is the elastic shortening prestress loss; AfPSR is the shrinkage prestress loss; and
AprR is the creep loss.

Relaxation prestress loss for low-relaxation strands at the ﬁnal stage of pre-

tensioning strands is based on AASHTO-LRFD section 5.9.5.4.4c-1 and is given by:

AfPR2 = 0-3 >< (20 - 0-4AfPES - 02(AfPSR + AfPCR )), (2-75)

81

where Aprz is the ﬁnal relaxation loss at the ﬁnal stage for low-relaxation pre-tensioning
strands.
Relaxation prestress loss at the ﬁnal stage for post-tensioning strands is based on

AASHTO-LRFD section 5.9.5.4.4c-2 and is given by:

AfPRz = 0-3 >< (20 - 0301\pr — O-4AfPES - 02(AfPSR + AfPCR )). (2-76)

where Apr2 is the ﬁnal relaxation loss at the ﬁnal stage for the post-tensioning strands.

2.11.6 Additional elastic shortening prestress losses due to post-
tensioning

The stressing of post-tensioning tendons causes additional, or secondary, elastic
shortening prestress losses in the girder segments and they are deﬁned as follows

(Castrodale and White 2004):

E P
AfPESZ : '1';- fcgpl , (2'77)

C

where N is the number of post-tensioning ducts; EC is the elastic modulus of the girder

concrete at the stage in question; Ep is the elastic modulus of prestress steel; and Afcgpl
is the summation of concrete stresses at center of gravity of the prestressing tendon due
to the post-tensioning forces and the removal of temporary supports as deﬁned in
Equation (2-65).

2.11.7 Friction prestress losses

Friction prestress losses from post-tensioning at the segment midspan are

computed by (AASHTO Article 5.9.5.2.2):

£3pr = f pj(1 - (“OH/1 (1)), (2-78)

82

where Apr is the friction prestress loss; k is wobble coefﬁcient; ,u is the friction

coefﬁcient; a is the angle between the post-tensioning tendon and the horizontal axis; x

is half of the girder length; and Afpj is the jacking stress.

2.11.8 Anchorage set prestress losses (NCHRP 517 report)

Anchor set prestress losses are those that occur on each post-tensioned tendon
when it is pulled into the anchorage seating. The constant rate of friction loss, RF, for
estimating friction losses from post-tensioning can be computed as recommended in the
NCHRP 517 report (Castrodale and White 2004) corresponding to the location at

midspan, x, as follows:

RF 2 AfPF . (2-79)

x
The stress at the anchorage is reduced based on the distance, XA, that the

anchorage is extended and is calculated as follows (Castrodale and White 2004):

A E
xA = I—SET P , (2-80)
RF

where A557 is the length of the anchorage set and Ep is the elastic modulus of the post-
tensioning strands.

The anchor prestress loss, AfA, is then calculated by using the following equation:

AfA =2xARF. (2-81)

2.12 Total Prestress Losses at Critical Construction Stages

The construction sequence of SGB can yield different prestress loss components

for the pre-tensioned and post-tensioned strands at each stage, and thus they need to be

83

individually considered for each critical construction stage case. This section presents
the critical construction stages for the different construction sequences that need to be
considered for the change in prestress losses.

The total prestress losses of the pre-tensioned and post-tensioned strands at the
critical construction stages of single-stage post-tensioning applied after deck casting (see

Section 2.11) are summarized in Table 2-6.

Table 2-6 Prestress losses during single-stage post-tensioning after deck casting

 

Pre-tensioned strands (AprEr)

 

Critical construction stage Components of prestress loss

 

Release of pre-tensioned strands AprSI + 21pr 1

 

Placement of deck and splice concrete AprS, + kCRIAprR + kSH/Afpsn + 41me

 

 

Stressing of post-tensioning tendons Afpgsj + kCRzAprR + kSHZAfPSR + Apr]
+AfPESZ
Final conditions after IOSSCS AprS] + A f PCR + AfpSR + Aprz +Afp552

 

 

Post-tensioned strands (Apr)

 

 

 

Critical construction stage Components of prestress loss
Stressing of post-tensioning tendons Afpp + Apr+ Afpgs
Final conditions after losses 21pr + Apr+ AprS + A prR+ Afp3R+ Aprz

 

 

 

 

The total prestress losses of the pre-tensioned and post-tensioned strands at the
critical construction stages of multi-stage post-tensioning applied after deck casting are

summarized in Table 2-7.

84

Table 2-7 Total prestress losses during multi—staged post-tensioning

 

Pre-tensioned Strands (AprET)

 

Critical construction stage Components of prestress loss

 

Release of pre-tensioned strands Aprs] + Apr,

 

Stressing of 1St post-tensioning tendons AprS] + kCRIAprR + kgmAbeR + Aﬁm

+AfP552

 

Placement of deck and splice concrete Afmssi + kCRgAprR + ksmAfPSR + Apr2

+AfP552

 

Stressing of 2nd post-tensioning tendons Afp55, + kCR3AprR + k5}{3Afp5R + Aprz

+Afpesz + Afr/553

 

Final conditions after losses Afpggl + AprR + AfPSR + Apr2 + Afpgsz

+Af P1353

 

 

Post-tensioned Strands (Afpr)

 

Critical construction stage Components of prestress loss

 

Stressing of lSt post-tensioning tendons Apr + Apr + Afpgg;

 

Stressing of 2nd post-tensioning tendons Apr + Apr + AprSl + Aprsz

 

Final conditions after losses Afpp + Apr + Afp551+ AprSZ + AprR +

AfPSR + Aﬁmz

 

 

 

 

The pre—tensioned stress at the ﬁnal stage after subtracting all prestress losses at

each critical stage is deﬁned as:

fpe,ﬁnal = fpi — AfPRE , (2-82)

85

where jig) is the initial pre-tensioning stress and Apr5 is the total prestress loss on the

pre-tensioning strands as deﬁned in Table 2-6 and Table 2-7.
The post-tensioned stress at the ﬁnal stage after subtracting all prestress losses at

each critical stage is given by:
fpe,ﬁnal = fpo “ AfPTa (2'83)

where fpo is the initial post-tensioning stress and Apr is the total prestress loss on the

post-tensioning strands as deﬁned in Table 2-6 and Table 2-7.

2.13 Life-Cycle Cost of Spliced Prestressed Concrete Bridges

Precast/prestressed concrete bridges have become the preferred type of bridge
construction, primarily for reasons of economy, savings in life-cycle costs and their fast
construction. Due to large and increasing costs associated with repairing concrete
structures, the durability of a structure should be considered during the design stage in
order to lengthen the structure service life. Normally, when structures are designed,
minimizing the initial cost is the ﬁrst priority and the durability of the structure is
considered as part of the design speciﬁcations, which is not clearly speciﬁed. However,
the overall economy of a structure is measured in terms of its life-cycle cost, which

includes the initial cost of the structure plus its maintenance cost.

There are several methods to lengthen the service life of concrete structures,
including using sealers, use of concrete admixtures, using coatings, and increasing
concrete clear cover. However, only parametric studies on the increase of concrete clear

cover on the slab and the precast/prestressed concrete girder are used herein since these

86

options must be deﬁned during the design process, unlike other parameters that can be

applied later when structure is complete.

Only limited research has been conducted concerning durability in combination with
life-cycle cost of concrete structures. Especially, the economical consequences of the
choice of parameters that can lengthen the service life of a structure in the design phase
has been neglected.

In order to provide cost-effective structures, it is necessary to predict the expected
service life of a structure in the design phase. When the expected service life is
calculated, and the maintenance and repair costs are assumed, the life-cycle-cost per year

can be determined.

2.13.1 Life-cycle cost criteria

The overall economy of a structure is measured in terms of life-cycle cost. This
includes the initial cost of the structure (1) plus the total expected maintenance costs (M)
and the removal cost (R). Precast/prestressed concrete bridges built in accordance with
speciﬁcation requirements should require very little, if any, maintenance. Fatigue
problems are minimal because trafﬁc loads induce only minor stresses. Prestressed
concrete structures built in the late 19505 have withstood heavy trafﬁc and severe

weathering for nearly 50 years and yet have required practically no major maintenance.

The life-cycle cost per year can be determined by dividing the total life-cycle
cost by the design life-span years. The life-cycle cost per year can be used to compare
design alternatives with different concrete covers. The cost for one-time maintenance

and the number of expected maintenance interventions before replacing the existing

87

bridge should also be part of the life-cycle cost analysis. Typically, maintenance cost is

0.5 to 1.2 times the structure’s initial cost.

Initial cost can be developed based on the design strength requirements, while
life-cycle cost incorporates all costs that could occur during the life time of the structure.
The initial cost of a new bridge is normally higher than that of an existing bridge, but its
maintenance cost should be much less. In addition, the life-cycle cost of new bridge

structures should be much smaller than that of old bridges.

2.13.2 Effect of chloride ingress on concrete structures

Chloride ingress is a common cause of deterioration in concrete bridges. The
presence of chloride ions initiates corrosion of the reinforcement (or rebar), which
reduces the service life of the structure as well as causing a reduction in its structural
capacity. The shortage of concrete cover is the primary cause of concrete deterioration

since the chloride ions can reach the steel faster (Y oshiki et. a1 2001)

Chlorides are well known for their ability to penetrate and destroy passive ﬁlms
on reinforcement. Micro pores and gel water, which are generally formed in concrete,
serve as a route for the movement of chloride ions. The pore structure in concrete
depends on the type of concrete, mix proportion and its quality. Corrosion in concrete
depends on the supply of oxygen and water as well as chloride content. Chloride
permeability depends on quality of concrete, concrete types, water cement ratio,
construction conditions, etc. Concrete may be exposed to chloride by the presence
seawater or deicing salts. Chlorides, either alone or combined with carbonation, are the

primary cause of concrete corrosion.

88

When chloride ions penetrate into the concrete and reach the surface of the steel
reinforcement, the chloride ions break the inert ﬁlm in the reinforcement surface and
starts corroding it. As corrosion continues, corrosion products build-up around the
reinforcing steel. These corrosion products occupy from 2 to 14 times the volume of the
original steel, creating an expansive force that is sufﬁcient to cause the concrete to crack.
On the surface around the crack of the concrete, a large quantity of chloride ions has
already penetrated into the concrete around the reinforce bar. Therefore, repair done by
coating and/or replacing the concrete cover is neither effective nor long lasting (Stephen

et. al 1998).

2.13.3 Reinforcement corrosion initiation period

The corrosion initiation period refers to the time during which steel is destroyed and the
reinforcement starts actively corroding. The time to initiation is a function of the quality
and the amount of concrete cover, the environment, the effective chloride diffusivity
properties of the concrete, the concentration of chlorides on the surface, and the chloride
threshold for the reinforcement.

Normally, structures can be classiﬁed as submerged components, splash-zone
components, and superstructure components (Stephen et al. 1998). For submerged
components there is a constant supply of water and chloride ions on the surface of the
concrete. However, corrosion of the steel is limited by the lack of dissolved oxygen in
the water. In splash-zone components there is a constant supply of chloride and ample
supply of oxygen. Thus splash-zone components have a higher corrosion rate than

submerged components. For superstructure components, chlorides come from deicing

89

salts. The corrosion rate for superstructure components depends on the surrounding
environment and will change over time.

If the environment boundary is kept at a constant chloride concentration, Co, the initial
concentration being zero throughout the medium, then the chloride concentration

solution at a distance x and time t can be determined by (Berke and Hicks 1994):

C(x,t) = Co(l — Elf (2-84)

x )
Zwcht
where erf is the error function; x is equal to the depth of concrete over the steel

reinforcement; C0 is the surface chloride concentration; Dc is the concrete diffusion

coefﬁcient; and t is the time in days. Equation 2-84 is most suited for evaluating
submerged and splash zone components.

The diffusion coefﬁcient of chloride ions in concrete, Dc, is known to depend on
the concrete’s pore structure, which is related to the water/cement ratio, cement type,
mix proportion and overall concrete quality. The water/cement ratio is a major factor
inﬂuencing the diffusion coefﬁcient. The smaller the water/cement ratio is, the smaller
the diffusion coefﬁcient (Ababneh et al. 2003).

If the environment changes with time, the surface chloride concentration also
changes and a new relation that allows surface build-up of chloride ions should be used.
When a linear build-up of chloride ions over time is assumed, the chloride concentration
solution at a distance x and time t can be obtained as follows (Stephen et al. 1998):

-x2

2 _
)e 40! } . (2-85)

C(x, t): kt{(1+2x-—)erfc( —7)c_b_—2)- (J—x—t

 

90

When a square root build-up of chloride ions over time is assumed, the chloride

concentration solution at a distance x and time t can be obtained as (Stephen et. al 1998):

C(x,t) = th{e4—D7 —[

J—
2J7

 

erfe(— (2-86)

ZJIBIW [C(ZJEH}.

2.13.4 Effect of concrete cover on service life of concrete components

Concrete cover is required to protect reinforcing bars from corrosion, to maintain
adequate bond between the concrete and the reinforcing bars, and for ease of casting. In
severe chloride environments the cover depth used for concrete structures should be
selected for corrosion protection. The required minimum cover depth for prestressed
concrete beams is normally less than that required for conventional reinforced concrete
beams. According to the ACI 318—02 recommendations (ACI 1992), the minimum cover
depth for precast/prestressed concrete beams should be at least equal to 38.1 mm (1.5
in).

In order to keep rebar and tendons away from chlorides an increase in concrete
cover depth is essential. Thus, a deeper concrete cover may be necessary when a long
service life period, such as more than 100 years, is set as a goal.

Some prestressed concrete bridges in coastal areas have been replaced due to
heavy corrosion after only a few decades of use, and in some cases the cost of
maintenance was more than the initial construction cost. When large cracks due to
corrosion are observed, it means that steel corrosion is already so severe that even
careful repair is no longer sufﬁcient. The design lifetime of a structure based on concrete

cover depth can be evaluated based on Fick’s 2nd law equation as speciﬁed in Equation

91

(2-84) for a constant chloride environment or Equation (2-86) for a build—up chloride

environment (Stephen et. a1 1998).

2.14 Summary

In this chapter, the fundamental assumptions, criteria and procedures for the design
and analysis of spliced precast/prestressed girder bridges were brieﬂy complied,
providing ample information that was used in this study. The major design criteria and
additional considerations in structural analyses were presented to show the differences
between conventional precast/prestressed concrete bridges and spliced girder bridges.
The load patterns from different construction sequence methods and the corresponding
prestress losses presented in this chapter were later used in the optimization and
sensitivity studies presented in Chapter 4 and Chapter 5. Equivalent post-tensioning
loads in non-prismatic sections were derived and setup in similar form to equivalent
uniform load patterns in prismatic sections with additional point loads at the splice
locations due to the non-prismatic section effects. In addition, the general concept and
methods to preliminarily asses durability and life-cycle cost of concrete structures were

reviewed are later integrated in the optimization studies presented in Chapter 4.

92

3 Design Optimization of Spliced Precast/Prestressed

Girder Bridges

3.1 Introduction

This chapter presents the approach used for the structural optimization and
sensitivity studies on Spliced Precast/Prestressed Girder Bridges. The formulation and
approaches for single and multicriteria structural optimization are presented. The single-
objective of cost minimization was chosen as the basic problem for the optimization
studies. Multicriteria methods address situations where in addition to minimum cost,
other criteria such as minimum post-tensioning, minimum concrete volume, and
minimum weight should also be satisﬁed. The constraint approach was used to generate
efﬁcient solutions and compromise programming is used for selecting the best solution.
Different levels of optimization, namely, system (or conﬁguration) and component
optimization were applied to SGB to determine prestress requirements and develop new
optimal pier/negative sections. The implementation of the optimization process in a

custom program in the Matlab environment is presented.

3.2 Structural Optimization

The conventional design process starts with some initial values for the design
variables thus deﬁning an initial design, which is then analyzed and checked against an
accepted in the form of performance limits. If the allowable values are exceeded, then

the design variables are adjusted and the process is repeated until the ﬁnal design

93

satisﬁes the allowable limits. Other design criterions are then checked one by one until
an acceptable design is obtained. This process usually yields conservative design results,
but it is an expensive process. In addition, when the parameters in the problem change,
the conventional process must be restarted and all calculations must be repeated. Clearly,
this process is improved, in both outcome and time investment, as experience is obtained
with the system being designed minimized the number of iterations since the values for
the design variables that lead to favorable solution can be arrived at quicker, sometimes
very early in the design process.

A computational optimal design process differs from the conventional one
described above by taking into account all the design variables and criteria
simultaneously and improving the design through automated iterations. The advantages
of using optimization are that once the problem has been formulated and deﬁned, it is
easy to solve it for a variety of different conditions and requirements. Furthermore,
changes to the design variables are directed towards the attainment of a speciﬁc
performance measure through the help of mathematical algorithms.

Structural design optimization has been widely used for the design of steel,
reinforced concrete, and prestressed concrete bridges. The objective of an optimal bridge
design problem is to determine the structural system design variables that satisfy the
behavior and constraints of the problem. Constraints can be assembled directly from
bridge design code speciﬁcations, while the design problem behavior can be the cost of

the bridge structure or other merit functions.

94

The standard mathematical form of an optimization problem is normally written

as:

Objective function: Min/(x)

Subject to: gi(x) 50 i = 1 to m

hJ-(x)=0 j=1top

XL _<x _<XU

where x is the design variables vector, XL and XU, are the lower and upper bound vectors

for the design variables, f(x) is the objective function, and g,-(x) and h,-(x) are the

inequality and equality constraints, respectively.

Optimization problems can be classiﬁed as unconstrained problems and
constrained problems. An unconstrained problem has no constraints restricting the
problem while the constrained problems contained constraints that restrict the design. In
a constrained problem, the problem needs to be transformed into unconstrained
subproblems that can be solved as the basis of an iterative automated process.

Optimization problems can be classiﬁed as linear programming problems and
nonlinear programming problems. When both the objective function and the constraints
are linear functions of the design variables, the problem is known as a linear
programming problem (LP). When either the objective function or the constraints are
nonlinear function of the design variables, the problem is known as a nonlinear
programming problem (N LP). Quadratic programming (QP) concerns the minimization
or maximization of a quadratic objective function that is linearly constrained. The

solutions from LP and QP problems are readily and reliably obtained while the solution

95

of NP problems require iterative procedures to establish a direction of the search at each
major iteration.

The formulation of an optimum design problem involves translating a descriptive
statement of the problem into a well deﬁned mathematical statement. The formulation of
a structural optimization problem can be deﬁned by the following steps (Arora 2004):

Step 1: Identiﬁcation of the problem statement.

Step 2: Collection of data and information.

Step 3: Identiﬁcation and deﬁnition of the design variables.
Step 4: Identiﬁcation and deﬁnition of an objective function.

Step 5: Identiﬁcation and deﬁnition of the design constraints.

3.2.1 Problem statement

The descriptive objective of the project needs to be identiﬁed by the designer or
project owner. The problem statement describes the overall objectives of the project and

the requirement criteria.

3.2.2 Data and information

The information of material cost, performance requirements, material properties,
and other relevant information needs to be gathered before developing a mathematical
formulation of the problem. Analysis design tools and their procedures must be

identiﬁed at this stage.

96

3.2.3 Design variables

Design variables are those that describe the system design in mathematical terms.
They are the variables in the problem that are allowed to vary or change in the design
process. Design variables indicate all of the unknowns in the optimization problem.
Optimization is the process of choosing values for the design variables that yield an
optimum design. In order to maximize the efﬁciency of the computational algorithm,
only key parameters that are important to the system design are chosen as design
variables. The system design is deﬁned once these variables are speciﬁed. Normally,
design variables are independent of each other.

Design variables can be classiﬁed as discrete (or integer), continuous, or a mixed
of both. Discrete variables must have a value from a speciﬁed discrete set (integer
values or just zero or one), while continuous variables can have any value from a
continuum. Mixed integer variables problem can have both discrete and continuous
design variables. In reality, design variables cannot be arbitrary due to manufacturing
and fabrication limitations. In most structural design problems the discrete nature of the
design variables tends to be disregarded due the expensive nature of the calculations
needed to deal with them. Rather, the optimum solution is typically obtained for
continuous variables and are then adjusted to the nearest available discrete value. This
assumption has proven to be acceptable since continuous-variable optimal solutions
always give conservative results while the discrete solutions normally lay in the

boundary of continuous solutions.

97

3.2.4 Objective functions

There can be many feasible designs for a system, and clearly some are better than
others. In order to compare different designs, an objective function is used as a criterion
to measure the “goodness” or “efﬁciency” of the system. The objective function is a
scalar value used as a merit criteria to obtain an optimal solution. A valid objective
function must be inﬂuenced directly or indirectly by the design variables of the design
problem; otherwise, it is not a meaningful objective function. The optimized design is
the one that yields the best value for the objective function. The objective function can
be maximized or minimized depending on the problem requirements. Possible objective
functions for the design of spliced girder bridges are the minimum superstructure cost,
the minimum amount of prestressing steel, the maximum girder spacing, the minimum
concrete volume, the maximum feasible span length, and the minimum superstructure

depth.

3.2.5 Constraints

Constraints are represented by function relations among the design variables.
Constraints are typically classiﬁed as either equality, h(x), or inequality, g(x),
constraints. Like the objective ﬁmction, constraints can be linear or nonlinear. Typically,
equality constraints are in the form of functions equal to zero while inequality

constraints are functions that can have values less/more than or equal to zero.

Inequality constraints: g,-(x) 50 (3-1)

Equality constraints: hj(x) = 0 (3-2)

98

Two types of constraints were considered for the presented optimal bridge design
studies. The ﬁrst constraint is a function of serviceability and ultimate state design
limits. The second type constraints are described in terms of practical design limitations
such as the available girder depth, practical girder spacing, and maximum number of

pre-tensioning strands per row in the prestressed concrete girder.

3.3 Optimization approach

Classical structural optimization combines techniques from different disciplines
including computer aided design, structural ﬁnite element analyses, structural sensitivity
analyses, and mathematical algorithms. The detail of each technique is described as

follows:

3.3.1 Computer aided design

Computer aided design refers to the computerized tools used in the design
optimization process for pre- and post-processing purposes. In this study, each bridge
girder was modeled as a beam line and divided into ﬁnite elements. The element length
of the beam was discretized into small elements to cover all critical locations, i.e., the
nridspan of middle segment, the splice location, and the midspan of the end segments.
Then later, the internal forces, displacements, and stresses were obtained at these
locations. The modeling and output retrieval was achieved through a custom frame

analysis program written in the Matlab environment.

99

3.3.2 Finite element analyses

The ﬁnite element method is the most widely used form of computer-based
engineering analysis. It is a numerical procedure for analyzing structures that are too
complex to be solved efﬁciently of satisfactorily by analytical methods. Finite element
methods are used to perform analysis in order to obtain the responses (i.e., internal
forces and deﬂections) caused by applied loads. Finite element analyses are normally
used in structural optimization to evaluate the structural responses required by the

objective and constraint functions, and also by the variable sensitivity analyses.

3.3.3 Sensitivities analyses

Sensitivity analyses provide gradient information on the objective and constraint
functions with respect to the design variables for formulating the optimization problem.
Sensitivity analyses allow ﬁnding the effect on the objective ﬁrnction if the design

variable values are changed.

Two types of techniques, numerical and analytical, can be used to calculate
structural sensitivity derivatives for structures [Adelman and Haﬁka, 1986]. Numerical
techniques use ﬁnite different methods, which are straightforward to determine since
derivatives are calculated by ﬁnite difference approximations. The accuracy of the
derivatives obtained from this technique depends on the perturbation step size for the
ﬁnite difference method. Analytical techniques include the direct differentiation method
and adjoint variable methods. Analytical techniques are suitable for applications that

require high computational efforts.

100

3.3.4 Mathematical algorithm

The selection of changes to the design variables such that they lead to an
improvement in the objective design function is controlled by some form of a
mathematical algorithm. Mathematical algorithms for solving constrained optimization
problem, are broadly classiﬁed as direct methods and transformation methods (Arora
and Belegundu, 1984 and 1985). Transformation methods include the barrier (interior)
and penalty (exterior) function methods as well as the augmented Lagrangian or
multiplier methods. Transformation methods were developed to solve constrained
optimization problems by transforming them into a sequence of unconstrained problems
whose collective solution converges to the solution of the original problem. The concept
of transformation methods is to construct a transformed function by adding a penalty
term for the constraints violations to the objective function. Transformation methods are
not widely used due to their limitation inherited from the transformation functions,

which have to be assumed. Direct methods were developed to solve the original

constrained optimization problem directly. The direction vector (fit) is calculated using

. . . . k . .
the function values and their gradients at pornt XI ). The step srze calculation needs only

the function values. Different methods can be generated depending on how the direction
d and step size are calculated. Many algorithms such as the gradient projection method,
the method of feasible directions, the generalized reduced gradient method, and the
constrained steepest descent method have been developed and successfully used for
engineering design problems. Compared to other methods, the constrained steepest
descent (CSD) method is one of the most commonly used since it is robust and effective

for solving nonlinear constrained optimization problems (Arora 1989).

101

3.4 Optimization Algorithm Types

In this section, three classical different optimization algorithm types, which
include gradient based algorithm, the direct search algorithm, and the branch and bound
combinatorial search algorithm, are brieﬂy explained. Later, optimal results obtained
from these methods are compared and the advantages and disadvantages of using these
methods, i.e. computation time, types of design variables, and types of constraints, are
discussed to provide the basis for the algorithm selection in this study.

Gradient based algorithms rely on the gradients derived from the objective and
constraint functions and can handle only continuous design variables. The direct search
algorithm is a global optimization algorithm that does not require derivative information
of the objective function or constraints. The branch and bound algorithm is a
combinatorial search approach that can handle both continuous and integer variables also
without the need of gradient information of the objective or constraint functions. Details

for each algorithm used in this study are presented in the following section:

3.4.1 Gradient based algorithms

Gradient based algorithms are commonly used for nonlinear, constrained,
continuous optimization problem. The algorithm is a sequential quadratic programming
approach. In the optimization of spliced girder bridges the objective functions and
constraints are nonlinear functions of the design variables. Therefore, the CSD method
using a gradient based algorithm was used. The four steps in the CSD method are as

follows:

102

a) Linearize the objective and constraint functions about the current design
estimate,

b) Deﬁne the subproblem formulation by the linearized objective and constraint
functions to determine a search direction,

c) Solve the subproblem to obtain a search direction, and

(1) Calculate a step size to minimize the descent function in the search direction.

Constrained steepest gradient-based algorithms rely on the gradients derived by
linearizing the objective and constraint functions with respect to the design variables.
Derivatives of the objective and constraints functions are used to construct an
approximate problem by redeﬁning them through Taylor series expansions. The
derivatives of the obtained linear Taylor series can be numerically evaluated by ﬁnite
difference approaches. For example, the forward difference technique is commonly used

to evaluate the numerical differentiation of a multi-variable function ﬁx) with respect to

variable xi, which is deﬁned as:

(if : f(x1,x,- +Axi,...,xn)—f(x1,...,x,-,...xn)
6x,- AX,‘ ’

 

 

(3-3)

where Ax,- is a small perturbation in the variable xi. The accuracy of function gradients

depend on the selection of the perturbation Ax).

The search direction is determined by a subproblem based on the linearized
objective and constraint functions of the current design. The descent function plays an
important role in constrained steepest descent (CSD) methods since it is used to monitor
the progress of the algorithm towards an optimum point and is used by the optimization

algorithm to determine an appropriate step size. The basic concept is to compute a step

103

size along the search direction such that the descent function is reduced. Figure 3-1
illustrates the steps of constrained steepest descent function for a gradient-based

optimization algorithm.

X2 ll

X(k,l)

\ Constraint
/ Correction steps

   
 

Constrained steepest
descent function

 

>X

 

I

Figure 3-1 Steps of constrained steepest descent gradient method

Pshenichny’s descent function (Arora, 1989, 1997) was used as the descent
function for the CSD method. The steepest descent function used the function gradient
vector (Equation 3-4) at each point to determine the search direction for each iteration.
Figure 3-2 schematically illustrates the steepest descent function directions. At any point

x where the gradient vector is nonzero, a negative gradient,-Vf (x), will point

“downhill” towards lower values of the objective function f(x). Generally, the negative
gradient is locally a direction of steepest descent as the objective function decreases

more rapidly along this direction than along any other path.

104

X1

Figure 3-2 Steepest descent function direction

The steepest descent method starts from an initial guess x0 and successive
approximate solutions are given by:
X(k+1) = X“) +aka(X(k)) =X(k) +akd(k) , (34)
where the superscript k represents the iteration number (k = 0, l, 2,. . .); Xlk) is the design
variables vector in the kth iteration; (1’9 is the search direction calculated using function

values and their gradients; Xlk W is the design estimate in the (k+1th) iteration. The step

size line search parameter (ak) determines how far to go in the chosen direction and is

given as follows:

T
gkgk

“k = (3-5)

T
g k H k gk
where gk is the gradient and Hk is the Hessian matrix of the objective function (Arora

1989)

105

The gradient vector of objective function is used to compute the search direction
as shown in equation (3-6). It is composed by the partial derivatives of the objective
function with respect to each of the independent variables as follows:

—_§f__
6x1
if-
Vf(x) 2 g1. 00 a 6x2 . (3-6)

1

Lfix" _

 

 

The Hessian matrix (H) represents the second-order derivatives of the objective
function with respect to the design variables. The Hessian matrix is also used to calculate
the search direction as shown in equation (3-7) and also used to determine the optimal
condition of the optimal point as shown later in equation (3-10). The Hessian matrix off

is a square and symmetric matrix of the second derivatives of objective function f(x) and

is given as follows:

 

 

 

 

 

I 62f 62f
6x12 axiaxn
Hks s 5 . (3-7)
62/ 92f
axnaxl 6x3

 

The concept behind this approach is to compute a step size along the search

direction d’k) such that the descent function is reduced at each iteration of the algorithm.

The step size direction is a line, or one-dimensional, search. For constrained problems it
is preferred to determine the step size by using an inexact line search since it is more

efﬁcient. However, this method involves a sequence of major iterations. Each iteration

106

requires the solution of linearly constrained subproblem where the nonlinearities are
conﬁned to the objective function. A design point is chosen to initiate the iterative
process. The iteration process is continued until a ﬁnal design point is reached; that is,
when there are no further changes in the value of the objective function.

The descent function used to solve constrained optimization problems requires a
searching direction towards the optimum design. A searching direction is established by
solving a subproblem. Quadratic programming (OP) is chosen to determine the
searching direction due to its high efﬁciency and quadratic convergence rate. The
sequential quadratic programming subproblem utilizes a Hessian matrix constructed by
a quadratic objective function and linear constrains, which are deﬁned as (Arora 1989):

f=%dTde+g/{d; (3-8)

where gk is the gradient; d is search direction; and Hk is the Hessian of the objective

function.

In large-scale optimization problems it is difﬁcult to calculate the Hessian
matrix. Quasi-Newton Methods have been developed to approximate the Hessian matrix.
The hessian matrix is updated by using design changes and the gradient vector of the
previous iteration. The modiﬁed BFGS (Broyden-Fletcher-Goldfarb-Shanno) (Powell
1978) method was thus implemented to approximate the Hessian matrix.

Perturbation vectors are used to determine a local optimum point as shown in

equation (3-9) and (3-10) and are deﬁned as: 6x 2 x — x0 and 8f 2 f (x) — f (x0) . As x is

to be considered as a local minimum, it must satisfy two optimality conditions termed:

the ﬁrst—order necessary condition and the second-order suﬂiciency. The ﬁrst-order

107

necessary condition is attained when f(x) has a minimum value f: at x: such that any
perturbations in x must result in higher value of f by deﬁnition of x:- to ﬁrst order.

Equation (3-9) represents the ﬁrst-order necessary condition.

6]; : Vf(X* )5)“. Z 0. (3’9)
If f(x), x e X C SR" , has a local minimum at an interior point x:- of the set X and if f(x) is

continuously differentiable at x*, then Vf(x*) = 0T. This result is a necessary condition

because it may also be true at points that are not local minima i.e., the gradient will be
zero at a local maximum and at a saddlepoint. All points at which the gradient is zero are
called stationary points and the necessary condition is called the stationarity condition.
The second-order sufﬁciency information as shown in Equation (3-10) helps make a
more ﬁrm conclusion of attaining a minimum/maximum condition since the ﬁrst-order

local derivative gives only useful information but it is not conclusive.

af, = édeH(x+)6x. 2 0 (3-10)

The sign of 6f+ depends on the sign of the differential quadratic form axT H (x+ )ax. If

this quadratic form (Equation 3-10) is strictly positive for all 6x at O , then x+ is deﬁnitely
a local minimum. If the Hessian matrix of f(x) is positive-deﬁnite at a stationary point x+,
then in is a local minimum.

An algorithm is considered to be convergent when it reaches a local minimum
point. An algorithm is considered as globally convergent when it converges from any
arbitrary starting initial design. To ensure that the optimized solution does not
correspond to a local minimum the optimization procedure should be performed more

than one time by changing the starting design point. If results still yield the same answer

108

irrespective of the starting design point, then the result can be considered as non-local
optimum.

The gradient based algorithm normally takes less computation time than other
algorithms since it uses the steepest descent function, which decreases more rapidly
along steepest direction and will point “downhill” towards lower values of the objective
function, for a search direction for each iteration. The disadvantage of using this
algorithm is that it is limited to problems with continuous design variables and thus
cannot handle integer or mixed integer variable problems. In addition, it requires
derivatives of the objective and constraints functions to construct the optimization

problem.

3.4.2 Direct search algorithms

Direct search algorithm methods do not require derivatives of the objective or
constraints functions. These methods also do not depend on the starting point because
the ﬁrst point is arbitrary selected, typically at the center of the design space. The
method starts by sampling the center of design space, and then the algorithm begins to
subdivide the design space into smaller rectangles. The algorithm iteratively continues
this process, selecting and subdividing those rectangles that have the highest likelihood
of producing an objective function value lower than the current lowest point. The direct
search algorithm is pure evaluation and thus its design variables can be discrete or
continuous variables.

An algorithm based on the above concept is the DIRECT search algorithm by
Jones (1994). It combines global and local searching. For example, small rectangles with

better sampled points at their centers are likely to be further subdivided (local search)

109

and larger rectangles have a greater likelihood of being further subdivided (global
search).

Direct search algorithms stop when the number of iterations performed by the
algorithm reaches the value of maximum iterations, when the total number of objective
function evaluations performed by the algorithm reaches the value of maximum function
evaluations, or when the change in the objective function from one point to next point is
less than a given tolerance.

The greatest advantage of direct search algorithms is the combination of global
and local searching. A distinct advantage over other optimization algorithms is that it
does not require ﬁnite difference step size and convergence tolerances. In addition, no
staring point is speciﬁcally required since the ﬁrst point is always at the center of the
design space.

Direct search algorithms also have few disadvantages, e.g., they cannot work
with equality constraints, they cannot terminate before the maximum number of function
evaluations and/or iteration have been exceeded, and they need bounds on the design
space in order to know how to divide the rectangles. The optimal result of this algorithm
is improved by a semi-directed random search; consequently, it is computationally more

compared to the gradient based algorithms.

3.4.3 Branch and bound combinatorial search algorithms

Branch and bound algorithms are based on exhaustive searches. The Operation of
combinatorial search algorithms can be visualized with search trees. The search tree has

member group levels corresponding to the numbers of design variables. Each member

110

group can be expanded into nodes. In the search tree for an exhaustive search a number
of branches emanates from each node number.

The branch and bound search begins by ﬁxing the value of each member group at
a time and treating other groups as continuous variables to ﬁnd their optimal values. The
lower bound of each group is obtained and the lowest bound for all groups is
determined. The last selection of the lowest bound for all groups represents a discrete
design since it has a lower value than the previous candidate in the groups.

The branch and bound method eliminates many of the design variables through
this exhaustive search. A nonlinear programming algorithm is executed to perform the
continuous Optimization to establish the lower bounds prior to determining discrete
solutions.

The branch and bound method has been used successfully to deal with discrete
and continuous design variable problems. However, for problems with a large number of
design variables, the number of subproblems (nodes) becomes large. Therefore, the
branch and bound method is not suitable for problems with many design variables since
they could lead to very expensive computation. In addition, branch and bound methods

guarantee to ﬁnd the global optimum only if the problem is linear or convex.

3.4.4 Comparison of optimal results

A single-span spliced girder bridge built through single-stage post-tensioning
with a total span length of 42.67 m (140 feet) was used as a case study to compare the
optimal results from different optimization algorithm types. Three different algorithm
methods, which included a gradient based algorithm (Matlab - fmincon function)

(Mathwork 2004), a direct search method (DIRECT) (Jones 1994), and a branch and

111

bound algorithm (OPTIMUS software) (STI Technologies, Inc. 2005), were used to
compare their optimal results.

Table 3-1 shows the comparison of optimum design variable values for each of
the optimization algorithm types. Preliminary results for the design variables obtained
from the sequential quadratic programming algorithm (MATLAB-fmincon function) are
continuous numbers, therefore some design variables need to be rounded off. However,
the design variables obtained from the DIRECT and OPTIMUS algorithms are a mixed
set of continuous and integer numbers. It can be seen that the objective value from
fmincon gave a lower objective function value than the DIRECT and the OPTIMUS
algorithms since all design variables are continuous. DIRECT is considered both a
global and local optimization algorithm and thus it led to a lower objective value than
the branch and bound method which depends only on an exhaustive search and does not
guarantee the attainment of a global optimum.

In order to improve the results from DIRECT and OPTIMUS closer to the
fmincon solution, the number of iterations or maximum number of evaluations used in
the algorithms should be increased. This could signiﬁcantly increase the computation

time compared to the fmincon method (days VS hours).

112

Table 3-1 Comparison of optimization versus algorithm performance

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fmincon fmincon DIRECT OPTIMUS
mixed integer
Design variables Sequential Sequential Global continous
Quadratic Quadratic nonlinear
pro g. pro g. optimization pro grammin
S 11.198 11.198 11.182 11.119
npr, 1 7.95 8 12 1 l
npry 7.95 8 3 2
n r31 2 2 2 l
npr4/ 2 2 2 2
n ru+npr21+npr31+npr41 19.9 20 19 16
nprlz 20.1 20 13 10
nprzg 4.1 4 8 8
npr32 0 0 1 0
npr42 0 0 0 0
npr12+npr22+npr32+npr42 24.1 24 22 18
npo 59.17 59 63 66
a 38.47 38.47 38.69 39.83
yb 10 10 10.09 10
71pm] 4 4 4 3
”d1 2 2 2 2
pm 0.25 0.25 0.25 0.25
f(x) 2.748e6 2.748e6 2.8589e6 2.86353e6
exceed 10
Constraint Satisﬁed ox psi ox o1<

 

 

When comparing results from the different algorithms, maximum beam spacing,
splice location, and the number of strands can be seen to be within the same range. Since
the objective of the present research was to understand the response and behavior of
SGB and to develop design charts for preliminary design, the use of nonlinear
continuous programming (fmincon in the MATLAB) can be considered adequate and
was thus used in this study. In addition, if designers were to use design charts developed
through optimization they would have to interpolate and interpret their solution from the

presented curves. Therefore, there is no guarantee that they will choose the exact

113

solutions obtained from the optimization solution. Nonlinear continuous programming
(fmincon function in MATLAB) was thus selected for this study to minimize

computation effort.

3.5 Multicriteria Optimization

Normally there can be several design solutions that satisfy all requirements of
safety and serviceability and a number of speciﬁed merit criteria imposed by a design
code. Thus, the designer needs to make a difﬁcult decision in selecting the best design
among the possible alternative solutions that adequately satisfy all the governing criteria.
The combination of single objective functions is considered as a form of multicriteria
optimization. Multicriteria optimization considers two, or more objective functions
simultaneously, and these functions may have conﬂicts. Multicriteria optimization
achieves a compromise between conﬂicting objectives and represents more rational
solutions than those obtained by optimizing individual objective functions (Arora 2004).
The resulting solution is expected to be feasible, efﬁcient, and non-dominant with
regards to all criteria and should ensure the best compromise between all competing

objective functions.

3.5.1 Mathematical formulation

The multicriteria optimization problem can be formulated as follows:

Objective function: Min F (x) = (f1(x), f2(x), fj-(x))
Such that: ﬁ-(x) _<cj j = 1, 2, m

Subject to: gk(x) S 0 k = 1 to n

114

h1(x) = 0 I: 1 to p;
where m is the number of objective functions, n is the number of inequality constraints,

p is the number of equality constraints, F (at) is the multicriteria objective functions and
has m-dimensional vector of objective functions, fJ-(x) is the single objective functions,

and gk(x) and h1(x) are the equality and inequality constraints, respectively.

When several objective functions are considered at the same time, many optimal

solutions can be obtained. For example, Figure 3-3 represents optimal design variables

(x,-) for two objective function f; and ﬂy. The feasible design space (S) is deﬁned as a

collection of all the feasible design points as shown in Equation 3-11. The inequality
constraint g1 and g; are the constraints that restricted the design for both objective

functions.

S = {xlhi(x) S 0; i =1t0p; and g,-(x) S 0;i=1to n}; (3-11)
where h,-(x) is equality constraints and g;(x) is the inequality constraints; and xi is design

variables

In Figure 3-3, the minimization of objective function f, cannot yield the
minimization of objective function f; and vice versa, thus the designer has to choose
which objective function should be used as the main criterion in a multi-criteria

optimization consideration.

115

 

 

Figure 3-3 Graphical representation of a two—objective optimization problem

3.5.2 s-constraint approach

Multicriteria problems are commonly solved by either the weighted objective
approach or e—constraint method, also known as the trade-oﬂ approach. The s-constraint
method has gained wide acceptance as compared to the weighted objective approach
because of its practicality and rationality (Lounis and Chon 1993).

The weighted objective approach can be formulated as follows:

min f(x) = W161 (x) + w2c2 (x) (3—12)

where w,- is the weight and c,~(x) is the single objective functions. The weights in

the weighted objective approach are rarely known precisely, so prefered values are
adjusted gradually until they become more evident. This approach is subjective and can

mislead the optimum results. Thus, the designer should trace the effect of these

116

subjective preferences on the optimal solution obtained after solving the problem
(Papalambros and Wilde 2000). The weighted objective approach was not used in this
study since the weight factors for each of the single-objective functions are subjective,
unlike the e—constraint approach which treats all single-obj ective functions equally.

The e-constraint method consists of transforming the multi-objective
optimization problem into a single scalar-primary objective optimization problem. All
secondary objectives are transformed into constraints bound by some allowable level 8,-
and are added into the transformed single-objective optimization problem. The most
important objective function is chosen as a primary objective and the other objectives are
then considered as secondary objectives. After the multi-objective optimization problem
is transformed into an equivalent single-objective optimization by the c-constraint
approach, each single objective can be solved by the single-objective optimization
approach stated in section 3.4.1.

Generally, there is no single optimal solution that simultaneously yields a
minimum value for all single-objective functions in a multi-objective problem. The
concept of Pareto optimum, which deals with non-dominated and non-inferior solutions,
(Carmichael 1980) is widely used as a solution approach to multi-objective optimization
problems.

A subset of the Pareto optima is generated by the s-constraint approach, which is
based on the adoption of one criterion as a scalar objective function and the
transformation of other criteria into constraints bound by some predetermined constants

ei. Predetennined constant bounds are determined from the minimum values of each

objective function. The values of c,- are then selected between the minimum and

117

maximum ranges of the predetermined constraint 8,: Each objective function thus yields
a set of an optimal design variables xj* in which f,-(xj*) is a minimum. Deﬁning fi(xj*) as

the lower bound and f,-(x,- *) as the upper bound on s,- , or:

fi(Xj*) 581 —< fi(x1*). ( 3-13)
The values of s; are then varied to generate the entire pareto optimal set. A vector x* is

considered a Pareto optimum if there is no feasible solution x that may yield a decrease
of some objective function without causing a simultaneous increase of at least another

objective function.

3.5.3 Compromised programming

Once a subset of Pareto optima has been generated the designer has to make a
decision to select the best solution from the subset. The technique of compromised
programming (Lounis and Cohn 1995) deﬁnes the best solution as the minimum distance
from the set of non-dominant solutions to the ideal solution. A solution to this
multicriteria optimization problem is achieved by the “Minimax and Minimum Euclidian
Distances Method” (Duckstein 1984). Euclidian distances are used as a criterion for
comparison in compromised programming and can lead to practical approaches for
decision making in the presence of multiple and conﬂicting criteria. This approach has
gained acceptance because it is both practical and rational (Lounis and Cohn 1993 and

1995).

The “minimax” criterion is implemented by minimizing the normalized deviation

from the ideal solution fmin measured by the family of distance metrics Lp, The value of

118

LP, indicates how close the solution is to the ideal solution. The satisfying solution can

be determined by selecting the minimum normalized deviation (LP) between the

minimum Euclidean distance (p = 2) and the minimum largest deviation (p = 00), or
“Minimax Criterion”

The minimum Euclidean distance (p = 2) is determined by normalizing minimum
distance from the set of non-dominant solutions to the ideal solution and is deﬁned as

follows:

. _ . m fi(X)-minfi(x) 21/2
mrn L2(X)—mm[i§1|maxfi(x)—minf,-(x) ] .

 

(3-14)

The minimum largest deviation of the distance from the set of non-dominant

solutions to the ideal solution is determined as follows:

I fi(X)—minfi(x) I.
lmaxﬁ(x)—minf,-(x)|

 

min L00 (x) = min max

(3-15)

3.6 Structural Optimization of Spliced Precast/Prestressed
Girder bridges

The design of spliced girder bridges depends on several parameters that
signiﬁcantly inﬂuence performance and cost. The most relevant are time dependent
effects, splicing locations, construction sequence, girder segment geometries, number of
beams, and the number and proﬁles of pre—tensioned and post-tensioned reinforcement.
Normally most design variables are determined based on the designer’s judgment

through a trial and error process. Consequently there is no guarantee of obtaining the

119

most economical design, which requires more time and effort to explore, and which

typical projects usually cannot afford.

Optimization for the design of spliced precast/prestress girder bridges were
implemented to make the design more efﬁcient and to eliminate the trial and error
approach and to provide a systematic approach to arrive at appropriate design variable
solutions, signiﬁcant cost savings, performance improvements and ease of design by
providing guidance and insight into optimal solutions.

Bridge systems may be associated with three levels of optimization: (1)
components, (2) transverse and longitudinal conﬁgurations, and (3) complete structural
systems. Only component and conﬁguration Optimization were considered in this work

for the design optimization of spliced girder bridges.

3.6.1 Component optimization

Component optimization deals with the study Of Optimal dimensioning and
section sizing of prestressed concrete girders. Component optimization for prestressed
concrete girders has broadly been used in research efforts sponsored by state highway

agencies for developing standard sections.

120

Presently, precast/prestressed concrete girders are being used extensively in
continuous multi-span bridges in which continuous segments over the pier (pier
segments or negative segments) are custom-designed to carry the high negative bending
moment and high shear demands. Consequently, deep sections using haunches are
typically used. Different haunch depths were considered to assess the sensitivity of the
structural system to the pier segment ﬂexural stiffness. These analyses were conducted

by using frame analyses that included construction staging and time-dependent effects.

A component Optimization process was performed to obtain the dimensions of
optimal negative sections for two-span continuous systems. This process was coupled
with the conﬁguration optimization problem in order to obtain the best geometry
solution for a given span length, beam spacing, and splice locations with the most

efﬁcient girder at the pier location.

Four methods to increase the section depth over the pier section were considered
in this study. The main goal was to assess the cost efficiency of different methods used
to obtain pier segments from standard sections. The study assessed optimal designs for
the following cases:

a) Use of existing standard positive sections with an additional concrete block
(sofﬁt) attached to the underside of the beam (Figure 3-4a).

b) A new optimal prismatic section to be used in both positive and negative
moment regions (Figure 3-4b).

c) A new optimal negative section with variable web (Figure 3-4c).

d) A new optimal negative section with variable sofﬁt and constant web

(Figure 3-4d).

121

3.6.2 Configuration optimization

Conﬁguration Optimization includes the effect of number of spans, pier locations,
number of girders (beam spacing), and pre-tensioning and post-tensioning requirements
(number of strands and layout). In addition, conﬁguration Optimization can also include
the substructure. Since the number of supports is commonly reduced for SGB, the total
cost of the system construction is anticipated to be reduced.

Structural optimization techniques was used to develop optimal conﬁguration
and sectional girder requirements for spliced girder bridges to improve understanding of
system behavior, design variable sensitivity, and serve as a guide to produce design aids.
Design aids can depict the relations between girder shape, girder spacing, pre-tensioning
and post-tensioning requirements (including proﬁles), splicing locations, pier segment
geometries, and initial and ﬁnal concrete compressive strengths. It is thought that bridge
engineers would ﬁnd beneﬁts from the presented optimization results since they provide
information on system performance as a function of multiple design variables and their

collection in the form of design charts can also help expedite the design process.

122

b2

 

 

 

 

 

 

 

 

 

 

t4
_ t3
4—
New Optimd
“‘“\ /”——J I/f—Standard
Prismatic
H Negative
Existing tw 3901100
’/r—Standara ‘——* ‘——-
Positive ___
Section
__22 LL_, I l/// \\7

 

 

 

 

 

(:AZZZQZE St L»—————-A "H I2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b1
(3) (b)
«— b2 r4 b2 +4
+3 3 '13
3 New Optimal C3 ,/”—New ODilmd
Q Nonprismatic g Nonprismatic
8 Negative +5) Negatrve
°C Section g LEA, ‘___Sect1on
g 1W L.)
_1L__ — __IL 8 —lI—— _—
I I rl2 g; ll 12
bl I I' :> I bl "
(C) (d)

Figure 3-4 Negative Segment Alternatives: (a) existing standard girder with attached
soﬂit (b) new optimal standard prismatic girder (c) new optimal standard non-prismatic
girder with variable web depth (d) new optimal standard non-prismatic girder with

variable soﬂit

123

3.6.3 Single objective optimization of spliced girder bridges

The single objective functions that were adopted as merit criteria for the optimal

design of longitudinally spliced precast/prestressed girder bridges are listed below:

a) Minimum superstructure construction cost

This objective can be considered as the most realistic objective in bridge design
since the superstructure cost represents about 70% of the total structure cost (Lounis
1993). A reliable construction cost is not easy to deﬁne because it often varies with time
and location. However, the assumed unit costs (see Section 4.5) can be considered

adequate for the comparison purposes of this study.

b) Minimum post-tensioning requirements
The post—tensioning strand cost is considered higher than the pre—tensioning
strand cost since it involves the Operation cost at the site as well as the cost Of anchorage

hardware. Thus, the goal of this objective function is to use the minimum number of

post-tensioning which will lead to the maximum number of pre-tensioning strands.

c) Minimum pre-tensioning requirements

Normally, pre-tensioning strand cost is included as part of the prestressed
concrete beam. Thus there is no separate cost between the beam and the pre-tensioning
cost. However, this objective was included to assess the effect of minimizing the

numbers of pre-tensioning strands for an achievable span length.

124

(1) Minimum girder concrete volume

This objective was used for developing non-standard pier segments for two-span
continuous spliced girder bridges since no cost data for the newly proposed girders is

available. This objective is equivalent to minimizing superstructure weight.

3.7 Multicriteria Optimization of Spliced Girder Bridges

A common design criterion for spliced girder bridge design is the minimization of
total superstructure cost. However, in some situations, other criteria, such as
minimization of post-tensioning requirements, minimization of concrete volume,
minimization of superstructure depth, maximization of span length, and reduction on the
number of piers may be of interest. When the combination of single-objective
optimization functions is considered, multicriteria optimization can be used to model a
compromising problem to determine an optimal solution. The proposed multi-Objective
function was the minimization of cost since it is the ﬁrst practical objective that designer
and owner will consider.

The economics multicriteria objective function F1(x) considered the
minimization of superstructure construction cost, the minimization of concrete volume,
and the minimization of pre-tensioning and post-tensioning requirements. F )(x) can then

be deﬁned by the combination Of the following single-objective functions:

fla) : Min. SUperstructure COSt : ”g Cg + Cc Vc + Apsto + tpCz + ers + Cf‘lf
[2(x) = Min. concrete volume = Vc

f3(x) = Min. post-tensioning strands = ApS

125

f4(x) = Min. pre-tensioning strands = Apre
where ng = number of girders; A [,5 = number of post-tensioning strands; A p5 = number of
pre-tensioning strands; tp = number of temporay supports; mr = mass of non-prestressed
steel; Vc = volume of concrete; A f = surface area of slab formwork; Cc = cost of
concrete; Cp = cost of prestressing; CPO = cost of post-tensioning; C t = cost of temporary
support; Cg = erection cost for girders (including cost of concrete and prestressing steel);

CS = cost of non-prestressing; and C j: cost of slab forming.

3.8 Implementation

The single- and multi-objective optimization of spliced girder bridges were
implemented in a custom automated program written in Matlab (MathWorks 2002).
Matlab is widely used in engineering and mathematics applications due to its robust
operation and useful built-in predeﬁned program code, which allows the user to easily
import and export data from external programs. Objective functions and constraints were

evaluated in conjunction with the mathematical optimization algorithms within Matlab.

126

A ﬂow chart for the implemented optimization approach is shown in Figure 3-5.
Custom computer aided design algorithms were used as preprocessing to discretize the
beam line into ﬁnite elements. Each beam line was divided into ﬁnite elements of size
between 305 mm (1 ft) and 1525 mm (5 ft) to cover all critical locations, i.e., the
midspan of middle segment, the splice location, the midspan of the end segments.

Sensitivity analysis is used to provide gradient information on the objective and
constraint functions with respect to the design variables to determine the effect on the
objective function if the design variable values are changed. The sensitivity analyses
were carried out through the analytical approach in a custom routine implemented in
Matlab.

Structural analyses of the SGB structure were based on 2-D elastic analyses
including the effect of non-prismatic section properties along the span, time-dependent
material behavior, and staged construction. A custom program using the stifﬁiess method
of analysis was used. Each span segment was divided into multiple elements to obtain
stresses, internal forces and deﬂection results at several locations along the span. Section
properties of non-prismatic elements were taken as the average from those at the end
nodes for a given element. Time dependent losses were determined based on the
AASHTO-LRFD bridge design speciﬁcations (AASHTO 2003). Each construction
method was analyzed separately and the maximum achievable span length for each

construction method was compared to determine their efﬁciency.

In the mathematical programming, the initial values for the design variables used
were predeﬁned within upper and lower bound limits. The search direction and a step

size toward the optimal solution were determined using built-in Matlab functions and the

127

constrained steepest descent method. The system geometry was then updated and

provided as input for the next analysis until the objective function value converged to a

unique solution and the optimality conditions are satisﬁed.

 

C Start 3

 

 

Computer Aided Design (CAD)
MATLAB

 

 

 

 

I

I Sensitivity Analyses I

 

 

 

 

I Structural Analysis

 

 

 

Mathematical
Programming

l

 

 

 

 

 

Evaluate objective function

 

 

      

 

 

Update
design

Yes
Optimal
solution

End

 

 

 

 

 

I Time-dependent analysis I

 

 

Allowable stress desigﬂ

 

 

 

 

Ultimate strength design

 

 

¢——I Gradient based algorithm I

 

Figure 3-5 Flow chart of computer program for optimization of spliced

precast/prestressed girder bridge systems.

128

3.9 Summary

This chapter presented an overview of the theory and methods used in this work taken
for single-obj ective and multicriteria Optimization studies on spliced prestressed concrete
girder bridges. The objective functions and constraints of spliced girder bridges are
nonlinear functions of the design variables. Therefore, the constrained steepest descent
method using a gradient based algorithm was selected since it is robust and effective for
solving nonlinear constrained Optimization problems (Arora 1989). For the multicriteria
optimization, the constraint approach was used to generate a set of pareto optima with
non-dominated solutions and compromise programming was utilized for selecting the
best solution. These approaches were implemented in the Matlab environment by
combining time-dependent and construction staging frame analyses with built-in Matlab

optimization functions in a custom automated program.

129

4 Optimum System Designs of Single-span Spliced

Precast/Prestressed Bridges

4.1 Introduction

Single-Span Spliced Precast/Prestressed Girder Bridges (SSGB) are of interest to
bridge engineers because they can signiﬁcantly increase the spanning capabilities of
conventional prestressed concrete bridges. Consequently, this system is favored when
limitation on pier placement exists. Currently, only custom deep beam cross-sections are
used for SSGB since they can span much longer than typical standard beams. The
objective here is to explore the beneﬁts of using standard precast/prestressed concrete
girder sections in SSGB construction since their fabrication and erection cost can be
much lower than for custom girders.

In order to obtain efﬁcient optimal solutions of SSGB and to eliminate trial and
error design iterations, structural optimization can be utilized in the design and analysis
process. Single objective functions as well as multi-criteria Objective functions are used
to determine the optimal results of design requirements in the optimization process. The
single objective optimization functions used in this study were the minimization of
superstructure cost, the minimization of post-tensioning strands, and the minimization of
pre-tensioning strands. A multi-criteria optimization approach combines these single
objective ﬁinctions to obtain compromise solutions to all these individual design

objectives.

130

Results from optimization studies were used to develop design aids for SSGB.
These results were for both single and multiple Objective optimization problems. The
maximum achievable span length for different beam types at different beam spacing and
splice locations was obtained from the optimizied design. Three types of construction
sequence methods were explored to evaluate the inﬂuence of construction sequence on
the maximum achievable span length as well as on prestresing requirements.

Finally, the effect of clear cover on prestressing strands and on reinforcement in
the deck were investigated to asses the service life of SSGB. This additional design

requirement was then used to determine the life-cycle-cost of SSGB.

4.2 Problem Statement

A three-lane single-span spliced girder bridge with a total width of 18.59 m (61
ft), as shown in Figure 4-1, and a span length (L)consisting of three segments, as
illustrated in Figure 4-2, was used in this study. The end segments were assumed to be
equal. The splice location, which is the ratio of the end segment length (L 1) to the total
span length (L), was used to deﬁne the mid-segment and end-segment length. The cast-
in-place concrete slab thickness was assumed to be 216 mm (8.5”). The beam spacing
varied from 1.8 m (6 it) to 2.7 m (9 ﬁ). Bulb-tee beams, AASHTO beams, and box
beams were investigated. The material properties are given in more detail in the design
parameter session (Section 4.3).

The design study was conducted using the simpliﬁed design method of the
AASHTO-LRFD Bridge Design Speciﬁcations (AASHTO 2003) and the
recommendations from the NCHRP 12-57 study (Castrodale and White 2004). Service

limit states and ultimate limit states were considered in the design.

131

18.59 m (61’-0"l Brige morn

 

17. 68 m (58’ -O”) Clear Roadway

40.5 m
l1'-6"l

 

 

3.6ml12’ -o"i36m112’ —0”)|3.6m112' -O")I3.'6m(12 -0" 13.0111110’ —0”1

 

 

Shoulder 'I‘ Lane Lane '1‘ Lone
Br 0 9 Coast CL
21., l ' 9

I Shoulder

 

l

 

 

 

 

 

 

 

 

 

'I Equal Space Spliced Beams

II I I I I

l

 

 

Figure 4-1 Bridge cross-section of prototype bridge

 

 

 

 

 

 

 

Ref. Ref.
Abutment Total Span Length (Ll Abutment
End Span (Lg I Midspon (L5 .1. End Span le
Splicing Joint—.\>>\\ I l/r—-Deck

\\r,/
Shoulder Lanes ”90'0” Lanes Shoulder
L- ‘14 = = = : ilﬁ ri

 

 

I<———Splice Location 2 (LﬁxL
L

Figure 4-2 T hree-segment sin gle-span spliced precast/prestressed bridge

132

 

 

4.3 Design Parameters

Values that remain ﬁxed in the optimization process are considered design
parameters. These parameters have a signiﬁcant role in deﬁning the problem and they
consequently signiﬁcantly inﬂuence the resulting value for the design variables.

The main geometry design parameters include beam spacing (S); total span

length (L); splice location (pm); and the girder sectional properties (for AASHTO I-

beams, PCI Bulb-tee beams, and Box—beams). The sectional properties include Ag =

beam area, Sb = bottom sectional modulus, and S, = top sectional modulus. Beam

spacing was varied from S = 1.8 m (6 ft) to 2.7 m (9 it) with 305 mm (1 ﬁ) increments.
The splice location (pra) was deﬁned as the ratio of the end span length (L1) over the
total span length (L). Three splice locations were investigated, namely 0.20L, 0.25L, and
0.30L. Optimization algorithms were then used to obtain optimal SSGB solutions for
each span length. The objective functions used in the optimization of SSGB to determine
the optimal solution were minimum construction cost, minimum post-tensioning, and
minimum pre-tensioning, respectively. The maximum achievable span length (L) was
determined by increasing the total span length to the point where it yielded an infeasible
solution. The problem formulation for the optimization of SSGB is given in detail in
Section 4.5.

Concrete properties for the girders, deck and splice joints are summarized in
Table 4-1. The pre-tensioning and post-tensioning strands were taken to be 15.24 mm
(0.6 in.) diameter and to be made from low-relaxation steel, with properties as given in

the AASHTO LRF D speciﬁcations (AASHTO 2003).

133

Table 4-1 Assumed material properties

 

 

Girder Deck & Splice Pretensioning Post-Tensioning
Concrete Concrete Strands Strands
SI Units
12:44.8 MPa f'cd=3l.0 MPa Am: 140 mm2 Am: 140 mm2
fa- = 34.5 MPa fcdi = 24.1 MPa ﬁn, = 1862 MPa [Du = 1862 MPa
EC = 33.7 MPa Ecd = 28.0 MPa fpy = 1675 MPa j},y = 1675 MPa

Ea- = 29.6 MPa Ecdi = 24.7 MPa fpo = 1396 MPa fpj = 1508 MPa
5,, = 196502 MPa 18,, = 196502 MPa

US Units
f’c = 6500 psi fcd = 4500 psi A p, = 0.217 in.2 AP, = 0.217 in.2
f’ci = 5000 psi fcdi = 3500 psi jjm, = 270 ksi fpu = 270 ksi
EC = 4888 ksi E“, = 4067 ksi [0, = 243 ksi fpy = 243 ksi
Ea- = 4287 ksi E6613: 3587 ksi 1;,0 = 202.5 ksi fpj = 218.7 ksi

5,, = 28500 ksi Ep = 28500 ksi

 

In this study, typical standard girder shapes used in SSGB were investigated,
which included AASHTO I-Beams, Box-Beams, and PCI-beams. Only deep beam
shapes were investigated since shallow section would not beneﬁt from the effects of
post-tensioning. Thus, only AASHTO Types 111 through VI was investigated. The depth
of box beams depends on the fabricator’s production capability. In the state of Michigan,
the maximum box beam depth used for conventional precast/prestressed concrete bridge
girders is 1220 mm x 1370 mm (48” x 54”). Three different size of box beams, which
include 1067 mm x 1220 mm (42”x 48”), 1220 mm x 1220 mm (48”x 48”), and 1524
mm x 1370 mm (54”x 48”), were investigated in the study. Two PCI bulb-tee girders,

namely the PCI-BT 72 and PCI-BT 96, were also used in the study.

134

4.4 Analysis of Spliced Precast/Prestressed Girder Bridges

Analysis of the spliced girder bridge structure was based on a 2-D elastic frame
analysis including the effects of time-dependent material behavior and staged
construction. A custom frame analysis program using the stifﬁiess method was used.
Each span segment was divided into multiple elements to obtain stresses, internal forces
and deﬂection results at the member ends. The beam-line model was subjected to the
demands of an interior girder according to the load distribution factors given in the
AASHTO-LRFD Speciﬁcations (AASHTO 2003). Interior girders were used in the
analysis since their demands usually control the superstructure design. In addition,
usually all beams are designed as interior beams to allow for future bridge widening.

Stressing losses were computed using the simpliﬁed AASHTO-LRFD
(AASHTO 2003) method with consideration of the effects of combined pre-tensioning
and post-tensioning on the girder segments, as recommended by a recent NCHRP study
on spliced girder bridge design (Castrodale and White 2004). Design lifetime of the
spliced girder bridge system was taken to be 75 years after all prestress losses have
occurred. The section design was performed using a custom program coded in
MATLAB (Mathworks 2002), which enforced compliance with service limit states as
stated in Section 4.5.

The analysis program was veriﬁed by comparing its results to those from a
Single-Span SGB Design Example (Example 1) from the NCHRP 517 report (Castrodale
and White 2004). The design example uses the modiﬁed 2.4 meter (96 in.) PCI bulb-tee
with a beam spacing of 2.7 m (9 it) for a span length of 59.7 m (196 ft). Single-staged

post-tensioning applied after pouring the deck was used as the constructor process for

135

the study example. All moment demands at each stage and stresses at the top and bottom
of the beam, and at the top of the deck, for all critical construction stages and load
combinations were checked along the length of the span. The only difference between
the analysis from this study and the NCHRP 517 example was the post-tensioning
proﬁle. The NCHRP example assumes straight line segments that approximate a
parabolic curve, whereas the developed analysis tool for this study uses an actual
continuous parabolic curve (see Section 2.5.30). The ﬁnal output results were found to

be within 5% of the NCHRP 517 results.

4.5 Optimum Design of Single-Span Spliced Girder Bridges

Three different construction sequence methods were investigated in this study to
determine their effect on the design of single—span spliced girder bridges. The
optimization formulation of each construction sequence method is described in detail in

the following subsections.

4.5.1 Single-stage post-tensioning after pouring deck (Method 1)

In this method, the precast/prestressed beams are ﬁrst installed on temporary
supports at the site. Spliced joints and diaphragms are then poured and cured. The deck
is added once the spliced and diaphragms have developed enough strength to resist
partial load. After the deck develops enough concrete strength, full longitudinal post-
tensioning is applied to connect all segments. The temporary supports are removed once
the deck develops enough strength to achieve composite action with the beams. The
optimization problem for this construction stage can be formulated in the following

sections.

136

a) Design variables

The design variables for the optimization problem are those deﬁning the amount and
layout of the pre-tensioning and post-tensioning strands in the girders as described in
Figure 4-3. The design variables for the SSGB problem are illustrated in Figure 4-3 and

include:

nprh- = number of bottom pre-tensioning strands on end segments at 1”” row (e. g.,

nprn represents the number of bottom pre-tensioning strands on the end-

segment at the 1St bottom row of the girder);

nprZi = number of bottom pre-tensioning strands on mid-span segments at 1"” row;

nprt; = number of top pre-tensioning strands on end-segments;
nprt2 = number of top pre-tensioning strands on mid-segment;

npo = total number of post-tensioning strands on beam;

ya = vertical distance from the bottom extreme ﬁber to the net centroid of the

post-tensioned duct at the end-segments;

ybb = vertical distance from the bottom extreme ﬁber to the net centroid of the

post-tensioned duct at the mid-segment.

137

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

nprt1 nprt2
r 50 lel 2”)
l (2.0.
_ _ __________________________________ _ _________________ __ rSPAG
50 mml2”)
(Typ.)
v
I - " :
L 1- 4
n
_nprml Typ.) anrzzﬂyp.) L—npg'z‘lZ _.npr41(Typ.|)
n r
anrnﬂyp.) ‘L npr12( Typ. ) p L— nDr'31(Typ. 1
‘ L >
(a) Pre-tensionning requirement
Post-tensioned
/ strands per duct
[CO C.
___:-::::::::::::;:;:

 

 

 

 

 

 

 

 

 

Y
a post-tensioned
bb ,
I profile
L a

(b) Post-tensioning requirement

 

Figure 4-3. Variables for the design Optimization of spliced girder bridges

These, and the remaining design variables, are noted in Figure 4-3. Variables ya

and Ybb are not direct design variables (i.e., not part of the objective function); however,

they have an inﬂuence on other design variables and should be included as design

138

variables. For example, when distances ya and ybb change the required numbers of post-

tensioned strands and numbers of pre-tensioned strands will be different due to their
coupled relationships.

Each pre-tensioning strand was layer taken to be located 50 mm (2 in.) apart,
including the distance of the ﬁrst row of strands to the beam bottom/top. The post-
tensioning tendon was assumed to follow a quadratic parabolic proﬁle. Normally, post-
tensioning strands require several ducts. In order to avoid having several duct proﬁles as
design variables, an effective post-tensioning proﬁle was used to represent a group of
ducts. Thus, every post-tensioning duct in the spliced girder section was assumed to have
the same post-tensioning proﬁle. The distance between individual post-tensioning ducts
was assumed to be uniform at 152 mm (6 in.) apart. The lowest point of the post-

tensioning proﬁle on the bottom row at the middle section was one of the design

variables, ybb, and it was assumed to be at least 152 mm (6 in.) from the extreme bottom

ﬁber. The effective post-tensioning proﬁle location was determined by the summation of
the vertical distances measured from the extreme bottom ﬁber to the centroid of each

duct and divided by numbers of the ducts at that location.

b) Objective function
Typically, there can be several design solutions that satisfy all requirements of
safety and serviceability imposed by a design code and a number of speciﬁed merit
criteria. A common design criterion in bridge design is the minimization of
superstructure construction cost (Cohn and Lounis 1994; Lounis and Cohn 1996; Lounis

et. a1 1997). For this study, the objective function was the minimization of superstructure

139

construction cost, which is deﬁned in Equation (4-1). Details of each material

component objective function cost are speciﬁed in Table 4-2.

f(xi)=CPS+CPT+CCD+CTS+CB+CRa (4-1)
where CPS is deﬁned as the pre-tensioning cost; Cpr is deﬁned as the post-tensioning
cost; CCD is deﬁned as the concrete deck cost; C TS is deﬁned as the temporary support
cost; and C3 is deﬁned as the beam cost.

A recent NCHRP project report on spliced girder bridges (Castrodale and White
2004) collected cost data for a sampling of projects around the US. However, detailed
construction cost information for post-tensioned spliced girder bridges was difﬁcult to
obtain since most states only track cost data on a project basis. In addition, the research
team found great variations in costs between regions, methods of project delivery, and
local consulting practices. As a result, it was difﬁcult to assign meaningful cost
parameters that are generally relevant.

Since it was difﬁcult to determine realistic cost values to use in the proposed cost
minimization objective function, cost estimates were based on values used for
conventional prestressed girders with supplemental costs for temporary supports and
post-tensioning. While this type of incremental approach is rarely valid (Castrodale and
White 2004), it was considered adequate for the comparative study of standard beam
sections being dealt with in the present study. The assumed unit materials cost are

summarized in Table 4-3.

140

Table 4-2 Components of objective function

 

 

Material Component Cost
Pre-tensioning, CPS C p - N g 'l("Pri1 + nprt1)-L1 -2 + (npriz + nprt2)-
Post-tensioning, CPT CPO -Ng -np0 -(L1 - 2 + L2)
Beam, C3 Cb~Ng-(L1-2+L2)
Concrete, CCD CC -L - W-tS
Reinforcement, CR C, -W - (L1 -2 + L2) - WS,
Temp support, C 73 C ts . Ng .2

 

c) Design constraints

The optimization constraints for the SSGB optimization problem were the
service and strength limit states speciﬁed in the AASHTO-LRFD Bridge Design
Speciﬁcations (AASHTO 2003) and the recommendations from NCHRP Report 517
(Castrodale and White 2004). Constraints for the service limit state were deﬁned in
terms of ﬂexural stress limits on the girder and deck. For ﬂexure, service limit state
requirements govern the required prestressed force. Constraints for the strength limit

state included checks on moment and shear capacities. The ductility requirement was

satisﬁed by ensuring that the relative neutral axis depth at the ultimate limit state c/de

was less than 0.42 (AASHTO 2003).
Stresses from applied (external) loads were computed from the bending moment

demands for composite and non-composite section properties, as appropriate. Stresses

141

were evaluated separately for each critical construction stage. The stress limits for each
construction sequence method are different since they depend on the material properties
at that stage as well as the loading conditions (i.e., full service load, partial service load
without live load, and partial service load with live load). For example, when the pre-
tensioning strands are applied to the girder, the girder still has not developed enough
strength; therefore, the concrete’s initial compressive strength is used for their stress
limit. On the other hand, when the deck is applied to the bridge, the girder concrete has
already developed its ﬁnal compressive strength; therefore the ﬁnal concrete

compressive strength is used for the limit state.

Table 4-3 Assumed unit cost of materials

 

 

Component Item Unit Cost
SI Units
Cost of Concrete (CC) 176 $/m3
Cost of Pre-Tensioning (C p) 50 $/strand-m
Cost of Post-Tensioning (C p0) 100 $/strand-m
Cost of Temporary Support (C temp) 1000 S/beam
Cost of PCI-BT 96 (Cb) 900 $/m
Cost of I-Beam: Type IV (C(,) 420 $/m
Cost of Epoxy Reinforcement (C r) 2 $/kg
US Units
Cost of Concrete (Cc) 230 $/yd3
Cost of Pre-Tensioning (C p) 15 $/strand-ft
Cost of Post-Tensioning (CPO) 3O $/strand-ﬁ
Cost of Temporary Support (C temp) 1000 S/beam
Cost of PCI-BT 96 (Q) 300 $/ft
Cost of I-Beam: Type IV (Cb) 140 $/ft

 

142

Flexural stresses for each critical construction stage as listed in Table 4-4 were
checked against the allowable compression and tension stresses. The allowable concrete
stress limits used are those prescribed by AASHTO LRFD (AASHTO 2003), which is
summarized in Table 4-5. Three locations across the composite girder-deck section were
checked along the span. Stresses on the bottom and top ﬁber of the girder and the top
ﬁber of the deck were checked at each critical stage to guarantee that all stresses
satisﬁed the constraints, i.e., they should be within the prescribed limits as listed in
Table 4-5 .

Table 44 Construction sequence for single-staged post-tensioning applied after pouring
deck in spliced girder bridges

 

 

Construction Time . . . PFC" tenIiiIdrtiIing
Action Descnptlon tens1on1ng . .
Stages (days) Critical Stage (28:12:21

1 - Stress pre-tensioning strands

2 0 Cast girder segments

3 1 Release pre-tensioning strands X

4 50 Erect girder segments

5 60 Place deck and splice concrete X

6 75 Stress post-tensioning strands X X

7 100 Add barriers X

8 140 Apply live load

9 15000 Add future wearing surface

10 27500 After all prestress losses X X

 

There are seven critical stage constraints for each location of the composite section. The
formulation of the bottom girder stress constraints (Constraints Iaa to 7aa) for each of

the critical construction stages is outlined in the next sections. Constraint laa refers to

143

the stress at stage 1 at the bottom extreme ﬁber of the girder after pouring the deck. The
nomenclature of Constraint Iaa is deﬁned as follows: 1 deﬁnes stage 1, the ﬁrst a
deﬁnes the girder bottom extreme ﬁber, and the last a deﬁnes that the construction
method is that of post—tensioning after pouring the deck. Similar constraint formulations
were implemented for the girder top extreme ﬁber (Constraints Iba to 7ba) and the top
deck stresses (Constraints 1 ca to 70a). Thus, Constraint Iba refers to the girder stress at
stage 1 at the girder top extreme ﬁber after pouring the deck and Constraint 1 ca refers to
the stress at stage 1 at the top of the deck with post-tensioning after pouring the deck.

Table 4-5 Allowable concrete stress limits for single-staged post-tensioning applied
after pouring deck at each construction stage

 

 

Construction Stages Stress Type Girder concrete Deck concrete
1)Prestressing Strand Release Campression 0-6fci
Tension -1.4 MPa
(-200 psi)
2) Deck / Splice Concrete Compression 0.6/2
Tension -1.4 MPa
(-200 psi)
3) Post-Tensioning Compression 0'6f’C 0.61% d

Tension -0.0914\/f'c(MPa) -0.0914\//’cd (MP3)
-024th (ksi) 024712,, (ksi)

4) Final Condition (after Compl’ﬁssion 1 0.6ew/c 0-69wfcd
losses)
Compression 2 0_45f’c 045de
Compression 3 0,4f’c 04de
Tension 1 00724th (MPa)
-0197]; (ksi)

 

144

Constraint Iaa: Stress Check at Pre-tensioning Stage

This construction stage considers the stresses due to the girder weight and the pre-
tensioning force. At this stage only prestress losses due to relaxation at release and
elastic shortening are deducted from the initial prestress stress of pre-tensioned strands.

The girder bottom ﬁber compression stress constraint is then deﬁned by:

Mg + Apsfpel _ Apsfpelep

Sbg A Sbg

 

 

 

s 0.6fc'i. (4-2)

Ulaa = _
g

Constraint 2aa: Stress Check at Placement of Deck and Splice Stage

Constraint 2 applies to the concrete casting operation of the deck, deck haunch,
and splice regions. The deck, haunch and splice are poured at this stage. Since the deck
is still not hardened, non-composite sectional girder properties are used. Prestress losses
for this stage include those from elastic shortening, creep, shrinkage, and steel
relaxation. Girder stresses at the bottom ﬁber are thus given by:

M8 _Md _Mh _Mc +Apsfp62 _APSfpez 91’

Sb Sbg

 

 

 

 

 

 

02m, = - s 0.6fc'.. (4—3)

Constraint 3aa: Stress Check at Post- T ensioning Stage

This constraint applies to the longitudinal splicing of the pre-tensioned girders
through a single post-tensioning operation. As previously presented, an equivalent
tendon proﬁle with a parabolic proﬁle, obtained according to the uniform dead load, is
assumed. At this stage, the deck is taken to be working compositely with the beam.
Prestress losses at this stage include those from elastic shortening, creep, shrinkage, steel
relaxation, and additional elastic shortening due to post-tensioning. The bottom ﬁber

girder stress constraint is thus deﬁned by:

145

Mg _Md _Mh _M, +Apsfpe3 +Apsfpe3ep +

Sb Sbg Sbg Sbc Ag Sbg

 

 

 

 

U3aa :—
g

"poapsquf + ”poapsqufee

$0.6 '. 4-4
Ac Sbc fc ( )

 

Constraint 4aa: Compression Stress Check under Full Service Load

This constraint applies to the service condition where the composite girders carry
full live loads and superimposed dead loads in addition to the loads applied in previous
stages. Compression stress limits under full service load are checked at this constraint.
Prestress losses for this constraint include elastic shortening, creep, shrinkage, relaxation

of steel, and additional elastic shortening due to post-tensioning. This constraint is

 

 

 

 

 

deﬁnedby:
0'4 =_y_§_Md _Mh _Mr _Ms __lllJL+Apsfpe4 +Apsfpe4ep +
aa
Sbg Sbg Sbg Sbc Sbc Sbc Ag Sbg
n a f n a f e .
p0 1” p0ff+ 1’“ 1” poff e £0.6fc. (4-5)
Ac Sbc

Constraint 5aa: Tension Stress Check with Reduced Live Load
The applied loads and prestress losses for this constraint are similar to those of
Constraint 4aa with the exception of a 20 percent reduction in the live load. The

objective is to check the tension limit state of the girder at the ﬁnal stage as deﬁned by:

M8 _Md _Mh _Mr _Ms _08M11+AP5fP84 +AP5fPe4eP +

Sbg Sbg Sbg Sbc Sbc Sbc Ag Sbg

 

 

 

 

USaa 2“

146

"poapsfpoff + "poapsfpoff'ee

s 0.07244]; (MP3) or -0.19\//’c (ksi). (4-6)
Ac Sbc

 

Constraint 6aa: Compression Stress Check without Live Load

This constraint check considers all loads and prestress losses as in Constraint 4aa
with the exception of live load. The compression limit state under these conditions is
then checked according to:

Mg _Md _Mh __Mr _MS +Apsfpe4 +Apsfpe4ep +
Sb Sbg Sbg Sbc Sbc Ag Sbg

 

 

U6aa : _
g

 

 

"poapsfpoff + "poapsfpotfee

s 0.45 '.
Ac Sbc fc

(4-7)

Constraint 7aa: Stress Check at Compression for the Partial Service Load
This critical stress check considers all of the loads of Constraint 4aa reduced by
50 percent except the live load. All the prestress losses considered for Constraint 4aa

apply here also. The bottom ﬁber girder stress check is then deﬁned by:

 

 

 

 

 

 

 

0'7 _05 _Mg __Md _Mh _Mr _MS +Apsfpe4 +Apsfpe4ep
aa" -
Sbg Sbg Sbg Sbc Sbc Ag Sbg
n a n a e M .
0.5 W pSfp0ﬂ+ 1” ”MM/I e ___” 30.41;. (4-8)
Ac Sbc tc

d) Results and discussion

Use of single-span S688 is commonly limited to speciﬁc medium to long-span

bridge projects; therefore, custom girders with deep sections are typically used. The use

147

of standard prismatic precast/prestressed girder in 8688 would make the system more
economical and, perhaps, more widely used. The efﬁciency of typical standard precast
prestressed concrete beams were thus investigated in this study for SGBs applications.
These standard sections were AASHTO I-beam Types 111 through Type VI, Michigan’s
DOT MI-1800 I-beam, PCI’s PCI-BT 72 and PCI-BT 96 bulb-tee beams, and box beams
of dimensions: 1067mm x 1220mm (42” x 48”), 1220mm x 1220mm (48” x 48”), and
1370mm x 1220mm (54” x 48”). The efﬁciency of the standard beams was determined
by the achievable span length that satisﬁed all of the design constraints.

The maximum span length of conventional precast prestressed concrete girders is

usually determined based on the initial concrete compressive strength ( fc'i) and the
service concrete compressive strength ( fc' ). These same concrete compressive strengths

were chosen to be consistent with the analyses on SGBs. Maximum beam spacing was
varied from 1.8 meters (6 ft) to 2.7 meters (9 ft) with 305 mm (1 ft) increments. Single-
span conventional prestressed concrete beams were analyzed by increasing the span
length using the commercial program CONSPAN (LEAP 2005) until they became
infeasible while SGBs were analyzed by custom design code written in MATLAB
(Mathwork 2002).

Comparison of maximum achievable span lengths and the percentage of span increase
between conventional precast prestressed concrete girders and simply-supported SGBs
for each beam type are summarized in Table 4-6 and Table 4-7. These results show that
the splice location at 0.25(L) is the most efﬁcient for single-span spliced girder bridges

with equal end-segment lengths.

148

Based on the results, AASHTO Type III beams, which were the shallowest beam
in the study, yielded the maximum percentage increase in span length compared with its
maximum conventional simple span length. These results show that longitudinal splicing
of precast beams is not only efﬁcient for deep beam sections; standard prestressed beams
with short to moderate depths, such as the AASHTO Type 111, can also be longitudinally
spliced to signiﬁcantly extend their spanning capability.

Shallow girder sections have higher efﬁciency in percentage increase of the span
length than deeper girder sections. When the depth of the AASHTO I-beam is increased,
the percent increase of spliced girder span length is less than that of a shallower
AASHTO I-beam. For example, spliced AASHTO Type VI beams have a percentage
increase in span length that is smaller than for the AASHTO Type V, Type IV, and Type
III beams, sequentially.

When the beam spacing is large, the percentage increase in the span length is
higher than with tight beam spacing. This result indicates that longitudinal splicing is
more efﬁcient when large beam spacing is used. When the vertical underclearance is not
an issue, using deeper beam sections with larger beam spacing can reduce the
construction cost dramatically. Normally the cost of standard beam types, such as
AASHTO Types III and IV, is not much different compared to the cost reduction of
eliminating beams. This outcome concurs with the current practice of using spliced
girder bridges to extend span length as well as to reduce the numbers of beams (larger

beam spacing) in the bridge system.

149

Table 4-6 Maximum span length of SGBs and conventional box-beams

 

 

 

 

B Max. Span Length (111) Percent Increase
s :23: “mammal Spliced Girder (s [iced Girder)
p g Girder L1/L p
0.2 0.25 0.3 0.2 0.25 0.3
ELM
1067mm X 1220mm
2.1 m 30.2 39.3 41.8 41.5 30% 38% 37%
2.4 m 28.7 39.0 40.2 41.1 36% 40% 44%
2.7 m 28.0 38.1 39.0 39.6 36% 39% 41%
1220mm X 1220mm
2.1 m 34.4 46.0 47.9 48.2 34% 39% 40%
2.4 m 33.8 45.4 46.3 46.9 34% 37% 39%
2.7 m 32.9 44.2 45.7 45.1 34% 39% 37%
1524mm X 1220mm
2.1 m 41.1 50.6 52.4 52.1 23% 27% 27%
2.4 m 39.0 49.1 50.6 51.8 26% 30% 33%
2.7 m 38.4 48.8 49.1 51.2 27% 28% 33%
Beam Max. Span Length (ft) Percent Increase
. Conventional Spliced Girder . .
Spacing Girder Ll/L (Sphced Girder)
0.2 0.25 0.3 0.2 0.25 0.3
US Units
Box-Beam 42”x48”
7 ft 99 129 137 136 30% 38% 37%
8 ft 94 128 132 135 36% 40% 44%
9 ft 92 125 128 130 36% 39% 41%
Box-Beam 48”x48”
7 ft 113 151 157 158 34% 39% 40%
8 ft 1 11 149 152 154 34% 37% 39%
9 ft 108 145 150 148 34% 39% 37%
Box-Beam 54”x48”
7 ft 135 166 172 171 23% 27% 27%
8 ft 128 161 166 170 26% 30% 33%
9 ft 126 160 161 168 27% 28% 33%

 

150

Table 4-7 Maximum span length of SGBs and conventional I- and bulb-tee beams

 

 

(SI Units)
Max. Span Length (m) Percent Increase
Beam Spacing Conventional Spliced Girder . .
Girder L1 IL (Spliced Girder)
0.2 0.25 0.30 0.2 0.25 0.3
Matti—0223.8
AASHTO Type III
1.8 m 25.9 37.5 41.1 39.9 45% 59% 54%
2.1 m 24.1 36.0 39.3 38.4 49% 63% 59%
2.4 m 22.9 34.7 37.5 36.6 52% 64% 60%
AASHTO Type IV
2.1 m 30.5 42.4 45.7 45.1 39% 50% 48%
2.4 m 29.6 40.8 44.5 43.6 38% 51% 47%
2.7 m 29.0 39.9 43.0 42.1 38% 48% 45%
AASHTO Type V
1.8 m 39.3 52.1 53.6 54.3 33% 36% 38%
2.1 m 37.5 50.9 52.4 52.1 36% 40% 39%
2.4 m 35.7 48.8 50.6 50.0 37% 42% 40%
AASHTO Type VI
2.1 m 44.5 53.6 55.8 55.8 21% 25% 25%
2.4 m 41.8 52.4 55.5 54.9 26% 33% 31%
2.7 m 39.3 51.5 54.3 53.6 31% 38% 36%
MI-1800
2.1 m 42.1 54.3 57.9 57.0 29% 38% 36%
2.4 m 40.2 52.1 57.3 55.5 30% 42% 38%
2.7 m 37.5 51.5 54.9 54.3 37% 46% 45%
0.2 0.223 0.25 0.2 0.223 0.25
PCI-BT 96
2.7 m 44.5 61.0 63.1 64.6 37% 42% 45%
3.0 m 42.1 60.4 61.6 64.0 43% 46% 52%
3.3 m 41.5 59.1 61.0 61.9 43% 47% 49%

 

151

Table 4-7 Maximum span length of SGBs and conventional I- and bulb-tee beams
(US Units - Cont.)

 

Max. Span Length (ft) Percent Increase

 

Beam Spacing Conventional Spliced Girder . .
Girder L1 IL (Sphced Girder)
0.2 0.25 0.3 0.2 0.25 0.3
English Units
AASHTO Type III
6 ft 85 123 135 131 45% 59% 54%
7 ft 79 118 129 126 49% 63% 59%
8 ft 75 114 123 120 52% 64% 60%
AASHTO Type IV
7 ft 100 139 150 148 39% 50% 48%
8 ft 97 134 146 143 38% 51% 47%
9 ft 95 131 141 138 38% 48% 45%
AASHTO Type V
6 ft 129 171 176 178 33% 36% 38%
7 ft 123 167 172 171 36% 40% 39%
8 ft 117 160 166 164 37% 42% 40%
AASHTO Type V]
7 ft 146 176 183 183 21% 25% 25%
8 ft 137 172 182 180 26% 33% 31%
9 ft 129 169 178 176 31% 38% 36%
MI-1800
7 ft 138 178 190 187 29% 38% 36%
8 ft 132 171 188 182 30% 42% 38%
9 ft 123 169 180 178 37% 46% 45%
0.2 0.223 0.25 0.2 0.223 0.25
PCI-BT 96
9 ft 146 200 207 212 37% 42% 45%
10 ft 138 198 202 210 43% 46% 52%
11 ft 136 194 200 203 43% 47% 49%

 

152

4.5.2 Single-stage post-tensioning before pouring deck (Method 2)

In this method, the precast/prestressed beams are ﬁrst installed on temporary
supports at the site. Then a full longitudinal post-tensioning is applied to connect all
segments. Spliced joints and diaphragms are then poured and cured. The deck is added
once the splice and diaphragms have developed enough strength to resist partial loads.
Temporary supports are removed once the deck develops enough strength to achieve
composite action with the beams. The optimization problem of this construction stage is

formulated in the following sub-sections.

a) Design variables

Design variables for this construction sequence method are similar to single-
stage post-tensioning applied after pouring the deck (see Section 4.5.1a). The only

difference is that the post-tensioning operation is applied before the deck is cast.

b) Objective functions

The objective function in this construction sequence method is the same as for
the case of single-stage post-tensioning applied after pouring the deck, namely the

minimization of structural cost (see Section 4.5.1b).

c) Design constraints

Flexural stresses at each critical construction stage as listed in Table 4-8 were
checked against the allowable compression and tension stresses. The allowable concrete
stress limits used are those prescribed by the MSHTO-LRFD speciﬁcations (AASHTO
2003), which are summarized in Table 4-9. Three locations across the composite girder-

deck section were checked along the span. The stresses on the bottom and top ﬂexural

153

ﬁbers of the girder and the top ﬁber on the deck were checked at each critical stage to

guarantee that all stresses satisﬁed the constraints.

Table 4-8 Construction sequence for single-staged post-tensioning applied before
pouring deck in spliced girder bridges

 

 

Construction Time Action Description Pre- Post-
d tensioning tensioning
Stages ( ays) Critical Stage Critical Stage

1 - Stress pre-tensioning strands
2 0 Cast girder segments
3 1 Release pre-tensioning X

strands
4 50 Erect girder segments

75 Stress post-tensioning strands X X

6 90 Place deck and splice X X

concrete
7 120 Add barriers X

140 Apply live load
15000 Add future wearing surface
10 27500 After all prestress losses X X

\0

 

For this construction option there are eight critical stage constraints on each location of
the composite section. Only the formulation of the girder bottom stress constraints
(Constraints 1 ab to 7ab) for each critical construction stage of the service limit state are
outlined in the next sections. Constraint 1 ab is refers to the stress at stage 1 at the girder
bottom extreme ﬁber before pouring the deck. The nomenclature of Constraint Iab is
defined as follows: 1 refers to the stress at stage 1, the ﬁrst a is deﬁned as at the extreme
bottom ﬁber, and the second b is deﬁned as post-tensioning before pouring the deck.
Similar constraint formulations were implemented for the girder top ﬁber (Constraint

1bb-7bb) and the deck top stresses (Constraint ch-7cb). Constraint Ibb is the stress at

154

stage 1 at the girder top extreme ﬁber after pouring the deck. Constraint 1cb is the stress

at stage 1 at the top of the deck.

Table 4-9 Allowable concrete stress limits for single—staged post-tensioning applied
before pouring deck at each construction stage

 

 

Construction Stages Stress Type Girder concrete Deck concrete
1) Prestressmg Strand Release Compressron 06va,.
Tension -l .4 MP3
(-200 psi)
2) Post-Tensronmg Compressron 0.6/dc
Tension -1.4 MP3
(-200 psi)
3) Deck /Sp11ce Concrete Compressron 0-6fc
Tension -1.4 MP8
(-200 psi)
4) Live loads/superimposed loads Compression 0.6/'0 06de

Tension -0.0914\/f'c(MPa) -0.0914\/f’cd (MPa)
-024th (ksi) 0247f“, (ksi)

5) Final Condition (after losses) Compression l 0.60“; 0.619ch
Compression 2 (145/Io 045de
Compression 3 0,4fc 04de
Tension 1 0.07247]; (MPa)
-019th (ksi)

 

Constraint Iab: Stress Check at Pre-tensioning Stage

This construction stage considers the stresses due to girder weight and the pre-
tensioning force similar to the construction Method 1 (see Section 4.5.10). At this stage
only prestress losses due to relaxation at release and elastic shortening are deducted from
the initial prestress stress of the pre-tensioned strands. The girder bottom stress

constraint is then deﬁned as:

155

 

 

 

S 0-6féi - (4-9)

Constraint 2ab: Stress Check at Post- T ensioning Stage

Post-tensioning is applied prior to pouring the deck, thus only the non-composite
girder section is used to resist additional loads due to post-tensioning forces. Prestress
losses at this stage are those from elastic shortening, creep, shrinkage, and relaxation of

the steel. The girder bottom stresses are given by:

M Af, Af.e naf
8+ pSch+ pSp62p+p0pSpr+

 

 

 

02ab : "
Sbg Ag Sbg Ag
n a f e .
p0 1” pof e $0.6fc. (4-10)
Sbg

Constraint 3ab: Stress Check at Placement of Deck and Splice Stage

The deck, haunch and splice are poured at this stage after longitudinal post-
tensioning in the previous stage. Since the deck has still not hardened, non-composite
sectional girder properties are used to determine the stresses. Prestress losses for this
stage include those from elastic shortening, creep, shrinkage, and steel relaxation.

Bottom girder stresses are thus:

M8 __Md _Mh _Mc +Apsfpe3 _Apsfpe3ep +

 

 

 

 

 

03ab = —

 

 

Sbg Sbg Sbg Sbg Ag Sbg
n a n a e o
’00 p5fp0f+ ”0 pSfpof e £0.6fc. (411)
Ag Sbg

156

Constraint 4ab: Compression Stress Check under Full Service Load

Compression stress limits under full service load are checked under this
constraint. Only forces from the applied superimposed dead loads, loads from the
removal of temporary support, and live loads are resisted by the girder and deck
composite section. The previous forces are still resisted by the non-composite girder
section. Prestress losses for this constraint include elastic shortening, creep, shrinkage,
relaxation of the steel, and additional elastic shortening due to post-tensioning. This

stress constraint is thus deﬁned by:

Mg _Md _M}, _Mr _Ms _M” +Apsfpe4 +Apsfpe4ep +

 

 

 

 

 

 

04ab = _

 

 

sbg Sbg Sbg S be Sbc Sbc Ag Sbg
npoapsqur + "poapsquree S 0.6 ft, (4-12)
Ag Sbg C

Constraint 5ab: Tension Stress Check with Reduced Live Load
The applied loads and prestress losses for this constraint are similar to Constraint
4ab except the live load is reduced by 20 percent in order to check the tension limit state

of the girder at the ﬁnal stage, as deﬁned by:

M8_Md_Mh _Mr _Ms
56 Sb

M A A e
-O.8 [1 + psfpe4 + psfpe4 p +
Sbg Sbc Sbc Sbc Ag Sbg

 

 

 

 

05ab = -

g g

n a . n a 6’
po :2:pr + P0 1'9S~*fi"’--ff 8 20.0724fo (MP8) or 0197/6 (ksi). (4-13)
8 b8

 

 

157

Constraint 6ab: Compression Stress Check without Live Load
This constraint considers all loads and prestress losses as in Constraint 4ab with
the exception of the live load. The compression limit state under these conditions is then

checked according to:

[Mg—Md _Mh _Mr_MS _{PApsfpe4+Apsfpe4‘9p+

S Sbg Sbg Sbc Sbc Ag Sbg

 

 

 

 

U6ab = T
be

"poapsfpnff + "poapsquffee

A Sbg

 

 

s 0.45 f5. (4-14)
g

Constraint 7ab: Stress Check at Compression for the Partial Service Load
This critical stress check considers all of the loads of Constraint 4ab reduced by
50 percent except for the live load. All of the prestress losses considered for Constraint

4ab apply here also. The bottom girder stress check is then deﬁned by:

Mg_Md _Mh _Mr_Ms +Apsfpe4+Apsfpe4ep+

 

 

 

0701, = 0.5[-

 

 

bg Sbg Sbg Sbc Sbc Ag Sbg
"poapsfm/f "poapsfPO/fee M11 1 '
-- + ~ ]- — s 0.4fc. (4-15)
Ag Sbg to

(I) Results and discussion
A single-span spliced girder bridge with a single-staged post-tensioning applied
before pouring the deck was investigated in order to compare its maximum achievable

span length with the construction sequences in Method 1 and Method 3. The same

158

standard beam sections used for Method 1 (see Section 4.5.1d) were used for these
analyses.

The achievable span lengths when using this construction method are much
shorter than when construction Method 1 is used. Post-tensioning was applied to the non
composite section; therefore, there is little gain from post-tensioning besides making the
segments continuous. When the three segments become one continuous piece, their
ﬂexural demands (which are used to determine the ﬂexural stresses due to the deck load,
haunch load, and construction load), increase dramatically due to the increase in span
length.

According to the results, a single-span spliced girder bridge with a single-staged
post-tensioning applied before pouring the deck is not an efﬁcient construction sequence
method as it hardly increases the maximum achievable span length when compared with
conventional precast concrete girder use as simple spans. Thus, this construction

sequence method is not recommended.

4.5.3 Multi-staged post-tensioning (Method 3)

In this method, the precast/prestressed beams are ﬁrst installed on temporary
supports at the site. Then the ﬁrst stage of full longitudinal post-tensioning is applied to
connect all segments. Spliced joints and diaphragms are then poured and allowed to
cure. The deck is added once the spliced and diaphragms have developed enough
strength to resist partial loads. After the deck develops enough concrete strength, the
second stage of full longitudinal post-tensioning is applied to the continuous segment.
The temporary supports are then removed. The optimization problem for this

construction stage is formulated in the following.

159

a) Design variables

Design variables for this construction sequence method are again similar to
construction Method 1 (see Section 4.5.1a). The only different design variables for this
construction method are the number of post-tensioning strands, which are applied twice:

ﬁrst before pouring the deck and then after the deck is cast.

b) Objective functions

The objective function for this construction sequence method is similar to construction

Method 1, which is the minimization of structural cost (see Section 4.5.1b).

c) Design constraints

Stresses at each critical construction stage as listed in

Table 4—10, are checked against the allowable compression and tension stresses. The
allowable concrete stress limits used are those prescribed by the AASHTO-LRFD
speciﬁcations (AASHTO 2003), which are summarized in Table 4-11. There are 10
critical stage constraints on each location of the composite section. Only the formulation
of the girder bottom stress constraints (Constraints Iam-8am) for each critical
construction stage of service limit state are outlined in the next sections. Similar
constraint formulations were formulated and implemented for the stresses at the girder

top ﬁber (Constraints Ibm-8bmb) and deck top (Constraints [cm-8cm).

Constraint 1am: Stress Check at Pre-Tensioning Stage
Girder loads and pre-tensioning forces for this stage are applied at the same time

as for the construction Method 1. Therefore, their girder bottom stress constraint

160

equation, girder top stress equation, and deck top stress equation are equal to Constraint

Iaa, Constraint Iba, and Constraint 1 ca, respectively. This constraint expression is:

 

Mg + Apsfpel _ Apsfpelep

 

 

01am = — S 0.6fc',-. (4-16)
S bg A g S bg
Table 4-10 Construction sequence for multi-stagedpost-tensioning
. . P Post-
Constructron Time _ . ' re- . . Tensioning
Actron Descrrpt1on Tensronrng C 't' 1
Stages (days) Critical Stage r1 rca
Stage
1 - Stress pre-
tensioning strands
2 0 Cast girder
segments
3 1 Release pre- X
tensioning strands
4 50 Erect girder
segments
5 75 Stress 1St post- X X
tensioning strands
6 90 Place deck and X X
splice concrete
7 120 Stress 2"d post- X X
tensioning strands
130 Add barriers X
9 140 Apply live load
10 1500 Add future wearing
0 surface
1 1 2750 Condition after all X X
0 prestress losses

 

161

Constraint 2am: Stress Check at First Post-Tensioning Stage

Post-tensioning is applied prior to deck casting; thus, only the non-composite
girder section is used to resist the additional loads due to post-tensioning. Prestress
losses at this stage are those from elastic shortening, creep, shrinkage, and relaxation of

the steel. The bottom stresses are given by:

M Af.,2Af2e naf
g+ PSPC+P5P9 P+P0PSP0f+

 

 

02am I; _

 

Sbg A8 Sbg Ag
71 a f e v
p0 [’5 pof e $0.6fc. (4-17)
Sbg

Table 4-11 Allowable concrete stress limits for multi-staged post-tensioning at each
construction stage

 

 

Construction Stages Stress Type Girder concrete Deck concrete
1) Prestressing Strand Release Compression 0-6fci
Tension -1.4 MPa
(-200 psi)
2) First Post-Tensioning Compression 0.6/Io
Tension -1.4 MPa
(-200 psi)
3) Deck /Sp1ice Concrete Compression 0-6fc
Tension -1.4 MPa
(-200 psi)
4) Second Post-Tensioning Compression 06/} 06de

Tension -0.0914\lf‘c(MPa) 009147de (MPa)
-0247}; (ksi) 0.247de (ksi)

5) Final Condition (after Compression 1 0.6a..f'c 0150141!" cd
losses)
Compression 2 0.45100 0.45de
Compression 3 Wife 04de
Tension 1 -0-0724‘/fc (MP8)
-019fo (ksi)

 

162

Constraint 3am: Stress Check at Placement of Deck and Splice Stage
The deck, haunch and splice are poured at this stage after longitudinal post-
tensioning in the second stage. Prestress losses for this stage include those from elastic

shortening, creep, shrinkage, and steel relaxation. Bottom girder stresses are limited

 

 

 

 

 

 

 

 

given by:
a _ M8 Md Mh Mc + Apsfpe3 Apsfpe3ep +
am ‘ _ — - — “
Sbg Sbg Sbg Sbg Ag Sbg
n p01“ psf pof "polapsfpofee ,
+ ~ g 0.6fc. (4-18)
Ag Sbg

Constraint 4am: Stress Check at Second Post- T ensioning Stage

The second post-tensioning operation is conducted after casting of the deck. Thus,
the composite girder section is used to resist the additional loads from the post-
tensioning forces. Only the forces from the second post-tensioning event are resisted by
the composite section of girder and deck. The previous forces are still resisted by the
non-composite section. Prestress losses for this constraint include elastic shortening,
creep, shrinkage, relaxation of the steel, and additional elastic shortening due to post-

tensioning. The bottom girder compression stress constraint is then deﬁned by:

Mg _Md _Mh +Apsfpe4 +Apsfpe4ep +

S Sbg Sbg A Sbg

 

04am : ‘

be 8

”polapsf 17017 + "polapsfpctffee + ”p02apsfp0f +

 

 

Ag Sbg Ac
’1 a 6
P02 ”Sfpof 6 30.6111. (4-19)
Sbc

163

Constraint 5am: Compression Stress Check under Full Service Load

Compression stress limits under full service load are checked at this constraint.
Only forces from the applied superimposed dead loads, removal of temporary supports,
and live loads are resisted by the girder and deck composite section. Forces prior to these
are still resisted by the non-composite section. Prestress losses for this constraint include
elastic shortening, creep, shrinkage, steel relaxation, and additional elastic shortening

due to post-tensioning. The bottom girder stress constraint is then deﬁned as:

 

 

 

 

 

 

Mg __Md _Mh _Mr _MS _M” +Apsfpe4 +Apsfpe4ep +

Sbg Sbg Sbg Sbc Sbc Sbc Ag Sbg

05am : _

"pol“psfpar + "polapsquree + "pozapsfpor +

 

 

 

Ag Sbg Ac

’1 a 8

P02 PSfPOff e £0.6fc'. (4-20)
Sbc

Constraint 6am: Tension Stress Check with Reduced Live Load
The applied loads and prestress losses for this constraint are similar to Constraint
4am except that the live load is reduced by 20 percent in order to check the tension limit

state of the girder at the ﬁnal stage as deﬁned by:

164

Mg_Md_Mh_Mr_MS_08M[[+Apsfpe4+
S Sbg Sbg Sbc Sbc Sbc Ag

 

 

06am : "
be

Apsfpe4ep + "polapsfpoff + "polapsfpoffee +

Sbg A Sbg

 

g

 

npozaifpoff + npozagsf‘DOffee Z-OO724\/fc (MP8) 01' -019\lfc (kSl). (4-21)
bc

Constraint 7am: Compression Stress Check without Live Load
This constraint check considers all loads and prestress losses as in Constraint
4am with the exception of live load. The compression limit state under these conditions

is then checked according to:

Mg _Md _Mh _Mr _Ms +Apsfpe4 +Apsfpe4ep +
Sbg Sbg Sbg Sbc Sbc Ag Sbg

 

 

07am I; -

"polapsfpnff + "polapsfpnffee + "p02apsfpntf +

 

Ag Sbg Ac

n a e

p02 psfpoff e S 0.45fé (4_22)
Sbc

Constraint 8am: Stress Check at Compression for Partial Service Load

This critical stress check considers all of the loads of Constraint 4am reduced by
50 percent except for the live load. All the prestress losses considered for Constraint

4am apply here also. The girder bottom stress check is then deﬁned by:

165

Mg _Md _M}, _Mr _Ms +Apsfpe4 +Apsfpe4ep +

 

 

 

 

08am = 0-5[‘

 

 

 

Sbg Sbg Sbg Sbc Sbc Ag Sbg

"polapsfpoli +"polapsfpnffee +"p020psfp0ﬂ +
Ag Sbg Ac

n a e M

[’02 ”f 1’fo " ] - —” s 0.4 f5. (4—23)
Sbc Stc

(I) Results and discussion

A single-span spliced girder bridge with multi-staged post-tensioning applied
before deck casting was investigated to determine the efﬁciency of this construction
method. These analyses used standard AASHTO I-beams Type 111 through Type VI.

The results of maximum achievable span lengths for this construction method
were found to be between those achieved for the methods of single-stage post-tensioning
applied before deck (Method 1) and after deck (Method 2) casting. The reason for the
reduced span lengths, compared to construction Method 1, was that the ﬁrst post-
tensioning event was applied to the noncomposite section; thus little beneﬁt is obtained
from it at this stage besides making the segments continuous. The second post-
tensioning event helps increase the span length because it is applied after the section
becomes composite (girder/deck). The deck is cast after the ﬁrst post-tensioning event;
therefore, large moment and ﬂexural stress demands are to be anticipated. Although
additional compression stresses were gained through the second post-tensioning operator

to aid resisting more ﬂexural loads, the beam can not span as long as when built using

166

construction Method 1. This is due to stress constraints at the deck casting stage, which
control the design of the multi-post-tensioning construction sequence.

According to the results, construction of single-span spliced girder bridges with
multi-staged post-tensioning is recommended when future deck replacement is
anticipated. By using this method, the beam would not be overstressed once the existing
deck is removed. Additionally, this construction method can allowed for longer spans
than those achievable through a single—stage post-tensioning operation before deck
casting. The disadvantage of this construction method is the cost of the second post-
tensioning operation since it requires the contractor to revisit the site once the deck is
poured. Also, the backwall cannot be cast until the second post-tensioning operation is
ﬁnished; thus increasing construction time.

Based on the results for the three construction sequence methods considered in
this study, a single-span spliced girder bridge with a single-staged post-tensioning
applied after deck pouring is the most efﬁcient construction sequence method since this
method provides the longest maximum achievable span length over other construction
sequence methods. However, other construction sequence methods were investigated
since they may be considered during the design phrase. For example, in areas where
future deck replacement is anticipated, a multi-staged construction sequence method

may be the best suitable.

4.6 Multicriteria Optimum Design of Single-Span Spliced
Girder Bridges

A three-lane single—span PCI-BT 96 spliced girder bridge with single-stage post-

tensioned applied after deck casting was chosen herein as a prototype structure to

167

demonstrate how multicriteria optimization can be applied for a spliced girder bridge. A
SSGB with a total width of 18.5 m (61 ft) and maximum beam spacing of 2.7 m (9 it)
was used in this study. The span length is 56.4 m (185 ft) and the system is to use three
longitudinally spliced segments with equal length end-segments. The splice location,
which is a ratio of the end-segment length (L 1) to the total span length (L), was part of
the design variables to determine the most efﬁcient splice location. The girders are to use
high-strength concrete with a ﬁnal compressive strength of 62 MPa (9 ksi) and an initial

compressive strength of 34.5 MPa (5 ksi).

4.6.1 Multicriteria objective functions

The selection of the best bridge system may be based on one or more merit
criteria, thus the selection of an optimal bridge system may be modeled as a multicriteria
objective function. Multicriteria optimization techniques are approaches that couple the
processes of design and decision making. A multicriteria objective function is composed
of _a set of different single objective functions. F1(x) is an economics multicriteria
optimization which can be deﬁned by the following combination: a minimum
superstructure cost f,(x), a minimum number of pre-tensioned strands f2(x), and a
minimum number of post-tensioned strands f3(x). These objective functions can be
deﬁned as follows:

F1(x)m,-,, = Economics multicriteria

I—I [f1(x)min f?(x)minf3(x)minl (4‘24)
f1(x) = Min. superstructure cost
= CPS + CPT + CCD + C75 + C8 + CR. (4'25)

168

f2(x) = Min. number of pre-tensioning strands
= npr11t npr21+ npr31+ ninr41+ npr12+ npr22+ npr32+ npr42+ nprtt + nprt2 (4-26)

f3(x) = Min. number of post-tensioning strands = npo. (4-27)
Each single objective function was formulated separately to determine its optimal
design solution. Details of minimum superstructure cost for the objective function in

Equation (4-25) were presented earlier in Table 4-2.

4.6.2 Comparison results of single-objective optimizations

Optimal design variable solutions for each of the single objective function were
obtained and the results are summarized in Table 4-12. Detailed design variable
notations were speciﬁed in Section 4.5.18 and are not repeated here.

The most efﬁcient splice location (pro) for all of the single objective functions
yielded the same result: 0.25(L). As expected, the minimization of the number of post-
tensioning strands (function f3) yields the maximum number of pre-tensioning strands.
Conversely, the minimization of pre-tensioning strands (function 1}) yields the maximum
number of post-tensioning strands. Minimization of construction cost yielded the same
post-tensioning requirements as that from the minimization of post-tensioned strands
(thirty seven strands). However, it should be noted that this objective function (minimum

construction cost) required less number of pre-tensioning strands than found from the

minimization of pre-tensioning strands (function fe), and a greater number of pre-

tensioning strands than found when minimizing the number of post-tensioning strands

(function fe).

169

Table 4-12 Optimization results for each objective function

 

L = 56.4 m (185 ft)

 

Design Variables £10") f2(x) (300

Min Const , .

cost m np '17 min npo

npr11 8 2 7
npr21 O 0 6
npr31 0 0 5
npr41 0 O l
n10r12 8 10 8
npr22 8 0 8
npr32 8 0 8
npr42 2 O 2
npo 37 61 37
ya 406 mm (16”) 965 mm (38”) 965 mm (38”)
”PH 2 2 4
nptz 6 2 6
pra 0.25 0.25 0.25
Objective $6,181,501 16 37
nprii (total) 43 I6 57

Construction cost $6,181,501 88,00,1793 $6,568,841

 

The discussion of results above shows that the optimal solutions for the objective
functions considered can be very different, especially when they contradict each other.
Thus, multicriteria optimization should be used to obtain compromised solutions with

the same given parameters (span length, beam spacing, and splice locations).

4.6.3 Pareto optima results and discussion

When all objective criteria are not equally important, or when there exist

conﬂicting criteria, vector optimization is used to select the preferred solution in a more

170

rational way. Multicriteria optimization techniques are used to generate a set of pareto
optima that are non-dominant solutions.

After obtaining solutions of each single objective function the range of minimum
and maximum values were identiﬁed. The numbers of total pre-tensioning strands were
thus varied from 16 to 57 strands while the numbers of post-tensioning strands were
varied from 37 to 61, as shown in Table 4-12.

Pareto optima sets were generated by the constraint approach, which is based on
the adoption of one criterion as a scalar primary objective function and the
transformation of other objective criteria into constraints bounded by some
predetermined constants. These scalar constraints are altered from the range of minimum
to maximum values, and the subset of pareto optima is generated.

As previously stated, the multicriteria objective herein (Equation 4-24) is
composed of three single objective functions. The combinations of single objective
functions were selected as pairs to determine the pareto optima set. There were four pairs
of objective functions selected in this study. Each pair of objective functions had one
primary objective function and the other objective functions were set as constraints.

The constraints were divided into arbitrary intervals. As the interval range
decreases, the accuracy of results is closer to the real compromised results. Since it is
impossible to generate the entire pareto optima set, a subset of the pareto optima
represented by six solutions was generated. Therefore, the predetermined constant

constraints in this study were divided into ﬁve intervals.

171

1. The ﬁrst pair of objective functions was to set a minimum total number of
pre-tensioning strands as the main objective function while varying the
total number of post-tensioning strands as constraints.

2. The second pair of objective functions was such that the minimization of
the total number of post-tensioning strands was the main objective
function while the varying number of total pre-tensioning strands was a
constraint.

3. The third pair of objective functions was to set minimum superstructure
construction cost as the main objective function while the numbers of
total pre-tensioning strands were set as a constraint.

4. The fourth pair of objective functions was such that number of total pre-
tensioning strands was the main objective function to be minimized while
minimum superstructure construction cost was set as a constraint.

Subsets of pareto optima were thus generated for the different objective function
pairs. The selection of the best solution from these subsets was decided based on

compromising programming. The criterion used in compromising programming is the

minimization of the normalized deviation from the ideal solution fmin measured by the

family of distance metrics LP _The value of LP indicates how close the solution is to the

ideal solution. The satisfying solution can be determined by selecting the minimum

normalized deviation (LP) between the minimum euclidean distance (p = 2) and the

minimum largest deviation (p = 00). This approach is also called the “Minimax

Criterion” and is formulated by:

172

min Lp(x) = minxemzl fi(X)-minf,-(x)

l/p ilS £00 . 4-28
i=1 maxfi(x)—minfi(x) p] ( p ) ( )

 

The ﬁrst set of pareto results having the number of pre-tensioning strands as the
main objective function to be minimized and the number of post-tensioning strands as a

constraint are given in Table 4-13. The Euclidian distance and maximum distance of
these pareto sets are best at L3(x) = 0.50 and Loc(x) = 0.37, respectively, which are the
minimum numbers of those values. These values correspond to Solution # 3 (see bold
values in Table 4-13) where npo = 45 and nprl-l- = 30.

Results for the second pareto set having number of post-tensioning strands as an
objective and numbers of pre-tensioning strands as constraints are given in Table 4-14.

The Euclidian distance and maximum distance of these pareto sets are best at L2(x) =

0.51 and Loo(x) = 0.39, respectively (see bold values in Table 4~14). These values

correspond to Solution # 3 where npo = 45 and nprii= 32.

Table 4—13 Pareto set of nprii as objective and npo as constraint (case 1)

 

 

 

 

 

 

 

 

 

Constraints Objective

function
501““ f30‘) f20‘) Qﬂnigl lip—mink) L610‘) L200
min no min am: “I’m“ “am“ “33:38:? 8311?."
1 37 57 0.00 1.00 1.00 1.00
2 42 37 0.20 0.52 0.52 0.55
3 45 30 0.37 0.34 0.37 0.50
4 51 23 0.59 0.16 0.59 0.61
5 56 20 0.80 0.09 0.80 0.81
6 61 16 1.00 0.00 1.00 1.00

 

173

Table 4-14 Pareto set of npo as objective and nprh- as constraint (case 2)

 

 

 

 

 

 

Constraints Objective
function
Solution 00) £300 Lf_2_:IIi_n_f21 113m) La00 L200
mannprﬁ minnpo «2,.... “an“ “58:23:” 8531?."
1 16 61 0.00 1.00 1.00 1.00
2 24 50 0.20 0.55 0.55 0.58
3 32 45 0.39 0.32 0.39 0.51 l
4 40 39 0.59 0.07 0.59 0.59
5 48 37 0.78 0.00 0.78 0.78
6 57 37 1.00 0.00 1.00 1.00

 

Third pareto set results having minimum construction cost as an objective and
numbers of pre-tensioning strands as constraints are given in

Table 4-15. The Euclidian distance and the maximum distance of these pareto

sets are minimum at L2(x) = 0.57 and Loo(x) = 0.41, respectively (see bold values in

Table 4-15). These values correspond to Solution # 3 where minimum construction cost

= $6,932,341 and nprii = 27.

Table 4-15 Pareto set of min construction cost as objective and nprii as constraint(case 3)

 

 

 

 

 

 

Constraints 2:111:33:
Solution 00) f100 Lf_2_-minf_21 01mm L800 L200
m... “2:222
1 16 88,001,793 0.00 1.00 1.00 1.00
2 22 87,209,841 0.20 0.56 0.56 0.60
3 L 27 $6,920,341 I 0.40 0.41 0.41 0.57
4 32 $6,682,591 0.60 0.28 0.60 0.66
5 38 86,32,184] 0.80 0.08 0.80 0.80
6 43 86,181,501 1.00 0.00 1.00 1.00

 

174

Results from the fourth pareto set having the minimization of pre-tensioning
strands as an objective and minimum construction cost as a constraint are listed in Table

4-16. The Euclidian distance and maximum distance of these pareto sets are minimum at

L 2(x) = 0.54 and Loo(x) = 0.40, respectively (see bold numbers in Table 4-16). These

values correspond to Solution # 3 where minimum construction cost = $6,909,618 and

nprii= 26.

Table 4-16 Pareto set of min nprii as objective and min construction cost as constraint

 

 

 

 

 

 

(case 4)
Constraints Objective
function
501“” f1(8) f2(X) 011111111 Mg) LAX) L2(X)
Maximum Euclidian
' - .. df , df ,
min cost min npru 1 max 2 max distance distance
1 86,181,501 43 0.00 1.00 1.00 1.00
2 $6,545,560 34 0.20 0.67 0.67 0.70
3 l 86,909,618 26 0.40 0.36 0.40 0.54
4 87,273,676 21 0.60 0.17 0.60 0.62
5 87,637,735 20 0.80 0.13 0.80 0.81
6 88,001,793 16 1.00 0.00 1.00 1.00

 

Results for the second pareto optima subset represented by six solutions is
illustrated in Figure 4-4 while Figure 4-5 illustrates the results for the third pareto
optima set. Figure 4-4 shows the relationship of pre-tensioning strands and post-
tensioning requirements when one of them is the main objective function and the other is

transformed to be a predetermined constant constraint and vice versa.

175

 

- f‘c = 62 MPa, f'ci = 34.5 MPa, 3 = 2.7 m, PCI-8T 96
65 — f'c=9ksi,f'ci=5ksi,s=9ft

 

—O— Pareto - Min npo
+ Pareto - Min nprii

 

 

    
 

 

Bast compromise
solution

 

No. of post-tensioned strands
3
1

 

 

p--------------

 

30 I I T I I I I I F I f I I I I I I r I
10 15 20 25 30 35 40 45 50 55 60

No. of pre-tensioned strands

Figure 4-4. Pareto set of pre-tensioned strand and post-tensioned strands

The same approach was applied for Figure 4-5, which shows the relationship of
pre-tensioning strand requirements and superstructure construction cost. The best
compromise solution of these objective functions is indicated by the dashed line in these
ﬁgures and also in bold font in Table 4-13 to Table 4-16.

Once the best compromise solution is obtained, the design variables that
correspond to this can be determined directly from the optimization outputs. These

design variables are illustrated in Table 4-17. Optimal solutions for the multicriteria
optimization problem are as shown in Figure 4-4 (npo = 45 and nprii = 32). This optimal

solution also yields the minimum superstructure construction cost.

176

 

- f'c = 62 MPa, f'ci = 34.5 MPa, 8 = 2.7 m, PCI-BT 96
50 — - ' I - o -
3 f'c - 9 ksr, f'cl - 5 ksr, S - 9 ft . Pareto _ Min "pm

45 - + Pareto - Min cost

 

 

 

 

 
   
    

Best compromise

No. of pro-tensioned strands

 

 

 

_ solution
3° 1.---N.R’_il.=.?.7 ........
20 — 5
‘ Cost = $6,920,341 1
15 — I
10 I I I I I I I II I I I I I I I I I I I ‘r
6.5e+6 7.0e+6 7.5e+6 8.0e+6

Minimum construction cost($)

Figure 4-5. Pareto set of pre-tensioned strand and minimum superstructure construction
cost

It can be noted that the design variable values for the optimal solution are
between the minimum and maximum range obtained in Section 4.6.2. Our multicriteria
optimal solution through compromising programming is non-dominated with regard to

all criteria, which ensures the best compromise between all of them.

177

Table 4-17 Design variables of best solutions of each pareto set

 

 

 

1st Pair 2nd Pair 3rd Pair 4th Pair
Main objective Min nprii Min npo Min cost Min nprh-
Constant
Constraint npo np r '7 np r ii C0“
npr1 1 3 4 2 2
n17'21 0 0 0 0
npr31 O 0 0 0
npr41 0 0 0 0
npr1 2 8 8 7 7
nprzz 7 7 6 6
nprsz 5 5 5 5
npr42 1 l l l
npo 46 45 48 48
ya 406 mm (16”) 406 mm (16”) 406 mm (16”) 483 mm (16”)
J’bb 178 mm (7”) 178 mm (7”) 178 mm (7”) 178 mm (7”)
nptI 2 2 2 2
nptz 4 4 4 4
"I” ii 30 32 27 26
Min const cost $6,766,113 $6,699,649 $6,869,535 $6,909,622

 

4.7 Parametric Studies

4.7.1 Assumption and procedure for parametric studies

The goal of the study was to compare the effectiveness of several design options

for extending the span range of longitudinally spliced precast/prestressed concrete

girders. Parametric studies were done by considering both material and geometry of the

structure to determine the maximum achievable span length.

A single-span spliced precast/prestressed girder bridge comprised of three

segments, with single-stage post-tensioning applied after pouring the deck and a

178

composite deck was selected for the comparative design study. The PCI-BT 96 girder,
with a spacing of 2.7 m (9 ft) and with 3 splice location at 0.25 of the total span length,
was used in this study. The spliced location at 0.25 of the total span length was chosen
since it was shown to be the most efﬁcient splice location in the system optimization
studies (see Section 4.6.2).

The base design of this study represents a typical design with normal weight

concrete 2403 kg/m3 (150 lb/ft3) for both deck and girder. The concrete compressive
strength for the girder at the ﬁnal condition was taken as fc' = 44.8 MPa (6.5 ksi) and
that at the initial state taken as fci’ = 34.5 ksi (5 ksi). The concrete compressive strength

of deck at the ﬁnal age was fed, = 31 Mpa (4.5 ksi). Pre-tensioned strands and post-

tensioned strands are designed based on 15.24 mm (0.6 in.) diameter. A single-stage
post-tensioning construction method was used in this comparative study since it was
shown to yield a longer span than any other of the investigated construction sequences

(see Section 4.5.1).

4.7.2 Material parametric studies for PCI-BT 96 beams

Maximum spans were determined for the single-span spliced girder bridge using
the selected material design variations for a comparative design study. Calculations were
performed using a custom program written in the MATLAB environment and included
the minimization of superstructure cost as the main objective in the design optimization

process.

179

Four major design variations on the base design were considered for the PCI-BT
96 spliced girder system. The variations ranged from examining the effect of concrete
compressive strength to modifying the unit weight of concrete for the girder and deck.

Following are the alternative designs that were developed through this study.

a) High strength concrete girders

The ﬁnal high-strength concrete compressive strength (fc') of precast girder was

varied from 44.8 to 82.7 MPa (6.5 to 12 ksi) in increments of 3.5 MPa (0.5 ksi), while

the concrete compressive strength at release (fci') of the precast girder was increased by

0.8fc'. The concrete compressive strength of deck at the ﬁnal age was fed = 31 Mpa (4.5

ksi). Normal weight concrete for both girder and deck was assumed. Table 4-1 8 losts the

percentage increase of PCI-BT 96 span length when the strength of concrete is increased

in 3.5 MPa (0.5 ksi) increments.

b) Lightweight concrete girders
The effect of reduced weight of the concrete in the girders was investigated by

varying its unit weight from 1602 to 2402 Kg/m3 (100 to 150 lb/ft3) in increments of 160

Kg/m3 (10 lb/ftI). The strength of the concrete girder remained unchanged from the base

design. Thus, this option simply reduces the weight of the girders. Table 4-19 lists the

percentage increase for the PCI-BT 96 spliced span length when the unit weight of

concrete for the girders is decreased in 160 Kg/m3 (10 lb/ftI) increments.

180

Table 4-18 Eﬂect of girder concrete strength on spanning increase of PCI—B T 96

 

 

. . o
kg/m kg/m a a a m m
(P00 (900 (kSi) (160 (160 (ﬁ) (ﬁ)
1 fC Base 2402 2402 44.8 35.9 31 .0 64 0 0%
design
(150) (150) (6.5) (5.2) (4.5) (210) (0) 0%
2fc Increased 2402 2402 48.3 38.6 31.0 64.3 0.3 0.5%
f'c
(150) (150) (7.0) (5.6) (4.5) (211) 1 0.5%
3fc Increased 2402 2402 51.7 41.3 31.0 64.6 0.6 0.95 %
f'c
(150) (150) (7.5) (6) (4.5) (212) 2 0.95 %
4fc Increased 2402 2402 55.2 44.1 31.0 64.8 0.7 1.1%
f'c
(150) (150) (8.0) (6.4) (4.5) (212.5 2.5 1.1%
)
5fc Increased 2402 2402 58.6 49.6 31.0 64.9 0.9 1.4%
f’c
(150) (150) (8.5) (55.2) (4.5) (213) 3 1.4%
6fc Increased 2402 2402 62.1 55.2 31.0 65.1 1.0 1.7%
f’c
(150) (150) (9.0) (8) (4.5) (213.5 3.5 1.7%
)
7fC Increased 2402 2402 65.5 60.7 31 .0 65.2 1.2 1 .9%
f'c
(150) (150) (9.5) (8.8) (4.5) (214) 4 1.9%
8fc Increased 2402 2402 68.9 66.2 31.0 65.4 1.3 2.1%
f'c
(150) (150) (10.0) (9.6) (4.5) (215) 4.5 2.1%

 

181

Table 4-19 Effect of girder concrete weight on spanning increase of PCI-BT 96

 

 

Case Description we chI fie fci fc d Max. InSpan Percent
kg/mI kg/m MPa MPa MPa splan ofse increase
(p00 (p00 (ksi) (ksi) (ksi) (ft) (0)
lwc Base design 2402 2402 44.8 34.5 31.0 64 0 0%
(150) (150) (6.5) (5) (4.5) (210) (0) (0%)
2wc Reducedwc 2243 2402 44.8 34.5 31.0 65.2 1.2 2%
(140) (150) (6.5) (5) (4.5) (214) (4) (2%
3wc Reduced Wc 2082 2402 44.8 34.5 31.0 65.8 1.8 2.8%
(130) (150) (6.5) (5) (4.5) (216) (6) (2.8%
4wc Reducedwc 1922 2402 44.8 34.5 31.0 66.8 2.7 4.2%
(120) (150) (6.5) (5) (4.5) (219) (9) (4.2%)
5wc Reduced Wc 1762 2402 44.8 34.5 31.0 67.4 3.3 5.2%
(110) (150) (6.5) (5) (4.5) (221) (11) (5.2%)
6wc Reduced Wc 1602 2402 44.8 34.5 31.0 68.3 4.2 6.7%
(100) (150) (6.5) (5) (4.5) (224) (14) (6.7%)

 

c) Lightweight concrete deck

The effect of reduced concrete deck weight was investigated by varying the concrete

unit weight from 1602 to 2402 Kg/mI (100 to 150 1b/ft3) in increments of 160 Kg/m3 (10

lb/ftI). The concrete strength for the deck remained unchanged from the base design.

Thus, this Option only reduces the weight of the deck placed on the bridge. Table 4-20

gives the percentage increase of PCI-BT 96 span length when the unit weight of concrete

in the deck is decreased in 160 Kg/m3 (10 lb/ft3) increments.

182

Table 4-20 Effect of lightweight concrete deck on spanning increase of PCI-B T 96

 

 

bridge
.1"; g;- ,qu 2222 22222. 22:22::
kg/m kg/m 3 a a m m
(PC0 (1300 (ksi) (ksi) (ksi) (ft) (ﬂ)
lwcd Base 2402 2402 44.8 34.5 31.0 64 0 0%
design
(150) (150) (6.5) (5) (4.5) (210) (0) (0%)
2wcd Decreased 2402 2243 44.8 34.5 31.0 64.9 0.9 1.4%
ch
(150) (140) (6.5) (5) (4.5) (213) (3) (1.4%)
3wcd Decreased 2402 2082 44.8 34.5 31.0 65.5 1.5 2.4%
ch
(150) (130) (6.5) (5) (4.5) (215) (5) (2.4%)
4wc Decreased 2402 1922 44.8 34.5 31.0 65.9 1.8 2.9%
ch
(150) (120) (6.5) (5) (4.5) (216) (6) (2.9%)
5wcd Decreased 2402 1762 44.8 34.5 31.0 66.8 2.7 4.3%
Wed
(150) (110) (6.5) (5) (4.5) (219) (9) (4.3%)
6wcd Decreased 2402 1602 44.8 34.5 31.0 67.1 3 .0 4.8%
ch
(150) (100) (6.5) (5) (4.5) (220) (10) (4.8%)

 

d) Lightweight concrete beam and deck

The effect of reduced weight in both girders and deck was investigated by

considering concrete with unit weights varying from 1602 to 2402 Kg/m3 (100 to 150

lb/ftI). The strength of the concrete for both girder and deck remained unchanged from

the base design. Therefore, this option only reduces the weight of the girder and deck

placed on the bridge. Table 4-21 gives the percentage increase of PCI-BT 96 span length

when the unit weight of the deck concrete (Wch) and the unit weight of the girder

concrete (WC) is decreased in 160 Kg/m3 (10 lb/ftI) increments.

183

Table 4-21 Span increase of lightweight concrete girder PCI-BT 96 and lightweight

 

 

deck
We. ,1; 22" g}; 22:: .2522. 2:22:22
kg/m kg/m 3 a a m m
(P00 (900 (ksi) (ksi) (ksi) (ﬁ) (ﬁ)
1w Base design 2402 2402 44.8 34.5 31.0 64.0 0 0%
(150) (150) (6.5) (5) (4.5) (210) (0) 0%
2w Decreased 2243 2243 44.8 34.5 31.0 65 .5 1.5 2.4%
We & ch
(140) (140) (6.5) (5) (4.5) (215) (5) 2.4%
3w Decreased 2082 2082 44.8 34.5 31.0 67.0 3 4.8 %
wc & ch
(130) (130) (6.5) (5) (4.5) (220) (10) 4.8 %
4w Decreased 1922 1922 44.8 34.5 31.0 68.3 4.3 6.7%
we & ch
(120) (120) (6.5) (5) (4.5) (224) (14) 6.7%
5w Decreased 1762 1762 44.8 34.5 31.0 70.1 6.1 9.5%
we & ch
(110) (110) (6.5) (5) (4.5) (230) (20) 9.5%
6w Decreased 1602 1602 44.8 34.5 31.0 71.3 7.3 11.4%
We & Wed
(100) (100) (6.5) (5) (4.5) (234) (24) 11.4%

 

e) Discussion of material parametric studies

The design results for all cases are listed in Table 4-18 to Table 4-21, which include
maximum span lengths, span length increase, and the percentage increase in maximum
span compared to the base design. The effect of girder concrete compressive strength
with achievable span length is illustrated in Figure 4-6. The base design for the PCI-BT
96 girder yielded a maximum span of 64 m (210 it), see Table 4-20 at the base design
row. Increasing the concrete strength of the PCI BT-96 girders, yields a 1.5 percentage
increase in span length, which is a minimal increase compared with the cost increase
associated with a higher strength concrete. The high-strength concrete girder has

increased the span length from 64 m (210 ft) to 65.4 m (214.5 ft). Since the design of

184

spliced girder bridge is normally not governed by stresses at the release stage, increasing
the girder concrete compressive strength would not be effective.

Span Length (ft)
6 7 8 9 10

I l I i I I I L i l I l l i I I l I

 

 

 

Q 75 - i l I I I i I '- s
E g s=2.7 m,PCI-BT96 L Q
" 70-5 wc=wcd=2402 Kg/m3 e~10 5
-= - r o:
'51 r n c
g 651 : 2
I3 60I “II 5
— _ o
.‘z’ i : '5
3 55‘: :‘8 g
2 1 ~ ..
3 r E
E 50- . o
o I —7 o
0 q I I.
h 45 I 0’
o _
IE I 5 g
33 4o . 0

 

 

l ' l ' l ' l ' l ' l ' l I
64.0 64.2 64.4 64.6 64.8 65.0 65.2 65.4

Span Length (m)

Figure 4-6 Effect of girder concrete compressive strength on span length

The effect of lightweight girder concrete on span length is illustrated in Figure 4-7.
The use of lightweight concrete on the girders is shown to increase the system span

length from 64 m (210 ft) to 68.3 m (224 ft) and provided an increase of 6.7 percent in

span length when the girder unit weight was reduced by 801 kg/m3 (50 lb/ft3).

185

Girder Unit Weight (lblft3)

100 110 120 130 140 150
70 I I J Li I 1 1 1 i l l I l i I I I I i I J 1 l 230

wcd = 2402 Kglm3, s = 2.7 m, PCI-8T 96
69 _. fc = 44.8 MPa, fci = 34.5 MPa, fcd = 31 .0 MPa

 

1

N
N
01

Span Length (111)
Span Length (ft)

N
4
0|

irrrlrrrrjrlrrjrrrr
N
N
O

 

 

 

 

64 T I I I I I I I I I ﬁ r I I I 210
1600 1700 1800 1900 2000 2100 2200 2300 2400

Girder unit weight (kg/m3)
Figure 4-7 Effect of girder unit weight on span length

Figure 4-8 illustrates the effect of utilizing lightweight concrete for the deck on the
system span length. The lightweight concrete deck is shown to increase the span length

from 64 m (210 it) to 67.1 m (220 ft) and provides an increase of 4.8 percent in span

length when the deck unit weight was reduced by 801 kg/m3 (50 lb/ftI). Although this

option may not increase the span as long as when using lightweight concrete for the
girders, the use of lightweight concrete in the deck generates less design issues of
concern than lightweight concrete in the girders. When lightweight concrete is used in
prestressed girders, different design issues need to be carefully considered, especially for
the prestress losses. Thus, the use of lightweight concrete on the deck only may be a

good option for increasing span length.

186

Deck Unit Weight (lblft3)

100 110 120 130 140 150
70 I I I l i 14 I I i I I L I i I I 4 L L I I I I 230

- wc = 2402 Kglm3, S = 2.7 m, PCI-BT 96
69 ‘ fc = 44.8 MPa, fci = 34.5 MPa, fcd = 31.0 MPa

 

 

 

 

 

.. - 5225..
s sa— — E
1.0
5 n I '5,
o: n c
g ""220 w
.1 : -'
= 5
'° ” o.
o. .- to
(D ~215
l-
“ I I I I I I I I I I I I I I I 210

1600 1700 1800 1900 2000 2100 2200 2300 2400

Deck unit weight (kg/m3)

Figure 4-8 Effect of deck unit weight on span length

The maximum span length increase was obtained when using a combination of
lightweight concrete for both girders and deck. Figure 4-9 shows the effect of utilizing
lightweight concrete for the girders and the deck and its inﬂuence on span length. The
use of lightweight concrete is shown to increase span length from 64 m (210 ii) to 71.3

m (234 ft) and thus provide an increase of 11.4 percent in span length when both the

girder and deck concrete unit weight was reduced by 801 kg/m3 (50 lb/ftI).

187

Girder and Deck Unit Weight (Iblft‘I')

 

 
  

 

   

 

100 110 120 130 140 150
74 I l l I i l l L #1 I l I l l I l I I i l I l I
_ s = 2.7 m, PCI-BT 96 E— 240
_ fc = 44.8 MPa, fci = 34.5 MPa, fcd = 31.0 MPa 3
72 _
5 235
’s‘ ‘ : s
5 70 — _— 230 5
o: : g
C - - 0
3 ;— 225 .1
c 68 I_l L- c
‘ N
8. . ’— 220 n
(I) E a)
66 — _
:‘215
64 I I ﬂ I I I I I I I I I I I I 210

 

1600 1700 1800 1900 2000 2100 2200 2300 2400
Girder and Deck Unit Weight (kg/m3)

Figure 4-9 Effect of girder and deck unit weight on Span length

The utilization of lightweight concrete for the beam, deck, or both provides an

increase of maximum span because of the 160-801 kg/m3 (10-50 lb/ft3) reduction in

concrete unit weight. The combination of options for increasing span ranges may be
signiﬁcantly different from the effect of individual options.

Analyses results were compared to give an indication of the effectiveness of each
design option for increasing the maximum span. Results from this study should be
considered as an indication of trends. In some cases, where girder section type, girder
spacing, splice locations, and other conditions are changed, the effect of different
alternatives may vary. Thus, the results from these comparisons must be considered as

speciﬁc cases.

188

4.7.3 Parametric studies on girder types

Different girder types with equal or nearly equal heights were considered in this
study to evaluate their maximum achievable span lengths. The girders under
consideration were the MI-1800 I-beam (Michigan DOT), AASHTO’s Type VI beam,
and PCI’s BT-72 beam. The maximum span length for each girder type with a beam
spacing of 2.4 m (8 ft) was obtained for the base design (i.e. unspliced simple span) and
four design variations. Non-composite section properties and general girder dimensions

are given in Table 4-22.

Table 4-22 Non-composite section properties of 1.8 m (6 ft) high girders

 

Girder H A I

ng
2 4

Type m m m m

tw Wb Wt

IT] If] m

(in.) (in.) (in?) (in.4) (in.) (in.) (in.)

 

PCI-72 1.829 0.930 0.495 227 0.152 0.660 1.067
(72) (36.60) (767) (545,857) (6) (26.00) (42.00)
MI-1800 1.829 0.844 0.565 260 0.152 0.902 1.200
(72) (33.23) (875) (624,700) (6) (35.50) (47.25)
AASHTO 1.829 0.924 0.700 305 0.203 0.711 1.067
TypeVI (72) (36.38) (1085 (733,319) (8) (28.00) (42.00)

 

The base design is based on normal weight concrete 801 kg/m3 (150 lb/ftI) in

both the girders and in the deck. The girder concrete compressive strength at release is

taken as 34.5 MPa (5 ksi) and the girder concrete compressive strength at ﬁnal

189

conditions is taken as 44.8 MPa (6.5 ksi). The parameters changed in the four design

variations were:

a) High strength concrete girder fc' = 68.9 MPa (10 ksi) and f' ct = 55.2 MPa (8 ksi),
b) Lightweight concrete deck at 1922 kg/m3 (120 lb/ftI),
c) Lightweight concrete girders at 1922 kg/m3 (120 lb/ftI),

d) Lightweight concrete girders and deck at 1922 kg/rn3 (120 111/03 ).

The increase in maximum span length for the four design variations over the base
design for each girder type indicates a different effect for the different girder types. The
MI—1800 girder is the most efﬁcient section compared with other beam types of equal
depth. Even though the MI-1800 I-beam cross-section area is smaller than that of the
AASHTO Type IV, it was shown to span longer for all design cases. This is due to its
large top and bottom ﬂange width, which allows it to contain more pre-tensioned strands
on each row and thus have increase efﬁciency over the AASHTO Type IV and PCI-BT
72 girders.

Comparison of base designs for different girder types indicates that the MI-1800
I-beams and the AASTHO Type VI girder achieve the largest and equal maximum span
length of 54.9 m (180 feet), while PCI-72 achieves a smaller maximum span of 50.9 m
(167 feet). These base maximum span lengths are used as comparison with the
maximum span length of design variation cases in Section 4.7.2.

The lightweight concrete MI-1800 girder with a lightweight concrete deck had
the greatest maximum span length for all design variation cases. This design increased

the achievable span length from 54.9 m (180 feet) of the base design to 60.7 m (199

190

feet), which is approximately an 11% increase. Use of lightweight concrete for both
girders and deck is the most efﬁcient way to increase span length for all design Options,
leading to improvements in maximum span length in the range of 7-11%.

Using either lightweight concrete girders or a lightweight concrete deck gives the
same percentage increase of maximum span length for the PCI BT-72 and The MI-1800

girders. However, use of lightweight concrete has a lesser effect on the maximum span

length of the AASHTO Type VI beams.

4.8 Life-cycle Analysis and Optimum Design of Single-span
SGBs

The aim of this section is to include the durability considerations, as it relates to
life-cycle cost, in the optimal design of longitudinally spliced girder bridges attacked by
chloride and describe the importance of durability considerations in extending their life
span from the design stage. Several methods to lengthen the service life of concrete
structures exist, which include using sealers, use of concrete admixtures, using coatings,
and increasing concrete clear cover. However, only parametric studies on the use of
increased concrete clear cover of the slab and precast/prestressed concrete girder were
considered since this approach must be deﬁned during the design process, unlike other

parameters that can be applied later once the structure is constructed.

4.8.1 Effect of concrete clear cover on chloride concentration

A single-span AASHTO Type III spliced girder bridge with an overall span of
36.3 m (119 ft) having a splice location at 0.25 of total span length was considered to

investigate the effect of concrete cover depth on the service life of the structure. The

191

bridge has a width of 18.6 m (61 ft) and a beam spacing of 2.1 m (7 ft). In this study, the
bridge was assumed to be located in a splash zone with a constant chloride concentration

(Anna et a1. 1993 and Stephen et al. 1998). The initial chloride concentration (C0) in

both cold and warm climates was assumed to be 19 kg/m3 (1.19 lb/ftI) (Stephen at al.

1998). This number was selected to represent a severe chloride environment (Stephen et
al. 1998). Therefore, the design service life of this structure may be very short compared
to a prestressed concrete bridge in a typical environment.

Chloride concentrations vary based on climate conditions (Dhir et al. 1991). The
diffusion coefﬁcient of concrete was assumed to be 0.0293 and 0.0728 for cold and
warm areas, respectively (Stephen et al. 1998). Chloride concentrations for different

concrete clear covers at time t were determined from Equation (4-29); where C0 is
chloride concentration at the surface; erf is the error function; x is the distance from the

concrete surface to the steel surface; Dc is the diffusion coefﬁcient, t is the considered

time. Chloride concentrations for different concrete clear covers for different climates
are shown in Figure 4—10 for warm weather and Figure 4-11 for cold weather.

" )
2,/Dct

The effect of chloride concentration on the service life of the structure in warm

C(x, t) = c, (1 - erf

 

(4-29)

and cold weather is shown in Figure 4-10 and Figure 4-11, respectively. These ﬁgures
contain the results of different clear cover values as well as the allowable threshold used
in the United States and Asian Regions (Stephen et al. 1998).

Service lifetime of a structure depends on the chloride threshold (Stephen et al.

1998). Normally, in cold areas of the United States, the chloride threshold is set at 0.7

192

kg/m3 (0.04 lb/ftI) while in warmer climates found in Asian countries the chloride

threshold is set at 1.2 kg/m3 (0.07 lb/ftI). The service lifetime of each climate structure

can thus be determined directly from Figure 4-10 and Figure 4-11. The service lifetimes

per concrete clear cover distance are listed in Table 4-23.

 

 

 

 

 

 

 

 

in?

g 12 'j + 50 mm Warm weather
x - + 60mm

I; — + 70mm

0 10 j + 80 mm

:3 -1 -O— 90mm

5 8 j -0'- USA Threshold

5 - Asian Threshold

2 I

o 6 —

o "l

.3 I

03 4 ‘:‘

E I

o 2 T c=12 4

c=07 ‘ )5/7 —o--o—-o—-o—-o-—o——o--o-o—o

 

 

O
_i
_
_
_.
.1
-—(
—I
u
—I
—l
—:
—(
—1

Service life of structure (years)

Figure 4-10 Effect of concrete cover on the service life of structures (Warm Area)

193

 

1

Cold weather

I

 

—O— 50 mm
—I- 60 mm
+ 70 mm
+ 80 mm
+ 90 mm
-0- - USA Threshold
Asian Threshold

0’
iIii

 

 

 

 

L

I

N
l

 

 

I

 

c 1.2
”w,
74 .44 4— _—-r g’
I I I I I I I I I
20 40 60 80
Service life of structure (years)

Chloride concentration (kg/m3)
h
1

 

O

 

Figure 4-11 Effect of concrete cover on the service life of structures (Cold Area)

Table 4-23 Service life time versus concrete cover in cold and warm areas

 

 

Desi Life rs
Concrete Cold Area gn (y )Warm Area
Cover Threshold Threshold
(mm) 3 3 3 3
0.7 kg/m 1.2 kg/m 0.7 kg/m 1.2 kg/m
(0.04 lb/ft3) (0.07 Ib/ftI) (0.04 lb/ft3) (0.07 lb/ft3)
50
(1.97”) 13 16 5 7
60
(2.36”) 19 23 8 10
70
(2.76”) 26 33 10 14
80
(3.15”) 40 48 16 19
90
(3.54”) 48 63 20 25

 

194

The service lifetime of a structure located in a warm area is shown to be less than
one-half that of a structure built in a cold area. In a low-temperature environment, the
chloride diffusion rate decreases, therefore the service life of a structure is increased. In
a warm environment, the chloride diffusion rate is increased, thereby decreasing the
structure’s service life. Temperature and surface characteristic normally yield
considerable different results in the service life of a structure.

From Table 4-23 it can be seen that the service life of the structure can been
increased by almost two and three times when the concrete cover is increased from 50

mm (2 in.) to 70 mm (2.76 in.) and 80 mm (3.15 in.), respectively.

4.8.2 Effect of concrete cover on initial cost

Conventional prestressed concrete beam designs were conducted to evaluate the
service limit state and the strength limit state of the girder with different concrete covers.
The initial construction cost of the girders and the deck were considered as the
superstructure cost. The initial construction cost is determined based on the strength and
serviceability design requirements. General bridge data used for the life-cycle-cost

analyses is listed in Table 4-24.

Normally, the initial construction cost of any type of prestressed concrete girder
is paid per meter, or foot, length and does not depend on the numbers of prestressing
strands in the section. Therefore, it is unclear as to how much the initial construction cost
would be increased by increasing the concrete cover depth. However, when the
requirement of prestressing strands is too high such that they can not be ﬁtted in the
beam, a deeper standard beam may be needed. In this latter case, the initial construction

cost would increase. Since precast/prestressed concrete beams are normally cast in

195

standard sections and paid for per linear length irrespective of the contentent, the initial
beam cost can be considered unique to each beam type. This differs greatly from the use
of reinforced concrete beams where the beam depth can be changed arbitrarily

depending on the design requirements.

Table 4-24 General bridge data information used in life-cycle-cost analyses

 

 

General Parameter Unit cost/Data
Bridge Width 18.6 m (61 ﬂ)
Bridge length 36.3 m (119 ft)
No. of Beams 8

Slab thickness w/o cover 152 mm (6 in.)
Approx. reinforcement 96 Kg/m3 (6 lb/ftI)
Lifetime estimate prior to replacement 90 Years

Superstructure concrete deck

Epoxy Steel Reinforcement

1,589 s/rn3 (45 8/ft3)

2.2 8/kg (1 Mb)

AASHTO Type 111 915mm (3 ft) 400 8/m (122 8/ft)
. 2 2
Deep Concrete Overlay 100mm (4 1n.) 183 8/m (l7 8/ft )
Shallow Concrete Overlay 38mm 2 2
(1.5 m.) 162 $/m (15 8/ft )
Pre-tensioned strand cost 53 8/m (16 8/ ft)
Post-tensioned strand cost 108 8/m (33 8/ft)

 

196

Parametric studies were thus conducted by varying the concrete clear cover on
standard precast/prestressed beams. From the analyses results in Section 4.8 (Table
4-25), AASHTO Type III beams can be used in single-span spliced systems for span
lengths up to 25900 mm (84.9 ft) with a beam spacing of 2100 mm (7 ft) and concrete
cover not exceeding 65 mm (2.5 in.). When the concrete cover exceeds 65 mm (2.5 in.),
a deeper beam, such as an AASHTO Type IV, needs to be used. Figure 4-12 illustrates
the effect of concrete cover on initial cost in cold weather climates. Table 4-25 illustrates
parametric studies of initial superstructure cost, maintenance cost, and life-cycle cost

versus concrete clear cover.

 

2.18e+6

I

AASHTO Type III, S = 2100 mm, pra = 0.25
2.16e+6 -—

  
    
  
     
   
 

—(

2.14e+6 —

.1

52.12936 —

'6 2.10e+6 4
8 _
_ 2.08e+6 —
N

d

E 2.06e-I-6 —

—1

2.04e+6 —

uni

2.02e+6 —

 

2.009+6 I r T l r r r l I r r l i r I

50 60 70 80 90
Clear cover (mm)

 

 

Figure 4-12 Effect of concrete clear cover on initial construction cost (Cold Area)

197

Table 4-25 Studies on effect of concrete cover on superstructure cost

 

Concrete Clear Cover

 

(in.) 1.97 2.36 2.76 3.15 3.54
Clear Cover (in.) 1.97 2.36 2.76 3.15 3.54
Clear Cover Difference

(in.) 2 0 0.39 0.79 1.18 1.57
Deck Area(ft ) 7259 7259 7259 7259 7259
Total Beam length (ft) 952 952 952 952 952
New slab thickness (in.) 8.0 8.4 8.8 9.1 9.5
De3ck concrete volume

(0 ) 4820 5058 5297 5535 5773
Reinforcement steel

(1b.) 28922 30351 31780 33208 34637
AASHTO Beam Type III III III H1 H1
No. of pre-tensioned

strands 27 26 26 27 26
No. of post-tensioned

strands 39 40 40 42 43
Initial Cost

Initial Deck Cost $245,834 $257,980 $270,126 $282,272 $294,418
Initial Beam Cost $1,755,908 $1,771,525 $1,771,525 $1,849,609 $1,865,225
Total Initial

superstructure cost $2,001,742 $2,029,505 $2,041,651 $2,131,881 $2,159,644
Maintenance and repair cost (Cold

Area)

Superstructure service

life (yrs) 16 23 33 48 63
No. of Deep Overlay 2 2 l 1

No. of Shallow Overlay 3 2 2 1 1
Deep Concrete Cost

Overlay per time $123,403 $123,403 $123,403 $123,403 $123,403
Shallow Concrete

Overlay Cost per time $108,885 $108,885 $108,885 $108,885 $108,885
Deep Concrete Overlay

Cost $246,806 $246,806 $123,403 $123,403 $0
Shallow Concrete

Overlay Cost $326,655 $217,770 $217,770 $108,885 $155,550
Total Maintenance and

Repair Cost $573,461 $464,576 $341,173 $232,288 $155,550
Life-Cycle-Cost $2,575,203 $2,494,081 $2,382,824 $2,364,169 $2,315,194
Percent increased

from the intial cost 29% 23 % 17% 1 l % 7%

 

198

4.8.3 Effect of concrete clear cover on life-cycle cost of single-span

spliced girder bridges

The life-cycle-cost per year of a bridge structure can be determined by dividing
the total life-cycle-cost by the design life-span (years). The life-cycle cost per year can
then be used to compare the design alternatives of structures with different concrete
cover depths. The life-cycle-cost also includes total maintenance cost during the service
life time of structure. The maintenance cost is estimated for the numbers of times that
need to have superstructure repair (i.e., shallow overlay, deep overlay, or deck
replacement). The cost of different repaired task can be estimated from the unit cost
database developed by state DOTS. Typically, maintenance cost is 0.5 to 1.2 times initial
cost and the removal cost is extremely great (Yoshiki et al. 2001). Figure 4-13 and
Figure 4-14 shows the relation of clear cover with maintenance and repair cost, and life-

cycle-cost, respectively.

 

7e+5
AASHTO Type III, S = 2100 mm, pra = 0.25

60+5 —

I

on

2

or
1

l

3e+5 -

Maintenance Cost (5)
i
1

2e+5 —

 

 

1e+5

 

 

50

Clear cover (mm)

Figure 4-13 Maintenance and repair cost versus concrete cover (Cold Area)

199

2.7e+6

 

 

 

 

AASHTO Type III, S = 2100 mm, pra = 0.25
2.6e+6 —
E
a
m _..
O 2.5e+6
0
I .1
2
‘5‘. 2.4e+6 —
9
g a
_l
2.3e+6 —<
2.29+6 I I I I I I I I I I I I I 7 I
50 60 70 80 90

Clear cover (mm)

Figure 4-14 Life-cycle-cost versus concrete clear cover (Cold Area)

4.8.4 Comparison of life-cycle cost with initial strength-based design
cost

Initial cost is governed by strength requirements, while life-cycle-cost
incorporates all costs that could occur during the lifetime of the structure. In the previous
study (Section 4.8.3), it was shown that for a structure in a cold area with 50 mm (2 in.)
clear cover the life-cycle-cost can be 143% of the initial cost. On the other hand, if the
clear cover is increased to 85 mm (3.35 in.) the service life of the structure will increase
to 80 years. In this case, no additional cost needs to be added to the life-cycle-cost value.
Figure 4-15 illustrates a comparison of initial, life-cycle, and maintenance cost as a

function of concrete cover depth.

200

 

AASHTO Type 3, S = 2100 mm, pra = 0.25

3e+6 3* ~I-\r
*7 2i

 

 

 

 

 

+
2e+6 k .— .
Cost ($) -
2e+6 -
_ '0' Initial cost
-I- Life-cycIe-cost
1e+6 - + Maintenance cost

 

 

 

 

 

 

50 60 7O 80 90
Clear cover (mm)

Figure 4-15 Comparison of initial, maintenance and life-cycle-cost versus concrete
clear cover

4.8.5 Discussion of life-cycle—cost of spliced girder bridges

The diffusion coefﬁcient for precast concrete components is normally much
smaller than in conventional concrete components due to the small water/cement ratio
and the high strength of the concrete used in precast/prestressed construction. Therefore,
deterioration of precast/prestressed concrete bridges is typically less than in other types
of concrete bridges. The increase in concrete cover depth is not always beneﬁcial in
terms of serviceability since there is a chance of developing shrinkage cracks. Therefore,
there must be a compromise between strength and serviceability requirements versus the

life-cycle-cost of the structure. For example, when the clear concrete cover in the beam

201

is increased, the eccentricity of the prestressing strands with respect to the centroid of
beam cross-section is reduced. Therefore, prestressed concrete beams would require
more prestressing strands to balance the superposed loads.

Experience to date has shown that when epoxy coated rebar is used in addition to
the added minimum concrete cover depth no major deterioration has been found.
However, minor problems due to poor workmanship have been observed. For example,
unintended reduction of cover depth usually causes the initiation of corrosion problems.
Thus, even bridges designed according to code requirements may have unintended
shallow cover depths. Quality control of cover depth is therefore very important.

While providing proper concrete cover can reduce the rate of corrosion of the
reinforcing steel, additional measures must be taken to ensure adequate and efﬁcient
protection in harsh environments. Such measures include the use of corrosion resistance
concrete (CRC), the use of epoxy-coated reinforcement (ECR), the use of low water to
cementitious material ratios, the use of surface coating, and the use of silica fume or ﬂy
ash to reduce the permeability of chloride ions in concrete.

Because the maintenance cost of existing structures is expected to continue
increasing in the future, reasonable and effective maintenance measures must be
developed. To achieve this goal, not only do technologies for the maintenance of
existing structures need to be improved, but it is also necessary for design methods of
newly constructed structures to take into account future maintenance as part of the life-

cycle-cost.

When the life-cycle cost is included during the design phase, the initial cost of a

new bridge could be higher than that of an existing bridge, but its maintenance cost

202

should be much less. This is due to the consideration of life-cycle-cost analysis during
the design stage. Life-cycle-cost analysis includes the maintenance cost, the initial

construction cost and the repair cost during service life the structure.

4.9 Effects of Erection Time Delay on Prestress Losses

The objective of this study was to investigate the variation of prestress losses due
to the delay of the spliced girder segments erection. This situation can happen when the
segments were fabricated and they sit in the yard for a long time prior to being
transported to the construction site. The delay in erection could affect the time-
dependent losses such as those due to creep and shrinkage.

A single-span spliced precast/prestressed girder bridge comprised of three
segments, with single-stage post-tensioning applied after deck casting and a composite
deck was selected for the comparative design study between the reference construction
sequence proposed and a simulated construction sequence with delay. The PCI-BT 96
girder, with a spacing of 2.7 m (9 ft) and a splice location at 0.25 of the total span length,
was used for the study.

Normally spliced segments are transported to the site and erected within 50 days
after the pre-tensioned strands are released. For a single-stage post-tensioning
construction method, the deck would thus be typically cast at day 60 and post-tensioning
would likely be applied at day 75. To simulate delays in construction the girder segments
were assumed to be transported to the site a month later than normal. The effect on
prestress losses at each critical construction stage for the reference and delayed
construction sequence are given in Table 4-26. The notations of each prestress loss

components were described in Chapter 2.

203

Table 4-26 Comparison of prestress losses at critical construction stage

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pre-tensioning strands (AprET)
Critical Components of Normal Delay
construction stage prestress loss Construction Construction
Release of pre- AfPES] +Apr/ 87.1 MPa 87.1 MPa
tensioned strands (12.63 ksi) (12. 63 ksi)
Placement of deck Afpgg + kCRIAprR + 209.9 MPa 223.6 MPa
and splice concrete kSHlAfPSR + Aprz (30.45 ksi) (32.43 ksi)
Stressing of post- AprS] + kaAprR + 281.7 MPa 292.5 MPa
tensioning tendons ksmAfPSR + .41pr2 (40.85 ksi) (42.43 ksi)

+AfPEsz

Final conditions Apr] + AprR + AfpsR 367.0 MPa 367.0 MPa
after losses +21me +Afp552 (53.23 ksi) (53.23 ksi)

Post-tensioning strands (A f")
Critical Components of Normal Delay
construction stage prestress loss Construction Construction
Stressing of post- Afpp + Apr+ Afpgs 8 7. 0 MPa 87.0 MPa
tensioning tendons (12.62 ksi) (12.62 ksi)
Final conditions 21pr + Apr+ Afpgs + 359.4 MPa 35 9.4 MPa
after losses AfPCR+ Afp5R+ AprZ (52.13 ksi) (52.13 ksi)

 

 

Comparison of prestress losses for different construction sequence times
indicates that the prestress losses at the ﬁnal stage are the same since the creep and
shrinkage prestress losses according to the simpliﬁed AASHTO method are calculated
based on the ﬁnal, service life time of the structure, which is assumed to be 75 years.

Only the time-dependent prestress losses during intermediate construction stages
between the release of pre-tensioning strands (stage 3) and the ﬁnal conditions stage

after all prestress losses (stage 10) are different. The creep and shrinkage prestress losses

204

at intermediate construction stages were calculated by multiplying the ﬁnal creep and
shrinkage prestress losses with corresponding factors give reference or refer to a section
in the dissertation where this is explained. These factors depend on the time at each
critical stage.

When construction is delayed only the prestress losses from the pre-tensioning
strands at the placement Of deck and splice (stage 5) and at post-tensioning (stage 6)
were different and found to increase by 13.8 MPa (2 ksi). The increase in the prestress
losses is considered very small and not to affect by the optimal solutions. This results
show that the Optimal solutions based on the reference construction time sequence
assumed in Table 4-4 can be used even when delays in girder placement could be

anticipated.

4.10 Summary

In this chapter, conﬁguration Optimization was applied to single-span spliced
girder bridges. A single Objective function, namely the minimization of construction
cost, was used to evaluate the efﬁciency of construction methods for this system. Single-
stage post-tensioning applied after deck casting was found to be the most efﬁcient
construction method since it yields longer spans than other construction methods.

The maximum achievable span length of longitudinally spliced standard
precast/prestressed concrete girders was determined based on the best construction
sequence method. Single-span spliced girder bridges lead to spanning increases Of 28-
60%, over conventional single-segment prestressed concrete bridge solutions. The

requirements of pre-tensioning and post-tensioning strands as well as their locations

205

along the bridge span length were Obtained from optimization studies and are provided
in the form of design aids in Chapter 6.

Material parametric studies were investigated to evaluate different Options to
increase the span length of single-span SGBs. The combination of lightweight concrete
for both girders and deck provide the greatest span length increase.

Results from multi-criteria Optimization yielded a compromise solution for
different single Objective functions and represents more rational solution than those
Obtained by optimizing individual Objective functions. Multicriteria optimization should
be used when decision making must be made for several merit criteria. The resulting
solution is expected to be non-dominant with regards to all criteria and ensures the best
compromise between all competing Objective functions.

Life-cycle-cost analyses were conducted to determine the increased initial
superstructure design cost when considering maintenance and repairs. The service
lifetime Of the structure was considered during the design process by varying the
concrete cover depth to increase the durability of girder components. Results showed
that with less than 25 mm (1 in.) increase in the clear cover of concrete, the service

lifetime of single-span spliced girder bridges can be increased more than twofold.

206

OPTIMAL DESIGN AND PERFORMANCE OF LONGITUDINALLY
SPLICED PRECAST/PRESTRESSED CONCRETE BRIDGES

VOLUME 11
By

Pimpida Surakomol

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Civil and Environmental Engineering

2005

5 Optimum System Design of Two-Span Continuous

Spliced Girder Bridges

5.1 Introduction

This chapter documents the Optimization design studies on continuous two-span
spliced precast/prestressed girder bridges (SGB). Single Objective and multi-criteria
Optimization were performed and compared to evaluate the inﬂuence of design
parameters on this continuous system. Results of optimal solutions were provided in
terms Of design aids. Example of design aid is provided in Chapter 6 and the complete
design aids of spliced precast/prestressed concrete girder bridges can be obtained from
the MSU-CE report (Surakomol and Burgueno 2005).

Currently, only custom designed pier girders are used in the deeper sections of pier
segments in continuous spans. Thus, the beneﬁts of using standard conventional
precast/prestressed concrete girder sections as well as newly optimized standard pier
sections for two-span continuous systems were studied.

Different construction methods for this system were explored to Obtain the most
efﬁcient construction method. The maximum achievable span lengths for different beam
types at different beam spacing and splice locations were obtained and the results were

compared.

207

5.2 Problem Statement

A two—span spliced girder bridge with a total width of 18.6 m (61 ft) and a span
length (L) consisting Of three segments was used in this study as a prototype continuous
bridge structure. The bridge cross-section is similar to the cross-section shown in Figure
4-1, and previously described in Section 4.2. The end segments were assumed to be
equal. The splice location, which is the ratio of the end (or ﬁeld/positive) segment length
(L1) to the total span length (L), was used to deﬁne the mid (or pier/negative) segment
and end-segment lengths. The pier segment can consist of a standard section (see Figure
5-1) or a non-prismatic section (see
Figure 5-2). The cast-in-place concrete slab thickness was assumed to be 213 mm (8.5
in.). The beam spacing was assumed to vary from 1.8 m (6 ft) to 2.7 m (9 ft). Standard
AASHTO beams were investigated and new Optimized non-prismatic beams were
developed.

The design studies were conducted using the simpliﬁed design method Of the
AASHTO—LRFD Bridge Design Speciﬁcations (AASHTO 2003) and the
recommendations from the NCHRP 517 study (Castrodale and White 2004). Service
limit states and ultimate limit states were considered in the design. A complete overview
of the design and analysis consideration for continuous spliced girder bridges has been

given in Chapter 2.

208

 

 

 

 

 

 

CL Abutment Total Span Length (Ll CL Abutment
A End Span (L1) J4 Midspon (L2) _L End Span (L1)
C P. l
Splice jointi L ller /—Deck
4" ) II Prestressed Beams
- v \ -l
\_ Positive segment or [:1 Prismatic
(Field segment) Negative segnent ‘
or Pier segment

 

 

 

 

 

 

 

l:_3 [‘__‘) L3

Figure 5-1 Two-span spliced precast/prestressed bridge with prismatic pier

 

 

 

segment
Ci. AbUImenI IOIOl Span LengIh (Ll CL AbUImenI
End Span (L1) ‘L Midspon (L21 _L End Span (L1)
. . . CL Pier I
SDllCG J01 my I Deck
/ l
I II ) "Prestressed Beams

 

 

   

 

 

Nonprismatic
Negative segnent
or Pier segment

   

Positive segment or
(Field segnent)

    

 

 

 

 

 

in

Figure 5-2 T wo-span spliced precast/prestressed bridge with non-prismatic pier
segment

209

5.3 Design Parameters

The main geometry design parameters for the prototype two-span SGB bridge
system are similar to those used for the single-span SGB study (see Section 4.2 and
Section 4.3). They are the beam spacing (S), the total span length of each span (L), the
splice location (pm) and the standard sectional properties Of the girders. Beam spacing
was varied from S = 1.8 m (6 ft) to 2.7 m (9 it) in 300 mm (1 ft) increments. The splice
location @ra) is deﬁned by the ratio of end segment (L1) over the entire bridge span
length L. Three splice locations were investigated, which were 0.35(L), 0.375(L), and
0.40(L). Concrete properties for the girders, deck and splice joints are summarized in
Table 5-1. The pre—tensioning and post-tensioning strands are assumed to be 15.24 mm

(0.6 in.) nominal diameter and made from low-relaxation steel.

5.4 Options to Negative Pier Segments in Continuous SGBs

Presently, precast/prestressed concrete girders are being used extensively in
continuous-span bridges in which the pier, or negative, segments are custom-designed to
carry the high negative-bending moment and high-shear demands. Consequently, deep
sections using haunches are typically used. The availability and use of standard negative
sections could reduce the cost of fabrication of pier segments; therefore, four different
Options to increase the section depth over the pier section were considered and studied.
Girder dimensions for the four Options considered are shown in Figure 5-3 through
Figure 5-5 and were part of the design variables in sectional optimization studies

detailed in Section 5.4.5.

210

Table 5-1 Assumed material properties

 

 

Girder Deck & Splice Pre-Tensioning TeriIinztriin
Concrete Concrete Strands g
Strands
SI units
2 2
=31,026 A =140 mm A = 140mm
f'c = 44,816 kN/m2 fed 2 1’5 1’5
kN/m
-= 24,132 = 1,861,585 =
f6,- =34,474 kN/mz fed‘ 2 9’“ fl“ 2
kN/m kN/m 1,861 ,585kN/m
E = 28,041 = 1,675,427 = 1,675,427
EC = 33,702 kN/m2 Cd 2 fpy 2 5’” 2
kN/m kN/m kN/m
E -= 24,732 = 1,396,189 -= 1,507,884
Ea- = 29,558 kN/m2 Cd’ 2 If)" 2 f” 2
kN/m kN/m kN/m

US units

f'c = 6,500 psi

f’c,- = 5,000 psi
EC = 4,888 ksi

EC,- = 4,287 ksi

fed = 4,500 psi
fag,- = 3,500 psi
Ecd = 4,067 ksi
Ecdi = 3,587 ksi

Ep = 196,500,660

kN/m2

A1,. = 0.217 in.2

f1,“ = 270 ksi
fay = 243 ksi

fp, = 202.5 ksi
E[) = 28,500 ksi

Ep =
196,500,660
kN/m2

Aps = 0.217 in.2

f1,“ = 270 ksi
fpy = 243 ksi

fp, = 218.7 ksi
E, = 28,500 ksi

The section Optimization processes were coupled with the conﬁguration Optimization
process to obtain Optimal sectional geometry solutions at the pier locations. Different
beam depths were considered to assess the sensitivity Of the structural system tO the pier
segment stifﬁiess. An overview on the Options to negative pier segments considered in

the study is given next.

211

5.4.1 Pier segment option 1: existing standard positive sections with
soﬁﬁt

The ﬁrst method explores the use of available existing standard positive sections
with an additional concrete block attached to the underside of the beam to make up the
negative pier segment, as shown in Figure 5-3. The sofﬁt tapers from a maximum on top
of the pier to zero depth at the splice location to match the height of the standard positive

segment.

 

C_\/__J

Existing
’//—IStandord
Positive
Section

 

 

 

 

 

L/ L
/M :: Iglbkhess \

Figure 5-3 Pier segment option 1: existing standard girder attached with soffit

 

 

 

 

5.4.2 Pier segment option 2: new optimal non-prismatic section with
variable web

A new optimal non-prismatic section was developed having a uniform ﬂange
thickness and a varied web height along the negative segment as shown in Figure 5-4.
The web height increases linearly from the prismatic positive section to a maximum web

height over the pier centerline. Due to the variable webs, this negative segment option is

212

lighter than solutions where the web is kept constant and the ﬂange thickness is
increased.

This new pier segment Option was studied for optimal design for beam spacing Of
1.8 m (6 ft), 2.1 m (7 ft) and 2.4 m (8 ft) and the span length was increased in increments
of 3.0 m (10 it), both given as parameters. The beam dimensions constituted the design
variables in the component Optimization and the upper and lower bounds to these
dimensions were deﬁned within practical ranges. The overall beam height over the pier
was set not to exceed 1.5 times the total beam height of the prismatic section. This
criterion was set based on current practice of custom designed pier segments. The
dimension limits for the web and ﬂanges were set to be in the range of available existing

standard beam dimensions.

 

'7TEII\\~ ///II
’//——

 

Varied New 0'pt1 mal
web Nonprismatm:
depth Negative
Section
_ll ..__

 

 

 

 

Constant

Flange

l \

Figure 5-4 Pier segment option 2: new optimal non-prismatic girder with
variable web depth

 

213

5.4.3 Pier segment option 3: new optimal non-prismatic section with
constant web height

A new optimal non-prismatic section having a uniform web height and a varied
bottom ﬂange was also developed as an Option for negative pier segments (see Figure
5-5). The bottom ﬂange thickness increases linearly from the prismatic positive section
to a maximum thickness at the pier centerline. This new section has a variable ﬂange
thickness and therefore its weight is heavier than other solutions where the bottom ﬂange
thickness is kept constant.

Beam dimensions were part of the component optimization process and the upper
and lower bounds on these dimensions were deﬁned within practical ranges. This new
Optimal girder was developed for a beam spacing of 1.8 m (6 ft), 2.1 m (7 ﬂ), and 2.4 m
(8 ft) and the span length was increased in the increments of 3.0 m (10 ft), both given as

parameters.

 

 

 

 

 

 

New Optimal
Constant Nonprismoth:
Web Negative
Height Section
V
__7r____
Varied
F I 9099 \

 

Figure 5-5 Pier segment option 3: new optimal non-prismatic girder with
variable bottom ﬂange thickness

214

5.4.4 Pier segment option 4: new optimal prismatic section for both
positive and negative Segments

A new Optimal prismatic section for use in both positive and negative segments
was also developed as shown in Figure 5-6. This section is considered to be more cost-
effective than having a girder Of varied height since fabrication is less complicated.

This new Optimal section Option was developed for a beam spacing Of 1.8 m (6
it), 2.1 m (7 it), and 2.4 m (8 it) and the span length was increased in an increments of
3.0 m (10 it) and given as a parameter. The overall height Of the beam was developed by
setting as upper bounds the results for the other pier segment Options. Beam dimensions
were also part of the component Optimization process coupled with the system

Optimization.

 

l k\\\\ _////J

New Optimm
New ;//IIStandard
Total Prismatic
Beam Negative
. Section
Height

 

 

 

 

 

,Kh r;

Figure 5-6 Pier segment Option 4: new optimal prismatic section for positive and
negative segments

 

215

5.4.5 Design variables for the development of pier segment options

Sectional Optimization studies required the deﬁnition Of design variables that
deﬁne the beam dimensions. For the option using an existing standard positive segment
with sofﬁt, only one design variable is needed, namely the maximum sofﬁt thickness (Sf)

at the pier centerline (see Figure 5-7).

 

{__\/__J

Existing
’/IIStondard

Positive
_2/ K__ Section

////, St

Figure 5-7 Design variable for existing standard girder with sofﬁt (pier segment
option 1)

 

 

 

 

 

 

 

The design variables for the new non-prismatic section with varied sofﬁt
thickness and constant web height are the bottom ﬂange thicknesses (t1 and t2), the top

ﬂange thicknesses (t3 and t4), the bottom ﬂange width (b1), the top ﬂange width (b2), the

web thickness (tw), the web depth (Dw), and the sofﬁt thickness (Sf) all deﬁned at the

pier centerline as shown in Figure 5-8.

216

b2 t4

7

 

 

 

 

Ex 4‘ t3
I/r-New Optimal
Nonprismotic
Negaﬁve

tw Section

__>. <——

7 m

bl

Constant Dw _J

 

 

 

Varied
St

Figure 5-8 Design variable for new non—prismatic section with constant web
depth (pier segment option 2)

Design variables for the new non-prismatic section with varied web height are

the bottom ﬂange thicknesses (t 1 and t2), the tap ﬂange thicknesses (t 3 and t4), the bottom

ﬂange width (b 1), the top ﬂange width (b;), the web thickness (tw), and the web depth

(DW), all deﬁned at the pier centerline as shown in Figure 5-9.

[)2

 

 

 

 

 

 

 

~ , i4
i3
3 New Optimal
Q [Nonprismotic
I8 Negative
'C Section
5 1L __
_n __ ____
l l 12
F——5r——4 ll

Figure 5-9 Design variables for new non-prismatic section with varied web
depth @ier segment option 3)

217

The design variables of a new prismatic section for both positive and negative

segments include the bottom ﬂange thicknesses (t, and t2), the top ﬂange thicknesses (t3

and t4), the bottom ﬂange width (b 1), the top ﬂange width (b2), the web thickness (tw),

and the web depth (DW), all deﬁned at the pier centerline as shown in Figure 5-10.

 

 

 

 

 

 

 

 

II II t3
ﬁr /_
New Optimal
Standard
Prismatic
H Negative
tw Section
6 ..
IIIEIIII ﬂ

Figure 5-10 Design variables for new prismatic section for both positive and negative
segments (pier segment option 4)

5.5 Analysis Continuous-Span Spliced Girder Bridges

The complete design and analysis considerations and procedures for continuous
SGB are presented in Chapter 2 and are only brieﬂy summarized in this section. The
analysis for the two-span continuous spliced girder bridge systems was based on 2-D
frame analyses including the effects of time-dependent material behavior and staged
construction. A custom program using the stiffness method was used. Each span

segment was divided into multiple elements to Obtain stresses, internal forces and

218

deﬂection results at the member ends. The beam-line modeling concept was used to
determine the demands of an interior girder according to the load distribution factors
speciﬁed in the AASHTO-LRFD Speciﬁcations (AASHTO 2003). Interior girders were
used in the analysis since their demands usually control the superstructure design. In
addition, all beams are typically designed as interior beams to allow future bridge

widening.

For all service limit states, analyses were performed using gross and uncracked
section properties while moment and shear capacity calculations were used to evaluate
the strength limit states. A two-span spliced girder bridge is statically indeterminate;
therefore, secondary effects from prestressing as well as the effect temporary support
removal were taken into account.

Prestress losses were computed using the simpliﬁed AASHTO-LRFD method
(AASHTO 2003) with consideration of the effects of combined pre-tensioning and post-
tensioning on the girder segments, as recommended by the recent NCHRP study on
spliced girder bridge design (Castrodale and White 2004). The design lifetime Of the
spliced girder bridge system was assumed to be 75 years after all pre-stress losses had
occurred. The analysis and section design was performed using a custom program coded
in MATLAB (Mathworks 2004), which enforced compliance with service and strength

limit states as deﬁned in Section 5.6 and 5.7.

219

5.6 Optimum Design of Two-Span SGB by Single-Stage Post-
Tensioning after Deck Casting

5.6.1 Design variables

The design optimization Of two-span continuous SGB requires more variables
that those considered for a single-span system (Chapter 4). Additional considerations
include design variables associated with the sectional optimization options and the
deﬁnition of additional variables to deﬁne the parabolic post-tensioning over the
haunched negative segment. The design variables are a mixed set of integer and
continuous geometric parameters deﬁning the dimension and shape of the standard and
non-prismatic precast/prestressed girder segments, the material layout, and the span
arrangement of the bridge system. Only key design variables were thus considered in the
design to maximize the efﬁciency of the computation algorithm.

The design variables for the sectional Optimization problems described in Section
5.4.5 were combined with the additional design variables Of prestress requirements and

prestress locations along the span for each construction method. These later design

variables are illustrated in Figure 5-11 and include: nprl-j = number of bottom pre-

. .th
tensroned strands on end-segments at row 1 (for example, nprjj represents the number

of bottom re-tensioned strands on the end-se ents at the lSt bottom row of the 'rder);
P gm 81

. . .th
nprig = number of top pre-tensroned strands on the mid-span segments at row 1 ; nprtj
= number Of top pro-tensioned strands on the end-segments; nprt2 = number Of bottom

pre-tensioned strands on the mid-segment; npo! = total number of post-tensioned strands

220

in the girder for single-stage post-tensioning construction or a ﬁrst stage post-tensioning
Operation for two-staged post-tensioning construction; ya = vertical distance from the
bottom extreme ﬁber of the girder to the net centroid of the equivalent post-tensioning

duct at the end-segments; yb = vertical distance from the neutral axis to the net centroid

of the equivalent post-tensioning duct at the middle of the end segment; yc = vertical

distance from the girder composite neutral axis to the net centroid of the equivalent post-

tensioning duct at the splice location; yd = vertical distance from the girder composite

neutral axis to the net centroid of the equivalent post-tensioning duct at the negative pier

location.

The sign convention for ya, yb, yc, and yd is such that they are positive when

measured down from the neutral axis. All Of the above-mentioned design variables, are

noted in Figure 5-11. Variables ya, yb, yc, and yd are not direct design variables (i.e., they

are not part of the Objective function). However, they have an inﬂuence on other design
variables and were considered in the design as part of the equivalent post-tensioned

loads.

The location of each pre-tensioned strand layer is taken to be 50 mm (2 in.) apart,
including the distance of the ﬁrst row of strands to the beam bottom and top. The post-
tensioning tendons were assumed to follow quadratic parabolic proﬁles. Normally, post-
tensioned strands require several ducts; however, only an effective post-tensioning
proﬁle was used to represent a group Of ducts in order to simplify the Optimization

computation by minimizing the number of design variables. Figure 5-1 la illustrates the

221

proﬁle of this effective post-tensioning duct for the construction method of single-stage
post-tensioning after deck casting. Figure 5-11b illustrates pre-tensioning requirements

for both positive and negative segments.

 

 

 

 

 

 

 

 

 

 

 

(a) Single-staged post-tensioning requirement

OPP npr
nprtl‘I / f—IFIPFZZ Iznpg, C (3
I 4+ .1 T ’/— I [I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

___________ \\ /_______[____.
III . 2
i \— '\\- \—
n r
r) r P
DPT} npr21 nprt2 p31 4]
l l l
..1 _2 ..1
l_

(b) Pro-tensioning requirement

Figure 5-11 Variables for the design optimization of two-span continuous SGB built
with single-staged post-tensioning after deck casting

222

5.6.2 Objective functions

Design optimization procedures rely on an objective function to evaluate the
“goodness” of a design and thus search for an improved solution that satisﬁes a set of
constraints, in this case the requirements of safety and serviceability imposed by design
codes and other merit criteria. A common design criterion in bridge design is the
minimization of total superstructure cost. Superstructure cost is typically chosen since it
commonly constitutes approximately 70% of the total bridge cost. The superstructure

cost objective function is then deﬁned as:

f(xi) = CPS + CPT + CCD + Crs + CB + CR. (5'1)

Details for each of the terms in the above cost objective ﬁmction (Equation 5-1)

are speciﬁed in Table 5-2.

Table 5-2 Components of objective function

 

Component Cost
Pre-Tensioning CPS C p - N g -[(npr,-1 + nprtl ) ' L1 - 2 + -(npr,-2 + nprt2) - L2]

 

Post-Tensioning CPT C p0 -Ng ~np0 -(L1 -2 + L2)
Concrete CCD Cc -L-W-ts

Reinforcement CR C, -W - (L1 -2 + L2) ~ WS,

Beang Cb-Ng-(L1-2+L2)
(Standard beam)

Beam CB va-Ng-CV

mew optimal beam)

Temp Support C73 Cts -Ng -2

 

223

Unit material costs for the two-span spliced girder bridge system were assumed
to be similar to the unit material costs for single-span spliced girder bridge. These costs
were described earlier in Section 4.5. Clearly, the costs of new prismatic and non-
prismatic beam sections are expected to be different from standard beams. The cost of a
standard pre-stressed concrete beam is deﬁned as a cost per unit length. However, there
is no clear guidance to estimate costs for new sections, whether prismatic or non-
prismatic. Therefore, the cost of fabrication the non-standard beams in these studies was

based on the concrete volume in each beam as speciﬁed in Table 5-3.

Table 53 Concrete volume of new optimal beams

 

Girder Section Girder Concrete Volume Component

New Prismatic Beam A g
CV: mX(2XL1 +L2)

New Non-prismatic

 

 

 

 

. . A A + A
Beam wzth Varzed CV: g1 x2xL1+(( g1 g2)x0.5xL2)
Web Depth 144
New Non-Prismatic
Beam with Constant A 3 (A 3 + A 4)
CV: g x2xL + g g x0.5xL
Web Depth 144 1 ( 144 2)

 

For the equations listed in Table 5-3: Ag is the area of the new prismatic beam; Ag] is the

area of the new non-prismatic beam with varied web depth in the positive section; A g2 is

the maximum area of a new non-prismatic beam with varied web depth over the pier

location; Ag3 is the area of a new non-prismatic beam with constant web depth in the

positive section; Ag4 is the maximum area of a new non-prismatic beam with constant

web depth over the pier location.

224

5.6.3 Design constraints

Constraints for the optimization problem were the service and strength limit
states speciﬁed in the AASHTO-LRFD Bridge Design Speciﬁcations (AASHTO 2003)
and the recommendations from NCHRP Report 517 (Castrodale and White 2004).
Constraints for the strength limit state included checks on moment and shear capacities.

The ductility requirement was satisﬁed by ensuring that the relative neutral axis depth at

the ultimate limit state, c/de, was less than 0.42 (AASHTO 2003).

Constraints for the service limit state were deﬁned in terms of flexural stress
limits on both the girder and deck. Stresses at each critical construction stage (see Table
5-4) were checked against the allowable compression and tension stresses recommended
by the AASHTO-LRFD Speciﬁcations (AASHTO 2003), see Table 4-5. Three locations
across the composite girder-deck section, namely, the girder bottom and top, and the top
of the deck were checked along the span length of beam at each critical construction
stage to guarantee that all stresses were within the prescribed limits. There are seven
critical stage constraints on each location of the composite section as noted by the ”X”
symbols in Table 5-4.

The formulation for the bottom girder stress constraints (Constraints lda to 7da)
for each critical construction stage of the service limit state are outlined below. Similar
expressions were deﬁned and implemented for the top girder (Constraints lea to 7ea)
and top deck (Constraints lfa to 7fa) stresses. The constraint expressions described
below illustrate the stress checks on the negative segment. Similar constraint

formulations were deﬁned and implemented for the positive segments.

225

Table 5-4 Construction sequence for single-staged post-tensioning after deck casting

 

 

. . Pre— P
Construction Time . . ' Tensioning OSt'. _
ACthIl DCSCI’lpthH C . . 1 Tenswnlng
Stages (days) r1t1ca Critical Stage
Stage
1 - Stress pre-tensioning
strands
2 0 Cast girder segments
3 1 Release pre-tensioning X
strands
4 50 Erect girder segments
60 Place deck and splice X
concrete
6 75 Stress post-tensioning X X
strands
7 100 Add barriers X
140 Apply live load
9 15000 Add future wearing surface
10 27500 After all pre-stress losses X X

 

Constraint lda: Compression Stress Check at Pre-Tensioning Stage

This construction stage considers the stresses due to the girder weight and the
pre-tensioning force. The gross area of the negative girder section varies along the
negative pier segment for the new non-prismatic sections and is uniform for the
prismatic sections. At this stage, only prestress losses due to relaxation at release and
elastic shortening were deducted from the initial prestress. Elastic shortening prestress
loss was determined by using the average cross-sectional area of the negative segment
(for non-prismatic elements) and this loss was assumed to be uniform along the entire

negative segment. The actual stresses along the span were evaluated from their varied

226

girder gross sectional properties (Agn and Sbgn) and eccentricity of the pre-tensioned

strands (epn). The bottom girder compression stress constraint was then deﬁned by:

 

M A f A f e
alda =_ gn + psn peln _ psn peln pn S O-6féi, (5_2)
S A S
bgn gn bgn

where Mg" is the girder moment along the negative segment; Apsn is the total pre-
tensioned area in the negative segment; fpeh, is the effective prestress in the pre-
tensioned strands after pre-tensioning; cm is the net eccentricity of pre-tensioning
strands along the negative segment measured from the neutral axis of the section; Sbgn is
the bottom section modulus along the negative segment; Ag” is the girder gross area

along the negative segment; and ab 1,, is the girder bottom stress along the negative

segment.

Constraint 2da: Compression Stress Check at Deck and Splice Placement Stage

 

Constraint 2da applies to the concrete casting of the deck, deck haunch, and
splice regions. Non-composite sectional girder properties were used since the deck has
not yet hardened. Prestress losses for this stage include those from elastic shortening,
creep, shrinkage, and steel relaxation.

Creep losses due to pre-tensioning were determined by using the average girder
gross section properties (relevant to non-prismatic elements). These average section
properties include the average cross-section area, the average gross moment of intertia,

and the average eccentricity of the pre-tensioned strands. This creep loss was assumed to

227

be uniform along the entire negative segment. Elastic shortening prestress losses were

assumed to be the same as deﬁned earlier on Constraint Ida. The actual stresses along

the span were also evaluated from their varied girder sectional properties (Agn and Sbgn)

and eccentricity of the pre-tensioned strands (epn). Bottom girder compressive stresses

were thus constrained by the following expression:

Mgn _ Md” _ Mhn _ Men + ApsnfpeZn _ ApsnfpeZnepn

 

 

aZda = ‘ 5 0-6fé ’

gn Sbgn

(5-3)
where Md" is the deck moment along the negative segment; Mhn is the haunch moment
along the negative segment; Mcn is the construction load moment along the negative
segment; fpezn is the effective prestress in the pre-tensioned strands after placing the

deck; and O'bzn is the girder bottom stress along the negative segment.

Constraint 3da: Compression Stress Check at Post-Tensioning Stage

This constraint applies to the longitudinal splicing of the pre-tensioned girders
through a single post-tensioning operation. A tendon proﬁle composed of multiple
quadratic parabolic curves was used to deﬁne the equivalent post-tensioning tendon
location along the girders. Equivalent multi-uniform transverse loadings with nodal
forces were used to represent the effect of post-tensioning forces. At this stage, the deck
is assumed to be working compositely with the beam. Prestress losses at this stage
include those from elastic shortening, creep, shrinkage, steel relaxation, and additional

elastic shortening due to post-tensioning. Details of prestress loss calculations were

228

described in Section 2.11 and are not re-stated here. The bottom girder stress constraint

is thus deﬁned by:

 

 

 

 

 

Mgn _Mdn _Mhn _Mrn + Apsnfpe3n + Apsnfpe3nepn +

03d“ z _ s s s s A s
bgn bgn bgn bcn gn bgn

 

n a f M M
P0 P5 Pof + 1’0“" _ sec” $0.6fc'. (5-4)
AC” Sbcn Sbcn

where Mposm is the primary moment due to post-tensioning; Msectn is the secondary
moment due to post-tensioning; Mm is the moment due to temporary support removal;
npo is the total number of post-tensioned strands; apsn is the total number of post-
tensioning strands; fpof is the effective prestress in the post-tensioned strands aﬁer post-
tensioning; fpem is the effective prestress in the pre-tensioned strands after post-
tensioning; and Ub3n is the girder bottom stress along negative segment.

Constraint 4da: Compression Stress Check Under Full Service Load

This constraint corresponds to the service condition where the composite girders
carry full live loads and superimposed dead loads in addition to the loads applied in the
previous stages. Prestress losses for this stage include elastic shortening, creep,
shrinkage, steel relaxation, and additional elastic shortening due to post-tensioning. The

compression stress limit under full service load is thus deﬁned by:

 

 

 

Mg"_Mdn_Mhn_Mrn [Vim_jwlln+AP5’7fPe4"+

J4da2" -
Sbgn Sbgn Sbgn Sbcn Sbcn Sbcn Agn

229

 

 

 

Apsnfpe4nepn + npoapsnfpoff + Mpostn _ Msec n

S 0.6 f '. , (5-5)
Sbgn Acn Sbcn Sbcn c

where M11" is the live load moment due to truck and lane loadings; [0017 is the effective
prestress in the post-tensioned strands after applying the live load; ﬁns," is the effective

prestress in the pre-tensioned strands after applying the live load; and 01,4" is the girder

bottom stress along the negative segment.

Constraint 5da: Tension Stress Check with Reduced Live Log!
The applied loads and prestress losses for this constraint are similar to Constraint
4da except that the live load is reduced by 20 percent. The objective is to check the

tension limit state of the girder at the ﬁnal stage as deﬁned by:

 

 

 

Mg" _Mdn _Mhn __Mrn _Msn —0.8XM“n +Apsnfpe4n +

05"“ = _ s s s s s s A
bgn bgn bgn bcn bcn bcn gn

Apsnfpe4nepn ”poapsnfpoﬂ + Mpost _ Msec
Sbgn Acn Sbcn Sbcn

 

 

2 —o.0724 fc' (MPa)

2—0.19 fc' (ksi) (5-6)

Constraint 6da: Compression Stress Check Without Live Load

This constraint considers all loads and prestress losses as deﬁned for Constraint
4da with the exception of live load. The compression limit state under these conditions is

then checked according to:

Mg" _ Mdn _ Mhn - Mm _ M5,, + Apsnfpe4 + Apsnfpe4nepn +
Sbgn Sbgn Sbgn Sbcn Sbcn Agn Sbgn

 

 

 

 

06da : -

230

 

”poapsn fpoff + M post _ M sec

S 0.45 fc' .
Acn Sbcn Sbcn

Constraint 7da: Compression Stress Check for Partial Service Load

(5-7)

This critical stress check considers all of the loads of Constraint 4da reduced by

50 percent except the live load. All prestress losses considered for Constraint 4da apply

here also. The bottom girder stress check is then deﬁned by:

_Alﬂ_Mdn _Mhn _Mrn _Msn +Apsnfpe4n +
Sbgn Sbgn Sbgn Sbcn Sbcn Agn

 

 

 

07da = 0.5 X [-

Apsnfpe4nepn "poapsnfpoff + Mpost _ Msec ]- Mlln
Sbgn Acn Sbcn Sbcn Sm.

 

 

 

s 0.4 fc'

(5-8)

5.7 Optimum Design of Two-Span SGB by Multi-Staged Post-

Tensioned Systems

The use of multi-staged post-tensioning where the non-composite girders are

stressed for continuity prior to placement of the deck is commonly used in multi-span

spliced girder bridges. Spliced girder bridges made continuous with multiple post-

tensioning operations were investigated herein to determine the efﬁciency of the system

in terms of construction cost as well as the resulting maximum achievable span length.

231

5.7.1 Design variables

The design variables for the multi-stage post-tensioning construction method are
similar to those used for the single-stage post-tensioned application. An additional
design variable in this construction method is the number of post-tensioning strands on
the second post-tensioning operation (npoz). Post-tensioning is applied twice - before
and aﬂer casting the deck. Figure 5-12 shows the post-tensioning design variables for the
multi-stage post-tensioning construction method. Pre—tensioning design variables for the
two-stage. post-tensioning method are the same as those used for the single-stage post-

tensioning method as shown in Figure 5—1 lb.

 

01 J

 

 

 

 

 

 

 

 

 

 

Figure 5-12 Post-tensioning requirement variables for the design optimization of
two-stagedpost-tensioning

5.7.2 Objective function

The same objective function as that used for the single-stage post-tensioning
construction method was used here, which was the minimization of structural cost. This
was chosen in order to be able to compare the results from this section with those of

Section 5.6.

232

5.7.3 Design Constraints

The Optimization constraints used for this construction method include the
service and strength limit states speciﬁed in the AASHTO-LRFD Bridge Design
Speciﬁcations (AASHTO 2003) and the recommendations from NCHRP Report 517

(Castrodale and White 2004).

Constraints for the strength limit state included checks of moment and shear
capacities and checks of the ductility requirement (c/de < 0.42) according to AASHTO-
LRFD (AASHTO 2003). Constraints for the service limit state were deﬁned in terms of
ﬂexural stress limits on both the girder and deck as outlined in Table 4-9. Stresses at
each critical construction stage were checked against the allowable compression and

tension stresses recommended by the AASHTO-LRFD Speciﬁcations (AASHTO 2003),

see Table 5-5.

233

Table 5-5 Construction sequence for multi-stage post-tensioning

 

 

Construction Time , . . P r ‘3' . . P05”. .
Action Description Tensromng Tensronrng
Stages (days) Critical Stage Critical Stage
1 - Stress pre—tensioning strands
2 0 Cast girder segments
3 1 Release pre-tensioning X
strands
4 50 Erect girder segments
75 Stress 1St post-tensioning X X
6 90 Place deck and splice X X
concrete
7 120 Stress 2nd post-tensioning X X
130 Add barriers X
9 140 Apply live load
10 15000 Add future wearing surface
1 1 27500 After all pre-stress losses X X

 

There are ten critical stage constraints on each location of the composite section.
Three locations across the composite girder-deck section, namely, the girder bottom and
top, and the top of the deck were checked along the span length of beam at each critical
construction stage to guarantee that all stresses were within the prescribed limits.

The formulation for the bottom girder stress constraints (Constraints Idm to
8dm) for each critical construction stage of the service limit state are outlined below.
Similar constraint formulations were deﬁned and implemented for the stresses at the top
of the girder (Constraints [em to 7em) and at the top of the deck (Constraints Ifm to
7fm). The constraint expressions outlined below are the stress checks on the negative
segment. Similar constraint formulations were deﬁned and implemented for the positive

segments.

234

Constraint ldm: Compression Stress Check at Pre-Tensioning Stage.

The stresses at this stage are similar to Constraint Ida in Section 5.6, which
include the girder weight and the pre-tensioning force. The section properties of the
negative section vary along the negative pier segment for the non-prismatic sections and
are clearly uniform for the prismatic sections.

Only prestress losses due to relaxation at release and elastic shortening were
deducted from the initial prestress. Elastic shortening losses were assumed to be constant
along the negative segment by using the average section properties of non-prismatic

elements; whereas the actual stresses along the span were evaluated from their varied

girder OSS SCCthII I'OpCI'thS (A and S ) and eccentrici Of re-tensioned strands
gr p gn bgn p
(e ). Ihe bottom gldeI' COITIpI‘CSSlOIl StI'CSS constraint iS thus CXpI'CSSCd as fOllOWSI

pn

M A f A f e
aldm :_ gn + psn peln _ psn peln pn S 0-6féi- (5-9)
Sbgn Agn Sbgn

 

Constraint 2dm: Compression Stress Check at First Post-Tensioning Stage

For this construction method post-tensioning is applied prior to pouring the deck.
Thus, only the non-composite girder section is used to resist additional loads due to post-
tensioning forces. A tendon proﬁle with multiple quadratic parabolic proﬁles was used
to deﬁne the equivalent post-tensioning tendon location along the girders. Prestress
losses for this stage include those from elastic shortening, creep, shrinkage, and steel
relaxation.

Creep and elastic shortening losses on non-prismatic pier segments were

determined by using the average girder gross section properties. These average section

235

properties include the average cross-sectional area, the average gross moment of intertia,
and the average eccentricity of the pre-tensioned strands. These losses were assumed to

be uniform along the negative segment. The actual stresses along the span were also

evaluated from their varied girder sectional properties (A gn and Sbgn) and eccentricity of

pre-tensioned strands (em). The bottom girder compressive stresses are thus constrained

by the following expression:

Mgn + Apsnfpe2n + Apsnfpe2nepn + "polapsnfpofl _
Sbgn Agn Sbgn Agn

 

 

02dm :-

MPOSII _ Msecl
Sbgn Sbgn

 

s 0.6/g. , (5-10)

where MPOS” is the primary moment due to ﬁrst stage post-tensioning; M5661 is the

seconda moment due to ﬁrst sta e ost-tensionin ; n 1 is the total number of ﬁrst
ry g p g p0

stage post-tensioned strands; fpoﬂ is the effective pre-stress in the ﬁrst post-tensioning

strands after ﬁrst post-tensioning; and O'bzn is the girder bottom stress along the negative

segment after ﬁrst stage post-tensioning.

Constraint 3dm: Compression Stress Check at Deck and Splice Placement Stage
The deck, haunch and splice are poured at this stage aﬁer longitudinal post-

tensioning in the second stage. These segments then become continuous along the span

length. Prestress losses for this stage include those from elastic shortening, creep,

shrinkage, and steel relaxation. The bottom girder stress constraint is thus deﬁned by:

236

Mgn _ Md" _ Mhn _ Men + Apsnfpe3n _ Apsnfpe3nepn
Sbgn Sbgn Sbgn Sbgn Agn Sbgn

 

 

 

U3dm : "

 

n a M
p01 psnfpofl + [10511 _ Msecl S 0.6fé, (5-11)
Ag" Sbgn Sbgn

where 01,3" is the girder bottom stress along the negative segment after deck and splice

placement.

Constraint 4dm: Compression Stress Check at Second Post-Tensioning Stage

A tendon proﬁle deﬁned by multiple quadratic parabolic curves, similar to the
ﬁrst post-tensioning, was also used to deﬁne the second equivalent post-tensioning
tendon location along the girders. Second post-tensioning is applied after casting of the
deck and splice, thus the composite girder section is used to resist additional loads due to
post-tensioning forces. Only forces from the second post-tensioning operation are
resisted by the composite section of the girder and deck. The previous forces are still
resisted by the non-composite section. Prestress losses for this constraint include those
from elastic shortening, creep, shrinkage, steel relaxation and additional elastic

shortening due to post-tensioning. The bottom girder stress constraint is thus deﬁned by:

Mgn _ Mdn _ Mhn + Apsnfpe4n + Apsnfpe4nepn + Mpost2 + Msec2 +
Sbgn Sbgn Sbgn Agn Sbgn Sbcn Sbcn

 

 

a4dm :—

npolapsnfpoffl + Mpostl _ Msecl + "po2apsnfpof2
A

 

 

S 0.6f' (5-12)
gn Sbgn Sbgn Acn c

237

where Mpost2 is the primary moment due to second stage post-tensioning; Msecz is the
secondary moment due to second stage post-tensioning; npog is the total number of
second stage post-tensioning strands; 5,017] is the effective prestress in the ﬁrst post-
tensioning strands after second post-tensioning; 5,012 is the effective prestress in the

second post-tensioning strands after second post-tensioning; and 01,4" is the girder

bottom stress along the negative segment after second stage post-tensioning.

Constraint Sdm: Compression Stress Check under Full Service Load

Compression stress limits under full service load are checked at this constraint.
Only forces from the applied superimposed dead loads, loads from the removal of
temporary supports, and live loads are resisted by the composite section of girder and
deck. The previous forces are still resisted by the non-composite section. Prestress losses
for this constraint include elastic shortening, creep, shrinkage, steel relaxation, and
additional elastic shortening due to post-tensioning. The compression stress limit under

full service load is thus deﬁned by:

238

Mg" _Mdn _Mhn _Mrn _Msn _Mlln +APS’7fPe4" +
Sbgn Sbgn Sbgn Sbcn Sbcn Sbcn Agn

 

 

 

USdm : —

Apsnfpe4nepn + "polapsnfpoffl + Mpostl _ Msecl +

Sbgn A

 

gn Sbgn Sbgn

 

 

n a f M M
p02 [)5]! p0ff2 + pOStz _ SCC2 S O-6fc" (5_13)
Acn Sbcn Sbcn
Constraint 6dm: Tension Stress Check with Reduced Live Load
The applied loads and prestress losses for this constraint are similar to those in
Constraint 4dm except that the live load is reduced by 20 percent. The objective is to

check the tension limit state of the girder at the ﬁnal stage as deﬁned by:

 

 

Mg" _Mdn _Mhn _Mrn _Msn —O.8x
Sbgn Sbgn Sbgn Sbcn Sbcn Sbcn Agn

M A f 4
06dm____ Iln + 12er pe n +

Apsnfpe4nepn + ”polapsnfpoffl + Mpostl _ Msecl +

Sbgn A Sbgn Sbgn

 

gn

"poZapsnfpoffZ + Mpost2 _ Msec2
Acn Sbcn Sbcn

 

 

2 —0.0724,/?C—' (MPa)

2 4.19775 (ksi) (5-14)

Constraint 7dm: Compression Stress Check Without Live Load

This constraint considers all loads and prestress losses as deﬁned for Constraint
4dm with the exception of live load. The compression limit state under these conditions

is then checked according to:

239

Mg" _ Mdn _ Mhn _ Mrn _ M5,, + Apsnfpe4n + Apsnfpeélnepn +
Sbgn Sbgn Sbgn Sbcn Sbcn Agn Sbgn

 

a7dm :—

 

npolapsnfpoffl +Mpostl _Msecl +
A

 

gn Sbgn Sbgn

npo2apsnfpoff2 + npoZapsnfpoﬂZeen

S 0.45f'. (5-15)
Acn Sbcn c

Constraint 8dm: Compression Stress Cl_1eck for Partial Service Load

This critical stress check considers all of the loads of Constraint 4dm reduced by
50 percent except the live load. All prestress losses considered for Constraint 4dm apply
here also. The bottom girder stress check is then deﬁned by:

Mgn _Mdn _Mhn _Mrn _Msn + Apsnfpe4n +Apsnfpe4nepn +

08dm=0.5x[-
Sbgn Sbgn Sbgn Sbcn Sbcn Agn Sbgn

 

 

"polapsnfpoffl + Mpostl _ Msec +

Agn Sbgn Sbgn

n a M M
p02 psnfpoffZ + postz _ sec2]__MI_1n S 0.4fC'H (5-16)
AC" Sbcn Sbcn Sbcn

 

 

5.8 Results and Discussion

5.8.1 Prestress requirements for single- and multi-stage post-
tensioning construction methods

An AASHTO Type IV standard beam was used to investigate the effect of

construction sequence on two-span continuous spliced girder bridges. The objective

240

functions were based on the minimization of superstructure construction cost. Beam
spacing at 1.8 m (6ft), 2.4 m (8ft), and 3.0 m (10 it) were investigated for three splice
locations: (0.35L, 0.375L, and 0.40L), where L is the total bridge span length of both
spans.

Figure 5-13 through Figure 5-19 show the prestressing requirements for different
span lengths. Single-stage post-tensioning and multi-stage post-tensioning results were
plotted in the same graphs in order to compare the requirements and the maximum

achievable span lengths.

Span Length (m)
32 34 36 38 40 42 44

 

 

    

 

 

 

 

 

 

 

 

 

 

 

30 1 EL 1 l l l l l l l l l l l l l l 1 LL 1 l l l l l l I l l l l l l l L 30
f f'c = 6.5 ksi, f'ci = 5 Ksi, s = 6 ft .9. Sing.e,o.35 E
25 _‘ ASSHTO Type IV * “ -B- Single-0375 ; 25
: + Single-0.40 :
_. o _
_‘ :- ‘”
g 20 _ _ 20 '2
C - -
w - ~ 8
b .1 - vo-i
(I) 15 _— 15 U)
“5 ”‘ : .5
§ 10 3 E— 10 2°
: + Mum-0.35 :
5 -; + Mum-0.375 :- 5
: + Mum-0.40 :
o T I T l I l l l l I l l l l I l I l l l I l l l o
100 110 120 130 140 150

Span Length (ft)

Figure 5-13 Pre-tensioning requirements on positive segments for single- and
multi—stagedpost—tensioning construction for S = 1.8 m (6ft)

241

Based on the results shown in Figure 5-13, the splice location at 0.35 of the total
span length is the most efﬁcient splice location for both single- and multi-staged post-
tensioning since it yields the maximum achievable span length in both cases. The results
show that when multi-staged post-tensioning was used it yielded a longer span than
when using a single-stage post—tensioning method. For example, at splice location of
0.35(L), the maximum achievable span length increased from 40.2 m (132 ft) to 42.4 m

(139 ft) when employing a multi-stage construction method. This is about a 5% increase

 

 

 

 

 

 

 

 

 

in span length.
Span Length (ft)
100 110 120 130 140
l l l I l l l l l l l l l l l l l l l l l
30 _ L 30
I f'c = 44.8 MPa, f'ci = 34.5 MPa, S =1.8m -
25 _- ASSHTO Type W -6- Single-0.35 L 25
I —B— Single-0.375 I
I -é-— Single-0.40 I m
g 20 j -v-- Mum-0.35 _- 20 1:
g 3 -¢-- Mum-0.375 ; 5
.5. - -o-- M ”-0.40 - 5
ca 15 — "t — 15 :75
- I —- c I “-
g a ,I :72 ::-E=~. : 9
1o ‘ ’7 X §=a — 10 °
/ / _
/
5’ "_ 5
0 I I I I I I I I I I I I I I I f I F I I I I I I 0
32 34 36 38 40 42 44

Span Length (m)

Figure 5-14 Pre-tensioning requirements on the negative-segment of single- and
multi—stagedpost-tensioning construction for S = 1.8 m (6 ft)

242

The requirements of pre-tensioned strands for the AASHTO Type IV with a
beam spacing of 1.8 m (6 ft) for the pier, or negative, segment are illustrated in Figure
5—14. It can be seen that pre-tensioned strand requirements at the positive segments for
single-stage post-tensioning construction are higher than when using multi—stage post-
tensioning; while the requirements of pre-tensioning strands at the pier segments for
multi-stage post-tensioning construction are higher than those for a single-stage post-
tensioning method. Requirements of pre-tensioned strands for a beam spacing of 2.4 m
(8 it) at the positive segments and at the negative segments are shown in Figure 5-15 and
Figure 5-16, respectively. Figure 5-18 and Figure 5-21 provide the pre—stressing for a
beam spacing of 3.0 m (10 ft). It can be seen that all of these graphs show the same

result trends as interpreted above for a beam spacing of 1.8 m (6 ft).

Span Length (ft)

 

 

 

 

 

 

 

 

 

100 110 120 130 140
35 q 1 1 1 1 L 1 1 1 1 l 1 1 1 1 l 1 1 1 1 __ 35
: f‘c = 6.5 ksi, f‘ci = 5 Ksi, S = 3 ft Single-0.35 I
_i u—
30 _- ASSHTO Type IV :- -B- Single-0.375 _— 30
1 . * ° + Single-0.40 :
—‘ + Mum-0.35 ;
w 25 1 + Mum-0.375 : 25 g
'2 3 + Mum-0.40 : 1:
p 20 g; __ 20 E
a t: _ 1H
m _ tn
“6 15 E- 15 “5
d : 6
2 10 :_ 10 Z
5 E— 5
0 I I I I I I I I I I I I I I I I I l I I I I I I I P- o
32 34 36 38 4o 42

Span Length (m)

Figure 5-15 Pre—tensioning requirements on positive segments of single- and
multi-stagedpost-tensioning construction for S = 2.4 m (8 ft)

243

Span Length (ft)

 

 

 

 

 

 

No. of Strands

 

 

 

 

100 110 120 130 140
35 .- 1 1 4 1 l 1 1 1 1 l 1 1 1 1 l 4 1 1 1 __ 35
: f'c = 6.5 ksi, f'ci = 5 Ksi, s = 3 ft E Slngle—0.35 :
30 ; ASSHTO Type Iv —a— Single-0.375 }- 30
l -é— Single-0.40 :
S —v— Mum-0.35 -_
25 3 + Mum-0.375 ; 25 .3
3 -e— Mum-0.40 ; :
ﬂ
3 : (D
15 - E— 15 “6
: 6
1o ___ 10 z
5 .' :— 5
c
o I ' I 0

 

4O 42

Span Length (m)

Figure 5-16 Pre-tensioning requirements on negative segment of single- and
multi-Staged Post-Tensioning Construction for S = 2.4 m (8 ft)

 

 

 

 

 

 

 

 

 

4O
f'c = 6.5 RSI, fCI = 5 KSI, S = 10 ft .9. Single-0.35
35 ASSHTO Type N —e— Single-0.375
-é- Single-0.40
30 a x a —*v-— Mum-0.35
a, _ + Mum-0.375
'g 25 a: ‘ = + Mum-0.40
E .3 v
5 20
Q-
o
0' 15
Z
—--—'
10 —v
5
0 I I I I I I W I I I I I
100 105 110 115 120 125 130

Span Length (ft)

Figure 5-17 Pre-tensioning requirements on positive segments of single- and
multi-stagedpost-tensioning construction for S = 3.0 m (1 0 ft)

244

 

 

 

 

 

 

 

 

 

 

 

 

20
f‘c = 6.5 kSl, f'Cl = 5 KSI, S = 10 ft _9_ Single-0.35
ASSHTO Type IV —6— Single-0.375
-é-— Single-0.40
15 + Mum-0.35
a, + Mum-0.375
1; + Mum-0.40
g
'0- H
o .,- -
0' 1 V —9
z 5
5
_ S H——€l
o I I I I I I I I I
100 105 110 115 120 125

Span Length (ft)

Figure 5-18 Pre-tensioning requirements on negative segment of single- and
multi-stagedpost-tensioning construction for S = 3.0 m (1 0 ft)

Post-tensioning strand requirements for single-stage and multi-stage post-
tensioning at 1.8 m (6 ft) beam spacing and different splice locations is illustrated in
Figure 5-19. Requirements of post-tensioning strands for a beam spacing of 2.4 m (8 ft)
and 3.0 m (10 ft) are also shown in Figure 5-20 and Figure 5-21, respectively.

The results in Figure 5-19 through Figure 5-21 show that more post-tensioning
strands are required when a multi-stage post-tensioning construction method was used.
Normally, the ﬁrst post-tensioning stage requires only a minimum of post-tensioning
strands to connect the segments together as a continuous piece. At this stage, the girder
section is still non-composite. Therefore, not many post-tensioned strands can be

applied. The ﬁrst post-tensioning operation helps reduce cracking in the deck due to the

245

pre-compression force of the post-tensioning. The second post-tensioning operation is
applied after the deck has already hardened and the girder/deck unit has become
composite. Therefore, it is really only the additional post-tensioning beyond the initial

stage that helps increase the span length of the system.

 

      
 

 

 

 

30
E f'c=6.5ksi, f'ci=5Ksi, S=6ft
25 _: ASSHTO Type IV
% 20 j
c —
g _.
if» 15 J
“_ 1 —e— Single-0.35
0. —e— Single-0.375
g 10 —a— Single-0.40
+ Mum-0.35
+ Mum-0.375
5 + Mum-0.40

 

 

 

 

 

o I I I I F I I I I
100 110 120 130 140 150
Span Length (ft)

Figure 5-19 Post-tensioning requirement of single- and multi-staged post-
tensioning construction for S = 1.8 m (6 ft)

246

No. of Strands

No. of Strands

 

 

 

 

 

 

 

 

35 _
: f'c=6.5ksi, f'ci=5Ksi, S=8ft
30 —: ASSHTO Type IV
25 —:
20 —] °
3 I”; '3
15 j / 5 -e— Single-0.35
‘ ___—e”; -B- Single-0.375
10 r "- 1 1 ’5 + Single-0.40
,, + Mum-0.35
2’: + Mum-0.375
5 -e— Mum-0.40
o I I I I I I I
100 110 120 130 140

Span Length (ft)

Figure 5-20 Post-tensioning requirement of single- and multi-staged post-
tensioning construction for S = 2.4 m (8 ft)

 

  

 

 

 

 

 

 

 

 

 

35
: fc=6.5 ksi, f'ci=5Ksi, S=10ft
30 —j ASSHTO Type IV
25 —:
20 —j
15 ‘ -e- Single-0.35
-a— Single-0.375
10 -ﬁ- Single-0.40
- + Mum-0.35
5 _- + Mum-0.375
: —o— Mum-0.40
o I I I I I I I I 1
100 105 110 115 120 125

Span Length (ft)

Figure 5-21 Post-tensioning requirement of single- and multi-stagedpost—
tensioning construction for S = 3.0 m (1 0 ft)

247

The advantage of using a multi-stage post-tensioning construction method is the
increase in the achievable span length. In addition, multi—stage post-tensioning is
advantageous on projects where future deck replacement is anticipated since the deck
can be removed without overstressing the existing girders. Although using a multi-stage
post-tensioning approach is advantageous from a design point of view, the construction
cost is more expensive. Staged post-tensioning would require that the post-tensioning
supplier returns to the site a second (or more) times to complete the job. All post-
tensioned strands need to be post-tensioned prior to having permanent connections.
Construction time will thus increase since a detailed survey would be required after the
ﬁrst stage is done to deﬁne the grades for matching the girder cambers.

Single-stage post-tensioning allows the post-tensioning to be applied to the
composite cross-section, thus making the system more efﬁcient. Applying all of the
post-tensioning after the deck is poured also has advantages over a multi-stage approach,
among them the reduction in construction time. The post-tensioned subcontractor does
not need to come back to the site since this method only requires a one-time application
of the post-tensioning. Also, a larger prestressing force can be applied from pre-
tensioning, resulting in reduced anchorage hardware costs. Thus, single-stage post-
tensioning seems to be the favorable staged construction for spliced girder bridges since
it is faster and less complex than multi-staged construction. However, this construction
method has drawbacks in projects where future deck removal is anticipated. Removal of
the deck during the lifetime of the bridge is complicated since the deck was present

when the post-tensioning was applied. Thus, removal of the deck can overstress the

248

girders. One way to avoid this problem would thus be to use a multi-stage post-

tensioning construction method.

5.8.2 Service moments for single— and multi-stage post-tensioning
construction

The service moment diagrams for two-span continuous spliced girder bridges
using AASHTO Type IV I-beams with a total span length of 68.6 m (225 ﬂ) and a beam
spacing of 1.8 m (6 ft) for a single-stage post-tensioning construction method are shown
in Figure 5-22. This graph shows the moments acting on the non-composite and
composite sections for comparison purposes. Moments acting on the non-composite
girder section include those due to the girder/deck, haunch, deck, and construction loads;
while moments acting on the composite girder section include those due to the

temporary support loads, superimposed dead load and live load.

 

 

 

 

 

 

 

 

 

-2000 _7 f'c = 6.5 RSI, f'CI = 5 KSI, S = 6 ft .9. Mgirder

_ Single-Staged Post-tensioning —e— Mtemp

3.: . AASHTO Type 4 “'9" Wee"
I
,5 2 + Mhaunch
,9. _ + Mconst
31‘. -1000 .1 + Msidl
‘5 ~ — MLL+
m
E
O
E
.3 °
2
0
(D
1000 -
I I I I I— I I I I
O 50 100 150 200 250
Distance (ft)

Figure 5—22 Service moment demands on two-span continuous SGB built with single-
staged post-tensioning with L = 68.3 m (224 ft) and S = 1.8 m (6 ft)

249

Moment diagrams for the single-stage construction method are shown in Figure
5-23. Non-composite behavior was assumed for the spans between the abutment and the
temporary supports for the positive segments and between the temporary support and a
middle pier for the pier segment. Since the deck is cast before the post-tensioning, each
segment still acts independently on the temporary supports. Consequently, moments
caused by deck load are high in the positive segment and low in the negative segment.
Temporary supports in single-stage post-tensioning construction have to carry all non-
composite loads including the deck load before post-tensioning. Thus, the moment due
to the removal of temporary supports is much higher than in multi-staged construction.

The moment demands along the 68.6 m (225 ft) span of a spliced girder system
built through multi-staged post-tensioning construction are shown in Figure 5-23.
Moments acting on the non-composite girder section include those due to the girder
load, temporary support removal and secondary moments due to the ﬁrst post-
tensioning. The latter moment is not shown in Figure 5-23 since it is not a moment due
to an external load. Rather, the secondary moment is an induced moment due to the ﬁrst
post-tensioning and was used in the stress calculations by considering it as part of the
moment in the limit state stress calculation.

Moments acting on the composite girder section include those due to the
girder/deck haunch, deck, construction loads, superimposed dead load and live load. For
a multi-stage construction, the ﬁrst post-tensioning operation makes all the separate
segments become a continuous piece before applying the deck. Thus, the loads from the
removal of temporary supports loads are much smaller than in the single-stage post-

tensioning method, which yields smaller support removal moments. Since the external

250

moments for the multi-stage construction method are smaller than the external moments
in single-staged construction, the ﬂexural stresses at each critical stage are smaller and
the maximum achievable span length should also be longer. This observations support
the design optimization results that were shown in Figure 5-13 through Figure 5-22.

For a single-stage post-tensioning construction method, the pre-stressing forces
are used to balance the non-composite dead loads; while the post-tensioning forces are
used to resist the effects of the composite superimposed dead loads and live loads, which
require larger pre-tensioning forces and a larger cross-section for the girder. For multi-
stage post-tensioning construction, the initial pre-tensioning force and ﬁrst stage post-
tensioning are used to balance only the self-weight of the girder and the fresh concrete
deck. The post-tensioning force from the second operation balances the additional dead
and live loads. Therefore, smaller pre-tensioning forces and smaller ﬁrst-stage post-
tensioning forces are required, leading to smaller cross sections compared with the

single-stage post-tensioning construction method.

251

Service Moment (kips-ft)

5.8.3

-2000

 

 

    
 
 
   

 

 

 

 

 

 

 

: f'c = 6.5 ksi, f'ci = 5 Ksi, S = 6 ft 8 Mgirder
-1500 — Multi-staged post-tensioning ‘3'- Mtemp
3 -e-— Mdeck
: ASSHTO Type W + Mhaunch
'1000 _— + Mconst
I + Msidl
‘ -— MLL'I'
500 I —‘— MLL-
0 _
500 —:
-l
1000 -E
I | I I I I I I j
0 50 100 150 200

Span Length (ft)

Figure 5-23 Service moment of multi-staged post-tensioning with L = 68.3 m
(224ft) andS = 1.8 m (6ft)

Performance of pier segment option 1: standard beam with soffit

The efﬁciency of using a standard girder for both positive and negative spans
(Option 1) in continuous two-span spliced girder bridges was evaluated by considering
AASHTO’s Type IV I-girder. The girder was provided with a tapering attached sofﬁt on
the negative segment. The maximum sofﬁt thickness (over the pier center line) was
increased from 0.1H to 0.51-I, where H is the total height of the standard girder. The
width of sofﬁt was assumed to be equal to the width of girder bottom ﬂange.
The objective function for the optimization problem was the minimization of
superstructure construction cost. The AASHTO Type IV I-girders were spaced at 1.8 m

(6 ft), 2.1 m (7 ﬂ) and 2.4 m (8 ft) on center. Three splice locations were evaluated at

252

0.20L, 0.223L, and 0.25L 0.223L, where L is the total system span length. Only the
results for maximum achievable span length for different beam spacing, the sofﬁt
requirement, and splice location are shown in this section. The prestressing requirements
and their vertical locations along the span are provided in the form of design charts.

The maximum achievable span lengths for different sofﬁt thicknesses at different
splice locations and beam spacing were determined from the design optimization process
(see Section 5.4) and the results are shown in Figure 5-24 through Figure 5-26. Figure
5-24 represents the relationship of maximum sofﬁt thickness to achievable span length
for 1.8 m (6 ft) beam spacing. The splice location at 0.35(L) was found to be the most
efﬁcient splice location since it yielded a longer span than any other splice locations
with the same sofﬁt thickness. Results show that span length is increased about 300 mm
(1 ft) for every 137 mm (5.4in.) of added sofﬁt thickness (maximum value at pier),
which is about 10% of the total beam height (height of Type IV beam is 1220 mm (54
in.). The percent increase in span length compared to the total span length is less than
1%, which is very small. The results for 2.1 m (7 ft) and 2.4 m (8 ft) beam spacing are
shown in Figure 5-25 and Figure 5-26, which illustrate the same trends as discussed

above.

253

 

      

 

 

 

 

 

 

 

: f'c=6.5 ksi, fci=5Ksi, S=6ft
: ASSHTO Type IV
135 -
5.: 4
g —
a:
c
e
.1
1:
co
5',- 125 .1
I —e— Splice Loc 0.35
- -B— Splice Loc 0.375
- -é-— Splice Loc 0.40
120 —
I I r l I l j l I l
0.0 0.1 0.2 0.3 0.4 0.5

Soffit over Beam Height Ratio

Figure 5-24 Maximum span length of ASSH T 0 Type I V standard beam with
soﬂit at beam spacing 1.8 m (6 ft)

140

 

j f'c=6.5ksi, f'ci=5Ksi, S=7ft
f ASSHTO Type IV

 

Span Length (ft)

 

  
 

-e— Splice Loc 0.35
—a— Splice Loc 0.375
120 ~ -é-- Splice Loc 0.40

 

 

 

 

 

 

I I I I I r I I I I
0.0 0.1 0.2 0.3 0.4 0.5
Soffit over Beam Height Ratio

Figure 5-25 Maximum span length of ASSH T 0 Type I V standard beam with
soffit at beam spacing 2.1 m (7 ft)

254

140

 

 

f‘c = 6.5 ksi, M = 5 Ksi. S = 8 ft —9— Splice Loc 0.35

ASSHTO Type IV -a— Splice Loc 0.375

135 -A— Splice Loc 0.40

 

 

 

Span Length (ft)
5
L1 1 l l l L l 1 l

E]
E

a

N

0|
l11m1J11m1

.3
N
O

 

 

 

l

i I
0.1 0.2 0.3 0.4 0.5
Soffit over Beam Height Ratio

9
c

Figure 5-26 Maximum span length of ASSH T 0 Type I V standard beam with
sofﬁt at beam spacing 2.4 m (8 ft)

5.8.4 Dimensions and performance of optimized pier segments- options
2, 3 and 4

The objective function used to develop new optimal pier, or negative, segments
was to minimize concrete volume, which is equivalent to minimizing girder weight. The
new optimal negative sections include a new optimal non-prismatic section with varied
web depth (option 2), a new optimal non-prismatic section with constant web depth
(option 3), and a new optimal prismatic section (option 4). Minimum construction cost

was not chosen as the main objective function here since there is no explicit cost

255

available for the modiﬁed girder. However, an objective function based on minimum
concrete volume can be directly deﬁned from the girder geometry.

The development of optimal prismatic pier sections (options 4 —- see Figure 5-6
and Figure 5-10) was done in accordance to Section 5.4 for different span lengths, beam
spacing, and the most efﬁcient splice location (at 0.35L). Geometric design variables
were those needed for the relevant component optimization together with those from the
system optimization. The obtained optimal girder dimension results are listed in Table
5-6. These new sections were developed by increasing the span length in 3050 mm (10
ft) increment to obtain the beam height range. Designers and producers can use these
values to develop standard beams by grouping them within desired span length and beam
spacing ranges. The prismatic sections were used for both positive and negative

segments.

256

Table 5-6 Dimensions of optimized prismatic beams (option 4) for two-span continuous
spliced girder bridges — SI units

 

S L H b1 b2 t1 t2 t3 t4 Dw tw
(m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m)
1.8 36.6 1.27 0.61 0.46 0.13 0.13 0.13 0.13 0.76 0.20
1.8 40.2 1.55 0.61 0.46 0.13 0.13 0.13 0.13 1.03 0.20
1.8 42.7 1.65 0.61 0.46 0.13 0.13 0.13 0.13 1.14 0.20
1.8 45.7 1.91 0.61 0.46 0.13 0.13 0.13 0.13 1.38 0.20
1.8 48.8 2.13 0.61 0.46 0.13 0.13 0.13 0.13 1.63 0.20
1.8 52.4 2.51 0.65 0.46 0.13 0.13 0.13 0.13 2.01 0.20
1.8 54.9 2.72 0.65 0.46 0.13 0.13 0.13 0.13 2.20 0.20

2.1 36.6 1.37 0.61 0.46 0.13 0.13 0.13 0.13 0.86 0.20
2.1 40.2 1.66 0.61 0.46 0.13 0.13 0.13 0.13 1.15 0.20
2.1 42.7 1.78 0.61 0.46 0.13 0.13 0.13 0.13 1.27 0.20
2.1 45.7 2.08 0.61 0.46 0.13 0.13 0.13 0.13 1.56 0.20
2.1 48.8 2.34 0.61 0.46 0.13 0.13 0.13 0.13 1.83 0.20
2.1 52.4 2.64 0.71 0.46 0.13 0.13 0.13 0.13 2.13 0.20
2.1 54.9 2.90 0.68 0.46 0.13 0.13 0.13 0.13 2.37 0.20

2.4 36.6 1.47 0.61 0.46 0.13 0.13 0.13 0.13 0.97 0.20
2.4 40.2 1.78 0.61 0.46 0.13 0.13 0.13 0.13 1.25 0.20
2.4 42.7 1.93 0.61 0.46 0.13 0.13 0.13 0.13 1.40 0.20
2.4 45.7 2.18 0.61 0.46 0.13 0.13 0.13 0.13 1.68 0.20
2.4 48.8 2.49 0.62 0.47 0.13 0.13 0.13 0.13 1.98 0.20
2.4 52.4 2.79 0.76 0.46 0.13 0.13 0.13 0.13 2.26 0.20
2.4 54.9 3.02 0.74 0.46 0.13 0.13 0.13 0.13 2.50 0.20

 

Based on the component and system optimization results, new proposed girder
dimensions for prismatic elements were obtained and are listed in Table 5-6, while the
requirements and location of pre-tensioning and post-tensioning strands are partially
shown in Chapter 6 in the form of design charts. For a complete design charts, please

refer to MSU-CE report (Surakomol and Burgueno 2005).

257

Table 5-6 Dimensions of optimized prismatic beams (option 4) for two-span continuous
spliced girder bridges — US units (Cont...)

 

 

S L H b1 1); t1 t2 t3 t4 Dw tw
(fr) (it) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.)
6 120 50 24 18 5 5 5 5 30 8
6 132 61 24 18 5 5 5 5 41 8
6 140 65 24 18 5 5 5 5 45 8
6 150 75 24 18 5 5 5 5 54 8
6 160 84 24 18 5 5 5 5 64 8
6 172 99 26 18 5 5 5 5 79 8
6 180 107 26 18 5 5 5 5 87 8
7 120 54 24 18 5 5 5 5 34 8
7 132 66 24 18 5 5 5 5 45 8
7 140 70 24 18 5 5 5 5 50 8
7 150 82 24 18 5 5 5 5 61 8
7 160 92 24 18 5 5 5 5 72 8
7 172 104 28 18 5 5 5 5 84 8
7 180 1 14 27 18 5 5 5 5 93 8
8 120 58 24 18 5 5 5 5 38 8
8 132 70 24 18 5 5 5 5 49 8
8 140 76 24 18 5 5 5 5 55 8
8 150 86 24 18 5 5 5 5 66 8
8 160 98 24 18 5 5 5 5 78 8
8 172 1 10 30 18 5 5 5 5 89 8
8 180 1 19 29 18 5 5 5 5 98 8

 

It can be seen that the bottom ﬂange width (b 1) has a larger value than the top

ﬂange width (b2). Results for the thickness of the top ﬂanges (t 1 and t2), the bottom

ﬂanges (t3 and t4) and the web thickness (tw) yielded the pre-deﬁned lower bound values

for these variables. The prismatic girder height varied from 1.27m (50.01n.) to 3.02m
(119.0in.) depending on the desired span length and beam spacing. The percent increase

in total beam height is about 8% of span length increase.

258

The same speciﬁed span lengths and beam spacing were used as parameters to
develop optimal non-prismatic beams with constant web height and varied bottom ﬂange
thickness: pier segment option3 as shown in Figure 5-5 and Figure 5-9. Their resulting
dimensions are given in Table 5-7.

Table 5-7 Dimensions of optimized non-prismatic beams with constant web height and
varied ﬂange (option 3) for two-span continuous spliced girder bridges — US Units

 

S L H b1 b2 t1 t2 t3 t4 Dw tw S
(m) Am) (“0 (m) (m) (m) (m) (m) (m) (m) (m) (m)
1.8 36.6 1.19 0.61 0.46 0.13 0.13 0.13 0.13 0.20 0.68 0.12
1.8 40.2 1.35 0.61 0.51 0.13 0.13 0.13 0.13 0.20 0.83 0.13
1.8 42.7 1.52 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.00 0.15
1.8 45.7 1.73 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.22 0.17
1.8 48.8 1.83 0.61 0.51 0.13 0.13 0.13 0.18 0.20 1.27 0.18
1.8 52.4 2.18 0.61 0.51 0.13 0.13 0.13 0.15 0.20 1.65 0.22
1.8 54.9 2.34 0.61 0.51 0.13 0.13 0.13 0.18 0.20 1.78 0.23

2.1 36.6 1.24 0.61 0.51 0.13 0.13 0.13 0.13 0.20 0.72 0.12
2.1 40.2 1.43 0.61 0.51 0.13 0.13 0.13 0.13 0.20 0.92 0.14
2.1 42.7 1.60 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.09 0.16
2.1 45.7 1.83 0.66 0.51 0.13 0.13 0.13 0.13 0.20 1.31 0.18
2.1 48.8 2.01 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.49 0.20
2.1 52.4 2.13 0.61 0.51 0.13 0.13 0.13 0.18 0.20 1.68 0.21
2.1 54.9 2.44 0.66 0.51 0.13 0.13 0.13 0.20 0.20 1.86 0.24

2.4 36.6 1.37 0.61 0.46 0.13 0.13 0.13 0.13 0.20 0.86 0.14
2.4 40.2 1.52 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.02 0.15
2.4 42.7 1.75 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.18 0.18
2.4 45.7 1.91 0.61 0.51 0.13 0.13 0.13 0.13 0.20 1.40 0.19
2.4 48.8 2.24 0.61 0.51 0.13 0.13 0.13 0.18 0.20 1.64 0.22
2.4 52.4 2.34 0.61 0.51 0.13 0.13 0.13 0.19 0.20 1.77 0.23
2.4 54.9 2.57 0.61 0.51 0.13 0.13 0.13 0.20 0.20 1.98 0.26

 

The obtained values for the bottom ﬂange width (b1) were similar to the

developed optimal prismatic section (see Table 5-6). However, larger values of the top

ﬂange width (b 2) were obtained for this new non-prismatic section. Again, the obtained

259

values for the thickness of the top ﬂanges (t 1 and t2) and the web thickness (tw) hit the
pre-deﬁned lower bound. As expected, the thickness values for the bottom ﬂanges (t3

and t4) do vary as noted in the results table. The maximum total height over the negative

span varied from 1.31 m (52.0 in.) to 2.82m (111.0 in.) for the different span lengths and
beam spacings. The percent increase of maximum total beam height for the negative
segment is about 10%.

Table 5-8 lists the maximum achievable span lengths and new girder dimension
for optimal non-prismatic beams with varied web height and constant bottom ﬂange
thickness: option 2 as shown in Figure 5-4 and Figure 5-8. The same previously
speciﬁed span lengths and beam spacings were used as parameters to develop these new

girders. Similar results to the new prismatic section (see Table 5-6) were obtained for the

bottom ﬂange width (b 1), but various values of top ﬂange width (b2) were obtained for
this new optimal non-prismatic section. The thickness of the top ﬂange (t 1 and t2) and

bottom ﬂange (t 3 and t4), and the web thickness (tw) were all bounded by the predeﬁned

lower value. The maximum total height over the pier varied from 1.8 m (69.0 in.) to 3.1
In (120.0 in.) for the different span lengths and beam spacings. The percent increase of

the maximum total beam height for the negative segment was approximately 10%.

260

Table 5-7 Dimensions of optimized non-prismatic beams with constant web height and
varied ﬂange (option 3) for two-span continuous SGB — US units (Cont...)

 

 

S L H b1 b2 t1 t2 t3 t4 Dw tw S
(ft) (ft) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.)
6 120 47 24 18 5 5 5 5 8 27 4.7
6 132 53 24 20 5 5 5 5 8 33 5 .3
6 140 60 24 20 5 5 5 5 8 39 6.0
6 150 68 24 20 5 5 5 5 8 48 6.8
6 160 72 24 20 5 5 5 7 8 50 7.2
6 172 86 24 20 5 5 5 6 8 65 8.6
6 180 92 24 20 5 5 5 7 8 70 9.2
7 120 49 24 20 5 5 5 5 8 28 4.9
7 132 56 24 20 5 5 5 5 8 36 5.6
7 140 63 24 20 5 5 5 5 8 43 6.3
7 150 72 26 20 5 5 5 5 8 52 7.2
7 160 79 24 20 5 5 5 5 8 59 7.9
7 172 84 24 20 5 5 5 7 8 66 8.4
7 180 96 26 20 5 5 5 8 8 73 9.6
8 120 54 24 18 5 5 5 5 8 34 5.4
8 132 60 24 20 5 5 5 5 8 40 6.0
8 140 69 24 20 5 5 5 5 8 46 6.9
8 150 75 24 20 5 5 5 5 8 55 7.5
8 160 88 24 2O 5 5 5 7 8 65 8.8
8 172 92 24 20 5 5 5 7 8 70 9.2
8 180 101 24 20 5 5 5 8 8 78 10.1

 

261

Table 5-8 Dimensions of optimized non-prismatic beams with constant ﬂange and
varied web (option 2) for two-span continuous spliced girder bridges -— SI units

 

s
(m)
1.8
1.8
1.8
1.8
1.8
1.8
1.8

2.1
2.1
2.1
2.1
2.1
2.1
2.1

2.4
2.4
2.4
2.4
2.4
2.4
2.4

L
(m)
36.6
40.2
42.7
45.7
48.8
52.4
54.9

36.6
40.2
42.7
45.7
48.8
52.4
54.9

36.6
40.2
42.7
45.7
48.8
52.4
54.9

H
(m)
1.17
1.30
1.42
1.60
1.78
2.06
2.26

1.22
1.42
1.52
1.68
2.11
2.16
2.54

1.30
1.60
1.70
1.93
2.03
2.49
2.59

b1
(m)
0.61
0.61
0.61
0.61
0.61
0.61
0.61

0.61
0.61
0.61
0.61
0.61
0.61
0.61

0.61
0.61
0.61
0.61
0.61
0.61
0.61

b2
(m)
0.46
0.46
0.46
0.46
0.46
0.51
0.51

0.46
0.46
0.46
0.46
0.46
0.51
0.51

0.48
0.46
0.46
0.48
0.51
0.46
0.51

t1
(m)
0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

t2
(m)
0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

t3
(m)
0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

t4
(m)
0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

0.13
0.13
0.13
0.13
0.13
0.13
0.13

Dw
(m)
0.20
0.20
0.20
0.20
0.20
0.20
0.20

0.20
0.20
0.20
0.20
0.20
0.20
0.20

0.20
0.20
0.20
0.20
0.20
0.20
0.20

tw
(m)
0.66
0.79
0.89
1.09
1.24
1.55
1.75

0.72
0.89
1.02
1.17
1.57
1.63
2.01

0.79
1.09
1.19
1.42
1.50
1.96
2.03

s
(m)
0.12
0.13
0.14
0.16
0.18
0.21
0.23

0.12
0.14
0.15
0.17
0.21
0.22
0.25

0.13
0.16
0.17
0.19
0.20
0.25
0.26

 

262

Table 5-8 Dimensions of optimized non-prismatic beams with constant ﬂange and
varied web(option 2) for two-span continuous spliced girder bridges — US units
(Cont...)

 

 

S L H b1 13; t1 t2 t3 t4 Dw tw S
(11) (ft) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.) (in.)
6 120 46 24 18 5 5 5 5 8 26 23
6 132 51 24 18 5 5 5 5 8 31 22
6 140 56 24 18 5 5 5 5 8 35 27
6 150 63 24 18 5 5 5 5 8 43 29
6 160 70 24 18 5 5 5 5 8 49 34
6 172 81 24 20 5 5 5 5 8 61 40
6 180 89 24 20 5 5 5 5 8 69 26
7 120 48 24 18 5 5 5 5 8 28 14
7 132 56 24 18 5 5 5 5 8 35 26
7 140 60 24 18 5 5 5 5 8 40 29
7 150 66 24 18 5 5 5 5 8 46 30
7 160 83 24 18 5 5 5 5 8 62 15
7 172 85 24 20 5 5 5 5 8 64 42
7 180 100 24 20 5 5 5 5 8 79 19
8 120 51 24 19 5 5 5 5 8 31 10
8 132 63 24 18 5 5 5 5 8 43 10
8 140 67 24 18 5 5 5 5 8 47 20
8 150 76 24 19 5 5 5 5 8 56 12
8 160 80 24 20 5 5 5 5 8 59 38
8 172 98 24 18 5 5 5 5 8 77 19
8 180 102 24 20 5 5 5 5 8 80 18

 

Correlation between the maximum total beam height, including the sofﬁt of
negative segments, at a beam spacing of 1.8 m (6 it) at different span lengths for all of
the newly proposed optimized negative sections are shown in Figure 5-27. The total
height of the optimal non-prismatic beam with varied web over the negative region
(option 2) is the deepest beam while the new non-prismatic beam with varied bottom
ﬂange (option 3) yields the shallowest beam at the same span lengths. The height of the

new prismatic girder (option 4) is bounded by the height of the new non-prismatic girder

263

beam with varied bottom ﬂange (option 3) and the height of the new non-prismatic

beams with varied web (option 2).

 

 

N
O

 

130 ,
3 f’c=6.5ksi, f'ci=5Ksi, S=6ft
120 1
A 110 —:
.E :
E 100 '3
.9 :
0 90':
'3 :
g 80%
.n -
E
o
I-

G
O

/ -O— Varied web

+ Varied ﬂange
+ Prismatic

0|
0
\

 

 

 

 

 

.h
C

IIIIIIIIIIIIIIIIIIITIIIIIIIIIIIIII

120 130 140 150 160 170 180 190
Span Length (ft)

Figure 5-27 Total beam height of new optimal negative segments for 1.8m (6ft)
beam spacing

When the same total height is used as a given parameter, the maximum achievable
span length for each of the newly developed negative sections can be obtained and
compared from Figure 5-27. For example, for a negative section at beam spacing of 1.8
m (6 ft) the new prismatic (option 4) and new non-prismatic beam with varied ﬂange
(option 3) can span longer than the new non-prismatic beam with varied web depth

(option 2).

The above optimal results were based on the minimization of concrete volume, or

equivalently to the minimum concrete weight of the girders. The proposed beam height

264

and dimension results can be used to develop standard negative pier segments for two-
span continuous spliced girder bridges. Although the new non-prismatic sections with
varied ﬂange thickness were shown to be more efﬁcient in the negative region than the
prismatic segment, in practice, the contractor may prefer to develop a standard prismatic

negative section since the manufacturing of a prismatic beam is most likely lower.

5.9 Multi-Criteria Optimum Design of Two-Span Continuous
SGB

A three-lane two-span continuous PCI-BT 96 spliced girder bridge with single-
stage post-tensioning aﬁer deck casting was chosen to study how multi-criteria
optimization can improve the design of two-span continuous spliced girder bridges. A
two-span continuous spliced girder bridge with a total width of 18.6 m (61 ﬂ) and a
maximum beam spacing of 2.7 m (9 ft) was used in this study. The total span length of
118.2 m (388 ft) consists of three segments: two equal end, or positive segments and a
middle, or pier, segment. The splice location, which is the ratio of the end segment
length (L,) and the total span length (L), was chosen to be at 0.35 (L) since it was
indicated by earlier studies to be the best splice location for the two-span system.
Concrete with a ﬁnal compressive strength of 62 MPa (9 ksi) and initial compressive

girder strength of 34.5 MPa (5 ksi) was assumed for the girders.

5.9.1 Multi-criteria objective functions

The multi-criteria objective function was composed of a set of different single

objective functions. F1(x) was a multi-criteria economics function deﬁned by the

combination of minimum superstructure cost f1(x), a minimum number of pre-tensioned

265

strands f2(x), and a minimum number of post-tensioned strands f3(x). Each single

objective function was set to obtain their optimal design solution for the speciﬁed

system. The objective functions are expressed as follows:

F1(x),,,,-,, = Economics Multi-criteria

: ”100mm f2(x)minf3(x)min]

f1(x) = Minimum superstructure cost

=CPS+CPT+CCD+CTS+CB+CR

f2(x) — Minimum number of pre-tensioning strands

= nprj 1+ nprz 1+ npr31+ npr41+ npr12+ "1902+

npr32+ npr42+ nprt1+ nprt2

f 3(x) = Minimum number of post-tensioning strands

= npo

(5-17)

(5-18)

(5-19)

(5-20)

Each of the single objective functions was formulated separately to determine its

optimal solution. Details of each of the parameters used to formulate the function of

minimum superstructure cost in Equation 5-18 were described earlier in Table 5-2, while

the objective function for the minimum number of pre-tensioned strands includes the

total number of all pre-tensioning strands on the top and bottom girder ﬂanges in all

segments.

266

5.9.2 Results and comparison to single-objective optimizations

Optimal design variable solutions for each of the single objective functions were

obtained for a total span length (L) of 118.2 m (388 ﬂ) and are given in Table 5-9.

Detailed design variable notations were speciﬁed in Section 5.6 and are not repeated

here.

Table 5-9 Optimization results for individual objective functions

 

L =118.2 m (388 ft)

Objective function

 

Design Variables
f1 (x) fzfx) f3(x)
min npo min nprﬁ min const cost
"m 1 12 13 T 10
npr21 11 5 10
npr31 5 5 5
"PHI 1 2 2
nprzz 4 2 7
nprzz 2 0 0
npr 32 2 O 0
"W42 2 0 0
npo 19 35 19
ya 79mm(3.1”) 25mm(1”) 25mm(l”)
J’b 1.45 m (57”) 736 mm (29”) 1.45 m (57”)
nptI 7 6 6
nptz 12 3 7
ye -229 mm (9”) -64 mm (2.5”) -211 mm (8.3”)
yd -813 mm (32”) -453 mm (18”) -813 mm (32”)
pm 0.35 0.35 0.35
Objective function 19 36 $9,160,000
57 36 48

nprii (total)

When the objective is to obtain a minimum number of post-tensioning strands,

the maximum number of pre-tensioning strands was obtained and vice versa. Thus,

267

minimum construction cost provides a compromise between the minimum number of
pre-tensioning strands and the minimum number of post-tensioning strands. Since these
objectives contradict each other, a compromised solution is necessary. Multi-criteria
optimization was thus used to obtain a compromised solution between these

contradicting objective functions (see Section 3.5).

5.9.3 Pareto optima results

Once different single-obj ective solutions were obtained, these results were used as
design constraints to determine a set of pareto Optima (see Section 3.5.3). When all
objective criteria are not equally important or conﬂicting, vector optimization can be
used to select the preferred solution in a more rational way. Multi-criteria optimization
techniques are used to generate a set of pareto optima, which are non-dominated
solutions (see Section 3.5.3).

The range of minimum and maximum values for each objective function was
determined after obtaining design variable solutions for each single objective function.
The number of total pre-tensioning strands varied from 36 to 57, while the number of
post-tensioning strands varied from 19 to 35, as shown in Table 5-9.

Different paired sets of objective functions were chosen to obtain the pareto sets.
Pareto optima sets are generated by a constraint approach that is based on an adoption of
one criterion as a scalar primary objective function and the transformation of other
objective criteria into constraints bounded by some pre-determined constants (see
Section 3.5.2). These scalar constraints are altered from the range of minimum to

maximum values, and the subset of pareto optima is generated.

268

The multi-criteria objective is composed of three single objective functions (see
Equation 5-17). The single objective functions were selected in pairs to determine the
pareto optima set. Two pairs of objective functions were selected for this study. Each
pair of objective functions had one objective function as a primary objective function
and the other objective ﬁlnction was set as a constraint. The constraints were divided
into arbitrary intervals. As the interval range reduces the accuracy of results is closer to
the real compromised results. The constraints for this study were divided into ﬁve
intervals.

The ﬁrst pair of objective functions was set to minimize the total number of pre-
tensioning strands as the main objective function while varying the total number of post-
tensioning strands as a constraint. Pareto sets of the number of pre-tensioning strands as
an objective and the numbers of post-tensioning strands as constraint are given in Table

5-10. The Euclidian distance and maximum distance of these Pareto sets is minimum at

L2(x) = 0.18 and Loo(x) = 0.23, respectively. These values correspond to Solution # 2,

where npo = 22 (number of pos-tensioning strands) and npri; = 39 (total number of pre-

tensioning strands).

269

Table 5-10 Pareto sets of nprii as an objective and npo as a constraint

 

 

 

 

 

 

Solution Constraints Objective
functron
f3(x) f2(X) (Lg-Mtg (tamer La(x) L2(x)
min npo min dfbmax dfznnax Maximum Euclidian
nprﬁ distance distance
1 19 57 0.00 1.00 1.00 1.00
2 L 22 39 I 0.18 0.14 I 0.18 0.23
3 25 36 0.37 0.00 0.37 0.37
4 28 36 0.57 0.00 0.57 0.57
5 32 36 0.83 0.00 0.83 0.83
6 35 36 1.00 0.01 1.00 1.00

 

The second pair of objective functions was set to minimize the total number of
post-tensioning strands as the main objective function while varying the number of total
pre-tensioning strands as a constraint. The resulting Pareto sets are given in Table 5-11.

The Euclidian distance and maximum distance of these Pareto sets is minimum at L2(x)

= 0.22 and Loo(x) = 0.30, respectively. These values correspond to Solution # 2 where

npo = 22 (number of post-tensioning strands) and npril- = 41 (total number of pre-

tensioning strands).

270

Table 5—11 Pareto sets of npo as an objective and nprii as a constraint

 

Solut Objective

 

 

 

 

 

. . Constraint
ron function
f2(x) f3(X) . . Lab!) L2(X)
min 11 o , (tr-"ml (Ll-M12 Maximum Euclidian
p m nprﬁ df2)ma‘x df1,max distance distance
1 35 36 1.00 0.00 1.00 1.00
2 l 22 41 l 0.20 0.22 [ 0.22 0.30
3 20 45 0.08 0.41 0.41 0.42
4 19 49 0.00 0.60 0.60 0.60
5 19 53 0.00 0.79 0.79 0.79
6 19 57 0.00 1.00 1.00 1.00

 

A subset of pareto optima represented by six solutions is illustrated in Figure
5-28. This ﬁgure shows the relationship of pre-tensioning and post-tensioning strand
requirements when one of them is the main objective function and the other is
transformed to be a pre-determined constant constraint and vice versa. The best
compromise solution of these objective functions is indicated by the dashed line in the

ﬁgure and also highlighted (bold text) in Table 5-10 and Table 5-11.

271

 

 

  
   
 
   

 

 

 

 

 

 

40
: f'c = 62 MPa, f'ci = 34.5 MPa, S = 2.7 m, PCI-BT 96

8 :rc=9ksi,rci=5ksi,s=9n
5 35 — -O- Pareto - Min npo
«a Z + Pareto - Min nprii
g I
c 30 —
.2 ‘
m _
c : B t '
3 25 J es compromise
J.- - solution
3 1.1192 .=. .23 ......
Q I
..- 1

20 — 1
°. - Nprii = 40;
O - i
z - l

15 I I I I I I I I I I 5 I I T I T I I I I I I I I I I I I I

30 35 40 45 50 55 60

No. of pre-tensioned strands

Figure 5-28 Pareto set of pre-tensioning and post-tensioning strands

A solution that corresponds to the best compromise between the optimized
solutions can be directly obtained since they are part of the optimization results. The
results for all design variables are given in Table 5-12. Thus, the optimal solution for the

multi-criteria optimization is obtained when npo = 22 (number of post-tensioning

strands) and npn-i = 40 (number of pre-tensioning strands) which is the best compromise
solution. This solution is between the minimum and maximum range that was obtained
in Section 5.9.2. The multi-criteria optimal solution from the compromised programming

is non-dominated with regard to all criteria and ensures the best compromise between all

competing criteria.

272

Table 5-12 Best solutions to pareto sets

 

 

1st Pair 2nd Pair
Main
objective Min nprii Min npo
Constant n 0
Constraint p np r ’7
ner 1 1 1 1
npr21 10 8
npr31 4 5
npr41 1 1
npr12 3 3
nprzz 0 0
npr32 0 0
npr42 0 0
npo 22 22
ya 25 mm (1”) 51 mm (2”)
yb 1448 mm (57”) 1422 mm (56”)
yc -191 mm (7.5”) -84 mm (3.3”)
yd -813 mm (32”) -760 mm (29.9”)
nptI 6 6
nptI 5 6
”t 40 40

 

5.10 Summary

In this chapter, an integrated component and conﬁguration optimization approach
was applied to the design and optimization of two-span continuous spliced girder
bridges. Different construction methods were investigated and the multi-staged
construction sequence method was found to be the most efﬁcient construction sequence
for two-span continuous spliced girder bridges due to the maximum increase in their
span length. Development of optimal pier segments was investigated and implemented.

The minimization of concrete volume was used as an objective function to develop new

273

optimal pier segments since their objective function could be directly deﬁned, unlike
fabrication cost. Existing standard beams with increased sofﬁts were used over the
negative segment to increase span length. However the increase in maximum span length
was found to be very small compared to the required increase in sofﬁt thickness. New
optimal prismatic and non-prismatic sections were thus developed to be used over the
negative moment region section. With the same speciﬁed span length, a newly proposed
optimal non-prismatic girder section with constant web depth yields the shallowest
section than other newly developed girder sections. Results from multi-criteria
optimization yielded a compromise solution for different single objective functions and
should be used when decision-making must be made for several competing merit

criteria.

274

6 Optimally-Derived Design Aids and Verification

Example

6.1 Introduction

The proposed design aids in this research were developed to simplify the analysis
and design of spliced girder bridges by providing guidance on variable sensitivity,
facilitate the typical trial and error design process, and to expedite the design of this type
of bridge systems. The availability of design aids for spliced girder bridges can be a
great asset to bridge engineers to have more alternatives during the bridge type selection
with less time effort, hopefully resulting in its wider use by state highway agencies.

In this chapter, the development of design aids for the preliminary design of
spliced prestressed concrete girder bridges is demonstrated. Optimal design variables
that were obtained from the nonlinear structural optimization techniques (see Section
4.5.1; Section 5.4.5; and Section 5.6.2) are shown as design aid charts. These optimal
solutions were used in design examples to demonstrate how the developed design aids
can be applied in the preliminary design of spliced girder bridges. In addition, the
optimal solutions from design aid charts were veriﬁed by examining results from the
NCHRP 517 report study (Castrodale and White 2004) and proven to be acceptable

solutions.

275

6.2 Preliminary Design Example 1

In this section, Design Example 1 is used to demonstrate the preliminary design of
a single-span spliced girder bridge. The optimal design variables from the optimization
design were used in these problems. Design Example 1, detailed calculations for both
service and the strength limit states are shown. For Design Example 2 only the service
limit state calculations are shown since the strength limit state calculations for two-span

continuous spliced girder bridges are similar to that of single-span spliced girder bridges.

6.2.1 Problem statement

A three-lane single-span spliced girder bridge with a total width of 18.59 m (61 ﬂ),
and a span length (L) of 60.35 m (198 ft) consisting of three segments, as illustrated in
Figure 6-1, was used as a prototype structure in this study. The PCI-BT 96 girder was
chosen in order to verify the present approach with the results documented for Design
Example 1 of the NCHRP 12-57 report. (Castrodale and White 2004). Girder dimensions

are shown in Figure 6-2.

 

 

 

 

 

2 Total Span Length (L)

End Span([_1) l MidsponlLZ) | End SpanlLll
SpliCing Jornt‘j>L\ I / Deck
1111 H
CL Splice

 

 

& CL Temp Support

Figure 6-1 T hree-segment single-span spliced precast/prestressed girder bridge

276

The end segments were assumed to be equal length (L1). The splice location is
deﬁned as the ratio of the end segment length (L 1) to the total span length (L) and was
assumed to be located at 0.223L. The length of the middle segment is thus deﬁned as L;
= L — L1. The cast-in-place concrete slab thickness was assumed to be 216 mm (8.5 in.)
with 50 mm (2 in.) built-up built on a 2.7 m (9 ft) girder spacing.

A single-staged post-tensioning construction was used in this evaluation.
Segments are assumed to be placed on temporary supports and post-tensioning is
assumed to be applied after the deck and splice joints are cast. A full depth diaphragm
cast with the splice was assumed to be provided at the splice locations. No other
intermediate diaphragms were assumed. The design study was conducted using the
simpliﬁed design method of the AASHTO-LRFD Bridge Design Speciﬁcations
(AASHTO 2003) and the recommendations of NCHRP 517 report (Castrodale and

White 2004). Both service limit states and ultimate limit states were considered in the

design.

 

 

 

 

 

 

 

 

 

i F‘ﬁL-b zl/Ti
E E A "040; m
o .
g g L0.050 m
E C; o‘
8 E 0.200 m
V O V ‘
- in
N CD
E
:2 _ L0.250 rn
CLA— ,
l ‘1 K \\1
l+—>l
0,150 m__l 0.700 m

Figure 6-2 System and girder cross sections for single-span spliced girder bridge study

277

The purpose of this design example is to illustrate how the developed design charts
can be used as an aid for the preliminary design of single-span spliced girder bridges.
The same prototype structure as in Section 6.2 is used for this preliminary design
example 1. The section at the mid-span section of Segment 2 (x = 30.175 in [99 11]

measured from the end span) of the spliced girder bridge is checked in this example.

6.2.2 Preliminary section properties

Non-composite PCI-BT 96 Girder Section Properties

ybg = 1231 m (48.47 in.)
3 . 3
Stg = 434,486,615 mm (26514 in. )
3 . 3
Sbg = 425,981,729 mm (25995 in. )
2 . 2
Ag=711,611 mm (1103 in.)
4 . 4
1g = 5.2449ell mm (1260081 in. )

Composite PCI-BT 96 Girder Section Properties

Ac = 1,204,514 mm2 (1867 111.2)
4 . 4
16: 1.203e12 mm (2474500 111. )
3 . 3
Sbc=582,117,675 mm (35523 In. )
3 . 3
5,6: 1,539,368,018 mm (93938 1n. )
3 . 3
5,6,: 1,398,767,009 mm (853581n. )

ACC = 1,447,739 mm2 (2244 inz)

278

ybc = 1,769 mm (69.66 in.)

ytc = 669 mm (26.34 in.)

6.2.3 Preliminary material properties
Girder Material Properties

fc' = 44.8 MPa (6.5 ksi)

fci' = 34.5 MPa (5 ksi)

Egi = 29,558 MPa (4287 ksi)

i:g = 33,702 MPa (4888 ksi)
3 3

WC = 2,400 kg/m (150 lb/ﬁ )

fcd' = 24.1 MPa (4.5 ksi)

Deck Material Properties

fcdi'= 31.0 MPa(3.5 ksi)
Ecdi = 24,731 MPa (3372 ksi)
Ecd = 28,041 MPa (3824 ksi)

ch = 2,400 kg/m3(150 lb/ﬁ3)
Pre—tensioning Strands Properties

aps = 140 mm2 (0.217 inz)

Ep = 196,502 MPa (28500 ksi)

279

1;,” = 1,861 MPa (270 ksi)
1;), = 0.91;,“ = 1,675 MPa (243 ksi)
fpo = 0.751;. = 1,396 MPa (202.5 ksi)

Post-tensioning Strands Properties

em = 140 mm2 (0.217 i112)

5,, = 196,502 MPa(28500 ksi)
fpu = 1,861 MPa (270 ksi)

1;), = 0.91;... = 1,675 MPa (243 ksi)

fpj = 0.9f,,y = 1,508 MPa (219 ksi)

6.2.4 General information
S = 2438 mm (8 ft)
t5 = 216 mm(8.5 in.)
Humidity = H = 75
t, = 1 day
td = 60 days
tpo = 75 days

Ase, = 9.53 mm (0.375 in.)

280

6.2.5 Use of design charts

The design variables in the optimization problem are those deﬁning the amount
and layout of the pre—tensioning and post-tensioning strands in the girders as described in
Figure 6-3. The location of each strand layer is assumed to be 50 mm apart, including
the distance of the ﬁrst row of strands to the beam top and bottom. Normally, the
number of post-tensioning strands requires the use of several ducts. To avoid having
several ducts deﬁned in the design variables, an effective post-tensioning proﬁle is used
to represent a group of ducts. The effective post-tensioning duct is assumed to follow the
proﬁle of a quadratic parabola.

Results from the optimization process were used to develop complete design aid
charts for PCI-BT 96 single-span spliced girder bridges as shown in Figure 6-4 through
Figure 6-7. These ﬁgures are used as an example to demonstrate how the developed
design aids can be used. These design charts were developed based on beam Spacing 2.7
m (9 ft), initial and ﬁnal concrete compressive strength are 35 (5 ksi) and 45 MPa (6.5
ksi), respectively. These results are direct output from the optimization procedure

(Section 4.5. 1 a), where the span length was kept as a design parameter.

281

Post-Tensioned

/ strands per duct

 

——‘—————

 

 

 

 

’

-’
—.——

‘—— ’l
_1— _ﬂ-

”--
_—
ﬂﬁ

 

 

 

 

 

 

y
o post-tensioned
Db ,
1 profile
.2 L 2

nprt1

npr t2

 

(a) Post-tensioning Requirement

50 0111(2”)
I—

 

-—-—-—-—-—-—---qp-—-—-—-—-—-—-—-—-—c—n—n—c—n—n—o—u—o

f 0.0.

PSPA@
50 111012”)

(Typ.)

 

 

 

 

 

 

 

 

 

 

 

 

_npr21lTyp.)
npq1(Typ.)

 

_nprzzt Typ. )
nprnl Typ. )

 

 

npr42

 

 

t_npr41(Typ.)
1
—npr31(lyp.)

‘1‘ ‘

—npr32

 

L1

 

 

 

 

(b) Pre-tensioning Requirement

‘AA‘A‘
V

7

 

Figure 6-3 Variables for the design optimization of single-span spliced girder bridges

Figure 6-4 and Figure 6-5 provide traces that relate the optimal requirements of

pre-tensioning strands per row in the end and middle girder segments, respectively. The

relation between the amount of continuous post-tensioning to the achievable span length

282

is illustrated in Figure 6-6. The eccentricity locations of the effective post-tensioning
duct at the bridge ends and at mid-span are shown in Figure 6-7. In addition to the solid
lines in the ﬁgures, which represent the optimized results, dotted lines are projected from
the x-axis which is the given Span length 60.4 m (198 ft) and extended into the desirable
trace in Figure 6-4 through Figure 6-7 to obtain the amount of prestress requirements
and eccentricity locations of the effective post-tensioning duct in each segment from the
y-axis. The resulting values (design variables) for the prototype structure are noted in the
ﬁgures.

From the charts (Fig 6-4 to 6-7) the required amount of pre-tensioning for the
end-segment is 2 strands on the ﬁrst bottom layer and 2 strands on the ﬁrst top layer, as
illustrated in Figure 6-4. The mid-segment requires 8 strands on the ﬁrst three bottom
layers, 2 strands on the forth bottom layer, and 6 strands on the ﬁrst top layer, as shown
in Figure 6-5. The requirement for post-tensioning strand is 52 strands, as shown in
Figure 6-6, which is assumed to be divided into 4 ducts as shown in Figure 6-8b. The
location of the centroid the post-tensioning strands at the end-segment is 770 m (30.31

in.) and is 400 mm (15.74 in.) at the mid-segrnent, as illustrated in Figure 6-7.

283

 

 

 

 

 

  

 

 
 

 

 

 

18
- f‘c =45 MPa, f'ci = 35 MPa Splice Loc. = 0.223L, S = 2.7 m
16 —
‘ -O— npr11
14 i + npr21
812 _ + npr31
1.: - + npr41
£10 — + nprt1
U) 2
"5 8
o'
z a l
.1
4 _l
4 npr11= nprt1= 2
2 t
npr21 = npr31 = npr41 = 0 /l\ L = 60.4 m
o TITII‘TTTTﬂﬁTTj

56 58 60 62 64
Span Length (m)

Figure 6-4 Number of pre-tensioning strands per row in end-segment for PCI BT-96
beam

 

 

 

 

 

 

 

 

 

 

 

18
‘ “‘45 MPa- f'ci=35 MPa. Splice Loc.=0. 223L,S=2.7m
16 —
‘ + npr12
‘4 "‘ + npr22
.812 __ + npr32
: - -'V'- npr42
"5 — —O— n rtz
$10 - p npr12=npr22=npr32=3
“' —k ' 4 ”..__...
O ‘ ' II\
0' nprt2-6
Z 5 1* 1 c $+__._.
‘ l
4 _ l
‘ n r42=2 I
2 p . 7 |
0 IL=60.4m
r I I I I j 1 I 1 I I I I I I 1 y r I r, ﬁr
52 54 62 64

Figure 6-5 Number of pre-tensioning strands per rows in mid-segment for PCI-B T 96
beam

284

 

75; f'c=45MPa, f'ci=35MPa S=2.7m

 

70 — -O— Splice Loc. = 0.20L
65 - —I— Splice Loc. = 0.223L
m 60 ﬂ -i-— Splice Loc. = 0.25L

 
  
  
    

 

 

 

C '1‘: ____________________
550 — ‘npo=52

 

 

P——-—---

L=60.4m
15 l l l I l l l I r l T I r l f rtf l

 

j l I l l I

I
56 58 60 62 64
Span Length (m)

Figure 6-6 Optimized total number of post-tensioning strands for PCI B T -96 beam for
different splice locations

 

l

f'c =45 MPa, f'ci = 35 MPa, Splice Loc. = 0.223L, S = 2.7

 
  

 

u-l
b
l

l

—O—ya

 

 

 

O
on
ILnglAI

P
c»

l l

1

Vertical location (m)

o
.p.
i [4*

l

l

l

l

I

I

I

l

l

I

l

l

|

l

 

 

 

 

Figure 6-7 Post-tensioning proﬁle at end and mid-segments for PCI BT-96 beam

285

The values for the variables used in this preliminary design calculation are
obtained from the optimized design charts as shown in Figure 6-4 through Figure 6-7.

These variable values are:

npr11: 2; npr12 = O; npr13 = 0;
npr/4 = O; nprt1: 2; npr21: 9;
nprgg = 8; npr23 = 8; npr24 =2;
nprt2 = 6; npo = 52; ybb = 165 mm (6.5 in.);

ya = 775 m (30.31 in.).

The service and ultimate limit states at the mid-span section of Segment 2 is
checked herein to verify that the results provided by the design aids do in fact satisfy all

serviceability and strength requirements.

6.2.6 Service limit state

Service limit states were deﬁned in terms of ﬂexural stress limits on the girder
and deck. For ﬂexure, service limit state requirements govern the required prestressing
force. The stress limit states for each critical construction stage of the single-span spliced
girder bridge when using a single-post-tensioning (see Section 4.5.1) are evaluated as

follows:

Stage 1: Stress check at pre-tensioning
The end-segment, or segment 1, length is based on the spliced location and can

be determined as:

L, = (splice location)(L) = (O.223)L = (0.223)(60.35) = 13.46 m.

286

End segments are assumed to be equal, therefore the mid-segment length can be
computed as:
L2 = L — (2)(L1) = 33.43 m.

The centroid of all pre-tensioned strands is computed as follows:

2 (nm 2 )(50) + ("W22 )(100) + (npr32)(150) + (npr42 )(200) + (nprt2 )(ybg + ytg - 50)

ye2
(mm + nprzz + npr32 + npr42 + nprtz)

yez = 522 mm.

The eccentricity of the pre-tensioned strands at the midspan of mid-segment is
computed as:
epg = J’bg —yez = 709 mm.

The total pre-tensioned strand areas include the area of all pre-tensioned strands

in the top and bottom girder ﬂanges:

2
Apsz = (apS)(npr12 + nprzz + npr32 + npr42 + nprt2) = 4,620 mm .

The girder dead load moment a the midspan of Segment 2 is computed as
_1 2 _
Mgz -§AngL2 — 2,313 kN-m
The stress at the centroid of the pre-tensioned strands of the mid-segment caused
by pre-tensioning and girder dead loads is calculated as:

2
0-7Ap32fpu + 0'7Ap32fpuep2 _ MgZepZ

fcgplZ =
Ag 1g Ig

=11.11MPa

Elastic shortening prestress losses due to the pretensioning at the release stage

can be determined as follows:

287

 

E
mm = fcgplz p = 73.84 MPa.
5g,-

Relaxation prestress losses at release of the pretensioned strands are deﬁned as:

 

lo (24: ) f 1'
dpr1 = g40 "1 [fp 4.55]ij 213.65 MPa.
P)’

The effective prestress at the centroid of the pre-tensioned strands after all losses

at release of pre-tensioning is calculated as follows:

fpelZ = fpei -de52 __dfPRl 21308.70 MPa.

The total top girder stress at the mid-span of the mid-segment at Stage 1 is

 

 

 

deﬁned as:
M A A e t
0,1 = g2 + pSZf “12 — pSZf p812 p2 =3.95 MPa< 0.6fci =20.68 MPa.
s,g Ag s,g

The total bottom girder stress at the mid-span of the mid-segment at Stage 1 is

 

calculated by:
M A A e I

01,1 =- g2 + ”We” — ”We” 1’2 =13.13 MPa< 0.6113,. =20.68 MPa.
Sbg Ag Sbg

Stage 2: Stress check at placement of deck and splice

Mdg = Deck load moment at midspan of Segment 2

= 1925 kN-m

M112 = Haunch load moment at midspan of Segment 2

= 356.58 kN-m

M52 = Superimposed dead load moment at midspan of Segment 2

288

= 2017.46 kN-m

M62 = Construction load moment at midspan of Segment 2
= 362 kN-m

M p 2 = Removal of support moment at midspan of Segment 2

= 10928 kN-m
Shrinkage prestress losses are assumed to be calculated at 75% humidity and are
determined according to AASHTO LRFD Equation 5.9.5 .4.2-1 (AASHTO 2003):

dfpSR = (17 — 0.15H) = 39.65 MPa.

The creep coefﬁcient during deck placement is based on the time at which the
deck is poured and can be calculated according to AASHTO LRFD Equation 5.4.2.3.2-1
(AASHTO 2003):

t0'6
: 0.5384.

 

KCth :10+t

The shrinkage coefﬁcient during deck placement is determined according to

(AASHTO LRFD Equation S.4.2.3.3-1(AASHTO 2003) :

 

[(5th = = 0.6316.

35+t

The initial post-tensioning stress is deﬁned as:

fp4 = 0.69 fpu = 1287 MPa.

The total post-tensioning forces are determined from all post-tensioned ducts and

can be calculated by:

FPT12 : ”poapsfp4 : 9371 kN-

289

The maximum number of strands in each duct is assumed to be 12. Therefore, the

number of post-tensioning ducts is determined as:
n [,0
NT = round (—12—-) = 4 ducts.

The location of the highest post-tensioning duct depends on the lowest centroid
of the effective post-tensioned duct and is calculated as follows:

Ybbb = Ybb + (N—1)x150= 622 mm.

The average location of all post-tensioning ducts is determined by the summation

of the lowest and highest post-tensioning duct locations as follows:

 

+
yb : (ybb 2Ybbb) 2394 mm.

The eccentricity of pretensioned strands for the composite section is deﬁned as:

ecz = ybc -ye2 = 1247 mm-

The average eccentricity of all post-tensioning tendons for the composite section
is calculated as:

ep12 = .Vbc ‘yb :1376 mm.

The stress at the centroid of the pre-tensioned strands on the mid-segment caused

by post-tensioning and temporary support removal loads can be determined as:

FPT12 + FPleepr-Zecz _ Mpzecz

fcngZ = Ac [c = 10.16 MPa.

 

 

C
The stress at the centroid of the pre-tensioned strands of the mid-segment caused

by the deck, girder/deck haunch, and superimposed dead loads can be calculated as:

M +M M
d2] h2 epz +ﬂ=553 Mpa,

g [C

 

dfcdp2 =

290

Creep prestressing losses at this stage are computed according to the AASHTO-
LRFD provisions (AASHTO 2003) and the recommendations from the NCHRP 517
report (Castrodale and White 2004). The ﬁrst two terms of the following equation are
determined according AASHTO-LRFD, Section 5.9.5.4 and the last term is added as

recommended by the NCHRP 517 report.

dfCRZ = lzfegplZ - 7dfcdp2 + 7fc = 16.57 MPa.

gch
Final relaxation prestress losses, which include the loss due to elastic shortening,

shrinkage, and creep, are determined according to AASHTO-LRFD Equation 5.9.5.4.4:

dfPR22 = 0.3[20 — 0481/1952 — 0.2(dfpSR + dprR2)] = 20.19 MPa.

The total prestress losses at the placement of the deck and splice are calculated as

follows:

dfpreTDZ = (#1352 + K (:de df CR2 + KSthdfpSR + dfPR22= 208-29 MP8-

The effective prestress at the centroid of the pre-tensioned strands after all losses

at placement of the deck and splice stage is calculated as follows:
f pe22 = f pei —dfp,.eTDz = 1188 MPa.

The total top girder stress at the mid-span of the mid-segment at Stage 2 is deﬁned as:

Mg2 + MdZ + MhZ + Mc2 + Aps2fpe22 _ ApstpeZZepZ

s,g s,g s,g stg A s,g

 

 

0t2 =
g

= 10.16 MPa < 0.6fc' = 26.89 MPa.

The total bottom girder stress at the mid-span of the mid-segment at Stage 2 is deﬁned

as:

291

MgZ _ Md2 _MlzZ _Mc2 + Apstpe22 ApstpeZZepZ

Sbg Sbg Sbg Sbg Ag Sbg

 

 

 

 

UbZZ"

= 5.21 MPa < 0.6fc' = 26.89MPa.

Stage 3: Stress check at post-tensioning

The creep coefﬁcient during the post-tensioning stage is based on the time at
which post-tensioning is applied and can be calculated as (AASHTO LRFD Equation
5.4.2.3.2-1):

t 0.6

[)0
K = = 0.571.
CR’p 10+tp0

 

The shrinkage coefﬁcient during post-tensioning is determined as (AASHTO LRFD

Equation 5.4.2.3.3-1):

t
K = J— = 0.682.
SH”) 35 + tpo

Stressing of the post-tensioning tendons causes secondary elastic shortening
prestress losses in the pre-tensioned strands in the girder segments. Additional elastic

shortening losses due to post-tensioning are computed as:
EP
dprSAZ = (22,—)fcgpc2: 5923 MP3:
g

Total prestress losses at the centroid of pre-tensioned strands from the post-

tensioning stage are calculated as follows:
dfpreTPZ = desz + KCRtpdfCRZ + KSHtpdfpSR + (#17822 + dprSA2= 274.96 MP3-

The effective prestress at the centroid of the pre-tensioned strands after all losses

at the post-tensioning stage is calculated as follows:

292

fpe32 = fpei _dfpreTPZ = 1121 MPa.

The eccentricity of the effective post-tensioning duct at the end span of the
composite section of Segment 1 is determined as:
eptl :ept3 :J’bc - Ya 2995 mm.

The eccentricity of the effective post-tensioning duct at the mid-span of
composite section of Segment 2 is determined as:

2
x x
ee =eptl+(-3ept1+4ep,2 —ept3)z+2(ept3 —2ep,2 +ep,1)22—= 1376mm.

The friction losses are computed according AASHTO LRFD Equation 5.9.5.2.2

as shown below:
dpr = fpj (1 —e‘(K"+W))= 46.33 MPa.
The stress at the centroid of all post-tensioned strands on the mid-segment caused

by the post-tensioning and temporary support removal loads can be determined as:

2
FPTO + FPTOeptZ _MrZeptZ
Ac 1c [c

= 10.49 MPa.

 

 

f cgpoZ =

Elastic shortening prestress losses at the centroid of the post-tensioning strands

due to post-tensioning is computed as:

(NT—1) Ep
2NT Eg

 

dfPTEsz = f egp02 = 2295 MP3-

Anchor set prestress loss is computed by estimating the loss that occurs when
each tendon pulls into the anchorage due to seating. The maximum anchorage set loss,
which is at the stressing anchorage, is computed using the NCHRP 517 report equation

7.3.1 .1.2-3 (Castodale and White 2004) as follows:

293

 

df A E x
de=2 pF 38’ 1’ =108.03MPa.
x dpr

The distance along which the anchor set loss extends is computed using the

NCHRP 517 report equation 7.3.1.1.2-2 (Castodale and White 2004) as shown below:

A E x
XA=i —Se—’—’—’—=2888mm.
12 dpr

The total prestress losses at the post-tensioning stage are calculated as follows:

(X A —0.5L)
A

 

dPTPT2= dpr +dfPTES2 +de = 84.65 MPa.

The effective prestress at the centroid of post-tensioned strands after all losses

from the post-tensioning stage is calculated as follows:
fPOf : fp012 = fpoi -dPTpT2= 1312 MPa.

The total top girder stress at the mid-span of Segment 2 at Stage 3 is deﬁned as:

M A A e
g2 + Md2 + Mh2 + Mr2 + pstpe32 _ pstpe32 p2 +
S, S, S, Ste Ag S

 

 

 

 

0t3 =

g g g
tg

"poapsfpof _ ”poapsfPOfee
Ac Ste

 

 

= 15.89 MPa < 0.6fc' = 26.89 MPa.

The total bottom girder stress at the mid-span of Segment 2 at Stage 3 is deﬁned

as:

294

Mg2 Mdz _Mhz _Mrz + Aps2fpe32 + Apszfpe326p2 + "poapsfpof +

 

 

O'b3 :—

 

S bg S bg Sbg S be A g S bg AC
npoapsfpofee
Sbc
= 16.84 MPa < 0.6/6' 2 26.89 MPa.
The total top deck stress at the mid-span of Segment 2 at Stage 3 is deﬁned as:
6,53 = Alﬁ- + ”00‘”fo — "poapSfpofee = 5.02 < 0.6fcd’ = 18.62 MPa.

 

SICS ACC St CS

Stage 4: Stress check at Compression 1 limit state

M52 = Superimposed dead load moment at midspan of Segment 2

= 2018 kN-m
M112 = Live load moment at midspan of Segment 2
= 6646 kN-m

Total prestress losses at the ﬁnal condition for Compression I are calculated as

follows:
dfpreTPF2 = desz + dfCRz + dfpSR + dfPRzz + dprSAz = 358-60 MP3-

The effective prestress at the centroid of the pre-tensioned strands after all

prestress losses for the ﬁnal condition for Compression l is calculated as follows:
f p842 = f pci - dfpreTPFZ = 1037-60 MP3-
The stress at the centroid of the post-tensioned strands of Segment 2 caused by

deck, girder/deck haunch, and superimposed dead loads can be calculated as:

295

M42 +M122 e + MSZePIZ

pt2
1g c

= 8.68 MPa.

 

dfcdp02 =

The stress at the centroid of the post-tensioned strands of Segment 2 caused by

post-tensioning and girder loads can be calculated as:

 

F F e e M e
fcgpcoz = p2 + [’2 p12 62 — g2 [”2 =22.09 MPa.
Ag 18 [g

Prestress loss due to creep at the centroid of the post-tensioning strands at this
stage is computed according to the AASHTO-LRFD speciﬁcations and the
recommendations from the NCHRP 517-12 report. The ﬁrst two terms of the following
equation are determined according to AASHTO-LRFD Section 5.9.5.4 (AASHTO 2003)
and the last term is added as recommended by the NCHRP 517 report (Castrodale and

White 2004).
dfpoCRZ =12fegp02 — 7dfedpo2 + 7fcgpcoZ = 21981 MP3-
Shrinkage prestress loss in the post-tensioning is calculated at 75% humidity and

according to AASHTO LRFD Equation 5.9.5 .4.2-2 (AASHTO 2003) and is deﬁned as:

dfPOSR = (13.5 — 0.123H)= 29.47 MPa.

Prestress loss due to ﬁnal relaxation of the post-tensioned strand is calculated

according to AASHTO-LRFD Equation 5.9.5.4.4 (AASHTO 2003):

dpr32 = 0.3(20 — 03(1pr — 0.4dprESZ — 0.2(dfp0CR2 + dfpo SR ) = 1949 MPa.

Total prestress losses at the Compression 1 limit state for the ﬁnal stage are
calculated as follows:

X A -0.5L)

 

(
dPTFz = dpr + dfPTEsz + df A + dfpoCRz + dfpoSR + dfPR32

296

= 354 MPa.
The effective prestress at the centroid of the post-tensioned strands after all

losses at the ﬁnal Compression 1 stage is calculated as follows:
f poff = f p622 = f p0,- — d m2 = 1043 MPa.

The total top girder stress at the mid-span of Segment 2 at Stage 4 is determined

 

 

 

 

 

 

 

 

 

 

 

 

as:

M M M M M M A f A f e
0.!4 : 82 + (12 + 112 + r2 + s2 + 112 + p52 p642 _ ps2 pe42 p2 +

Stg Stg Stg Ste Stc Ste Ag Stg

"poapsfpeff __ "poapsquffee

AC Ste
= 21.73MPa < 0.6fc' = 26.89 MPa.

The total bottom girder stress at the mid-span of Segment 2 at Stage 4 is deﬁned
as:
0.1)4 :_M82 _ Md2 _MhZ _Mr2 _Msz _MlIZ + Aps2fpe42 + Apstpe428p2 +

Sbg Sbg Sbg Sbc Sbc Sbc A g S bg

"maps/140.17 + "poapsfpeﬂee
Ac Sbc

 

= —5.48MPa < 0.6fc' = 26.89 MPa.

The total top deck stress at the mid-span of Segment 2 at Stage 4 is deﬁned as:

Mr2 +Ms2 +M112 +"poapsfp9ff _"poapsfpeffee
Stcs Stcs Stcs A cc S

 

 

0'ts4 :

(CS

= 11.78 < 0.6de = 18.62 MPa.

297

Stage 5: Stress check at tension limit state

The total top girder stress at the mid-span of Segment 2 at Stage 5 is deﬁned as:

 

 

 

 

o'ts =Mg2 +Md2 +Mh2 +Mr2 +M52 +0.8M112 +AP32fpe42 _-
Stg Sig Stg Stc StC SIC Ag

 

 

 

Apstpe42‘3p2 + "poapsfpoff _ "poapsfpoffee

Stg Ac Stc

= 20.87MPa > —0.19 fc' = —3.34 MPa.

The total bottom girder stress at the mid—span of Segment 2 at Stage 5 is obtained

as:

_MgZ _Mdz __Mh2 _Mr2 _M

S Sbg

M112 + Apstpe42 +
Sbg Sbe Sbe Sbc Ag

 

 

 

 

065 =
be

’41?st P6426P2 + "poapsfpaff + "poapsf p017 ee

Sbg Ac Sbc

 

=—3.199MPa > —0.l9>< fc' = —3.34 MPa.

The total top deck stress at the mid-span of Segment 2 at Stage 5 is calculated by:

Mr2 + M32 + 0 8 M112 + ”poapsfpotf _ "poapsfPOffee
Stes Stcs Stcs Ace Stcs

 

 

 

0355 =

= 10.83 MPa < 0.6fcd' =18.62 MPa.

Stage 6: Stress check at Compression 2 limit state

The total top girder stress at the mid-span of Segment 2 at Stage 6 is deﬁned as:

M M M M M Af A~f 6
82+ (12+ 122+ r2+ 52+ p52p642_ p42p642p2+

Stg Sig Sig Ste Ste A Sig

 

 

 

 

 

0t6 =
g

298

"poapsfpoﬁ’ _"po "aps xfpeffxee
Ac Ste

 

 

= 17.42MPa < 0.45 fc' = 20.17 MPa.

The total bottom girder stress at the mid-span of Segment 2 at Stage 6 is

computed as:

Mg2 _ Md2 _ Mh2 _ Mr2 _ M52 + Apstpe42 + Apstpe429p2 +
Sbg Sbg Sbg 56c Sbc Ag Sbg

 

 

 

 

"poapsfpoff + ”poapsfpeffee
Ac Sbe

= 5.94MPa < 0.45fc' = 20.17 MPa.

The total top deck stress at the mid-span of Segment 2 at Stage 6 is deﬁned as:

Mr2 + M32 + "poapsfpoff "poapsfpeffee
Stes Stcs Ace Stcs

 

 

0756 =

= 7.03 < 0.45 fcd' = 13.96 MPa.

Stage7: Stress check at Compression 2 limit state

The total top girder stress at the mid-span of Segment 2 at Stage 7 is deﬁned as:

 

 

 

 

 

 

M A
a =0_5( 82 +Md2 +Mhz +Mr2 +M32 + pstpe42 _
’7 s s s s A
’8 lg lg ’0 Stc g
ApSpre‘lzepz npOapSfpoff npoapsfpoﬂee A4”2
+ - )+__
Stg AC SIC SIC

= 13.02 MPa < 0.4fc' = 17.92 MPa.

The total bottom girder stress at the mid-span of Segment 2 at Stage 7 is

determined as:

299

Mgz _Mdz _Mhz _Mrz _Msz + Apszfpe42 +

Sbg Sbg Sbg Sbc Sbc Ag

 

 

 

07,7 = 0.5(-

ApstPe‘lZePZ + ”poapsfpaff + "poapsfpoffee ]_ M112

Sbg Ac Sbc Ste

 

= —8.45MPa < 0.4fc' = 17.93 MPa.

The total top deck stress at the mid-span of Segment 2 at Stage 7 is deﬁned as:

 

 

 

 

0: 7 _0 5(M r2 +M32 + ”poapsfpatf _"poapsquff ee)+M112
s - -
Stcs Stcs Ace Stcs Stcs

fed "

= 8.27 < 0.4 = 12.41 MPa.
1000

 

6.2.7 Strength limit state

After the design is completed for the service limit state, each critical section of
girder needs to be checked for strength limit state requirements. The strength limit state

included checks on moment and shear capacities. The ductility requirement was satisﬁed

by ensuring that the relative neutral axis depth at the ultimate limit state c/de was less

than 0.42 (AASHTO 2003).

a) Moment capacity at the midspan of segment 2

The ﬂexural capacity at the midspan of the single-span spliced girder bridge is
evaluated in this section. Optimal design variables at this stage include the number of
pre-tensioned strands at the midspan of Segment 2, locations of the effective post-

tensioning at the end span and at the midspan, and number of post-tensioning strands.

300

The total number of post-tensioning strands (npo) obtained from the design chart
(see Figure 6-6) is:
npo = 52

The maximum number of strands in each duct is assumed to be 12. Therefore, the

number of post-tensioning ducts is determined as:
n W
NT = round (—-—)= 4
12
The location of the effective post-tensioning duct at the end of Segment 2

obtained from the design chart (see Figure 6-7) is:

ya = 775 mm

The location of effective post-tensioning duct at the midspan of Segment 2

obtained from the design chart (see Figure 6-7) is:

Ybb = 165 mm.

The space between each post-tensioning duct is assumed to be 150 mm (6 in.)
apart from each other. Therefore, the location of the highest post-tensioning duct
depends on the lowest centroid of the effective post-tensioning duct and is calculated as

follows:

ybbb = ybb +150(NT — 1) = 622 mm.

The average location of all post-tensioning ducts is determined by the summation

of the lowest and highest post-tensioning duct locations as follows:

+
yb : (.Vbb .Vbbb) = 394 mm.

 

The total area of post-tensioned strands is determined as:

301

A PT = npoaps = 7280 mm2_

The total area of pre-tensioned strands at the midspan of Segment 2 is calculated

as:
2
ApreT = ("17’12 + n.0r22 + "W32 + npr42)aps =3780 mm .

The distance from the location of effective post-tensioning duct to the extreme
top ﬁber of composite section is calculated as:

dppT =H+ts -y,, =2254 mm.

The centroid of all pre-tensioned strands at the mid-span of Segment 2 is deﬁned

as:

50 + 100 + 150 + 200
ycm :( n.0'12 "W22 npr32 n10r42) 2107mm.

(npr12 + npr22 + npr32 + npr42)

 

The distance from the centroid location of all pre-tensioning strands to the
extreme top ﬁber of composite section is calculated as:

d t=H+tS—ycm=2541 mm.

ppre
The centroid of prestress strands due to pre-tensioning and post-tensioning is

determined as:

_ dPPTAPT + dppret Apret
APT + Apret

= 2352 mm.

 

P

The prestress stress in the post-tensioned strands is deﬁned by:

fps = 0.92fpu = 1715 MPa.

The total tension forces in the girder section due to the pre-tensioned and post-

tensioned forces is computed as:

302

T = (APT + APRET)fpS = 18965 kN.

The concrete compression force in the top ﬂange is deﬁned as:

C1 = 0.85fcdbehf = 15160 kN.

The concrete compression force in the web is determined as:

C2 = 0.85fcbfhw = 6175hw.

The total concrete compression forces in the concrete girder must be equivalent
to the total tension forces of the prestress steel and is computed as:

C = C1 + C2 = T.

The web height that is in the compression zone is obtained by trial and error until

the tension and compression forces are equivalent. The portion of the web height under
compression is:

hw = 89.38 mm.

The equivalent concrete compression stress block depth is then deﬁned as:

a =hf +hw =299 mm.

The equivalent stress block coefﬁcient used to determine the neutral axis of the
girder section depends on the girder concrete compressive strength. The coefﬁcient

value of this example is:
,8] = 0.725.

The neutral axis of the girder section is determined as:
c = £— = 412 mm.
.51

The nominal moment resistance of the T-section is calculated as:

303

h h
Mn = 0.85fcdbehf(dp ——-2’:) + 0.85fcbfhw(dp —hf —% .

Mn = 42056 kN—m.

The required moment must be less than the nominal moment resistance as shown
below:

Mn =42056kN—m>Mu =33853kN—m.

The maximum reinforcement limit according to AASHTO LRFD Article

5.7.3.3.1 (AASHTO 2003) is checked as:

C

—=0.175 < 0.42.
e

The moment capacity is thus adequate.

b) Shear resistance of the interface at spliced locations

There is a potential plane of weakness at the face of the splice, therefore shear at
the splice must be considered. The interface shear capacity at the splice location is
determined according to AASHTO LRFD Article 5.8.4 (AASHTO 2003). In this
evaluation, the interface is considered to be a concrete to hardened concrete surface that
has not been intentionally roughened.

The permissible stress at the surface is according to AASHTO LRFD Article
5.8.4.2 (AASHTO 2003) as:

c = 0.517 MPa.

The friction coefﬁcient for normal weight concrete according to AASHTO
LRFD Article 5.8.4.2 (AASHTO 2003) is calculated as

p2062=06.

304

The area of concrete engaged in shear transfer is determined as:

AC, = tw(h + z, — 61,) = 486451 mmz.

The area of shear reinforcement crossing the shear plane is calculated as:

Avf = tW(h +ts —d,) = 7280 m2.

The permanent net compressive force normal to the shear plane is computed as:
PC :fpeAvf = 7028 kN

The nominal shear resistance at the spliced is determined as:

V" = CAcv + ,u(Avffy + PC) = 6276 kN

The factored nominal shear resistance at the splice most include the strength

reduction factor of 0.90 and then compared with the ultimate shear requirement at the

splice location as shown below:

¢V,, = 0.9V" = 5649 kips > V, = 1899 kN

c) Vertical shear at H/2

Calculation of the shear capacity of spliced girder bridges is as that for
conventional prestressed concrete bridges. Nevertheless, the vertical shear capacity at

H/2 from the support is also documented herein.

The computed vertical shear resistance from the concrete due to external loads is

calculated as:

. V-M
VC,_CO,,, = 0.6 fc b'd'+Vd + ii = 1966 kips = 8745 kN

max

305

The minimum vertical shear resistance from the concrete is based on the modulus

of rupture of concrete and can be determined as

VC,_min =MinVCi =17 fc'b'd = 101.65 kips = 452 kN.

The maximum shear resisting force from concrete is determined as:

Vci = Max(Vc,-_mm , Vci—min ) = 8745 kN.

The total prestress forces due to the pre-tensioned and post-tensioned forces can be
deﬁned as:

FZFpOSl+F

pm, =91 l +9012 = 9924 kN.

The sine of the angle of the post-tensioned duct proﬁle at the end with respect to

the horizontal axis is equivalent to the tangent angle as shown below:

—y“ _y” =0.012
L/2

sina =tana=

The vertical shear force resistancecomponent due to the post-tensioning force is

calculated as:

Vp = Fsina = 9924(0.012) =119kN

The nominal vertical shear resistance at H/2 is determined as:

¢Vn (H/2) = ¢(Vc + VS + Vp) = 0.9(8745 +119) = 7980 kN.

The shear due to stirrup reinforcement is ignored in this calculation since the shear
from the post-tensioning is much larger than the shear force from the stirrup

reinforcement.

6.2.8 Design evaluation

The layout arrangement of pre- and post-tensioning requirements for design

306

Example 1 obtained from the design charts and those from the NCHRP 12-57 report
(Castrodale and White 2004) are depicted in Figure 6-8 for comparison. Total prestress
results obtained from the design charts and the NCHRP 12-57 for each segment are
compared in Table 6-1. The provided nominal capacities of shear at the splice location
and at H/2 as well as moments from the design charts and the NCHRP results are

compared against the calculated demands also in Table 6-1.

Table 6—1 Comparison of design requirements and capacities from NCHRP 51 7 and
optimization chart results

 

Prestressing Requirements

 

 

”p0 Npr; Nprz
NCHRP 517 60 10 20
Design Charts 52 4 32

 

Design Capacities and Demands

 

M (kN-m) V_Sp1ice (kN) V_h/2 (kN)

 

NCHRP 517 45,990 8,265 NA
Design Charts 41,115 5,440 7,980
Demand 33,858 1,900 3,648

 

307

 

 

 

 

 

 

 

 

 

 

 

 

1 2 Strands 1L 1 4 Strands 11
K 1 KO’J E g°1°K°J E
i O 1 o
1 8 I 8
1 o’ 1 0'
! 12 Strands !
-JL——-— an//_per ductlTyp. ) 1
0.330 111’ (1) 1
— i
0.330 m/ﬁ i i
---- (i) E mJL— (D‘t-12 Strands
0.330 m/ﬁ ! 3 8j1§gmq: 8 Der duct (Typ 1
—_ G) O 0.125 m——:_
0.225 m’ __ | O. 0,125m—‘j_ 0.050 m
0.125 m/':_1<.$.h1 0.1215726, D1
0.088 m: 71 1 11 0.088 m11‘___.10.050 m
8 Strands 16 Strands
SEGMENT 1 8 3 SEGMENT 2

a) NCHRP Report 51 7-12 Example 1 result

6 Strands
HL

1 2 Strands 1L

 

 

 

 

 

 

 

 

 

 

 

 

 

L~4L.\ 1,»~4*'] E L““‘\ 1/—""'“ E
1 o i o
1 2‘3 ! 8
1 cs ' o‘
1 i
i 1
i 1
i i
1 11 Strands !
1L__ 1 11 Strands
0.188 m/‘i_ elf/11 Strands J,— (94/11 Strands
/15 Strands 0.150 m-—— 15 Strands
0.188 mﬁ 15 Strands 0.150 m:_ (9‘ 15 51- d
0.188 m/‘ib ?‘<L 0.150 m:_ 3% orggosm
[___ K r— Gum: 0.050 m
0.225 m 11%—>11 10 (mm 0175 mil, .1 0.050 m
2 Strands 25 Strands
SEGMENT 1 8 3 SEGMENT 2

b) Optimized Design Aids result

Figure 6-8 Pre- and post-tensioning strands requirement comparison between NCHRP
and optimized aids results

308

As one more measure of comparison between the results from the NCHRP report
and those from the optimized charts, the ﬂexural stresses from each critical construction
stage were normalized in Figure 6-9. The normalized stresses for service limit states
show that all results from the design charts are within the allowable limits for every
construction stage. The required moment and shear of strength limit states were also
compared with the nominal moment and shear results from the design charts, and they

were found to be adequate.

 

 

.......................................... E NCHRP 517 1
Design Charts

.............................................

 

 

 

 

.............................

 

............................................................

uuuuuuuuuuuuuu

---------------------

       

Demand to Allowable Ratio
///_//////////////////////////////////2

 

 

 

 

 

”ll/ll/I/[l/l/I/l/I/I/I/I/I/I/I/I/l/I/I/I/l/I/I/I/I/A

 

 

”/l/l/I/l/l/l/l/I/I/I/I/l/I/I/I/I/I/I/I/l/M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a 8 8 8
.§.. ..2.. ..2.. \. ...2.
a a a N a
a a 1:. § :2.
..< ..... 4 .. ”(u \. ...<.
"' ‘6 ‘6 \ 3
\ g z z N z
0-0 11 1’11 111111111111

51 1 $12 $21 $22 531 S32 333 S41 s42 $43 $51 $52 $53 861 S62 see 371 $72 $73
Limit State [ Sij = Stress at stage i, location j ]

Figure 6-9 Normalized stress results comparison between NCHRP and optimized aids
designs

The layout arrangement of pre- and post-tensioning requirements for design
Example 1 obtained from the design charts and those from the NCHRP 12-57 report

(Castrodale and White 2004) are depicted in Figure 6-8 for comparison.

309

6.3 Preliminary Design Example 2

6.3.1 Problem statement

A three-lane two-span continuous pliced girder bridge with a total width of 18.59
m (61 ft), and a span length (L) of 80.47 m (264 11) consisting of three segments, as
illustrated in Figure 6-10, was used as a prototype structure in this study. The new
optimized girder was chosen in order to show how the service limit state of two—span
continuous spliced girder bridge is different from single-span SGB. In addition, the total
height of the new proposed beam obtained from the component optimization is used to

compare with a prismatic spliced girder prestressed girder which is proposed by Lounis

 

 

(1997).
C1. AbUlmenl 10101 Span Lenglh (1.) CL AbUlmenl
CL Pier I

Splice joi nt—x'1\ l /—— Deck
I

l 11 ) “Prestressed Beams

Positive segment or
(Field segment)

 

 

 

    

 

 

 

Nonprismotic
Negative segnent
or Pier segment

 
   
 

   

 

 

 

 

[___]

Figure 6-10 Two-span spliced precast/prestressed bridge with non-prismatic pier
segment

The end segments were assumed to be equal length (L 1). The splice location is

deﬁned as the ratio of the end segment length (L I) to the total span length (L) and was

310

assumed to be located at 0.35L. The length of the middle segment is thus deﬁned as L 2 =
L —- L 1. The cast-in-place concrete slab thickness was assumed to be 216 mm (8.5 in.)
with 50 mm (2 in.) built-up built on a 1.8 m (6 it) girder spacing.

The design study was conducted using the simpliﬁed design method of the
AASHTO-LRF D Bridge Design Speciﬁcations (AASHTO 2003) and the
recommendations of NCHRP 517 report (Castrodale and White 2004). Both service limit
states and ultimate limit states were considered in the design.

The purpose of this design example is to illustrate how the developed design charts
can be used as an aid for the preliminary design of two-span spliced girder bridges as
shown in Figure 6-8. The non-prismatic negative segment with constant web height
(option 3) obtained from the component optimization (see Section 5.4.3) for a bridge
with total span length of 80.46 m [264 ft] was used in this example. A single-stage post-
tensioning applied after deck casting construction is used in this study. The section at
the mid-span section of negative segment (x = 40.23 m [132 it] measured from the end

span) of the spliced girder bridge is checked in this example.

6.3.2 Preliminary material properties

Girder Material Properties

fc' = 44.8 MPa (6.5 ksi)
fci' = 34.5 MPa (5 ksi)
15g,- = 29,558 MP8 (4287 ksi)

Eg = 33,702 MP3 (4888 ksi)

311

wc = 2,400 kg/m3(150 lb/ﬁ3)
fcd' = 24.1 MPa (4.5 ksi)

Deck Material Properties

fab”: 31.0 MPa(3.5 ksi)
Ecdi = 24,731 MPa (3372 ksi)
Ea, = 28,041 MPa (3824 ksi)
3 3
Wed = 2,400 kg/m (150 lb/ﬁ )

Pre—tensioning Strands Properties

ap, = 140 mm2 (0.217 in2)

5,, = 196,502 MPa(28500 ksi)

fl,“ = 1,861 MPa (270 ksi)

1;” = 0.9fp. = 1,675 MPa (243 ksi)
5,0 = 0.751;... = 1,396 MPa (202.5 ksi)

Post-tensioning Strands Properties

ops = 140 mm2 (0.217 in.2)
Ep = 196,502 MPa (28500 ksi)
ﬁn, = 1,861 MP8 (270 ksi)

fpy = 0.91;... = 1,675 MPa (243 ksi)

312

fpj = 0.9fpy = 1,508 MPa (219 ksi)

6.3.3 Calculated section properties of optimized negative section

Non-composite Girder Section Properties (Negative Segments)

ybg = 637 mm (25.08 in.)
Mg = 749 mm (29.48 in.)
3 . 3
S,g = 122,902,980 mm (75001n. )
3 . 3
Sbg = 146,418,417 mm (89351n. )
2 . 2
Ag: 431,805 mm (669.3 1n. )

1g = 9.32899e10 mm4 (224,130 111.4)

Composite Girder Section Properties (Negative Segments)

Ac}, = 760,644 mm2(1179 111.2)

Id, = 2.3155787e1 1 mm4 (556320 in“)
3 . 3

Sbch = 229,861,347 mm (14027111. )
3 . 3

5,6,, = 611,466,906 mm (37314111. )
3 . 3

SW = 468,047,322 mm (285621n. )

2 . 2
AM, = 913,547 mm (1416111. )
ybch = 1007 mm (39.66 in.)

ytch = 378 mm (14.91 in.)

313

ytcsh = 595 mm (23.41 in.)

6.3.4 General information
8 =1800 mm (6 8)
ts = 216 mm (8.5 in.)

Humidity = H = 75

t, = 1 day
td = 60 days
tpo = 75 days

Ase, = 9.53 mm (0.375 in.)

6.3.5 Use of design charts

The design optimization of two-span continuous SGB requires more design
variables than those considered for the single-span system in Design Example 1. These
include variables to deﬁne the sectional optimization options as shown in Figure 6-11,
pre- and post-tensioning requirement and the deﬁnition of parabolic post-tensioning over

the haunched negative segment as shown in Figure 6-12.

314

b2 14

 

 

 

 

’/r-New Optimal
Nonprismotic
Negative
tw Section

——>-¢—

W tl 12

bl

 

 

Sf 2L Constant 0w

ﬂ

 

Varied

Figure 6-11 Design variables for new non-prismatic section with constant web(option 3)

_\5 /‘”P°1r_cs.

 

 

 

 

 

 

 

 

 

 

 

(a) Post-Tensioning Requirement

nprt1—1 npr12 1‘1ng
3 npr.22 ”P22 C13.
1 _1/ f , r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

!_
___________ \\\ //__________Z___.
\_//

1 i
K k\_ \ \\
\> , Lnor
» nor. 0 1
”pm 5 2: \——0P*tz p 1
1 1
L1 L2 1.
L

(b) Pre-Tensioning Requirement

Figure 6-12 Variables for the design optimization of two-span continuous spliced girder
bridge

315

Design variables were obtained from the developed optimized design aid charts in
the same way as explained for the single-span SGB (Example 1). Using design aid

charts, the design variables are:

npr11: 5; npr12 = 5; npr13 = 3;
npr14 = 2; nprt1: 2; npr21: 0;
nprzg = 0; npr23 = 0; npr24 =0;

nprt1 = 3; nprt2 = 7; npo; = 20;

t1: min 127 mm (5 in.) — max 178 mm (7 in.);
t2 = 127 mm (5 in.); t3 = 127 mm (5 in.);

t4=127mm(5in.); b1=610mm(24in.);

b; = 457 mm (18 in.); tw= 200 mm (8 in.);

Dw=818 mm (32.2 in.);

ya = 25 mm (1 in.); yb = 635 mm (25 in.);

yc = 50 mm (2 in.) upward from neutral axis;

yd = 200 mm (8 in.) upward from neutral axis.

The service limit state and ultimate limit state at the mid-span section of the negative

segment are now checked to verify that the results provided by the design aids do in fact

satisfy all serviceability and strength requirements.

316

6.3.6 Service limit states

Service limit states were deﬁned in terms of ﬂexural stress limits on the girder
and deck. For ﬂexure, service limit state requirements govern the required prestressing
force. The stress limit states for each critical construction stage of the single-span spliced
girder bridge when using a single-post-tensioning (see Section 4.5.1) are evaluated as
follows:

Stage 1: Stress check at pre—tensioning

The end-segment, or segment 1, length is based on the spliced location and can
be determined as:

L; = (splice location)(L) = (0.35)L = (0.35)(80.47) = 28.16 m.

End segments are assumed to be equal, therefore the mid-segment length can be
computed as:

L2 = L — (2)(L1) = 24.15 m.

The centroid of all pre-tensioned strands measured from the top of the negative

segment is computed as follows:

y (npr12)(50) + ("W22 )(100) + (npr32)(150) + (npr42)(200) + (nprt2)(ybg + ytg - 50)
2 =
6 (mm + nprzz + nW32 + n191212 + nprtz)

 

yez = 1003 mm.

The eccentricity of the pre-tensioned strands at the midspan of negative segment

is computed as:

epz = (ytgm — yez) = 310 mm (measured above neutral axis).

The total pre-tensioned strand areas include the area of all pre-tensioning strands

in the top and bottom girder ﬂanges:

317

2
Apsg = (aps )(nprlz + npr22 + nprgz + npr42 + nprt2) = 1,260 mm .

The girder dead load moment at the midspan of the negative segment is

computed as
Mgg = -186.4 kN-m

The cross-section area of the non-prismatic negative segment varies along its
length. Therefore, the average cross-section area is used for the non-prismatic section

and can be determined as:

(A +A )
= g 2 gm =413548mm2.

 

g8

Similarly, the average moment of inertia is used for the non-prismatic section and

can be determined as:

_(Ig +Igm)

ge = 8.53610 mm4.

The stress at the centroid of the pre-tensioned strands of the negative segment

caused by pre-tensioning and girder dead load is calculated as:

0.7A p52 f p“ + 0-7Aps2f 101461262 _ 144821181221

2 = 5.143 MPa.
fcgp12 A I

 

ge ge 189
Elastic shortening prestress losses due to the pre-tensioning at the release stage

can be determined as follows:

E
des2 = fcgplz i = 34.2 MPa.
Eg,

Relaxation prestress losses at release of the pre-tensioned strands are deﬁned as:

318

j
[fp 4.55]ij =13.65 MPa.

py

__ log(24t,.1)
40

 

df PR1

The effective prestress at the centroid of the pre-tensioned strands after all losses

at release of the pre-tensioning is calculated as follows:
f [7812 = f pei - desz - dme = 1348 MPa.

The total top girder stress at the mid-span of the mid-segment at Stage 1 is

deﬁned as:

Mg2 + ApstpelZ _ Aps2fpe123p2

 

 

 

6,1: =6.76 MPa < 0.6fc', =20.68

Stge Age Stge
MPa.

The total bottom girder stress at the mid-span of the mid-segment at Stage 1 is

calculated by:

 

Mg2 + ApstpelZ _ ApstpelZep2

= 1.59 MPa < 0.6fc', = 20.68 MPa.
Sbge Age Sbge

0b] :—

Stage 2: Stress check at placement of deck and splice

Md; = Deck load moment at midspan of negative segment

= -174.14 kN-m

th = Haunch load moment at midspan of negative segment
= -29.91 kN-m

MS; = Superimposed dead load moment at midspan of negative segment
= -663.92 kN-m

MC; = Construction load moment at midspan of negative segment

319

= -32.78 kN—m

M1,; = Moment due to removal of supports at midspan of negative segment

= -3227 kN-m
Mpn = Moment due to post-tensioning uniform loads
= 2027 kN-m

war = Moment due to post-tensioning uniform loads

= 283 kN-m
M550 = Moment due to secondary effect of post-tensioned loads
= 233 kN-m
Shrinkage prestress losses are assumed to be calculated at 75% humidity and are

determined according to AASHTO LRFD Equation 5.9.5.4.2-1 (AASHTO 2003):

dfpSR = (17 — 0.15H) = 39.65 MPa.

The creep coefﬁcient during deck placement is based on the time that deck is

poured and can be calculated according to AASHTO LRFD Equation 5.4.2.3.2-1

 

(AASHTO 2003):
0.6
K — td - 0 5384
Cth 10+td . .

The shrinkage coefﬁcient during deck placement is determined according to

(AASHTO LRFD Equation 5.4.2.3.3-1(AASHTO 2003) :

 

t I
K = C — 0.6316.
Sth 35 Id

The initial post-tensioning stress is deﬁned as:

fp4 = 0.69 fpu = 1287 MP8.

320

 

 

The total post-tensioning forces are determined from all post-tensioning ducts
and can be calculated by:

FpTlZ = npoapsfp4 = 5587 kN.

The maximum number of strands in each duct is assumed to be 12. Therefore, the

number of post-tensioning ducts is determined as:
"p0
NT 2 round (—12—) = 2 ducts.

The maximum eccentricity of the pre-tensioned strands for the composite section
over the pier centerline is deﬁned as:

ec2m : —(ytch " yeZ ) = 625 mm.

The eccentricity of the pre-tensioning strands for the composite section at the
splice location is deﬁned as:

ecz = "(ytc - ye2)= 674 mm.

The average eccentricity of the pre-tensioning strands for the composite section

at the negative segment is deﬁned as:

6 +8
6626 = M: 650 mm‘

The average top centroid of the haunched section with composite behavior for

the negative segment is determined as:

. + .
ytche : lehz ytc :354 mm.

The average cross—sectional area of at the negative segment with composite

behavior is determined as:

321

A +A
M: 741934 mmz.

Ace =

The average moment of inertia of the composite section for the negative segment
is determined as:

1 +1
Ice =—Ch———c=2.13219e11mm4.
2

The stress at the centroid of the pre-tensioned strands on the negative segment
caused by post-tensioning, secondary effects, and temporary support removal loads can

be determined as:

 

_ FPTIZ + 10071280126626 _ 1M RDLZIecze + M SECZ lecZel

f: c2 _
cgp Ace Ice Ice Ice

+ M wPTIeczel

 

Ice
= 6.72 MPa.
The average moment of inertia of the non-composite section at the negative
segment is calculated as:

I

+ I
1 — iii—51: 8.5307610 mm4

86 “
The eccentricity of the pre-tensioned strands for the non-composite section over
the pier centerline is deﬁned as:
ep2 =—(y,g —y62) = 310 mm.
The stress at the centroid of the pre-tensioned strands of the negative segment

caused by the deck, girder/deck haunch, and superimposed dead loads can be calculated

as:

322

lMdthhzl

 

1e p21 + Ill-ii—II—ec—zc—L 2.758 MPa.

dfcdp2 : 11g61 ce

The creep prestressing losses at this stage are computed according to the
AASHTO-LRFD provisions (AASHTO 2003) and the recommendations from the
NCHRP 517 report (Castrodale and White 2004). The ﬁrst two terms of the following
equation are determined according to AASHTO-LRFD Section 5.9.5.4 and the last term

is added as recommended by the NCHRP 517 report.

dfCRZ :12fcgp12 — 7dfcdp2 + 7fcgp62 = 34.0 MPa.
Final relaxation prestress losses, which include those due to elastic shortening,
shrinkage, and creep, are determined according to AASHTO-LRFD Equation 5.9.5.4.4:

dprzz = 0.3[20 — 0.4df552 — 0.2(dfpSR + dprRz )] = 32.84 MPa.

The total prestress losses at the time of deck and splice placement are calculated

as follows:

dfpreroz = (#552 + KCthdfCRz + KSthdfpSR + dfPRzz = 110-45 MP3-

The effective prestress at the centroid of the pre-tensioned strands after all losses

at the time of deck and splice is calculated as follows:
fpezz = fpe; _dfpreTDZ = 1282 MPa.

The total top girder stress at the mid-span of the negative segment at Stage 2 (current

stage) is deﬁned as:

Mg2 + Md2 + Mh2 +M02 + Aps2fpe22 _ ApstpeZZepZ

 

 

 

 

0 t 2 :-
Stge Stge Stge Stge Age Stge

= 4.4MPa < 0.6fc' = 26.89 MPa.

323

The total bottom girder stress at the mid-span of the negative segment at Stage 2 is

deﬁned as:

 

 

 

 

_Mg2 _Mdz _Mhz _Mcz + ApstpeZ2 _ Apszfpezzepz

0b2 =
Sbge Sbge Sbge Sbge Age Sbge

= 3.19MPa < 0.6fc' = 26.89 MPa.

Stage 3: Stress check at post-tensioning

The creep coefﬁcient during the post-tensioning stage is based on the time at
which post-tensioning is applied and can be calculated as (AASHTO LRFD Equation
5.4.2.3.2-1):

0.6
p0
K =_= 0.571.
CR’p 10+tp0

The shrinkage coefﬁcient during post-tensioning is determined as (AASHTO LRFD

Equation 5.4.2.3.3-1):

t
p0
K = ————-—= 0.682.
SHIP 35 + tpo

Stressing of the post-tensioning tendons causes secondary elastic shortening
prestress losses in the pre-tensioned strands in the girder segments. Additional elastic

shortening losses due to post—tensioning are computed as:

E
dprSAZ = ('Ep")fcgpc2: 39°21 MPa:
g

Total prestress losses at the centroid of the pre-tensioned strands at the post-

tensioning stage are calculated as follows:

dfpreTPZ = dfesz + KCRtpdfCRZ t KSHtpdfpSR + dfPRzz + dprSAZ = 152-71 MP3-

324

The effective prestress at the centroid of the pre-tensioned strands after all losses

at the post-tensioning stage is calculated as follows:

f pe32 = f pei -dfpreTP2 = 1243 MP3-

The friction losses are computed according AASHTO LRFD Equation 5.9.5.2.2
as shown below:

21pr = fly-(1 4419414”): 55.6 MPa.

Anchor set prestress loss is computed by estimating the loss that occurs when
each tendon pulls into the anchorage at seating. The maximum anchorage set loss, which
is at the stressing anchorage, is computed using the NCHRP 517 report Equation

7.3.1.1 .2-3 (Castodale and White 2004) as follows:

df A E x
de=2 pF 5‘” 1’ =101.7MPa.
x dpr

The distance along which the anchor set loss extends is computed using the

 

NCHRP 517 report Equation 7.3.1.1.2-2 (Castodale and White 2004) as shown below:

A E x
XA =—1— iii—IL: 1449mm.
12 211ij

The average eccentricity of the composite section over the pier is determined as:

e pt2 = kkk(4 — thhe) — 2 = 303 mm (measured above neutral-axis).

The total post-tensioning forces are determined from all of the post-tensioned

ducts and can be calculated by:

FpTO = npoapspri = 5587 kN.

325

The stress at the centroid of all post-tensioned strands on the negative segment
caused by the post-tensioning and temporary support removal loads can be determined

as:

 

 

 

2
F Fproe ,2 er2|8 :2 MSECZet2
fcgpozz ”0+ p — I” l+ P =2.18MPa.

Ace Ice Ice Ice
Elastic shortening prestress losses at the centroid of the post-tensioning strands

due to post-tensioning are computed as:

(NT—115p
2NT Eg

 

dfPTESZ : fcgpoZ= 5-44 MPa-

The total prestress losses at the post-tensioning stage are calculated as follows:

(X A —0.SL)
A

 

dPTPTZ: (1pr +dfPTESZ +de = 51.55 MPa.

The effective prestress at the centroid of post-tensioned strands after all losses

from the post-tensioning stage is calculated as follows:
f.DOf = fp012 = fpoi —dPTPT2 = 1344 MPa.

The total top girder stress at the mid-span of the negative segment at Stage 3 is

deﬁned as:

 

 

 

_ Mg2 Md2 M112 Mr2 Apstpe32 _ Aps2fpe329p2

O't3 -— + + '1' + +

Stge Stge Stge Stce Age St
ge

 

 

"poapsfpof +MPT2 +MSEC2 +MwPT
Ace Stce Stce Stce

= 8.677 MPa < 0.6fc' = 26.89 MPa.

326

The total bottom girder stress at the mid-span of the negative segment at Stage 3

is deﬁned as:

0b3z‘

 

Mg2 _ Mdz _Mhz _ Mrz + Apstpe32 + Apszfpe326p2 +
S S S A S
bge bge Sbge bce ge bge

”poapsfpof _ _ M PT2 _ M szcz _ M wPT
Ace Sbce Sbce Sbce

 

=11.11MPa < 0.6fc' = 26.89 MPa.

The total top deck stress at the mid-span of negative segment at Stage 3 is

deﬁned as:

Mr2 + ”poapsfpof _ MPT2 __ MSECZ __ MwPT =
Sftcs Accf Sftcs Sftcs Sftcs

6,53 = 2.55 MPa

< 0.6fcd' =18.62 MPa.

Stage 4: Stress check at Compression 1 limit state
M 52 = Superimposed dead load moment at midspan of negative segment
= -664 kN-m
M112 = Live load moment at midspan of negative segment
= -1945 kN-m

Total prestress losses at the ﬁnal condition after all prestress losses for

Compression 1 limit state are calculated as follows:

dfpreTPFz = desz + dfCRz t dfpSR + dfPR22 + dprSA2 = 147-3 MP3-

327

The effective prestress at the centroid of the pre-tensioned strands after all
prestress losses for the ﬁnal condition for the Compression 1 limit state is calculated as

follows:
fpg42 '—‘ fpei - dfpreTPF2 = 1249 MPa.

The stress at the centroid of the post-tensioned strands on the negative segment
caused by the deck, girder/deck haunch, and superimposed dead loads can be calculated

as:

= 1.669 MPa.

 

M +M M e
dfcdpoz :I d2 h2118p21+1 s2|| c2e|
11881 “3

The stress at the centroid of the post-tensioned strands of the negative segment

caused by post-tensioning and girder dead loads can be calculated as:

0-7Ap52fpu + 0-7Ap32fpuepZeC26 _ 1114821181721
A I

= 0.482 MPa

 

fcgpc02 = I
ge ge

ge

Prestress loss due to creep at the centroid of the post-tensioning strands in this
stage is computed according to the AASHTO-LRFD speciﬁcations and the
recommendations from the NCHRP 517-12 report. The ﬁrst two terms of the following
equation are determined according to AASHTO-LRFD Section 5.9.5.4 (AASHTO 2003)

and the last term is added as recommended by the NCHRP 517 report (Castrodale and

White 2004).
dprCR2 : 12fcgpoZ - 7dfcdpoZ + 7fcgpcoZ : 34-4 MPa-

Shrinkage prestress loss in the post-tensioning is calculated at 75% humidity and

according to AASHTO LRFD equation 5.9.5.4.2-2 (AASHTO 2003) and is deﬁned as:

dfPOSR = (13.5 — 0.123H) = 29.47 MP8.

328

Prestress loss due to ﬁnal relaxation of the post-tensioning strands is calculated

according to AASHTO-LRFD Equation 5.9.5.4.4 (AASHTO 2003):

dpr32 = 0.3(20-0.361pr -0°4dfPTESZ _0-2(dfpoCR2 +dfpoSR)= 31.87 MPa.

Total prestress losses at the Compression 1 limit state for the ﬁnal stage are

calculated as follows:

(X -0.5L)
dPTFz = dpr +dfPTES2 +de AXA +dfpoCR2 +dfpoSR +dfPR32

 

= 259 MPa.

The effective prestress at the centroid of the post-tensioned strands after all

losses at the ﬁnal Compression 1 stage is calculated as follows:
f poff = f p022 = f poi - a11mm: 1137 MPa-

The total top girder stress at the mid-span of the negative segment at Stage 4 is

determined as:

 

M M M M M M A f A f 6
82+ d2+ 122+ r2+ 52+112+p32pe42_p52pe42p2+

Stge Stge Stge S tce Stce Stce Age Stge

014 =

”poapsfpoff + M PTZ + M SECZ + M wPT
Ace Stce S tce SIce

= 3.17MPa < 0.6fc' = 26.89 MPa.

The total bottom girder stress at the mid-span of the negative segment at Stage 4

is deﬁned as:

Mg2 _ Mdz _ Mhz _Mrz _Msz _ M112 + Apstpe42 + Apszfpe426p2 +

 

 

 

 

 

 

‘7 b4 3 "
Sbge Sbge Sbge S bce Sbce Sbce Age Sbge
"poapsfpoff __ M PT2 _ M SECZ _ M wPT
Ace Sbce Sbce Sbce

329

= 23.9MPa < 0.6fc' = 26.89 MPa.

The total top deck stress at the mid-span of the negative segment at Stage 4 is

deﬁned as:

Mr2 + M32 + M112 + ”poapsfpoff _Mprz _MSEC2 _MwPT
Stcs Stcs Stcs A cc Sftcs Sftcs Sftcs

 

 

 

 

 

0354 :
= —4.254MPa < (we'd =18.62 MPa.

Stage 5: Stress check at tension limit state
The total top girder stress at the mid-span of the negative segment at Stage 5 is
deﬁned as:

M M M M
82 + d2 + h2 +Mr2 +
Stge Stge Stge Stce Stce Stce Age

M112 + Aps2fpe42 _

 

 

 

0t5 =

Apst 126426122 + ”poapsfPOff + M PT2 + M SEC2 + M wPT
Stge Ace Stce Stce Stce

 

= 5.85MPa > -0.19 fc' = —3.34 MPa.

The total bottom girder stress at the mid-span of the negative segment at Stage 5

is obtained as:

M82 _Mdz _MhZ _Mr2 _Msz —0 8M112 + Aps2fpe42 +
Sbge Sbge Sbge Sbce Sbce Sbce Age

 

 

 

 

065 =-

 

ApSZfPe‘lzePz + ”poapsfpoff _ M PT2 _ M SEC2 __ M wPT
Sbge Ace Sbce Sbce Sbce

=— 2.99 MPa > —0.19 x ,/fc' = —3.34 MPa.

The total top deck stress at the mid-span of the negative segment at Stage 5 is

calculated by:

330

M M M n a f

SICSC Stcse Stcse ACCC
_MPT2 _MSEcz _MWPT
Sftcs Sftcs Sftcs

 

 

 

UtsS =

 

 

= 11.25 MPa < 0.6fcd'=18.62 MPa.

Stage 6: Stress check at Compression 2 limit state
The total top girder stress at the mid—span of the negative segment at Stage 6 is

deﬁned as:

M M M M M A f A f e
82 (112+ 112+ r2+ 32+ .032 pe42_ p52 p842 122+

 

 

 

 

0t6 =
Stge Stge Stge S tce Stce Age Stge

”poapsfpoff +MPT2 + M SEC2 + M wPT
Ace Stce Stce Stce

 

= 6.38MPa < 0.45 fc'. = 20.17 MPa.

The total bottom girder stress at the mid-span of the negative segment at Stage 6

is computed as:

 

 

 

Mg2 _Md2 _Mh2 _Mr2 _Msz + Aps2fpe42 + Apstpe423p2 +

 

 

0% = — Sbge Sbge Sbge Sbce Sbce Age Sbge
"poapsfpoﬂ _ M PT2 _ M SECZ _ M wPT
Ace Sbce Sbce Sbce
= 15.4MPa < 0.45fc' = 20.17 MPa.
The total top deck stress at the mid-span of the negative segment at Stage 6 is
deﬁned as:

Mr2 + Ms2 + ”poapsfpoff _ MPT2 _ MSEC2 _ MwPT
Stcse Stcse Acce Sftcs Sftcs Sftcs

 

 

0256 =

331

= —0.097 < 0.45 fcd' =13.96 MPa.

Stage7: Stress check at Compression 3 limit state

The total top girder stress at the mid-span of the negative segment at Stage 7 is

 

 

 

 

 

 

 

 

deﬁned as:
M M M M M A f A f e
017 :015( 82 + d2 + h2 + r2 + 52 + p52 p642 __ p52 p842 p2 +
Stge Stge Stge Stce S tce Age Stge
"poapsfpoff ) +M112 +MPT2 +MSEC2 +MwPT
AC6 Stce SICC Stce Stce

= 0.63 MPa < 0.4fc' =17.92 MPa.

The total bottom girder stress at the mid-span of the negative segment at Stage 7

is determined as:

Mg2 _Md2 _MhZ _MrZ _Ms2 + Aps2fpe42 + Aps2fpe4zeP2 +

 

 

 

 

 

 

 

O'b7 = 0.5(-
Sbge Sbge Sbge Sbce Sbce Age Sbge
”poapsfpoff ] _ M112 _ M PT2 _ M SEC2 _ M wPT
Ace Sm Sbce S bce Sbce

=17.88MPa < 0.4fc' =17.93 MPa.

The total top deck stress at the mid-span of the negative segment at Stage 7 is

deﬁned as:

 

 

 

M M n a f n a f e M
0’57 :0.“ r2 + s2 + .00 ps p0ff _ [)0 ps pOff e)+ 112 _
Stcse Stcse Acce Stcse Stcse

_MPT2 __MSECZ _MwPT
Sftcs Sftcs Sftcs

 

 

= 3.39 < 0.4 fed =12.41 MP8.
1000

332

The strength limit state calculations for design Example 2, which include
moment and shear capacities as well as shear resistance of the interface at the splice
locations, are not shown here since their procedures are similar to the strength limit state

calculations shown for Design Example 1.

6.3.7 Design evaluation

The comparison of negative girder depth of SGB obtained by our optimal design
charts is compared with the prismatic girder depth proposed by Lounis (1997) as
illustrated in Table 6-2. The total height of our negative girder segment is in the same
range that obtained by Lounis (1997). In Lounis study, the prismatic girder section is
used for both positive and negative segment while our optimized girder illustrated in this
example is used for the nonprismatic negative girder section with a constant web depth.

Table 6-1 Comparison of negative pier segment between our method and Lounis method

 

 

Results t1 t7- t3 t4 dw tw H
(Inn!) (mm) (Hun) (Hun) (turn) (Inn!) (turn)
Lounis 180 150 100 100 870 150 1400
Our results 127 127 127 127 818 203 1326

 

Based on the presented design and analysis for the SGB with PCI BT-96 girders,
it has been shown that the developed optimized design charts for single-span spliced
girder bridges provide satisfactory results and can be efﬁciently utilized as preliminary
design aids. The developed design charts can result in less time effort in the design, can
lead to performance improvements of the bridge system and can provide guidance and

insight to bridge designers into optimal solutions.

333

7 Conclusion and Future Research Needs

7.1 Overview

The research presented in this dissertation was possible due to the combination of
elastic time-dependent analyses with structural optimization algorithms. This permitted
understanding the behavior and performance of spliced girder bridges in terms of
different component and system design variables. In addition, the use of standard precast
girders and the used of newly proposed optimized sections were studied for improved
system efﬁciency considering system geometry, construction staging and time-
dependent effects.

The present study allowed the determination of efﬁcient construction methods
and splice locations. The efﬁciency of standard I-, Box, PCI-BT Beams and new optimal
girder sections were determined to use in positive and/or negative segments. The
developed optimal negative sections were found to be more efﬁcient than using existing
standard sections with attached sofﬁt.

A signiﬁcant research contribution to the bridge engineering industry is the
development of design aids for the preliminary design of spliced girder bridge systems.
These design tools can help expedite the design process and eliminate trial and error
procedures. With the availability of achievable span lengths, the information in these
charts can be a great asset to bridge engineers so that more alternatives during the
bridge-type selection process can be available with less effort. The availability of
alternative bridge types (i.e., spliced girder bridges) to steel plate girders, can lead to

increased competition, which can result in lower cost to owners.

334

7 .2 Conclusions

Conventional precast/prestressed concrete bridges have become the preferred
type of bridge construction, primarily for reasons of economy, savings in life-cycle-cost,
fast construction and low maintenance. However, the use of precast/prestressed concrete
bridges for spans greater than 48.8 m (160 it) has been limited due to girder
transportation and handling weight limitations. Consequently, for medium-span bridges,
steel plate girders are typically the chosen solution, although they have the drawback of
higher material, construction, and maintenance costs.

In order to overcome these drawbacks and to have an alternative option to
compete with steel superstructures, methods to achieve continuity with precast
prestressed girders have become of great interest to increase the spanning capabilities of
precast concrete bridges. With the advancement in design and material technologies,
longitudinal splicing is allowing designers to overcome the limitations of fabrication,
shipping, and erection that have prevented the use of very long precast/prestressed
concrete girder bridges in the past. Much progress has been made over the. past two
decades to facilitate the analysis of spliced girder bridges by the development of research
and commercial programs (Karim and Tadros 1993, LEAP 2004a). Unfortunately, in
spite of these powerful tools, design aids of spliced girder bridges to help designers are
not available.

Longitudinally spliced girder construction is not often considered as an option in
the preliminary development of design alternates for conventional highway bridges. The
reason primarily stems from the limited experience with this typically project-speciﬁc

type of bridge as well as limited availability of preliminary analysis and design tools.

335

The availability of design aids for spliced girder bridges can be a great asset to bridge
engineers to expedite the design process and promote this bridge system as an alternative
during the preliminary design phase. Design aids can simplify the analysis and design of
spliced girder bridges by providing guidance on variable sensitivity and reduce effort in
the design process. In addition, the availability of achievable spliced span lengths for
standard beam types would help the bridge engineer have more alternatives during the
bridge type selection with less effort and time investment. Having more choices can lead
to increased competition between bridge types and materials, which can result in lower
cost to owners. Furthermore, the replacement of an existing multi-span structure with a
single-span spliced prestressed girder bridge can reduce overall costs due to the
reduction of substructure units.

Implementation of mathematical optimization algorithms can eliminate trial-and-
error design procedures and thus lead to the design of more efﬁcient systems.
Optimization techniques have been used effectively before to develop design charts for
conventional pre-tensioned girders. Computational optimization techniques (Arora 2004)
can be used as a tool to produce design aids, charts, and tables based an optimal
solutions to expedite the design process. Use of structural optimization for spliced girder
bridge design is limited. For single-span spliced girder bridges, only the maximum
achievable span lengths of standard 1800 mm (72 in.) girders with three-equal segment
lengths by using custom spreadsheets and the program CONSPLICE were investigated
(Collett and Saliba 2002). For two-span continuous spliced girder bridges, only studies
on newly proposed sections for use in both pre- and post-tensioned construction and

achievable span lengths have been performed (Lounis et al. 1997). Based on the authors’

336

knowledge, at this time there are no available design aids to determine the pre- and post-
tensioning requirements and maximum achievable span lengths corresponding to
different beam spacings for spliced girder bridges, which was thus the aim of the present
study.

This study illustrated the potential of using nonlinear structural optimization
approaches integrated with sequential time-dependent design and analysis to develop
design aid charts for the preliminary design of spliced precast/prestressed girder bridges.
The optimization procedure allowed the determination of design solutions for girder
systems in a manner that satisﬁed a common single objective or a multi-objective when
two or more objectives were to be simultaneously satisﬁed. Multicriteria optimization
techniques were used to generate a set of pareto optima with non-dominated solutions to
yield a compromise solutions for different single objective functions.

Longitudinal splicing of prestressed girders for single- and two-span continuous
bridges were considered in this study. Existing standard sections used for simply
supported bridges were used for section lengths under positive moment. Pier or negative
segments normally consist of custom-designed sections for beam segments over
intermediate supports for continuous bridge solutions. In addition to custom-design
sections, existing standard sections with increased sofﬁt were also investigated to use
over intermediate supports.

A system optimization approach was used in this study since it provides a
systematic approach to arrive at appropriate design solutions. Values that remain ﬁxed in
the optimization process, which included girder segment geometries, splice locations,

construction sequences, and beams spacing, were considered as design parameters. The

337

design variables in the optimization problem of spliced girder bridges were those
deﬁning the amount and layout of the pre-tensioning and post-tensioning strands in each
girder segment. The requirements of pre-tensioning strands, post-tensioning strands, and
their locations along the span length were obtained from the optimization studies and
presented in terms of design charts. Optimal solutions from design charts were also
veriﬁed by comparing to the results from a recent NCHRP study (Castrodale and White
2004) and proven to be acceptable solutions.

In the exploration of using standard precast/prestressed girders in single-span
spliced girder systems, results showed that shallow girder sections have higher span gain
in percentage increase than deeper girder sections. Also, large beam spacing was
determined to be more efﬁcient than tight beam spacing for use in a spliced girder
system. The use of standard girder sections with increased sofﬁt as the negative segment
in two-continuous span bridges was also evaluated. However, the increase in maximum
span length is small compared with the increase in the sofﬁt thickness.

In addition to the investigation of standard beams, component optimization was
coupled with the system optimization and used as a tool to develop optimal girder shapes
for the negative segment of two-span continuous systems. New optimal pier sections
were developed for different span lengths, beam spacings, and splice locations. These
new optimal negative sections included prismatic sections, non-prismatic sections with
varied web depth, and non-prismatic sections with constant web depth.

The developed optimal prismatic girder section has uniform cross-section for
both the positive and negative segments. This girder section was developed since the

fabrication cost of prismatic girders is lower than for non-prismatic sections when

338

dealing with negative segments. As compared to the other developed girder sections, the
new prismatic girder has a deeper section over the positive moment region and its
section height over the negative moment region is bounded by the heights of the optimal
non-prismatic girder sections.

The developed optimal non-prismatic girder section with varied web height for
the negative segment in continuous systems has a uniform bottom ﬂange thickness and
varied web height. The web height increased linearly from the depth of the prismatic
positive section to a maximum web height over the pier. Based on the comparison of
different developed girder sections, the non-prismatic girder section with varied web
depth used over the negative region yields the shallowest section over the positive
moment region and the deepest section over the negative moment region.

The developed optimal non-prismatic girder section for the negative segment
with uniform web height and a variable bottom ﬂange thickness has a bottom ﬂange
with variable thickness that increases linearly from the prismatic positive girder section
to its maximum value over the pier. This cross-section approach yields the shallowest
negative section as compared to the other developed girder sections. The height of the
positive section developed for this non-prismatic negative section is deeper than those
required for the girder section with varied web height and shallower than that required
for the optimal prismatic girder solution.

The effects of construction sequence for single- and two-span continuous spliced
girder bridges were also studied. It was learned that single-staged post-tensioning

applied after pouring the deck is the most efﬁcient construction method for single-span

339

spliced girder bridges, while multi-staged post-tensioning was determined to best for
two-span continuous systems due to their longer span length.

Furthermore, the maximum achievable span lengths for spliced girder bridges
using existing standard girders and developed optimal girders at different splice
locations and beam spacing were obtained and provided as a part of design charts. The
most efﬁcient splice locations were determined to be at 0.25(L), where L is the total
system span length, for a single-span three-segment spliced girder bridge with equal end
segments; and at 0.35(L) for two-span continuous spliced girder bridge systems.

Material parametric studies were investigated to determine the most efﬁcient
option to increase the spanning length of spliced girder bridges. The maximum span
length increase was obtained when using lightweight concrete for both girders and deck.
However, using a lightweight concrete deck generates less design issues of concern than
when using lightweight concrete on the girders, especially regarding prestress losses.
Therefore, using lightweight concrete on the deck might be a better option to increase
span length for these systems.

The structure’s service lifetime in combination with the life-cycle cost of a
spliced prestressed girder bridge was integrated into the design optimization process by
considering the variation of concrete cover to increase the structure’s durability to
chloride attach. The studies showed that with an increase in the clear cover of concrete
of less than 25 mm (1 in.) the service lifetime of a single-span spliced girder bridge

structure can be increased more than twice.

340

Given the added complexity in the design and analysis of spliced girder bridges,
the availability of design aids for spliced prestressed concrete girder bridges could
facilitate a reduction of the typical trial and error process in design and be a great asset to
bridge engineers to expedite the design process of this bridge type, potentially resulting

in its wider use by state highway agencies.

7 .3 Future Research

The research presented in this dissertation for new optimal negative sections of
two-span continuous spliced girder bridges has not yet been veriﬁed by means of
experimental testing. Only analytical models were developed and investigated in this
study. Thus, the proposed optimal negative sections presented here attempt only to serve
as new feasible standard sections for this bridge system. Veriﬁcation of the analytical
results with experimental evaluation can lead to the full development and use of new
standard negative sections for continuous spliced girder bridges.

The investigation of using lightweight concrete on the girders could also increase
the span length dramatically. However, the execution of a full-scale experimental
research program would be needed before accepting the use of this material in a complex
design concept such as spliced bridges, especially considering prestress losses and
transfer length issues.

The proposed design aids in this research were developed for use only in regions
where seismic effects are not an issue. However, the same optimization concept can be

used to develop design aids for spliced prestressed girder bridges in seismic zones.

341

REFERENCES

Chapter 1

American Association of State Highway and Transportation Ofﬁcials 2003, Bridge
design speciﬁcations, 2nd edition

Anderson, A. R 1973, “ Stretched-out AASHTO-PCI beams types 111 and IV for longer
span highway bridges,” PCI Journal, V.18, No. 5, pp 32-49

Arora, J. S. 1997, “Guide to structural optimization,” ASCE Manuals and Reports on
Engineering Practice No. 90, Reston, VA, American Society of Civil Engineer

Arora, J .S. 1989, “Introduction to optimum design,” New York, McGraw-Hill

Arora, J. S. and Haug, El. 1979, “Methods of design sensitivity analysis in structural
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348