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INNOVATIVE PRE-CAST CANTILEVER CONSTRUCTED BRIDGE CONCEPT
by
Brent Tyler Visscher
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied ScienceGraduate Department of Civil Engineering
University of Toronto
© Copyright by Brent Tyler Visscher (2008)
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Innovative Pre-cast Cantilever Constructed Bridge Concept
Brent Tyler VisscherMaster of Applied ScienceDepartment of Civil EngineeringUniversity of Toronto, Canada2008
ABSTRACT
Minimum impact construction for bridge building is a growing demand in modern urban
environments. Pre-cast segmental construction is one solution that offers low-impact, economical,
and aesthetically pleasing bridges. The standardization of pre-cast concrete sections and segments
has facilitated an improved level of economy in pre-cast construction. Through the development
of high performance materials such as high strength fibre-reinforced concrete (FRC), further
economy in pre-cast segmental construction may be realized. The design of pre-cast bridges using
high-strength FRC and external unbonded tendons for cantilever construction may provide an
economical, low-impact alternative to overpass bridge design.
This thesis investigates the feasibility and possible savings that can be realized for a single cell
box girder bridge with thin concrete sections post-tensioned exclusively with external unbonded
tendons in the longitudinal direction. A cantilever-constructed single cell box girder with a
curtailed arrangement of external unbonded tendons is examined.
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ACKNOWLEGEMENTS
This work has been partially supported by the National Science and Engineering Research
Council of Canada.
I would like to thank my supervisor Dr. Paul Gauvreau for his guidance, support, and
fellowship throughout the course of this design project.
Many thanks are owed to my friends and colleagues, especially those within our bridge
engineering research group, for their helpful discussions and insightful contributions to the
development of this project: Billy Cheung, Jamie McIntyre, Kris Mermigas, Talayah Noshiravani,
Graham Potter, Carlene Ramsay, Jason Salonga, and Jimmy Susetyo. Additional thanks go out to
other graduate students who I have had the pleasure of spending time with throughout my graduate
study: Jeff Erochko, Hyungjoon Kim, Nabil Mansour, Michael Montgomery, and Lydell Wiebe.
Special thanks to Jimmy Susetyo for his generous contributions in concrete material design and for
his laboratory assistance in test specimen preparation. I would like to thank John MacDonald for
his contribution during cylinder testing.
Finally, I thank my family for their constant encouragement and support throughout this task.
TABLE OF CONTENTS
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Scope and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.0 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2.1 Constitutive Laws for High Performance FRC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Compressive Stress-Strain Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Tensile Stress-Strain Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Typical Modern Highway Overpass Bridge Design Currently in Ontario . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Standardization of Precast Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Description of the Proposed Segmental Box Girder Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Visual Comparison of Proposed Box Girder with Typical Slab-on-Girder Overpass . . . . . . . . . . . . . . . . . 222.5 Benefits and Drawbacks of the Proposed Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.2 Drawbacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.0 LONGITUDINAL FLEXURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Moment-Curvature Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Behaviour of Unbonded Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Reference State of Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Long-Term Effective Prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Tendon Stress Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Preliminary Design Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Preliminary Design Methodology for Cantilever Tendons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1 Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6.2 Ultimate Moment Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6.3 Sizing of Cantilever Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 Change in Structural System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.8 Preliminary Design Methodology for Continuity Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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3.8.1 Ultimate Moment Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8.2 Spreading of Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.8.3 Sizing of Continuity Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.8.4 Secondary Prestress Moment due to Continuity Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.9 Bottom Flange Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.10 Prestress Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.11 Ultimate Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.12 Serviceability Limit States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.12.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.12.2 Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.12.3 Global Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.13 Refinement in Tendon Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.13.1 Ductility of Fibre-Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.13.2 Deformation Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.13.3 Example of Tendon Stress Increase for Negative Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.13.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.14 Global System Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.14.1 Span-to-Depth Ratio for Constant Depth Box Girder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.14.2 Range of Spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.14.3 Alternative Girder Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.0 TRANSVERSE FLEXURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Typical Transverse Prestressing for Segmental Bridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 Prestressing Concept for Light Weight Slab Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Post-Tensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3.2 Pretensioning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Bond Strength of 15mm Pretensioning Strands in FRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.1 Mechanisms of Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.2 Development of Stress in a 15mm Pretensioning Strand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4.3 Improvement of Bond Strength due to High Strength FRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5.1 Grillage Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5.2 Frame Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6 Comparative Study for Transverse Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.6.1 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6.2 Material Definitions used in Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6.3 Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.6.4 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 Prestress Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.8 Service Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.9 Cross Section Design and Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.9.1 Transverse Rib Proportions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
v
4.9.2 Web Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.10 Design for Barrier Impact Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.0 SHEAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
5.1 Introduction to Strut-and-Tie Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Funicular Load Path for Girders Prestressed with Curtailed Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3 Parallel Chord Truss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4 Arching Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4.1 Tied Arch Model (Noshiravani, 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5 Preliminary Design Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.6 Opening of Joints in Segmental Bridges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.7 Design of Web Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7.1 Comparison to CAN/CSA-S6-06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.0 ANCHORAGE ZONE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 Flow of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3 Strut-and-Tie Model for Local Spreading of Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.1 Anchorage of External Unbonded Tendons in Slab Blisters (Wollman, 1993) . . . . . . . . . . . . . . . . . . 1276.3.2 Design Model for External Tendon Blisters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Detailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5 Reinforcement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.0 FIXED END ABUTMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.2 Bras de la Plaine Bridge, France (Tanis, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.3 Flexible Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.4 Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.1 Preliminary Design Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.4.2 Recommended Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Sizing of Flexible Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.0 MATERIAL UTILIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
8.1 Reference Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.1.1 Windward Viaduct, Interstate Route H-3, Hawaii, USA (Hawaii DOT, 1991) . . . . . . . . . . . . . . . . . . 1458.1.2 Hwy 407 - Islington Avenue Underpass, Toronto, Canada (MTO, 1990) . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Concrete Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.3 Mild Reinforcing Steel Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.4 Longitudinal Post-Tensioning Utilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.5 Transverse Post-Tensioning Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.0 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155
vi
9.1 Longitudinal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.2 Transverse Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.3 Shear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.4 Anchorage Zone Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.5 Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.0 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159
APPENDIX A - DESIGN DRAWINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
APPENDIX B - MATERIAL TESTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175
APPENDIX C - CAN/CSA-S6-06 SHEAR PROVISIONS . . . . . . . . . . . . . . . . . . . . . . . .178
CURRICULUM VITAE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180
vii
LIST OF FIGURES2.0 BACKGROUND ..................................................................................................................7
Figure 2-1. Variation in modulus of elasticity with the ultimate compressive strength of concrete............... 8Figure 2-2. Compressive stress-strain relationship for nominal 80MPa FRC test specimens and simplified
model ...................................................................................................................................... 9Figure 2-3. Photo of intensively cracked compression cylinder after failure and significant post-peak
straining ................................................................................................................................ 10Figure 2-4. a) Dogbone-shaped specimen: dimensions and sensors, b) test set-up of the uniaxial tensile test,
showing the front side of the specimen (adapted from Habel et al., 2006) .......................... 11Figure 2-5. Tensile stress-strain relationship for nominal 80MPa FRC test specimens and simplified model:12Figure 2-6. Typical slab-on-girder highway underpass built in Markham, Ontario on Highway 407.
Photograph courtesy of Scott Steves, 2006........................................................................... 13Figure 2-7. Recommended girder spacing with respect to span length for standard CPCI precast I-girders
(adapted from CPCI, 1996)................................................................................................... 14Figure 2-8. Plan and elevation view of a possible cantilever-constructed overpass bridge.......................... 16Figure 2-9. Cross section and details of the constant depth box girder shown in Figure 2-8 ....................... 17Figure 2-10. Typical standard segment components of proposed box girder concept .................................. 18Figure 2-11. Top flange supported by steel struts, a) example shown of Shibakawa Viaduct, Japan (photo
courtesy of Takashi Kosaka, 2006), b) rendering of proposed 90m overpass structure ....... 19Figure 2-12. Second Severn Bridge, a) total external unbonded prestressing tendons, b) cantilever
construction showing bulkhead face of segment with no continuous bonded steel across the joint and no internal bonded tendons (adapted from Mizon, 1997)...................................... 20
Figure 2-13. Possible tendon arrangement shown for half the span (symmetrical about span centreline) ... 21Figure 2-14. Numbering scheme for segment labels..................................................................................... 22Figure 2-15. Rendered perspective views on typical highway overpass construction using precast CPCI
girders (left) and proposed cantilever-constructed single cell box girder (right) as seen on an overcast day........................................................................................................................... 23
Figure 2-16. Deck cross sections for a) two-span precast CPCI girder overpass and b) proposed cantilever-constructed single cell box girder. ........................................................................................ 23
Figure 2-17. Visual slenderness attained through shadow casting for a superstructure with a a) small deck overhang and b) large deck overhang. .................................................................................. 25
Figure 2-18. Collapse slab-on-girder bridge after being struck by tractor-trailer (adapted from El-Tawil, 2005) ..................................................................................................................................... 26
Figure 2-19. Elevation (1:2500 scale) showing the result of elevating the overpass road grade.................. 27
3.0 LONGITUDINAL FLEXURE..........................................................................................28Figure 3-1. Idealization of box cross section for analysis of longitudinal flexure........................................ 29Figure 3-2. Internal lever arm from tendon elevation to the compression flange, a) for a continuously guided
tendon, b) for a tendon attached only at discrete locations................................................... 30Figure 3-3. Strain Distribution at ultimate limit state (adapted from Naaman, 2006) .................................. 31Figure 3-4. Material strains due to equilibrium in the reference state and in the ultimate state ................... 31Figure 3-5. Family of moment-curvature responses for a specific cross section and varying prestress force
assuming a concentric prestress ............................................................................................ 35Figure 3-6. Iterative method behaviour for tendon stress calculation for a) girder impending plastic hinging,
b) girder experiencing plastic hinging .................................................................................. 36Figure 3-7. Iterative method solution for obtaining tendon stress by incrementing applied load................. 37
viii
Figure 3-8. Typical segment cross section .................................................................................................... 40Figure 3-9. Comparison of live load models defined in CAN/CSA-S6-06 .................................................. 41Figure 3-10. ULS bending moments on continuous girder........................................................................... 42Figure 3-11. Cantilever prestress concepts for haunched and constant depth girders .................................. 44Figure 3-12. Preliminary ULS demand and capacity for cantilever PT........................................................ 45Figure 3-13. Layout of cantilever tendons .................................................................................................... 46Figure 3-14. Redistribution of forces through concrete creep due to change in structural system (adapted from
Podolny et al., 1982) ............................................................................................................. 48Figure 3-15. Time-dependent creep coefficient assuming t0 = 28 days, a) for 75 year period, b) for 5 year
period .................................................................................................................................... 49Figure 3-16. Sectional forces due to prestressing with top and bottom tendons........................................... 51Figure 3-17. Moment development length, la, due to anchorage of continuity tendon (adapted from Menn,
1990) ..................................................................................................................................... 52Figure 3-18. Required moment development length for a girder post-tensioned with unbonded tendons ... 53Figure 3-19. Layout of continuity tendons.................................................................................................... 54Figure 3-20. Preliminary ULS demand and capacity for continuity tendons................................................ 55Figure 3-21. Secondary prestress moment due to prestressed continuity tendon ......................................... 57Figure 3-22. Ultimate state of stress for section with undersized bottom flange.......................................... 59Figure 3-23. Appropriately sized bottom flange thickness for negative bending at ULS............................. 59Figure 3-24. Primary prestress force distribution.......................................................................................... 61Figure 3-25. Girder response for varying prestress force.............................................................................. 63Figure 3-26. Girder response at ULS (half span shown)............................................................................... 64Figure 3-27. Girder response at SLS (half span shown) ............................................................................... 66Figure 3-28. Stress-strain response of FRC and typical normal-weight concrete......................................... 68Figure 3-29. FRC comparison for total moment-curvature response in negative bending ........................... 69Figure 3-30. Plastic hinging in negative moment regions............................................................................. 70Figure 3-31. 90m span with midspan hinge when load is close to failure .................................................... 71Figure 3-32. Stresses in unbonded tendons for curtailed prestressing. a) change in tendon stresses during
construction due to the addition of segments and post-tensioning, b) change in tendon stresses due to the application of increasing uniform load until failure............................................. 72
Figure 3-33. Ultimate behaviour for a continuous girder using unbonded draped tendons (adapted from Muller et al., 1990) ............................................................................................................... 73
Figure 3-34. Material consumption for varying span/depth ratios................................................................ 74Figure 3-35. Concept comparison of haunched girder and constant depth girder elevation......................... 76
4.0 TRANSVERSE FLEXURE ..............................................................................................78Figure 4-1. Possible pretensioning arrangement ........................................................................................... 82Figure 4-2. Bond mechanisms between FRC and 15mm pretensioning strand (adapted from Chao, 2006) 83Figure 4-3. Idealized strand stress profile in a pretensioned strand under applied load (adapted from Kahn,
2002) ..................................................................................................................................... 83Figure 4-4. Generalized frame element properties for grillage analysis (adapted from Menn, 1990).......... 87Figure 4-5. 3-dimensional grillage model ..................................................................................................... 88Figure 4-6. Comparison of top flange thicknesses of AASHTO-PCI-ASBI standard box girder examples
(adapted from Prestress/Precast Concrete Institute, 1997) ................................................... 90Figure 4-7. Cross section dimensions for typical segment............................................................................ 91
ix
Figure 4-8. Assumed stress-strain material properties for investigation of transverse stiffness................... 91Figure 4-9. Live load models for transverse load analysis............................................................................ 92Figure 4-10. Vertical slab deflection envelope due to live load, AASHTO-PCI-ASBI box girder .............. 93Figure 4-11. Edge beam stiffening ................................................................................................................ 93Figure 4-12. Vertical slab deflection envelope due to live load of proposed box girder: a) thin slab stiffened
with transverse ribs only, b) thin slab stiffened with transverse ribs and longitudinal edge beam...................................................................................................................................... 94
Figure 4-13. Moment distribution due to non-concentric prestress applied at the transfer length ............... 95Figure 4-14. Total transverse bending moments for transverse flexure ........................................................ 96Figure 4-15. SLS top slab stress ranges at the extreme fibre due to service loading.................................... 97Figure 4-16. Longitudinal section for a typical segment .............................................................................. 98Figure 4-17. Web spacing requirements........................................................................................................ 99Figure 4-18. Truss model for basis of barrier reinforcement ........................................................................ 99Figure 4-19. PL-2 barrier detail for impact loading .................................................................................... 100
5.0 SHEAR..............................................................................................................................101Figure 5-1. 45 degree truss model (adapted from Ritter, 1899) .................................................................. 102Figure 5-2. Possible truss models for girders with unbonded and bonded prestressing steel (adapted from
Gauvreau, 1993).................................................................................................................. 102Figure 5-3. Alternative girder designs to resist applied load Q .................................................................. 104Figure 5-4. Effect of prestress arrangement on funicular compression chord ............................................ 105Figure 5-5. Funicular shape of compression spine for continuous cantilever girder with curtailed tendons106Figure 5-6. Strut-and-tie model for a fully prestressed flanged section. a) simplified model; b) through d)
detailed model of web, top flange and bottom flange, respectively (adapted from Schlaich et al., 1989) ............................................................................................................................. 108
Figure 5-7. Variable angle truss model (adapted from Collins et al., 1997) ............................................... 109Figure 5-8. Arch spreading of forces for a rectangular section (adapted from Schlaich et al., 1989) ........ 110Figure 5-9. Comparison of tied arch model for a rectangular section and a flanged section (adapted from
Noshiravani, 2006).............................................................................................................. 112Figure 5-10. Girder dimensions (adapted from Noshiravani, 2007) ........................................................... 113Figure 5-11. Girder cross section and web reinforcement (adapted from Noshiravani, 2007) ................... 113Figure 5-12. Tied arch model for load stage corresponding to flexural cracking (adapted from Noshiravani,
2007) ................................................................................................................................... 113Figure 5-13. Tied arch model for load stage corresponding to ULS (adapted from Noshiravani, 2007) ... 114Figure 5-14. Crack pattern and alternative parallel chord model for load stage corresponding to flexural
cracking............................................................................................................................... 115Figure 5-15. Crack pattern and alternative parallel chord model for load stage corresponding to ULS .... 116Figure 5-16. Truss model for shear design. a) fully prestressed state for continuous beam; b) modified model
to account for flange decompression (formation of plastic hinge) ..................................... 117Figure 5-17. Prestress forces applied to truss model in Figure 5-16 for each applied anchorage force ..... 117Figure 5-18. Incorrect hanger reinforcement model for shear stresses crossing an open joint (adapted from
Virlogeux, 1993) ................................................................................................................. 118Figure 5-19. ULS Combination 1 maximum shear demands...................................................................... 119Figure 5-20. ULS transverse steel demand and chosen capacity ................................................................ 120Figure 5-21. Web reinforcement ................................................................................................................. 120
x
Figure 5-22. Shear panel tests for normal reinforced concrete and FRC concrete (adapted from Susetyo, 2007) ................................................................................................................................... 122
Figure 5-23. Shear parameter predictions according to CAN/CSA-S6-06 ................................................. 123
6.0 ANCHORAGE ZONE.....................................................................................................125Figure 6-1. Top and bottom corner anchorage blisters................................................................................ 126Figure 6-2. Flow of forces at an intermediate anchorage............................................................................ 127Figure 6-3. Strut-and-tie model for corner blister (Adapted from Wollman, 1993) ................................... 129Figure 6-4. Strut-and-tie model for corner blister satisfying static equilibrium; a) 3d isometric view, b) front
view, c) side elevation......................................................................................................... 130Figure 6-5. Jack clearances necessary for installation (adapted from VSL, 2007)..................................... 132Figure 6-6. Minimum anchorage eccentricities to box girder corner.......................................................... 132Figure 6-7. Anchorage blister lengths ......................................................................................................... 133Figure 6-8. Strut-and-tie anchorage model for a bottom tendon................................................................. 133Figure 6-9. Tension ties linking deviation force to box reinforcement (adapted from Beaupre et al., 1990)134Figure 6-10. Flow of forces for different reinforcement details ................................................................. 135Figure 6-11. Anchorage zone reinforcement for critical 19 strand bottom tendon..................................... 136
7.0 FIXED END ABUTMENT .............................................................................................138Figure 7-1. Bras de la Plaine Bridge, Reunion Island, France (adapted from Tanis, 2003) ....................... 139Figure 7-2. Ballasted abutment of Bras de la Plaine Bridge (adapted from Tanis, 2003)........................... 139Figure 7-3. Counterweight abutment concept (superseded)........................................................................ 141Figure 7-4. Tie-down abutment concept (superseded) ................................................................................ 142Figure 7-5. Proposed conceptual abutment design...................................................................................... 143
8.0 MATERIAL UTILIZATION .........................................................................................145Figure 8-1. Partial elevation view of Windward Viaduct (adapted from Hawaii DOT, 1991) ................... 146Figure 8-2. Typical cross section of Windward Viaduct (adapted from Hawaii DOT, 1991)..................... 146Figure 8-3. Post-tensioning arrangement for typical span of Windward Viaduct (adapted from Hawaii DOT,
1991) ................................................................................................................................... 147Figure 8-4. General arrangement of Islington Avenue Underpass (adapted from MTO, 1990) ................. 148Figure 8-5. Typical cross section of Islington Avenue Underpass (adapted from MTO, 1990) ................. 148
xi
xii
LIST OF TABLES2.0 BACKGROUND ..................................................................................................................7
Table 2-1. Reference concrete mix quantities (Susetyo, 2007)....................................................................... 9Table 2-2. Simplified constitutive law for the nominal 80Mpa stress-strain behaviour used in analysis ..... 12
3.0 LONGITUDINAL FLEXURE..........................................................................................28Table 3-1. Procedure for calculating the stress in an unbonded tendon due to girder deflections ................ 34Table 3-2. Procedure for calculating uniform applied load corresponding to a desired tendon stress.......... 37Table 3-3. Estimation of dead load for typical segment................................................................................ 40Table 3-4. Estimation of superimposed dead load for typical segment ........................................................ 40Table 3-5. Preliminary prestressing scheme for cantilever tendons based on ultimate capacity .................. 46Table 3-6. Calculation of long-term bending moment redistribution............................................................ 50Table 3-7. Preliminary prestressing scheme for continuity tendons based on ultimate capacity.................. 54Table 3-8. Secondary prestress moment calculation ..................................................................................... 56Table 3-9. Tendon unit sizes (15mm strands) used for cantilever post-tensioning ....................................... 70
6.0 ANCHORAGE ZONE.....................................................................................................125Table 6-1. Area of steel required for local spreading reinforcement for top tendons (in mm2).................. 137Table 6-2. Area of steel required for local spreading reinforcement for bottom tendons (in mm2) ........... 137
8.0 MATERIAL UTILIZATION .........................................................................................145Table 8-1. Concrete consumption for one span of the Windward Viaduct (Hawaii DOT, 1991) ............... 149Table 8-2. Concrete consumption for Islington Avenue Underpass (MTO, 1990) ..................................... 149Table 8-3. Concrete consumption for proposed cantilever box girder ........................................................ 149Table 8-4. Mild reinforcing steel utilization for one span of the Windward Viaduct (Hawaii DOT, 1991)150Table 8-5. Mild reinforcing steel utilization for Islington Ave Underpass (MTO, 1990) ........................... 151Table 8-6. Mild reinforcing steel utilization for proposed box girder......................................................... 151Table 8-7. Post-tensioning utilization for one span of the Windward Viaduct (Hawaii DOT, 1991) ......... 152Table 8-8. Post-tensioning utilization for Islington Ave. Underpass (MTO, 1990) .................................... 152Table 8-9. Post-tensioning utilization for proposed cantilever box girder .................................................. 152Table 8-10. Transverse prestressing steel utilization for one span of the Windward Viaduct (Hawaii DOT,
1991) ................................................................................................................................... 153Table 8-11. Transverse prestressing steel utilization for Islington Avenue Underpass (MTO, 1990) ........ 153Table 8-13. Relative material consumption of proposed box girder ........................................................... 154Table 8-12. Transverse prestressing steel utilization for proposed box girder............................................ 154
LIST OF SYMBOLS
Ap area of unbonded prestressing steel
Aps area of prestressing strand
As area of bonded steel
b width of compression flange
bbf width of bottom flange
bc concrete width function for idealized cross section
btf width of top flange
c distance from neutral axis to extreme compression fibre
d depth of cross section; centre-to-centre distance between tension and compression piles
db diameter of bonded steel
dp distance from centroid of prestressing steel to extreme compression fibre
dv effective shear depth taken as the greater of 0.9d or 0.72h (CSA 23.3-04)
e eccentricity of prestressing steel
eb continuity tendon eccentricity from girder axis
Ec compressive modulus of elasticity of concrete
Ep elastic modulus of prestressing steel
Et tensile modulus of elasticity of concrete
et cantilever tendon eccentricity from girder axis
fc stress in concrete
Fc force in concrete
f’c cylinder compressive strength of concrete
fci initial concrete strength at transfer
long-term effective prestress
fps stress in bonded prestressing strand
fpu ultimate stress of prestressing steel
fpy yield stress of prestressing steel
fse effective stress in bonded pretensioning steel after losses
fsi stress in bonded prestressing strand just prior to transfer
fsy yield stress of bonded reinforcing steel
f’t dogbone tensile strength of concrete
fp∞
xiii
H horizontal force in pile caused by deflection
Ic uncracked sectional moment of inertia
K torsional constant
kb distance from girder axis to centroid of compression block in bottom flange
kt distance from girder axis to centroid of compression block in top flange
L span length
la moment development length
Lb length between anchorage blister ties for spreading of forces
Lc length of cantilever
lf flexural bond length for 14mm pretensioning strand
lt transfer length for 15mm pretensioning strand
Lten length of continuity tendon
M bending moment
fully redistributed bending moment
Mclo dead load moment applied to statically determinate cantilever at time of closure
Mf factored total moment at a section
Mfal bending moment obtained assuming entire structure was cast simultaneously on falsework
live load moment applied to statically indeterminate continuous girder
MP primary prestress moment,
Mpb prestress moment due to bottom continuity tendons
Mq moment due to external load
Mr factored moment resistance
superimposed dead load applied to statically indeterminate continuous girder after closure
Mtot total moment due to dead load plus external loads plus prestress loads
N axial force in concrete
P force in prestressing steel
P0 initial force in prestressing steel after jacking
long-term effective prestress force after all losses
Pb bottom prestress force of continuity tendons
pps perimeter of pretensioning strand
Pt top prestress force of cantilever tendons
∆
M∞
MLLindet
P e×
MSDLindet
P∞
xiv
Ptot total prestress force of unbonded tendons for top and bottom prestressing steel
q uniform load
Q applied load
Qu applied load corresponding to the ultimate state
tbf thickness of bottom flange
ttf thickness of top flange
uf average flexural bond stress of pretensioning strand
V shear
Vc shear resistance attributed to concrete
Vf factored shear demand
Vs shear resistance attributed to transverse reinforcement
w width of web; uniform dead load
x coordinate along longitudinal axis of girder
y elevation in cross section (where bottom fibre elevation is y = 0)
yclo cantilever tip deflection at closure
midspan deflection of continuous system due to long-term creep
ratio of the average stress in compression block to the specified concrete strength
live load factor
prestress load factor
ratio of the depth of compression block to the neutral axis depth
girder deflection
centre-to-centre spacing of transverse ribs
change in beam length due to prestress, dead load, and applied load
elongation of unbonded prestressing steel relative to initial prestress
initial change in beam length due to prestress and dead load
increase in tendon force due to girder deformations
loss in prestress due to superimposed dead load
increase in prestress due to superimposed dead load
change in strain in prestressing steel relative to strain due to initial prestress
strain
bottom fibre strain
ultimate strain of concrete
initial longitudinal strain in concrete at level of prestressing steel
yφ
α1
αL
αP
β1
∆
∆b
∆lcp
∆lcp ∆lp0–
∆lp0
∆P
∆P∞
∆Pdl
∆εp
ε
εb
ε'cεc0
xv
average initial strain in concrete due to prestress and dead load
strain caused by concrete stress fc
strain in concrete at level of prestressing steel
ultimate compressive strain
strain in prestressing steel
initial strain in prestressing steel
top fibre strain
cracking strain of fibre-reinforced concrete
strain of concrete at mid depth of member
girder slope; angle of principle compressive stress in the web; angle of force resultant trajectory for arching shear
angle of compression force in slabs due to anchorage bursting forces
angle of compression force in anchorage blister projected onto flange
angle of compression force in anchorage blister projected onto web
coefficient of friction
fully redistributed concrete stress
bottom fibre stress
concrete actual stress at time of closure
concrete stress obtained assuming entire structure is built simultaneously on falsework
stress in prestressing steel
long-term effective prestress
top fibre stress
curvature
resistance factor for concrete
resistance factor for prestressing steel
resistance factor for steel
cantilever tip rotation at closure
εc0 avg,
εcf
εcp
εcu
εP
εP0
εt
ε'tεx
θ
θb
θf
θw
µ
σ∞
σb
σclo
σfal
σp
σp∞
σt
φ
φc
φp
φs
ωclo
xvi
1
1.0 INTRODUCTION
In modern urban environments, providing increased capacity on existing roadways and bridges
is a significant challenge. Engineers are faced with several demanding factors including tight
construction scheduling, the maintenance of traffic in highly travelled corridors, and the creation
of structures that enhance the urban landscape. Precast segmental construction is one solution that
can provide aesthetically pleasing structures that can be erected on difficult construction sites. By
the method of match-cast precasting, segments can be fabricated and prepared off-site and erected
rapidly with minimal effect on traffic.
Traditional methods of bridge construction over high-volume highways generally require lane
closures, traffic delays, and in some cases total road closures and traffic detours. Faced with a
growing economy and increased need for traffic flow reliability, minimum impact construction and
rapid erection for bridge building is becoming more crucial especially in the dense urban
environment. This necessity is realized not only for commuter traffic but also industries that rely
on just-in-time delivery. In Europe, new technologies have been implemented exclusively for the
purpose of fast erection in order to minimize impact of construction. Recent implementations of
fast construction in Ontario has been the use of full-width precast panels for overpass bridge
construction over Highway 401 in the Municipality of Chatham-Kent (Rapoport, 2006), and the
recent use of a self-propelled modular transporter for replacement of an overpass bridge on
Highway 417 in Ottawa (Tinkess, 2007). The implementation of new rapid bridge building
technology in Ontario is a testament to the increasing need of new construction methods for fast,
low-impact construction.
New opportunities for efficient and aesthetically pleasing designs exist through the
development of high performance materials such as high strength concrete. Due to the low water-
to-cement ratio made possible by the inclusion of water reducing admixtures, high strength
concrete exhibits favourable design properties such as higher ultimate stress and stiffer modulus of
elasticity. The impact on design due to these improved material properties is the ability to create
more elegant structural designs that make better use materials through the use of thinner sections
that result in light weight solutions. The increased stiffness of new materials allows engineers to
design more slender shapes that add to the aesthetic transparency of the structure.
Light weight girders with thin sections for prestressed concrete structures can be made possible
through the use of external unbonded tendons. Flange and web sections with large amounts of
internal bonded steel generally require a significant amount of concrete clear cover solely for the
purpose of durability and corrosion protection. This sacrificial layer of concrete is necessary to
protect the bonded tendons from environmental exposure, but it is not designed as an efficient use
of materials from a strength perspective. By removing the tendons from the concrete section, the
concrete thickness may be reduced to make more efficient use of concrete material, while reducing
the overall weight of the superstructure.
1.1 Statement of Problem
A new minimum-impact solution for rapid highway overpass construction is desired which
makes efficient use of high-performance concrete and external unbonded tendons. The use of
external unbonded tendons in prestressed concrete structures allows for the minimization of
concrete consumption, reduction in dead load due to thinner cross sections, and simplified
construction.
The cantilever method is one type of segmental construction that is self-supporting as it allows
erection of the bridge from the fixed end with no interference with the ground below and no
requirement for falsework. This method is attractive for minimum impact highway overpass
construction since it allows the bridge to be built overhead of the traffic with no requirement for
equipment and machinery below the bridge. For prestressed concrete cantilever-constructed
bridges, cantilever tendons are traditionally designed as internally bonded within the top flange.
Flange thickness is governed by the space required for tendons inside the concrete section, and the
minimum reinforcing steel required for adequate crack control and durability.
Through the development of high-performance materials such as high strength fibre-reinforced
concrete (FRC), improved mechanical properties have been observed to allow engineers greater
possibilities for the design of prestressed concrete structures. Of interest are the benefits to be
gained by the adaptation of high strength FRC and unbonded tendons for use with precast
segmental structures. Quality control of concrete is maximized for precasting in comparison to
cast-in-place concrete, as it is fabricated, cast and cured in a controlled environment. The use of
high strength FRC in combination with external unbonded tendons in concrete structures appears
to be a complimentary pairing of materials as external tendons allow for the minimization of
2
concrete consumption and high strength concrete allows for large precompression stresses from
external prestressing.
Precast segmental cantilevered-construction using exclusively external unbonded tendons with
no continuous reinforcing steel is traditionally not done in standard practice. One rare example of
the use of total external unbonded tendons for precast segmental cantilever construction is the
Second Severn Bridge across the River Severn between England and Wales. Although the use of
total external unbonded prestressing for this bridge was primarily due to a Government durability
requirement, this design also simplified deck unit production and enabled thinner webs and flanges
(Mizon, 1997). Extending the use of external unbonded tendons to cantilever construction
potentially provides an opportunity for designers to erect medium span overpass bridges faster, off
the right-of-way, with little or no interference with traffic demands.
1.2 Scope and Objective
The purpose of this study is to develop and validate a design concept using a new application
of external unbonded tendons intended for rapid overpass bridge construction requiring a low-
impact construction technique.
The objectives of this study are as follows:
• develop valid design assumptions for consideration of the proposed bridge type
• determine the global member response of the superstructure under serviceability and ultimate limit states
• incorporate the use of high performance (80MPa characteristic compressive strength) fibre-reinforced concrete (FRC) for the design of thin sections to reduce dead load and minimize reinforcing steel
• validate the use of external unbonded tendons as a new application in segmental cantilevered construction
The member response investigated for this study examines explicitly the effects of longitudinal
flexure, longitudinal shear and transverse flexure due to the force effects of dead load, live load,
and prestress.
Several actions have not been considered explicitly within the scope of this thesis. These
effects include: a) thermal action, b) torsional loads, and c) combined effects of moment, shear, and
torsion. These effects are important to the final design and for the full understanding of structural
behaviour, and could possibly have an impact on the proposed design concept. Future work is
3
necessary for the complete validation of the superstructure concept. In addition, the primary focus
of this thesis has been on the design and development of the superstructure system for a light
weight box girder design using high strength FRC. The design and development of the substructure
system is briefly discussed but a detailed design of the substructure is beyond the scope of this
work.
1.3 Thesis Structure
This document consists of 9 chapters. Chapter 1 introduces the topic and establishes the design
motivation for using high performance concrete and external unbonded tendons for the application
of rapid highway bridge construction. Chapter 2 presents background information on material
properties and the proposed structural system for discussion. Chapter 3 describes the behaviour of
unbonded tendons in a system with curtailed prestressing and the design for longitudinal flexure of
the proposed bridge. Chapter 4 discusses the motivation for minimizing the thickness of the top
flange and provides a feasible solution for transverse bending using a thin top flange and transverse
pretensioning. Chapter 5 presents a variation of design models for the analysis of shear behaviour
in concrete structures prestressed with unbonded tendons, and provides the design for shear for the
proposed bridge. Chapter 6 describes the importance of anchorages for girders prestressed with
external tendons, and provides the design developed for anchorage reinforcement. Chapter 7
briefly discusses a design concept for a substructure design. Chapter 8 reviews the material
consumption for the proposed bridge and compares the material consumption of other
conventionally constructed bridge designs. Conclusions for this work are summarized in Chapter
9.
Chapter 2 begins by describing the benefits of FRC and the improved constitutive laws that are
distinctive of the material. The results of a mini experimental study are presented and used for the
basis of a simplified design model for the behaviour of high strength FRC in compression. Typical
highway overpass bridges are discussed and the proposed alternative solution is described and
compared.
Chapter 3 first describes the flexural response of concrete girders prestressed with unbonded
tendons and convenient analysis techniques for the preliminary design of girders. For concrete
girders that are prestressed under the action of self-weight, a reference state of strain is described
that corresponds to the deformations of the girder after jacking. To begin the design process,
conservative preliminary design assumptions are described for the state of stress of the unbonded
4
tendons. Following this, the design methodology for cantilever tendons is discussed for a
preliminary estimate of tendon arrangement. The change in structural system due to closure of the
span is discussed and an estimation for long-term redistribution of forces is presented. The design
methodology for continuity tendons is then presented and the effect of the tendon layout on
secondary prestress moments is described. Based on the designed prestress loads, the bottom
flange of the girder is designed for negative bending at the support. The girder behaviour is then
analyzed under factored loads at the ultimate limit state, and unfactored loads for seviceability
performance. Following this, a refinement in tendon stress at the ultimate limit state is investigated
to improve the economy of prestress strand consumption. Finally, the effect of span-to-depth ratio
of the proposed box girder is presented, and a practical range of spans for the proposed box girder
is identified.
Chapter 4 describes the design challenges of satisfying prestress requirements for the thin top
slab in transverse bending due applied vertical loads and prestress forces. The design assumption
of small concrete covers is addressed. The chosen pre-tensioning prestress system is justified based
on the available studies that demonstrate the improved bond strength characteristics between high
strength FRC and 15mm pretensioning strands. The method of analysis to determine the live load
distribution of forces is described. A comparative study between the proposed box girder and an
AASHTO-PCI-ASBI standard box girder (Precast/Prestressed Concrete Institute, 1997) is then
carried out for the validation of the thin top flange design in transverse bending on the basis of
maximum allowable deflections. The deflections and computed stress ranges of the top flange at
service loads is shown for the final design. Finally, a design detail for a proposed PL-2 barrier
defined in the CAN/CSA S6-06 (Canadian Standards Association, 2006) is presented for the
resistance of impact loads.
Chapter 5 begins with a discussion on strut-and-tie models and the description of the funicular
load path in a prestressed girder with curtailed unbonded prestressing tendons. Various approaches
to shear design are discussed based on alternative decompositions of the funicular load path in
concrete structures. A parallel chord model is discussed and the variable angle parallel chord model
used in the CAN/CSA-S6-06 shear provisions is described. Based on the results of a previous
study, an arching model by Noshiravani (2007) is described and compared to the parallel chord
model. For the proposed box girder, a parallel chord model is decided to be most appropriate to
describe the flow of forces in the box girder to transfer vertical loads. The proposed girder is then
5
analyzed for the shear demands based on CAN/CSA-S6-06 load configurations. Finally, the design
for web reinforcement is presented based on the parallel chord model.
Chapter 6 begins with a discussion of anchorages for prestressed concrete girders with external
unbonded tendons. The fundamentally different flow of forces within the anchorage zone for an
unbonded tendon is then described in comparison to the anchorage zone of an internal tendon. A
three-dimensional strut-and-tie model is developed for the design of reinforcement required to
anchor an external unbonded tendon at a corner blister. Possible reinforcement detailing based on
the developed truss model is proposed to enable the required flow of forces.
Chapter 7 briefly describes a conceptual abutment design for the proposed bridge. The
abutment must: a) adequately provide anchorage of all cantilever prestressing, b) resist large fixed
end moments due to negative flexure of the superstructure, and c) provide longitudinal flexibility
to allow for thermal movements since the proposed bridge is designed to be monolithic at the
midspan with no expansion joints. A detailed analysis of substructure behaviour is beyond the
scope of this work, however.
Chapter 8 first describes two reference bridges which are used to determine the material
utilization of conventional construction in comparison to the proposed box girder bridge. One
reference bridge is representative of traditional cantilever construction, and the other reference is
representative of a typical highway overpass structure in Ontario. Concrete usage, mild steel
reinforcement utilization, longitudinal prestressing and transverse prestressing steel consumption
quantities are compared.
Chapter 9 concludes with a summary of findings for the design components investigated within
this study. Recommendations for the design of cantilever-constructed bridges prestressed with
unbonded tendons are provided. Advantages of using thin concrete sections with unbonded
tendons are described, and the disadvantages are identified. Necessary future work is discussed for
the full validation of the proposed design concept.
6
7
2.0 BACKGROUND
This chapter begins by discussing the behaviour of fibre-reinforced concrete (FRC) under axial
compression and axial tension. The results of a small experimental program are presented and
briefly discussed for test specimens fabricated with nominal 80MPa strength concrete, similar to
the concrete intended for design. A simple stress-strain model is developed for the FRC specimens
tested which will be used for the behaviour of concrete throughout the work of this study.
Following this, conventional highway overpass bridge construction using precast components is
described and a description of the proposed bridge concept using precast segments is presented. A
visual comparison of the proposed bridge to a conventional overpass is made to evaluate the
aesthetic qualities of the proposed bridge. Finally, benefits and drawbacks of the proposed bridge
concept are identified.
2.1 Constitutive Laws for High Performance FRC
2.1.1 Compressive Stress-Strain Behaviour
The innovation of material technology has introduced a widespread variety of high
performance concrete materials that have improved mechanical properties over traditional 35MPa
concrete mix designs. High performance concretes (HPC) range to approximately 100MPa and
ultra-high-performance fibre-reinforced concrete (UHPFRC) can be as high as 200MPa. Research
programs have been carried out in many countries to study the behaviour high performance
concrete, leading to significant changes to design codes in several countries. (Paultre et al., 2003).
Design codes are continually evolving documents using a consistent philosophy and the latest
research results, often reflecting the prior state of the art and tradition of the country of origin
(Paultre et al., 2003). However, because concrete is a heterogeneous material without standardized
mixture designs, the compressive stress-strain responses of different concrete mixtures exhibit
significant scatter (Graybeal, 2007). As a result, the empirical definitions of compressive stiffness
of HPC is not entirely consistent among design codes. A comparison of elastic modulus code
definitions and recent results of some researchers are displayed in Figure 2-1 below.
Figure 2-1. Variation in modulus of elasticity with the ultimate compressive strength of concrete
While these design values for elastic stiffness are useful for the calculation of structural
behaviour at the serviceability limit state, very little information is available in the literature that
discusses the post-peak descending branch of FRC. The post peak behaviour is important since it
has a considerable effect on the flexural ductility of a member at the ultimate limit state. Due to the
large amount of scatter in the elastic stiffness constitutive laws, and the lack of information
published on the compressive post-peak response of FRC, compression cylinder tests were
performed to develop a full range stress-strain material model for use in the analysis of the
proposed design.
A concrete mix design was adapted from Susetyo (2007) for a nominal 80MPa FRC containing
1.5% hooked steel fibre volume ratio. The reference concrete mix design is outlined in Table 2-1.
Two compression cylinder specimens were tested for the basis of the simplified stress-strain model
shown in Figure 2-2. The average peak stress was observed to be 74.5MPa, and the average
uncracked elastic modulus was observed to be 43680MPa. A more thorough description of the
experimental program is given in Appendix B. The compression cylinder test results were
compared with a previous study by Chao et al. (2006), who performed cylinder tests for a 75MPa
FRC mortar matrix containing no coarse aggregate and 1% steel fibres, and found there was good
agreement in the results. The curves are comprised of a stiff initial ascending branch, followed by
a ductile post-peak descending branch.
0
00001
00002
00003
00004
00005
00006
00007
00008
00009
00108060402
80Mpa concrete strength intended for design
Ec [MPa]
f‘c [MPa]
Graybeal 2007 (f’c ≤ 200MPa)CHBDC 2006 (f’c ≤ 85MPa)ACI 1992 (f’c ≤ 83MPa)ACI 2005 (f’c ≤ 40MPa)
Ma et al. 2004 (f’c ≤ 200MPa)CEB-FIP 1990 (f’c ≤ 88MPa)CSA 2004 (f’c ≤ 80MPa)CSA 2004 (f’c ≤ 40MPa)
8
Figure 2-2. Compressive stress-strain relationship for nominal 80MPa FRC test specimens and simplified model
The presence of fibres in the concrete matrix provides excellent confinement which facilitates
the sustained compressive stresses after concrete crushing and allows for significant ductility in the
post-peak response. Figure 2-3 displays a failed FRC compression cylinder after the test. Even
after significant post-peak straining, the concrete remains intact and very little spalling from the
surface is observed. The resistance to spalling not only contributes to the strength of FRC at the
ultimate limit state, but it also provides an important benefit for durability of structures as it
improves the performance for frost resistance during freeze-thaw cycles (Xu et al., 1998).
Table 2-1. Reference concrete mix quantities (Susetyo, 2007)
Material Measured Unit Quantity Equivalent Volume
HSF Cement kg 600 0.191
Sand (SSD) kg 1133 0.419
10mm limestones (SSD) kg 802 0.292
Water L 162 0.162
Water Reducer mL 4200 -
Superplasticizer mL 9600 -
Entrapped Air 0.02
Total Volume m3 1.083
0
01
02
03
04
05
06
07
08
09
001
800.0700.0600.0500.0400.0300.0200.0100.00
Chao (2006)
UofT testscompression model developed for design
# specimens tested = 2Eavg = 43680 MPaε’c avg = 0.00209 mm/mmf’c avg = 74.6 MPa
fc [MPa]
εc [mm/mm]
9
Figure 2-3. Photo of intensively cracked compression cylinder after failure and significant post-peak straining
2.1.2 Tensile Stress-Strain Behaviour
Since fibre-reinforced concrete exhibits a more ductile tension stress-strain response in
comparison to the brittle nature of normal concrete, the standard ASTM C496 cylinder splitting
test is insufficient to measure the tensile response of FRC. In lieu of the ASTM C496 test,
deformation-controlled dogbone-shaped test specimens loaded in uniaxial tension have been
performed by previous researchers to provide a tensile stress-strain relationship of FRC. Susetyo
(2007) performed a dogbone tension test on the nominal 80MPa FRC similar to the test set-up by
Habel et al. (2006), shown in Figure 2-4 below.
10
Figure 2-4. a) Dogbone-shaped specimen: dimensions and sensors, b) test set-up of the uniaxial tensile test, showing the front side of the specimen (adapted from Habel et al., 2006)
The result of the dogbone tension test performed by Susetyo (2007) is presented in Figure 2-5.
The FRC mix design predominantly exhibits tension softening following the occurrence of the first
crack. In general, the formation and opening of one primary crack dominates the post-cracking
response due to the relatively low fibre ratio of 1.5%. Although tension stiffening behaviour is
possible for some FRC mix designs (which generally requires a larger amount of fibres), an
increased consumption of steel fibres to achieve this behaviour is not necessary or economical for
the application of FRC in precast segmental construction. For either dry joints or epoxy joints
between precast units, there is no embedment length of fibres across the joint; therefore, the
smeared tension behaviour of FRC can not be relied upon at the joint interface undergoing flexural
tension forces. For epoxied joints, a brittle tensile behaviour would be expected that is related to
the cracking strength of the concrete, with no contribution of the fibres since internal equilibrium
of tension stresses in the fibres can not be established at the joint. Thus, the inclusion of steel fibres
in the concrete mix is included predominantly for the benefit of controlling plastic shrinkage
cracking which replaces typical temperature and shrinkage reinforcement.
Sivakumar (2007) has provided a study which indicates that for a high strength 60MPa silica
fume cement, the inclusion of only 0.5% steel fibres reduces the plastic shrinkage cracking by 49%
compared to the same mix with no fibres. Some hybrid mix designs including steel fibres and
polyester fibres provided a better reduction in shrinkage cracking by 95% while maintaining
a) b)
1005050
100
5050
100
100200
LVDT
U4
Front Backspecimendepth = 50mm
U4 - deformation transducersLVDT - linear variable differential transformer
11
reasonable workability. For the material considered in this report, the fibre ratio of 1.5% steel fibres
should provide reasonable control over plastic shrinkage cracking; however, no tests have been
performed to measure the cracking behaviour of this mix design.
A simplified model for the tension behaviour has been estimated using a tri-linear response
(Figure 2-5). Based on the dogbone test described above, a simplified constitutive material model
has been created for design of the proposed bridge concept. A summary of the response for both
tension and compression is given in Table 2-2 below, where tension stress is positive and
compression stress is negative
Figure 2-5. Tensile stress-strain relationship for nominal 80MPa FRC test specimens and simplified model:
Table 2-2. Simplified constitutive law for the nominal 80Mpa stress-strain behaviour used in analysis
Strain[mm/mm]
Stress [MPa]
-0.016 0
-0.00209 -74.6
-0.0008539 -37.3
0 0
0.00012 5.2
0.0025 2.2
0.01 0
0
1
2
3
4
5
6
10.0800.0600.0400.0200.00
Susetyo (2007)
# specimens tested = 1Et = 63600 MPaε’t = 0.00012 mm/mmf’t = 5.15 MPa
tension model developedfor design
ft [MPa]
εc [mm/mm]
12
2.2 Typical Modern Highway Overpass Bridge Design Currently in Ontario
Since the 1950’s, prestressed concrete bridges have become increasingly popular, and
comprise about two-thirds of all bridges with spans between 18 and 36m (Lounis et al., 1997). The
precast concrete I-girder system represents about 50% of all prestressed concrete bridges built up
until the early 1990’s (Dunker et al., 1992). Precast I-girder systems have expanded their use even
further in the last thirty years through the concept of spliced girders to extend the span capability
(Rabbat et al., 1999; CPCI, on-line). Although transportation equipment and available cranes
limited the length of precast pretensioned girders to around 34m in the 1960’s, precast I-girders
now can be fabricated and transported in lengths of 40m to 50m and weights up to 75 to 90 tonnes
(CPCI, on-line). The standardization of girder sections has led to simplified designs, speed of
construction, and resulted in economy (Rabbat et al., 1999).
Since slab-on-girder bridges represents such a large proportion of bridges of short to medium
spans, precast slab-on-girder bridges will be regarded as a typical highway overpass design for
comparison to the proposed precast segmental box girder design. Both of these systems are precast
and are comparable in terms of low impact of construction solutions. A visual example of a typical
modern slab-on-girder structure is shown in Figure 2-6. This bridge depicts a typical continuous
two-span overpass structure supported by a centre pier in the median of the highway. The
abutments have parallel wingwalls oriented parallel to the axis of the bridge. A concrete parapet
lines the edges of the deck slab along the length of the bridge and continues to the furthest extents
of the wingwalls.
Figure 2-6. Typical slab-on-girder highway underpass built in Markham, Ontario on Highway 407. Photograph courtesy of Scott Steves, 2006.
13
To maximize economy, bridge girder sections have been standardized as it provides a basis for
consistency and allows for the multiple reuse of standard forms for several bridge projects (Figg,
1997). The Canadian Precast/Prestressed Concrete Institute (CPCI) provides a range of standard
sections that are available in Canada. A suggested range of spans is provided for the use of each
standard section and a recommended transverse girder spacing is provided in relation to the girder
span length (shown in Figure 2-7). Although this recommendation is given only as a guideline, it
provides engineers with a launching point in which to form a slab-on-girder bridge concept.
Figure 2-7. Recommended girder spacing with respect to span length for standard CPCI precast I-girders (adapted from CPCI, 1996)
In addition to precast concrete girder sections being standardized, the design of slab-on-girder
bridges are also somewhat standardized as they must conform to several geometric constraints for
evaluation using the Simplified Method of Analysis detailed in the CAN/CSA-S6-06 (cl. 5.6.1, cl.
5.7.1). As the Simplified Method outlines an empirical approach to provide safe designs for a wide
range of span lengths and bridge widths, the design of bridges using these methods in general
produces conservative designs.
2.2.1 Standardization of Precast Components
Following the standardization of precast I-girder sections, an evolution for prestressed precast
concrete applications emerged in 1997 for standardized segmental bridge construction (Rabbat et
al., 1999). The AASHTO-PCI-ASBI Segmental Box Girder Standards provided a new product for
grade separations and interchange bridges for span lengths up to 61m (Freyermuth, 1997). Two
families of standard box girder sections exist: one intended for span-by-span construction, and
another intended for cantilever construction. The standard box girder for span-by-span
construction, which has a maximum intended span of 45.7m, exhibits thinner flanges since all post-
14
tensioning steel is comprised of external unbonded tendons. The standard box girder for cantilever
construction has thicker top and bottom flange components to incorporate the internal bonded
cantilever and continuity post-tensioning tendons and a thicker web since cantilevering can achieve
longer spans.
Segmental spans can be erected very quickly, typically in a few days, minimizing disruption of
traffic (Figg, 1997). For bridge projects intended for the standard precast box girders, it is generally
anticipated that segments are erected by crane (Freyermuth, 1997). The large amount of work that
can be done off site under a controlled environment is favourable as it reduces the amount of on-
site construction time and permits the best curing conditions for the concrete to achieve a higher
standard of durability. Quality control can be maintained as segments fabricated with high-quality
precast concrete can be erected during non-peak traffic hours when disruption of traffic is kept to
a minimum.
The recommended minimum concrete strength for the standard segmental sections is 34MPa.
In some cases, concrete with a greater compressive strength can be used; however, no changes to
the standard AASHTO-PCI-ASBI cross sectional dimensions and cross sectional thicknesses can
be made to exploit the full potential and economy of high-strength concrete mix designs. These
standard sections have been developed for the application of traditionally reinforced normal
concrete mixes. For the application of high-performance FRC and total external unbonded post-
tensioning, a new structural section, respecting the inherent advantages of FRC, needs to be
developed to achieve economical designs and efficient use of the material.
2.3 Description of the Proposed Segmental Box Girder Concept
The proposed bridge system for precast segmental cantilever construction is a constant depth
single cell box girder with a span-to-depth ratio of 25:1. The box girder cross section is designed
to have thin flanges and thin webs to make efficient use of high-strength concrete and to reduce
dead load of the superstructure. The maximum bridge span considered for this study is 90m for
practical and visual considerations. As a practical consideration, the maximum depth of the girder
segments should be limited to accommodate ease of transportation which is constrained by
standard vertical clearances for existing overpass structures. As a visual consideration, the depth
of the girder should be limited to maintain a reasonable level of slenderness for the design. At a
25:1 span-to-depth ratio, a 90m span length corresponds to a girder depth of 3.6m.
15
For the transportation of tall segments, a “low-boy” trailer is necessary which has a lowered
section of the trailer to reduce the top of the bed to within 610mm of the roadway surface (Precast/
Prestressed Concrete Institute, 1997). The minimum vertical clearance provided under highway
bridges in Ontario is 4.7m for existing structures and 4.8m for new structures (Ontario Ministry of
Transportation, 2002). The barrier walls of the precast segments are intended to be cast-in-place
after the cantilever structure has been closed; therefore, the 3.6m deep girder may be transported
along any major highway in Ontario with a minimum vertical clearance of 490mm for protruding
reinforcing steel for the cast-in-place barrier wall.
A conceptual plan and elevation of the proposed cantilever girder concept for the longest span
considered of 90m is shown in Figure 2-8. The girder is erected segmentally from the abutments,
and each cantilevered segment during construction is post-tensioned to the structure with
permanent external unbonded post-tensioning tendons. The typical precast segment length is 3m
to accommodate standard traffic lane widths for delivery of segments. The minimum vertical
clearance provided over traffic lanes is 5000mm, which is consistent with standard policy for new
structures over roadways for this bridge type (Ontario Ministry of Transportation, 2002) and a
700mm construction tolerance has been provided for safety during construction. The 90m
mainspan is continuous across the abutment face and the short 15m sidespan is ballasted by backfill
material to maintain rotational restraint at the abutment face and to prevent uplift of the backspan.
Figure 2-8. Plan and elevation view of a possible cantilever-constructed overpass bridge
16
Assuming the overpass bridge structure services an arterial undivided urban roadway, the
minimum side clearance from the edge of the travelled way to the face of the barrier is 2.0m
(Ontario Ministry of Transportation, 2002). The arrangement of traffic lanes intended for the box
girder bridge consists of two centrally located lanes with a width of 3.6m each, two exterior
shoulders with a width of 2.0m each, and an allowance for traffic barriers with a width of 400mm
each. Therefore, the total cross sectional width of the top flange considered for this design is 12.0m
(shown in Figure 2-9). As there is no thickness requirement to place internally bonded cantilever
post-tensioning tendons, the top flange has been designed to minimize the usage of high-
performance FRC. The continuous top flange is 85mm thick, which is stiffened with transverse ribs
equally spaced at 1500mm centre-to-centre. The transverse ribs are cast monolithically with the top
slab and have a total depth of 200mm from the top of the flange surface. For each typical precast
segment, there is a 500mm wide transverse rib located in the middle of the segment, and 250mm
wide transverse ribs located at the both edges of the segment. The 250mm wide transverse ribs
provide a larger face for bulkhead detailing of match-cast joints, and it provides a seamless
transition of joined segments since the combination of two adjacent edge ribs are of the same size
as the centre rib.
Figure 2-9. Cross section and details of the constant depth box girder shown in Figure 2-8
The webs of the box girder are 120mm in thickness and are tapered inward towards the bottom
flange. For a 3600mm deep girder, the height-to-thickness slenderness ratio is 30:1. Dilger et al.
(2003) have shown that slender webs for box girder bridges are stable in the uncracked state. Box
girder bridges with webs of comparative slenderness are generally cantilever-constructed variable
depth girders where the web height is greatest in the high shear region near the support. As an
example, the Confederation Bridge in Eastern Canada has a girder depth of 14m at the support and
17
a web thickness of 400mm, resulting in a slenderness ratio of 35:1 (Dilger et al., 2003). The webs
of the proposed girder are connected to the bottom flange which is 4000mm wide. The bottom
flange has a uniform thickness of 150mm for typical segments, and increases in thickness to
305mm at the support.
Figure 2-10. Typical standard segment components of proposed box girder concept
A three-dimensional representation of a typical segment is shown in Figure 2-10. The main
components of the box girder segment are identified. On the exterior of the box, steel struts are
installed to provide stiffness to the thin top flange. Each segment has a pair of steel struts, one on
each side of the box girder located in the mid-plane of the segment and coincident in plan with the
middle transverse rib of the top flange. When adjacent segments are assembled, the steel struts are
equally spaced at 3000mm centre-to-centre. Steel struts have proved to be an economical and
aesthetically pleasing solution for box girder bridges requiring a light weight design. The
Shibakawa Viaduct, which is part of the Second Tomei Expressway in Japan, incorporated the use
of inclined steel struts to support the cantilever slab. Through the use of these inclined struts, the
dead load of the superstructure was reduced to approximately 80% of the original box girder design
concept that did not use inclined struts (Mutsuyoshi, 2004). In addition, further savings were
realized in the substructure due to the innovation of this strutted box girder section. Figure 2-11
displays the successful design of steel struts used on the cantilever-constructed Shibakawa
steel strut
top corner blistersfor cantilever tendonanchorage
bottom corner blisterfor continuity tendonanchorage
continuous transverse rib through box(500mm width)
continuous transverse ribs through box(250mm width)
18
Viaduct, and the adaptation of steel struts to the proposed box girder design for overpass
construction.
Figure 2-11. Top flange supported by steel struts, a) example shown of Shibakawa Viaduct, Japan (photo courtesy of Takashi Kosaka, 2006), b) rendering of proposed 90m overpass structure
On the interior of the box girder, the external unbonded post-tensioning is anchored at formed
anchorage blisters at the inside corners. During construction, each segment is attached to the
cantilever girder by stressing a pair of external unbonded post-tensioning tendons at the top corners
of the box girder. In a similar fashion, continuity of the box girder is achieved by stressing a pair
of external unbonded post-tensioning tendons at the bottom corners of the box girder. External
unbonded post-tensioning for precast segmental cantilever construction has been successfully
designed for the Second Severn Bridge across the River Severn between England and Wales
(Mizon, 1997). Figure 2-12 displays the adaptation of total external prestressing inside the viaduct
box girder of the Second Severn Bridge, and the bulkhead detailing showing no continuous bonded
reinforcement at the joint. The top cantilever tendons are placed at a minimal distance below the
underside of the top flange in order to maximize the flexural lever arm for negative bending.
Likewise, the bottom continuity tendons are placed at a minimal distance above the bottom flange
to maximize the flexural lever arm for positive bending.
a) Shibakawa Viaduct, Japan b) Proposed overpass structure
steel strut steel
strut
19
Figure 2-12. Second Severn Bridge, a) total external unbonded prestressing tendons, b) cantilever construction showing bulkhead face of segment with no continuous bonded steel across the joint and no internal bonded tendons (adapted from Mizon, 1997)
The tendon arrangement proposed for the constant depth overpass structure involves the use of
curtailed post-tensioning tendons. The curtailment of post-tensioning tendons refers to the
termination of prestressing steel at discrete locations for every precast segment. Thus, unlike box
girders built by the span-by-span method, the area of prestressing steel is not constant at all cross
sections along the length of the girder. The result of this curtailment of tendons is an efficient
arrangement of prestressing steel since the flexural capacity of the box girder closely matches the
flexural demand. This point will be illustrated further in Chapter 3. One possible arrangement of
prestressing tendons is shown in Figure 2-13. The cantilever post-tensioning is anchored at the
inside corner blisters of the box and deviated towards the middle of the box girder at sections closer
to the abutment. At the abutment wall, the cantilever tendons are deviated in the vertical plane and
splay into three distinct trajectories. At an internal deviator wall 5m from the abutment wall, the
tendons are deviated and splayed horizontally to provide adequate space for all tendon anchorages
at the back face of the abutment. The continuity tendons follow a similar arrangement as the
cantilever tendons. Each pair of continuity tendons are anchored at a bottom anchorage corner
blister inside the box girder and deviated towards the middle of the box girder for sections closer
to the midspan. A deviator beam is located near the midspan to provide deviation of tendon units
a) b)
20
in the horizontal plane for the continuity tendons. Similarly, a deviator beam is also located near
the abutment to provide deviation of tendon units in the horizontal plane for the cantilever tendons.
Figure 2-13. Possible tendon arrangement shown for half the span (symmetrical about span centreline)
As a way to provide reference to specific segments in the structure, the segments have been
numbered in the order in which the segments are erected. Thus, Segment 1 is the segment that is
21
closest to the abutment wall; the segment which is match-cast against Segment 1 is numbered
Segment 2, and so forth. This numbering scheme is represented in Figure 2-14.
Figure 2-14. Numbering scheme for segment labels
2.4 Visual Comparison of Proposed Box Girder with Typical Slab-on-Girder Overpass
As a basis for visual comparison, we shall contrast the unique aesthetic characteristics between
the proposed box girder design and more traditional slab-on-girder highway overpass bridge using
CPCI precast I-girders. For the purpose of this section, only a visual comparison of aesthetic
qualities will be made; implications on economy will be addressed in Chapter 9 of this thesis. As
shown in Figure 2-7, the recommended transverse girder spacing for a span length of 45m using
standard 2300mm deep CPCI I-girders ranges from 1.61m to 1.85m (CPCI, 1996). Assuming that
the reference bridge has been designed using the simplified method of analysis defined by the
CAN/CSA-S6-06, the deck overhang must not exceed 1.8m or 60% of the typical girder spacing
(cl. 5.6.1, cl. 5.7.1). Therefore 7 girders are required at a typical spacing of 1.67m. For a deck slab
thickness of 225mm and a slab haunch of 75mm, the total structural depth of the slab on girder
superstructure is 2.6m. This corresponds to a span-to-depth ratio of 17.3:1 for two equal 45m spans
comprising the 90m total bridge length.
The single span cantilever-constructed segmental box girder is designed at a more slender
span-to-depth ratio of 25:1. For the 90m span, the structural depth of the box girder is 3.6m. The
box girder maintains a constant span-to-depth ratio for the entire length of the bridge. A minimum
vertical clearance of 5m is provided for each of the overpass bridge concepts.
A 3-dimensional solid model for each conceptual design under comparison has been developed
to illustrate the visual experience of the observer for selected vantage points. These vantage points
portray realistic perspective views of each structure to provide a credible basis for visual
comparison of each type of highway overpass. The views shown in Figure 2-15 are indicative of
the sights experienced on an overcast day where direct sun and shadow casting is not present.
Figure 2-16 provides the cross sections for the superstructures under comparison.
22
Figure 2-15. Rendered perspective views on typical highway overpass construction using precast CPCI girders (left) and proposed cantilever-constructed single cell box girder (right) as seen on an overcast day.
Figure 2-16. Deck cross sections for a) two-span precast CPCI girder overpass and b) proposed cantilever-constructed single cell box girder.
a) elevation view
b) side profile
c) view from abutment
a) Two-span slab on CPCI girders b) Single span segmental box girder
23
The first comparison to be made is shown in Figure 2-15a), where the entire bridge is viewed
from abutment to abutment at an angle which is orthogonal to the axis of the bridge direction. As
the eye of the observer tends to trace the entire continuous length of the structure, the most
dominant shape that is observed is the clearance window below the superstructure from abutment
to abutment. This clearance window is defined by the 90m horizontal opening and the 5m vertical
clearance. For the CPCI girder design, the clearance window is broken visually due to the presence
of the centre pier; however, notwithstanding this discontinuity the overall appearance of the
superstructure remains relatively slender as the presence of the pier appears to have a minor impact
on the overall transparency of the bridge at this vantage point. If the observer pays minimal
attention to the presence of the pier, the perceived span-to-depth ratio of the superstructure
effectively has the appearance of being thinner than the actual slenderness ratio of 17.3:1. The box
girder design provides a clearance window that is completely unbroken from the outermost
supports with no intermediate supports between abutments. From this vantage point, the structure
provides an open unobstructed view throughout the full length of the bridge; however, the cross
section in contrast to traditional designs is noticeably deeper due to the low proximity to the
ground. This absolute depth is most pronounced on an overcast day under ambient lighting;
however, on a bright day when directional light creates shadows on the structure, the perceived
depth is more slender.
Visual slenderness can be achieved by taking advantage of shadow casting that results from
overhead sunlight. The shadow created on the webs of I-girders and box girders reduces the visual
prominence of the structural depth, and therefore portrays a lighter appearance to the viewer.
Larger deck overhangs produces a deeper shadow, thus a larger benefit due to shadow casting is
gained through the incorporation of a large deck overhang. As the overhang is restricted to 60% of
the typical girder spacing for slab-on-girder bridges designed using the simplified method, the
maximum overhang allowed following this restriction is 1000mm from the centre of the web for
the reference bridge. The deck overhang to girder depth ratio for this structure is 0.44, providing
only a small benefit in shadow casting to enhance the visual slenderness of the structure. For the
box girder shown in this example, the deck overhang is approximately 3500mm from the centre of
the web at the elevation of the top flange which provides a deck overhang to girder depth ratio of
0.97. Shadow casting is enhanced even more for a box girder with inclined webs in comparison to
a girder with vertical webs as it provides a deeper shadow projection from the top flange. Figure 2-
17 displays the reduction in the dominance of the girder that can be attained through the
24
incorporation of a larger deck overhang inherent in box girder superstructures. This reduction in
dominance is clear when comparing to Figure 2-15b which shows the same view under ambient
lighting.
Figure 2-17. Visual slenderness attained through shadow casting for a superstructure with a a) small deck overhang and b) large deck overhang.
Lastly, Figure 2-15c provides a vantage point that is seen when the observer is up close to the
structure, viewing from a pedestrian level at the foot of one abutment and viewing along the axis
of the bridge. From this vantage point, the traditional 2-span slab-on-girder structure exhibits an
array of parallel girder lines along the axis of the structure. These girders are supported by the
centre pier which has an increased visual prominence when seen from a more acute angle. The
centre pier detracts from the overall transparency of the structure, and creates a visual barrier when
the observer is in the vicinity of the bridge. The CPCI girders are pin-supported at the abutments
where bearings and expansion joints are normally present; the abutment walls are prone to staining
and delamination spalling due to drainage runoff and corrosion from de-icing salts. In contrast,
from this vantage point the box girder design provides a unique visual experience as the steel struts
exhibit a cascading effect along the continuous flange lines. The depth of the box girder is not
prominent when viewing the bridge from this angle. The observer also has an unobstructed view
for the full width of the right-of-way with no intermediate visual barriers. The box girder is
designed to be integrally connected to the abutments; therefore, no expansion joints are present at
the abutments which provides protection to the abutment from staining and deterioration.
2.5 Benefits and Drawbacks of the Proposed Design
2.5.1 Benefits
1) The absence of a pier in the median of the mainline highway provides motorists with a
greater level of safety. Piers in the vicinity of the right-of-way present a risk for vehicular collision.
a) small deck overhang b) large deck overhang
25
Catastrophic vehicular impacts with bridge piers have occurred in the past, causing loss of the
structure and loss of life (El-Tawil et al., 2005). As an example, Figure 2-18 displays the
catastrophic failure that was caused to a slab-on-girder structure in Nebraska, USA due to the
impact of a heavy truck. Following the expensive traffic delays and demolition and cleanup of the
failed structure, a new structure was designed with fewer bridge piers to improve safety (ENR,
2003).
Figure 2-18. Collapse of slab-on-girder bridge after being struck by tractor-trailer (adapted from El-Tawil, 2005)
2) Box girder sections for overpass structures in comparison to slab-on-girder bridges also
provide enhanced durability due to a smaller surface of concrete exposed to salt spray. For the cross
sections that were considered in Section 2.4 and shown in Figure 2-16, the slab-on-girder bridge
has a surface area of 76.8m2 per meter of superstructure directly exposed to the atmosphere, while
the box girder bridge only has a surface area of 34.4m2 per meter of superstructure directly exposed
to the atmosphere.
3) As box girder sections are integral units comprising the deck surface itself, the entire
structural section is fabricated with high-quality precast concrete that is cast and cured in a
controlled environment. Typical slab on girder bridges are constructed using precast girders with
a cast-in-place concrete deck. The concrete deck, which is arguably the most damage prone
component of the bridge superstructure, provides enhanced durability due to the improved quality
of the exposed concrete. Improved quality translates to lower life-cycle costs and longer life (Ralls,
2003).
4) Not only does the absence of the centre pier provide improved safety for motorists, but it
also allows for the maximum flexibility in lane arrangement possibilities for the mainline highway.
This may be especially significant in dense urban areas where express and collector lanes are
required.
26
5) Accelerated bridge construction inherently provides greater service to the public by reducing
the amount of time required to erect a structure. In addition, greater savings in labour can be
realized due to the accelerated scheduling. Ralls (2003) describes that traffic control can run
anywhere from 20 to 40 percent of construction costs and user delays are priced at thousands of
dollars per day in heavy traffic areas.
2.5.2 Drawbacks
1) In comparison to the two-span slab-on-girder alternative, the single-span box girder design
requires a deeper structural section. The larger bridge depth has implications on the approach
design as the road elevation is necessarily higher at the abutment to accommodate the depth of the
section. The top of roadway elevation of the approach must match the top of roadway of the
structure. Therefore, for an increase in structure height, two situations may exist for the approach
road: either the approach grade must be steeper for a fixed length of approach, or the approach must
be elongated to meet the desired elevation for a fixed approach grade (shown in Figure 2-19). In
both cases, an increase in sub-grade material is necessary for the approach road.
Figure 2-19. Elevation (1:2500 scale) showing the result of elevating the overpass road grade
2) The general understanding of the behaviour of external unbonded post-tensioning is not well
understood with respect to the application of cantilever construction with curtailed tendons. To the
authors knowledge, no published material exists that describes the behaviour of curtailed external
unbonded tendons both at serviceability limit states (SLS) and at ultimate limit states (ULS). The
increase in tendon stress at ULS through the application of FRC and the potential benefits of the
ductility of FRC is presently unknown. It is the intent of the following chapter to explore and
determine the potential benefits of using FRC in conjunction with external unbonded tendons for
cantilever girders and to determine the level of efficiency that can be obtained using the proposed
cantilever-constructed bridge as a vehicle to uncover these questions.
box girder elevation slab-on-girder elevation
27
28
3.0 LONGITUDINAL FLEXURE
In this chapter, the design for longitudinal flexure of a cantilever-constructed girder prestressed
exclusively with external unbonded tendons will be discussed. Sections 3.2 and 3.3 discuss the
calculations involved for determining the response of unbonded tendons. Section 3.4 outlines two
procedures for the calculation of tendon stress due to global deformations of the box girder.
Following this, Section 3.5 describes simplifying design assumptions made to develop a
preliminary design concept for the box girder. Sections 3.6 to 3.8 discuss the strategies for
designing cantilever and continuity external post-tensioning tendons. Section 3.9 describes the
design method for sizing the bottom slab thickness in negative moment regions. Sections 3.10 to
3.12 describe the girder behaviour at SLS and ULS. Section 3.13 investigates the possible
refinement in tendon stress at ULS to achieve greater economy in prestressing steel. Finally,
Section 3.14 provides a discussion on the effect of span-to-depth ratio on material consumption,
and the practical range of spans of the proposed box girder.
3.1 Introduction
The design of tendons for a cantilevered bridge achieves significant economy in steel
consumption since the area of prestressing steel at any section generally corresponds to the flexural
demands of the bending moment diagram. Classical cantilever-constructed bridges are built out
symmetrically from pier tables using cast-in-place construction having segments ranging from 3 to
5 meters in length (Menn, 1990). These classical cantilever bridges are normally haunched, where
the depth of the cross section varies as the bottom flange follows a parabolic profile to increase the
internal lever arm at the support. The cantilever tendons are typically internal and bonded in the
top slab, and the continuity tendons are typically internal and bonded in the bottom slab. In general,
prestressed concrete cantilever type bridges carry a significant amount of dead load compared to
live load, typically in the order of 4:1 (Menn, 1990). This ratio is indicative of the total load
proportions on a box girder, although this ratio may decrease some for shorter spans since the unit
weight of a shallower box girder is decreased, all other things being equal. A common measure
taken in cantilever-constructed concrete bridges is the reduction of weight for segments placed
closer to the midspan in order to reduce the negative bending moment at the support. The result of
this design effort is a haunched girder profile, where the depth of the section is reduced at the
midspan. The bottom slab of cantilever girders also varies in thickness in some cases, being thicker
at the support for strength requirements and thinner near the midspan to reduce dead load. In light
of this design strategy, the reduction of dead load for a cantilever-constructed bridge presents great
value when designing for the ultimate limit state.
3.2 Moment-Curvature Response
Using a sectional analysis, a typical concrete box girder can be idealized as a simple wide
flanged section when considering the global flexural response. The idealized cross sectional
geometry can be described as:
(3-1)
where bbf is the width of the bottom flange, w is the web thickness of one web, btf is the width
of the top flange, tbf is the thickness of the bottom flange, ttf is the thickness of the top flange, and
d is the depth of the section. This sectional idealization is illustrated in Figure 3-1 below.
Figure 3-1. Idealization of box cross section for analysis of longitudinal flexure
Assuming plane sections remain plane, the strain distribution over the depth of the section and
corresponding curvature are given as
(3-2)
(3-3)
where are the top and bottom fibre strains, respectively. Internal equilibrium of
forces for axial load and internal moment are given by
bc y( )
bbf
2wbtf⎩
⎪⎨⎪⎧
=
if 0 y tbf <≤
if tbf y d ttf–<≤
if d t– tf y d ≤ ≤
wtbf
ttf
btf
d
bbf
tbf
ttf
btf
d
bbf
2w
a) actual cross section b) idealized cross section
y
εcf y εb εt, ,( ) εbεt εb–
d---------------y+=
φ εb εt,( )εb εt–
d---------------=
εb and εt
29
(3-4)
(3-5)
where P is the axial prestress load of the unbonded tendons at the section under consideration,
fc is the concrete stress, and are the concrete and prestress material resistance factors,
respectively.
For structures prestressed with external unbonded post-tensioning where tendons are attached
only at discrete locations (such as anchorages and deviators), a change in tendon eccentricity
occurs due to deformation of the girder. This concept is illustrated in Figure 3-2, where an example
of a continuously guided tendon and an example where the tendon is attached only at the cantilever
tip and at the support are shown. For the tendon attached only at discrete locations, a decrease in
internal lever arm from the elevation of the tendon to the compression flange results in a decrease
in ultimate resistance. In this thesis, the external unbonded tendons are assumed to behave as
internal tendons (that is, like the continuously guided tendon). This estimation is reasonable
provided that tendon guides are located at sufficient intervals along the length of the girder.
Treating the tendons as internal tendons, total moment can be calculated based only on applied
loads and prestress force when examining the response of a girder in flexure.
Figure 3-2. Internal lever arm from tendon elevation to the compression flange, a) for a continuously guided tendon, b) for a tendon attached only at discrete locations
3.3 Behaviour of Unbonded Tendons
In addition to satisfying equilibrium with external loads, internal forces must correspond to
deformations that satisfy conditions of geometrical compatibility appropriate to the materials and
structural components used. In concrete girders reinforced entirely with bonded steel, internal
forces at a given cross section can be computed using the compatibility conditions at that section
only. For beams of dimensions commonly used in bridge design practice, compatibility is defined
by the assumption that plane sections remain plane after deformation and are orthogonal to the
N εb εt P, ,( ) φcfc εcf y εb εt, ,( )( )bc y( ) yd0
d
∫ φpP+=
M εb εt P, ,( ) φcfc εcf y εb εt, ,( )( )bc y( )y yd0
d
∫ φpPyp+=
φc and φp
a) continuously guided tendon b) tendon attached only at discrete locations
yp(x) = const
yp(x=a)
yp(x=a) < yp(x=0) x x
30
deformed member axis, and that perfect bond exists between the bonded steel and the concrete. In
girders prestressed with unbonded tendons, however, the plane sections compatibility is no longer
valid at a specific section. This is illustrated in Figure 3-3 where for the unbonded system is
shown to be less than for that of a system with bonded steel. The total strain in the unbonded tendon
is therefore the summation of the effective prestress strain, , and the overstress strain, , due
to deformation of the girder system. The effective prestress strain is shown as a broken line as it is
generally much larger than the initial concrete strain, . This is illustrated more clearly by
comparing material strains, shown in Figure 3-4.
Figure 3-3. Strain Distribution at ultimate limit state (adapted from Naaman, 2006)
Figure 3-4. Material strains due to equilibrium in the reference state and in the ultimate state
3.3.1 Reference State of Strain
When a beam with unbonded tendons is stressed, assuming deflections are free to occur during
the stressing operation (which is true for a cantilevered beam), there will be an initial elastic
deflection due to the axial prestress load, P0, the induced prestress moment, Mp, and the dead load
∆εp
εp0 ∆εp
εc0
εcu
Ultimate State(P0 + ΔP), Mu
P0, Md Reference State
εc0
εp0
CompressionTension
εcp
dp
(∆εp)bonded
(∆εp)unbonded
c
-80
-60
-40
-20
0-0.02-0.015-0.01-0.0050
0
400
800
1200
1600
0 0.005 0.01 0.015 0.02
fc [MPa]
εc [mm/mm]
fp [MPa]
εp [mm/mm]
εp0
εc0
εcu εp0+ Δεp
a) concrete strains
Δεp
b) prestressing steel strains
εc0 corresponds toinitial strain aftertransfer
εp0 corresponds toinitial effectiveprestress strain
31
moment due to the girder self-weight, Md. The integration of elastic straining in the concrete at the
level of the prestressing tendon when the prestress force is anchored gives the initial change in
length of the beam at the elevation of the tendon, and is given by
(3-6)
The average straining in the concrete at the level of the prestressing tendon is given as
(3-7)
The reference state of deformations is chosen to be the initial length of the tendon after transfer,
where the initial concrete deformation due to the jacking effort under dead load is . This
choice of reference state takes into account structural behaviour, and particularly, the construction
sequence of sequential cantilevering.
For a cantilever-constructed bridge, each tendon set has its own unique reference state of elastic
strain in the cantilever beam that corresponds to the jacking stress after transfer (which is assumed
to be 0.7fpu). The state of elastic strain that defines the reference point for one set of cantilever
tendons depends on the dead load applied to the girder during the stressing operation, and the force
in tendons overlapping the tendon being stressed. The addition of dead load and the addition of
subsequent pairs of unbonded tendons due to the successive placement of segments will affect the
state of deformations of the previously stressed tendons. Since dead load is active during the
stressing operation of tendons, the deformations due to dead load are included in the calculation of
for the tendon pair being stressed and not in the calculation of , which corresponds to a
change in length after the stressing operation. The calculation for is the same as for as
described in Equation 3-6, except all loads applied to the girder are considered. Any changes in
tendon stress after the initial state is determined by the tendon deformations relative to the initial
deformation, .
3.3.2 Long-Term Effective Prestress
The effective initial prestressing force is defined as , which includes the force jacked into
the tendon and the effect of dead load during prestressing. The long-term effective prestress that
includes the added effect of permanent dead load added after prestressing, plus any long term
losses due to creep and shrinkage is given as:
∆lp0
∆lp0 εcp P Md L x,( ) x,,( ) xd0
L
∫=
εc0 avg,∆lp0
L----------=
∆lP0
∆lP0 ∆lcP
∆lcP ∆lP0
∆lcP ∆lP0–
P0
32
(3-8)
where is the increase in tendon stress due to additional permanent dead load, and is
the change in prestress force due to long-term losses. is required in design for the calculation
of ultimate strength of girders with unbonded prestressing steel. For concrete girders of typical
proportions and reinforcing ratios prestressed with bonded steel, large increases in stress causing
yield of steel in regions of high moment can be attained at the ultimate limit state. However, for
girders prestressed with unbonded tendons, the increase in tendon stress is generally far lower since
the stress increase is averaged out over the tendon length between anchorages (Collins and
Mitchell, 1997). The ultimate force in unbonded tendons can be calculated by the summation of
the long-term effective prestress force plus additional tendon force gained due to girder
deformations from short-term applied loads, .
The ultimate strength of girders in general is proportional to two main parameters: the force in
tension steel, and the lever arm from tension steel to the compression flange. For a box girder with
a predetermined depth having tendons inside the cross section, the maximum flexural lever arm
effectively is also predetermined. Therefore, the ultimate strength of girders prestressed with
unbonded tendons is dependent on the maximum force that can be attained in the prestressing steel.
Since tends to be smaller for unbonded tendons, the effective prestressing force is important
for the ultimate strength of the girder. To achieve efficient and safe use of unbonded tendons in
design, a reasonable estimate of ultimate stress must be determined.
3.4 Tendon Stress Calculation Methods
To calculate the response of concrete girders prestressed with unbonded tendons, a general
procedure is given in Table 3-1 below. The state of stress in an unbonded tendon corresponding to
a known arrangement of applied loads can be determined by first assuming the effective prestress
in the tendon and iterating the tendon stress until an acceptable convergence is achieved. This
procedure returns an ‘exact’ tendon stress for the given arrangement of load and state of
deformations of the girder. When the girder is in a linear elastic state of stress, convergence is
generally achieved within 4 iterations.
P∞ P0 ∆Pdl ∆P∞+ +=
∆Pdl ∆P∞
P∞
P∞ ∆P+
∆P
33
This method requires the iteration of tendon stress, which consequently requires the
recalculation of moment-curvature response corresponding to the state of stress in the tendon. If
we are interested to determine the associated response due to a known arrangement of applied loads
on a girder, an interpolation scheme between known moment-curvature responses (corresponding
to constant prestress forces) is a convenient method to save time and computational effort for the
iterative calculation of tendon stress. If the prestress force is assumed to be concentric to the girder
(where Mp = 0) then the linear elastic region of all curves will be coincident along
and the post cracking responses will deviate from the linear elastic state toward the ultimate
resistance and ultimate curvature (shown in Figure 3-5). Interpolating between these lines is a fast
calculation and it accelerates the speed for each iteration step since it eliminates the need to
reproduce moment curvature responses for every unique tendon stress. Several chosen responses
may be calculated and stored in matrix form to serve as a database for the automated interpolation
of responses, illustrated by the following matrices below:
Table 3-1. Procedure for calculating the stress in an unbonded tendon due to girder deflections
Step Calculation Comment
1 Guess tendon stress to be the effective prestress
2 Calculate total moment on girder including applied loads, plus primary and secondary prestress loads
3Calculate curvature and tendon level strain distribution over the length of the girder using member response corresponding to lines of equal force
4 Calculate absolute tendon elongation by integrating concrete strain due to girder deformations
5 Calculate difference in tendon elongation from the reference state
6 Calculate difference in tendon strain from the reference state
7 Calculate new tendon strain
8for linear elastic
Calculate new tendon stress with respect to known constitutive material laws
9 Check Check to see if the difference between the new calculated stress and the assumed stress is sufficiently small. If so, finish. If not, return to step 1 with
σp σp∞=
Mq x( )
Mp σp x,( ) P σp( ) e x( )( )=
Mtot σp x,( ) Mq x( ) Mp σp x,( )+=
+ Mps σp x,( )
φ σp x,( )
εcp σp x,( )
∆lcP εcp σp x,( ) xd0
L
∫=
∆ldiff ∆lcP ∆lP0–=
εpdiff∆ldiff
L-------------=
εpnew εp0 εpdiff+=
σpnew εpnew( )
σp Epεp=
σpnew σp–σp σpnew=
M φ EcIc( )⁄=
34
The vector n contains the prestress force information, where n is the number of prestress
loads chosen. The vector m contains the x-ordinate information for the moment-curvature
response and is consistent for all calculated responses, where m is the number of data points in each
curve. The matrix m,n contains the y-ordinate information for all calculated responses, where
the column i corresponds to prestress load Pi. As can be seen in Figure 3-5, the variation
between responses is approximately linear, thus, an linear interpolation should yield sufficient
accuracy in design. A similar interpolation scheme is also necessary for the variation of
with moment.
Figure 3-5. Family of moment-curvature responses for a specific cross section and varying prestress force assuming a concentric prestress
In some cases when the girder response is approaching the ultimate limit state, problems with
convergence of tendon stress may arise due to the high sensitivity of the response in the vicinity of
the ultimate resistance. As illustrated previously in Figure 3-5, the moment-curvature response is
nearly bi-linear, where the slope of the response in the linear elastic range is EI, and the slope of
the response when the girder is undergoing plastic deformation is nearly flat. This drastic change
in response over a very small range of bending moment causes the tendon stress calculation to have
a diverging iterative solution. If a girder undergoes significant plastic behaviour for a given
arrangement of load, the first iteration through the calculation will indicate that a large stress
[ ]
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
nm,m,1
2,22,1
n1,1,21,1
m
2
1
n 21
MM
MMMMM
P P P
LL
MOM
M
L
M
L
φ
φφ
P[ ]
φ[ ]
M[ ]
M{ }
εt and εb
-60000
-50000
-40000
-30000
-20000
-10000
0
10000
20000
30000
40000
2MN6MN10MN14MN18MN22MN
d = 3600mm
EI
Linear Elastic Response
M [kNm]
Ф [1/m]-0.002 -0.001 0 0.001 0.002
35
increase in the tendon has occurred which is indicated by the large integrated strain in the hinge
region. For the second iteration, a higher stress will be guessed to perform the calculation;
however, this new guess causes the response to be primarily linear elastic indicating that the tendon
stress is much smaller than the guessed stress. Regardless of the accuracy of the guess value, the
method still renders divergent behaviour. A typical example of this behaviour is represented in
Figure 3-6 showing both the convergent and divergent solutions.
Figure 3-6. Iterative method behaviour for tendon stress calculation for a) girder impending plastic hinging, b) girder experiencing plastic hinging
A more stable iterative method has been observed to fix a desired target tendon stress, and
iterate the applied load until convergence is achieved. This procedure is given in Table 3-2, in
which the applied load must be approximated by a uniformly distributed load. The advantage of
this method is that only one moment-curvature response needs to be computed (for the assumed
tendon stress), and iterating of applied load is a straightforward process. This is typically not done
in design since the applied loads are known; however, this method provides a stable method to
determine the tendon stress increase for a range of applied loads up until crushing of the
compression flange. This method requires one to choose an increment value of load to iterate
towards a solution; therefore, the speed and accuracy of this solution is dependent on the
coarseness of the load increment and the closeness of the guess value to the actual value. A typical
example of this method is represented in Figure 3-7 showing termination after reaching the target
value.
100010501100115012001250130013501400
0 5 10 15 20 25 30100010501100115012001250130013501400
0 5 10 15 20 25 30
guess fp = 1190MPa
guess fp = 1220MPa
guess fp = 1240MPa
guess fp = 1020MPa
guess fp = 1350MPa
guess fp = 1240MPa
fp [MPa]
iteration
fp [MPa]
iteration
no solution foundconvergence obtained
36
Figure 3-7. Iterative method solution for obtaining tendon stress by incrementing applied load
3.5 Preliminary Design Assumptions
The previous section has described methods which can be used to determine the stress in an
unbonded tendon due to girder deformation. These iterative methods allow for the accurate
Table 3-2. Procedure for calculating uniform applied load corresponding to a desired tendon stress
Step Calculation Comment
1Guess
Set target Set a target tendon stress. Guess an load to correspond to this tar-get stress
2 Calculate target tendon strain corresponding to the set target stress using known constitutive material law
3 Calculate a target tendon elongation measured from the reference state
4 Calculate total moment on girder including assumed applied load, plus primary and secondary prestress loads.
5Calculate curvature and tendon level strain distribution over the length of the girder using member response corresponding to lines of equal force
6 Calculate absolute tendon elongation by integrating concrete strain due to girder deformations
7 Calculate difference in tendon elongation from the reference state
8 Check
Check if the change in tendon elongation due to the assumed load is greater than or less than the change in tendon elongation due to the target stress. Increment the applied load
to get closer to the target stress and return to step 4. When target stress is reached, finish.
0
1.0
2.0
3.0
4.0
5.0
0 5 10 15 20 25
guess wf = 143.1kN/m
guess wf = 139.6kN/m
Δltarget = 1.92mm
Δldiff [mm]
iteration
criteria met whereΔldiff < Δltarget
criteria met whereΔldiff > Δltarget
q q0=
σp σpt etarg=
εpt etarg σpt etarg( )
∆lt etarg εpt etarg εp0–( )L=
Mq q x,( )
Mp σp x,( ) P σp( ) e x( )( )=
Mtot σp x,( ) Mq x( ) Mp σp x,( )+=
+ Mps σp x,( )
φ σp q x, ,( )
εcp σp q x, ,( )
∆l εcp σp x,( ) xd0
L
∫=
∆ldiff ∆l ∆lP0–=
∆ldiff ∆lt etarg or>
∆ldiff ∆lt etarg< q q qincrement+=
37
calculation of tendon stresses by accounting for the distribution of strain along the girder at the
elevation of the prestressing tendon. The determination of this strain distribution is a
computationally intensive task as it requires the accurate calculation of stresses for all unbonded
tendon sets for a given arrangement of applied load. The stress in one tendon set is not independent
from the stress in another tendon set, rather all overlapping tendon sets that share a union of girder
length are coupled in the calculation of tendon stresses. The effect of increasing stress for a long
tendon that envelops the entire structure will place increased axial prestress, P, and its associated
prestress moment, , on the system. This change in prestress force will directly affect the state
of strain for any shorter tendons that are embodied within the length of the longest tendon.
Conversely, any change in tendon force of short tendons will have an effect on the average
straining in the longest tendon. It is not at all clear what increases in unbonded tendon stress can
be relied upon at the ultimate limit state due to member deformation for a girder with tendons of
varying lengths.
In order to construct a working bridge concept, the design process is simplified initially by
making conservative assumptions for the state of stress in the external unbonded tendons. At
serviceability limit states (before cracking), practice has been to assume the tendon stress is
constant using the long-term effective prestress, . This assumption is always
conservative (Menn, 1990) since the increase in tendon stress that is gained due to any deformation
of the girder is ignored. At the serviceability limit state, this assumption is reasonably accurate
since the average elastic straining of concrete over the length of the tendon due to deformation of
the girder is in general much smaller than the effective prestress strain in the prestressing steel.
For prestressed structures that have permanent unbonded tendons, it is economical to calculate
the ultimate resistance with the actual value of as it will minimize the consumption of
prestressing steel (Menn, 1990). The stress increase in unbonded prestressing due to girder
deformation typically is significant since flexural cracking and opening of joints is expected at the
ultimate limit state. In addition, the nominal compressive stress of 80MPa is relatively high
compared to conventional construction; therefore, the girder strain at the level of the prestressing
tendon at ULS is expected to be larger than a box girder using conventional strength concrete for
a cross section at the critical bending location. A refined treatment of is preferable at ULS for
an accurate prediction of ultimate strength; however, this refined treatment will be dealt with after
a working concept has been established using conservative assumptions. For the proposed box
girder which incorporates a complex arrangement of overlapping unbonded tendons of varying
P e×
σp σp∞=
∆σp
∆σp
38
lengths, the tendon stress at ultimate limit state will also initially be assumed to be equal to the
long-term effective prestress .
Post-tensioned tendons are limited by the CAN/CSA-S6-06 to a tendon stress of at
jacking to provide some reserve capacity in case of unanticipated strand breakages (cl. 8.7.1). The
stress limit specified at transfer, which accounts for anchorage set loss, friction, initial relaxation
and elastic shortening, reduces the tendon stress to (cl. 8.7.1). Because the straight external
tendons have practically no angle break in the vertical plane, frictional losses will be small.
Regardless, this stress limitation at transfer of 0.7fpu still applies since the intent of this
specification is to reduce the probability of breakage of strand wires at the grips (cl. C8.7.1).
After jacking, long-term losses are realized which lower the effective prestress even further. In
lieu of considering long term losses explicitly, for the purpose of preliminary design and validation,
long term losses in tendon stress are incorporated into a reduced effective tendon stress. For design
of the proposed girder, is conservatively assumed to be 0.6fpu, which includes immediate
losses due to friction and anchorage set, and long term losses due to creep, shrinkage and
relaxation.
3.6 Preliminary Design Methodology for Cantilever Tendons
The following section will discuss recommended design considerations for the initial design of
a precast segmental cantilever bridge prestressed with unbonded tendons and no longitudinal
continuous bonded steel. First, the loads considered for the continuous girder will be described and
computed. Following this, the design for ultimate bending resistance will be discussed.
3.6.1 Loading
For a single span continuous system, it is relatively straightforward to calculate the longitudinal
factored bending moments, ignoring redistribution due to long-term effects. The total factored
load, , is considered as the following summation:
(3-9)
where is the cantilever dead load which is applied to the statically determinate
cantilever system, is the secondary prestress moment due to stressing of the continuity
tendons, and are the superimposed dead load and live load, respectively,
which are applied to the continuous statically indeterminate structure.
σp σp∞=
0.8fpu
0.7fpu
σp∞
Mf x( )
Mf x( ) αDMDdetx( ) αPMps αDMSDLindet
x( ) αLMLLindetx( )+( )+ +=
MDdetx( )
Mps x( )
MSDLindetx( ) MLLindet
x( )
39
For the typical cross section shown in Figure 3-8, the unfactored dead load smeared over the
length of the segment is 88.2kN/m and the unfactored superimposed dead load of 28.7kN/m which
includes traffic barriers and an 80mm asphalt layer. These dead load derivations are summarized
below in Table 3-3 and Table 3-4.
Figure 3-8. Typical segment cross section
The secondary prestress moment, Mps, is a redundant moment of the statically indeterminate
system applied to the ends of the girder at the fixed end. Mps is constant along the entire length of
Table 3-3. Estimation of dead load for typical segment
Item Quantity[m3]
Density[kN/m3]
Weight[kN]
Continuous concrete 7.29 24.5 179
Transverse ribs 1.35 24.5 33.1
Anchorages 1.64 24.5 40.2
Subtotal [kN] 252
5% Allowance for Tendons and Struts 12.6
Total segment weight [kN] 265
Smeared weight over 3m segment [kN/m] 88.2
Table 3-4. Estimation of superimposed dead load for typical segment
Item Quantity[m3]
Density[kN/m3]
Weight[kN]
Traffic barriers 2.12 24.5 51.9
80mm asphalt layer 2.69 12.7 34.2
Total weight per segment [kN] 86.1
Smeared weight over 3m segment [kN/m] 28.7
40
the girder. It is necessary to begin the cantilever design with an initial estimate of Mps since it can
not be determined until the continuity prestressing is designed. Mps is assumed to be equal to 56%
of the total factored positive moment at midspan due to and applied to the
continuous system plus factored positive moment due to long-term redistribution of dead load. The
validity of this assumption for the proposed box girder will be demonstrated later on in the design
process when the continuity prestressing steel is designed.
Following the live load model defined in the CAN/CSA-S6-06, for a 12m wide bridge deck, 3
design lanes must be considered for the maximum effect (cl. 3.8.2). For the span of 90m, the lane
load model (cl. 3.8.3.2) tends to govern the design at the maximum and minimum moment
locations since the uniform applied load becomes more critical for midspan moment and end
moment than the CL-625-ONT truck model (cl. 3.8.3.3). However, the CL-625-ONT truck model
produces significant moment demands in other regions of the girder since it is a moving load
amplified by the Dynamic Load Allowance (DLA) (cl. 3.8.4.5.3). It is observed that the bending
moment diagram due to loads from the truck model produces critical locations for the positive
moment envelope from the abutment to approximately the quarter points of the span, and the truck
model is also critical for the negative moment envelope near the third points of the span. The
factored live load bending moment envelopes for the CL-625-ONT truck model and the lane load
model are shown on the tension side in Figure 3-9.
Figure 3-9. Comparison of live load models defined in CAN/CSA-S6-06
MSDLindetMLLindet
-60000
-50000
-40000
-30000
-20000
-10000
0
10000
20000
30000
40000
0 10 20 30 40 50 60 70 80 90
truck load governsnegative envelope
truck load governspositive envelope
truck load governspositive envelope
lane load governspositive envelope
M [kNm]
distance [m]
lane load governsnegative envelope
truck load governsnegative envelope
lane load governsnegative envelope
Max Lane Load (cl. 3.8.3.3)Min Lane Load
Max Truck Load (cl. 3.8.3.2)Min Truck Load
CL-625-ONT TruckNo. Lanes = 3MLF = 0.8
41
Considering dead load, live load, and secondary prestress moment for the ultimate limit state,
the ultimate demand is constructed using the load factors specified in ULS Combination 1 (cl.
3.5.1). The critical factored live load for the single span girder is taken as the maximum and
minimum demand envelope that encompasses the demands for both the truck load case and the lane
load case. The dead load and superimposed dead load demands are based on the estimates
previously outlined in Table 3-3 and Table 3-4. Secondary prestress moment is temporarily
assumed, and it will be calculated directly during the preliminary design of continuity tendons. If
the actual value Mps is found to be significantly different than the assumed value, the tendon design
must be refined with a subsequent iteration. The total loads following ULS Combination 1 to
produce the largest negative moment are illustrated in Figure 3-10 below.
Figure 3-10. ULS bending moments on continuous girder
3.6.2 Ultimate Moment Resistance
In preliminary design, an initial estimate for the sizing of flexural steel can be based on the
flexural resistance of the girder at the ultimate limit state, given by the equation below,
(3-10)
where is the long-term effective prestress of the unbonded tendons at the section, and z is
the internal flexural lever arm which can be assumed to be the height taken at the centroid of the
Mps
DL+Mps
DL+SDL+Mps
DL+SDL+LL+Mps
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40 50 60 70 80 90
LL envelope
Assumed Mps
M [kNm]
distance [m]
Mr φsAsσsy φpApσp∞+( )z=
σp∞
42
thin compression slab to the level of the flexural steel. Since there is no continuous bonded
reinforcing steel crossing the joints for the proposed girder, the term goes to zero, and the
sectional moment resistance is based purely on the unbonded tendons.
Unbonded cantilever tendons are anchored to top corner blisters of the box girder segments.
Using standard 3m length segments, the 90m span continuous girder will be erected by two
adjacent 45m cantilevers consisting of 15 segments each. Therefore, with the cantilever length
discretized over 15 segments, the curtailment of cantilever tendons may be chosen in several ways.
Cantilever tendons are typically 12, 15, or 19 strand tendons composed of 0.6” diameter strands for
traditional cantilever-constructed bridges of typical weights (LoBuono, 1997). These tendon sizes
require the use of two anchorage unit sizes. In general, it is advantageous to minimize the variety
of anchorage units when designing the tendon arrangement so that the number of required stressing
jacks is also minimized.
For traditional cantilever construction, it is sometimes possible to design the cantilever
prestressing with only one size anchorage for all tendon units. This can be attributed to the classical
haunched profile which has an increase in internal lever arm as the section becomes deeper near
the support. For a parabolically haunched girder, the elevation of the girder axis also approximately
follows a parabolic profile, which allows the linearly increasing prestress force to provide adequate
resistance for the bending moment demands at serviceability. The parabolically haunched girder
axis is significant for the behaviour of the girder at serviceability, and the parabolically haunched
bottom flange is significant for the lever arm at the ultimate limit state. This concept is illustrated
in Figure 3-11.
For a constant depth girder, the girder axis is horizontal with a constant eccentricity to the
flexural steel. Thus, to satisfy for all x and if , an economical tendon
arrangement which minimizes steel consumption varies parabolically from the cantilever tip to the
φsAsσsy
Mr x( ) Mf x( )> Mf x( ) x2∝
43
support since the flexural lever arm is constant. This arrangment, however, requires a larger variety
of tendon anchorages.
Figure 3-11. Cantilever prestress concepts for haunched and constant depth girders
3.6.3 Sizing of Cantilever Tendons
The maximum bending moment at the support will govern the design of the cantilever tendons.
The tendon arrangement should be chosen to satisfy a reasonable trade-off between the variation
of anchorage units used and the amount of prestressing steel consumed in the design. As discussed
previously, a varied arrangement of tendon units increasing in size from the midspan to the support
will provide the greatest economy in strand consumption. However, for the preliminary design, the
emphasis has been placed on the minimization of anchorage unit variation at the expense of
consuming more prestressing strand. With the motivation for fast construction and repetition of
standard details, only one anchorage unit (VSL ES6-12) has been used for the cantilever post-
tensioning which requires only one prestressing jack for all tendons. The VSL ES series multi-
strand post-tensioning anchorage has been chosen as it is more compact than the E series
anchorage. This design provides a reasonable compromise between required equipment and tendon
utilization. In comparison to an alternative tendon arrangement designed for minimum steel
consumption (shown in Figure 3-12), this design requires approximately 24% more cantilever
P(x)
Mr(x) > Mf(x)
z(x) z = const
Mf(x) Mf(x)
Pa
PbPa >Pb
Pi = const
haunched girder constant depth girder
Mr(x) = P(x)z(x) Mr(x) = P(x)z
44
post-tensioning steel. The design for minimum steel, however, comes at the expense of the need
for several prestressing jacks and is a less than perfect rationalization of precasting.
Figure 3-12. Preliminary ULS demand and capacity for cantilever PT
The range in number of strands for a 12-strand anchorage is between 8 and 12 strands (Collins
et al., 1997). Using only one anchorage size for cantilever prestressing, the strand consumption can
still be optimized effectively by reducing the number of strands in the tendon and provide spares
in the anchorages. Therefore, to provide economy in prestressing strand consumption, the
minimum number of strands will be included for the longer tendon sets.
Mps
DL+Mps
DL+SDL+Mps
DL+SDL+LL+Mps
Capacity (minimum anchorage units)
Capacity (minimum steel consumption)critical location fornegative bending(governs design)
LL
DL+SDL+Mps
LL/(DL+SDL+Mps) = 0.55
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40 50 60 70 80 90
LL envelope
Assumed Mps
M [kNm]
distance [m]
45
Segments are numbered according to the order of placement. (i.e. segment 1 is placed closest
to the support, and segment 15 represents the cantilever tip). The initial prestressing scheme is
illustrated below.in Figure 3-13 and given in Table 3-5.
Figure 3-13. Layout of cantilever tendons
One important point to consider is the impact on prestressing steel consumption when
designing light weight cantilever bridges having thin slabs and thin webs. In general, the dead load
moment of cantilever-constructed bridges consumes a large majority of the steel capacity. Menn
(1990) states that the ratio of prestress moment to dead load moment near the supports for
cantilever-constructed bridges is typically lower than for conventionally built bridges. This is a
consequence of a low ratio of live load moment to dead load moment, which is typically equal to
0.2 for cantilever constructed bridges. In this design example, under the initial approximation of
, the ratio of factored live load moment to dead load moment is 0.55 (which includes
superimposed dead load). The unfactored live load moment to dead load moment ratio is computed
as . Since the weight of the cantilever during construction is a large
contributor to the consumption of prestressing steel, the overall consumption of materials is
expected to be less than a classical cantilever-constructed bridge. Assuming that live load moment
is constant between the traditional girder and the proposed box girder, for the ratios given above,
Table 3-5. Preliminary prestressing scheme for cantilever tendons based on ultimate capacity
Segment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Strands 8 12 12 12 12 12 12 12 12 12 12 8 8 8 8
Mps
0.55 1.2 1.7⁄( )× 0.39=
46
a traditional cantilever-constructed bridge consumes 1.95 times more concrete material. In light of
this difference, it is clear that a relative design ratio of tendon utilization in terms of kilograms of
steel per cubic meter of concrete becomes less meaningful when comparing the proposed design
concept to existing structures. An explicit quantitative comparison of material consumption of the
proposed girder and other reference bridges is given later in Chapter 8.
3.7 Change in Structural System
A change in structural system occurs after forming a monolithic closure at the centre segment
and stressing the continuity tendons. This change in structural system modifies the erected
structure from two statically determinate cantilevers to a statically indeterminate continuous
girder. The force effects which undergo a redistribution are the permanent loads which are initially
applied to the statically determinate structure. For the proposed bridge under consideration, these
forces include the self-weight of the girder at the time of erection, and the effects of cantilever
prestressing tendons. The permanent load on the final structure due to superimposed dead load
does not undergo any long-term redistribution since it is placed on the continuous structure after
closure of the midspan. Figure 3-14 describes the change in structural system and the associated
force effects. The self-weight of the cantilevers produces a moment:
(3-11)
at both ends with an initial deflection and rotation due to the combined effect of dead load and
prestressing, , respectively. Since the prestress moment generally offsets the dead
load moments, this girder deflection tends to be small. In some cases, the designed cantilever
tendons cause prestress moments that exceed the dead load moments so a positive cantilever tip
deflection is possible. Through the match-casting of segments and adequate geometry control, the
initial deflection of the cantilever tips at the time of closure can be controlled within a reasonable
tolerance (Post-Tensioning Institute, 2006). Proper geometry control can also minimize the angular
rotation at the tip, , however a small discrepancy in tip angle can be easily accommodated by
the cast-in-place closure segment. Through long-term concrete creep, additional deformations
MclowLc
2
2---------- wL2
8----------= =
yclo and ωclo
ωclo
47
occur in the structural system due to self-weight, , as creep has a softening effect on the long-
term elastic modulus of concrete..
Figure 3-14. Redistribution of forces through concrete creep due to change in structural system (adapted from Podolny et al., 1982)
Considering the concrete strain at any point of the structure, Podolny et al. (1982) describes the
total strain to be the sum of two terms: the strain before continuity is achieved, and the strain after
continuity is achieved. Menn (1990) describes this sum more explicitly as:
(3-12)
where is the initial concrete stress due to permanent load, is the elastic modulus of
concrete, is the time-dependent creep function, and is the aging function for which a typical
value is 0.8. Based on this relationship, Menn (1990) provides a useful engineering approximation
for the calculation of redistributed stress based on an average creep coefficient for dead load plus
prestressing:
(3-13)
where is the fully redistributed stress, is the actual stress at the time of closure, and
is the stress obtained assuming the entire structure was cast simultaneously on conventional
falsework. The distribution factor 0.8 is a typical value for cast-in-place concrete structures.
yφ
Mclo
Lc Lc
L = 2Lcyclo
yφ2ωclo
MφMφ
Mclo - Mφ
Mφ
(a)
(b)
(c)
(d)
ε t( )σ0Ec------ 1 φ t( )+( )
σ t( ) σ0–Ec
---------------------- 1 µφ t( )+( )+=
σ0 Ec
φ t( ) µ
σ∞ σclo 0.8 σfal σclo–( )+=
σ∞ σclo
σfal
48
As a general rule of thumb, the long term creep coefficient is approximated to be for
precast structures. In comparison to creep expressions given in the CAN/CSA-S6-06 (cl. 8.4.1.6.3),
this approximation appears to be reasonable and perhaps even conservative. Assuming that the
precast segments are 28 days old before erection, a creep coefficient of 1.5 corresponds to an
average relative humidity (RH) of 32% over 75 years. For higher relative humidities, the creep is
even less. The expression for creep defined in the Code is illustrated in Figure 3-15.
Figure 3-15. Time-dependent creep coefficient assuming t0 = 28 days, a) for 75 year period, b) for 5 year period
A modified relationship which is similar to Equation 3-13 has been given that is acceptable for
the determination of redistributed stress for precast structures (Post-Tensioning Institute, 2006).
For serviceability, bending moment is linearly proportional to the stress in a girder, and so a similar
relationship can be formulated for the calculation of fully redistributed bending moment. Thus, the
distribution equation is given as:
(3-14)
where is the fully redistributed moment, is the actual bending moment at the time of
closure, and is the bending moment obtained assuming the entire structure was cast
simultaneously on conventional falsework. The distribution factor 0.68 is typical for precast
concrete structures. Using this relationship, it is possible to determine the moment redistribution
due to creep, equal to the distribution term in Equation 3-14 as .
φ∞ 1.5=
00.20.40.60.8
11.21.41.61.8
2
0 500 1000 1500 20000
0.20.40.60.8
11.21.41.61.8
2
0 5000 10000 15000 20000 25000daysdays
φφ
5 years1 year1 month
RH = 32% a = 10 MPa rv = 82mmf’c = 80MPa t0 = 28 days
φ∞ = 1.5φ∞ = 1.5
1 yr
5 yrs
75 yrs
a = 10 MPa rv = 82mmf’c = 80MPa t0 = 28 days
RH = 70%RH = 50%RH = 32%
a) b)
M∞ Mcloφ∞
1 µφ∞+------------------- Mfal Mclo+( )+=
Mclo1.5
1 0.8 1.5⋅+---------------------------- Mfal Mclo–( )+=
Mclo 0.68 Mfal Mclo–( )+=
M∞ Mclo
Mfal
Mφ 0.68 Mfal Mclo–( )
49
For the single span continuous structure we are considering, the redistribution of bending
moment is a constant value for all locations on the girder since Mclo and Mfal are offset only by the
redundant moment that provides continuity at midspan. Therefore, the redistribution can be easily
computed by considering the end moments of Mclo and Mfal. Using the uniform dead load of wD
= 88.2kN/m derived previously in Table 3-3, the redistributed moment is shifted towards positive
bending by 24300kNm. This simple calculation is demonstrated in Table 3-6 below.
3.8 Preliminary Design Methodology for Continuity Tendons
The following section will discuss recommended design considerations for the initial estimate
of continuity tendons for a monolithic closure at the midspan.
3.8.1 Ultimate Moment Resistance
Similar to the preliminary design of cantilever tendons, an initial estimate for the sizing of
flexural steel can be based on the behaviour of the system at the ultimate limit state. For sections
which have both cantilever and continuity prestressing tendons, a simple calculation to account for
both top and bottom prestressing is:
for positive flexure (3-15)
for negative flexure (3-16)
where Ptot is the total prestressing force on the section, kt is the distance from the girder axis to
the centroid of compression in the top flange, kb is the distance from the girder axis to the centroid
of compression in the bottom flange, and Mpt and Mpb are the prestress moments due to top and
bottom prestress tendons, respectively. These parameters are illustrated in Figure 3-16 below. In
general, the prestressing tendons are located as close as possible to the flanges to maximize the
Table 3-6. Calculation of long-term bending moment redistribution
Item αD wD [kN/m]
L [m]
End Moment Equation Moment [kN/m]
Mfal 1.2 88.2 90 107000
Mclo 1.2 88.2 90 71400
0.68(Mfal - Mclo) - - - - 24300
wDLc2
2--------------wDL2
8--------------=
wDL2
12--------------
Mr φp Ptotkt M+ pt Mpb+( )=
Mr φp Ptotkb M+ pt Mpb+( )=
50
internal lever arm. The ultimate resistance of the section is determined by moving the eccentrically
applied loads to the centroid of the cross section and considering the equivalent concentric
sectional forces.
Figure 3-16. Sectional forces due to prestressing with top and bottom tendons
3.8.2 Spreading of Forces
For structures prestressed with external unbonded tendons, moment resistance is essentially a
product of the tension steel force and the internal lever arm between the tension steel and the
centroid of compression stress in the concrete. Since the increase in steel stress due to the formation
of cracks for structures with unbonded tendons is in general less sensitive than those with bonded
tendons, it is especially important to ensure that the internal lever arm may be developed at a
section where the resistance is required. The development of internal lever arm for the box girder
is dependent on an induction length which takes into account the spreading of forces in the concrete
from the point of origin of the applied prestress force (Canadian Standards Association, 2006).
Menn (1990) illustrates this concept clearly using a three dimensional truss model to describe the
flow of forces due to an anchorage force located at the bottom corner of a box section (Figure 3-
17). This model depicts an extruded cross section of a concrete box girder by considering only the
closed box shape. On the left side, the diagram illustrates the state of stress in the concrete, fc, at
various distances from the applied prestress force. Menn (1990) shows that the plane sections
hypothesis is not valid in the vicinity of the applied load where there is a significant peak stress at
the the corner. On the right side, the diagram depicts a simple truss model that represents the
spreading of force in the cross section. Menn (1990) assumes a spreading angle of which
et
eb
Applied Forces Equivalent Sectional Forces
Ptot = Pt + PbPt
Pb
Mpt = Ptet
Mpb = Pbeb
Ptot
kt
kb
Cross Section
45°
51
results in a required development length of la = h+b/4 from the bottom corner where the prestress
force is assumed to originate.
Figure 3-17. Moment development length, la, due to anchorage of continuity tendon (adapted from Menn, 1990)
The CAN-CSA-S6-06 requires for segmental cantilevered construction that continuity tendons
be anchored at least one segment beyond where they are theoretically required since the prestress
force requires an induction length before it may be assumed to be effective over the whole section.
This requirement is consistent with the model by Menn (1990) for girders that have a web height
less than the segment length minus b/4. For precast segmental girders that have a limited segment
length of only 3m and have a web height that is deeper than the segment length, providing only one
segment length for moment development is unconservative. The moment development length
specified by the Code also does not distinguish between the required development length for
girders post-tensioned with internal bonded or external unbonded tendons.
For the segmental design we are considering, the external unbonded tendons applies a force to
the cross section at external anchorage blisters located on the inside of the box near the junction of
the web and flange. However, since this load is applied from within the box itself, the origin of this
point load is eccentric to the centroid of the web and flange intersection. It is necessary for the
prestress force to spread towards the corner before it can begin to develop within the section. This
additional spreading of forces due to the eccentric anchorage of prestress force is an additional
consideration that must be taken into account for the spreading of forces in the box girder. For the
purpose of this chapter, only the longitudinal spreading length of the anchorage force is important
when designing the post tensioning arrangement for global bending. The spreading length is
approximately equal to twice the length of the anchorage blister (shown in Figure 3-18). This
spreading length has been determined based on a local spreading truss model which is discussed in
V
MP = const
V
P
PFcFc
fc
hb/4
P
MPM
la = h +b/4
52
detail in Chapter 6. The total required moment development length is illustrated below in Figure 3-
18.
For the girder considered in this study, the anchorage length, girder depth, and top flange span
between webs is 1495mm, 3600mm, 5000mm, respectively. Therefore, the required moment
development length for the local spreading of forces to the corner of the box is approximately
7840mm. This corresponds to over two segment lengths required for the spreading of forces to
develop the compressive stresses in the top flange of the girder.
Figure 3-18. Required moment development length for a girder post-tensioned with unbonded tendons
3.8.3 Sizing of Continuity Tendons
The maximum effect for positive bending is greatest following long-term redistribution of
forces; therefore, the full redistribution of bending moment previously calculated in Table 3-6 will
be assumed to consider the worst case for positive bending. Loads which are applied to the girder
immediately after construction and before much redistribution has taken place will result in smaller
positive bending demands; therefore, it does not present a critical load case for the design of
positive bending resistance. Up until now, an initial assumption of Mps has been used for the design
of cantilever tendons since it is undefined without a continuity post-tensioning arrangement.
Following the preliminary design of continuity tendons, this initial assumption of Mps must be
checked by calculating Mps directly. Afterward, the tendon design must undergo a subsequent
design iteration based on the Mps calculated with the preliminary continuity tendon design if the
assumed value is unconservative or uneconomical.
For consistency with the tendon selection of the cantilever post-tensioning, the continuity
tendons have been designed to minimize the variety of anchorage sizes and the associated
MP = const
2Lb h b/4
M
V V
P
53
prestressing jacks required for construction while satisfying a reasonable utilization of prestressing
steel. Using the same VSL ES6-12 and VSL ES6-19 size anchorages, the prestressing schedule is
shown in below in Table 3-7.
Figure 3-19. Layout of continuity tendons
This arrangement of tendons provides reasonable economy in steel consumption for the design
of continuity tendons while maintaining minimum anchorage variety. The adjusted factored
demands including long-term creep and the factored capacity envelope is shown in Figure 3-20.
When considering the long-term effects of the structure, the live load moment to dead load moment
at the support is further increased from 0.55 to 0.74 due to the redistribution of forces. At midspan
the live load moment to dead load moment is 0.69. For long term redistribution of forces, a savings
in continuity tendons can also be realized due to the design of light weight box girders since there
is a lesser proportion of the total load on the bridge that has been applied to the statically
determinate system during erection.
To satisfy the safety requirements for longitudinal flexure, the factored demands must be
smaller than the factored resistance at all locations along the girder. Factored demands have been
calculated according to ULS Combination 1 (cl. 3.5.1). It is seen in Figure 3-20 that there is a
significant amount of reserve capacity in negative bending for most locations along the girder, and
Table 3-7. Preliminary prestressing scheme for continuity tendons based on ultimate capacity
Segment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Strands - - - 19 19 19 12 12 12 8 8 - - - -
54
only one critical location in negative bending at the support. The demand for positive flexure shows
a smaller margin for reserve capacity with respect to the positive bending capacity envelope. The
moment development length due to spreading for positive bending has been accounted for by
shifting the demand envelope to provide the required development length determined previously
as 7840mm (Figure 3-20).
Figure 3-20. Preliminary ULS demand and capacity for continuity tendons
3.8.4 Secondary Prestress Moment due to Continuity Tendons
Once a preliminary prestressing scheme for continuity tendons is determined, the initial
estimate made for must be adjusted to reflect the preliminary continuity tendon arrangement.
Following this adjustment, the system must be checked so that the initial estimate did not result in
an unconservative design, or alternatively, an overly conservative design. For a constant depth
girder with equal stiffness along its length, the secondary prestress moment can be calculated
directly due to the influence of each tendon using a frame section analysis. This analysis assumes
that the abutments are longitudinally flexible, meaning that there is no axial restraint provided by
the abutments due to the stressing of continuity tendons.
A frame analysis consisting of a single beam element with constant stiffness having moment
fixity at the supports is considered. As shown previously in Figure 3-16, the application of
prestress forces on a section can be decomposed into an equivalent prestress force acting at the
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40 50 60 70 80 90
Mps DL+Mps+Mφ
DL+SDL+Mps+Mφ
DL+SDL+LL+Mps+Mφ
ULS Capacity
DL+SDL+LL+Mps+Mφ (shifted)Mps+Mφ
LL
DL+SDL+Mps+Mφ
LL/(DL+SDL+Mps+Mφ) = 0.74
LL envelope
Assumed Mps
M [kNm]
distance [m]
shifted demandenvelope to accountfor spreading
Mps
55
girder axis and a corresponding prestress moment equal to . For a continuity tendon that is
stressed symmetrically about the midspan of the structure, the secondary prestress moment due to
the deformations of the indeterminate system are proportional to Lten/L where Lten is the length of
the tendon and L is the span length (Figure 3-21). Since this design includes only straight tendons,
no deviation forces are considered for Mps since deflections are assumed to be small.
The secondary prestress moment breakdown that results from the preliminary continuity
tendon design is summarized in Table 3-8 below.
Table 3-8. Secondary prestress moment calculation
Segment Tendon Unit Pba
[kN]
a. Prestress force assumes a constant tendon stress of 0.6fpu.
Mp [kNm]
Lten [m]
Lten/L Mps [kNm]
4 ES-19 5640 10600 72 0.800 8450
5 ES-19 5640 10600 66 0.733 7750
6 ES-19 5640 10600 60 0.667 7040
7 ES-12 3560 6670 54 0.600 4000
8 ES-12 3560 6670 48 0.533 3560
9 ES-12 3560 6670 42 0.467 3110
10 ES-8 2370 4450 36 0.400 1780
11 ES-8 2370 4450 30 0.333 1480
Σ = 37200
P e×
56
Figure 3-21. Secondary prestress moment due to prestressed continuity tendon
The secondary prestress moment due to the summation of all continuity prestressing tendons
described in Table 3-8 is 37200kNm. This corresponds to 56% of the factored positive moment due
Lten
tendon
eP P
a) elevation of actual structureLLside Lside
Mp Mp
Mps Mps
b) assumed fixity at abutment
c) simplified beam model for analysis of moment only
d) primary system and applied prestress moment
Mp Mp
e) primary system bending moment Mp
f) primary system deflection
g) redundant moments to restore continuity
Mps Mps
Mps
Mp(1-Lten/L)
h) redundant bending moment, Mps
i) total prestress moment, Mptot = Mp + Mps
Mps = Mp(Lten/L)
57
to MSDL, MLL, and . Therefore, the initial assumption used for the proposed box girder is valid
for the designed prestress arrangement. There is no need to adjust the loads on the system for a
second iteration of tendon arrangement.
Figure 3-21 shows that the rigid abutment attracts more positive bending moment for tendons
that are anchored further from the midspan. Thus, the longest continuity tendon lengths have a
greater influence on the secondary prestress moments than the shorter tendon lengths anchored
closer to the midspan. Since Mf varies directly with Mr due to Mps, it is often necessary to compute
a few iterations of tendon arrangement designs before a final design is selected where Mr > Mf for
all sections.
3.9 Bottom Flange Sizing
In preliminary design, the box girder cross section is assumed to have a typical geometry with
a constant bottom flange thickness of 150mm (shown in Figure 3-22 below). For cantilever-
constructed bridges, large negative moments exist at the supports for both dead load moments and
live load moments. In Section 3.6.3 the cantilever tendons were designed based on the ultimate
limit state capacity which assumes an internal lever arm, z. Now that the tendon arrangement is
known and the demands on the structure have been determined, the capacity must be checked for
negative flexure at the critical bending location.
The internal lever arm at the ultimate limit state is directly related to the axial prestress force,
P, applied by the unbonded cantilever tendons. At the support, the axial prestress force in the girder
is due to the accumulation of all cantilever prestressing tendons. At the ultimate limit state, P =
47.5MN (under the initial assumption that all tendon stresses are 60% of ultimate strength). A
bottom flange thickness of 150mm is seen to be insufficient for the large prestress force since
premature crushing of the bottom flange occurs at a very small curvature (Figure 3-22).
Mφ
58
Figure 3-22. Ultimate state of stress for section with undersized bottom flange
For design, the bottom flange thickness should be sized based on an idealized stress block at
ultimate limit state. Sizing the bottom flange so that at ultimate will maximize the internal
lever arm and the flexural ductility of the girder in negative bending. Internal equilibrium of forces
provides the design equation for sizing of the bottom flange due to negative bending:
(3-17)
where tbf is the bottom flange thickness, P is the prestress force, is the concrete material
factor, are the stress block factors, is the ultimate material stress, and b is the bottom
flange width.
Figure 3-23. Appropriately sized bottom flange thickness for negative bending at ULS
Using the FRC constitutive material law model developed for analysis, the ultimate moment
resistance has been observed to occur when . The associated stress block factors
derived for this state of stress are and (Figure 3-23). The stress block
y c
P
Stress Distribution
fc(y)
UltimatePlane ofStrain
Cross Section
P
Force Distribution
fc(y)ּb(y)
z < zassumedΔy
tbf c=
tbfP
φcα1f'cβ1b---------------------------=
φc
α1 and β1 f'c
y c β1c
α1f’c
P P
IdealizedStress Block
Stress Distribution
fc(y)
Ultimate Plane ofStrain
Cross Section
P
Force Distribution
fc(y)ּb(y)
z
εb 1.6ε'c=
α1 0.903= β1 0.774=
59
factor is a value which is used to match the elevation of the actual compression stress centroid
to the centroid of an idealized equivalent rectangular stress block. Knowing that the centroid of a
rectangle is the half depth, the value is a multiplier to give the height of the rectangular stress
block expressed as a scalar multiple of the height of compression, c. The stress block factor is
determined by equating the centroids of the actual stress distribution to the rectangular distribution:
Once the centroids of the compression stress distributions have been matched, the stress block
factor , which is a scalar value to determine the equivalent stress of the idealized stress block,
is determined by equating the compression force in the concrete:
Using Equation 3-17, the bottom flange at the support should be 305mm thick. It is
recommended for design that the bottom flange thickness varies linearly from 305mm at the
abutment to the typical thickness of 150mm at the point of inflection.
3.10 Prestress Distribution
Axial prestress force is introduced at every anchorage location along the length of the girder.
Cantilever tendons introduce an axial force on girder segments that are located between the
cantilever tendon anchorage and the abutment. The prestress force on the girder due to cantilever
tendons is accumulative from the midspan region towards the support. Continuity tendons
introduce an axial force on the girder between symmetrical anchorages across the midspan region.
Similarly, the prestress force on the girder due to continuity tendons is accumulative from the
continuity tendon anchorages towards the midspan region. Considering the total prestress effects
which combines the cantilever and continuity tendons, the resultant prestress force distribution
over the length of the girder obtained from the post-tensioning design shows that the axial force is
β1
β1
β1
yφc fc y( )y yd
0
c
∫
φc fc y( ) yd0
c
∫---------------------------------
β1c2--------==
β1
2 fc y( )y yd0
c
∫
c fc y( ) yd0
c
∫-------------------------------=
α1
y φc fc y( ) yd0
c
∫ b y( ) φcα1f'cβ1cb=⋅=
α1
fc y( ) yd0
c
∫f'cβ1c------------------------=
60
relatively constant over the length of the girder. The primary prestress moment that is induced due
prestress forces applied eccentrically to the concrete centroid varies along the length of the girder
in a shape that corresponds to the inverse of the applied bending moments.
Figure 3-24. Primary prestress force distribution
a) girder elevation
-80000
-60000
-40000
-20000
0
20000
40000
60000
80000
0 5 10 15 20 25 30 35 40 45
0
10000
20000
30000
40000
50000
60000
70000
80000
0 5 10 15 20 25 30 35 40 45
P [kN]
Ptot
Pt
Pb
cantilever tendonprestress, Pt
continuity tendonprestress, Pb
total prestressforce, Ptot = Pt + Pb
b) girder prestress force, P
c) girder prestress moment, Mp = Pּe
Mp [kNm]
continuity tendonprestress moment,Mpb = Pbeb
cantilever tendonprestress moment,Mpt = Ptet
total primary prestress moment, Mp = Mpt + Mpb
Mp
Mpt
Mpb
61
The deformation response and stress distribution along the length of the girder due to applied
loads and prestress forces is determined through the unique moment-curvature relationships that
pertain to each segment. As shown in Figure 3-24, the sectional prestress forces vary along the
length of the girder where each segment corresponds to a specific prestress force, P, and a total
primary prestress moment, Mp. For the typical box girder cross section, these two values determine
the member response for the segment under consideration which corresponds to the constant
prestress force, P. Using the interpolation method that was described previously in Section 3.4, the
distribution of sectional effects along the length of the girder can be determined. This procedure is
illustrated below in Figure 3-25 for the curvature distribution along the girder length. Only two
examples are shown for the girder at segments 3 and 14. A similar procedure is performed to
determine the distribution of stresses and strains along the length of the girder.
62
Figure 3-25. Girder response for varying prestress force
3.11 Ultimate Limit State
Figure 3-26 displays the girder response at ultimate limit state due to permanent factored load
(dead load plus prestressing) and maximum factored live load. Flexural cracking is observed to
-150000
-100000
-50000
0
50000
100000
0 10 20 30 40
-0.0005-0.0004-0.0003-0.0002-0.0001
00.00010.00020.00030.00040.0005
0 10 20 30 40
-150000
-100000
-50000
0
50000
100000
-0.002 -0.001 0 0.001 0.002
-150000
-100000
-50000
0
50000
100000
-0.002 -0.001 0 0.001 0.002
M [kNm]
Φ [1/m]
interpolated curvesconstant force datums
P14
Φ [1/m]
M [kNm]
Segment 14 P14 = 37.1MN
Mp14 = -39.8MNm
Mp3
Mp14
P3
M [kNm]
Segment 3 P3 = 41.6MN
Mp3 = -66.0MNm
Φ [1/m]
a) girder elevation b) interpolation between known curves of constant force
P3,Mp3
P14,Mp14
c) ULS bending moments d) primary prestress moment as internal resistance
e) cuvature distribution along girder length
interpolated curvesMp internal resistance
63
occur at the support due to negative bending. Localized tension strains on the top surface are
observed to occur in the vicinity of the support.
Figure 3-26. Girder response at ULS (half span shown)
3.12 Serviceability Limit States
The following section will describe some of the key components of serviceability required for
adequate durability, crack control, and deflection limit states.
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40
-0.0005-0.0004-0.0003-0.0002-0.0001
00.00010.00020.00030.00040.0005
0 10 20 30 40
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0 10 20 30 40
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0 10 20 30 40
-80
-70
-60
-50
-40
-30
-20
-10
00 10 20 30 40
-80
-70
-60
-50
-40
-30
-20
-10
00 10 20 30 40
M [kNm] Φ [1/m]
εt [mm/mm] εb [mm/mm]
σt [MPa] σt [MPa]
CapacityDL+SDL+LL+Mps
DL+SDL+Mps
critical location for negative bending
concrete model assumes zero tensile capacityaccounting for no tensile strength across the joint
64
3.12.1 Construction
For the erection of segmental cantilever girders with unbonded tendons, large prestressing
forces are applied to the top corner of the box girder at the location of anchorage blisters. The
prestress moment, , can be significant and often exceeds the dead load bending moment due
to the light weight girder design, especially in the vicinity of the cantilever tip where the dead load
moment is zero. For segmental bridges, epoxied joints are necessary to prevent corrosive de-icing
salts from leaking into the box girder. At the time of erection, a minimum compressive stress of
350kPa must be applied to the entire cross section for adequate adhesion of joints (cl. 8.22.6.1.2).
Temporary bottom post-tensioning is necessary for full compression of the joint face in the vicinity
of the cantilever tip. The critical loading condition for each joint face corresponds to the jacking
force of the cantilever tendon immediately ahead of the joint.
The largest cantilever tendon unit considered in this analysis consists of 12 strands of 0.6”
diameter. As the jacking force corresponds to 80% ultimate stress of the tendon, the temporary post
tensioning must be designed for the associated jacking moment. The DSI Threadbar system
(Dywidag-Systems International, 2006) can be easily adopted for the temporary post tensioning of
the cantilever during construction. Typically, only the first three to four segments at the cantilever
tip are critical - past this point, the dead load of the cantilever applies sufficient dead load moment
to provide full compression on the joint face. Temporary post-tensioning is assumed to be reusable,
so the maximum temporary jacking stress is only 0.6fpu. The temporary post-tensioning steel can
be optimized by using couplers and leapfrogging bars for the progressive erection of the cantilever.
Assuming a linear elastic plane section at the joint, and neglecting the benefit of dead load, the
temporary post-tensioning to provide the required compressive concrete stress at the critical joint
is two 36mm diameter Threadbar post-tensioning rods stressed to 60% of ultimate stress.
3.12.2 Cracking
The SLS girder response due to unfactored loads is determined based on SLS Combination 1
(cl. 3.5.1). It is observed that the girder is fully compressed for serviceability using the tendon
arrangement designed to satisfy strength requirements at the ultimate limit state. The girder
response is shown in Figure 3-27 below.
P et×
65
Figure 3-27. Girder response at SLS (half span shown)
-0.0005-0.0004-0.0003-0.0002-0.0001
00.00010.00020.00030.00040.0005
0 10 20 30 40
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0 10 20 30 40
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0 10 20 30 40
-80
-70
-60
-50
-40
-30
-20
-10
00 10 20 30 40
-80
-70
-60
-50
-40
-30
-20
-10
00 10 20 30 40
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
0 10 20 30 40
-15-10-505
1015202530
0 10 20 30 40
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40
M [kNm] Φ [1/m]
εt [mm/mm] εb [mm/mm]
σt [MPa] σt [MPa]
Θ [rad] Δ [mm]
DL+SDL+LL+Mps
DL+SDL+Mps
66
3.12.3 Global Deflection
The vibration limit state defined in CAN/CSA-S6-06 for SLS Combination 2 does not govern
the design of the box girder. For the 90m span with a 25:1 span-to-depth ratio, the vertical
deflection due to longitudinal bending for one passing truck (cl. 3.8.4.1c) is 11.4mm. The
additional sidewalk deflection as a result of transverse slab bending at the barrier face has been
computed as 2.8mm. This result is obtained from a grillage analysis which is described later in
Chapter 4. The girder twist caused by torsion in the box girder due to a truck placed eccentrically
from the bridge centreline has been computed as 3.1mm from a linear elastic analysis. The
combined deflection of 17.3mm is well within the deflection limit of 42mm for bridges with
occasional pedestrian use corresponding to the first-order vibration computed as 1.9Hz. For typical
highway overpass bridges, this deflection is acceptable.
3.13 Refinement in Tendon Stress
The proposed box girder bridge concept has been developed under the assumption that
at the ultimate limit state. The consequence of this assumption is an increase in
consumption of required prestressing steel. It is of interest to investigate the stress increase in the
unbonded prestressing steel that is possible at the ultimate limit state due to girder deformation for
the purpose of achieving greater economy prestressing steel consumption.
3.13.1 Ductility of Fibre-Reinforced Concrete
With reference to the demand and capacity envelopes shown previously in Figure 3-12 and
Figure 3-20, the critical flexural regions of the girder are observed to be at the midspan and at the
supports with the least reserve capacity. Due to the large concrete area provided in the 12m wide
top flange and the relatively small factored axial prestress of 34.7MN, there is a substantial
capacity for rotational ductility for positive bending at the midspan. At the support, however, the
bottom flange has a considerably smaller width and a higher factored axial prestress force of
47.5MN. At the ultimate limit state, the rotational capacity is limited by the crushing of concrete.
In comparison to a typical compressive stress-strain curve for concrete with a nominally similar
compressive strength, FRC exhibits far greater ductility in compression for the post-peak response
(Figure 3-28). This ductility in compression becomes abundantly important when considering the
ultimate flexural response of girders prestressed with unbonded tendons. The increment of strain
σp σp∞=
67
in unbonded tendons is generally lower than bonded tendons (Ramos et al, 1996); therefore,
yielding of steel at the ultimate limit state is less likely to take precedence over crushing of the
compression flange. Typically, the failure of normal concrete in crushing is a brittle mechanism,
allowing little flexural ductility and plastic redistribution of forces. The capability of FRC to
sustain significant compressive stress for large strains following the peak stress allows structural
members to undergo larger curvatures before failure. For this reason, it is expected that girders cast
with high strength FRC can maintain equilibrium under large curvatures due to a ductile crushing
of the compression flange.
Figure 3-28. Stress-strain response of FRC and typical normal-weight concrete
The use of FRC in combination with structures prestressed with unbonded tendons appears to
be complementary since FRC allows for increased girder deformation to produce higher strains in
the tendon at ultimate limit state. The moment-curvature response shown in Figure 3-29 compares
the behaviour of identical box girder sections with normal and fibre reinforced concrete for
negative bending. For highly prestressed sections, the rapid degradation in compressive strength of
normal concrete when causes crushing failures to occur at small curvatures. For
geometrically identical box girders, the bottom flange of the girder fabricated with regular concrete
fails when whereas the bottom flange of the girder fabricated with FRC fails when
. Since negative bending at the abutment is expected to control the failure mechanism
at the ultimate limit state, it is prudent to maximize the ductility of this failure mode for safety. For
maximum curvature when , the idealized stress block factors have been computed to
be and . The calculation method for stress block factors has been
0
01
02
03
04
05
06
07
08
210.010.0800.0600.0400.0200.00
FRC model (from tests)
normal-weight concrete (Popovics, 1970)ε’c = 0.00271Ec = 37100
post-peak ductility of FRCdue to fibre confinement
fc [MPa]
εc [mm/mm]
ε ε'c>
εb 1.4ε'c=
εb 3.07ε'c=
εb 3.07ε'c=
α1 0.812= β1 0.921=
68
described previously in Section 3.9. Maximum ductility, however, can be achieved using the
bottom flange concrete thickness designed for maximum strength.
Figure 3-29. FRC comparison for total moment-curvature response in negative bending
3.13.2 Deformation Capacity
The rotational ductility of the concrete box girder allows the structure to undergo plastic
deformation to produce large cracks at the ultimate limit state. The available rotation capacity can
be derived from the curvature distribution over the length of the girder. Large curvatures result
when the bending moment demands approach the ultimate capacity of the box girder. Under this
condition, the centroid of compression stress block is entirely in the compression flange and large
tensile strains result at the level of the prestressing tendon (shown previously in Figure 3-23). If the
girder is loaded to failure, the bottom flange crushing governs the failure mode at the support due
to negative bending.
It was found that for cantilever girders with straight unbonded cantilever tendons are not well
adapted to getting large tendon stress increases at the ultimate limit state. Although higher
curvatures are possible using high strength FRC, the rotational capacity for girders in negative
bending regions is still somewhat small. This is because large curvatures only exist over a small
plastic hinge length since the bending moment demand in negative moment regions approaches the
member capacity at a fast rate (shown in Figure 3-30). The integration of curvature over the small
plastic hinge length results in a small rotational capacity before failure. Therefore, the tendon stress
increase due to girder deformation is not significant at the ultimate limit state due to the limited
girder deformation capacity. Since the abutment location is the only critical negative bending
region, only one predominant crack (or joint opening) occurs at the abutment face to provide strain
00008-
00006-
00004-
00002-
00500.0-10.0-510.0-20.0-
normal concrete girder failure
FRC girder failure
M [kNm]
φ [1/m]
responses shown are for a constant prestress force, P
69
increases for the unbonded tendons. This single crack has the largest influence on tendon stress
increase for the shortest tendon, and a diminishing influence for longer tendons.
Figure 3-30. Plastic hinging in negative moment regions
3.13.3 Example of Tendon Stress Increase for Negative Bending
An investigation was carried out to consider an alternative 90m span bridge design which is
hinged at the midspan and uses a tendon design which more closely matches the moment demand
envelope for cantilever bending. The intent of this design is to promote several joint openings at
the ultimate limit state to provide larger tendon stress increases at the ultimate limit state. The
tendon arrangement given in Table 3-9 is assumed for the cantilever system.
For simplicity, only a uniform distributed load is applied to the hinged cantilever system to
investigate the structural response at failure. As can be seen in Figure 3-31, the capacity and
demand envelopes are closely matched which causes several locations along the girder to approach
Table 3-9. Tendon unit sizes (15mm strands) used for cantilever post-tensioning
Segment No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Strands per tendon 19 19 19 12 12 12 12 7 7 7 7 4 4 4 4
Φ [1/m]
small plastichinge length
-200000
-150000
-100000
-50000
0
50000
100000
150000
0 10 20 30 40
M [kNm]
CapacityDemand
rapidly approaching demand to the ultimate resistance in negative bending regions
Segment 1Segment 1
70
the ultimate resistance of the cross section. Therefore, it is observed that plastic deformation occurs
at multiple segments at the ultimate limit state to increase the stress in the unbonded tendon.
Figure 3-31. 90m span with midspan hinge when load is close to failure
Cantilever tendon stresses have been computed for each construction stage, and for short-term
loading (shown in Figure 3-32). During the construction process, it can be seen that there is an
initial decrease in cantilever stresses due to the addition of a new segment and corresponding
prestressing force. The tendon stresses reach a minimum as the effect of additional prestressing is
offset by the additional dead load moment due to the weight of the cantilever. In the last stage of
construction, the cantilever is assumed to be closed with a hinge at the midspan.
The change in tendon stresses of unbonded tendons due to girder deformations was then
examined by applying additional load to the girder after closure. It is seen that the increase in
tendon stress for curtailed unbonded prestressing is linear for the uncracked state. Following the
first cracking, the rate of change in tendon stress increases due to large local strains in the vicinity
of the crack at the level of the prestressing. Even though this system provides the most optimal
conditions for increases in tendon stress at the ultimate limit state, the average increase in tendon
stress before failure was only 0.07fpu. For the proposed box girder designed with cantilever and
-200000
-150000
-100000
-50000
00 10 20 30 40
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0 10 20 30 40
M [kNm]
Φ [1/m]
capacity envelope
uniform distributed load
plastic rotation at several locations
71
continuity post-tensioning which allows only one segment joint to open, the average increase in
tendon stress at the ultimate limit state is negligable.
Figure 3-32. Stresses in unbonded tendons for curtailed prestressing. a) change in tendon stresses during construction due to the addition of segments and post-tensioning, b) change in tendon stresses due to the application of increasing uniform load until failure
3.13.4 Discussion
While it has been observed that the use of FRC provides a higher capacity for curvature at the
ultimate limit state, it appears that there is no great benefit in the increase in tendon stress at the
ultimate limit state. For cantilever type systems with curtailed prestressing tendons, the critical
moment location is seen to occur at the support where the bending moment is greatest. Since the
bending moment approaches the critical section at a fast rate, the region in which the plastic
rotation occurs is very small. Although the absolute curvatures that can be realized for structures
with fibre reinforced concrete is large, the total crack opening determined by integrating the strain
distribution over the small plastic region is still quite small. It appears that a cantilever system
using straight external unbonded tendons for the cantilever and continuity post-tensioning is not
well adapted to provide multiple crack opening to give larger stress increases at the ultimate limit
state. Tuning the capacity envelope provides a marginal benefit to the increase in tendon stress at
ultimate limit state; however, this is neither a practical nor a desirable approach to realistic designs
for constant depth girders.
Unbonded tendons are more efficiently and appropriately used for beams in positive moment
regions where the capacity envelope is more closely matched to the demand which allows for
multiple discrete cracks to open, providing larger strain increases in the tendon. For the proposed
box girder with straight tendons, the resulting prestress force distribution for the combination of all
prestressing steel is relatively constant and the resulting prestress moment is inversely related to
12001250130013501400145015001550160016501700
0 5 10 15 0 20 40 60 80 100w [kN/m]Stage of Construction
girder failure at w = 99.5kN/m
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
yield stress yield stress
first cracking
a) b)fp [MPa]
72
the applied loads to the structural system. This resulting distribution resembles that of a girder
designed using draped tendons that are anchored at the supports. An example of a draped tendon
arrangement is shown in Figure 3-33. For the draped system described by Muller et al. (1990), it
can be seen visually that tendon stress increases for this arrangement is largely due to the multiple
crack openings that occur in the positive bending region, shown by the large opened cracks at the
midspan.
Figure 3-33. Ultimate behaviour for a continuous girder using unbonded draped tendons (adapted from Muller et al., 1990)
It can be concluded that the shape of the bending moment diagram for negative bending
produces a small plastic hinge length which greatly reduces the rotational capacity of the girder.
The straight cantilever tendons which are at a constant elevation below the top flange do not
experience an appreciable tendon stress increase since the plastic rotation capacity in negative
bending is small. To this end, the proposed box girder design is not revised further to optimize the
tendon steel consumption.
3.14 Global System Considerations
3.14.1 Span-to-Depth Ratio for Constant Depth Box Girder
A range of span-to-depth ratios were considered for the design of the proposed constant depth
box girder to investigate the effect of span-to-depth ratio on material economy. The general trend
for material consumption showed that the concrete usage varied slightly for a wide range of span-
to-depth ratios but overall the concrete consumption is relatively constant for regardless of girder
depth. The reason for this is because the top flange design, which accounts for a large portion of
the total concrete usage, is governed by the design for transverse bending. The bottom slab is also
chosen to have a typical constant thickness for varying girder depths. Internal corner blisters for
73
cantilever and continuity tendon anchorages must remain the same size since minimum clearance
requirements for stressing jacks must be satisfied for construction and the anchorage details are
intended to be standardized for precast construction. Therefore, only the height of the web is the
predominant factor which affects the overall concrete consumption due to varying girder depth.
Since the webs are slender, this change is relatively small (shown in Figure 3-34).
The tendon consumption varies approximately linearly with span-to-depth ratio. Since the
ultimate resistance of the cross section is essentially a the product of tendon force and internal lever
arm, the depth of the girder has a significant effect on the prestressing steel demand. As the internal
lever arm is decreased due to a shallower box girder, the tendon consumption increases.
Figure 3-34. Material consumption for varying span/depth ratios
For the 90m span under consideration, a 25:1 span-to-depth ratio was chosen as a reasonable
compromise of economy and visual girder slenderness while using efficient use of high strength
fibre reinforced concrete. Deeper box girders have increased visual prominence which detracts
from the transparency of the structure. Thinner box girders require large prestress forces which
decrease the tendon utilization economy. The need for large prestress forces also requires larger
tendon units. Due to anchorage detailing (which is discussed in Chapter 6), the maximum tendon
unit size presents a limitation for girder slenderness.
3.14.2 Range of Spans
Assuming that alternative span lengths would also be designed at a span-to-depth ratio of 25:1,
the change in total concrete consumption for varying girder depths is small since the only changing
variable for the typical section is the web depth (as shown for shallower sections in Figure 3-34).
Due to this, the uniform dead load is similar for any span length. For shorter spans, the
0.0
20.0
40.0
60.0
80.0
100.0
120.0
16 18 20 22 24 26 28 30 32 34 360.00
1.00
2.00
3.00
4.00
5.00
6.00Tendon consumption
Concrete consumption
span/depth ratio
Concrete Consumption[m3/m]
TendonConsumption[kg/m3]
74
superimposed dead load is unchanged for equal bridge widths. Therefore, the effects of dead load,
superimposed dead load and long-term redistribution of dead load are proportional to L2.
The live load model applied to the box girder defined by the CAN/CSA-S6-06 is also the same
for all span lengths for equal bridge widths. Since the top flange is governed by transverse bending
due to live load, the top flange design for all span lengths does not change. The continuous slab
thickness is governed by the chosen concrete clear cover to the transverse post tensioning. Since
the proposed bridge design concept corresponds to only one single continuous span, there are no
influences due to multiple span effects. The prestressing steel arrangement may be kept similar for
alternative span lengths, and all other things being equal, the quantity of steel consumed for shorter
span lengths is expected to be scaled proportionally to the span length since:
The 90m span considered for this concept is an upper boundary for the range of spans since
deeper cross sections required for longer spans is not recommended due to the visual prominence
of the girder. The maximum depth of segments is also governed by ease of transportation as the
minimum vertical clearance for existing bridges on Ontario highways is 4.7m (Ontario Ministry of
Transportation, 2002). With the consideration for trailer bed height and protruding reinforcing steel
for the cast-in-place barrier, the maximum girder depth for segments is 3.6m.
The smallest span for this concept is anticipated to be 45m due to practical and economical
reasons. Maintaining a 25:1 span-to-depth ratio results in a 1.8m deep box girder. To allow
sufficient working space for continuity tendon stressing inside the box girder and to provide ease
of inspection, a minimum clearance must be provided for workers and inspectors to walk inside the
box girder. Also, at a span length of 45m, precast I-girders can also offer a single span solution
since I-girders can be transported in lengths up to 50m (CPCI, on-line) and may provide a more
economical solution.
Mr Mf>
PdvwL2
const-------------->
P 0.9L25-----------⎝ ⎠
⎛ ⎞ wL2
const-------------->
P L const×>
75
3.14.3 Alternative Girder Designs
The choice to use a constant depth girder over a haunched girder for a light weight design
prestressed with unbonded tendons has significant advantages: a) the extra premium in dead load
due to the full depth of the web is a small increase from the haunched profile, b) a significant
increase in internal lever arm is gained at the midspan (typical 50:1 span-to-depth for haunched
girders (Menn, 1990), 1:25 span-to depth for constant depth) which provides a considerable
increase in flexural resistance for positive bending, and c) the constant girder depth facilitates
repetitive and simpler detailing for the strutted connections. Figure 3-35 illustrates the alternative
haunched girder concept, using span-to-depth proportions suggested by Menn (1990).
Figure 3-35. Concept comparison of haunched girder and constant depth girder elevation
Several girder profiles were investigated for the proposed cantilever-constructed bridge. A
constant depth girder was found to be the most structurally efficient girder shape, unlike traditional
haunched girders designed for cantilever construction. Since dead load typically contributes a
substantial bending moment demand at the pier for traditional cantilever-constructed bridges,
reducing the dead load of segments nearing the midspan relieves the flexural demands at the
support. However, for light weight girders with slender webs, reducing the depth of the girder has
an insignificant reduction of dead weight since most of the concrete volume is in the flanges. The
savings in cantilever post-tensioning due to the reduction in dead load of the cantilever do not
outweigh the additional continuity prestressing required to compensate for the loss in lever arm at
midspan. Since unbonded tendons do not experience any appreciable increase in tendon stress at
76
the ultimate state, the midspan flexural resistance is reduced significantly by the loss in lever arm
to the compression flange. For a haunched girder with span-to-depth ratio of 17:1 at the support
and 50:1 at the midspan, the required demand for prestressing steel was found to be approximately
15% more than for the 25:1 constant depth girder.
Unlike typical balanced cantilever construction, the supports of the proposed bridge must resist
a significant overturning moment in the final stage due to factored dead load and live load. Large
flexural demands at the support introduce large forces in the substructure due to the overturning
effect; therefore, it is advantageous to maximize the redistribution of moments to the midspan by
making a constant depth girder monolithic at the closure segment. The installation of straight
continuity tendons in the superstructure is uncomplicated and provides a straightforward and
efficient solution to resist flexural demands. It is more economical to provide a monolithic
connection at midspan to relieve negative bending moment at the abutment. The elimination of a
joint at midspan also simplifies the detailing at closure and provides added protection against
corrosion due to de-icing salts.
Since the top slab has been designed for minimum weight and minimum thickness, a strutted
connection is necessary at the cantilever tip to provide adequate strength. The strutted connection
is standardized for a constant depth girder since the length of the strut and the angle in which it
connects to the web and top flange is consistent. This repetition of details provides further economy
in the design concept. For a haunched girder, the strutted connection becomes more complicated.
Either the connection of the strut to the box frames into the bottom flange which varies in elevation
at every segment, or the strut is connected to the box at a constant elevation below the top flange.
In the latter configuration, a compression strut inside the box must be installed to resist the
horizontal forces of the external diagonal struts. In both cases, the detailing required for the strutted
connection becomes more complex and is less economical from a standardized precasting
perspective.
77
78
4.0 TRANSVERSE FLEXURE
This chapter begins by describing the main objective for the transverse design, and discusses
typical transverse prestressing in existing structures. The motivation for the design of a thin top
slab is described, and the feasibility of post-tensioned or pre-tensioned transverse prestressing
alternatives are identified. Following the selection of the pretensioning concept, a grillage analysis
is selected as the method of analysis for determining the live load distribution of transverse bending
moments on the top slab. The sizing of concrete slab dimensions is designed with respect to
maximum allowable transverse bending deflections which is determined by comparing the linear
elastic deflections of a standard AASHTO-PCI-ASBI girder (Precast/Prestressed Concrete
Institute, 1997). The behaviour of the structure is described for service loads and for the ultimate
limit state. A traffic barrier design is presented which satisfies the CAN/CSA-S6-06 Performance
Level 2 (PL-2) requirements (cl. 12.4.3.2) for factored impact loading.
4.1 Introduction
The structural components of the box girder cross section that provide vertical support
reactions for the top slab are the webs which support the top flange continuously along the web
lines and the diagonal struts which support the top flange at discrete points along the cantilever
wing. Live load is applied vertically to the top slab which must carry the forces in transverse
bending to the support reactions. The bending moments in the top slab that act in the transverse
direction vary depending on the location of the vertically applied live load forces. In general, the
maximum negative moments occur above the web supports since the top flange is continuous over
the webs. For girders with a free cantilever wing and no struts, the maximum positive moments
occur between the webs where there is a large influence for positive bending. However, for the
proposed box girder with struts, the boundary conditions for the top slab are modified which
produce a significant positive bending moment between the webs and the strut connections. The
critical transverse bending effects can be determined by accounting for all possible load
configurations on the bridge deck and by taking the maximum and minimum live load distribution
envelopes for all cases.
For the given number of traffic lanes to design for, the distribution of live load in the transverse
direction can be modified by altering the boundary conditions which support the top slab.
Maximum positive bending moment between the webs can be reduced by moving the webs closer
to the centreline of the bridge; however, the maximum negative bending moment at the web is
increased due to this repositioning. The bending moment distribution is also affected by the relative
stiffnesses of the box girder structural components, such as the top flange stiffness and the web
stiffness. Since the top flange is rigidly supported at the web, a distribution of moments occurs due
to frame action. However, the bending stiffness of the slender webs is much smaller than the
required bending stiffness of the top flange, so very little moment is distributed into the webs. This
result is presented later using a frame analysis of the cross section. A favourable distribution of
bending moment demands is found to design the top slab for transverse bending which satisfies
strength requirements and accounts for geometrical constraints.
The design intent of the transverse prestressing is to make the top slab intensively prestressed
transversely for adequate crack control under service loads. Since it is expected that the girder will
behave elastically for SLS conditions, a linear elastic analysis of forces is appropriate for analysing
service loads and structural response.
4.2 Typical Transverse Prestressing for Segmental Bridges
Typical box girder bridges are transversely post tensioned using multi-strand flat post-
tensioning ducts. The VSL Type SO anchorage is one type that allows for the bundling of four
15mm strands placed in a flat duct for slab prestressing (VSL, 2007). The east approach girders of
the Confederation Bridge has a deck width of 11.64m that is post-tensioned using transverse
tendons containing three 15mm strands, with an even tendon spacing at 1000mm and a minimum
concrete cover to the top surface of 85mm (Straight Crossing, 1994). The advantage of flat post-
tensioning ducts is that they can be deviated in the slab vertically to provide deviation forces
opposite in direction to the applied live loads in addition to maximizing the internal lever arm for
flexural resistance. For post-tensioning, the casting operation and stressing of the tendons are
uncoupled processes.
4.3 Prestressing Concept for Light Weight Slab Design
The main objective of the deck design is to reduce the dead weight. In fulfilling this design
motivation, smaller concrete covers are assumed to be valid for the application of high-
performance FRC. The CAN/CSA-S6-06 requires that the top concrete cover for transverse post-
tensioning be mm and the soffit concrete cover for pretensioning strands of deck slabs less 80 10±
79
than 300mm thick to be mm (cl. 8.11.2.2). Following these strict provisions for the use of
FRC in bridge decks results in overdesigned strength and poor efficiency of materials. For the
efficient utilization of high-performance concrete and the minimization of dead load, the transverse
design has been based on strength requirements for transverse flexure and serviceability
requirements for acceptable crack control and deflection. The concrete cover recommendations
given in the CAN/CSA-S6-06 have not been used. A top clear cover of 40mm to the pretensioning
strand and a bottom clear cover of 30mm to the pretensioning strand has been chosen for the thin
slab design using high strength concrete. High-strength concretes with low water-to-cement ratios
generally are less porous and have increased durability properties due to low permeability (Neville,
1995). The concrete mix tested for this study contained 8% silica fume of the total cementitious
material. The influence of silica fume upon the permeability of concrete is very large - a 5% content
of silica fume was reported by Khayat and Aïtcin (1993) to reduce the coefficient of permeability
by 3 orders of magnitude. In addition, the low water-to-cement ratio of silica fume concretes, using
water reducing and superplasticizing admixtures, can result in total elimination of capillary
porosity in the concrete mixture provided sufficient curing had been ensured (Mehta, 1986). With
the combination of a high-performance waterproofing membrane, smaller concrete covers may be
acceptable for design; however, a definitive proof for this assumption is not investigated in this
thesis.
4.3.1 Post-Tensioning
A typical transverse post-tensioning arrangement, using flat multi-strand slab tendons, was
originally investigated for the transverse prestressing concept. It was found that using the largest
flat slab tendon of 4-15mm strands did not provide sufficient resistance for the design of a thin
concrete section under consideration. The closest specified VSL SO6-4 tendon spacing is 352mm
centre-to-centre which is governed by the wide anchorage unit that must be cast at the cantilever
wing tip (VSL, 2007). However, the practical limitation for minimum tendon spacing is larger
because the tendons can only be placed within transverse ribs since the tendon duct diameter is
larger than what can be reasonably placed within the thin slab section.
In addition, the prestress force applied by the anchorages requires a development length for the
prestress force to spread over the width of the deck slab. Within the spreading length and between
the anchorages, there are dead zones of deck slab which are not prestressed. For a thin slab that
relies on the precompression force due to prestressing for good performance at serviceability, it is
60 5±
80
necessary to provide a more uniform application of transverse prestress force along the length of
the girder. Therefore, to provide sufficient transverse bending strength and uniform prestressing at
serviceability and to avoid the use of large diameter tendons, post-tensioning is not a feasible
solution for the design of transverse prestressing in thin slabs. A pretensioning design will be
investigated for the transverse prestressing concept.
4.3.2 Pretensioning
The use of regularly spaced pretensioning strands for the design of transverse flexure in precast
segmental construction appears to be well-adapted for the design of standardized segments with
thin slabs. For the design of a thin top slab strengthened by transverse ribs, only straight
pretensioning strands are considered. Since prestressing strands are not deviated in the slab to
accommodate negative and positive flexural regions, a prestressing arrangement that results close
to a concentric prestress has been decided to be the most effective arrangement for constructability
and structural behaviour. For a prestress arrangement that is not concentric to the top slab,
secondary prestress moments will be produced due to the indeterminate structural system of the
strutted box girder. Prestressing strands are assumed to be evenly spaced in the slab, and in the
bottom of the transverse ribs to maximize the ultimate strength. The advantage of evenly spaced
strands is the uniform introduction of prestress force on the section. The top concrete cover has
been chosen to be 40mm and the bottom concrete cover is chosen to be 30mm. These covers
provided are only half of what the CAN/CSA-S6-06 requires, however this assumption is
maintained in design for high performance concrete notwithstanding the Code recommendations.
Using these assumed concrete covers, the required slab thickness including the diameter of a 15mm
strand is 85mm. A possible arrangement of pretensioning strands and the clear covers to the strands
is shown in Figure 4-1. Since pretensioning does not use any mechanical anchorage at the face of
the slab, an investigation of bond strength is required to determine the required transfer and
development length which is necessary for the prestress to be effective.
81
Figure 4-1. Possible pretensioning arrangement
4.4 Bond Strength of 15mm Pretensioning Strands in FRC
4.4.1 Mechanisms of Bond
The bond resistance of prestressing strands embedded in concrete depends primarily on
frictional resistance and mechanical interlock. The chemical adhesion bond, if any, fails at very
small slips (Chao, 2006). A slip is the movement of the bonded steel relative to the surrounding
concrete. Due to the differential slip between steel and concrete, a shear stress due to frictional
bond is activated. Frictional bond provides initial resistance against loading and further loading
mobilizes the mechanical interlock between the concrete and strands. The helical outer wires
around the straight centre wire of a seven-wire strand are responsible for the mechanical action in
a pretensioning strand (den Uijl, 1998).
In seven-wire strands embedded in a FRC matrix, the confining effect of the fibres enhances
the frictional and mechanical interlocking significantly (Figure 4-2). As strands slip in a FRC
matrix, the pullout mechanism is formed by the helical path of the outer wires. Crack widths due
to the radial tension field for seven-wire strands are generally smaller than crack widths for
deformed bars, due to both the fibre confinement and due to the helical pullout path (since
deformed bars pull out directly). Thus, the strand exhibits higher bond strength and maintains bond
stresses for longer slip distances (Chao, 2006). A ductile stress-slip response is exhibited for
82
strands embedded in FRC, which is characteristically different from the response of reinforcing
bars (Chao, 2006).
Figure 4-2. Bond mechanisms between FRC and 15mm pretensioning strand (adapted from Chao, 2006)
4.4.2 Development of Stress in a 15mm Pretensioning Strand
The initial function of bond is to transfer the prestress force from the pretensioned strand to the
concrete member. The bonded length of strand needed for such transfer is known as the transfer
length, lt. Since the concrete is cast around the strand when it is pretensioned, the strand is slightly
thinner due to the Poisson's ratio. When the strand is released after the concrete has cured, there is
a swelling of the strand along the transfer zone which gives rise to considerable radial pressure and
thus larger friction forces against slip. This increase in frictional bond is termed the Hoyer's effect.
When a member is subjected to loads in the range from SLS to the ULS, the stress in the
prestressed strand increases substantially beyond the prestress level and a bonded embedment
length beyond the transfer length is required to develop the increased steel stress. This length is
referred to as the flexural bond length, lf. The sum of the transfer and the flexural bond length is
known as the development length (Figure 4-3).
Figure 4-3. Idealized strand stress profile in a pretensioned strand under applied load (adapted from Kahn, 2002)
BondStress
Slip
without fibers
with fibers
strand pulloutdirection
twistdirection
mechanical interlocking
mechanical interlocking
friction
narrowlongitudinal
concrete crack
SteelStress
Distance from free endlt ld
transferlength
flexural bond lengthdevelopment length
increase in strand stressdue to applied loads
fps
fse
83
The CAN/CSA-S6-06 (cl. 8.15.4) defines the transfer length and the flexural bond length based
on the work done by Zia and Mostafa (1977):
(4-1)
(4-2)
where fsi is the stress in the prestressing strand just prior to transfer (which is typically 0.78fpu),
and fse is the effective stress in prestressing steel after losses (which is typically 0.74fpu). A
conservative estimate of development length is obtained by adopting these empirical equations
from the Code where lt = 436mm and lf = 803mm assuming fci = 59Mpa (based on 7-day cylinder
strength) and fps = fpy.
4.4.3 Improvement of Bond Strength due to High Strength FRC
Previous research (Hota and Naaman, 1995) has shown that the addition of hooked steel fibres
into a cementitous matrix can improve the bond properties of bonded steel. The presence of fibres
increases the confinement to resist the radial forces which increases the frictional bond on the
strand.
Recent studies have been carried out to investigate the increased bond properties on 15mm
prestressing strands due to the use of high-strength concrete and the influence of fibre
reinforcement. Kahn (2002) has investigated the influence of high-strength concrete ranging from
70 MPa to 100 MPa on the transfer and development lengths of 15mm pretensioning strands, and
showed that they are less than calculated by the AASHTO (1996) and ACI (1999) code provisions.
Chao (2006) has studied, among many other variables, the influence of hooked steel fibres on the
flexural bond of 15mm pre-tensioning strands for 76 MPa (11 ksi) concrete, but did not perform
tests to measure the transfer length.
It has been shown by Russell et al. (1993) that the average flexural bond stress based on ACI
(AASHTO) development equations can be calculated by solving equilibrium on a strand:
(4-3)
lt 1.5fsif'ci------db 117–=
lf 0.18 fps fse–( )db=
lf uf pps⋅ ⋅ fps fse–( )Aps=
uffps fse–( )Aps
lf pps⋅--------------------------------=
84
where is the perimeter of a seven-wire strand, and is the cross
sectional area of a seven-wire strand.
Chao (2006) performed several monolithic pull-out tests for specimens containing a concrete
matrix strength of 76 MPa (11 ksi) and 1% fibre volume fraction. The composite material
containing hooked steel fibres exhibited a peak bond strength of approximately 6.89 MPa (1000
psi). Assuming that the hooked steel fibre specimen responds similarly to the concrete being
considered for design, the flexural bond length may be determined on the basis of these tests. Since
the work done by Kahn (2002) investigated only the use of high-strength concrete with no fibres,
the transfer length determined in his study is too conservative for FRC. Thus, for analysis, the
benefit due to the Hoyer’s effect will be conservatively ignored, and the flexural bond strength
determined by Chao (2006) for FRC will be used for the transfer length. Assuming a bond strength
of 6.89MPa, the strand length to develop the effective stress of 0.74fpu and the yield stress of 0.9fpu
is 440mm and 535mm, respectively (Chao, 2006). Assuming a linear increase in the effective
prestress over the transfer length, the top slab prestressing is sufficient on the inside face of the
barrier for service loads with truck loading next to the barrier. It is now reasonable to proceed with
a pretensioned design with a detailed analysis of transverse loads.
4.5 Methods of Analysis
The lateral distribution of live load forces in the girder cross section must first be determined
in order to obtain the governing transverse loads. The maximum load effect can be obtained by
applying various load arrangements on the deck and then taking the maximum and minimum
envelope for all load arrangements. The distribution of load can be obtained in several ways:
conservative two-dimensional hand calculations using Pucher charts (Pucher, 1977) or Homberg
charts (Homberg, 1968), two-dimensional frame analyses of a cross section, three-dimensional
folded plate analyses, or three-dimensional finite element analyses (LoBuono, 1997). Pucher and
Homberg charts provide elastic plate influence surfaces for a variety of load and support cases.
Most support and load combinations are available that are typical for bridge design; however, the
boundary conditions which correspond to the cantilever overhang for a strutted box girder is not
available in these charts. This case pertains to a continuous plate that is supported continuously at
the web and supported only at discrete locations at the struts. In this design, a three-dimensional
finite element model using a grillage of beam elements is used for analysis to accurately model the
boundary conditions of the top flange.
pps43---πdb= Aps
736------πdb
2=
85
4.5.1 Grillage Analysis
In bridge engineering, grillage analyses have primarily been used to determine overall bridge
behaviour (Sotelino et al., 2004). The grillage analysis method is inexpensive and simple to
implement and comprehend, thus it has been favoured over finite element analyses in the field of
bridge engineering (Sotelino et al., 2004). Menn (1990) states that grillage models will yield
reasonably accurate sectional forces and deformations provided the member stiffnesses in flexure,
shear, and torsion have been properly chosen. Results of acceptable accuracy are always obtained
when the member stiffnesses are calculated from properties of the homogeneous uncracked
section.
The anticipated design of the top flange includes a thin continuous top slab that is stiffened by
discrete monolithic transverse ribs. The composite action of the slabs and ribs acting together is
similar to that of a multiple web T-girder. For the developed grillage model, the transverse beam
elements are assigned smeared sectional properties that are obtained directly from the cross
sectional geometry of a transverse strip of slab. The longitudinal beam elements are assigned
sectional properties due to a similar procedure, assuming a fixed number of longitudinal beams.
Menn (1990) provides a guideline to accurately model beam stiffnesses within a grillage model,
given in Figure 4-4.
86
Figure 4-4. Generalized frame element properties for grillage analysis (adapted from Menn, 1990)
For the table provided in Figure 4-4, G is the shear modulus of concrete, E is the elastic
modulus of concrete, A is the shear area of the beam, K is the torsional constant of the beam, ttf is
the thickness of the top slab, and EItot is the total flexural stiffness of the whole cross section before
dividing into discrete beam elements.
A three-dimensional finite element model was developed using the SAP2000 software to
determine the transverse bending moment distribution due to lane loading on the bridge. It was
found that the slab acts primarily as a one-way slab since the ribs provide a much higher stiffness
in the transverse direction. The presence of equally spaced strut supports at the cantilever tip also
diminishes the transfer of longitudinal bending in the slab with respect to a cantilever wing without
struts. Therefore, it is not necessary to model the full length of the bridge to determine the
Stiffness Transverse Beam (TB) Longitudinal Beam (LB)
Flexural where
Shear
Torsion
Δb Δb Δb Δb Δb Δb
beam width
longitudinalcross section
transversecross section
m transverse beams
n longitudinal beams
EITB EItotm-----------≅ EILB 1
12------E ∆l ttf( )3⋅ ⋅≅ ∆l ln 1+------------=
αGATB ∞≅ αGALB ∞≅
GKTB GKTB
m---------------≅ GKLB 0≅
87
transverse bending effects in the transverse direction. It was observed that modelling a short bridge
span with only 5 segments is sufficient to obtain accurate results in transverse bending for the top
slab. To provide sufficient resolution for the transverse bending moment demands due to moving
load cases, longitudinal beams were chosen to be 500mm centre-to-centre. The SAP2000 model is
illustrated below in Figure 4-5.
Figure 4-5. 3-dimensional grillage model
4.5.2 Frame Analysis
For static loads that are evenly distributed over the section such as dead load and prestress load,
a simple two-dimensional frame analysis of the cross section can be performed. By assuming a
transverse strip of box girder, sectional properties can be calculated and assigned to the appropriate
cross section components. Anchorage and deviation forces acting on the box girder due to the
prestress can be applied to the frame model as external applied loads to determine the distribution
of forces in the box girder. A frame model showing the transverse moment distribution due to a
non-concentric prestress is discussed later.
4.6 Comparative Study for Transverse Stiffness
Using the method of analysis described above, a grillage model is developed for the design of
the top slab in transverse flexure due to the application of live load. The CAN/CSA-S6-06 provides
section at strut
section between struts
longitudinal beamsspaced at 500mm
edge stiffening member along cantilever tip
transverse beamsspaced at 750mm
pin-ended strutsspaced at 3000mm
shaded area indicates web meshed withshell elements
88
no specific guidelines pertaining to maximum acceptable deflections for deck slabs in transverse
bending. To provide a basis for validation, the design for transverse stiffness is carried out by
allowing similar maximum deflections as that of an existing reference structure. The reference
structure chosen for comparison is the standardized AASHTO-PCI-ASBI precast box girder
(Prestressed/Precast Concrete Institute, 1997). This reference structure allows for a direct
comparison of live load deflections for the same arrangement of design lanes; however, it is
possible that this chosen standard is conservative. A similar grillage model, making consistent
modelling assumptions, is created for the AASHTO-PCI-ASBI box girder for direct comparison
of transverse stiffness to the proposed segmental strutted box girder.
4.6.1 Cross Sections
There are two types of standard AASHTO-PCI-ASBI box girder sections - one type is intended
for span-by-span construction, and the other is intended for cantilever construction (shown in
Figure 4-6). With regard to the top slab proportions, the box girder intended for cantilever
construction has a much deeper haunch at the junction of the webs. This haunch tapers to the
typical thickness of 225mm near the quarter point of the interior slab span and approximately the
midpoint of the cantilever wing. The depth of the top slab for this design is not governed by the
required transverse stiffness of the deck slab to satisfy minimum slab deflections due to transverse
bending, rather it is governed by the spatial requirement for internal bonded tendons.
The cross section chosen for comparison is the 2400mm deep box girder intended for span-by-
span construction (Prestressed/Precast Concrete Institute, 1997). This section is chosen since the
top flange thickness for span-by-span constructed girders is not governed by the inclusion of
bonded tendons as it is with cantilever girders with internal tendons. Since the span-by-span girder
top flange is thinner, it provides a better comparison for minimum transverse stiffness. The 12m
wide standard section has an interior span of 6.54m and a cantilever span of 2.73m.
89
Figure 4-6. Comparison of top flange thicknesses of AASHTO-PCI-ASBI standard box girder examples (adapted from Prestress/Precast Concrete Institute, 1997)
The proposed box girder has an interior span of approximately 5m from centre-to-centre of web
at the centroid elevation of the top flange. The cantilever span is approximately 3.5m from the
centre of the web to the cantilever tip. The cantilever span is continuously supported at the web and
supported by discrete struts equally spaced at 3m centre-to-centre at the cantilever tip. The slab
thickness is chosen to be 85mm and the depth of the ribs is chosen to be 200mm from the top
surface. Transverse ribs are 500mm wide and are spaced at 1500mm centre-to-centre. Figure 4-7
displays the top flange dimensions for a typical segment.
b) AASHTO-PCI-ASBI standard section for span-by-span construction
a) AASHTO-PCI-ASBI standard section for cantilever construction
90
Figure 4-7. Cross section dimensions for typical segment
4.6.2 Material Definitions used in Analysis
The recommended minimum strength concrete for standardized AASHTO-PCI-ASBI
segmental box girders is 5000psi (34.5MPa) (Precast/Prestressed Concrete Institute, 1997).
Normal concrete of this strength corresponds to a modulus of elasticity of approximately
26400MPa (Collins et al., 1997). The standard PCI section is assumed to use the recommended
minimum strength concrete for the determination of minimum transverse stiffness. The proposed
girder is expected to be fabricated with high-strength FRC similar to the concrete cylinders tested
at the University of Toronto which were described previously in Chapter 2. The modulus of
elasticity observed in the laboratory tests for the FRC is approximately 43680MPa. Figure 4-8
displays the concrete models considered for the comparative study. A linear elastic analysis
considering only the uncracked stiffness is performed for the SLS investigation.
Figure 4-8. Assumed stress-strain material properties for investigation of transverse stiffness
08-
07-
06-
05-
04-
03-
02-
01-
0400.0-300.0-200.0-100.0-0
FRC compression cylindersf’c = 74.6 MPaEc = 43680 MPa
PCI recommended minimumstrength concretef’c = 34.5 MPaEc = 26400 MPa
fc [MPa]
εc [mm/mm]
91
4.6.3 Loading
The girder considered for the proposed design has a total top slab width of 12m. The CAN/
CSA-S6-06 requires that a deck width of 10m to13m must be checked for 2 or 3 design lanes,
whichever causes the worst effect (cl. 3.8.2). This distribution of design lanes is displayed in
Figure 4-9. The 2 lane model developed comprises two design lanes side-by-side centrally located
on the bridge to maximize the positive bending moment and deflection in the interior span of the
box girder. The 3 lane model comprises a centre lane coincident with the bridge axis, and two
exterior lanes. The exterior lanes are located so that the centre of the truck tire load is 0.5m from
the cantilever tip. The CAN/CSA-S6-06 requires that the centre of the truck tire load be positioned
0.3m from the inside face of the barrier. Thus, the live load analysis assumes a barrier width is
0.2m, which is conservative.
The truck load used for design considers a CL-625-ONT truck defined in the CAN/CSA-S6-
06 (Annex A3.4). A Dynamic Load Allowance (DLA) of 0.25 was used in the analysis which
corresponds to multiple-axle loading. The CAN/CSA-S6-06 Lane Load (cl. 3.8.3.3) comprises a
3m-wide uniform load of 9kN/m superimposed with 80% of the CL-625-ONT truck load (cl.
3.8.3.2) without the DLA multiplier. Figure 4-9 displays the transverse load arrangements
considered in the analysis. The load arrangement shown is applied to both the proposed box girder
and the standard PCI girder.
Figure 4-9. Live load models for transverse load analysis
4.6.4 Deflections
Maximum live load deflections were determined through the use of a three-dimensional
grillage analysis and moving load cases corresponding to the live load models described
previously. The maximum downward deflection due to vertical applied load for the AASHTO-
PCI-ASBI was observed to be approximately 7.0mm cantilever tip and 5.9mm in the middle of the
a) 3 design lane model b) 2 design lane model
92
interior span. The maximum deflection in the interior span corresponds to the two-lane model
where both traffic lanes are placed as close as possible to the centre of the bridge within the design
lanes. The maximum tip deflection of the cantilever also corresponds to the two-lane model the
traffic lanes are acting on the furthest extents of the bridge deck. The centres of the traffic wheels
should be placed 0.3m from the face of the barrier (cl. 3.8.4.3).
Figure 4-10. Vertical slab deflection envelope due to live load, AASHTO-PCI-ASBI box girder
Strictly using the transverse and longitudinal beam properties for the typical ribbed cross
section yields large deflections at the cantilever tip between the struts. As the thin top slab is
flexible in the longitudinal direction, the slab transfers load primarily in the transverse direction
but large deflections occur for loads applied near the cantilever tip between the strut supports.
Therefore, the inclusion of a stiff longitudinal member at the cantilever tip improves the flexural
stiffness in the longitudinal direction significantly. The edge stiffening beam is given dimensions
250mm thick by 400mm wide (shown in Figure 4-11).
Figure 4-11. Edge beam stiffening
For a girder with no continuous edge stiffening beam, the slab deflection due to live load at the
cantilever edge between struts is observed to be approximately 8mm (shown in Figure 4-12).
However, with the edge stiffening beam incorporated into the cantilever tip, the live load
deflections are reduced markedly between struts from 8mm to approximately 1.5mm. The
deflection envelope for the improved system is shown in Figure 4-12 below. For the proposed box
Max/Min deflection
8-
4-0
4Δ [mm]
93
girder, the maximum vertical deflection at the middle of the interior span is 7.2mm and the
maximum deflection on the cantilever wings is 3.5mm.
Figure 4-12. Vertical slab deflection envelope due to live load of proposed box girder: a) thin slab stiffened with transverse ribs only, b) thin slab stiffened with transverse ribs and longitudinal edge beam
The maximum deflection of the proposed box girder is comparable to that of the standard
AASHTO-PCI-ASBI box girder. Therefore, based on this reference structure, the transverse
bending deflections of the proposed box girder is acceptable.
4.7 Prestress Moments
The prestressing arrangement for the transverse design includes equally spaced 15mm strands
placed in the slab with 40mm top clear cover, and additional 15mm strands equally spaced in the
rib with 30mm bottom clear cover. This arrangement of steel was shown previously in Figure 4-1.
The resultant of the prestress force applied to the section is not exactly concentric, so a prestress
moment is induced in the transfer length of the strands. The prestress moment causes negative
curvature at cantilever tip and induces a compressive reaction on the struts since the struts provide
a redundant reaction to the indeterminate system. This arrangement of steel counteracts the live
load bending moments in the cantilever wing when load is placed on the cantilever, and it provides
a clamping force on the strut in the unloaded condition after fabrication. Since the strutted box
girder is a statically indeterminate frame system, the secondary prestress moments due to slab
prestressing must be determined to evaluate the total moments on the slab section. Assuming a
large deflections between strutsMax/Min between strutsMax/Min at strut
8-
4-0
4Δ [mm]
8-
4-0
4Δ [mm] improved stiffness
at the cantilever tipMax/Min between strutsMax/Min at strut
a) no edge beam
b) with edge beam
94
linear elastic response, a frame analysis is conducted to determine the secondary prestress moments
due to a unit load. The prestress moment is assumed to vary linearly over the transfer length.
Figure 4-13. Moment distribution due to non-concentric prestress applied at the transfer length
4.8 Service Loads
Using the live load demands from the grillage analysis, and the applied prestress moments from
the frame section analysis, the prestress design can be determined to resist service loads. By
placing 25-15mm strands equally spaced at 120mm c/c in the slab, and a total of 12-15mm strands
equally spaced at 60mm c/c in the ribs, adequate capacity is obtained for transverse flexure. The
transverse design is not fully prestressed for service loads. For the interior span, the maximum
moment exceeds the cracking stress at the extreme fibre of the cross section which is at the
elevation of the bottom fibre of the transverse ribs. Cracking will occur within the inside of the box
girder where the concrete is protected from atmospheric conditions, and the crack depth is
superficial since the cracking stress, fcr = 5.15MPa, is only exceeded within the bottom 9.3mm of
the transverse rib. The CAN/CSA-S6-06 limits the prestressing steel stress range to 125MPa to
satisfy fatigue requirements (cl. 8.5.3.2). The maximum and minimum stresses experienced by the
bottom prestressing strands at the critical bending location causes a stress range of only 83.7MPa;
therefore, fatigue does not present a problem for the tendon design. The total moments on the
transverse section for serviceability are shown in Figure 4-14. The transverse design based on
serviceability performance has been found to be adequate for ultimate limit states.
-0.120Mp
-0.265Mp
-0.385Mp
Mp Mp-0.265Mp
-0.385Mp
-0.024Mp -0.024Mp
lt lt
MpMp
a) applied prestress moment b) bending moment distribution
95
Figure 4-14. Total transverse bending moments for transverse flexure
The linear elastic stress ranges in the thin slab due to the total moments applied to the cross
section are displayed in Figure 4-15. The maximum stress experienced by the slab occurs at the
bottom face of the slab at the web support, where the compression stress is 38.4Mpa. The average
transverse stress in the top slab due to transverse prestressing is 15.6Mpa, assuming a tendon stress
of 0.6fpu to account for long-term losses.
dead load
bottom fibre cracking
live load
prestress
250
200
150
100
50
0
-50
-100
-150
-200
-250
M [kNm]
96
Figure 4-15. SLS top slab stress ranges at the extreme fibre due to service loading
4.9 Cross Section Design and Layout
4.9.1 Transverse Rib Proportions
The typical transverse cross section for one segment is shown in Figure 4-16. The layout of ribs
has been chosen to include one central rib of 500mm in width, and two exterior ribs at the segment
faces where are 250mm in width. This geometry facilitates a local thickening at the joint where
shear keys may be incorporated in the bulkhead detailing. Also, the layout allows for a seamless
transition between segments when constructed in the final state since the two exterior ribs placed
-40
-30
-20
-10
0
10
-40
-30
-20
-10
0
10
a) top fibre transverse stressesfc [MPa]
fc [MPa]
c) cross section
-22.6MPa -15.1MPa
-0.32MPa
-14.4MPa
-38.4MPa
b) bottom fibre transverse stresses
-5.15MPa
-29.6MPa
-0.63MPa
97
together at the joint form an equivalent 500mm rib. The visual appearance in the final state is
evenly spaced ribs of 500mm in width, spaced at 1500mm centre-to-centre.
Figure 4-16. Longitudinal section for a typical segment
4.9.2 Web Spacing
The maximum and minimum transverse bending moments that have been shown in Figure 4-
14 indicate that the web spacing of the box girder can be further optimized since there is a
substantial reserve capacity before cracking in negative bending, and a smaller reserve capacity in
positive bending. By shifting the location of the webs towards the centreline of the bridge, the
maximum positive moment at middle would be decreased and the maximum negative moment at
the webs would be increased. This alternative was considered; however, for the span of 90m a
practical limitation governs the design of the web spacing. As the girder is constructed by the
cantilever method with 3m length segments, 15 pairs of tendons are required to be placed inside
the box girder at the support. All cantilever tendons are desired to be placed in one layer below the
top flange to maximize the internal lever arm. To sufficiently guide the cantilever tendons inside
the box girder, a minimum width of the box is required at the elevation of the cantilever tendons.
For internal bonded post-tensioned tendons, the minimum clear distance between tendons is
40mm (cl. 8.14.2.2.2). For external unbonded tendons, the minimum clear distance has been
chosen to be 60mm since high-density polyethylene (HDPE) sheathing is required to pass through
rigid steel pipe at tendon guide locations. The web spacing requirement to fit all cantilever tendons
in the box is approximately 5m (Figure 4-17). For shorter spans with less cantilever tendons, the
98
interior span and cantilever span of the top flange can be optimized better with respect to the
transverse bending moment distribution due to live load.
Figure 4-17. Web spacing requirements
4.10 Design for Barrier Impact Loads
The barrier design provided satisfies the ultimate limit state barrier impact for a Performance
Level 2 (PL-2) barrier defined in the CAN/CSA-S6-06 (cl. 12.4.3.2). The detail adequately ties the
barrier into the structural system with sufficient ultimate capacity. The barrier detailing provides a
feasible flow of forces that is described by the truss model in Figure 4-18.
Figure 4-18. Truss model for basis of barrier reinforcement
The moment produced by the horizontal impact load on the barrier is assumed to be taken
primarily by flexure of the transverse ribs. Between the ribs, the concentrated moment due to
impact is transferred to the transverse ribs by torsion in the edge beam. The torsional resistance of
the edge beam is provided by the closed stirrup geometry of the barrier reinforcement (Figure 4-
19). The barrier should be cast to encapsulate the edge of the deck slab, forming a drip nose at the
edge of the deck. A similar detail for barrier design was used for the Confederation bridge. This
detail provides concrete cover to protect the pre-tensioning strands against corrosion and exposure
a) inclined webs to increase interior span b) smaller interior span insufficient
F
99
to de-icing salts. The drip nose also offers a better solution compared to the typical drip groove for
proper water management to protect the superstructure.
Figure 4-19. PL-2 barrier detail for impact loading
15M @ 250required for impact resistance
Cross Section
drip nose detail for properwater runoff and end protectionfor pretensioning strands
100
101
5.0 SHEAR
This chapter begins by briefly discussing a background on strut-and-tie models. Section 5.2
describes funicular load path of compression stresses in a traditional prestressed concrete beam and
describes the funicular load path for the proposed segmental girder with curtailed prestressing
tendons. Section 5.3 describes one possible method of decomposing the resultant compression
stress in the girder into a parallel chord truss model. Section 5.4 describes an alternative method of
decomposing the resultant compression stress in a girder to account for the inclined spreading of
forces which contribute to the arching effect of carrying shear forces. Section 5.5 describes the
preliminary shear model chosen for design of the web reinforcement in the proposed box girder.
Finally, the designed web reinforcement is presented. For the design of web reinforcement, there
are two components to be considered: a local spreading of forces due to the intermediate anchorage
of external unbonded tendons, and the global spreading of forces to safely transfer the applied
vertical forces to the supports. Within this chapter, only the global spreading of forces within the
web are discussed. The design for web reinforcement due to local spreading at anchorages is
discussed later in Chapter 6.
5.1 Introduction to Strut-and-Tie Models
The strut-and-tie model, originally introduced by Ritter (1899), is a powerful engineering tool
developed to describe the complex flow of forces in a simple way for cracked reinforced concrete
beams (Figure 5-1). Strut-and-tie models comprise an arrangement of discrete compression struts
and tension ties connected at pin-ended nodes to describe a state of internal force equilibrium.
Compression struts and tension ties are representative of physical concrete elements and
reinforcing steel, respectively. This simple engineering model provides a great benefit to the
designer as it gives physical significance to the calculation of forces in concrete structures. Truss
models provide a visual flow of forces and can be applied to a wide range of multi-dimensional
geometries for varying load conditions (Marti, 1985).
Figure 5-1. 45 degree truss model (adapted from Ritter, 1899)
Schlaich et al. (1989) provided a comprehensive treatise to the application of truss models in
reinforced concrete and prestressed concrete structures. In the uncracked state, when elastic
concrete strains are small, a concrete girder reinforced with bonded prestressing steel behaves like
the system with unbonded tendons since there is no significant local increases in tendon stress due
to bond action (shown in Figure 5-2). For any cross sectional shape, the resultant compressive
force must assume the funicular shape of the net load to satisfy internal equilibrium (Gauvreau,
1993). After the formation of a crack, bonded reinforcing steel bridging the crack increases in
stress as the mechanical bond is activated. In truss models, this increase in steel stress is
represented by a tension tie at the level of the bonded steel which is equilibrated by diagonal
compression struts and vertical tension ties that connect to the compression chord (shown in
Figure 5-2). The effect of these diagonal and vertical forces on the compression chord is a
modification to the funicular shape to equilibrate the internal forces.
Figure 5-2. Possible truss models for girders with unbonded and bonded prestressing steel (adapted from Gauvreau, 1993)
For structural systems with unbonded steel and no bonded longitudinal reinforcement, in which
tendons are attached only at discrete anchorage locations, there is no mechanical bond present to
provide a local increase in tendon stress at the crack. Since the sectional compatibility of strain in
Q Q
P P
Q Q
Q Q
P P
Q Q
a) beam reinforced with unbonded prestressing only
b) beam reinforced with longitudinal bonded steel
cracked regionuncracked uncracked
tension tie due to bond
resultant funicular shape
102
the concrete and strain in the steel are uncoupled for structures with unbonded tendons, there is no
mechanism for a local transfer of axial force at any location away from the anchorages of an
unbonded tendon. Given this, the classical two chord truss model that depicts an increasing tension
force in the tension chord due to bond action can not reasonably explain the force equilibrium in a
girder prestressed only with unbonded tendons.
5.2 Funicular Load Path for Girders Prestressed with Curtailed Tendons
One unique characteristic of prestressed segmental concrete bridges built by the cantilever
method is the curtailed arrangement of prestressing tendons. As the intrinsic nature of the
construction scheme demands, prestressed tendons must be anchored at each stage of construction
when a new segment has been placed. The implication of this tendon arrangement is that the axial
prestress force in the girder is not constant along the length. To the authors knowledge, this type
of structural system is not extensively described in the literature as it is with prestressed girders
with a constant prestress force.
As a system with curtailed external unbonded prestressing is uncommon, it is easiest to first
envision the funicular load path through the simple familiar example of a simply supported beam
(Figure 5-3). The funicular shape of the compression chord illustrates that the horizontally applied
prestress force is inclined due to the deviation of this force at the support. The inclined compression
force in the concrete provides capacity for the girder to transfer shear from the applied loads to the
support reaction. The implication of the inclined force displayed by this simple model is that there
is no requirement for vertical reinforcing steel in the girder.
To compare the effect of a varying prestress force along the length of the girder, the simply
supported girder is examined for two cases: full-length unbonded tendons, and curtailed unbonded
tendons. It can be seen that the compression force that is applied by the intermediate anchorages at
the level of the prestressing steel must spread through the web and resolve at the compression chord
of the funicular load path. This flow of forces presents a load path that is fundamentally different
from the case with full-length tendons as it requires vertical steel reinforcement to satisfy
equilibrium. This vertical reinforcement is not required for the action of shear, rather it is required
to provide adequate spreading of forces to move the concentrated load applied at the anchorage to
the funicular compression chord. The vertical steel reinforcement required for the anchorage of
curtailed unbonded tendons is analogous to corbel reinforcement that is required to tie a heavy
point load into a support. The accumulation of prestress forces applied at the anchorages increase
103
the compression force in the compression chord at node locations where the diagonal compression
members meet the compression chord. Since all girders shown in Figure 5-3 have the same
dimensions and Q is the same for all girders, the force in the compression chord at midspan must
be equal to P to provide necessary member resistance.
When comparing the curtailed unbonded prestressing arrangement to the case with bonded
prestressing steel (shown in Figure 5-3b and Figure 5-3c), the models display a similar flow of
forces to satisfy internal equilibrium. In both cases, diagonal compression struts form due to the
increase in steel force at the level of the prestressing tendons which must be equilibrated with
vertical reinforcing steel in the web. One major difference between these models is that there is no
tension chord at the level of the prestressing steel for the case with unbonded tendons since there
is no local bond action between steel and concrete.
Figure 5-3. Alternative girder designs to resist applied load Q
Another major difference between the case with bonded steel and unbonded tendons is that the
funicular shape of the compression chord for the case with unbonded tendons is dependent on the
arrangement of horizontal prestress forces. For the case with bonded steel, the elevation of the
compression chord is chosen for the cracked region of the beam; however, for the case with
unbonded tendons the elevation of the compression chord is defined by the resultant location of
compressive force in the section. This concept is illustrated in Figure 5-4 for two different girders
with a different arrangement of prestressing forces. It can be seen that the shape of the compression
Q Q
0.66P 0.66P
Q Q
c) beam reinforced with longitudinal bonded steelincrease in member resistance due tolocal increase in steel stress
Q Q
0.66PQ Q
b) beam reinforced with curtailed unbonded tendons
0.21P 0.13P 0.66P0.21P0.13P
increase in member resistance due to intermediate anchorage force
Q Q
P P
Q Q
a) beam reinforced with full-length unbonded prestressing tendons
resultant funicular shape
104
chord, or ‘spine’, is modified for a girder with different applied anchorage forces for the same
vertical applied load Q. Note that for both girders to provide the same member resistance at
midspan, the compression chord at the midspan has a compressive force of P. The truss model for
this girder is statically determinate, and the geometry of the compression chord can be computed
by solving for required forces and strut angles through the use of method of joints for trusses or
through the use of graphic statics.
Figure 5-4. Effect of prestress arrangement on funicular compression chord
The extension of this concept to the proposed 90m cantilevered girder with curtailed cantilever
and continuity prestressing tendons is given in Figure 5-5. During the cantilevering process, the
compression spine is developed through the progressive building of segments and anchorage of
tendons. As it was with the simply supported case, the geometry of the truss model is not constant,
rather the geometry is variable depending on the prestress and applied forces. Thus, the shape of
the truss model is unique for every stage of construction and for all combinations of applied load.
Uniformly distributed dead load is assumed to be lumped into discrete point loads based on the
tributary width and positioned at anchorage locations.
For the continuous girder in the final stage (after stressing continuity tendons), the structure is
statically indeterminate due to continuity at midspan. To solve for the truss geometry, the total
bending moment at midspan must first be computed using a frame analysis to provide the boundary
condition for the truss model at midspan. The elevation of the compression spine at midspan can
be calculated by determining the eccentricity of the resultant compression force from the neutral
axis of the girder. This is calculated by dividing the total bending moment at midspan by the
effective prestress force in the girder at midspan. Once this boundary condition is solved, the truss
geometry can be computed using the same methods as previously performed for the simply
Q Q
0.81PQ Q
b) beam reinforced with alternate arrangement of curtailed unbonded tendons
0.10P 0.09P 0.81P0.10P0.09P
spine
Q Q
0.66PQ Q
a) beam reinforced with one arrangement of curtailed unbonded tendons
0.21P 0.13P 0.66P0.21P0.13P
spine
105
supported girder. A half-span elevation of the funicular truss model immediately after installation
of the continuity tendons is shown in Figure 5-5b.
Figure 5-5. Funicular shape of compression spine for continuous cantilever girder with curtailed tendons
Assuming a symmetrical uniform distributed load is applied to the continuous girder system,
the ultimate limit state for the chosen prestress arrangement is shown in Figure 5-5c. A rotation of
the compressive ‘spine’ about the inflection point is observed. The limiting mode of failure is
crushing of the bottom flange at the abutment due to negative flexure (assuming sufficient web
AbutmentQ/2 Q Q Q Q Q/2
Cross Section
a) Construction stage - 5 segments cantilevered from the abutment
b) Half span elevation with all cantilever and continuity tendons installed
Q/2 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q/2Abutment CL Midspan
Pspine
Pspine
P1t P2t P3t P4t P5t P6t P7t P8t P9t P10t P11t P12t P13t P14t P15t
P6b P7b P8b P9b P10b P11b P12b
P1t P2t P3t P4t P5t
V
Pspine
V
V
1.0
c) Half span elevation at ultimate load
Qu/2 Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu Qu/2Abutment CL Midspan
Pspine
Pspine
P1t P2t P3t P4t P5t P6t P7t P8t P9t P10t P11t P12t P13t P14t P15t
P6b P7b P8b P9b P10b P11b P12b
(Vs/Vu)
d) Transverse demand envelope for reinforcing steel at the ultimate limit state shown in (c) { { Cantilever tendons govern Continutity tendons govern
Jump caused by change in tendon unit at P2t Change in tendon
unit at P9b
Crushing of bottom flange
106
reinforcement has been provided to deviate the prestress forces). The transverse demands given by
the model combine an influence of the applied loads and the prestress forces. For the tendon
arrangement assumed in this shear example, the prestress tendon units increase in size for segments
that are nearer to the abutment. This change in prestress force is reflected in the transverse demand,
since a larger tension tie is required to pull the applied prestress force into the section at the location
of the resultant force (Figure 5-5d).
One major deficiency in simple truss models that consider only the resultant funicular
compression force in the concrete is that there is total disregard for the cross sectional shape of the
girder. For the examples discussed above, all truss models have been determined purely on the
basis of graphic statics and equilibrium, and these models give no consideration for the spreading
of forces into predominant flanges of the box girder cross section. To achieve a more realistic
representation of the location of forces within the cross section, the resultant funicular compression
chord should be decomposed into a more refined model having two or more chords that provide a
better representation of the location of forces within the concrete section.
5.3 Parallel Chord Truss Models
One method of decomposing the resultant force in a concrete section with known flexural
compression and tension regions is to divide the resultant compression force into two parallel
chords forming a classical truss with a horizontal upper and lower chord with diagonal struts and
vertical ties in the web. For box girders and other members with predominant flanges, the chords
are typically chosen to be located at the flange elevations. A shortcoming of this chosen geometry
is that the benefit of arching action due to the continuous inclined forces following the funicular
shape is dismissed since the horizontal chords do not provide any component of force in the vertical
direction to resist shear. Rather, shear in the girder is resisted through the discontinuous load path
of diagonal compression struts and vertical tension ties in the web. This model provides one valid
explanation for the possible flow of forces in a girder to resist shear, but it does not account for the
benefit of arching action in girders to resist shear.
Schlaich et al. (1989) have presented a possible flow of forces for flanged sections prestressed
with unbonded tendons where the top and bottom chord are fixed in position at the level of the
flanges (Figure 5-6). According to Schlaich et al. (1989), the truss model consisting of a top and
bottom compression chord already develops when the resultant force is within the kern zone of the
girder section due to the large concentrated forces in the flanges. For the fully prestressed section
107
shown in Figure 5-6, both the top and bottom chord are in compression for regions where there is
no flexural cracking. As the forces in the truss must resolve to the funicular load path shown, a
change in force must take place in the top and bottom chord. Unlike a classical truss model which
depicts a changing tension force in the bottom flange due to bond action, Schlaich et al. (1989)
demonstrates that a change in force of the bottom chord is also possible through the exchanging of
compression forces between flanges.
Figure 5-6. Strut-and-tie model for a fully prestressed flanged section. a) simplified model; b) through d) detailed model of web, top flange and bottom flange, respectively (adapted from Schlaich et al., 1989)
The required web steel that is determined for parallel chord models depends on the angle of
diagonal compression struts in the web. Steeper angles of compression in the web result in larger
vertical forces in the tension ties and shallower angles of compression result in smaller vertical
forces in the tension ties. For non-prestressed concrete members, Menn (1990) recommends that
the constant angle of compression in the web can be chosen by the designer to be between and
, which is within of the principal stresses for the elastic stress state. This choice of angle
for web compression stress is consistent with current Swiss design practice which is based on
plasticity (SIA 262, 2003). Since the plastic redistribution of forces may require large crack
openings to occur at the ultimate limit state, the shear capacity is determined purely on the shear
resistance of the steel reinforcement.
Centroid of forces coming from the web
Q
P P
Q
Q
P P
Q
P P
a)
b) web
c) top flange
d) bottom flange
30°
60° 15°
108
Alternatively, the Modified Compression Field Theory (MCFT) provides a rational method to
predict the angle of diagonal compressive stresses, θ, from the strain conditions in the web (Bentz,
2006). Through the use of this method to determine θ, a variable-angle truss model can be
developed to more accurately describe the state of stress in the web (shown in Figure 5-7). This
model displays a varying angle of compression stress in the web which is equilibrated by vertical
tension ties. The pattern of diagonal forces in the web is regular everywhere except near the
supports and at concentrated loads (Gauvreau, 1993). In these regions, this local flow of forces
which have a radial spreading of compression stress are called compression fans (shown in
Figure 5-7). The diagonal compression struts require a change in force in both the top and bottom
flange. For the case with internal bonded steel, the bottom flange can take a tension force due to
the action of bond. For the case with only external unbonded tendons, the bottom flange can only
take compression stresses; therefore, this model is only valid in regions where the full depth of the
section is compressed and an exchange of compression forces between flanges can take place, as
shown in the previous model by Schlaich et al. (1989) in Figure 5-6.
Figure 5-7. Variable angle truss model (adapted from Collins et al., 1997)
Shear design using the MCFT is not based on plasticity, but rather it predicts an ultimate
resistance that reflects the actual state of deformation in the web. In this way, the shear resistance
can be determined due to the reistance provided by web reinforcement and aggregate interlock of
the concrete at a crack. The shear resistance of aggregate interlock is significant when the crack
widths in the web are small, and it diminishes as the crack widths in the web increase. Therefore,
the design of web reinforcement must correspond to the diagonal stress state that is related to the
shear resistance of aggregate interlock in the concrete. This requirement provides a method to
determine the shear capacity of reinforced concrete sections, but prevents the designer from
choosing the angle of web stress for the ultimate limit state. The CAN/CSA-S6-06 shear provisions
follow a design approach using the MCFT which is presented in Appendix C.
The parallel chord model used for the CAN/CSA-S6-06 shear provisions does not account for
an arching contribution due to the flow of inclined compression forces following the funicular
shape, but rather it accounts for the presence of prestressing by simply offsetting the force in the
θ compression fan
109
tension flange by including the compression force due to all prestressing tendons that are on the
flexural tension side of the girder. Alternative models exist that do consider the arching effect of
compressive stress through a modified decomposition of the funicular compression chord. These
model types will be described next.
5.4 Arching Models
The decomposition of the resultant funicular compression chord into the parallel chord model
involved predetermining the distance between the chords which means that the magnitude of force
in the chords must change to satisfy equilibrium. Another possible way of decomposing the
resultant compression chord is to split the resultant force into two compression chords with
predetermined values of compression force for each chord. By setting the force in each of the
chords, the elevation and slope of the compression chords must not be fixed at the flanges to satisfy
equilibrium, rather the geometry of the compression chords must be positioned at elevations that
are statically equivalent to the resultant force.
It is possible to show this spreading of forces through the use of a two-chord model that
demonstrates the bottle shaped flow of forces due to applied point loads. Schlaich et al. (1989) have
presented this possible flow of forces for a rectangular shaped cross section (Figure 5-8). This
model gives account for the benefit of arching action in concrete due to the continuous inclined
compression stresses in the chords. The vertical applied loads to the girder are transferred to the
reactions by the inclined component of force in the compression chords. The model also
demonstrates that vertical web reinforcement is necessary to provide tie forces for the compression
chords to deviate from the line of resultant force.
Figure 5-8. Arch spreading of forces for a rectangular section (adapted from Schlaich et al., 1989)
F
P P
Fa)
F
P P
Fb)
110
Although this model illustrates the arching effect of inclined compression stresses in the
concrete due to prestress forces applied at the girder ends, Schlaich et al (1989) give no explicit
discussion on how to choose the chord geometry of this spreading model.
5.4.1 Tied Arch Model (Noshiravani, 2007)
Noshiravani (2007) has recently performed a study that describes a rational approach to
defining the geometry of a similar spreading model for concrete girders prestressed with external
unbonded tendons without any bonded continuous longitudinal reinforcement. By definition, the
external unbonded tendons must have a constant force between the anchorage locations. Thus,
since there is no mechanical bond between steel and concrete, no local increase in steel stress can
occur in the prestressing tendons. Therefore, in her analysis, the flow of forces in the girder does
not include a bottom chord at the elevation of the tension flange.
The sectional model proposed by Noshiravani (2007) involves splitting the resultant
compression force into two chords, giving each chord an equal magnitude of force. The geometry
of the tied arch model is determined on the basis of linear elastic stress diagrams at several
locations along the length of the girder. Assuming the plane section hypothesis, Noshiravani
(2007) locates the elevation of the compression chords by analyzing several planes of strain at
various sections of the girder and calculates the centroid of force for each chord. Schlaich et al.
(1989) originally proposed a two-chord arching model for the use with rectangular sections only;
however, Noshiravani (2007) extends the application of this concept of a two-chord arching model
to other sectional shapes including predominant flanges. Since sectional force is determined as a
product of the cross sectional stresses and the concrete area of the girder, the shape of the cross
section can be accounted for using this method of plane sections. An illustration of the cross
sectional influence on the chord geometry and the resulting transverse demands are shown below
in Figure 5-9.
Noshiravani (2007) applies the tied arch model between the supports within the B-region of the
simply supported girder. Outside of the support within the D-region of the anchorage zone,
Noshiravani (2007) provides a simple strut-and-tie model which satisfies St. Venant’s principle.
At the support, the elevation of the compression chords are determined on the basis of the elastic
stress diagram which provides the boundary conditions for the truss model within the anchorage
zone. Providing boundary conditions for D-regions based on elastic stress diagrams is typical for
the general design of truss modelling (Schlaich et al., 1989).
111
.
Figure 5-9. Comparison of tied arch model for a rectangular section and a flanged section (adapted from Noshiravani, 2006)
This new application of a two-chord arching model based on the elastic stress diagram using
plane sections for the B-region provides a straightforward method to determine the shape of the
flow of forces to describe spreading in a girder. However, to provide a complete consideration for
the possible flow of forces in a prestressed section with wide flanges, it is of interest to compare
the results of Noshiravani (2007) to an alternative parallel chord model. On this basis of
comparison, a design model for the proposed box girder may be decided.
For the experimental study performed by Noshiravani (2007), the girder was loaded in static
four-point bending by means of a force-controlled machine head. The load on the girder was
increased incrementally at several load stages to assess the condition of the girder. Strain gauge
measurements in the vertical stirrups along the length of the girder and crack formations in the
webs of the girder were recorded at each load stage. The two load stages under consideration
correspond to: a) the load corresponding to the occurrence of flexural cracking in the bottom flange
of the girder (Load Stage 4), and b) the load corresponding to a load close to the ultimate strength
of the girder (Load Stage 7). The dimensions of the girder tested are shown in Figure 5-10 and the
typical cross section is shown in Figure 5-11. For a full description of the experimental program,
the reader is referred to Noshiravani (2007). The strain gauge measurements of the transverse steel
(Vs / Vu)
Flanged sectionRectangular section
1.0
P P
Q
P P
Q
CompressionTension
Model applied only in B-regionD-region D-region
boundary conditions for D-region provided by model
112
and the predictions of this two-chord model for these two load stages are presented in Figure 5-12
and Figure 5-13.
Figure 5-10. Girder dimensions (adapted from Noshiravani, 2007)
Figure 5-11. Girder cross section and web reinforcement (adapted from Noshiravani, 2007)
Figure 5-12. Tied arch model for load stage corresponding to flexural cracking (adapted from Noshiravani, 2007)
D4 Cold Drawn Wire Propertiesfy = 530MPaAs = 24.2mm2
05
1015
0200400600800
1000
εv [mm/m]
fv [MPa]
tied arch model predictionactual transverse stress
unaccounted demand
c) Calculated transverse stress and tied arch model prediction
b) Measured transverse strains in vertical stirrups
a) Noshiravani tied arch model
320kN 320kN
2835kN 2835kN
320kN 320kNindicated stirrup yield
yield strain
yield stress
113
Figure 5-13. Tied arch model for load stage corresponding to ULS (adapted from Noshiravani, 2007)
The observed relationship between the predicted transverse stress of the tied arch model and
the actual transverse stress measured during the test for the load stages shown in Figure 5-12 and
Figure 5-13 show that the tied arch model underestimates the transverse demand for steel. This is
especially evident at ULS where the model predicts virtually no transverse stress at a location
where the stirrups are yielding. One possible explanation for this discrepancy is provided by
Schlaich et al. (1989) where top and bottom parallel compression chords develop for flange
sections when the resultant force is within the kern zone of the girder section with predominant
flanges. To investigate the possibility of this alternative flow of forces, a comparison of
Noshiravani’s (2007) test results will be performed in relation to the parallel chord model described
by Schlaich et al. (1989).
Since the girder has been already designed by Noshiravani (2007), it is possible to compare the
crack patterns of the test to the crack direction that would be predicted by a parallel chord model
assuming yielding of the stirrups at ULS (which was observed in the test). Based on the CAN/CSA-
S6-06 equation for unfactored shear capacity based only on the capacity of the stirrups (Equation
C-3 in Appendix C), the expected web crack angle can be determined by:
tied arch model predictionactual transverse stress
unaccounted demand
05
1015
0200400600800
1000
εv [mm/m]
fv [MPa]
c) Calculated transverse stress and tied arch model prediction
b) Measured transverse strains in vertical stirrups
a) Noshiravani tied arch model428kN 428kN
3585kN 3585kN
428kN 428kNindicated stirrup yield
no strain gauge reading
yield strain
yield stress
114
(5-1)
For the 2-leg D4 deformed wire stirrups equally spaced in the webs at 100mm centre-to-centre,
the angle of web cracks expected for the load stage at flexural cracking and the load stage near
failure is and , respectively. An alternative parallel chord model and the corresponding
transverse stress demand for the two load stages are presented in Figure 5-14 and Figure 5-15.
Comparing the crack patterns observed during the test to the parallel chord predictions for the
diagonal compressive stresses in the webs shows a remarkable similarity. Since the shear capacity
of concrete has been neglected for the prediction of web crack angle, it is expected that the
predicted angle is slightly shallower than the observed crack angle. For the two load stages
considered, the observed web crack angle is within of the predicted angle at flexural cracking,
and within of the predicted angle near ULS. Based on these observed crack patterns and the
measured stress readings in the vertical steel, the girder tested by Noshiravani (2007) appears to be
symptomatic of the flow of forces described by the parallel chord model proposed by Schlaich et
al. (1989) within the central region of the shear span. However, fan cracking in the vicinity of
concentrated vertical loads was not observed in this test. In these regions where there is flexural
cracking, it seems likely that arching action exists which decreases the demand for transverse steel.
There remains some uncertainty to explain the flow of forces in girders with unbonded tendons;
however, further truss model development is beyond the scope of this work.
Figure 5-14. Crack pattern and alternative parallel chord model for load stage corresponding to flexural cracking
θAvfydv
Vs-----------------⎝ ⎠⎛ ⎞atan=
20° 15°
1°
4°
a) Crack pattern and alternative parallel chord model
θ = 20˚
web crack 21˚320kN 320kN
2835kN 2835kN
320kN 320kN
compression fan
compression fandetailed trusssimplified truss
0200400600800
1000fv [MPa]
parallel chord model designactual transverse stress
conservative capacityin compression fan regions
b) Calculated transverse stress and alternative parallel chord design
yield stress
115
Figure 5-15. Crack pattern and alternative parallel chord model for load stage corresponding to ULS
It appears that a parallel chord model consisting of two compression chords for fully
compressed girder regions provides a lower bound solution for the design of transverse steel in the
web to resist shear forces for a girder prestressed with unbonded tendons. Due to the intensive
prestressing arrangement provided by the curtailed prestressing tendons of the proposed cantilever
box girder, this fully-compressed state exists for a large portion of the girder which was previously
shown in Chapter 3. It seems reasonable to proceed with a parallel chord model to describe the flow
of forces in the web for the proposed girder and to design the vertical web reinforcement based on
this approach.
5.5 Preliminary Design Model
The truss model developed for the design of the 90m cantilevered segmental girder is shown in
Figure 5-16. This truss has two parallel compression chords that are assumed to be located at the
centroid of the flanges, where there are large force concentrations. In the uncracked state, the truss
is continuous at the supports, since the cantilever prestressing causes the top flange to be fully
compressed under service loads. For the ultimate limit state, flexural decompression of the top
flange occurs at the supports, which modifies the model as the top compression chord at the support
vanishes since it can not carry tension. Although this change in the truss model impacts the flexural
behaviour of the girder, in general it does not change the way the truss model carries shear. The
forces in the chords of the model can be determined by analyzing the truss due to the vertical
applied loads, and including the effects of prestressing in the flanges shown in Figure 5-17.
a) Crack pattern and alternative parallel chord model
WestWest
428kN 428kN
3585kN 3585kN
428kN 428kNθ = 15˚
compression fan
compression fandetailed trusssimplified truss
extended web crack 19˚
parallel chord model design
actual transverse stress
conservative capacityin compression fan regions
0200400600800
1000fv [MPa]
b) Calculated transverse stress and parallel chord model design
yield stress
116
Figure 5-16. Truss model for shear design. a) fully prestressed state for continuous beam; b) modified model to account for flange decompression (formation of plastic hinge)
The truss model chosen for design is based on plasticity, where the redistribution of web stress
is assumed to occur at the ultimate limit state to satisfy equilibrium based on a chosen angle of web
stress. To ensure a ductile failure mechanism at the ultimate limit state, Menn (1990) recommends
that the shear resistance of concrete members should be taken based only on the shear capacity of
the vertical stirrups. In this regard, yielding of the stirrups will govern the design instead of
compression strut failures due to factored transverse loading. For prestressed members the angle
of shear stress in the web is generally smaller than that of reinforced concrete members (Canadian
Standards Association, 2006). The typical angle of compression stress in the web that is observed
for fully compressed concrete members is (demonstrated by Equation C-5 in Appendix C).
For the purpose of designing the preliminary web reinforcement for the proposed box girder, a
typical compression stress angle will be assumed to be . This angle of stress in the web is
supported by Menn (1990).
Figure 5-17. Prestress forces applied to truss model in Figure 5-16 for each applied anchorage force
θ
θ
a)
b)
Q Q Q Q Q Q Q Q Q Q Q Q Q
Q Q Q Q Q Q Q Q Q Q Q Q Q
V V
V VFlexural decompression of top flange causes chord to vanish from model (no tension capacity across the joint due to no continuous bonded steel)
29°
30°
= = =
QPt
Pb
Pe
Mp=Pe
P
ete
eb
Peb/d Mp/d
Pet/d Mp/d
det
eb
Pt
Pb
WherePt = P (eb/d) + Mp/dPb = P (et/d) - Mp/d
θ
VAppliedprestress
Equivalentsectional forces
d
Model geometry
Distributedchord forces
117
5.6 Opening of Joints in Segmental Bridges
For critical flexure regions of the girder at the ultimate limit state, the total decompression of a
chord on the flexural tension side occurs. Following this decompression of the flange, an opening
of segmental joints in precast concrete bridges can form since there is no continuous bonded
reinforcing steel across the joint. This has been a concern for designers as the mechanism for
carrying shear stresses across an open dry joint in the vicinity of a support is uncertain since the
assumed diagonal compression strut in the web cannot be established across the open joint.
Virlogeux (1993) has claimed that the opening of joints near the support must not be allowed
for the ultimate limit state design since there is no clear load path for the web shear stresses to be
reacted by the support. In order to bring the shear forces from the superstructure to the support
across an open joint, he states that vertical reinforcing steel or prestressing must be installed in the
vicinity of the open joint to lift the shear forces to an elevation where the shear force can pass
through the support. This additional vertical reinforcing steel, also termed as ‘hanger
reinforcement’, is displayed in Figure 5-18. This explanation of the flow of forces; however, is
fundamentally flawed since horizontal equilibrium is not satisfied at the top node next to the open
joint. Turmo (2005) has conducted a finite element study to show that, indeed, inclined
compressive arching stresses crossing an open joint occurs to transfer the force across an opened
joint. As the girder experiences distress due to joint opening, vertical equilibrium in the vicinity of
the open joint is maintained due to the large inclined forces within the slab and the lower region of
the web in compression. Therefore, it is not necessary to provide additional vertical web
reinforcement in regions where joint opening is possible since the compressive arching of forces
for rotated joints will satisfy vertical equilibrium.
Figure 5-18. Incorrect hanger reinforcement model for shear stresses crossing an open joint (adapted from Virlogeux, 1993)
118
5.7 Design of Web Reinforcement
To determine the maximum shear demands on the bridge, ULS Combination 1 defined in the
CAN/CSA-S6-06 was considered. For the span of 90m, the lane load model defined in cl. 3.8.3.3
governed the live load demands. For a deck width of 12m, three design lanes are considered to
produce the worst effect, shown in Figure 5-19 below.
Figure 5-19. ULS Combination 1 maximum shear demands
Based on these demands, the arrangement of vertical reinforcing steel in the web can be
quantified that is required to provide the necessary spreading of compression stresses in the web.
The critical sectional force for the design of vertical reinforcing steel has been taken at a distance
of dv from the support (cl. 8.9.3.1). To provide economy in web steel consumption, the bar spacing
has been varied at several regions on the girder. The required vertical steel demand and the chosen
arrangement is shown below in Figure 5-20. At locations where the bar spacing changes, the
capacity envelope can be transitioned over a length equal to the height of the girder, where the
transition of capacity is centred at the location where the bar spacing changes.
V [kN]
distance [m]
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
12000
0 10 20 30 40 50 60 70 80 90
DL
DL+SDL
DL+SDL+LL
Uniform distributedlane load (9kN/m per lane)
Shaded area:Envelope due to CL-625-ONT Truck
119
Figure 5-20. ULS transverse steel demand and chosen capacity
Due to the slender web design, only one leg of reinforcement is placed at the centre plane of
the web. Equally spaced 20M headed reinforcing bars are placed in the web. The use of headed
reinforcing bars allow for reduced bar congestion in the vicinity of the flange junction.
Figure 5-21. Web reinforcement
5.7.1 Comparison to CAN/CSA-S6-06
To compare the reinforcement design provided by the simple design model to the CAN/CSA-
S6-06, the angle of compression stress in the web needs to be determined based on the code
equations for girders in shear. In order to calculate the angle of compression stress, θ, both the
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40 45
dv
h
h
Av/s
calculated transversesteel required so that Vs = Vf
transverse steelprovided
120
sectional strain at the mid-height of the web, , and the equivalent crack spacing parameter, sze,
must be known (given in Equation C-5 of Appendix C). The mid-height web strain can be
computed directly based on sectional forces and cross section geometry. The equivalent crack
spacing parameter, however, is an empirical equation which is suitable for use in reinforced
concrete beams designed using standard practice methods. Strictly following the Code, a deep box
girder reinforced only with one single direction of transverse steel will produce a very large
equivalent crack spacing parameter which consequently results in steep predicted angles of
diagonal web compression. This is because the code does not take into account crack control
provided by fibre reinforced concrete. To illustrate this point, consider the case for the proposed
3600mm deep box girder with nominal 20mm aggregate and only vertical steel in the webs:
If the box girder is fully compressed, the strain in the web will be a small compressive value.
Conservatively assuming , the predicted angle of web compression is:
The use of fibre-reinforced concrete in girder webs as a replacement of typical minimum crack
control reinforcement is not current practice; however, current studies are in progress to investigate
the performance of fibre-reinforced concrete panels in shear with only one direction of bonded
reinforcing steel (Susetyo, 2007). Below in Figure 5-22 is a visual comparison of two shear panel
tests performed by Susetyo (2007) which have similar compressive concrete strength, but are
reinforced with different steel arrangements. A high strength 80MPa normal concrete panel is
reinforced with 3.3% reinforcement in one direction (representative of vertical web steel), and
0.4% reinforcement in the orthogonal direction (representative of crack control horizontal steel).
εx
sze 35sz 15 ag+( )⁄=
35(0.9dv ) 15 20+( )⁄=
0.9 3600( )=3240mm=
εx 0=
θ 29 7000εx+( ) 0.88sze
2500------------+⎝ ⎠⎛ ⎞=
29 7000 0( )+( ) 0.88 32402500------------+⎝ ⎠
⎛ ⎞=
63°=
121
A high strength 80MPa FRC panel containing 1.5% hooked steel fibres has also been tested with
3.3% reinforcement in one direction only.
Figure 5-22. Shear panel tests for normal reinforced concrete and FRC concrete (adapted from Susetyo, 2007)
It is seen that the FRC test specimen has a more distributed crack pattern than the normal
reinforced concrete test specimen prior to failure. The FRC test specimen cracks are more closely
spaced and the average crack width is smaller. These preliminary results do not conclude that this
behvioiur is reliable for every case, but they provide some evidence to speculate the cracking
behaviour of webs in shear made of high strength FRC. For the purpose of calculation using the
Code provisions, the web of the box girder is treated as a typical minimum reinforced web with
adequate crack control reinforcement and the crack spacing parameter, sz, is assumed to be equal
to 300mm (cl. 8.9.3.7). Under this assumption, for the proposed box girder design is evaluated
under the load condition corresponding to ULS Combination 1. The following distribution of
diagonal compression stress and the corresponding parameters can be seen below in Figure 5-23.
a) 80MPa concrete; two orthogonal steel reinforcement directions
b) 80MPa FRC; one reinforcement direction
122
Figure 5-23. Shear parameter predictions according to CAN/CSA-S6-06
The prediction of the CAN/CSA-S6-06 provisions show that the angle of web compression for
the vast majority of the beam is close to for the entire length of the girder since the girder is
fully compressed at the ultimate limit state for all locations at a distance further than dv away from
the support. Notwithstanding the fact that resistance provided by aggregate interlock has been
neglected for the design of web reinforcement, there appears to be good agreement between the
chosen truss model for design and the variable angle truss model predicted by the Code. For the
global spreading of forces, the designed web reinforcement is sufficient to transfer the compressive
-200000
-150000
-100000
-50000
0
50000
100000
0 10 20 30 40
0
2500
5000
7500
10000
12500
15000
0 10 20 30 40
-0.0002
-0.0001
0
0.0001
0.0002
0 10 20 30 40
05
10152025303540
0 10 20 30 40
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 400
10000
20000
30000
40000
50000
60000
0 10 20 30 40
Mf [kNm]
Vf [kN]
fpeAps [kN]
εx [mm/mm]
θ [deg]
βcantilever post-tensioning
continuitypost-tensioning
minimum envelope
maximumenvelope
critical point ofinflection
fpoAps for Mf < 0fpoAps for Mf > 0
constant θ = 30˚assumed for designtruss model
εx > -0.0002
dv
all εx < 0 at sections> dv from support
DL+SDL+LL
dv
bottom tensionflange
top tensionflange
CAN/CSA-S6-06
30°
123
stresses in the web. Web reinforcement corresponding to local spreading at the anchorage will now
be considered in the following chapter.
124
125
6.0 ANCHORAGE ZONE
This chapter begins by introducing the importance of anchorages for concrete structures
prestressed with external unbonded tendons in Section 6.1. In Section 6.2, the fundamentally
different flow of forces is described within the anchorage zone for an external unbonded tendon
anchorage in comparison to the flow of forces for an anchorage for an internal tendon. A three-
dimensional truss model is developed in Section 6.3 to describe the flow of forces in the anchorage
zone for the corner blisters which is consistent with the previous model used for global transfer of
shear force in the box girder. Section 6.4 discusses the detailing requirements for the anchorage
steel and Section 6.5 presents a possible arrangement of reinforcing steel for a typical anchorage
zone.
6.1 Introduction
The proper anchorage of external unbonded tendons to anchorage blisters is of paramount
importance for the structural integrity of the proposed system during the whole lifetime of the
structure. Unlike internal bonded systems, the transfer of longitudinal force for external unbonded
tendons into a box girder is only facilitated at discrete anchorages located at the ends of the tendon.
For structures with bonded prestressing, there is an inherent redundancy in the system for the
transfer of prestress force. In the case of damage to the original anchorage, the prestressing force
in a bonded tendon may be partly taken over by the activation of mechanical bond of the grouted
tendon to the structure. However, damage and loss of an anchorage for an external unbonded
tendon results in the loss of the entire tendon. Since prestressing of a cantilever girder with no
continuous bonded steel relies exclusively on the anchorage blister, special attention must be paid
to ensure adequate anchorage can be provided.
Anchorages should be located at the corner of the box section for minimum development length
(Menn, 1990). The temporary external unbonded post-tensioning required during the cantilever
construction stage should also be provided near the corners of the cross section for adequate crack
control. Post-tensioning forces applied to the section at the middle of slab components can cause
significant longitudinal cracks due to strain compatibility of local deformations (Moon, 2004).
Also, anchorages located away from the corner can cause significant out-of-plane bending in the
slab element due to the eccentricity of the external anchorage blister (Eibl, 1990). For the proposed
precast segmental girder, all anchorages have been located at corner blisters (shown below in
Figure 6-1).
Figure 6-1. Top and bottom corner anchorage blisters
6.2 Flow of Forces
The flow of forces for an external unbonded tendon anchored at an intermediate corner blister
is fundamentally different than the flow of forces for an internal tendon that is guided into the slab.
For an internal tendon, the curvature of the tendon profile in elevation creates a vertical deviation
force due to the tendon bearing against the surrounding concrete. The self-equilibrating state of
forces for a girder with internal tendons shows that the resultant prestress force follows the profile
of the tendon. As such, the resultant prestress force is deviated sufficiently into the bottom slab for
the overall equilibrium of the box girder. Figure 6-2 shows the anchorage forces, deviation forces,
and boundary forces for a girder segment prestressed with an internal tendon and a girder segment
prestressed with an external tendon at intermediate anchorages located on the bottom slab. It can
be seen that for the internal tendon, the forces on the tendon and the forces on the concrete are equal
and opposite and act in the same plane.
For the case with the external unbonded tendon, the boundary forces on the segment which
satisfy a statically equivalent set of loads are not coincident with the tendon force. Since the tendon
exits the concrete section, the compression forces in the concrete are not guided into the bottom
flange the same way as it is with the guided tendon. Therefore, a different state of internal
equilibrium must exist to deviate the applied anchorage force into the concrete box girder. The
boundary forces shown in Figure 6-2 follows a consistent method of analysis with regard to the
parallel chord model for the global spreading of forces in the web. Large concentrated forces are
assumed to be carried in the flanges; therefore, the statically equivalent set of loads to equilibrate
the applied anchorage force are shown to be distributed in both the top and bottom flange. For the
126
anchorage force to be distributed in this way, a local spreading of forces must take place within the
box girder to provide this state of equilibrium.
Figure 6-2. Flow of forces at an intermediate anchorage
6.3 Strut-and-Tie Model for Local Spreading of Forces
To visualize the local spreading of forces in a box girder that is prestressed with intermediate
external unbonded post-tensioning at corner blisters, a simple strut-and-tie model desired. Since
the centroid of the anchorage force and the centroid of the web are not in coincident planes, the
strut-and-tie model required to sufficiently describe the spreading of forces must be in three
dimensions.
6.3.1 Anchorage of External Unbonded Tendons in Slab Blisters (Wollman, 1993)
Wollman (1993) performed a laboratory study to investigate the behaviour of external tendon
anchorages in box girder diaphragms and intermediate slab blisters. His study included eight blister
and rib specimens, one of which was an external tendon corner blister. The load path for a corner
blister is complex as it requires consideration in three dimensions to correctly represent the flow
of forces. He offers a strut-and-tie model to describe the flow of forces for a box girder due to the
anchorage of external unbonded tendon at a corner blister, shown in Figure 6-3. Strong corbel
reinforcement (shown as force T1) is required to tie the blister to the slab, and diagonal tension
reinforcement (shown as force T2) is required at the end of the blister to pull the forces into the
plane of the web and the flange. Web and flange bursting reinforcement (shown as force T3) is also
a) internal bonded tendon
b) external unbonded tendon
forces on concrete forces on tendon
forces on concrete forces on tendon
local spreading of forces into both flangesis required to satisfyequilibrium
deviation forces due to tendon curvature bring prestress force into the flange
127
provided to take into account the spreading of forces. Wollman (1993) also provides longitudinal
reinforcement in the vicinity of the corner for the truss model to satisfy equilibrium due to the
longitudinal component of force of the diagonal compression struts in the slabs. According to his
additional finite element analysis performed for the corner blister, the highest tension forces occur
behind the anchor at the re-entrant corner, while the slab bursting stresses are much smaller. The
test specimen was reinforced according to the strut-and-tie model, and failure was observed due to
concrete crushing in the local zone, which was typical for all blisters. Wollman (1993) states that
the most consistent predictor of blister failure is the calculation of capacity of local zone based on
the ratio of the area of supporting concrete to plate area (A/Ab) and the confinement provided by
the spiral.
Although the strut-and-tie model given by Wollman (1993) represents a reasonable flow of
forces due to the anchorage force at a corner blister, it appears to be incomplete as it does not satisfy
equilibrium at all nodes. The strong corbel reinforcement, T1, provides a tension force to deviate
the longitudinal anchorage force, P, but the inclined tensile force is not in transverse equilibrium
at node A. In addition, the spreading of compressive forces at the end of the anchorage blister can
not take on the trajectories shown in Figure 6-3 since the force needed in tension tie T2 to pull the
compressive forces into the plane of the slabs requires the resultant of the compressive forces to
converge at the corner. By moment equilibrium of global forces, the equal distribution of the
resultant P/2 forces in each slab shown in this strut-and-tie model can only be true if the application
of the prestress force, P, is equidistant from the web and the bottom flange.
The boundary forces provided on this model may be representative of the specimen tested by
Wollman (1993). The equal boundary forces provided at the mid-height of the web and at the mid-
width of the flange are reasonable assumptions for a specimen with only a vertical web slab and a
horizontal flange slab. However, for the application of this model into a continuous box girder,
some modification is necessary to adapt the idea of this spreading of forces into the full cross
section.
128
Figure 6-3. Strut-and-tie model for corner blister (Adapted from Wollman, 1993)
6.3.2 Design Model for External Tendon Blisters
Using model described previously by Wollman (1993) as a reference, a revised strut-and-tie
model was developed that resolves equilibrium at all nodes, shown in Figure 6-4. The isometric
profile, shown in Figure 6-4a, outlines the centreline of the web in the vertical plane, the centreline
of the bottom flange in the bottom plane, and the anchorage blister at the level of applied force in
the corner. The equilibrium of the tension tie, T1, is equilibrated vertically and horizontally by
inclined compressive struts in the web and in the flange, respectively. The longitudinal component
of force associated with the inclined web and flange compressive struts is tied forward with the
tensile force T3. Compressive struts at the end of the anchorage blister follow the trajectory path
that is required by equilibrium, which resolves in the corner to meet the longitudinal tension tie T3.
The prestressing force, P, resolves into three discretized boundary forces, F1, F2, and F3 at a
distance 2Lb past the face of the anchorage blister, where Lb.is the longitudinal length between ties
T1,and T2. Figure 6-4b displays the front view of the three dimensional model, showing the
eccentricity of the anchorage force, P, to the web and the flange, ew and ef, respectively. Figure 6-
4c displays the angle of the compression struts in elevation. The angle of compression in the slabs
is labelled θs. The angle in the anchorage blister measured from the flange, θf, is defined by the
eccentricity of the force to the slab, ew, and the distance between the centroids of tension ties, Lb.
The top view to define θf is geometrically similar. For longer anchorage lengths, the tensile
demand in transverse ties is decreased since the angle of compression is flatter. However, from a
practical design perspective, the maximum length of the anchorage block limits the maximum
length of Lb for bottom corner blisters due to the required clearance to place a jack between
anchorages.
T1
T2
P
T3
T3
P/2
P/2
ACompressionTension
129
Figure 6-4. Strut-and-tie model for corner blister satisfying static equilibrium; a) 3d isometric view, b) front view, c) side elevation
For the strut-and-tie model shown in Figure 6-4, equilibrium of joints gives the following
tension demands:
(6-1)
(6-2)
P
F3
F2
F1
T4
T5T1
T2
T3
CompressionTension
LbLbew
ef
θf
θfθs
P
F2+F3
F1
Front View Side Elevation
dv
T1P2--- θwtan2 θftan2+ P
2---ewLb------⎝ ⎠
⎛ ⎞2 ef
Lb------⎝ ⎠
⎛ ⎞2
+= =
T2 T1P2---
ewLb------⎝ ⎠
⎛ ⎞2 ef
Lb------⎝ ⎠
⎛ ⎞2
+= =
130
(6-3)
(6-4)
(6-5)
The reaction forces satisfying global equilibrium of the free body are given as:
(6-6)
(6-7)
(6-8)
It is observed from the demand equations that the demand for steel is largely influenced by the
eccentricity of the anchorage force to the corner and the length of the anchorage blister. To
minimize the demand for steel, the tendon should be anchored as close as possible to the corner of
the box girder.
For the top anchorages, all cantilever post-tensioning is installed during the construction
process at the cantilever tip where there is an open face. Therefore, ample clearance is available for
the stressing of tendons. The largest tendon unit used for the cantilever post-tensioning is a VSL
ES6-12. The spiral that is required for these tendon units is 350mm in diameter. The vertical
limiting factor for placing the anchorage close to the corner is the to ensure that the anchorage
spiral and the transverse pretensioning strands do not interfere with each other. The horizontal
limiting factor is to ensure that the anchorage head clears the inside face of the web.
For the bottom tendons, the construction clearance governs the placement of the anchorages.
Minimum jack clearances for stressing tendons must be provided to allow installation of tendons.
The necessary jack clearances are illustrated in Figure 6-5. The largest tendon units that are used
for the design of longitudinal post-tensioning are VSL ES6-19. The stressing jack type required to
stress these tendons is a ZPE-460/31. For this jack type, the edge clearance required (shown as
T3 P θs θwtan θftan+( )cos P2Lb
4Lb2 dv
2+----------------------------
⎝ ⎠⎜ ⎟⎛ ⎞ ew ef+
Lb----------------⎝ ⎠
⎛ ⎞ 2Pew ef+
4Lb2 dv
2+----------------------------
⎝ ⎠⎜ ⎟⎛ ⎞
= = =
T4P2--- θftan
Pef2Lb---------= =
T5P2--- θwtan
Pew2Lb---------= =
F1P2---
θftanθssin------------- P
2-------
efLb------= =
F2 P T3– P P2
-------ew ef+
Lb----------------⎝ ⎠
⎛ ⎞–= =
F3P2---
θwtanθssin-------------- P
2-------
ewLb------= =
131
dimension ‘E’) is 305mm and the minimum longitudinal distance required for installation (shown
as dimension ‘C’) is 1505mm.
Figure 6-5. Jack clearances necessary for installation (adapted from VSL, 2007)
Satisfying these clearance criteria, the top anchorages result in a vertical eccentricity of 270mm
and a horizontal eccentricity of 278mm. The bottom anchorages result in a vertical eccentricity of
380mm and a horizontal eccentricity of 305mm (shown below in Figure 6-6).
Figure 6-6. Minimum anchorage eccentricities to box girder corner
According to the truss model shown previously in Figure 6-4, it is observed that the demand
for steel is also minimized by maximizing the longitudinal length of the anchorage blister. For the
bottom tendon anchorages, the maximum length of the anchorage is governed by the longitudinal
jack clearance requirement of 1505mm shown by Dimension ‘C’ in Figure 6-5 above. For a
segment length of 3000mm, the maximum anchorage length for the bottom tendons is limited to
1495mm. For the top tendons, a similar anchorage length has been chosen which also takes into
account the integration of anchorage blister and transverse ribs formed in the top slab. To minimize
the number of surfaces to form, the anchorage blister has been chosen a length of 1250mm which
corresponds to the vertical face of the transverse rib (as shown in Figure 6-7).
B
C
D
E
E
132
Figure 6-7. Anchorage blister lengths
Therefore, adapting the anchorage model to the minimum anchorage eccentricity and the
3600mm deep proposed box girder, the local spreading of forces to provide the statically
equivalent boundary conditions at the flange is shown below in Figure 6-8 for a bottom tendon.
Figure 6-8. Strut-and-tie anchorage model for a bottom tendon
6.4 Detailing
Beaupre et al. (1990) investigated the design of deviator saddles for externally post-tensioned
structures. The saddles studied are generally used to deviate external unbonded tendons vertically
and horizontally for typical span-by-span type structures. Unlike an anchorage blister which must
transfer a large longitudinal force into the structure, these deviation saddles need only provide
resistance in the vertical plane. Notwithstanding this difference, it is useful to note the various
arrangements of reinforcement possibilities to tie a horizontal or vertical load into a box girder due
to deviation forces applied by tendons. Beaupre et al. (1990) show that a typical saddle detail used
in straight-span bridges consists of a direct tension link bar to efficiently tie the deviation force into
the corner node of a box girder section, shown in Figure 6-9. He also proposes a modified
reinforcement scheme to simplify and standardize reinforcement patterns and saddle geometry.
P
F2+F3
F1
133
The closed loop stirrup arrangement relieves reinforcement congestion at the corner node by
placing stirrups around the tendons being deviated.
Figure 6-9. Tension ties linking deviation force to box reinforcement (adapted from Beaupre et al., 1990)
Deviation forces are resisted by combined shear action in the plane of the web and in the plane
of the flange. Since the increase in tendon force is small for the proposed structure at the ultimate
limit state, the critical design force for the deviator reinforcement occurs due to initial jacking of
the tendon. Under CAN/CSA-S6-06 specifications, the maximum allowable initial jacking force is
limited to 80% of the ultimate strength of the tendon (0.8fpuAps).
The truss model shown previously in Figure 6-8 displays diagonal tension tie members at T1
and T2 which is similar to the efficiently oriented steel links shown by Beaupre et al. (1990). A
direct tension link is the most efficient arrangement of steel since it provides two legs of
reinforcement per stirrup that are oriented in the direction of the optimum load path. However, as
an alternative arrangement for simplified and standardized detailing, the closed loop reinforcement
will be used for design instead of the direct tension links. The closed loop arrangement does not
effect the forces on the global system, however, the local load path with the anchorage blister is
somewhat modified (Figure 6-10). The direct tension link ties the force to the corner of the box
directly; however, the closed hoop reinforcement provides a load path where web and flange forces
are met with a diagonal compression strut at the corner.
The quantity of steel required for each detail is different. For the direct tension link, area of steel
which resists the direct load is 2Ab for each link. For the closed loop, the area of steel which resists
an orthogonal component of the direct load is 1Ab in both directions. Therefore, since legs in both
directions have equal area, the number of required closed loops is governed by the leg which
attracts more force. The proportion of forces in the legs can be determined using the geometrically
similar eccentricities shown above in Figure 6-6. The size and shape of hoop is consistent along
the length of the anchorage blister, with concentrated reinforcement at the ends to satisfy the
demand of the truss model given by T1 and T2.
FLinkbar
Direct tension link stirrups
inner loopbarclosed stirrup
Loop stirrups
134
Figure 6-10. Flow of forces for different reinforcement details
Haroon (2006) investigated the benefits of using FRC for anchorage zones in post-tensioned
structures and found that there are significant enhancements in performance. The best performing
anchorage specimen configuration satisfying AASHTO acceptance criteria included 1% hooked-
end fibres and no spiral and no skin reinforcement. The minimum compressive strength of concrete
was 40.7MPa. With the use of high-strength concrete and 1% steel fibres, this allows for 100%
total elimination of the secondary reinforcement. The test specimens in this study were for VSL
type EC5-7 anchorage units; however, Haroon (2006) concludes that the results are expected to be
valid for larger anchorage units. The anchorage blister reinforcement for the proposed girder
design will include a spiral at the anchorage zone to provide confinement at the bearing plate and
the necessary steel required to tie the anchorage force into the box girder, but no secondary skin
reinforcement will be designed since adequate crack control has been observed in tests due to the
benefit of FRC.
6.5 Reinforcement
The anchorage blister reinforcement and all necessary tension steel for local spreading
described by the truss model in Figure 6-4 was used as a basis for the design of anchorage
reinforcement in the box girder. All reinforcing steel has been designed to a limited stress of
240MPa (cl. 8.6.2.7.4). It was found that the required tension forces to deviate the large anchorage
b) closed hoop
a) 2-leg link
truss model reinforcement detail
H
H
direction of forcesat H
magnitude of forces
As = 2Ab
As = 1Ab
135
forces is significant within the anchorage blister and in the web. The closed loop reinforcement
within the blister has been chosen to be not larger than 20M reinforcing bars to accommodate the
minimum inside bar bend radius of 4db(cl. 8.4.2.1.1). Limiting the bar size, the steel required for
the deviation within the blister has been chosen to be bundled to allow for a larger Lb between
deviation forces.
For the critical bottom anchorage corresponding to a VSL ES6-19 tendon, the design force
corresponding to 80% of the ultimate strength of the tendon is 3958kN. For a bottom flange
thickness and web thickness of 150mm and 120mm, respectively, the eccentricities ef and ew are
380mm and 305mm, respectively (as shown previously in Figure 6-6). Nine hoops of 20M bars are
required to satisfy each of the tie forces, T1 and T2. Three sets of bundled bars consisting of 3-20M
bars each have been provided at the locations of T1 and T2. The blister length between the centroid
of steel providing the deviation forces is Lb = 1160mm which allows for 30mm cover for the
outermost hoop. Nine 20M bars are required for the bursting force in the web, which have been
evenly distributed at 50mm behind the anchorage (shown in Figure 6-11). Longitudinal tension
steel is required to resist the longitudinal component of the inclined compressive struts in the slabs
at the segment face. The longitudinal reinforcement described by T3 is required to be effective at
the location of the anchorage, which has been designed to be at the segment face. Due to congestion
difficulties, headed reinforcement has been provided in the longitudinal direction. Headed
reinforcement reduces congestion of bars in the concrete, and provides full development of the bar
at the location of the head.
Figure 6-11. Anchorage zone reinforcement for critical 19 strand bottom tendon
For a constant anchorage blister geometry, the change in required anchorage reinforcement and
the change in anchorage force for smaller tendon units are linearly related (since tension forces are
proportional to P). A similar reinforcing arrangement is required for all other anchored tendons.
The area of steel required for anchorage zone reinforcement on the basis of the truss model for all
136
chosen prestressing arrangements is summarized below. Table 6-1 and Table 6-2 give the area of
required reinforcement steel corresponding to the major tension demands of the truss model in
Figure 6-4 for top cantilever tendons and bottom continuity tendons, respectively. All values in the
table have been generated using the formulas given from Equation 6-1 to Equation 6-5.
Table 6-1. Area of steel required for local spreading reinforcement for top tendonsa (in mm2)
a. corresponds to: jacking stress in tendon = 0.8fpu, allowable steel reinforcement stress = 240MPa
Tendon Size T1 T2 T3 T4 T5
8 strand 980 980 1920 960 980
12 strand 1470 1470 2880 1430 1470
Table 6-2. Area of steel required for local spreading reinforcement for bottom tendonsa (in mm2)
a. corresponds to: jacking stress in tendon = 0.8fpu, allowable steel reinforcement stress = 240MPa
Tendon Size T1 T2 T3 T4 T5
8 strand 1140 1140 2290 1140 910
12 strand 1710 1710 3430 1710 1370
19 strand 2700 2700 5440 2700 2170
137
138
7.0 FIXED END ABUTMENT
In this chapter, a conceptual abutment design will be presented for the validation of the
cantilever-constructed bridge. The chapter begins by discussing the design decisions made for the
superstructure and the implications they have on the substructure design. A reference bridge for a
similar structural system is discussed which provides design guidelines on the conceptual layout
of the proposed abutment. Several conceptual design alternatives are briefly presented and
described. A conceptual layout of the recommended system is briefly discussed; however, the
design and analysis of the abutment structure is outside the scope of this thesis.
7.1 Introduction
For the proposed constant depth cantilever-constructed bridge, a monolithic connection at the
midspan was found to be most economical as it promotes the redistribution of moment to the
midspan region. Large positive moment at the midspan is desirable since a) the constant depth box
girder has significant capacity for positive bending resistance due to the large internal lever arm
and wide compression flange, and b) the redistribution of moment relieves large negative moments
at the support which decreases the demands on the substructure. Alternatively, providing an
expansion joint at the midspan while promoting the redistribution of positive moment in the
midspan region is less desirable since a) a continuity beam and interior diaphragms are required
for a moment connection across the joint, and b) detailing of continuity tendon anchorages in the
vicinity of the midspan expansion joint is crowded and complex. Thus, a monolithically connected
superstructure with no expansion joints is desired since it allows for the most favourable
distribution of forces, simplifies detailing, and provides better serviceability performance with
respect to durability and driver comfort. A jointless bridge, however, requires longitudinal
flexibility for the expansion and contraction of superstructure due to change in temperature.
7.2 Bras de la Plaine Bridge, France (Tanis, 2003)
The Bras de la Plaine Bridge, situated in the south of Reunion Island, France, is a modern
example of a light weight cantilever-constructed bridge erected from fixed abutments. The long
span of 280m is embedded in ballasted abutments, with a typical abutment length of 45m exterior
of the main span (Figure 7-1). This results in a mainspan to backspan ratio of approximately 6.2.
The abutments are 11.9m wide which matches the width of the bridge deck and the abutment height
ranges from the girder depth of 18m to a maximum height of 21m. The abutments economically
place counterweight material where the lever arm is greatest, as it is composed of a back
compartment 11m long filled with ballast, and a hollow front compartment (Figure 7-2). The
bottom slab of the abutment has a constant thickness of 1m to transfer the large compression force
due to the cantilever moment of the superstructure to the counterweight at the back compartment.
Figure 7-1. Bras de la Plaine Bridge, Reunion Island, France (adapted from Tanis, 2003)
Although the superstructure of the Bras de la Plaine Bridge has some dissimilarities from the
proposed cantilever girder, the rotational resistance at the abutment provided by the foundations is
similar to the boundary conditions required for the proposed cantilever girder. The top slab of the
Bras de la Plaine Bridge is prestressed with a combination of both internal and external prestressing
tendons; however, the prestress arrangement is similar since the cantilever tendons are stressed at
the face of each cantilevered segment and anchored at the back of the abutment. This prestress
arrangement and abutment proportions provides valuable information to use for reference in the
proposed bridge structure for the conceptual design of the abutment.
Figure 7-2. Ballasted abutment of Bras de la Plaine Bridge (adapted from Tanis, 2003)
3,0119%
1000
LEST
700
4000
joint
45,347
Struts
0,600
19,7
94
0,500
B AC27
section
11,900 35,973waterpipe5,950 5,950
Elevation
139
7.3 Flexible Supports
One main advantage of cantilever-constructed bridges is that they are self-supporting during
erection, and do not require the placement of falsework below the superstructure for construction.
This advantage is useful for the proposed bridge since it requires no temporary supports which
would interfere with traffic below the structure, and for many cantilever bridges this advantage is
useful since it allows for ease of construction on tall slender piers. The Felsenau bridge, in Bern,
Switzerland, is one example of a cantilever-constructed bridge built on tall slender two-leg piers to
provide adequate moment resistance during construction, and longitudinal flexibility for service
due to thermal movements of the superstructure in the final state. For smaller span jointless bridges
that are built on abutments, longitudinal flexibility is achieved by the incorporation of integral
abutments.
Bridge structures with no intermediate joints and integral abutments do not contain traditional
expansion joints between the abutments and the superstructure (Abendroth, 1989). Instead, the
superstructure and the abutments are monolithically connected and founded on flexible steel piles.
The piles are subject to horizontal movement as the bridge contracts and expands due to variation
in temperature. This horizontal displacement of the superstructure induces a moment in the piles
in addition to the axial load applied from the vertical applied loads. The piles may be designed as
beam-columns, either from a conventional elastic design approach, or an inelastic design approach
that accounts for redistribution principles (Abendroth, 1989).
7.4 Design Approach
The abutment system requires the following features: a) anchorage of all external unbonded
cantilever tendons at the back of the abutment, b) capacity to resist the large negative bending
moment of the cantilever-constructed main span due to factored dead load and live load, and c)
longitudinal flexibility to allow for thermal expansion and contraction of the 90m main span. A
large force couple in the abutment is required to establish equilibrium for the resistance of the
cantilever moment. The Bras de la Plaine bridge provides this force couple due to the heavy ballast
placed at the back of the abutment which levers against the front abutment wall. For the Felsenau
bridge, this force couple is achieved by a force differential in the two-leg pier system.
140
7.4.1 Preliminary Design Alternatives
Several preliminary abutment design alternatives were considered in the development of a
feasible abutment concept for construction. A ballasted abutment design provides the cantilever
moment resistance that is needed; however, longitudinal flexibility can not be provided. Due to the
large counterweight force of the ballast material, a significant normal stress is present on the
bottom surface of the abutment. Using polyethylene sheeting as an anti-friction coating between
the abutment and soil reduces the friction coefficient, , to 0.5 (VSL, 1990); however, the sliding
force required to cause movement in the abutment is still excessive.
Figure 7-3. Counterweight abutment concept (superseded)
A similar problem exists when considering a tie-down abutment system, where the
counterbalancing force is provided by soil anchors on the backspan instead of ballast material
(Figure 7-4). The post-tensioning soil anchors are required to provide a proof load to ensure that
adequate capacity can be provided to resist the ultimate moment demand. Thus, the large normal
stresses between the bottom abutment surface and the soil are large and do not allow for
longitudinal sliding. This concept presents additional concerns since the stability of the structure
for the entire life of the bridge is dependent on soil anchor reliability.
µ
141
Figure 7-4. Tie-down abutment concept (superseded)
7.4.2 Recommended Design
The recommended design uses ballast material to provide uplift resistance while incorporating
the longitudinal flexibility of a two-leg system to allow for horizontal movement due to expansion
and contraction. Similar to the Bras de la Plaine bridge, the backspan to mainspan ratio has been
designed to be 0.6, which results in an abutment backspan of 15m. The abutment system requires
the excavation of soil on each side of the highway to an adequate depth for the casting of a strong
slab to provide a firm foundation for bridge piles. The piles are intended to be encased in
corrugated steel pipe (CSP) filled with loose sand to allow longitudinal expansion/contraction of
the bridge with minimum passive soil pressure acting on the pile. Granular backfill with a density
of is assumed to provide ballast for the backspan. The calculated depth of soil, H,
required to resist uplift due to the maximum factored load on the superstructure is 8.5m. The
backspan of the abutment includes a solid anchorage zone to allow spreading of forces required for
cantilever tendons, and a front hollow compartment to minimize material consumption (Tanis,
2003). The backspan is tied down by using flexible HP piles which act as massive headed
reinforcing bars. Mechanical bond of the HP section to the concrete should be provided by a
sufficient number of shear studs welded to the flanges. The studs should be designed to effectively
develop the full cross sectional strength of the tension piles above the cantilever prestressing
anchors.
All cantilever tendons are arranged in a single horizontal plane in the superstructure, and
deviate vertically at the abutment wall. At a distance 5m from the abutment wall, all tendons are
arranged on three distinct elevations with adequate vertical clearance between tendon ducts. A
vertical wall is required at this location to provide horizontal deviation of tendons. The tendons
20kN/m3
142
continue on the same vertical trajectory but are arranged horizontally into vertical columns
consisting of three anchorages each at the back face of the abutment. Providing three rows of
anchorages at the back of the abutment minimizes the angle breaks in the tendon as it minimizes
the amount of fanning of tendons required to provide sufficient anchorage space. The centrally
anchored tendons also provides a more direct flow of forces into the components of the box girder.
Figure 7-5. Proposed conceptual abutment design
7.5 Sizing of Flexible Piles
Abendroth (1989) has provided a rational approach for the design of integral abutment bridge
piles. One of the two design alternatives he discusses is based on elastic behaviour assuming no
plastic redistribution of internal forces. For HP piles in compression, the steel sections can be
designed as beam-columns undergoing axial force due to weight of the applied vertical loads and
moment caused by horizontal displacement due to thermal variations. Following the maximum and
minimum design temperatures defined in the CAN/CSA-S6-06 for a structure of this type located
near Toronto, Ontario, the design temperature range is approximately . For a 90m span, the
total horizontal displacement due to temperature, assuming a coefficient of thermal expansion to
be / , is approximately 52mm. Assuming that the bridge is built in temperature
γ
58°C
10x10 6– °C 15°C
143
and that each abutment deflects equally, the maximum expansion and contraction at each abutment
is 8mm and 18mm, respectively. The elastic moment and horizontal force for a pile fixed at the
ends due a horizontal deflection, , is given by:
(7-1)
(7-2)
The maximum factored shear force and the maximum factored bending moment at the
abutment are conservatively assumed to occur simultaneously for the design of the compression
piles. For Vf = 9810kN, Mf = 144000kNm and d = 15m, the maximum compression force required
at the abutment wall is 20280kN. Assuming that the section used for compression piles is
HP310x125, the bending moment and horizontal force induced due to the 18mm thermal deflection
is 63kNm and 23kN, respectively. For an unbraced length of 5500mm and an effective length
factor of 0.65 for design, 6 - HP310x125 piles are required to resist the axial compression due to
the total vertical load and moment due to maximum thermal contraction, assuming the loads are
evenly distributed over the piles. For the backspan tie, 4 - HP310x95 piles are required to provide
the necessary tie down force, limiting the steel stress to 240Mpa.
7.6 Future Work
A feasible abutment concept for the proposed cantilever-constructed bridge design has been
described; however, several design considerations are yet to be considered. To name a few: a) the
design of the end zone reinforcement must be investigated for the spreading of forces for all
cantilever tendons, b) the webs must be designed to transfer the constant shear force to the
abutment wall, c) the top slab must be checked for live loads, d) the strong slab must be designed
to rigidly support the factored uplift load, and e) the front toe of the strong slab must be designed
as a spread footing for the large compression force at the abutment wall.
∆
M 6EI∆
L2-------------=
H 12EI∆
L3----------------=
144
145
8.0 MATERIAL UTILIZATION
This chapter begins by describing two reference bridges. These bridges provide a basis for
comparison to evaluate the economy in material consumption for the proposed box girder bridge.
Material quantity estimates are presented and compared for the reference bridges and the proposed
box girder bridge. Concrete consumption, mild steel utilization, longitudinal prestressing strand
utilization, and transverse prestressing strand utilization will be identified. Since the abutment
presented in this thesis has not been designed, only the superstructure quantities will be presented
and compared within this chapter.
8.1 Reference Bridges
In this section, a brief description will be given on the chosen bridge structures used for a
comparison on material consumption. The bridges are representative of two different types of
bridge design: a classical precast segmental cantilever-constructed bridge, and a typical box girder
highway overpass bridge built using current practice of the Ontario Ministry of Transportation. The
precast segmental bridge reference will provide a comparative basis for the savings in material that
may be realized by building a cantilever-constructed bridge using external unbonded tendons
instead of traditional internally bonded tendons. The highway overpass bridge reference will
provide a comparative basis for the savings in material that may be realized by adapting a new
cantilever-constructed type of overpass bridge in place of the current methods used.
8.1.1 Windward Viaduct, Interstate Route H-3, Hawaii, USA (Hawaii DOT, 1991)
The Windward Viaduct is a precast segmental cantilever-constructed multiple span bridge with
typical span lengths of 85.3m (280ft) and is located in the Haiku Valley of Hawaii. The elevation
of a typical span is shown below in Figure 8-1. The precast segments are made with typical
37.9MPa (5500psi) concrete and reinforced with ASTM 615 Grade 60 reinforcing steel. The
haunched girder has a depth of 4.88m (16ft) at the pier and a depth of 2.44m (8ft) at the mid-span.
These girder depths correspond to span-to-depth ratios of 17.5:1 and 35:1, respectively.
Figure 8-1. Partial elevation view of Windward Viaduct (adapted from Hawaii DOT, 1991)
The cross section is a single cell box girder with a vertical web and a constant web thickness
of 381mm (15in). The typical deck width is 12520mm (41ft 1in) and the web spacing is 6320mm
(20ft 9in) centre-to-centre. The top flange is constant for all girder depths. It has a typical thickness
of 229mm (9in) and tapers to a haunch of 508mm (20in) at the webs. The bottom flange varies in
thickness from 229mm (9in) at the midspan to 406mm (16in) at the pier. The typical cross section
is shown below in Figure 8-2.
Figure 8-2. Typical cross section of Windward Viaduct (adapted from Hawaii DOT, 1991)
All cantilever prestressing tendons are internal within the top flange thickness and all
continuity prestressing tendons are internal within the bottom flange thickness. In the top flange,
146
the cantilever tendons are anchored at the junction of the top flange and the web; therefore, the
haunch in the top flange is present to accommodate the anchorage and spacing of tendon ducts. The
continuity tendons are anchored at corner blisters on the bottom flange and follow the vertical
profile of the bottom slab haunch. The prestressing arrangement consists of 17-strand and 19-
strand tendons using 13mm (0.5in) strands. A typical prestressing arrangement is shown in
Figure 8-3 which displays a half plan for the top flange prestressing tendons and a half plan for the
bottom prestressing tendons.
Figure 8-3. Post-tensioning arrangement for typical span of Windward Viaduct (adapted from Hawaii DOT, 1991)
8.1.2 Hwy 407 - Islington Avenue Underpass, Toronto, Canada (MTO, 1990)
The Islington Avenue Underpass is a two-span structure with equal span lengths of 50.5m built
over Highway 407 in Toronto, Canada. The general arrangement elevation is shown below in
Figure 8-4. The superstructure is a cast-in-place post-tensioned multiple cell box girder. The
superstructure is cast with 35MPa concrete and reinforced with Grade 400 reinforcing steel. The
147
constant depth box girder has a girder depth of 2m which corresponds to a span-to-depth ratio of
25.2:1.
Figure 8-4. General arrangement of Islington Avenue Underpass (adapted from MTO, 1990)
The cross section is a multiple cell box girder with 6 webs having a typical web thickness of
840mm each. Each web contains 6 multi-strand draped post-tensioning tendons. The deck width is
20.41m, and the typical top flange thickness is 225mm. The cantilever wing extends 2m beyond
the web and tapers to a flange thickness of 450mm at the web. The typical bottom flange thickness
is 175mm. A typical cross section is shown below in Figure 8-5.
Figure 8-5. Typical cross section of Islington Avenue Underpass (adapted from MTO, 1990)
8.2 Concrete Consumption
The concrete consumption for the bridges described above and the proposed segmental box
girder have been summarized in Table 8-1, Table 8-2, and Table 8-3 below. Since all bridges
considered have different bridge widths and bridge lengths, the concrete consumption for each
bridge has been normalized to the volume of concrete consumed per square meter of bridge deck.
The concrete consumption for the Windward Viaduct has been estimated based on the variable
depth typical cross section shown in Figure 8-2, and the pier segment volume which is substantially
heavier. The concrete consumption for the Islington Avenue Underpass has been estimated based
on the typical cross section shown in Figure 8-5, and solid concrete sections which are located over
148
the abutments and the centre pier. The concrete consumption for the proposed box girder bridge is
based on the typical 3.6m deep segment.
It is observed that there is greater economy in concrete consumption for traditional cantilever-
constructed bridges than conventionally constructed bridges built by the Ontario Ministry of
Transportation (MTO). Comparing the unit volume consumption of the Windward Viaduct to the
Islington Avenue Underpass, the Islington Avenue Underpass consumes approximately 44% more
concrete per square meter of bridge deck than the Windward Viaduct. The design of multiple cell
girders with thick webs for typical MTO overpass bridge construction has a significant impact on
the overall concrete consumption. The proposed box girder bridge consumes approximately 44%
of concrete compared to classical cantilever constructed bridges, and approximately 31% of
concrete compared to traditional MTO overpass bridge construction per square meter of bridge
deck.
Table 8-1. Concrete consumption for one span of the Windward Viaduct (Hawaii DOT, 1991)
Item Area[m2]
Width[m]
Length[m]
Volume[m3]
Unit Volume[m3/m2]
Typical Segment varies from 7.18 to 9.61 12.5 81.1 637 0.628
Pier Segment 13.7 12.5 4.20 58 1.10
Total 85.3 695 0.651
Table 8-2. Concrete consumption for Islington Avenue Underpass (MTO, 1990)
Item Area[m2]
Width[m]
Length[m]
Volume[m3]
Unit Volume[m3/m2]
Typical Section 16.2 20.41 84 1360 0.794
Solid sections over supports 33.6 20.41 17 571 1.65
Total 101 1931 0.937
Table 8-3. Concrete consumption for proposed cantilever box girder
Item Area[m2]
Width[m]
Length[m]
Volume[m3]
Unit Volume[m3/m2]
Typical Section 3.43a
a. smeared area accounts for transverse ribs and corner blisters
12 90 308 0.286
149
8.3 Mild Reinforcing Steel Utilization
The mild reinforcing steel utilization for the reference bridges and the proposed segmental box
girder have been summarized in Table 8-4, Table 8-5, and Table 8-6 below. Since concrete
consumption per square meter of bridge deck is vastly different between the structures compared,
the classical reinforcement utilization ratio expressed on the basis of mass of steel per cubic meter
of concrete is less meaningful. For a more consistent comparison, reinforcing steel utilization on
the basis of mass of steel per square meter of bridge deck has also been given. The Windward
Viaduct mild steel reinforcement has been estimated based on the reinforcing schedule for the
typical segment. For the pier segment, the quantity of steel has been estimated by assuming the
calculated reinforcing ratio, ρ = 0.0089, based on the typical section. The Islington Avenue
Underpass mild steel reinforcement has been estimated based on the reinforcing schedule for the
typical cross section. For the solid sections over the supports, the quantity of steel has been
estimated by assuming the calculated reinforcing ratio, ρ = 0.0088, based on the typical section.
The mild steel reinforcement for the proposed box girder bridge accounts for the required web
reinforcement for global spreading of forces and the anchorage zone reinforcement required for
local spreading.
Table 8-4. Mild reinforcing steel utilization for one span of the Windward Viaduct (Hawaii DOT, 1991)
Item SteelVolume
[m3]
SteelMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Top Flange (typ.) 2.22 17500 25.2 16.4
Webs (typ.) 1.13 8880 12.8 8.30
Bottom Flange (typ.) 2.23 17500 25.2 16.3
Pier Segment c
c. assumes equal reinforcing ratio as the typical segment, ρ = 0.0089
0.52 4070 5.8 3.81
6.10 48000 69.0 44.9
150
On the basis of mild steel consumption per square meter of bridge deck, it is observed that there
is greater economy in steel consumption for classical cantilever construction than for
conventionally constructed bridges built by the MTO. However, both structures are reinforced with
approximately the same reinforcing ratio on the basis of steel consumption per cubic meter of
concrete. Since the overall concrete consumption for the Windward Viaduct is less than the
Islington Avenue Underpass, the steel consumption for the Windward Viaduct is also less. For the
proposed box girder bridge, the low concrete consumption results in a high steel utilization factor
on the basis of steel consumption per cubic meter of concrete. However, the economy in the
proposed girder design can be seen when comparing the material utilization factor based on the
bridge deck area. The proposed box girder bridge consumes only 50% of mild reinforcing steel in
comparison to the Windward Viaduct, and only 29% mild reinforcing steel in comparison to the
Islington Avenue Underpass on the basis of bridge deck area.
Table 8-5. Mild reinforcing steel utilization for Islington Ave Underpass (MTO, 1990)
Item SteelVolume
[m3]
SteelMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Top Flange (typ.) 5.49 43100 22.3 25.1
Webs (typ.) 1.63 12800 6.63 7.46
Bottom Flange (typ.) 4.85 38100 19.7 22.2
Solid sections over supportsc
c. assumes equal reinforcing ratio as the typical section, ρ = 0.0088
5.02 39400 20.41 23.0
17.0 133000 69.0 77.8
Table 8-6. Mild reinforcing steel utilization for proposed box girder
Item SteelVolume
[m3]
SteelMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Web Reinforcement 1.17 9190 29.8 8.51
Anchorage Zone Reinforcementc
c. includes bursting reinforcement in slabs
1.90 14900 48.4 13.8
3.07 24100 78.2 22.3
151
8.4 Longitudinal Post-Tensioning Utilization
The longitudinal post-tensioning strand utilization for the reference bridges and the proposed
segmental box girder have been summarized in Table 8-7, Table 8-8, and Table 8-9 below. For
consistency in steel utilization, the amount of strand consumed has been expressed in terms of mass
of strand per square meter of bridge deck. The Windward Viaduct strand consumption and number
of required anchorage units has been estimated based on one typical span. The Islington Avenue
Underpass strand consumption and number of required anchorages has been determined based on
the full bridge span. The proposed box girder bridge strand consumption and number of required
anchorage units accounts for all anchorages and strand within the span and the abutment.
Table 8-7. Post-tensioning utilization for one span of the Windward Viaduct (Hawaii DOT, 1991)
Item Number of Anchorages
Strand Volume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Cantilever Tendons 60 2.30 18100 26.0 16.9
Continuity Tendons 32 1.14 8920 12.8 8.35
92 3.44 27000 38.8 25.3
Table 8-8. Post-tensioning utilization for Islington Ave. Underpass (MTO, 1990)
Item Number of Anchorages
Strand Volume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Draped Tendons 72 13.2 104000 53.8 50.4
Table 8-9. Post-tensioning utilization for proposed cantilever box girder
Item Number of Anchorages
Strand Volume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Cantilever Tendons 120 1.03 13300 43.2 12.3
Continuity Tendons 32 1.69 13200 42.9 12.2
152 2.72 26500 86.1 24.6
152
The longitudinal post-tensioning strand utilization for the proposed girder is comparable to the
strand utilization of a traditional cantilever constructed bridge per square meter of bridge deck.
Within the single span of the proposed girder, the strand utilization is less than the traditional
cantilever girder; however, this economy is offset by the significant extension of cantilever strand
length that is required to the back face of the abutment. The required number of anchorage units
for the proposed box girder is also significantly increased since the proposed box girder requires
twice the number of anchorage units for cantilever post-tensioning for one span (accounting for
anchors at the stressing face and at the abutment). In comparison to conventional overpass
construction, the proposed box girder consumes approximately 49% post-tensioning strand, but
requires approximately 2.1 times the number of anchorages.
8.5 Transverse Post-Tensioning Utilization
The transverse post-tensioning strand utilization for the reference bridges and the proposed
segmental box girder have been summarized in Table 8-10, Table 8-11, and Table 8-12 below. The
Windward Viaduct top flange is transversely post-tensioned with 4-15mm multi-strand post-
tensioning flat slab tendons. The Islington Avenue Underpass is not designed with a transversely
post-tensioned deck; however, transverse post-tensioning tendons are used in the solid cross
section regions in the vicinity of the supports. The proposed box girder is transversely prestressed
with 15mm pretensioning strands.
Table 8-10. Transverse prestressing steel utilization for one span of the Windward Viaduct (Hawaii DOT, 1991)
Item StrandVolume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Post-tensioning tendons 1.17 9210 13.2 8.62
Table 8-11. Transverse prestressing steel utilization for Islington Avenue Underpass (MTO, 1990)
Item StrandVolume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
a. per cubic meter of concrete
Utilization Ratiob
[kg/m2]
b. per square meter of deck
Post-tensioning tendonsc
c. transverse post-tensioning is only at the supports. deck is not transversely prestressed.
0.346 2720 1.41 1.32
153
Comparing the transversely prestressed deck of the proposed box girder to the Windward
Viaduct, it is observed that the proposed box girder requires approximately 58% more prestressing
strand. The deck slab design for the proposed girder has been carried out using high strength FRC
and thin section components. The major design requirements for the top flange proportioning
includes that: a) the slab thickness satisfies strength requirements for longitudinal global bending,
b) the slab thickness provides a minimum of 40mm top cover to bonded steel and 30mm bottom
cover to bonded steel, c) the transverse stiffness provides an acceptable maximum allowable
deflection based on a chosen reference structure, and d) the SLS cracking and fatigue requirements
are satisfied. This design minimizes the consumption of high strength concrete; however, the
minimization of concrete consumption and the resulting light weight design comes at the cost of
an increased need for transverse prestressing steel.
8.6 Summary
Using the material utilization factors derived above for material consumption per square meter
of bridge deck, a relative savings in materials can be determined for the proposed bridge design.
For example, the savings in material of the proposed girder with respect to the Windward Viaduct
may be expressed as:
The relative material savings are given in Table 8-13 below.
Table 8-12. Transverse prestressing steel utilization for proposed box girder
Item StrandVolume
[m3]
StrandMass[kg]
Utilization Ratioa
[kg/m3]
Utilization Ratiob
[kg/m2]
Pretensioning strands 1.86 14600 47.5 13.6
a. per cubic meter of concreteb. per square meter of deck
Table 8-13. Relative material consumption of proposed box girder
Item Windward Viaduct Islington Avenue Underpass
Concrete 0.44 0.31
Mild Reinforcing Steel 0.50 0.29
Longitudinal Prestressing 0.97 0.49
Transverse Prestressing 1.58 n/a
Utilization FactorproposedUtilization FactorWindward Viaduct-----------------------------------------------------------------------------
154
155
9.0 CONCLUSIONS
This chapter summarizes the principal findings related to the bridge concept described in this
thesis and the application of external unbonded prestressing for cantilever construction.
9.1 Longitudinal Design
The increase in stress for external unbonded tendons in a cantilever-constructed girder is small
at the ultimate limit state. The increased rotational capacity achieved through the ductile nature of
high-strength fibre reinforced concrete does not significantly improve the economy of prestressing
steel consumption. This is due to the fact that only a small plastic hinge length forms in negative
bending regions. However, economy in prestressing steel consumption is achieved by reducing the
dead load of the girder through the minimization of concrete consumption. The minimization of
concrete consumption is achieved through the design of thin section components. In comparison
to conventional precast cantilever construction, the demand for longitudinal prestressing steel is
similar. However, in comparison to conventional overpass bridge construction, the proposed box
girder requires only 49% of prestressing strand per square meter of bridge deck.
The moment development length for a box girder with external unbonded prestressing tendons
anchored at intermediate corner blisters is much larger than a girder with internal tendons. In
addition to the typical spreading of compressive stresses in a cross section to achieve moment
resistance, a local spreading of forces is first required to deviate the eccentrically applied prestress
force into the box section. While the CAN/CSA-S6-06 requires that continuity tendons should be
anchored one segment beyond where they are theoretically required, external unbonded tendons
should be anchored two segements beyond where they are theoretically required.
A constant depth girder appears to be the most economical solution for the intended
standardized precast box girder cantilevered from fixed abutments. There are four reasons for this
result. First, the reduction in cantilever dead load moment at the fixed end is small due to a decrease
in web depth at midpsan. Second, reducing the internal lever arm to the continuity tendons at
midspan increases the demand for continuity prestressing steel. Third, a larger redistribution of
positive bending moment to the midspan is desirable since it relieves the overturning demand on
the substructure. Lastly, for the intent of precasting and standardization, economy is optimized by
maintaining repetitive and standard detailing of struts for a constant depth girder. A span-to-depth
ratio of 1:25 appears to be a reasonable girder slenderness for design using nominal 80MPa FRC.
This girder slenderness satisfies strength requirements, deflection requirements, and provides
reasonable visual slenderness.
9.2 Transverse Design
For the design of thin-slabs, a pretensioning transverse prestress concept is necessary.
Pretensioning strands can be placed more compactly in the top slab and the well-distributed
arrangement of prestressing strands provides a more uniform introduction of prestress force at the
cantilever tip. The high tensile strength of FRC facilitates excellent bond strength on 15mm
strands, reducing the transfer length to allow for a sufficient effective prestress at the inside face
of the barrier.
The design of thin slabs using high-strength FRC markedly reduces overall concrete
consumption and reduces dead load; however, the decrease in flange thickness results in an
increased demand for transverse prestressing steel to satisfy serviceability requirements. A
comparison of transverse prestressing utilization showed that the proposed box girder requires 58%
more prestressing strand (per square meter of bridge deck) than a conventional box girder with free
cantilever wings.
9.3 Shear Design
The flow of forces in the webs of box girders prestressed with curtailed tendons at intermediate
anchorages at the top and bottom flanges is complex. The flow of forces can be separated into a
local spreading of forces in the vicinity of the anchorage and a global flow of forces in the cross
section to transfer vertical applied load to the supports. As a consistent methodology for modelling
the flow of forces in the girder, a parallel chord truss model with chords at the elevation of the
flanges is developed for the design of web reinforcement. For the design of thin webs, a single layer
of web reinforcement is placed in the central plane of the web axis.
9.4 Anchorage Zone Design
The flow of forces in anchorage zones for girders prestressed with external unbonded tendons
at corner blisters is fundamentally different from girders prestressed with internal tendons at corner
blisters. Additional mild steel reinforcement is required for girders with unbonded tendons to
156
deviate the prestress force into the cross section. This anchorage zone steel reinforcement can be
reduced by minimizing the distance from the anchorage to the junction of the flange and web. The
anchorage zone steel reinforcement can also be reduced by increasing the length of the anchorage
blister which allows the deviation to occur over a longer distance. An increase in deviation length,
however, also increases the moment development length for the global bending resistance of the
cross section. The anchorage blister length for continuity tendons is governed by jack clearance
requirements for the tendon stressing operation.
As a practical limitation, the maximum tendon unit size for use in external post-tensioned
bridges is 19 strands of 15mm diameter. Larger tendon units result in an increased demand for local
spreading reinforcement which increases congestion of reinforcing bars.
9.5 Economy
Based on material consumption estimations, a large savings in material for the superstructure
may be realized for overpass bridge construction by adapting a cantilever-constructed box girder
design using external unbonded tendons. In comparison with conventional construction for
overpass bridges in Ontario, the proposed bridge design consumes only 31% of concrete, 29% of
mild reinforcing steel, and 49% of longitudinal post-tensioning strand per square meter of bridge
deck. The proposed bridge, however, requires a significant amount of prestressing steel for the
transverse prestressing system. In comparison to a conventional box girder bridge with a
transversely prestressed concrete deck, 58% more prestressing strand consumption is required for
the proposed box girder.
The material consumption for the proposed box girder superstructure indicates that a savings
in material is possible; however, a full bridge design including abutment and approach materials is
necessary to understand the overall savings of this concept.
9.6 Future Work
The current work has identified the primary design considerations for longitudinal bending,
transverse bending, shear, and anchorage zone detailing. However, additional force effects due to
thermal loads and torsion have not been considered in this work. For final design, creep and
shrinkage should be considered in a more explicit manner. The use of small concrete covers has
been assumed in this study, but this assumption has not been supported with a definitive proof.
Further work is required to validate the use of small concrete covers for high-performance
157
concrete. Finally, a complete substructure analysis needs to be performed to validate the proposed
box girder concept for cantilever construction.
158
159
10.0 REFERENCES
Abendroth, R.E., and Greimann, L.F. 1989. Rational Design Approach for Integral Abutment Bridge Piles. Transportation Research Record. 1223: 12-23.
American Association of State Highway and Transportation Officials. 1996. Standard Specifications for Highway Bridges, 16th Ed. Washington, D.C.
American Concrete Institute. 1999. Building Code Requirements for Structural Concrete, ACI 318-99. ACI, Detroit, Mich.
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APPENDIX A DESIGN DRAWINGS
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APPENDIX B MATERIAL TESTING
This appendix describes the concrete mix design adapted from Susetyo (2007) and the
compression cylinder test results obtained for the 7-day strength tests and the 28-day compressive
behaviour tests. The stress-strain relationship obtained from these material tests was implemented
as the design material model for the proposed bridge concept.
B.1 Reference Concrete Mix
A nominal 80Mpa fibre-reinforced concrete mix design developed by Susetyo (2007) was
adapted for the specimen design. The concrete mix has a water-to-cement ratio of 0.27 and uses
high silica fume (HSF) cement. The hooked steel fibre volume ratio is 1.5%. The coarse and fine
aggregates are measured by weight in the saturated surface dry (SSD) condition, where the
aggregates neither absorb water from nor contribute water to the concrete mixture. The total
volume of concrete required for one test batch (for casting of two cylinders) is 0.013m3, which
includes a 20% allowance for spillage and losses. Scaling the proportions directly for the reference
mix and the batch mix, the mix proportions are described in Table B-1 below:
Since the reference design mix is calibrated for fine and coarse aggregate in the SSD condition,
the sand and limestone aggregates in the Galbraith Building concrete lab were tested for moisture
content and absorption rates. The moisture content test results are given below in Table B-2:
Table B-1. Reference concrete mix quantities (Susetyo, 2007) and scaled batch quantities
MaterialMeasured
Unit Ref. QuantityRef. Equivalent
VolumeScaled Batch
Quantity
HSF Cement kg 600 0.191 7.20
Sand (SSD) kg 1133 0.419 13.59
10mm limestones (SSD) kg 802 0.292 9.62
Water L 162 0.162 1.94
Water Reducer mL 4200 - 50.40
Superplasticizer mL 9600 - 115.20
Entrapped Air 0.02
Total Volume m3 1.083
The water absorption rate of the sand and the stone are 0.99% and 1.22%, respectively. Based
on the measured moisture content and absorption rates of the fine and coarse aggregate, the batch
quantities were adjusted to match the designed water-to-cement ratio of 0.27. The adjusted batch
quantities are given below in Table B-3:
B.2 Material Batching and Preparation
Six compression cylinders 150mm diameter by 300mm in height were desired for testing the
FRC behaviour - three cylinders were intended to measure the 7-day strength, and three cylinders
were intended to measure the 28-day compressive stress-strain response. The small mixer in the
Galbraith Building concrete lab was used for the batching and mixing of the concrete. This required
three separate batches (for casting of two cylinder specimens each) for mixing due to the volume
limitation of the mixer. Specimens were cast in the cylinder moulds and left to harden for 1 day,
and then the cylinders were demoulded and placed in the moist-curing room to cure. The cylinders
were removed temporarily after 6 days for end grinding, and then replaced in the moist-curing
room.
Table B-2. Measured moisture content of fine and coarse aggregates
Sand Stone
Tray 200g 195g
Tray + Material (wet) 1590g 1700g
Tray + Material (oven dry) 1580g 1685g
Moisture Content 0.72% 1.01%
Table B-3. Adjusted batch quantities for compression cylinders
MaterialMeasured
Unit Single BatchTotal Material
(3 Batches)
HSF Cement kg/m3 7.20 21.60
Sand kg/m3 13.55 40.66
10mm limestones kg/m3 9.60 28.80
Water L/m3 2.00 6.00
Water Reducer mL/m3 50.40 151.20
Superplasticizer mL/m3 115.20 345.60
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B.3 7-day Compressive Strength Tests
One cylinder from each batch was tested for 7-day strength. No displacement measurements
were taken for the load deflection response. The cylinders were loaded at a constant rate of 4kN/s.
The test cylinders were observed to have a slightly lower compressive 7-day strength than expected
based on previous results obtained by Susetyo. The 7-day strength results are summarized below
in Table B-4, where the equivalent stress has been calculated based on the gross cross sectional
area:
B.4 28-day Compressive Strength Tests
The remaining cylinders from each batch was tested for 28-day strength. Results from the
analog plot based on the displacement of the load head of the test machine have been presented in
Chapter 2. The peak stress and peak strain results are tabulated below in Table B-5:
Table B-4. 7-day compressive strength results
Cylinder Peak Load (kN) Peak Stress (Mpa)
FRC 1b 979 55.4
FRC 2b 1164 65.9
FRC 3b 999 56.5
Average 1047 59.3
Table B-5. 28-day compressive strength results
Cylinder f’c (Mpa) (mm/mm) Ec (Mpa)
FRC 1a 72.1 0.002139 48200
FRC 1b n/a n/a n/a
FRC 1c 75.9 0.002050 39160
Average 74.6 0.00209 43680
ε’c
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APPENDIX C CAN/CSA-S6-06 SHEAR PROVISIONS
The sectional design approach of the CAN/CSA S6-06 shear provisions calculates the shear
resistance of concrete members based on an idealized cross section that is composed of a flexural
compression flange, a web, and a flexural tension flange. The total shear resistance of the web is
due to the additive contribution of the shear resistance provided by tensile stresses in concrete, Vc,
shear resistance provided by the transverse steel reinforcement, Vs, and the shear resistance
provided by the component of force in a prestress tendon acting in the direction of applied shear,
Vp.
(C-1)
The factored shear resistance due to the contribution of tensile stresses in the concrete is given
as:
(C-2)
where is the factor that accounts for the shear resistance of cracked concrete, fcr is the
cracking stress of concrete, bv is the width of the structural component resisting shear, and dv is
taken as 0.9d. The cracking stress is limited to 3.2MPa since observed tests have shown no
increased benefit in shear resistance with concrete strengths above 65MPa (CSA S6.1-06).
The factored shear resistance due to the contribution of transverse steel reinforcement is given
as:
(C-3)
where Av is the cross sectional area of each stirrup, fy is the yield stress of reinforcing steel,
is the angle of compressive stresses in the web, and s is the spacing of stirrups.
The angle of compressive stress, , and the ability of cracked concrete to transfer stress across
a crack, , are determined through convenient empirical equations based on analytical results of
MCFT elements. These parameters are given as:
(C-4)
Vr Vc Vs Vp+ +≥
Vc 2.5βφcfcrbvdv=
β
VsφsAvfydv θcot
s----------------------------------=
θ
θ
β
β 0.41 1500εx+--------------------------⎝ ⎠
⎛ ⎞ 13001000 sze+-------------------------⎝ ⎠
⎛ ⎞=
(C-5)
where sze is the crack spacing parameter and is defined as . The parameter
accounting for nominal size of coarse aggregate, ag, is taken as zero for high strength concrete
above 70MPa and the spacing factor, sz, is taken as dv for girders with no longitudinal steel in the
webs.
Most importantly, the parameters required for the calculation of shear resistance are dependent
on the longitudinal strain in the concrete member taken at the mid-height of the section, due to
the combined actions of shear, moment, and prestressing. Conservative assumptions for the
calculation of are built into the code for a safe approximation that can be calculated in lieu of a
more accurate iterative calculation, which are described in cl.8.9.3.8. The strain at mid-height is
given as:
(C-6)
where Mf, Vf, and Nf are applied sectional forces due to moment, shear, and axial force,
respectively, and taken as positive quantities. The area of prestressing steel is given as Aps, and fpe
represents the long-term effective prestress for unbonded tendons, taken as 0.6fpu in this study. For
compressive mid-height strains, the denominator is modified to account for the axial stiffness of
the concrete flange on the flexural tension side.
The parallel chord model used for the CAN/CSA-S6-06 shear provisions does not account for
an arching contribution due to the flow of inclined compression forces following the funicular
shape, but rather it accounts for the presence of prestressing by simply offsetting the force in the
tension flange by including the compression force due to all prestressing tendons that are on the
flexural tension side of the girder. This is represented by the Apsfpe term that appears in the
numerator of the equation for .
θ 29 7000εx+( ) 0.88sze
2500------------+⎝ ⎠⎛ ⎞=
35sz 15 ag+( )⁄( )
εx
εx
εxMf dv⁄ Vf Vp– 0.5Nf Apsfpe–+ +
2 EsAs EpAps+( )------------------------------------------------------------------------------------=
εx
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180
CURRICULUM VITAE
Brent Tyler Visscher
Born September 6, 1981 in St. Thomas, Ontario, Canada
Education
Master of Applied Science (2008)University of Toronto, Toronto, Ontario, Canada
Bachelor of Engineering Science (2005)The University of Western Ontario, London, Ontario, Canada
Honours and Awards
2006, Excellence in Graduate Studies Award2005, Natural Sciences and Engineering Research Council, Canada Graduate Scholarship2005, Ontario Graduate Scholarship (Declined)2005, Dr. James A. Vance Gold Medal in Civil Engineering2005, 1st Place Civil and Environmental Engineering Thesis2005, 1st Place Team of the City of London Bridge Design Competition2004, London and District Construction Association Award2004, Dr. L. Stuart Lauchland Scholarship2003, Natural Sciences and Engineering Research Council Undergraduate Student Research Award2000, Governor General's Bronze Medal
Related Work Experience
September 2005 to December 2007Teaching Assistant, University of Toronto, Toronto, Canada
September 2003 to August 2004Engineer-In-Training, Buckland & Taylor Ltd. Bridge Engineers, North Vancouver, Canada
May 2003 to August 2003Project Researcher, Alan G. Davenport Wind Engineering Group, London, Canada
Publications
Refereed Journal Papers
Visscher, B.T., Kopp, G.A. 2007. Trajectories of Roof Sheathing Panels Under High Winds, Journal of Wind Engineering and Industrial Aerodynamics, 95, 8: 697-713.
Conference Proceedings
Visscher, B.T., Kopp, G.A. 2005. Wind Tunnel Measurements of Trajectories of Roof Sheathing Panels Under High Winds, Proceedings of the 10th Americas Conference on Wind Engineering, Baton Rouge, USA (CD-ROM).
Other Publications - Reports
Visscher, B.T. 2005. Trajectories of Roof Sheathing Panels Under High Winds. Graduation Thesis, Department of Civil and Environmental Engineering, The University of Western Ontario, London, Canada.
Visscher, B., Kopp, G.A., and Vickery, P.J. 2004. Wind Loads on Houses: Destructive Model Testing of a Residential Gable Roofed House, Institute for Catastrophic Loss Reduction Research Paper 37, London, Canada.
Visscher, B. 2003. Destructive model of a low-rise residential gable roofed house, Alan G. Davenport Wind Engineering Group Rep. No. BLWT-1-2003, The University of Western Ontario, London, Canada.
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