Behaviour and Design of
EXTRADOSED BRIDGES
by
Konstantinos Kris Mermigas
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Civil Engineering University of Toronto
© Copyright by Konstantinos Kris Mermigas (2008)
Behaviour and Design of Extradosed Bridges Konstantinos Kris Mermigas Master of Applied Science Graduate Department of Civil Engineering University of Toronto 2008
ABSTRACT The purpose of this thesis is to provide insight into how different geometric parameters such as tower height, girder depth, and pier dimensions influence the structural behaviour, cost, and feasibility of an extradosed bridge. A study of 51 extradosed bridges shows the variability in proportions and use of extradosed bridges, and compares their material quantities and structural characteristics to girder and cable-stayed bridges. The strategies and factors that must be considered in the design of an extradosed bridge are discussed. Two cantilever constructed girder bridges, an extradosed bridge with stiff girder, and an extradosed bridge with stiff tower are designed for a three span bridge with central span of 140 m. The structural behaviour, materials utilisation, and costs of each bridge are compared. Providing stiffness either in the girder or in the piers of an extradosed bridge are both found to be effective stategies that lead to competitive designs.
ii
ACKNOWLEDGEMENTS I would like to thank Professor Gauvreau for the opportunity to study under his guidance and be part of a dynamic and ambitious bridge research group. Professor Gauvreau has served as an inspiration and mentor to me in my career. Thanks to my research colleagues for their insightful discussions and feedback: Mike Montgomery, Jimmy Susetyo, Ivan Wu, Jeff Erochko, Lydell Wiebe, Brent Visscher, Jamie McIntyre, Lulu Shen, Eileen Li, Sandy Poon, Davis Doan, and especially Jason Salonga. Finally, thanks to my family and Mary Jane for encouraging and supporting me through my graduate studies.
iii
TABLE OF CONTENTS ii
ABSTRACT
iii
ACKNOWLEDGEMENTS
iv
TABLE OF CONTENTS
viii
LIST OF FIGURES
xiii
LIST OF TABLES
1
1 WHAT MAKES A BRIDGE EXTRADOSED?
1
1.1 Introduction
2
1.2 Objectives and Scope
3
1.3 Historical Context
13
1.4 General Studies on Extradosed Bridges
15
2 REVIEW OF EXISTING EXTRADOSED BRIDGES
15
2.1 Study of Extradosed Bridges
15
2.2 Trends in Extradosed Bridges to Date
29
2.3 Characteristics of Extradosed Bridges
29
2.3.1 Materials Usage
31
2.3.2 Girder Stiffness
32
2.4 Detailed Descriptions of Extradosed Bridges
32
2.4.1 Odawara Port Bridge, Japan
33
2.4.2 Tsukuhara Extradosed Bridge, Japan
34
2.4.3 Ibi and Kiso River Bridges, Japan
35
2.4.4 Shin-Meisei Bridge, Japan
36
2.4.5 North Arm Bridge, Canada
37
2.4.6 Pont de Saint-Rémy-de-Maurienne, France
38
2.4.7 Viaduc de la ravine des Trois Bassins, Réunion
38
2.4.8 Sunniberg Bridge at Klosters, Switzerland
40
2.5 Concluding Remarks
iv
41
3 DESIGN AND CONSTRUCTION OF EXTRADOSED BRIDGES
41
3.1 Loads
41
3.1.1 Live Load
44
3.1.2 Temperature
46
3.2 Design Concepts
46
3.2.1 Stiffness of Cables and Girder
48
3.2.2 Stiffness of Superstructure and Substructure
50
3.2.3 Prestressing Methodology
52
3.3 Conceptual Design
52
3.3.1 Fixity of the Girder to the Piers
53
3.3.2 Side Span Length
53
3.3.3 Tower Height and Girder Depth
56
3.4 Stay Cables and Anchorages
56
3.4.1 Cable Arrangement
60
3.4.2 Stay Cable Protection
60
3.4.3 Anchorages in Girder
61
3.4.4 Cable Anchorage in Towers: Saddles or Anchorages?
64
3.4.5 Equivalent Modulus of Elasticity for Stay Cables
65
3.4.6 Preliminary Design of Stay Cables at Serviceability Limit States
67
3.4.7 Verification of Stay Cables at the Fatigue Limit State
68
3.4.8 Verification of Stay Cables at Ultimate Limit States
69
3.4.9 Stay Cable Tensioning
74
3.5 Towers and Piers
75
3.6 The Girder Cross-Section
76
3.6.1 Centrally Supported Box Girder Cross-Section
78
3.6.2 Laterally Supported Single Cell Box Girder Cross-Section
78
3.6.3 Multiple Cell Box Girder Cross-Section
80
3.6.4 Laterally Supported Stiffened Slab Cross-Section
81
3.6.5 Composite Cross-Section
82
3.7 Tendon Layout
83
3.8 Erection
85
3.9 Verification at the Ultimate Limit States
86
3.10 Concluding Remarks v
88
4 DESIGN OF CANTILEVER CONSTRUCTED GIRDER BRIDGE, EXTRADOSED BRIDGE WITH STIFF GIRDER, AND EXTRADOSED BRIDGE WITH STIFF TOWER
89
4.1 Design Assumptions
89
4.1.1 Material Properties and Detailing
90
4.1.2 Analysis and Limit States Verification
91
4.1.3 Temperature Gradient
92
4.1.4 Construction Sequence and Segment Construction Cycle
93
4.2 Cantilever Constructed Girder Bridge
93
4.2.1 Layout and Cross-Section
93
4.2.2 Longitudinal Prestressing
96
4.2.3 Verification at SLS and ULS
97
4.3 Extradosed Bridge with Stiff Deck
97
4.3.1 Layout and Cross-Section
98
4.3.2 Longitudinal Prestressing
100
4.3.3 Comparison with Bending Moments in a Girder Bridge
102
4.3.4 Detailed Model
102
4.3.5 Verification at SLS and ULS
104
4.4 Extradosed Bridge with Stiff Tower
104
4.4.1 Layout and Cross-Section
106
4.4.2 Detailed Model
108
4.4.3 Verification at SLS and ULS
108
4.5 Design Comparison
108
4.5.1 Girder Cross-Section
109
4.5.2 Material Quantities
110
4.5.3 Cost Comparison
112
5 CONCLUSIONS
112
5.1 Review of Extradosed Bridges
112
5.2 Design Considerations
113
5.3 Comparison between Extradosed and Cantilever Constructed Girder Bridges
114
6 REFERENCES
vi
122
DRAWINGS
123
CANT-S1. Cantilevered PT Bridge - General Arrangement
124
CANT-A-S2. Cantilevered PT Bridge, Mixed Tendons - P.T. Tendon Duct Locations
125
CANT-A-S3. Cantilevered PT Bridge, Mixed Tendons - P.T. Layout
126
CANT-B-S2. Cantilevered PT Bridge, Internal Tendons - P.T. Tendon Duct Locations & Typical Reinforcement
127
CANT-B-S3. Cantilevered PT Bridge, Internal Tendons - P.T. Layout
128
EXTG-S1. Extradosed Bridge, Stiff Girder - General Arrangement
129
EXTG-S2. Extradosed Bridge, Stiff Girder - P.T. Tendon Duct Locations & Typical Reinforcement
130
EXTG-S3. Extradosed Bridge, Stiff Girder - P.T. Layout
131
EXTT-S1. Extradosed Bridge, Stiff Tower - General Arrangement
132
EXTT-S2. Extradosed Bridge, Stiff Tower - P.T. Tendon Duct Locations & Typical Reinforcement
133
EXTT-S3. Extradosed Bridge, Stiff Tower - P.T. Layout
134
APPENDIX A Chapter 2 Supplementary Information
139
APPENDIX B Chapter 4 Supporting Calculations
140
B.1 Girder Bridge Preliminary PT Design A - Mixed Tendons
144
B.2 Girder Bridge PT Design A - Detailed Model SLS Stress Checks
145
B.3 Girder Bridge PT Design A - Detailed Model ULS Moment Capacity Check
146
B.4 Girder Bridge Preliminary PT Design B - Internal Tendons
150
B.5 Girder Bridge PT Design B - Detailed Model SLS Stress Checks
151
B.6 Girder Bridge PT Design B - Detailed Model ULS Moment Capacity Check
152
B.7 Stiff Girder Extradosed Bridge PT Design - Detailed Model SLS Stress Checks
153
B.8 Stiff Girder Extradosed Bridge PT Design - Detailed Model ULS Moment Capacity Check
154
B.9 Stiff Girder Extradosed Bridge - Cable Forces
155
B.10 Stiff Tower Extradosed Bridge PT Design - Detailed Model SLS Stress Checks
156
B.11 Stiff Tower Extradosed Bridge PT Design - Detailed Model ULS Moment Capacity Check
157
B.12 Stiff Tower Extradosed Bridge - Cable Forces
158
APPENDIX C Chapter 4 Quantities
159
C.1 Breakdown of Prestressing Quantities of Chapter 4 Bridges
160
APPENDIX D Presentation Handout
vii
LIST OF FIGURES 1
1 WHAT MAKES A BRIDGE EXTRADOSED?
1
1-1 Comparison between cantilever-constructed girder, extradosed, and cable-stayed bridge types.
2
1-2 Finback, cable-panel and extradosed bridge types.
4
1-3 Ganter Bridge (Vogel & Marti 1997)
5
1-5 Barton Creek Bridge (Gee 1991).
5
1-4 Arrêt-Darré Viaduct (Mathivat 1988).
6
1-6 Odawara Blueway Bridge (Kasuga 2006).
7
1-7 Saint-Remy-de-Maurienne Bridge (photo by Jacques Mossot, Structurae).
7
1-8 Concept for Usses Viaduct (Virlogeux 2002b).
8
1-9 Santiago Calatrava’s concepts for crossing deep Alpine valleys. From left to right: Variant 1, Variant 2 model, Variant 7 sketch and detail presented by Menn at the IABSE Symposium in Zurich in 1979 (Calatrava 2004).
8
1-10 Response of cable-stiffened, girder-stiffened, and tower-stiffened cable-stayed bridge to live load.
9
1-11 Poya Bridge, Switzerland: a) Menn’s 1989 proposal (Menn 1996) and b) cable-stayed design selected in 2006 for construction (Mandataire Projet Poya 2005).
10
1-12 Millau Viaduct tower options (Virlogeux 2004). Drawn by Sir Norman Foster after discussions with Virlogeux.
11
1-13 Sunniberg Bridge, Switzerland.
11
1-14 Arrêt-Darré and Sunniberg Bridges (see drawings in Figure 2-1).
12
1-15 Golden Ears Hybrid Extradosed Bridge, Vancouver (Bergman 2007).
13
1-16 North Arm Bridge, Canada Line LRT, Vancouver (photo courtesy of Stephen Rees)
13
1-17 Pearl Harbor Memorial Bridge in New Haven, Connecticut (Stroh et al. 2003)
15
2 REVIEW OF EXISTING EXTRADOSED BRIDGES
23
2-1 Drawings of extradosed bridges.
27
2-2 Extradosed Bridges separated span to depth ratio at a) pier and b) midspan.
28
2-3 Span to depth ratios of extradosed bridges at midspan and pier.
28
2-4 Haunching in extradosed bridges shown a) in groups by span length and b) as the pier to midspan depth ratio by span.
29
2-5 Span to tower height ratio of extradosed bridges.
viii
30
2-6 Average girder concrete thickness of cantilever-constructed girder, extradosed and cable-stayed bridges.
30
2-7 Average girder concrete thickness of extradosed bridges.
31
2-8 Mass of steel in cantilever constructed girder bridges: a) longitudinal prestressing steel, and b) reinforcing steel (plots are based on data from SETRA 2007, Lacaze 2002, DEAL 1999).
32
2-9 Moment of inertia of girder at midspan for extradosed and cable-stayed bridges (per 10 m width).
33
2-10 Odawara Extradosed Bridge details of tower saddle and arrangement of prestressing bars in tower from FEM analysis (Kasuga et al. 1994).
33
2-11 Odawara Extradosed Bridge: a) strand supply system; b) saddle structure at the pier top, and c) anchorage structure at the main girder. (Toniyama & Mikami 1994).
34
2-12 Ibi River Bridge Prestressing Tendon Layout in Cross-Section (Kutsuna et al. 1999).
35
2-13 Nonlinear Behaviour of the Ibi River Bridge up to ultimate load (Kutsuna et al. 2002).
35
2-14 Shin-Meisei Birdge construction of side spans (Iida et al. 2002).
36
2-15 Shin-Meisei Birdge a) photo of steel shell of tower; b) elevation of composite tower and c) details of composite tower (drawings: Iida et al. 2002, photo and rendering: Kasuga 2006).
36
2-16 North Arm Birdge a) deck level extradosed cable anchorage; b) precast tower, and c) tower anchor segment (from Griezic et al. 2006).
38
2-17 Trois Bassins Viaduct main pier (Frappart 2005).
39
2-18 Sünniberg Bridge a) deck cross-section and b) prestressing and reinforcement (adapted from Tiefbauamt Graubünden 2001).
39
2-19 Anchorages in towers (adapted from Tiefbauamt Graubünden 2001).
39
2-19 Anchorages in towers (adapted from Tiefbauamt Graubünden 2001).
40
2-20 Sunniberg Bridge a) bending moments and deflections of the edge beam through one stage of construction, and b) forces and deflections of the main span inner edge beam of the final structure due to permanent and live loads (adapted from Figi et al. 1998).
41
3 DESIGN AND CONSTRUCTION OF EXTRADOSED BRIDGES
42
3-1 CL-625 Live Loading: Maximum of CL-625 Truck (including DLA) or CL-625 Lane Load.
42
3-2 ASCE Loading (adapted from Buckland 1981)
44
3-3 multiple lane loading effect by deck width according to CHBDC 2006 and ASCE 1981, for two planes of cables and for single plane central cable suspension.
45
3-4 Comparison of Temperature Gradients (adapted from Priestley 1978, AASHTO 2004).
52
3-5 CHBDC CL-625 Live load envelopes for a main span of 100 m.
ix
54
3-6 Comparison of span to depth ratio and effect of the roadway height above ground on the overall proportions of 3 span cantilever, extradosed, and cable-stayed bridges.
55
3-7 Extradosed bridge geometry studied by Chio Cho (2000).
55
3-8 Girder and extradosed bridge proportions recommended by others.
56
3-9 Bridge proportions used for design in Chapter 4.
57
3-10 Effect of cable inclination on the force components in a cable for a) a constant total force and b) a constant vertical force.
57
3-11 Quantity of cable steel as a function of relative height of towers - Comparison between fan and harp cable configurations in a) 1970 (Leonhardt & Zellner 1970) and b) 1980 (Leonhardt & Zellner 1980).
58
3-12 Quantity of cable steel as a function of relative height of towers - comparison between semifan and harp cable configurations for 140 m main span.
59
3-13 Influence of partial cable support (adapted from Tang 2007).
61
3-14 DSI Extradosed Anchorage Type XD-E (Dywidag 2006).
65
3-15 Ratio of equivalent to initial modulus of elasticity showing the influence of a cable’s sag on its stiffness (plot adapted from Leonhardt & Zellner 1970).
66
3-16 Allowable stress in cable stays as a function of the stress range due to live load at SLS
70
3-17 Cable pre-strain and maximum moments for 25 iterations of the zero displacement method applied in two staged process: a) towers fixed, main span cable strains adjusted and b) towers released and side span cable strains adjusted. Cable 1 is anchored closest to the pier.
72
3-18 Cable force corresponding to dead load moment distribution (adapted from Gimsing 1997).
73
3-19 Tensions of main span cables, at each stage of construction up to midspan closure, resulting from a) backwards analysis and b) Staged construction including time-dependent effects (form traveller not considered).
74
3-20 Tower and pier configurations. From left to right: Barton Creek Bridge, North Arm Bridge (LRT), Kiso and Ibi Bridges, Sunniberg Bridge, Odawara Blueway Bridge, Tsukuhara Bridge, Shin-Karato Bridge, Hozu Birdge, Miyakodagawa Bridge and Domovinski Bridge (LRT and road). See Table 2-1 for drawing sources.
76
3-21 Arrêt-Darré Viaduct, France (concept 1982-83): main span 100 m, span to depth ratio 27, cantilever construction with precast segments with voided webs (Mathivat 1988).
77
3-22 Barton Creek Bridge, United States (completed 1987): main span 103.6 m, span to depth ratio 27 at midspan, cantilever construction, with the fin poured progressively after completion of 3 segments (Gee 1991).
x
77
3-23 Kiso and Ibi River Bridges, Japan (completed 2001): 275 m maximum spans, span to depth ratio 39 at piers and 69 at midspan, cantilever construction with precast segments lifted with 600 tonne barge mounted cranes, and central 95 to 105 m steel sections strand-lifted from barges and made continuous (Casteleyn 1999, Kasuga 2006).
77
3-24 Shin-Meisei Bridge, Japan (completed 2004): 122.3 m main span, span to depth ratio 35, castin-place cantilever construction (Kasuga 2006).
77
3-25 North Arm Bridge, Vancouver, Canada (completed 2008): 180 m main span, span to depth ratio 53, cantilever construction with precast segments (Griezic et al. 2006).
77
3-26 Trois Bassins Viaduct, Reunion (completed 2008): three main spans of 126 - 104.4 - 75.6 m, with cables overlapping through the middle span, effective span to depth ratio 30 at tallest pier and 50 at midspan, cantilever construction of central box, and construction of deck cantilevers and struts with mobile carriages (Frappart 2005).
78
3-27 Tsukuhara Bridge, Japan (completed 1997): main span of 180 m, span to depth ratio of 33 at piers and 60 at midspan, cantilever construction in 6 m long segments, transverse tendons in deck (Kasuga 2006).
78
3-28 Himi Bridge, Japan (completed 2004): main span of 180 m, span to depth ratio of 45, cantilever construction (Kasuga 2006).
78
3-29 Korror-Babeldoap (Japan-Palau Friendship) Bridge, Palau (completed 2002): main span of 247 m, span to depth ratio of 35 at the piers and 70 at midspan, cantilever construction of concrete portions of spans, and central 82 m steel section lifted from barges and made continuous (Oshimi et al. 2002).
79
3-30 Odawara Bridge, Japan (completed 1994): main span of 122.3 m, span to depth ratio of 35 at the piers and 55 at midspan, cantilever construction (Kasuga 2006).
79
3-31 Shin-Karato Bridge, Japan (completed 1998): main span 140 m, span to depth ratio of 40 at piers and 56 at midspan, cantilever construction (Tomita et al. 1999).
79
3-32 Domovinski Bridge, Croatia (completed 2006): 840 m total length, spans of 60 m typical with a main extradosed span of 120 m, span to depth ratio of 30, cantilever construction in 4 m segments (Balić & Veverka 1999).
80
3-33 Rittoh Bridge, Japan (completed 2006): main span of 170 m, effective span to depth ratio of 37 at pier and 61 at midspan, cantilever construction (Yasukawa 2002).
80
3-34 Pyung-Yeo Bridge, South Korea (completed 2007): main span of 120 m, span to depth ratio of 34 at piers and 30 at midspan, cantilever construction with one pair of travellers (Masterson 2006).
80
3-35 Pearl Harbor Memorial Bridge, United States (under construction): main span of 157 m, span to depth ratio of 31 at piers and 45 at midspan (Stroh et al. 2003).
xi
81
3-36 Saint-Rémy-de-Maurienne Bridge, France (completed 1996): spans of 52.4 and 48.5 m, effective span to depth ratio of 35, cast-in-place on falsework (Grison & Tonello 1997).
81
3-37 Sunniberg Bridge, Switzerland (completed 1998): main spans 128, 140, and 134 m, span to depth ratio of 127, cantilever construction in 6 m stages (Figi et al. 1997).
81
3-38 Third Bridge over Rio Branco, Brasil (completed 2006): main span of 90 m, span to depth ratio of 36 at piers and 45 at midspan, cantilever construction (Ishii 2006).
82
3-39 Golden Ears Bridge, Canada (completion 2009): main span of 242 m, span to depth ratio of 54 at piers and 80 at midspan, cantilever construction with precast deck panels (Bergman et al. 2007).
83
3-40 Possible types of tendons in an Extradosed Bridge
85
3-41 Sunniberg Bridge form traveller (adapted from Figi et al. 1998).
88
4 DESIGN OF CANTILEVER CONSTRUCTED GIRDER BRIDGE, EXTRADOSED BRIDGE WITH STIFF GIRDER, AND EXTRADOSED BRIDGE WITH STIFF TOWER
88
4-1 Roadway cross-section.
92
4-2 Construction sequence.
93
4-3 Girder cross-section of cantilever constructed girder bridge.
95
4-4 Bending moment in cantilever girder bridge (SAP2000 diagrams at the same relative scale).
98
4-5 Girder cross-section of stiff girder extradosed bridge.
101
4-6 Maximum bending moment in a typical girder bridge of 12.4 m width: a) as a function of longest span, and b) as a function of girder depth.
102
4-7 Bending moment in extradosed bridge (SAP2000 diagrams at the same relative scale).
104
4-8 Tension in cables of stiff girder extradosed bridge.
105
4-9 Simplified model of main span used to obtain the maximum live load stress range in the cables.
105
4-10 Two basic girder cross-sections considered for the stiff tower extradosed bridge.
106
4-11 Girder cross-section of stiff tower extradosed bridge.
107
4-12 Bending moment in stiff tower extradosed bridge (SAP2000 diagrams at the same relative scale - bending moments in the tower and rigid links have not been shown for clarity in all diagrams except temperature).
108
4-13 Tension in cables of stiff tower extradosed bridge.
110
4-14 Average girder concrete thickness of Chapter 4 bridge designs compared with Chapter 2 study bridges.
xii
LIST OF TABLES 15
2 REVIEW OF EXISTING EXTRADOSED BRIDGES
16
2-1 Summary of Extradosed Bridges.
41
3-1 Load factors and load combinations (CSA 2006a).
41
3-2 Permanent loads - maximum and minimum values of load factors α for ULS (CSA 2006a).
41
3 DESIGN AND CONSTRUCTION OF EXTRADOSED BRIDGES
44
3-3 Comparison of multiple lane load effects according to CHBDC (2006a) and ASCE (Buckland 1981) for the same basic lane load.
48
3-4 Comparison between Sunniberg Bridge and North Arm Bridge response to live load.
49
3-5 Comparison between monolithic and released connnection at main piers of the North Arm Bridge.
62
3-6 Minimum saddle radii for strand based cables to prevent fretting fatigue.
88
4 DESIGN OF CANTILEVER CONSTRUCTED GIRDER BRIDGE, EXTRADOSED BRIDGE WITH STIFF GIRDER, AND EXTRADOSED BRIDGE WITH STIFF TOWER
89
4-1 Material Characteristics assumed for design.
89
4-2 Concrete Covers and Tolerances specified in the CHBDC (CSA 2006a).
90
4-3 Friction Coefficients (per metre length of prestressing tendon)
94
4-4 Critical sections for design and corresponding load cases to produce the maximum load effect.
95
4-5 Preliminary and final tendons for cantilever constructed girder bridge.
96
4-6 SLS Forces and Maximum Stresses in the Girder - Internal and External Tendons
96
4-7 SLS Forces and Maximum Stresses in the Girder - Internal Tendons
99
4-8 Evaluation of alternative cross-sections for lateral support by planes of cables.
100
4-9 Preliminary and final tendons for extradosed bridge.
101
4-10 Maximum Bending Moments in a Typical Box Girder Bridge
103
4-11 SLS Forces and Maximum Stresses in the Girder
103
4-12 ULS Forces in the Girder
109
4-13 Material quantities in girder and cables.
110
4-14 Average material quantities in girder and cables.
111
4-15 In-place costs of materials in girder and cables.
111
4-16 Cost estimate of bridges (costs in $1000).
xiii
134
APPENDIX A
135
A-1 Extradosed Bridges in Chapter 2 Study.
137
A-2 Cantilever Constructed Girder Bridges in Chapter 2 Graphs.
138
A-3 Cable-Stayed Bridges in Chapter 2 Graphs.
158
APPENDIX C
159
C-1 Prestressing quantities in cantilever-constructed girder bridge with internal prestressing.
159
C-2 Prestressing quantities in cantilever-constructed girder bridge with internal and external prestressing.
159
C-3 Prestressing quantities in stiff girder extradosed bridge.
159
C-4 Prestressing quantities in stiff tower extradosed bridge.
xiv
1
WHAT MAKES A BRIDGE EXTRADOSED?
1.1
Introduction
From 1994 to 2008, over fifty extradosed bridges have been constructed worldwide, and the preferred proportions and cable arrangements have evolved. While there are many articles available on the design of specific extradosed bridges, very little has been published on their design from a general perspective. The intrados is defined as the interior curve of an arch, or in the case of cantilever-constructed girder bridge, the soffit of the girder. Similarly, the extrados is defined as the uppermost surface of the arch. The term ‘extradosed’ was coined by Jacques Mathivat (1988) to appropriately describe an innovative cabling concept he developed for the Arrêt-Darré Viaduct (shown in Figure 1-4), in which external tendons were placed above the deck instead of within the cross-section as would be the case in a girder bridge. To differentiate these shallow external tendons, which define the uppermost surface of the bridge, from the stay cables found in a cable-stayed bridge, Mathivat called them ‘extradosed’ prestressing. There is some debate over the boundary between cable-stayed and extradosed bridges. Visually, extradosed bridges are most obviously distinguished from cable-stayed bridges by their tower height in proportion to the main span, as shown in Figure 1-1. Extradosed bridges typically have a tower height of less than one eighth of the main span, corresponding to a cable inclination of 17 degrees, as observed from the bridges considered in Section 2. In this thesis, the term ‘extradosed bridge’ will be used to describe all bridges that have a tower that is shorter than that of a conventional cable-stayed bridge, which is widely accepted to be around a fifth of the span, as will be explained in Section 3.4.1. The reduced cable inclination in an extradosed bridge leads to an increase in the axial load in the deck and a decrease in vertical component of force at the cable anchorages. Thus, the function of the extradosed cables is also to prestress the deck, not only to provide vertical support as in a cable-stayed bridge. Extradosed bridges are characterised by a low live load stress range in the stay cables. The definition of an extradosed bridge adopted in this thesis, based on geometry alone, disregards the live load stress range in the cables, which is done purposefully to consider a range of structures with any distribution of live load between the axial force resisting system (axial force couple between cable and deck) and the girder. Extradosed bridges are sometimes criticised for being inefficient structures due to the reliance on this secondary girder system, because the lever arm between the cable and deck is larger than the lever arm within the girder. Girder Bridge
Extradosed Bridge
Cable-Stayed Bridge
H ~ L/18 to L/15 h ~ L/50 to L/30
H ~ L/15 to L/8 h ~ L/50 to L/30
H ~ L/5 to L/4 h ~ L/100 to L/50
Variable Depth
Constant or Variable Depth
Constant Depth
Internal and external prestress Maximum cable stress 0.70 fpu
External prestress Maximum cable stress 0.60 fpu
Cable stays Maximum cable stress 0.45 fpu
Figure 1-1. Comparison between cantilever-constructed girder, extradosed, and cable-stayed bridge types.
1
2
The detailing and technology found in extradosed bridges is taken directly from externally prestressed girder bridges and from modern cable-stayed bridges. Modern cable-stayed bridges have a fifty year history and have been constructed with span lengths from 15 m to over 1000 m. As compared with cable-stayed bridges, the advantages of extradosed bridges for spans less than approximately 200 m are numerous. Since the live load stress range is typically small (Mathivat 1988), the cables can be deviated at the piers by means of a saddle, allowing for a more compact tower, especially in the case of a fan cable arrangement. The stay cables can be anchored near the webs and the vertical component of the stay cable force (which is small in comparison to a cable-stayed bridge) is transferred directly to the girders without the need for a transverse diaphragm at the anchorage location. As with external prestressing, extradosed bridges can use normal prestressing anchorages instead of the high stress range type used for cable-stayed bridges. Given a stiff girder, the extradosed bridge can be constructed without any need to adjust the tension in the cables (Chio Cho 2002). Finback Bridge
Cable-Panel Bridge
Extradosed Bridge
Figure 1-2. Finback, cable-panel and extradosed bridge types.
The development of the extradosed bridge has evolved with and may have been influenced by other types of unconventional cantilevered bridges in which the top tendons rise above the deck level in the negative moment regions, as shown in Figure 1-2. The ‘fin-back’ bridge has a wall containing the negative moment tendons that is monolithic with the deck creating a single section, whereas a ‘cable-panel’ bridge has a wall that is detached from the deck section, serving more as passive protection for the cables. Section 1.2 will describe the development of the extradosed bridge in more detail.
1.2
Objectives and Scope
The purpose of this thesis is to provide insight into how different geometric parameters such as tower height, girder depth, and pier dimensions influence the structural behaviour, cost, and feasibility of the extradosed design concept. The objectives are as follows: 1. Clarify what is meant by an extradosed bridge and how this structure type has evolved; 2. Provide a comprehensive summary of extradosed bridges constructed to date; 3. Determine the loads that govern the design of an extradosed bridge; 4. Find a strategy for the design of an extradosed bridge; 5. Determine if providing stiffness in the girder of a cable-supported structure is an efficient structural solution; 6. Determine if an extradosed bridge is competive against a cantilever constructed girder bridge; It is desirable to have a sense of judgment over how each of the components of the extradosed bridge affect the structural behaviour. It can be difficult to develop an intuitive feel for an extradosed structure due to the complex relationship between its components. Chapter 2 provides an in-depth summary of extradosed bridges constructed to date that shows the large variability and great potential that exists within this form. This information is also useful as a starting
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point for initial dimensioning of the overall structure and its components, and for estimates of material quantities. Chapter 3 discusses the primary factors that define the design of an extradosed bridge, from where it fits into the realm of bridge types, to solutions for critical details that must be worked out in the final design. Many of the issues discussed are of general applicability to any type of medium to long-span bridge, as they represent important considerations in the conceptual design process, and are brought together here in one document. The analysis and comparisons, however, are more specific in their findings, and may be limited to the ‘typical’ extradosed bridge problem assumed. In Chapter 4, designs of a cantilever-constructed girder bridge, a stiff girder extradosed bridge, and a stiff tower (slender girder) extradosed bridge are presented for a three span bridge, with a central span of 140 m. in length. Variations in prestressing approach are also considered. A materials and cost comparison is presented to highlight the main differences and overall cost-effectiveness of each design. Finally, Chapter 5 concludes the thesis by summarizing the primary findings of the preceding chapters and gives suggestions for future studies. This thesis will provide enough detail on the designs undertaken to allow a practicing engineer to understand the key design steps involved in designing an extradosed bridge, in accordance with the Canadian Highway Bridge Design Code (CHBDC - CSA 2006a) and other relevant codes.
1.3
Historical Context
Jacques Mathivat is most commonly credited for ‘inventing’ the concept of extradosed prestressing, which he published in a journal article in 1988 (Mathivat 1988). Mathivat presents extradosed prestressing as a natural progression from cantilever construction, prompted by the desire to have a construction scheme with fully replaceable tendons. In France since the 1980s, cantilever constructed girder bridges have mainly used two groups of tendons: internal horizontal tendons for cantilevering, and external tendons draped from one pier table to another and deviated by two segments at the third points of the span. The effectiveness of external tendons over the piers is proportional to the lever arm between the tendon and the compression block. In a girder bridge, this increase in lever arm comes at the expense of additional dead load in the webs. To avoid this, the tendons are placed above the deck and deviated by means of a stub column to create a force couple between the tension in the tendons and the compression in the entire concrete deck, as is the case in a cable-stayed bridge. The first modern concrete multi-cable-stayed bridge was the Main Bridge near Frankfurt, designed by Ulrich Finsterwalder and completed in 1973, only 10 years prior to Mathivat’s extradosed concept. This modern cable stayed bridges rely entirely on the force couple between the stay cables and the deck to resist all load. Initial designs for concrete cable-stayed bridges featured prestressed concrete decks supported at discrete locations by internal tendons encased in concrete, such as Morandi’s Maracaibo Bridge (Billington 1991), completed in 1962. In these structures, the load was carried in bending to the discrete supports. Extradosed prestressing makes use of a stiffer girder to distribute live load to multiple cables. Chio Cho (2002) suggests the idea of extradosed prestressing may have come from these long-span
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concrete cable-stayed structures (around 200 m), which combined with temporary stays for cantilever construction of constant depth girder bridges of medium span length (around 80 m), result in a hybrid form where the temporary stays are made permanent. Mathivat (1988) points out that cable-panel bridges and fin-back bridges may have been inspired by the same desire to reduce the self-weight of cantilever constructed girder bridges. By locating the prestressing cables in walls above the deck, the capacity of deck slab in compression can be utilised in negative moment regions (over the piers) leading to a more efficient structure than a conventional cantilever constructed box girder bridge. These structures bear some resemblance to extradosed bridges, but they differ in appearance and in their stiffness, and the cables cannot be easily replaced since they are encased in a concrete wall. Nevertheless, the proportions of this type of bridge had a significant impact on the development of the extradosed bridge. The Ganter Bridge, completed in 1980, was the first bridge of this type, is the most well-known, and inspired the concept for the Arrêt-Darré Viaduct (Virlogeux 2002c). The Ganter Bridge, located in Switzerland, is a cablepanel bridge with a main span of 174 m that takes a roadway over a deep valley at heights of up to 140 m above the valley floor. The bridge is designed by Christian Menn, former Professor of Structural Engineering at the ETH in Zurich and the designer of many elegant prestressed concrete bridges in Switzerland. The roadway runs parallel to the valley on either side, while the bridge crosses at a skew, which necessitates sharp curves at both ends of the bridge. The bridge had two Figure 1-3. Ganter Bridge (Vogel & Marti 1997) unique design requirements: tall, stiff piers in light of the high winds through the valley, and a very narrow roadway for a bridge of this maximum span length. David Billington, a Professor of Engineering and director of the Program in Architecture and Engineering at Princeton University, explains that Menn’s design decisions were made with aesthetics in mind. While a conventional cantilever constructed box girder would have been technically feasible, it would have had a “visually weak horizontal profile compared to the powerful vertical elements required” (Billington 2003). This does not however explain why Menn encased the walls in concrete: the walls do not extend into the curved part of the roadway, and visually the cables would have provided the same perception of strength as the concrete walls while opening up the view through the valley.
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Figure 1-4. Arrêt-Darré Viaduct (Mathivat 1988)
The first application of extradosed prestressing was Mathivat’s proposal for the Arrêt-Darré viaduct with precast box girder sections (Mathivat 1987), developed in 1982-1983. The extradosed prestressing along with voided box girder webs resulted in a material savings of 30% compared with a conventional cantilever constructed box girder bridge. Mathivat’s proposal substituted the internal tendons in the top flange of a box girder for external cables above the running surface, deviated over the piers by stub columns and anchored inside the box girder, which he called ‘extradosed cables’. The low eccentricity of the cables over the piers allowed them to be stressed to the same level as traditional prestressing since the cables’ primary role was to provide horizontal prestress, and they were subject to a low fatigue stress. Virlogeux explains (1999) that the concept was partially motivated by a ‘distortion of code specifications’ to use stay cables more efficiently, since an allowable stress of 0.65 fpu could be used for design of the cables instead of the value of 0.45 fpu typically adopted for cable-stayed bridges. Unfortunately, the proposal was not selected for construction and a conventional cantilever post-tensioned structure was constructed instead. The Barton Creek Bridge is one of a few prestressed concrete fin-back bridges constructed. The bridge connects Austin, Texas to the Estates of Barton Creek over an environmentally sensitive gorge. Preliminary design estimates found the fin-back bridge with a main span of 104 m to be comparable in cost to a conventional cantilever box girder, both least cost options for the crossing (Gee 1991). The fin-back bridge design was chosen as it would be a visible landmark into the Figure 1-5. Barton Creek Bridge (Gee 1991). Estates from above the bridge and it would consequently attract publicity to the development. The developer’s architect noted (Gee 1991) that the “flat triangular fins ideally complemented the peaks of the gently rolling hills on the horizon forming the backdrop against which the bridge would be seen.” Gee explains that his firm was aware of previous proposals for fin-back bridges and was conscious of the fact that the low material quantities of these designs, as compared with conventional box girders, were not resulting in competitive bids for construction. A few notable proposals were the Kessock Bridge in Scotland, the Foyle Bridge in Northern Ireland and the Gateway Bridge in Australia. The Gateway Bridge was in fact tendered as a fin-back bridge in 1980 but the contract was
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awarded based on an alternative design of a conventional box-girder bridge (Gee 1991). Tony Gee and Partners, having been involved with previous fin-backed proposals, designed the Barton Creek Bridge with constructibility as the most important objective. The cross-section consists of a single box girder below the deck, with constant dimensions and no internal diaphragms (even over the piers) and webs that incline inwards from the bottom slab to merge at the deck slab into a central fin that rises above the deck with constant width. The segments were poured with the median barrier, which allowed the fin to be built up progressively and off the critical path, as three segments could be cast before increasing the height of the fin. The bridge was constructed with a form traveller supported laterally outside the webs and on the ribs, and was completed in 1987 at a cost that was 20% above the original estimate (Gee 1991). The increase was accounted for in additional items (lighting, approach railings) not included in the initial estimate. In terms of durability and maintenance, the fin-back bridge has the advantage over a conventional cantilevered bridge that the main tendons are encased in the massive fin, away from the deck surface which is exposed to traffic. As well, there is no decrease in longitudinal prestress in the deck under live load near the piers, because the neutral axis of the section lies in the deck slab. Despite the success of the Barton Creek Bridge, there have been very few fin-back bridges built since. Virlogeux (1999) claims that the concrete walls in cable-panel and fin-back bridges have two drawbacks: the tendons cannot be replaced and there is a cost to construct the concrete walls. The designers of the Barton Creek Bridge took measures to reduce the cost of the single concrete wall and it is conceivable that the additional cost of the protection system for stay cables would have exceeded the cost of the walls. However, since the concrete walls add dead load to the bridge, their use is only economical in shorter spans. In terms of aesthetics, the stay cables of extradosed bridges offer a lighter appearance than the heavy concrete fins of fin-back bridges. Akio Kasuga, the Chief Bridge Engineer at Sumitomo Mitsui Construction Co. Ltd., was the first to apply Mathivat’s concept to the first extradosed bridge to be constructed, the Odawara Blueway Bridge. The construction was completed in 1994. Kasuga claims that his designs follow Mathivat’s theory that the tower height of extradosed bridges should be half of the tower height for a cable stayed bridge of equivalent span Figure 1-6. Odawara Blueway Bridge (Kasuga 2006)
(Ogawa and Kasuga 1998), but Mathivat actually suggests using a tower height closer to one third of
that for a cable-stayed bridge (Mathivat 1988). The single plane of cables proposed for the Arrêt-Darré viaduct is lighter than the two planes on the Odawara Bridge. However, the towers of the Odawara Bridge, as shown in Figure 1-6, are well integrated with the pier columns below the deck, while the Arrêt-Darré superstructure rests on an overly massive round pier column, which visually dominates the bridge’s view
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in profile. Kasuga (2006) states that “practical experience has induced great admiration for the incisiveness of Mathivat’s proposal.” Several articles on extradosed bridges in Japan credit and praise Mathivat for inventing the extradosed bridge (Ogawa and Kasuga 1998; Hirano et al. 1999; Kato et al. 2001; Kasuga 2006), but these bridges are more similar in appearance and proportions to the Ganter Bridge than to the Arrêt-Darré concept, with the difference that they do not have cables encased in concrete walls. All extradosed bridges in Japan to date have cables arranged in a semi-fan configuration, with the first cable offset about a fifth of the span from the pier. Most of these bridges are cast in place and have variable depth girders that are 50% deeper at the piers than in midspan. The Japan Highway Public Corporation, the owner of several extradosed bridges including the Odawara, Tsukuhara, and Kiso and Ibi extradosed bridges, allows only external tendons to be used in their bridges (Chilstrom 2001). Mathivat’s bridge had six spans with a continuous girder supported on bearings whereas the first extradosed bridges constructed in Japan were three span structures, with monolithic connections at the piers that result in frame action. This distinction allowed the first extradosed bridges to be more slender than Mathivat originally proposed. In France, the first extradosed bridge was constructed on a much smaller scale than those in Japan, as a means of spanning further with minimal structural depth available below the roadway. The Saint-Remyde-Maurienne Bridge over the A43 highway, shown in Figure 1-7, was tendered in 1993 and completed in 1996 and has a maximum span of only 54 m (Grison and Tonello 1997). On this small scale, it is difficult to imagine that extradosed tendons could be more economical than constructing concrete walls up to a maximum height of 3 m, but there is little doubt that the constructed bridge is attractive. Architect Charles Lavigne, who developed the Arrêt-Darré concept with Jacques Mathivat, was also involved in the SaintRemy Bridge.
Figure 1-7. Saint-Remy-de-Maurienne Bridge (photo by Jacques Mossot, Structurae)
Figure 1-8. Concept for Usses Viaduct (Virlogeux 2002b)
In the mid 90s, an extradosed bridge was considered for the A41 viaduct over Usses valley in France (Virlogeux 2002b). The concept was developed by Jean Tonello, Charles Lavigne and Daniel Vibert and brings together elements from the Arrêt-Darré Viaduct and the Barton Creek Bridge. The cross-section consists of a single-cell box girder with constant dimensions, with wide cantilevers supported by struts, as seen in Figure 1-8. The project was delayed for many years, and ultimately a steel twin girder structure with composite concrete deck was constructed at the site in 2008. However, a similiar cross-section to that
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of the Usses viaduct was adopted for the viaduct over Trois-Bassins in Réunion, completed in 2008 and shown later in Figure 3-26.
Figure 1-9. Santiago Calatrava’s concepts for crossing deep Alpine valleys. From left to right: Variant 1, Variant 2 model, Variant 7 sketch and detail presented by Menn at the IABSE Symposium in Zurich in 1979 (Calatrava 2004).
In Switzerland, the Ganter Bridge led to a different path for the extradosed bridge. Santiago Calatrava, while studying at the ETH in 1979, produced a series of sketches of alternatives of the Ganter Bridge, some of which were presented by Menn at the IABSE Symposium in Zurich in 1979 (Tzonis and Donadei 2005). All sketches show a slender deck, suspended by cables from a stiff pier. In some alternatives, the two sides of the tower flare outwards from the roadway in order to accommodate the cables along a curved horizontal profile. Eight years later, Menn (1987) presented his ideas on the advantages of stiffer, lower towers for cable-stayed bridges, which “enable the use of the full range of effective depth of crosssection”. Since the short towers act as cantilevers, effectively prestressed by the dead load of the girder acting through the cables, they require relatively little reinforcement to resist bending due to live load. Neither a flexurally stiff girder nor backstays are required in order to provide adequate system stiffness to control deformations due to live load. With short towers, larger stay cables are required, but the towers are more economical than the tall towers normally found in cable-stayed bridges (Menn 1987). Stiffness from Backstay Cable
Stiffness in Deck
Stiffness in Tower
LL resisted by backstay cable
LL resisted by bending in deck
LL resisted by bending in tower
LL offset by dead load in main span; Short side span prevents backstay from going slack
LL resisted by bending in deck
Magnitude of bending in tower is not affected by decrease in tower height
Figure 1-10. Response of cable-stiffened, girder-stiffened, and tower-stiffened cable-stayed bridge to live load.
There are three different approaches to providing stiffness in cable-stayed bridges, as shown in Figure 1-10, which determines how stability of the bridge will be assured under live load. Each approach
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provides stiffness primarily in one of the three load-bearing elements of the cable-stayed bridge: the stays, the deck, or the towers. Most early designs, 6 out of the 58 cable-stayed bridges constructed by the end of 1976 (Billington and Nazmy 1991) have cable spacings of 20 to 40 m and provide stiffness in the girder, with span to depth ratios below 70 (Walther et al. 1999). Since the 1980s, however, almost all cablestayed bridges are multiple-stay bridges with cable spacings of less than 10 m, which, combined with an increased understanding of aerodynamic stability and buckling safety of slender girders, has led to very slender girders. In these bridges, the fan cable configuration is used to load the tower in axial compression only, and backstay cables stabilise the tower and control girder deflections due to live load (Menn 1994). With this system, girder span to depth ratios of up to 500 are possible (Bergermann and Strathopoulos 1988). With stiff towers, the girder can still be made very slender, with span to depth ratios of up to around 200, but backstay cables are no longer required. A harp cable configuration favours stiff towers, since live load at the quarter points of the main span will cause significant bending in towers, regardless of tower stiffness.
Figure 1-11. Poya Bridge, Switzerland: a) Menn’s 1989 proposal (Menn 1996) and b) cable-stayed design selected in 2006 for construction (Mandataire Projet Poya 2005).
In 1989, Menn proposed an extradosed concept for the Poya Bridge as the ideal solution to integrate the bridge with the deep valley crossing in Fribourg, Switzerland (Billington 2003). Menn was serving on the jury for the design competition for the bridge, but was not satisfied with any of the designs and proposed his own concept. Menn (1991) felt that a cable-stayed bridge, which would require towers with a total height of 120 m, 45 m of which would be above the roadway, would rise above the city and detract from its historical character. In this same article, Menn elaborates on his ideas for a stiff tower and slender girder, explaining that the live load is resisted directly by the towers, and thus side span lengths can be increased to half of the main span length. In most modern cable-stayed bridges, the tower must be stabilised by backstay cables which require shorter side spans (around 0.4 of the main span) to remain adequately stressed when they are loaded, as shown in Figure 1-10. For the Poya Bridge concept, Menn maintained the slender cross-section throughout the viaduct through the use of external prestressing in an
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under-deck cable-stayed configuration. His proposal was not accepted and a cable-stayed bridge is now under construction after many years of public consultation. Both concepts are shown in Figure 1-11. In 1993, Menn proposed his extradosed concept for the Sunniberg Bridge (Honnigmann and Billington 2003). Menn, again serving on the jury for the design competition and not satisfied with the three designs submitted, presented his concept to the highway department’s architectural consultant who endorsed it and helped convince the
Alternatives to provide adequate rigidity between pier, tower, and deck
department to accept it. The 526 m length of the bridge and 60 m depth of the valley below the roadway favoured multiple piers with spans of around 150 m. Menn favoured the proposal for the same aesthetic reasons that he claimed for the Poya Bridge: if the towers
Pier options to accommodate longitudinal deformations
were designed with conventional proportions, they would rise 35 m
Figure 1-12. Millau Viaduct tower options (Virlogeux 2004). Drawn by Sir
above the roadway, and would overpower the surrounding landscape
Norman Foster after discussions with Virlogeux.
(Figi et al. 1997). A multiple span cable-stayed bridge with conventional proportions would still require stiff towers, as illustrated by the conceptual drawings shown in Figure 1-12 for the towers of the Millau Viaduct, an 8 span cable-stayed bridge (Virlogeux 2004). Cross cables between towers have been proposed as another means of stabilising them in the horizontal direction, but this solution has questionable aesthetics and is difficult to construct (Walther et al. 1999) if not infeasible on a curved roadway. The designs submitted for the design competition did not include a cable-stayed bridge but were of a more conventional nature: a continuous below deck truss bridge, a cantilever-constructed girder bridge, a composite box girder bridge, and a box girder viaduct. Given the “unusually high demand” on aesthetics at this site, the extradosed bridge was selected to fit into the landscape with elegance, but also stands out as a memorable crossing and technical achievement (Figi et al. 1997). As constructed, the Sunniberg Bridge towers appear in proper proportion with the surrounding landscape when viewed from the driver’s perspective. Compared with the Ganter Bridge in Figure 1-3, the Sunniberg Bridge appears much lighter and more transparent, not only because the cables are light and provide unobstructed views of the valley, but also because of the open views through the pier legs, as seen in Figure 1-13. Menn clearly knew about Mathivat’s ideas for the extradosed bridge, since Mathivat presented an earlier version of his 1988 article at Menn’s 60th birthday celebration in 1986 (Mathivat 1986). Despite this, Menn’s concept provides stiffness in the piers, while Mathivat’s concept provides stiffness in the girder. The two solutions reflect the trends and visions of engineers in their respective countries of origin. French bridge engineers were using precast segmental construction for many medium span bridges to encourage mechanisation and rationalize formwork reuse. Precast segmental construction favours a constant depth cross-section, a nominal girder depth for easier assembly and geometric control from the relative geometry of segments (Virlogeux 1994a), use of bearings, and allows for future replacement of tendons. In contrast, Swiss engineers preferred cast-in-place construction for durability and the economy provided by partial prestressing. Cast-in-place construction allows monolithic connections to be made as required. Flexible decks are more sensitive to local deformations from transverse bending and thermal
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Figure 1-13. Sunniberg Bridge, Switzerland.
effects of concrete hardening, but are also more accommodating of field adjustments to their final geometry since the deck profile can be readjusted after completion without introducing considerable bending moments (Virlogeux 1994a). From an aesthetic perspective, Menn’s Sunniberg Bridge is much lighter overall than Mathivat’s Arrêt-Darré concept, since the deck section is dimensioned as the minimum required to span Figure 1-14. Arrêt-Darré and Sunniberg Bridges (see drawings in Figure 2-1).
between two adjacent cables. Although the pylons1 are nearly 7 metres wide in the longitudinal direction, they integrate
seamlessly into the pier legs below the deck, and the total surface area observed from an elevation view is much less than if a stiff girder was used, as shown in Figure 1-14. Seen from afar, the deck looks as though it is cradled between the pylons, as seen in Figure 1-13. The pylons, when viewed from the transverse perspective, are extremely light because the overall member thicknesses are kept to a minimum through frame action in the crossbeams. The crossbeams are set back from the pier legs which emphasize the continuous vertical lines from bottom to top which form surfaces of slender proportions at the outside edges of the pier legs. With a two lane structure on a horizontal curve, supporting the cross-section from both sides is the only reasonable solution from an operations perspective, and a harp cable arrangement is a natural choice to avoid the chaotic appearance of cables crossing each other at different angles.
1.
Throughout this thesis, the term tower is used to denote anything structural that supports the cables above the deck; the term pylon is used to denote the two columns of a tower in a stayed bridge with lateral cable planes; the term mast is used to denote a single column tower, and pier is used to refer to any substructure beneath the deck, supporting the tower.
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Extradosed bridges are becoming increasingly popular for spans from 50 m to 250 metres. Over 25 extradosed bridges have been completed in Japan and 15 are underway in South Korea (Bd & e 2006). Many extradosed bridges constructed to date cross waterways where there is a navigational clearance requirement as well as interest in minimizing the roadway grade raise at the approaches. This favours an extradosed bridge over a cantilever-constructed girder bridge, which would have a girder depth at the supports of two to three times that of the extradosed bridge of equivalent span. While a cable-stayed bridge is sometimes a feasible option, an extradosed bridge has been selected in many cases because of overhead glidepath clearance requirements imposed by nearby airports (Stroh 2003; Griezic et al. 2006).
Figure 1-15. Golden Ears Hybrid Extradosed Bridge, Vancouver (Bergman 2007).
In 2001, the Ibi and Kiso Bridges set the record for longest extradosed viaducts with total lengths of 1145 and 1400 m respectively, and extradosed spans of up to 275 m. This was achieved with a hybrid girder arrangement: a variable depth concrete girder was supported by extradosed cables from the piers and 100 m central steel box girder was connected to the concrete girder to transfer moment and shear but allow longitudinal expansion. In 2002, the Japan-Palau Friendship Bridge was completed with the same hybrid configuration with a main span of 247 m. The Golden Ears Bridge, under construction in Vancouver and scheduled for completion in 2009, has main spans of 242 m and will be the first composite extradosed bridge constructed. The concrete deck is constructed from precast panels and the bridge has a main span to tower height ratio of 6, between that of an extradosed and cable-stayed bridge (Bergman et al. 2007). The bridge’s engineers have described the bridge as a hybrid extradosed bridge, with a portion of the load near the piers supported by bending in deck, but load at midspan resisted almost directly by the cables (Trimbath 2006). The Wuhu Bridge in China, completed in 2000, is the only extradosed bridge to carry heavy rail. Extradosed cables allow this bridge to span 312 m over the Yangtze River with the same deep truss cross-section that is used for the other 144 m spans (Fang 2004). Recently, Man-Chung Tang, who was responsible for the design and construction engineering of many cable-stayed bridges in North America, has extolled the advantages of partially cable-supported structures for the potential to fully utilize the capacity of a girder cross-section (Tang 2007). He explains that there is a tradeoff between girder depth and the size of the stay cables. Live load is shared between the girder and stay cables based on the relative stiffness of each, while the cable forces under dead loads can be adjusted as desired. Furthermore, each cable can be adjusted to carry a different amount of dead load, and the cable layout can be adjusted to achieve greater efficiency. According to Tang, for bridges with medium span lengths, using the maximum capacity of the girder can result in savings in stay cables and towers. From some of the proposed concepts, partially cable-supporting creates possibilities for new forms of towers,
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some of which are sculptural but structurally inefficient and would be unable to support the tension forces from the stay cables if the girder did not share some of the load.
Figure 1-16. North Arm Bridge, Canada Line LRT, Vancouver (photo courtesy of Stephen Rees)
Figure 1-17. Pearl Harbor Memorial Bridge in New Haven, Connecticut (Stroh et al. 2003)
SETRA (2001) published recommended allowable stress limits that cover the full range of external cables. In that document, external prestressing tendons are defined as being subjected to a stress range of up to 15 MPa under live load while stays for cable-stayed bridges are subjected to a stress range of around 100 MPa and above. Extradosed cables are characterised as being subjected to a live load stress range between 30 MPa and 100 MPa and are not sensitive to wind vibrations. These specifications resulted from a need for design recommendations for bridges that do not fall into distinct categories, and they propose design limits and approximations based on rational principles. These recommendations were used for the design of the North Arm Bridge in Canada (Griezic et al. 2006), and they influenced the allowable stress limit for the Pearl Harbor Memorial Bridge in the USA (Stroh et al. 2003). It is important to have guidelines like those published by SETRA (2001) to encourage creative and innovative use of stay cables in new structural systems, since it is difficult in the initial stages of design to assess the behaviour of a bridge at ultimate and fatigue limit states.
1.4
General Studies on Extradosed Bridges
Komiya (1999), of the Japan Bridge & Structures Institute, carried out a series of analyses on an extradosed bridge of 74 - 122 - 74 m spans, with span to depth ratio of 35 at the piers and 55 at midspan. These are the same span dimensions and proportions as the Odawara Bridge. Parameters that were varied include the tower height, the anchoring position of the extradosed cables, the stiffness of the girder to account for cracking, and the stiffness of the tower. As a result of this study, Komiya presents five reasons why an extradosed bridge might be preferable over a cantilever constructed or cable-stayed bridge: 1. The girder depth is around 2 to 4 m, about half that of a cantilevered bridge and twice that of a cablestayed bridge, which has better constructibility than the other types for laying out post-tensioning and placing reinforcing steel and concrete; 2. A stiff deck allows the structure to be continuous over multiple spans, and appropriate for rail loading where deflection limits must be adhered to; 3. The presence of towers can create a symbolic structure as compared to a conventional girder bridge;
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4. Extradosed cable can use conventional anchorages instead of the expensive anchorages with a high fatigue strength as used for cable-stayed bridges. Extradosed cables are less sensitive to vibration, and they do not need to be restressed during construction. On this basis, Komiya concludes that extradosed cables are more economical than conventional stay cables and lead to better constructibility; 5. Extradosed bridges are less costly than cable-stayed bridges but more costly than cantilever constructed girder bridges, based on materials consumption. Extradosed bridges could be more economical than girder bridges in the case where girder depth is limited by traffic or navigational constraints, or in the case where poor soil conditions provide an incentive to reduce the structure’s self-weight. Chio Cho (2000) carried out a parametric study on an extradosed bridge with similar characteristics to the Odawara Blueway Bridge, with span lengths of 74 - 122 - 74 m. The results of his study on the structural behaviour of extradosed prestressing during construction and in service lead to a few important design recommendations (Chio Cho 2002). The first cable should be anchored between 0.18 and 0.25 of the main span from the tower as the cables closest to the tower are ineffective. The deck should be proportioned with a span to depth ratio of 35 at the piers and 45 at midspan to keep all live load stress range in the cables below 80 MPa, the limit for conventional prestressing anchorages. A compensation of all permanent loads, with the cables stressed as the bridge is constructed in balanced cantilever, results in high tensile forces in the bottom fibre of the deck, making a single stressing operation impractical. A compensation of 80% of the permanent load is preferred in order to control the stresses during construction and avoid restressing after the final structural configuration is acheived. Since it is not possible to completely balance all permanent loads by stressing the cables during construction only, consideration of creep effects is essential. It is necessary to precamber the deck for long term deflections due to creep. Santos (2006) carried out a parametric analysis on an extradosed bridge with a 150 m main span with simple supports at the piers, which looked at the influence of tower height, depth of deck, and length of the side span on the total extradosed cable area, the stress variation in the cables, and the bending moment in the deck. For span to tower height ratios of 15, 10 and 5, he found that span to depth ratios of 37, 34, and 27 respectively were necessary in order to keep the live load stress range to below 50 MPa. Santos did not explicitly consider the forces in the girder at each stage of construction, and assumes that superimposed dead load is added at each stage of cantilevering, which is unrealistic for construction. Santos does not consider any redistribution of forces in the final state due to creep and shrinkage. Due to these simplifications, some of the conclusions are of limited use.
2 REVIEW OF EXISTING EXTRADOSED BRIDGES This chapter presents a study 51 extradosed bridges designed to date, with a focus on specific bridges that have had an influence on subsequent designs, based on their proportions and aesthetic merit. A review of previously constructed or conceived extradosed bridges provides insight into where and how extradosed bridges have been used. Section 2.1 summarizes the physical dimensions of the bridges. Section 2.2 describes the typical reasons extradosed bridges were chosen over other bridge types and analyzes the proportions to find trends. Section 2.3 evaluates the materials usage and stiffness of the girder in the bridges and compares them to those of typical cantilever constructed box girder and cable-stayed bridges. Section 2.4 discusses significant details of selected extradosed bridges which show how their designers have made key conceptual and detailed design decisions.
2.1
Study of Extradosed Bridges
Extradosed bridges provide an economical means of crossing spans of 100 to 250 m with new aesthetic opportunities relative to cantilever constructed girder bridges and cable-stayed bridge. Table 2-1 shows the extradosed bridges considered in this study in chronological order by date of construction. Some of these bridges are described in greater detail in Section 2.2. Figure 2-1 shows elevation and pier section drawings for the extradosed bridges identified in Table 2-1 as having detailed drawings. The figure shows the wide variety of forms, spans, and cable configurations that have been selected. For the two span extradosed bridges considered in this study, the main span has been taken as an effective main span equal to 0.8 of the sum of the two spans adjacent to the pier, in order to study the dimensions and proportions of both two and three span extradosed bridges together in Section 2.2. This is done to account for the influence of positive moment regions at the ends of the span where the girder resists load in bending only.
2.2
Trends in Extradosed Bridges to Date
Extradosed bridges have been used for one or more of the following reasons: 1. A shallow structural depth below the roadway is preferable, either to meet clearance requirements or when there is interest in minimizing the approach grades. 2. In the opinion of some designers (Menn 1991), tall piers over a deep valley do not permit a cablestayed tower to be aesthetically pleasing when the portion of the tower above the deck is around half of the height between the deck and the ground. 3. There are height restrictions imposed by a nearby airport that limit the height of the towers overhead. 4. The cross-section of the approach spans on a long viaduct can be made to span further with extradosed prestressing. Extradosed prestressing can be kept to a minimum by using as many internal and external tendons in the girder of the extradosed span as in the approach spans. There are only a few bridge types that meet these restrictions: trusses, tied arches, extradosed bridges, and cantilever post-tensioned bridges (when condition 1 is not a requirement - the roadway elevation can be raised). Trusses may have been an option in the past, but the rising cost of labour has made them uneconomical for medium and long spans, and their aesthetics can make other structural types preferable. 15
16 Table 2-1. Summary of Extradosed Bridges Name and Location
Operational Date
-Deck Depth x Width -Span Lengths -Deck Description
Picture
1980 2.5 - 5 x 10 127.0 + 174.0 + 127.0 Wide single cell concrete box girder, cablepanel stayed.
1
Ganter Bridge, Switzerland
2
Proposed 3.75 x 20.5 Arrêt-Darré 60.0 + 100.0 + 100.0 + 100.0 + 100.0 + 52.0 Viaduct, France Single cell concrete box girder with voided webs and struts supporting deck cantilevers.
Vogel & Marti 1997, Kasuga 2006
Photo from Virlogeux 1999 Mathivat 1986, Mathivat 1988, Virlogeux 2002a
3
Barton Creek Bridge, Austin, USA
1987 3.7 - 10.7 x 17.7 47.6 + 103.6 + 57.9 Single cell concrete box girder with webs inclined inwards into a central fin above the deck level, and transverse struts supporting the deck slab. Photo from Gee 1999 Gee 1990
4
Socorridos Bridge, Madeira, Portugal
1993 3.5 x 20 54.0 + 85.0 + 106.0 + 86.0 Single cell concrete box girder, cable-panel stayed. Photo from Reis & Pereira 1994 Reis & Pereira 1994
5
Odawara Blueway Bridge, Japan
1994 2.2 - 3.5 x 13 73.3 + 122.3 + 73.3 Wide double cell concrete box girder.
Photo from Kasuga 2006 Taniyama et al. 1994, Kasuga et al. 1994
6
Saint-Rémy-deMaurienne Bridge, Savoie, France
1996 2.2 x 13.4 52.4 + 48.5 U shaped concrete deck with transverse ribs between edge beams. Photo by Jaques Mossot, Structurae Grison & Tonello 1997, Kasuga 2006
7
Tsukuhara Bridge, Japan
1997 3 - 5.5 x 12.8 65.4 + 180.0 + 76.4 Wide single cell concrete box girder.
Photo from Ogawa et al. 1998 Ogawa et al. 1998
Detailed Drawing
17 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location 8
Operational Date
Kanisawa Bridge, Japan
-Deck Depth x Width -Span Lengths -Deck Description
Picture
1998 3.3 - 5.6 x 17.5 99.3 + 180.0 + 99.3 Concrete box girder.
Photo from Cho 2002 Kasuga 2006
9
Shin-Karato Bridge, Kobe, Japan
1998 2.5 - 3.5 x 11.5 74.1 + 140.0 + 69.1 Two and three cell concrete box girder.
Photo from Tomita et al. 1999 Tomita et al. 1999
10 Sunniberg Bridge, Switzerland
1998 1.1 x 12.375 59.0 + 128.0 + 140.0 + 134.0 + 65.0 Concrete slab with edge stiffening beams.
Photo by author Figi et al. 1997, Figi et al. 1998, Menn 1998, Baumann & Däniker 1999, Honigmann & Billington 2003
11 Santanigawa (Mitanigawa) Bridge, Japan
1999 2.5 - 6.5 x 20.4 57.9 + 92.9 Double cell concrete box girder.
Nishimura et al. 2002, Stroh et al. 2003
12 Second Mandaue Mactan (Marcelo Fernan) Bridge, Mactan, Philippines
1999 3.3 - 5.1 x 18 111.5 + 185.0 + 111.5 Three cell concrete box girder.
13 Matakina Bridge, Nago, Japan
2000 3.5 - 6 x 11.3 109.3 + 89.3 Single cell concrete box girder.
Photo from www.jsce.or.jp/committee/tanaka-sho/jyushou Kasuga 2006
Photo from www.dywidag-systems.com Kasuga 2006
14 Pakse (LaoNippon) Bridge, Laos
2000 3 - 6.5 x 13.8 52.0 + 123.0 + 143.0 + 91.5 + 34.5 Single cell concrete box girder.
Photo from www.dywidag-systems.com Nakamura 2001, Kikuchi et al. 2002, Kasuga 2006
15 Sajiki Bridge, Japan
2000 2.1 - 3.2 x 11 60.8 + 105.0 + 57.5
Kasuga 2006
Detailed Drawing
18 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location
Operational Date
16 Shikari Bridge, Japan
-Deck Depth x Width -Span Lengths -Deck Description
Picture
2000 3 - 6 x 23 94.0 + 140.0 + 140.0 + 140.0 + 94.0 Concrete box girder.
Photo from www.jsce.or.jp/committee/tanaka-sho/jyushou Stroh et al. 2003
17 Surikamigawa Bridge, Japan
2000 2.8 - 5 x 9.2 84.82
Kasuga 2006
18 Wuhu Yangtze River Bridge, Wuhan, China
2000 15 x 23.4 180.0 + 312.0 + 180.0 Double-decker steel truss with composite deck slab on top roadway, two rail lines on bottom level. Photo from Fang 2004 Fang 2004
19 Yukisawa-Ohashi Bridge, Japan
2000 2 - 3.5 x 15.8 70.3 + 71.0 + 34.4 Two cell concrete box girder with wide sidewalks on deck cantilever overhangs outside of cable planes. Nunoshita et al. 2002, Kasuga 2006
20 Hozu Bridge, Japan
2001 2.8 x 15.3 33.0 + 50.0 + 76.0 + 100.0 + 76.0 + 31.0 Single cell concrete box girder.
Photo from www.jsce.or.jp/committee/tanaka-sho/jyushou Sumida et al. 2002, Kasuga 2006
21 Ibi River Bridge, Japan
2001 4.3 - 7.3 x 33 154 + 271.5 + 271.5 + 271.5 + 271.5 + 157 Hybrid cross section: four cell concrete box girder near piers and steel box girder in central 100 m with moment and shear connection.
Photo from Mutsuyoshi 2004 Hirano et al. 1999, Casteleyn 1999, Kutsuna et al. 2002, Kasuga 2006
22 Kiso River Bridge, Japan
2001 4.3 - 7.3 x 33 160.0 + 275.0 + 275.0 + 275.0 + 160.0 Hybrid cross section: four cell concrete box girder near piers and steel box girder in central 100 m with moment and shear connection. Photo from Mutsuyoshi 2004 Hirano et al. 1999, Casteleyn 1999, Kasuga 2006
Detailed Drawing
19 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location
Operational Date
23 Miyakodagawa Bridge, Japan
-Deck Depth x Width -Span Lengths -Deck Description
Picture
2001 4 - 6.5 x 19.9 134.0 + 134.0 Parallel double cell box concrete box girders.
Photo from www.jsce.or.jp/committee/tanaka-sho/jyushou Kato et al. 2001, Terada et al. 2002
24 Nakanoike Bridge, Japan
2001 2.5 - 4 x 21.4 60.6 + 60.6
Kasuga 2006
25 Fukaura Bridge, Japan
2002 2.5 - 3 x 13.7 62.1 + 90.0 + 66 + 45.0 + 29.1
Kasuga 2006
26 Korror Babeldoap Bridge, Palau
2002 3.5 - 7 x 11.6 82.0 + 247.0 + 82.0 Hybrid cross section: wide single concrete box girder near piers and steel box girder in central 82 m. Photo courtesy of Tony Jones (www.onhiatus.com) Oshimi et al. 2002, Ewert 2003
27 Sashikubo Bridge, Japan
2002 3.2 - 6.5 x 11.3 114.0 + 114.0 Concrete box girder.
Kasuga 2006
28 Shinkawa (Tobiuo) Bridge, Hamamatsu, Japan
2002 2.4 - 4 x 25.8 38.5 + 45.0 + 90.0 + 130.0 + 80.5 Three cell concrete box girder.
Photo from www.dywidag-systems.com Kasuga 2006
29 Deba River Bridge, Gipuzkoa, Spain
2003 2.7 x 13.9 42.0 - 66.0 - 42.0 U shaped concrete deck with transverse ribs between edge beams. Photo from www.eipsa.net Jaques 2005
30 Himi Bridge, Japan
2004 4 x 12.45 91.8 + 180.0 + 91.8 Single cell doubly composite box girder with corrugated steel webs. Photo from bd&e Second Quarter 2004 Kasuga 2006
Detailed Drawing
20 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location
Operational Date
31 Korong Bridge, Budapest, Hungary
-Deck Depth x Width -Span Lengths -Deck Description
Picture
2004 2.5 x 15.85 52.26 + 61.98 Three cell concrete box girder stiffened with transverse ribs. Photo from Becze & Barta 2006 Becze & Barta 2006
32 Shin-Meisei Bridge, Japan
2004 3.5 x 19 89.6 + 122.3 + 82.4 Three cell concrete trapezoidal box girder.
Photo from Mutsuyoshi et al. 2004 Iida et al. 2002, Kasuga 2006
33 Tatekoshi Bridge, Japan
2004 1.8 - 2.9 x 19.14 56.3 + 55.3
Kasuga 2006
34 Sannohe-Boukyo Bridge, Aomori, Japan
2005 3.5 - 6.5 x 13.45 99.9 + 200.0 + 99.9 Concrete box girder.
Photo from www.dywidag-systems.com Kasuga 2006
35 Domovinski Bridge over the River Sava, Croatia
2006 3.55 x 34 48 + 6x60 + 72 + 120 + 72 + 2x60 + 48 Five cell concrete box girder supports light rail between cable planes. Photo from Structurae Balić & Veverka 1999
36 Kack-Hwa First Bridge, Gwangju, South Korea
2006
- x 31.1 55.0 + 115.0 + 100.0 Multiple cell concrete box girder.
Structurae
37 Nanchiku Bridge, Japan
2006 2.6 - 3.5 x 20.55 68.1 + 110.0 + 68.1
Kasuga 2006
38 Rittoh Bridge, Japan
2006 4.5 - 7.5 x 19.6 140 + 170 + 115 + 70 (Tokyo bound) Three cell doubly composite box girder with corrugated steel webs. Photo from Masterson 2004 Wilcox et al. 2002, Yasukawa et al. 2002, Masterson 2004
Detailed Drawing
21 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location
Operational Date
-Deck Depth x Width -Span Lengths -Deck Description
Picture
39 Tagami Bridge, Japan
2006 3 - 4.5 x 17.8 80.2 + 80.2
40 Third Bridge over Rio Branco, Brasil
2006 2 - 2.5 x 17.4 54 + 90 + 54 Deck slab with L shaped edge beams (appears as single box girder with incomplete bottom slab) that taper to I beams at midspan.
Kasuga 2006
Photo from Ishii 2006 Ishii 2006
41 Tokuyama Bridge, Japan
2006 4 - 6.5 x 17.4 139.7 + 220.0 + 139.7
Stroh et al. 2003
42 Yanagawa Bridge, Japan
2006 4 - 6.5 x 17.4 130.7 + 130.7
43 Brazil-Peru Integration Bridge, Brazil
2007 2.35 - 3.35 x 16.8 65.0 - 110.0 -65.0 Wide single cell concrete box girder.
Kasuga, 2006
Photo from Ishii 2006 Structurae
44 Gum-Ga Grand Bridge, Chungcheongnamdo, South Korea
2007
- x 23 85.4 + 125.0 + 125.0 + 125.0 + 125.0 + 125.0 + Mulitple cell concrete box girder.
Structurae
45 Pyung-Yeo 2 Bridge, Yeosu, South Korea
2007 3.5 - 4 x 23.5 65.0 + 120.0 + 65.0 Four cell concrete box girder.
Photo from Masterson 2006 Masterson 2006
46 Second Vivekananda Bridge over the Hooghly River, Calcutta, India
2007 3.5 x 28.6 55.0 + 7 x 110.0 + 55.0 Wide single cell trapezoidal box girder with internal struts (Bang Na cross section). Photo from www.ibtengineers.com Binns 2005
Detailed Drawing
22 Table 2-1. Summary of Extradosed Bridges (continued) Name and Location
Operational Date
-Deck Depth x Width -Span Lengths -Deck Description
Picture
47 Cho-Rack Bridge, Dangjin, South Korea
2008
48 North Arm Bridge (Canada Line Extradosed Transit Bridge), Canada
2008 3.4 x 10.31 139.0 + 180.0 + 139.0 Single cell concrete box girder for LRT.
49 Trois Bassins Viaduct, Reunion, France
2008 4 - 7 x 22 18.6 - 126.0 - 104.4 - 75.6 - 43.2 Single cell concrete box girder with steel struts supporting long deck cantilevers.
- x 14 70.0 + 130.0 + 130.0 + 130.0 + 70.0 Multiple cell concrete box girder.
Structurae
Photo from bd&e 2004 Griezic 2006
Photo from Frappart 2005 Frappart 2005, Boudot et al. 2007
50 Golden Ears Bridge, Canada
2009 2.7 - 4.5 x 31.5 121.0 + 242.0 + 242.0 + 242.0 + 121.0 Steel box girders at edge of deck with transverse floor beams composite with precast concrete deck. Photo from Trimbath 2006 Trimbath 2006, Bergman et al. 2007
51 Pearl Harbor Memorial (Quinnipiac) Bridge, New Haven, USA
2012 3.5 - 5 x 33.7 75.9 + 157.0 + 75.9 Parallel five cell concrete box girders with inclined exterior webs. Photo from Stroh et al. 2003 Stroh et al. 2003
Detailed Drawing
23
24
25
26
27
Tied arches and extradosed spans are two good options for urban environments. Girder bridges are not visible to the driver and as stated by Menn (1991): “the general public was never captivated by modern bridge construction. Beam bridges were largely perceived as boring.” For a ‘signature bridge’, girder bridges do not have the visual elegance that is desired by governing authorities and designers alike. The choice between a tied arch and an extradosed structure might be guided by the span configuration. If only one long span is required, a tied arch may prove to be economical, but in other cases a cablesupported bridge will be a clear choice due to the potential for cantilevered construction, which has a relatively low impact on the terrain below. As the cost of concrete increases and the incremental cost premium for higher strength decreases, there is more incentive to use materials efficiently. The extradosed bridge presents a way to make use of the compressive capacity of the concrete, while maintaining conventional girder cross-sections and common construction methods. \ Extradosed bridges allow for unequal span lengths, unsymmetrical span arrangements, multiple stayed spans and approach spans with the same cross section as the main spans. Of the extradosed bridges in this study, 41 of 51 have main spans between 75 and 200 m; 11 have two extradosed spans (one tower), 30 have three extradosed spans (2 towers), and 10 have more than three extradosed spans. From Figure 2-4b, it is observed that multiple span bridges are viable for all span lengths. Mathivat’s concept for a span to depth ratio of between 30 and 35 at the piers has been followed for 22 of the extradosed bridges, but the span to depth ratio at midspan has been increased to over 50 in 23 bridges, as observed from Figure 2-2. There are 13 extradosed bridges with a constant depth cross-section, 10 of which have a span to depth ratio between 30 and 35. The 3 exceptions are the Himi Bridge, the North Arm Bridge, and the Sunniberg Bridge. 20
20 Constant Depth
Constant Depth
Variable Depth
Variable Depth 15
Bridges
Bridges
15
10
10
5
5
0
0 0
15
20
25
30 35 40 45 50 Span : Depth at Pier
55
60 65+
0
15
20
25
30 35 40 45 50 Span : Depth at Midspan
55
60 65+
Figure 2-2. Extradosed Bridges separated span to depth ratio at a) pier and b) midspan.
Around half (26) of the extradosed bridges have girders that are embedded (fixed in rotation) at the piers, which reduces the live load stresses in both the girder and the cables. Extradosed bridges that have a span to depth ratio above 40 at the pier are always embedded. It can be seen from Figure 2-3 that girders that are embedded generally have a higher span to depth ratio at midspan, and the span to depth ratio increases with increasing span length. As well, the girder is almost always embedded at longer span lengths (only 3 of 19 bridges above 150 m are on simple supports).
28 140
Constant Depth Simple Supports Constant Depth Embedded Variable Depth Simple Supports Variable Depth Embedded
120
Span : Depth
100
at midspan at pier
80 60
1:55
40
1:30
20 0 50
66
100
150
200
250
275
Longest Span, m
Figure 2-3. Span to depth ratios of extradosed bridges at midspan and pier.
Variable depth girders are more frequently used at longer span lengths, as shown in Figure 2-4a. A variable depth cross-section, with the depth at the pier more than 1.5 times the depth at midspan, is used in 26 extradosed girders. As the span length increases, the degree of haunching, or pier to midspan depth ratio, also increases as observed from Figure 2-4b. 3 >1.5 1-1.5 =1
Bridges
10
5
0 25
Depth at Pier : Depth at Midspan
15
2
1 2 Spans 3 Spans 4 Spans + 0
50
75
100 125 150 175 Span, m
200 225 250+
0
50
100 150 200 Longest Span, m
250
300
Figure 2-4. Haunching in extradosed bridges shown a) in groups by span length and b) as the pier to midspan depth ratio by span.
Tower height does not seem to be affected by girder depth or the fixity between the girder and piers. It can however be observed from Figure 2-5 that in general, the span to tower height ratio decreases with increasing span length. Most bridges have a span to tower height ratio between 8 and 12, and there are very few bridges (3 of 51) with a span to tower height ratio close to 15, as suggested by Mathivat (1988).
29 15 Mathivat (1988)
14
Span : Height of Tower
12 10
8
8 6
5 Constant Depth Simple Supports Constant Depth Embedded Variable Depth Simple Supports Variable Depth Embedded
4 2 0
50
66
100
150
200
250
Cable-Stayed Typical
275
Longest Span, m
Figure 2-5. Span to tower height ratio of extradosed bridges.
2.3
Characteristics of Extradosed Bridges
2.3.1
Materials Usage
The average girder thickness, the volume of concrete in the girder divided by the deck surface area, can be used to compare the material usage of different bridge types. In Figure 2-6, the average girder thickness of extradosed bridges and a selection of cantilever-constructed girder bridges and cable-stayed bridges, is plotted against the longest span. Information on the girder and cable-stayed bridges considered in this section is included in Appendix A. Also shown in Figure 2-6 are estimates of average girder thickness tg as a function of average span lm suggested by Menn (1990) and SETRA (2007), for cantilever constructed girder bridges, repeated below. Menn’s Estimate: t g = 0.35 + 0.0045l m
SETRA Estimate: t g = 0.4 + 0.0035l m
As observed from Figure 2-6, the average girder thickness of an extradosed bridge will lie somewhere between that of a girder bridge and a cable-stayed bridge of the same span length. For a main span between 80 and 100 m, there can be very little difference between the average girder thickness in a girder bridge and an extradosed bridge, but the difference increases rapidly, indicating a greater savings in concrete with an extradosed bridge as the span increases. In contrast, the average thickness of a cablestayed bridge increases very gradually, with the longest cable-stayed bridge having an average thickness of not more than 60% more than the shortest of comparable cross-section. For a 200 m span, the average girder thickness of a cable-stayed bridge is around 30% of that of a girder bridge. It appears that a minimum average thickness of around 0.4 m for a cable supported (extradosed or cable-stayed) bridge and of around 0.6 m for a girder bridge is required regardless of span length. This minimum thickness is dictated by transverse behaviour and practical requirements. For the cable-stayed bridges considered in Figure 2-6, stiffened slabs are always lighter than box girders, which can generally be explained by the absence of a bottom slab. A closer look at the average girder thickness of the extradosed bridges only, as seen in Figure 2-7, shows a huge spread of values, especially for spans of around 100 m. The single cell box girders and
30 Menn (1990) Estimate Cantilevered Regression
1.17
SETRA (2007) Estimate
1.1
Cantilever Constructed Girder Extradosed Cable-Stayed
Average depth of concrete, m 3/m2
1.0
0.9 Extradosed Regression
0.8
0.7
0.6
Cable-Stayed Regression
0.5
0.4 0.34 0.3 0
100
200
300 Longest Span, m
400
500
530
Figure 2-6. Average girder concrete thickness of cantilever-constructed girder, extradosed and cable-stayed bridges. 0.91 0.9
Miyakodagawa Pyung-Yeo
Average depth of concrete, m 3/m2
Regression
Shin-Karato
0.8
Ganter
Domovinski
Korong
Hozu YukisawaOhashi
Odawara Pearl Harbor
Tsukuhara
Ibi Kiso
Shinkawa
0.7
Pakse
Himi
Saint-Remy Shin-Meisei
0.6
Rittoh
Socorridos
0.5
Barton
Brazil-Peru
North Arm
Trois Bassins
Sunniberg
Arret-Darre
0.4 0.39
Rio Branco
0.3 50
80
100
150
200
250
275
Longest Span, m
Figure 2-7. Average girder concrete thickness of extradosed bridges.
stiffened slab cross-sections always have a lower average thickness than multiple box girder cross-sections for the same span. In terms of total consumption of longitudinal prestressing steel, from available data on internal prestressing in extradosed bridges (Grison & Tonello 1997; Becze & Barta 2006; Boudot et al. 2007), it appears that the weight of total prestressing per unit volume of concrete is at best comparable to that of
31
cantilever constructed girder bridges. Therefore, any reduction in self-weight of concrete in the deck should reduce the prestressing steel. Figure 2-8 shows the mass of longitudinal prestressing and reinforcing steel, per unit volume of concrete, in a selection of cantilever constructed girder bridges. The quantity of longitudinal prestressing steel is fairly consistent, especially for shorter spans. The quantity of reinforcement on the other hand varies considerably, but most of this variation comes from the transverse system of the deck slab (SETRA 2007). The upper values correspond to reinforced decks, wheras the lower values correspond to transversly prestressed and ribbed decks. Shear reinforcement also affects the results since there is a large difference between the web reinforcement if the webs are vertically prestressed or if external tendons are used, as found for the Chapter 4 girder bridge designs. Regression lines are shown for the available data, as are lines for preliminary estimates of superstructure costs suggested by Menn (1990) based on an analysis of 19 bridges: Prestressing Steel: m P = 0.35l m
Reinforcing Steel: m s = 90 + 0.35l m
The parameter lm is the geometrical average span length (Σli2/Σli) in metres while the quantities mP and ms are the mass per unit volume of concrete (kg/m3). For a cantilever-constructed girder bridge, the average span can be taken as the longest span since the concrete and reinforcement in each side span is essentially equivalent to half of the main span. 200 Menn (1990) Estimate
60
40
20
Mass of Reinforcing Steel, kg/m 3
Mass of Longitudinal Prestressing Steel, kg/m3
80
180 Menn (1990) Estimate
160 140 120 100 80
0
0
50
100
150
Longest Span, m
200
0
50
100 150 Longest Span, m
200
Figure 2-8. Mass of steel in cantilever constructed girder bridges: a) longitudinal prestressing steel, and b) reinforcing steel (plots are based on data from SETRA 2007, Lacaze 2002, DEAL 1999).
The quantity of reinforcing steel in an extradosed bridge can be estimated from the above charts by using an equivalent girder span length. The equivalent girder span length is calculated by multiplying the extradosed span by the average depth of the extradosed cross section divided by the average depth of the cantilever construction girder cross section. For an extradosed bridge with constant span to depth ratio of 50, the equivalent cantilever constructed girder bridge span would be 65% of the extradosed span.
2.3.2
Girder Stiffness
The moment of inertia of the girder of extradosed bridges at midspan varies considerably, especially for bridges with a span between 100 and 150 m, as observed in Figure 2-9. This variation can be explained by
32
the variation in span to depth ratio at midspan observed in Figure 2-3. At the upper end, the moment of inertia of some extradosed bridges is higher than that of cantilver constructed girder bridge of equal span, while on the lower end, it is barely higher than that of a cable-stayed bridge of equal span, as seen in Figure 2-9. There is however one extradosed bridge, the Sunniberg Bridge, which has a girder that is considerably more flexible (lower moment of inertia) than most extradosed bridges. Its structural behaviour appears to be more like a cable-stayed bridge, despite having a cable inclination of an extradosed bridge. It can also be observed that there is a distinct difference between the moment of inertia of cable-stayed bridges that have box-girders and those that have slabs stiffened by edge beams. In this small sample of cable-stayed bridges, all box girders are centrally suspended, while all slab or stiffened slab bridges are laterally supported. 25
Span/50
Span/40
Extradosed Cable-Stayed Typical Cantilever Constructed Box Girder
Midspan I per 10 m width, m 4
20
15
10
5
Sunniberg
0
0
80 100
200
300 Longest Span (m)
400
500 530
Figure 2-9. Moment of inertia of girder at midspan for extradosed and cable-stayed bridges (per 10 m width).
2.4
Detailed Descriptions of Extradosed Bridges
2.4.1
Odawara Port Bridge, Japan
The Odawara Bridge was the first extradosed bridge in the world to be constructed. The bridge has a two cell box girder cross-section that is haunched near the piers, and supported in three spans by extradosed cables in a semi-fan arrangement. The deck is monolithically connected to the two lateral tower legs, which have a hexagonal shape. There are 2 lateral planes of 8 extradosed 19-15 mm dia strand tendons per half span. The stress range in the extradosed cables due to live load is between 15 and 38 MPa (Kasuga et al. 1994). The extradosed cables are carried over the piers in saddles spaced vertically at 300 mm, with the internal sheath anchored outside the saddles to prevent slip, as seen in Figure 2-10. After the extradosed cables enter into the deck,
33
Section through top of pylon
Figure 2-10. Odawara Extradosed Bridge details of tower saddle and arrangement of prestressing bars in tower from FEM analysis (Kasuga et al. 1994).
they curve inwards to provide the necessary clearance for jacking them from within the box girder. This is a unique solution to conceal the cable anchorages from view above or beneath the bridge. The sheaths are installed in steel recess tubes both over the towers and through the girder to allow for future replacement of the complete cable (Taniyama & Mikami 1994). Details of the cable anchorages and installation are shown in Figure 2-11. The strands are epoxy coated and grouted inside an FRP sheath. High damping rubber dampers are installed at the stay anchorages to decrease rain and wind vibrations. a)
b)
c)
Figure 2-11. Odawara Extradosed Bridge: a) strand supply system; b) saddle structure at the pier top, and c) anchorage structure at the main girder. (Toniyama & Mikami 1994).
The bridge was constructed in cantilever with cast-in-place concrete. Temporary cables were used for cantilever construction of the girder out to the first extradosed cables. The bridge features only external tendons within the girder cross-section, although internal 12-13 mm dia strand tendons were included for cantilevering.
2.4.2
Tsukuhara Extradosed Bridge, Japan
Twin parallel extradosed structures cross the Tsukuhara Bridge as part of a road network with the AkashiKaikyo Bridge (Ogawa et al. 1998). Due to the steep embankments and deep river, the side spans are very short at 65.4 m (0.36 of main span) and 76.4 m (0.42 of main span). Fill concrete in the box girder is used as a counterweight at the end of the each side span in order to reduce the overturning moment at the piers.
34
The deck slab spans 9 m between webs and is post-tensioned with 28.6 mm dia monostrand tendons with an after-bond pregrouted epoxy that does not require conventional grouting. There are 2 lateral planes of 8 extradosed 27-15 mm dia strand tendons per half span. The maximum stress range in the extradosed cables due to live load is 37 MPa (Ogawa et al. 1998). The strands are individually sheathed with polyethylene, bundled and encased in an HDPE pipe which is filled with a polyethylene filler (Chilstrom 2001). There are 12 external 19-15 mm dia strand tendons inside the box girder across the main span to resist positive bending moments, and internal 12-13 mm dia strand tendons that are mainly used for cantilevering of 7 m segments (Ogawa et al. 1998). The Tsukuhara Bridge has a span to depth ratio of 60 at midspan, which is very shallow compared with others studied in this chapter. Despite this fact, the live load stress range in the cables is very low.
2.4.3
Ibi and Kiso River Bridges, Japan
The Ibi and Kiso River Bridges are 5 and 6 span continuous structures that have total lengths of 1397 and 1145 m respectively, with maximum spans of 271.5 and 275 m. The towers are integral with the deck and the superstructure rests on force distributing rubber bearings on the piers (Chilstrom 2001). The superstructure has a hybrid construction, with each span consisting of cable-supported precast concrete segments for the first 90 m from the piers, and a central steel box girder which is continuous with the concrete sections. The continuity is achieved with shear studs, internal prestressing, and external prestessing which is installed across the concrete segments, deviated at the piers and in span, and anchored at the ends of the steel girders (Hirano et al. 1999). The precast concrete segments are of 60 MPa concrete and have dimensions of 5 m in length, 33 m in width, and up to 7 m in height. The segments weigh up to 400 tonnes, and were erected using a 600 tonne crane. The segments were cast using two fabrication lines for each bridge, and transported 10 to 15 km by barge to the site (Casteleyn 1999). The steel spans of 95 to 105 m weigh up to 2000 tonnes and were strand-lifted and joined to the concrete segments with prestressing and shear studs. Figure 2-12. Ibi River Bridge Prestressing Tendon Layout in CrossSection (Kutsuna et al. 1999).
Kutsuna and Kasuga (2002) simulated the nonlinear behaviour of the structure up to its ultimate limit state, to account for material and geometrical nonlinearity. The model included the effects of the girder, internal and external tendons, and extradosed cables. The total load was increased gradually until the concrete reached an ultimate strain of 0.0025. The target load was established as (D+L)×1.7. As the load increased, the precompression stress in the upper fibre of the girder section at the pier increased until it exceeded the cracking stress at (D+L)×1.5. Beyond this point, the increase in the tensile force was taken by the internal and external tendons. The tension in the external tendons was almost constant up to (D+L)×1.5, then increased by 100 MPa at (D+L)×1.7. The uppermost extradosed cable
35
Lower fibre concrete stress at the support section
External tendon stress at the support section
Extradosed cable stress at the tower
Figure 2-13. Nonlinear Behaviour of the Ibi River Bridge up to ultimate load (Kutsuna et al. 2002).
reached the yield stress at (D+L)×1.4, and by (D+L)×1.7, all cables had yielded. Since all the cables yield at ULS, Kutsuna and Kasuga conclude that the extradosed cables are used effectively. While the magnitude of the live load is not given, the target load of (D+L)×1.7 is already very conservative. Given a ratio of maximum live load (unfactored) to dead load of 0.2, as calculated for the extradosed bridges in Chapter 4 and typical of concrete cable-stayed bridges (Walther et al. 1999), a total lead of (D+L)×1.5 equates to 1.2D+3LL which already exceeds the ULS1 requirement of the CHBDC (CSA 2006a).
2.4.4
Shin-Meisei Bridge, Japan
The structure was chosen both for economy and aesthetics, and carries two lanes of traffic in each direction across the river. The trapezoidal box girder cross-section has two interior webs close to the cable anchorages centred in the cross-section, and tapered exterior webs with shallow ribs to add visual interest when observed from beneath the bridge. The tapered exterior webs were used to limit the width of the bottom slab to give the bridge a lighter appearance (Kasuga 2006). The cross-section was cast-in-place in balanced cantilever from the piers, and the side spans were assembled by overcantilevering with precast core segments erected from a crane. The precast core segments measured 6 m wide by 1.8 m in length to keep their weight below 25 tonnes. After the core segments acheived continuity across the side spans, the wings were cast-in-place with the same traveller used for the full cast-in-place segments (Iida et al. 2002).
Core parts erected by truck crane
Wings constructed by form traveller
Figure 2-14. Shin-Meisei Birdge construction of side spans (Iida et al. 2002).
The tower has a composite steel and concrete cross-section which allows the transfer of tensile stresses across the tower without heavy prestressing, and allows for assembly without bolting or welding on-site. The steel shell is assembled from box sections each weighing less than 5 tonnes. The post-cast concrete
36
joint down the centre of the tower, shown in Figure 2-15, alleviates any cracking that would occur from the elastic strain across the tower as the cables are installed.
a)
b)
c)
Figure 2-15. Shin-Meisei Birdge a) photo of steel shell of tower; b) elevation of composite tower and c) details of composite tower (drawings: Iida et al. 2002, photo and rendering: Kasuga 2006).
2.4.5
North Arm Bridge, Canada
The North Arm Bridge is an extradosed bridge carrying the Canada Line LRT from the Vancouver Airport into the City across the Fraser River North Arm. The extradosed bridge type was chosen to keep the track profile as low as possible to cross the navigational clearance envelope, while keeping the towers below the glidepath clearance envelope (Griezic et al. 2006). The main span is constructed from precast segments of 2.8 m maximum length, with a maximum weight of 70 tonnes for transportation and lifting. A deck level extradosed cable anchorage segment is shown in Figure 2-16a. There are 6 centrally positioned extradosed 58-16 mm diametre strand tendons per half span, installed with monostrand jacking equipment. The strands are galvanized, individually sheathed and waxed, and enclosed in an HDPE pipe. The maximum stress range in the cables due to live load is 73 MPa. a)
b)
c)
Figure 2-16. North Arm Birdge a) deck level extradosed cable anchorage; b) precast tower, and c) tower anchor segment (from Griezic et al. 2006).
The towers are assembled from precast composite sections, as shown in Figure 2-16b, and posttensioned vertically with 4 internal 19-15 mm diametre strand tendons. The cables are anchored in a central steel box to avoid post-tensioning across the towers. Small HSS sections are welded between the steel web plates to prevent vertical stresses, due to creep and shrinkage strains in the concrete, from loading the steel web plates over time.
37
The designers of this bridge made two important decisions based on economy. A detailed comparison was made and anchorages were chosen over saddles in the towers. Secondly, a constant depth girder was chosen over a variable depth girder at the piers.
2.4.6
Pont de Saint-Rémy-de-Maurienne, France
The bridge crosses over the A43 highway and a river on a curve, at a location with tight geometry where a very shallow clearance was required, since the roadway surface is only 0.9 m above the highway clearance envelope. The bridge has two spans of 52.5 and 48.5 m, and is cast-in-place on falsework and posttensioned. The bridge cross-section is U shaped and consists of edge girders with transverse cross-beams at 2.27 m spacing supporting a 220 mm thick concrete deck. There are 6 34-15 mm dia strand tendons in each girder, which are internal through most of the span and rise above the girder only around the deviators to become extradosed cables (Grison & Tonello 1997). With an average girder thickness of 0.65 m, the Saint-Rémy Bridge has a longitudinal prestressing mass per unit volume of concrete of 52 kg/m3 and a mass per unit area of deck surface of 30 kg/m2. This is comparable to another short extradosed bridge, the Korong Bridge (Becze & Barta 2006), which has two spans of 62 and 52 m, an average girder thickness of 0.785 m and a longitudinal prestressing mass of 43 kg/m3 and 33 kg/m2. However, this is quite alot more prestressing than in a conventional box girder bridge of equivalent spans, which would be expected to have a similar average girder thickness of around 0.6 m but a longitudinal prestressing mass of only 20 to 30 kg/m3, based on Section 2.3.1. Compared with other channel bridges, the material quantities in the Saint-Rémy Bridge seem more reasonable. The Route 302 Bridge over Route 17 in New York State is a channel bridge (Allen & Naret 1998; Shepherd & Gibbens 2004) assembled from precast segments that carries two lanes of traffic over two spans of 34.1 m. The bridge has an average girder thickness of 0.48 m and a longitudinal prestressing mass of 51 kg/m3 and 23 kg/m2. Given that the span to depth ratio of the Saint-Rémy Bridge is 24 and that of the Route 302 Bridge is 22, it is clear that the Saint-Rémy Bridge could have been designed with internaly prestressing only. While the extradosed prestressing may lead to an optimal alignment for the prestressing, as claimed by Grison & Tonello (1997), it is unlikely that the cost of the building towers and providing external protection details for the cables would have been less than the cost of additional prestressing strand. From this comparison, it is clear that the extradosed form was primarily chosen for the second reason stated by its designers, that of aesthetics that closely mimic the structural behaviour of the bridge.
38
2.4.7
Viaduc de la ravine des Trois Bassins, Réunion
The extradosed bridge form was found to be a good solution for the ravine both in terms of technological and architectural considerations, and was thought to integrate harmoniously into the environment (Frappart 2005). A concrete bridge was favoured due to the local availability of the material and a labour force that was not familiar with structural steel. The bridge has spans of 126 - 104.4 - 75.6 - 43.2 m with a counterweighted span of 18.6 m adjacent the longest span to stabilise the bridge transversely against cyclone winds of up to 49 m/s. The unusual span arrangement was chosen partly because access was restricted to one side of the ravine only. The bridge was cast-in-place with 60 MPa concrete in segments of 3.6 m length. For each segment, the form traveler was used to cast Figure 2-17. Trois
the main box section, while two smaller mobile formwork travelers were used to Bassins Viaduct main pier install the struts and cast the deck overhang cantilevers in 7.2 m segments, off the (Frappart 2005). critical path. The bridge was constructed starting with the short spans and ending with the closure of the longest span. The webs are inclined outwards to achieve the necessary torsional resistance of the section without increasing the web thickness, and concurrently limiting the offset between the extradosed anchorages and the webs (Frappart 2005). There are double planes of 7 extradosed 37-16 mm dia strand cables across the main tower, as shown in Figure 2-17, and one plane of 3 extradosed cables of the same size extending across the secondary pier (Boudot et al. 2007). The strands of the extradosed cables are individually greased and sheathed and insulated inside an HDPE external sheath, and are deviated across the towers by saddles. Internal 19 and 12-16 mm dia strand tendons were used for cantilevering, with one pair of cables anchored in most segments, and were also used for continuity between cantilevers. External 19-16 mm dia strand continuity tendons were draped across two or more spans. The mass of longitudinal prestressing (extradosed cables, internal and external prestressing) per volume of concrete in the girder is 52 kg/m³ and the mass per unit area of deck surface is 46 kg/m². The deck is transversely prestressed with 3 4-16 mm dia strand tendons at the struts and one 4-16 mm dia strand tendon per metre length of deck elsewhere.
2.4.8
Sunniberg Bridge at Klosters, Switzerland
The Sunniberg Bridge spans a total of 526 m, with a maximum span of 140 m, over the Landquart River valley before entering the Gotschna tunnel which is part of a new bypass to Klosters. As a highly visible structure along the bypass, the owner desired a high aesthetic standard and minimum impact to the valley
39
below during construction (Figi et al. 1997). The deck cross-section consists of a solid slab with longitudinal edge beams, with a 7% roadway superelevation. a)
1m
3.5 m
3.5 m
1m
b)
Stay cable
4 o 22
5 o 26
o 14 e=20
o14/o16 e=15
Longitudinal prestressing
o 30 e15/12.5
1.9 m
1.15 m
Cable anchorage
0. 4 m
4 o 22 4 o 22
Figure 2-18. Sünniberg Bridge a) deck cross-section and b) prestressing and reinforcement (adapted from Tiefbauamt Graubünden 2001).
The bridge is curved in plan and connected monolithically at the abutments, which provides full longitudinal restraint and allows the bridge to deform as a horizontal arch under deformation due to temperature range. The abutments were designed as earth filled containers to anchor the horizontal reaction forces (Baumann and Däniker 1999). The pier columns have a parabolic variation in depth, and flare outwards from the base so that the towers are leaning outwards in order to provide the required clearance for the cables. There are two planes of 8 to 10 stay cables per half span. Each cable consists of 125 to 160 galvanised 7 mm dia wires, prefabricated to length and anchored by means of button heads in BBR DINA bonded anchorages, for a high fatigue Steel construction Stay cables with 125 to 160 wires of 7 mm diametre - cable forces 3850 to 4900 kN
Figure 2-19. Anchorages in towers (adapted from Tiefbauamt Graubünden 2001).
resistance. The cables have an ultimate tensile strength of 1600 MPa and were designed for a maximum allowable stress of 0.50 fpu. Each group of four cables were stressed simultaneously. There are 3 12-15 mm dia strand tendons (1900 kN each) in each edge beam through the midspan areas to compensate for the decrease in axial force from the cables (Baumann and Däniker 1999).
The flexible deck results in large deflections throughout each stage of construction, and correspondingly large variations in bending moment, as seen in Figure 2-20. Under permanent loads, the moment distribution across the deck resembles that of a continuous beam on simple supports, while under live load a point load is distibuted into nearby cables (Figi et al. 1997). The vertical deflection of the girder at midspan is 225 mm under a point load of 360 kN with and uniformly distributed load of 2 kN/m3, approximately L/600 which is under the L/400 limit that was agreed upon with the owner (Tiefbauamt Graubünden 2001). Of the total deflection, 40% comes from the rotation of the piers while 60% results from the elastic extension of the cables, resulting in an upwards displacement of 60 mm in the adjacent spans, 25% of the deflection of the loaded span. The total cost of the Sunniberg Bridge was SFr. 20 M, corresponding to SFr. 3075 per m2 (Baumann and Däniker 1999). The major cost items are the foundations, piers, and pylons (20.6%), bridge deck including form traveller (33.5%) and the stay cables (23.1%). The total mass of the stay cables is given as 320 tonnes, while the mass of longitudinal internal prestressing in the deck is 13 tonnes. The stay cable system cost around SFr. 14 400 per tonne. The mass of all prestressing (stay cables and internal PT) per
40 Form traveler advanced to Stage 5
Tower CL
Midspan
Displacement
Axial force from Stay Cables
Deck poured Displacement
Permanent Variable
Axial force from Internal Prestressing Permanent Variable
Concrete
Moments in inner edge beam
Stay Cables installed and stressed Displacement
Displacements due to live load Live load
Figure 2-20. Sunniberg Bridge a) bending moments and deflections of the edge beam through one stage of construction, and b) forces and deflections of the main span inner edge beam of the final structure due to permanent and live loads (adapted from Figi et al. 1998).
volume of concrete in the deck is 100 kg/m3 and the mass per unit area of deck is 97 kg/m2. These values are higher than are typically found in extradosed bridges, such as the viaduc de la ravine des Trois Bassins described in Section 2.4.7. The deck has an average thickness of only 0.51 m, which is lighter than all extradosed bridges in this span range, but the mass of reinforcement per unit volume of concrete is over 200 kg/m3.
2.5
Concluding Remarks
This chapter presented the results of a study of 51 extradosed bridges constructed or proposed to date. Most of the bridges in the study have proportions that are between those suggested by Mathivat (1988) and those used in the first few extradosed bridges constructed with girders embedded on the piers, designed by Menn and Kasuga. The girder concrete usage of the extradosed bridges is between typical values of cantilever constructed girder and cable-stayed bridges, although a large variation was observed amongst the extradosed bridges. A close examination of prestressing quantities in selected extradosed bridges indicates that the prestressing usage is around the same as a cantilever girder bridge of the same span, while reinforcing steel usage is lower. The examples of this chapter illustrate what is typical and what is achievable for extradosed bridges; they are equally well suited to cast-in-place and precast construction. In the next chapter, the insight gained from the study of existing extradosed bridges in this chapter provides a basis for recommendations on the design of cable configuration, girder cross-section, towers, and piers.
3
DESIGN AND CONSTRUCTION OF EXTRADOSED BRIDGES
Chapter 3 presents a comprehensive review of loads, design methodology, extradosed bridge proportions, stay cable technology, girder cross-sections, cable and tendon layout, erection and analysis. The purpose of this chapter is to explore the structural behaviour of an extradosed bridge, and determine what should be considered for its design. This is necessary in order to develop realistic bridge designs in Chapter 4.
3.1
Loads
The CAN/CSA-S6-06 Canadian Highway Bridge Design Code (CSA 2006a hereafter CHBDC) has been used throughout this thesis as the basis for all analysis of example extradosed bridges, bridge designs, and parametric studies. The CHBDC uses the limit states philosophy to satisfy the requirements for serviceability and fatigue, and to ensure the structure has adequate factored resistance to meet the factored load effect at ultimate limit states. Similar to cable-stayed bridges, the extradosed bridge is designed with service loads and allowable stresses in the stay cables and tendons. In the final stages of design, the capacity of the sections are verified at the ultimate limit state. The purpose of this section is to assess whether the CHBDC loads are adequate for the design of extradosed bridges. The load combinations of relevance to this thesis are summarised in Tables 3-1 and 3-2. Table 3-1. Load factors and load combinations (CSA 2006a). Loads SLS Combination 1 ULS Combination 1 ULS Combination 2 ULS Combination 9
Permanent D P 1.00 1.00 αP αD αD αP 1.35 αP
Transitory L K 0.90 0.80 1.70 0 1.60 1.15 0 0
Table 3-2. Permanent loads - maximum and minimum values of load factors α for ULS (CSA 2006a). Dead Load Factory produced components Cast-in-place components Wearing surfaces Prestress Secondary prestress efffects
Maximum 1.10 1.20 1.50 Maximum 1.05
Minimum 0.95 0.90 0.65 Minimum 0.95
Legend: D - dead load P - secondary prestress effects L - live load (including the dynamic load allowance) K - all strains, deformations, and displacements and their effects - includes those due to temperature change, temperature differential, concrete shrinkage, differential shrinkage, and creep.
3.1.1
Live Load
The live loading prescribed by the CHBDC has been adopted throughout this thesis. For long spans, the CHBDC live load is based on traffic loading for long span bridges recommended by the American Society of Civil Engineers Committee of Loads and Forces on Bridges (Buckland 1981), which was established after traffic studies were conducted on the Second Narrows Bridge in Vancouver, BC which is considered to be typical of traffic on most North Americal long span bridges (Buckland 1991). However, there are differences in vehicle positioning in the lane and multiple lane loading between the two codes which are described and quantified in this section. The CHBDC live load model consists of the CL-625 Truck, which consists of a series of axle loads which total 625 kN, and the CL-625 Lane Load which is made up of the CL-625 Truck reduced to 80% and 41
42
superimposed with a uniformly distributed load of 9 kN/m. Under serviceability limit state (SLS) and ultimate limit state (ULS), the CL-625 Truck load effect is increased by the addition of a dynamic load amplification (DLA) factor to account for impact, which varies depending on how many axles are loading the component under consideration as shown in Figure 3-1. For all axles acting on the bridge, the DLA is 25%. A DLA is not applied to the CL-625 Lane load, as this maximum load condition is assumed to occur with stationary vehicles on the bridge (CSA 2006b). The CL-625 live load is shown in Figure 3-1.
CL-625 Truck Dynamic Load Allowance
CL-625 Truck
Clearance Envelope
Modification factor for multiple lane loading
CL-625 Lane Load
Figure 3-1. CL-625 Live Loading: Maximum of CL-625 Truck (including DLA) or CL-625 Lane Load.
For spans up to approximately 50 m, the CL-625 Truck will govern the loading, but at spans beyond approximately 90 m, the CL-625 Lane load will govern. In between these span lengths, the Truck will govern in positive moment regions, while the Lane load will govern in negative moment regions. At SLS1, the live load is reduced to 0.9 of the value of the CL-625 Live load. Fractions of Basic Lane Load
1
P = Concentrated Load per Lane U = Uniform Load per Lane %H.V. = Average Percentage of Heavy Vehicles in Traffic Flow
3 4 5 6 2 Multiple Lane Distribution
Lane Loading to Produce Maximum Torque at Location X
Figure 3-2. ASCE Loading (adapted from Buckland 1981)
The CL-625 Lane load is based on the ASCE traffic loading for long span bridges, shown in Figure 32, with an average 30 % heavy vehicles in traffic flow, despite the fact that the Second Narrows Bridge was found to have an upper limit of 7.5 % average heavy vehicles in traffic flow (Buckland 1981). The 30 % loading was proposed by the ASCE Committee as an upper limit for some routes with a large number of
43
trucks, and was selected by the CHBDC Subcommittee on Loads to account for an increase in truck weights in the last 25 years and to allow for future growth in truck traffic (CSA 2006b). The loading of multiple lanes is not a simple matter, but it is important because it has a significant effect on the stress range of the cables due to live loading. Buckland (1991) presents a comparison of live loading between British Standards (BD 37/88, BS 5400 1978) and North American Standards (ASCE, AASHTO 1983, CAN/CSA-S6-88 1988) with a section on multiple lane loading. While the current CHBDC CL-625 Lane load is based on the ASCE 30% heavy vehicle curves, the multi-lane factors do not follow the ASCE recommendations. The multi-lane loading factors in the CHBDC follow the North American practice of reducing the lane loads uniformly across all lanes. The multi-lane loading takes into account the reduced probability of more than one lane being loaded simultaneously, due to actions such as traffic distribution, traffic volume, traffic speed, accident situations, and decrease of dynamic loads since it is unlikely that vehicles in multiple lanes vibrate in harmony (CSA 2006b). The CHBDC positions the vehicles biased towards one side of the design lane to produce the maximum load effect. In contrast, the ASCE loading positions the vehicles centrally in the design lanes, based on the assumption that over a long span, vehicles will be randomly spaced with the average position close to the centre of the lane (Buckland 1981). European practice has been to keep one or two lanes fully loaded while reducing the load on all others. As noted by Buckland, there “is merit to this idea as there is no reason to suppose the most heavily loaded lane will be less loaded simply because other lanes are open” and he emphasizes the importance of accurately representing the effects of the traffic (Buckland 1991). The ASCE Loading recommends one lane loaded to the maximum, the second lane loaded to 0.7 of the maximum, and all other lanes loaded to 0.4 of the maximum. The maximum torque on a bridge that is centrally supported is produced by a point load applied at location X leading in front of the uniform live load with multiple lane load factors applied, and a uniform load with multiple lane load factors applied up to the point X in lanes of the opposite direction. This corresponds to the maximum torque that could be produced with vehicles stationary blocked by an incident at location X. . Although the basic lane load may be similar, the two codes differ in their treatment of multiple lane loading, as illustrated in Figure 3-3. Table 3-3 shows the difference in load effect between the CHBDC and ASCE treatment of multiple lane loading and vehicle biasing in the lane. For a bridge suspended by a single plane of cables, the CHBDC live load effect is anywhere between 3% and 14% higher than that of the ASCE loading, for the same basic lane load. For a bridge with two planes of cables, the CHBDC live load effect is the same as the ASCE load effect on average, but between 8% higher and 6% lower. The total live load on the cables when a bridge is laterally supported, neglecting torsional redistribution, is up to 14% higher for CHBDC multiple lane loading, and up to 20% higher for ASCE multiple lane loading, than if the bridge is centrally supported. As a bridge deck increases in width, one might expect the live load to dead load ratio to decrease, since the live load per metre width of deck is lesser. This would in fact be the case if we assume the dead load of the cross-section to be linearly proportional to the width. From the extradosed bridges examined in
Multiple of Basic Lane Load
44 5
5
4
4
Lateral Suspension
3
3 CHBDC Lateral
Central Suspension
CHBDC Central 2
2
ASCE Lateral ASCE Central
1
1 2 lanes
3 lanes
4 lanes
5 lanes
6 lanes
7 lanes
0 6.0
10.0
13.5
17.0
20.5
24.0
CHBDC 2006
8 lanes
27.5
0 31.0
Wc, m
ASCE 1981
Figure 3-3. multiple lane loading effect by deck width according to CHBDC 2006 and ASCE 1981, for two planes of cables and for single plane central cable suspension. Table 3-3. Comparison of multiple lane load effects according to CHBDC (2006a) and ASCE (Buckland 1981) for the same basic lane load. Comparison CHBDC/ASCE Total Load on single cable plane CHBDC/ASCE Total Load on two planes of cables CHBDC - Ratio of total load of two planes to total load on single plane ASCE - Ratio of total load of two planes to total load on single plane
2 Lanes 1.06
3 Lanes 1.14
4 Lanes 1.12
5 Lanes 1.03
6 Lanes 1.00
7 Lanes 1.04
8 Lanes 1.07
1.08
1.04
1.03
0.99
0.96
0.92
0.94
1.10
1.07
1.10
1.14
1.14
1.04
1.02
1.08
1.18
1.20
1.20
1.19
1.18
1.17
Chapter 2, the wider bridges do appear to have effective thicknesses that are less than the narrower bridges (average effective thickness of decks 18 m or wider is 13% lower), however the live load to dead load ratio decreases faster. In a well designed cross-section, the dead load should be a function of the transverse system and the stiffening system, where only the transverse system is affected by and increase in deck width, as will be discussed further in Section 3.6. Two important observations can be made from this discussion of live load: 1. The multiple lane loading factors and lane biasing of the CHBDC will result in larger live load effects for than those prescribed in the ASCE, specifically for centrally suspended girders and laterally supported girders with fewer than 5 lanes. 2. The total live load effect on two planes of cables will be between 10 and 20% higher than on a single plane of cables, which implies that central suspension lead to a lower live load demand in cable supported bridges.
3.1.2
Temperature
At SLS1, the CHBDC (CSA 2006a) requires that the effects of temperature be considered in combination with live load. There are three temperature effects that cause forces in an extradosed bridge: temperature
45
gradient in the girder, temperature differential between the cables and girder, and a uniform temperature range applied to the entire structure. The effects of temperature gradient and temperature differential on the extradosed bridge become more significant as the stiffness of girder increased and must be considered and will be discussed in greater detail, while a temperature range mainly affects the piers. Temperature Gradient in Girder The CHBDC (CSA 2006a) specifies a linear temperature gradient which is a function of the section depth. This may provide a reasonable approximation of the curvature induced by the sun shining on the surface of a bridge deck for short spans, but is overly conservative for deeper cross-sections where corresponding curvature is primarily due to the strain in the deck slab and its distance to the centroid of the girder cross-section. One of the earlier rational models for temperature gradient was proposed by Priestley (1978) based on experimental and analytical research conducted in the early 1970s. A design gradient was proposed that would accurately predict the critical conditions for seven bridge sections investigated and was adopted for all major concrete bridge design in New Zealand (Priestley 1978). This design gradient is specified by a fifth-order curve with the point of zero temperature difference at 1200 mm below the deck surface. T, deg C 0.0 0
Depth, m
0.4
10.0
20.0
30.0
ilevers webs, cant voids bove deck a
0.8 New Zealand concrete deck
1.2
New Zealand 90mm asphalt
Zone 1 2 3 4
T = 32 - 0.2h ty = T (y/1200)5 t’y = 5 - 0.05h h: asphalt thickness
T1 30 25 23 21
T2 7.8 6.7 6.0 5.0
AASHTO Zone 1
1.6
AASHTO Zone 3 CHBDC
New Zealand Gradient
AASHTO LRFD Gradient
2
Figure 3-4. Comparison of Temperature Gradients (adapted from Priestley 1978, AASHTO 2004).
The current AASHTO (2004) LRFD temperature gradient is based on a model proposed in the 1985 NCHRP Report 276, which was based on work initially done by Potgieter and Gamble (1983). RobertsWollman, Breene and Carson (2002) describe the development of the current code provisions in greater detail, and compare the 1994 AASHTO temperature gradient with field measurements of two segmental concrete bridges over the course of 2.5 years. Their results indicate that the 1994 AASHTO LRFD positive gradients are conservative for an exposed concrete deck, and appropriate for a deck with 50 mm asphalt topping, while the negative gradients are slightly conservative. As well, they found the shape of the temperature gradient to be most similar to the trilinear form specified in the 1989 AASHTO Guide Specifications for Thermal Effects in Concrete Bridge Superstructures. The current 2004 AASHTO LRFD temperature gradient is very similar to the 1994 version, except that the reduction of the temperature gradient for asphalt wearing surface (to 0.8 of the untopped value) has been eliminated. Negative temperature gradients are obtained by multiplying the positive gradient by -0.3 for a concrete wearing surface, and by -0.2 for an asphalt wearing surface.
46
The AASHTO (2004) LRFD code specifies a load factor γTG for temperature gradient, to be taken as 1.0 at SLS when live load is not considered, and as 0.5 when combined with live load. Since the AASHTO temperature gradient is a rational model that has been shown to be conservative in its prediction of strains in concrete bridges, its use and use of partial load factor in combination with live load should be adopted for any bridge where the effects of temperature gradient are likely to influence the design of prestressing and reinforcement, such as for an extradosed bridge. Mondorf (2006) and O’Brien and Keogh (1999) provide good explanations of how to calculate the effects due to nonlinear temperature gradient. Priestley (1978) notes that the effects of temperature gradients should not be be considered as equivalent forces at ULS, because the force-deformation response of the girder section is no longer linear, and equivalent forces would incorrectly predict failure, wheras the only significance of thermal gradient at ULS is a reduction in ductility of the section. Temperature Differential between Stays and Girder Sun shining on a bridge will cause steel above the deck to heat up more rapidly than a concrete girder. Thus the stays will lengthen due to the temperature differential with respect to the girder, and will cause bending in the girder. The CHBDC (CSA 2006a Clause 3.9.4.1) states that Type A structures (steel superstructure above the deck) will be subject to temperatures of 25˚ above the maximum mean daily temperature, while Type C structures (concrete systems with concrete decks) will be 10˚ above. This would imply a temperature differential of 15˚ between stays and concrete girder, although there is a further reduction in the temperature of the concrete girder due to depth that would increase this differential. Eurocode 1-1-5 (CEN 2002a) specifies a temperature differential of 10˚ for light coloured stays and 20˚ for dark coloured stays, which does not take into account girder materials or depth. A temperature differential of ±15˚ was used in the design of the Pasco-Kennewick Bridge (Mondorf 2006). The effect of temperature differential indicates a preference for light coloured stays in extradosed bridges.
3.2
Design Concepts
3.2.1
Stiffness of Cables and Girder
The extradosed bridge form allows the designer to select the distribution of live load between the stay cables and the girder, by changing the stiffness ratio of these two elements. Ogawa and Kasuga (1998) compare this to the choice a designer has when designing an arch as deck-stiffened or arch-stiffened. Thus there are two different approaches that have been taken in the design of extradosed bridges. The following examples of the Pearl Harbor Memorial Bridge that has a stiff deck, and the Sunniberg Bridge that has a flexible deck, illustrate these two extremes. For the concrete design of the Pearl Harbour Memorial Bridge (51 in Table 2-1) to be built in New Haven, Connecticut (Stroh et al. 2003), the designers proportioned the girder section based on the maximum depth available given the grade and navigational clearance constraints, as well as transverse bending requirements. The girder was then dimensioned with the “maximum desirable amount of internal longitudinal posttensioning” for the section (Stroh et al. 2003). Extradosed tendons were used to reduce the bending moment demand to meet the available moment resistance of the box section. This process
47
resulted in a bridge with a span to depth ratio of 31 at the piers and 45 elsewhere. For a 5 lane bridge of 157 m main span, the cable mass required to support half of the main span was 41 tonnes, a mass of 18 kg/ m² of the deck surface. For the design of the Sunniberg Bridge (10 in Table 2-1) (Honigmann & Billington 2003), Menn has used the same approach as for the design of a cable-stayed bridge: a cable arrangement is selected, cables are sized according to maximum load for the allowable cable stress, and the girder is designed to resist the bending moment between cables under dead load, and compatibility moments under live load, caused by the distribution of axles loads to several adjacent cables. Finally, the cross section was checked for buckling at the pier under combined bending and axial compression in the deck. This process resulted in a bridge that has a span to depth ratio of 127. For a 2 lane bridge of 140 m main span, the cable mass required to support half of the 140 m main span was 43 tonnes, a mass of 49 kg/m² of the deck surface, more than double that of the Pearl Harbour Memorial Bridge. Most extradosed bridges built to date lie somewhere between these two examples, as is the case for the North Arm Bridge (48 in Table 2-1), a 562 m long, five span LRT bridge with a 180 m extradosed main span, recently completed in Vancouver (Griezic et al. 2006). A precast concrete segmental box girder cross-section was being used on other parts of the project and was a logical choice for the extradosed span. In addition, the approach spans could be cantilever constructed by varying the depth of the box girder. In the main span, extradosed tendons were added to extend the useable span of the box girder while meeting navigational and glidepath clearance requirements above and below the bridge. These constraints resulted in a bridge with a span to depth ratio of 53. For a bridge with two LRT lines, the cable mass required to support half of the main span was 22 tonnes, a mass of 23 kg/m² of the deck surface. The designers reported a maximum stress range of 73 MPa due to service live load (Griezic et al. 2006), while an analysis done for this thesis found a maximum of 61 MPa at SLS under CHBDC live load. While the North Arm Bridge is an LRT bridge, it can be compared with the other two road bridges based on the maximum live loading stress in the extradosed cables. Ogawa and Kasuga (1998) define an index β as the distribution of the live load to the stay cables, and claims that this index also represents the stiffness ratio between stay cables and girders.
β=
Load carried by stay cables Total vertical load
The boundary between extradosed bridges and cable-stayed bridges is suggested to occur at β = 0.30, corresponding to a live load stress range in the cables of around 50 MPa. In practice, the distribution index is not easily determined since all common live load models consist of both a uniform lane load and point loads representing vehicle axles. The distribution index is higher for point loads than for uniform loads, because the girder locally distributes the point load to cables surrounding the point of loading, not to all cables in the span. The vertical stiffness of a cable anchored at the deck is related to the inverse of its length and decreases as the length increases (i.e. as we move away from the pier towards the midspan). Therefore, a point load applied at midspan will distribute more evenly to adjacent cables than a point load applied closer to the pier. This is demonstrated in the comparison between the Sunniberg Bridge and the
48
North Arm Bridge in Table 3-4, each subjected to point load, uniform load, and CL-625 live load. The β ratios are higher for the Sunniberg Bridge than for the North Arm Bridge, and the β ratio for point load at midspan is higher than for a uniform load in each bridge. Table 3-4. Comparison between Sunniberg Bridge and North Arm Bridge response to live load. Sunniberg Bridge, 140 m main span North Arm Bridge, 180 m main span Axial force and bending moment due to 9 kN/m uniform load across main span
β = 0.72
Mmax = 670 kNm
β = 0.23
Mmax = 9140 kNm
Axial force and bending moment due to 625 kN point load applied at midspan
β = 1.00
Mmax = 3570 kNm
β = 0.43
Mmax = 11700 kNm
Axial force and bending moment envelopes due CL-625 live loading
σL= 198 MPa* maximum LL stress in cables
Mmax = 4500 kNm* Mmin = -7930 kNm*
σL= 61 MPa* maximum LL stress in cables
Mmax = 28000 kNm* Mmin = -39000 kNm*
* Values given are for 2 lanes loaded including multilane reduction and service load factors. Note: Model geometry and load shading for the two bridges are to the same relative scale.
3.2.2
Stiffness of Superstructure and Substructure
The girder, cables, and tower form the superstructure load resisting system. For all extradosed bridges considered in Chapter 2, the tower is fixed to the girder, but the superstructure is fixed to the substructure (piers) in only half of the bridges. In bridges with side spans of less than half of the main span, as is almost always the case for cable-stayed bridges, the tower can be stabilised by backstay cables. Backstay cables will not be discussed in this section because very few extradosed bridges rely on backstay cables, but the topic is explained thoroughly by Leonhardt and Zellner (1980), Menn (1994), Gimsing (1997) and Walther et al. (1999). When the superstructure rests on simple supports at the piers (free in rotation), as opposed to being embedded (fixed in rotation) at the piers, a live load in any one span causes bending in the girder, which
49
causes a downwards displacement in the loaded span and an upwards displacement in the adjacent span(s). To resist the bending moment in the girder and control displacements, the girder alone must have adequate bending resistance and stiffness. For these two conditions to be jointly met, a certain section depth is required to provide stiffness to the system, since the cables simply transfer the load in one span to the adjacent span(s). A tensile force in a cable due to a point load in one span is distributed through the tower to multiple cables in the adjacent span. In the case of a superstructure embedded at the piers, any rotation of the superstructure at each pier will be partially restrained by the substructure. This will decrease the bending moment in the girder due to live load, since some of the moment is resisted by the pier. The corresponding displacements are also reduced. If the girder is flexible, the substructure must provide enough stiffness to control deflections of the girder due to live load. Table 3-5. Comparison between monolithic and released connnection at main piers of the North Arm Bridge. Bending moment due to 625 kN point load on main span Monolithic
Mmin = -8290 kNm
Mmax = 11700 kNm Released
Mmin = -5500 kNm
Mmin = -4760 kNm
Mmax = 6550 kNm
Mmax = 14100 kNm
Bending moment due to 9 kN/m uniform load across main span Monolithic
Mmin = -15300 kNm
Mmax = 9140 kNm Released
Mmin = -10500 kNm
Mmin = -8120 kNm Mmax = 10700 kNm
Mmax = 13300 kNm
Bending moment envelopes due to CL-625 live loading* on main span Mmin = -39000 kNm
Monolithic
Mmax = 28000 kNm Released
Mmin = -26200 kNm
Mmin = -19800 kNm
Mmax = 37800 kNm
Axial force envelopes due CL-625 live loading* on main span Monolithic
PL= 270 kNm
PL= 390 kNm
Released
PL= 660 kNm
PL= 440 kNm
Mmax = 26400 kNm
50 Table 3-5. Comparison between monolithic and released connnection at main piers of the North Arm Bridge. Deflected shape envelope due CL-625 live loading* on main span Monolithic
d = 20 mm
d = -155 mm Released
d = 67 mm
d = -300 mm
* Values given are for 2 lanes loaded including multilane reduction and service load factors.
Table 3-5 shows the forces and displacements in the North Arm Bridge resulting from live load across the main span, for both the superstructure embedded on the piers (monolithic as it was constructed) and the superstructure simply supported (released against rotation) at the piers. The monolitic connection causes a shift in the moment diagram from positive to negative moment regions, and virtually eliminates any bending in the back spans. In the released condition, the live load in the main span is reflected in the back spans, the effect of which is pronounced since the side spans are very long in this bridge.
3.2.3
Prestressing Methodology
From the previous two sections, we observe that there are two separate factors which influence the magnitude of the bending moments in the girder due to live load. Firstly, the relative stiffness of the cables and girder which affects the distribution of forces between these two systems, and secondly the connection between the superstructure and the substructure, which affects the moment distribution between the superstructure and the piers. Long-term effects lead to changes in the magnitude and distribution of bending moments in the extradosed bridge, and warrant further discussion before explaining the prestressing methodology for an extradosed bridge. Axial shortening of the girder due to creep and shrinkage will cause a decrease in the cables’ pretensions that causes long term bending moments, as will be discussed in Section 3.4.9. For a concrete cable supported structure, it is always desirable to have no net bending moment in the girder under permanent loads (a bending moment distribution in the girder equivalent to that of a continuous beam on simple supports) to reduce creep-induced deflections and uncertainties in the deflections over the lifetime of the structure. This is especially important for cable-stayed bridge with flexible decks, where the live load moment is a much greater proportion of the total moment than permanent moments. Undulating internal tendons are installed to exactly balance the self-weight of the slab between anchorages, to eliminate any net moment under permanent loads. This is sometimes referred to as centred forces under permanent loads and is the preferred means of keeping geometrical nonlinear effects to a minimum under permanent loads (Virlogeux 1994). It is also desirable to have centred forces during construction, as creep deformations will be accelerated due to the early age of the concrete. This is only possible with cable tensions adjusted to balance the construction loads: the self-weight of the deck and the weight of the construction equipment. This creates an apparent contradiction, which is often solved by first stressing the cables to balance
51
construction loads, then restressing the cables to balance the permanent loads, after the superimposed load is applied to the continuous structure. This is efficient in terms of limiting the bending moment in the deck at all stages but it requires a laborious restressing operation. The aforementioned factors lead to the following prestressing methodology for cable-stayed bridges. The cables are dimensioned to resist all permanent loads, all uniform live load, and concentrated live loads reduced to account for some distribution into adjacent cables. The cables are first pretensioned to balance construction loads during construction in balanced cantilever, then they are retensioned to balance all permanent loads after construction of barriers and asphalt paving. Internal prestressing tendons are straight and centred to limit creep effects, geometrical nonlinear effects, and uncertainties in the magnitude of bending moments. Partial prestressing is used in the girder to limit crack widths at SLS (Hansvold 1994; Jordet & Svensson 1994; Wheeler et al. 1994), typically to 0.2 mm. Additional bending capacity at ULS is provided by reinforcing steel. Full prestressing to keep the girder uncracked at SLS, especially at midspan where there is no axial force induced in the girder from the cables, would require a prohibitively high quantity of prestressing. Sometimes, the span is required to remain fully prestressed for a ‘typical’ truck, and only partially prestressed for full SLS loads (Bergermann & Stathopoulos 1988). The challenge of designing an extradosed bridge with stiff girder lies in proportioning the girder, cables and substructure to control the stress range in the cables due to live load, in order to take advantage of a higher allowable stress for the extradosed cables. Since the girder is stiff, axial shortening of the girder due to creep and shrinkage causes long term bending moments of similar magnitude to those due to live load, which cannot be avoided. Prestressing is required in the girder to resist bending moments due to both long term effects and live load. This leads to the following prestressing methodology, suggested by Komiya (1999) and Chio Cho (2000). The cables of an extradosed bridge could be dimensioned to balance the girder self-weight with an allowance for live load, and pretensioned to balance construction loads (girder self-weight only). Superimposed dead loads are applied to the continuous structure, but are resisted mostly by the girder causing only a marginal increase in cable tensions, between 3 and 6% for the bridge designed in Section 4.3. Instead of restressing the stay cables, these bending moments can be balanced by internal bottom and/or external draped continuity tendons installed in the final structure. These tendons provide resistance for the bending moment due to live load and long term effects. Tendon layout will be discussed in Section 3.7, and bending moment diagrams describing the above load cases can be found in Section 4.3.2. For an extradosed bridge with flexible girder, the girder must be as flexible as possible to limit longterm bending moments due to creep and shrinkage. At the same time, the superstructure and substructure together must provide enough stiffness to limit deflections due to live load. The prestressing methodology is the same as for cable-stayed bridges.
52
3.3
Conceptual Design
3.3.1
Fixity of the Girder to the Piers
Fixity of the girder, both at the side span supports and on the main piers, has a significant effect on both the bending moment in the deck and on the stress range in the cables due to live load. Fixing the girder at the piers allows the bridge to resist live load as a frame, causing a shift in bending moment in the loaded span from positive to negative moment regions, where the moment is distributed into the piers. Fixing the girder decreases the total bending moment in the deck and decreases the displacements, especially in the spans adjacent to the applied load. Of the extradosed bridges in the Chapter 2 study, 26 out of 50 have girders that are embedded on the piers. Figure 3-5 shows the moment envelopes due to the CHBDC (CSA 2006a) CL-625 Live load for a three span girder bridge of constant cross-section, with a main span of 100 m and side spans of different length. The envelopes on the left side of the figure are for a case where the girder is fully restrained at the inside piers, whereas those on the right side are for a girder on simple supports at the interior piers. For the fixed condition, the moment range at the section of maximum moment in the side spans (difference between maximum and minimum moment) is 55% of the value for the simply supported condition. When the girder is embedded on the piers, the moment envelope will be somewhere between the two extremes shown in Figure 3-5. -25
Moment, MNm
-20
Girder fixed at interior supports
Girder on simple supports
-15 -10 Distance from CL Pier, m 100 50
0
-5 -50 0
0
50
100
5 10 15 20
Figure 3-5. CHBDC CL-625 Live load envelopes for a main span of 100 m.
In a girder bridge, the live load is a small portion of the total moment in the bridge, and embedding the girder on the piers does not significantly affect the design, since the decrease in total moment is relatively small (for the cantilever constructed girder bridge in Chapter 4, which is simply supported at the piers, the live load at midspan is 23% of the total moment demand at SLS). In an extradosed bridge however, the live load makes up a significant portion of the total moment in the girder (for the stiff girder extradosed bridge in Chapter 4 which has a girder that is embedded on the piers, the live load at midspan is 44% of the total moment demand at SLS which is a higher proportion of the total moment than in the girder bridge). Since the live load is shared between the cables and the girder, any decrease in live load moment is doubly beneficial since the total moment in the girder is reduced, and the stress range in the cables due to live load is decreased.
53
The height and configuration of the piers will influence the bending moment at the level of the foundations, especially due to resp. Piers that are fixed at their base deform in double bending, and thus the moment at the base will be similar to that at the girder, unless there is a variation in the pier’s crosssection. The piers can be proportioned to resist the bending moment due to live load without significant reinforcement. The lever arm of the pier can be increased more easily than the deck, through a wider pier or twin pier legs, without detriment to the aesthetics of the bridge. If footing dimensions are constrained, a simply supported deck will be preferred to eliminate bending at the foundation level due to live load on the superstructure. The proportioning of the girder and piers are interrelated and cannot be treated independently. The decision of whether to fix the girder to the piers in rotation or not should be made early on in the design process as this significantly affects the forces in the bridge under live load. Both scenarios present no difficulties in construction, and it appears preferable keep the girder embedded on the piers.
3.3.2
Side Span Length
When the girder is stiff, side spans should be proportioned similar to ordinary girder bridges (Kasuga 2006), generally between 0.6 and 0.8 of the main span, to balance the maximum moments in the side spans and main span. Chio Cho (2000) found that side spans of less than half of the main span decrease the bending moment in the main span, but recommends the use of side spans longer than 0.60 of the main span to produce a positive bending moment in the side span due to live load that is similar in magnitude to that of the main span.
3.3.3
Tower Height and Girder Depth
Mathivat (1988) suggests using a constant depth girder with a span to depth ratio of 30 to 35 and a tower with a span to height ratio of 15. Based on Mathivat’s semi-fan cable arrangement, with the cables anchored at each segment and resting on saddles at the deviators, the ratio of span to lever arm is around 15. These span to depth ratios correspond to those typically found in cantilever constructed girder bridges in France, most of which use external continuity tendons and are simply-supported by bearings on the piers, and have a span to depth ratio between 30 and 35 at midspan. Upper and lower bounds for the proportions of an extradosed bridge can be established by looking at current practice for cantilevered and cable-stayed bridges. Current practice in France, as recommended by SETRA (2007), is to use a span to depth ratio of 16 to 18 at the supports and 30 to 35 at midspan, with a minimum cross-section depth of 2.2 m for movement through the box-girder. SETRA provides formulas for suggested span l to depth h ratios, based on cantilevered bridges constructed in France: Over Pier:
l l ----- = 14 + -----hp 45
At Midspan:
l l ------ = 19 + --hm 7
The fib Guidance for good bridge design (2000) claims the that most economical span to depth ratio for a cantilever constructed girder bridge is approximately 15, but that an increase from 15 to 20 will not affect the cost signifcantly. The fib guide recommends span to depth ratios at midspan of 35 to 40 for
54
continuous spans simply-supported on the piers, and 40 to 45 for continuous spans embedded (fixed in rotation) at the piers. As a point of comparison, for cantilever bridges with internal tendons, Menn (1990) suggests a span to depth ratio of 50 at midspan and 17 at the piers, based on aesthetic and economic considerations. The fib (2000) guide also suggests that the pier depth to midspan depth ratio of 3 is aesthetically pleasing for bridges that are low to the ground but should be closer to 2 for tall structures. There is merit to this recommendation; tall bridges with large variation in girder depth can look weak at midspan and out of proportion with their wide piers, as shown in Figure 3-6. However, this awkwardness can be diffused with twin piers columns. Cantilever Bridge Span to Depth: 17:1at piers 50:1 at midspan
Satisfactory appearance
Good appearance
Good
Satisfactory
Good
Good
Towers too tall
Good
Cantilever Bridge Span to Depth: 17.5:1at piers 35:1 at midspan Extradosed Bridge 10:1 Span to Tower Height 50:1 Span to Depth
Cable-Stayed Bridge 4.5:1 Span to Tower Height 100:1 Span to Depth
Figure 3-6. Comparison of span to depth ratio and effect of the roadway height above ground on the overall proportions of 3 span cantilever, extradosed, and cable-stayed bridges.
In an article on the conceptual design of cable-stayed bridges, Menn (1996) points out that the portion of towers above the deck, which form the lever arm between cables and deck, cannot be reduced to the same depth as that required for a cantilever box girder, because the cables have a much lower axial stiffness than the prestressed deck slab of a box girder which is the tension chord in that system. In the case of a classical cable-stayed bridge where the girder is slender, the deflections under live load would be large. Menn claims that a main span to tower height ratio of 7 is possible, “provided the towers are stiff and the girder is restrained longitudinally”, which is a reasonable claim given that the Sunniberg Bridge, with a main span to tower height of 10, was under construction at that time. A few researchers have studied extradosed bridges and have made recommendations for selecting the tower height and proportioning the deck cross-section. Komiya (1999) suggests a span to tower height ratio of 8 to 12. The taller tower results in not more than 10% savings to the combined cost of the cables and tower. The span to depth ratio of the girder should be 35 at the piers and 55 at midspan, for girders embedded at the piers. Komiya notes that reducing the girder stiffness to 0.50 and 0.25 of the gross stiffness results in increases to cable forces of 3 and 8% on average, while the deflections due to live load increase by a factor of 1.5 and 2.3.
55
Chio Cho (2000) recommends that towers not exceed a span to tower height ratio of 10, in order to limit the stress range due to live load in the extradosed cables to 80 MPa. Chio Cho claims that the purpose of a variable depth cross-section is to reduce the cost of the bridge by reducing the girder’s self-weight, without reducing the height at the supports. Haunching the deck at the piers reduces the quantity of both internal and extradosed prestressing. Chio Cho suggests a span to depth ratio of 30 at the piers and 45 at midspan for girders simply supported at the piers, with the transition occurring over a distance of 0.18 of the span from the pier, as shown in Figure 3-7b. Increasing the girder depth at the pier reduces the stress range in the extradosed cables, and increases the bending moment at the supports with a small reduction in moment at midspan. A larger girder depth at pier to depth at midspan ratio, as shown in Figure 3-7c,allows the first set of extradosed cables to be anchored farther away from the pier. a)
c)
b)
Constant Depth
Variable Depth hpier : hmid = 1.5
Variable Depth hpier : hmid = 2.0
Figure 3-7. Extradosed bridge geometry studied by Chio Cho (2000).
The aforementioned recommended proportions for span to tower height and span to girder depth ratios are drawn in Figure 3-8 for the Chapter 4 bridge crossing, with a main span of 140 m. When compared to the cantilever constructed girder bridges, the girder of the extradosed bridge dimensioned with Mathivat’s proportions appears heavy at midspan, but the visual effect of the cables is subtle and unobtrusive. The haunching at the piers recommended by Chio Cho and Komiya, combined with the fan cable configuration that anchors the first cables beyond the haunch, draws the eye to the piers and emphasizes the verticalilty of the bridge at this location, giving it a sense of strength. The girder in the extradosed bridge dimensioned with Komiya’s proportions is noticeably more slender than the other two, which shifts the focus onto the cable system. Cantilever Constructed Girder Bridge SETRA/fib recommended Span to depth ratios: 17:1 at piers, 40:1 midspan Cantilever Constructed Girder Bridge Menn recommended Span to depth ratios: 17:1 at piers, 50:1 midspan Extradosed Bridge - Mathivat recommended Span to tower height: 15:1 Span to depth ratio: 35:1 Extradosed Bridge - Chio Cho recommended Span to tower height: 10:1 Span to depth ratios: 30:1 at piers, 45:1 midspan Extradosed Bridge - Komiya recommended Span to tower height: 10:1 Span to depth ratios: 35:1 at piers, 55:1 midspan
Figure 3-8. Girder and extradosed bridge proportions recommended by others.
The span to depth ratios proposed by Mathivat (1988), Komiya (1999) and Chio Cho (2000) can be used as a starting point for proportioning an extradosed bridge but should not be regarded as hard and fast
56
rules, since they are based primarily on limits imposed on the maximum live load stress range in the cables of the bridges studied by those individuals. The variety in proportions of the bridges in Figure 2-1 should be drawn upon for inspiration and to assess the feasibility in developing concepts for an extradosed bridge. It seems necessary and rational to develop a computer model at an early stage in the design to account for the interaction between tower, girder, substructure and cables to determine whether a given system stiffness is adequate to limit the live load stress range in the cables to the desired level. This was the approach taken in Chapter 4 which resulted in the bridges shown in Figure 3-9. Cantilever Constructed Girder Bridge - Chapter 4 Span to depth ratios: 17.5:1 at piers, 44:1 midspan
Extradosed Bridge with Stiff Girder- Chapter 4 Span to tower height: 10:1 Span to depth ratio: 50:1 Extradosed Bridge with Stiff Tower - Chapter 4 Span to tower height: 10:1 Span to depth ratio: 140:1
Figure 3-9. Bridge proportions used for design in Chapter 4.
In the Chapter 4 designs, a constant depth girder and harp cable configuration were chosen for constructability and appearance. The constant depth girder provides the greatest continuity across the entire bridge, while the parallel cables and simple tower shapes give the bridge a uniform texture. Repetition and consistency of local components, such as ribs and anchorages, give the bridge an orderly appearance from all vantage points.
3.4
Stay Cables and Anchorages
3.4.1
Cable Arrangement
In a cable supported bridge, the vertical component of force in the cables lifts the continuous girder, while the horizontal component prestresses the girder. The cable configuration and the height of the tower are the two factors that influence the cable inclination, and hence the action of the cable on the deck. These two factors will be discussed in relation to extradosed bridges. The effect of cable inclination on the force components in a stay cable is examined. For a constant cable force, the vertical component of force increases almost linearly with an increase in inclination from 0˚ to 30˚, but the horizontal force decreases only 13%. If the stay cables are designed to resist 100% of the dead load at each anchorage, as if the girder were a continuous beam on simple supports at the cables, the force in each cable for an extradosed bridge will be two to three times larger than is typical for a cablestayed bridge, as can be seen in Figure 3-10b. Additionally, the maximum compression force in the girder of the extradosed bridge (at the piers) will be equivalent to that in a cable-stayed bridge two to three times as long. The fan cable configuration consists of cables anchored at a single point at the top of the tower, while the harp cable configuration consists of parallel cables anchored over the full height of the tower. In practice, it is difficult to achieve a pure fan configuration because all cables cannot be anchored at one
57 a)
0.07
1
Tower Height h / Span Length l 0.13 0.20 0.25
b)
12
Total Horizontal
Tower Height h / Span Length l 0.13 0.20 0.25
10
0.8 Extradosed Typical
0.07
Cable-Stayed Typical
Extradosed Typical
8
Cable-Stayed Typical
Force
0.6 Force
Vertical
6
0.4 4 0.2
Total
2 Horizontal
Vertical 0
0 5
9
16
25 30 Angle, deg
45
5
9
16
25 30 Angle, deg
45
Figure 3-10. Effect of cable inclination on the force components in a cable for a) a constant total force and b) a constant vertical force.
point, and thus the cables are often anchored at a constant offset down the tower in a semi-fan configuration. The semi-fan cable configuration is more effective than a harp configuration in providing a vertical component of resistance to the deck, but each cable anchorage at the deck level will be at a slightly different angle and must be detailed separately. With a harp configuration, more cable steel is required, but the cable anchorages have a common design which is advantageous, since the formwork and reinforcement are consistent for all anchorage segments. For shorter cables, such as found in extradosed bridges, the anchorages represent a greater portion of the total cost of the cable system, and they should contain the maximum number of strand for which they are designed. The additional horizontal force from the harp cable configuration prestresses the girder near the piers and thus substitutes internal prestressing. The cost of additional cable steel in the harp configuration is partially offset by the savings from common detailing of the anchorages. a)
b)
Figure 3-11. Quantity of cable steel as a function of relative height of towers - Comparison between fan and harp cable configurations in a) 1970 (Leonhardt & Zellner 1970) and b) 1980 (Leonhardt & Zellner 1980).
58
Leonhardt and Zellner (1970) published a chart comparing the stay cable steel consumption of harp and fan (radiating) cable configurations. From this chart, shown in Figure 3-11a, it can be seen that the optimum ratio of tower height to main span is around 0.30 considering stay cable steel consumption alone. In 1970, the optimum height considering the cost of a tower was suggested to be between 0.16l and 0.22l where l is the main span length, while in a similar chart published in 1980 (Leonhardt & Zellner 1980), shown in Figure 3-11b, the optimum height was 0.20l to 0.25l. This difference is attributed to the changes in material preference and relative material costs. In the 1970 article, the tower is assumed to be steel, whereas by 1980, concrete had become the economical material of choice. For the Brotonne Bridge, completed in 1977, the cost of stay cable system was 29% of the total cost of the bridge, while the cost of the tower was only 4% (Mathivat 1983). A simple model was used to investigate the effect of tower height on steel consumption in stay cables for a 140 m main span, the results of which are shown in Figure 3-12. There are 10 cables in a half-span spaced at 6 m, with the first cable 13 m from the pier, and each cable is assumed to resist an equal vertical load of 0.05 units, for a total of 1 unit of load per span. For the semi-fan configuration, the cable anchorages are spaced 0.3 m vertically on the tower. The graph bears a close resemblance to those of Leonhardt and Zellner in Figure 3-11. It is apparent that the harp configuration leads to a larger total cable force, especially for a tower height below 0.1L.
Total Cable Force Per Span (multiple of vertical load per span)
10
8
6 Semi-Fan Cable Configuration 4
Semi-Fan
2 Extradosed Cable-Stayed Typical Typical
Harp
Harp Cable Configuration
0
0
0.1 0.2 0.3 0.4 Tower Height H / Span Length L
Figure 3-12. Quantity of cable steel as a function of relative height of towers - comparison between semi-fan and harp cable configurations for 140 m main span.
In practice, each stay cable will not be detailed with an optimised number of strands as required to load every cable to its allowable stress. According to Mathivat (1983), the theoretical values should be increased by 10% because the demand of the cables must be met with real stay cable cross-sections. This is especially true for the fan configuration, where the theoretical cross-section of the cable decreases towards the pier as the cable inclination increases. The furthest cable from the pier will be a comparable size for both harp and fan cable configurations, establishing a maximum anchorage size. In a cable-stayed bridge, the harp cable configuration presents two main disadvantages: a higher compressive force in the deck (around 60% higher than the semi-fan configuration) and increased bending in the towers. In an extradosed bridge these two factors are not problematic: the higher compressive stress in the girder
59
increases the moment resistance of the girder, and the short towers can be easily proportioned to resist the increased bending. Based on estimates from the aforementioned model, a harp cable configuration with a tower height of L/8 requires a similar cable quantity to a semi-fan configuration with a tower height of L/12. For tower heights of L/8 and L/12, the theoretical cable steel consumption will be 41 and 46 % more for the harp configuration than the semi-fan while the maximum compression force in the deck will be 58 and 54% higher. Since the extradosed bridge has two load carrying systems, it is possible to provide cable support to only a portion of the span. Figure 3-13 illustrates the effectiveness of providing partial cable support to the deck, by plotting the ratio of fixed end moments of a partially loaded span to a fully loaded span (Tang 2007). For a section with constant weight and stiffness, it is most efficient to provide the cable support closest to the midspan, as indicated by the upper line in Figure 3-13. Many of the extradosed bridges studied in Chapter 2 have cables distributed across 60% of the span. It can be determined from the upper line Figure 3-13 that cables across 60% of the span will offset 80% of the moment of cables supporting 100% of the span. 1.0
Fixed End Moment Ratio, M partial/M full
Mfull 0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4 0.6 Loaded length, b/L
0.8
1.0
Figure 3-13. Influence of partial cable support (adapted from Tang 2007).
The idea of reducing the length of deck supported by the cables is not new. Mathivat (1983) suggests offsetting the first cables from the tower by 0.10 of the main span, as was done on the Brotonne Bridge, to achieve a savings in cable steel of up to 20%. Chio Cho (2000) suggests that an offset of 0.18 of the main span is optimal in the case of an extradosed bridge with variable depth cross-section since beyond this length there is no significant savings in extradosed cable material quantity. Komiya (1999) considered first cable offsets of 0.14, 0.20, and 0.24 of the main span, and found the combined cost of extradosed and internal tendons for all three arrangements to be within 2% of each other. An offset of 0.20 of the main span was most economical, with extradosed tendons accounting for 60% of the cost, and internal tendons for 40%.
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3.4.2
Stay Cable Protection
There are many systems for protecting the prestressing steel of the stay cables. Most codes and specifications now require that the corrosion protection system consist of a minimum of two nested barriers (PTI 2001, SETRA 2001; FIB 2005). The external barrier provides direct protection from corrosive elements, while the interior barrier provides a backup system if the external barrier is breached. In certain cases, the external sheath is not considered to be the external barrier. Several sources (Walther et al. 1999, SETRA 2001; FIB 2005) provide a comprehensive discussion of protection systems and the advantages and susceptibilities of each. Grouting with cementitious grout is no longer considered to be a good method for filling the void between the external sheath and prestressing strand in stay cables. The CHBDC (2006a Clause 10.6.4.3) specifies that the strands of stay cables must be hot-dip galvanized and encased in a sheath filled with grease or wax. The sheath can be high density polyethylene (HDPE) or stainless steel. Within the last 10 years, the international trend has beeen to provide further redundancy in protection by greasing and sheathing the strands individually, and then bundling them in an external sheath. The external sheath serves as a barrier to rain and UV rays, while the cable protection occurs at the strand level. This also facilitates replacement of individual strands. Strands can be coated with epoxy or polyethylene after galvanizing for an additional layer of protection. Epoxy coated strands have much higher friction coefficients than bare strand that must be accounted for in the design (Taniyama & Mikami 1994).
3.4.3
Anchorages in Girder
Many sources (Gimsing 1997; Walther et al. 1999; SETRA 2001; PTI 2006), present comprehensive explanations of cable types and anchorage details. Despite the advantages that are attainable with prefabricated cables, those of superior quality control and a higher fatigue resistance, the current trend is towards strand-based cables that can be assembled on site and tensioned strand-by-strand with compact jacking equipment. In the case of stay cable anchorages, restressing is accomplished by stressing the entire cable unit with a gradient jack and adjusting a ring nut at the bearing plate. In extradosed bridges where the live load stress range is limited to 80 MPa, conventional prestressing anchorages can be used to anchor the cables instead of stay cable anchorages that are typically designed for a stress range of 200 MPa to 250 MPa. The cable’s anchorage in the girder is almost always the live end anchorage. Dywidag-Systems International markets an Extradosed Anchorage, Type XD (-E for epoxy coated strands), shown in Figure 3-14, that is purposefully designed to combine the anchor head of an external tendon with the protection details found on their DYNA-Grip and DYNA-Bond Stay Cables (Dywidag
61
2006). For the three aforementioned cable types, an elastomeric bearing is contained within the recess pipe to prevent bending stresses at the anchor head. Cap
Wedges
Filling Material
Sealing
Shim
Spacer
Anchor Body
Wedge Plate
Recess Pipe
HDPE Liner
Epoxy Coated Strands
Clamp
Exit Pipe
Filler Material Connection Pipe
Elastomeric Bearing
HDPE-Sheathing
Figure 3-14. DSI Extradosed Anchorage Type XD-E (Dywidag 2006).
Some bridges in Chapter 2 (Odawara Bridge, Tsukuhara Bridge, Domovinski Bridge) were constructed with stay cable anchorages even though the stress range due to live load in the cables would have allowed the use of conventional prestressing anchorages. For these bridges, extradosed anchorages may not have been available, but they should be used in the future and should reduce the cost of the cable system.
3.4.4
Cable Anchorage in Towers: Saddles or Anchorages?
A cable can either pass through the tower over a saddle or terminate at an anchorage. If the cable is anchored in the tower, the horizontal force must be transferred through the tower to the opposite cable, either by overlapping the cables, or installing post-tensioning or steel plates across the tower. Saddles provide a simple alternative to anchorages, but must be designed to resist differential forces from opposing sides of the tower, either through friction, bond or mechanical means. Cables passing over saddles must be replaceable and corrosion protection must be ensured through the saddle. The cable strands are subject to flexural stresses and fretting fatigue. The Japan Prestressed Concrete Engineering Association, the PTI, the SETRA, and fib have published recommendations on the use of saddles that are compared below, along with a discussion of the advantages and disadvantages of using saddles instead of anchorages in a tower. Cables that are passed through saddles must be dimensioned to account for the bending stress due to curvature fC determined as follows: r f C = E --R where E is the elastic modulus of the wire or strand; r is the radius of the wire, strand or bundle, and R is the radius of the saddle bend. Therefore, the total stress in the cable is: Er P f = f C + f A = ------ + --R A For a saddle radius of 5 m and a seven-wire strand of 15.7 mm diameter (area of 150 mm²), commonly used for strand-based stay cables, the maximum bending stress due to curvature is 314 MPa. For an individual wire, the maximum stress is around 100 MPa. The stress due to axial force in a stay cable is typically around 0.35 fpu under dead load only, or 650 MPa for strand. Therefore, the curvature of the
62
strand over a saddle with radius of 5 m increases the stress by 48% to 0.52 fpu. The stress due to curvature in a prefabricated cable would be impractically large when passed over a saddle, but strand-based cables are normally assembled on site. Fretting fatigue results from differential strains in strand bundles. Variable axial loads at the cable ends (due to a fatigue load) cause the strands to slip when the friction between the individual wires and the saddle under radial pressure is exceeded. The saddle radius must be large enough to avoid the onset of fretting fatigue. Minimum radii R for common cable sizes (of bundled bare strand) are summarised in Table 3-6, determined from equations given in the PTI1 (2001) and fib (2005) recommendations: – 0.4
n R > F ----------R ≥ 30D fib: 90 where F is the mean cable force under fatigue conditions (kN); n is the number of strands in the bundle, and D is the diameter of the stay pipe. PTI:
Table 3-6. Minimum saddle radii for strand based cables to prevent fretting fatigue. Number of Strand in Cable Typical diameter of external sheath (mm) Minimum R* suggested in PTI (m), calculated assuming F = 0.35 fpu x Ap Minimum R required by fib (m)
12 110 2.6
19 140 3.4
31 160 4.6
37 180 5.1
55 200 6.5
73 250 7.7
91 280 8.8
109 315 9.8
127 315 10.7
3.3
4.2
4.8
5.4
6
7.5
8.4
9.5
9.5
156 12.1
Individual sheathing or epoxy coating of strand could be used to reduce interstrand fretting, but any reduction must be proven by prototype testing of full scale specimens (PTI 2001). Figg Engineering Group (Figg et al. 2005) designed a cradle system in which each individual strand passes through a seperate saddle pipe which is contained within an outer centering pipe. Since each strand is deviated over its own saddle, fretting between strands is eliminated, and the minimum saddle radius is that of an individual strand, even for large cables made up of 156 strands. In this system, it is possible to remove a single strand for inspection or replacement. The SETRA Recommendations (SETRA 2001) discourage the use of saddles because of the difficulties with future replacement of the cables. If saddles are used, the radius of curvature must be larger than 125 times the exterior diametre of the strand (2.0 m radius for 15.7 mm diametre strand), and the ultimate strength of the deviated cable, as determined by test, must be reduced by no less than 5% unless detailed calculations show that a greater reduction has been allowed for in the design (§14.7). The fib recommendations (fib 2005) require the minimum saddle radius to be larger than 400 times the diametre of the individual wire of strand (2.1 m radius for 15.7 mm diametre strand) when individual strands are inside individual tubes. The PTI Recommendations (PTI 2001) provide specific requirements for saddles. Cable saddles must have a minimum radius of 3 m if supporting individual strands, and 4 m if supporting bundled multi-strand cable unless fatigue testing is done on the saddle with axially stressed cables. Saddles must be designed to 1.
Tests were conducted at the University of Texas on bare strands in ducts of 2.75 times the area of the cables: “Fretting Fatigue in Post Tensioned Concrete,” Center for Transportation Research, University of Texas at Austin, TX, March 3 to April 3, 1998.
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preclude slip and fretting for 1.25 times the maximum load differential at the strength limit state (AASHTO equivalent of ULS). The Japan Prestressed Concrete Engineering Association’s Specifications1 allow saddles to be used when the live load stress range is less than 50 MPa, based on testing of fretting fatigue done on 37-15 mm diametre strand tendons (Kasuga 2006). The minimum radius required for the saddles is the same as for external tendons. Goñi (1999) advocates the use of cable saddles to reduce the number of anchors and stressing operations, to facilitate installation, and to avoid the horizontal force transfer through the pylon. Montens (1998) points out that saddles are half the cost of anchorages, and saddles become more economical as the span decreases because the anchorages represent a higher proportion of the total cost of the stay cables. According to the Figg brochure (Figg 2004) for the stay cradle system, its unit cost is lower than the two anchorages it replaces. The system was first used for the Maumee River Bridge (Bonzon 2008), a two span centrally supported cable-stayed bridge with spans of 186.7 m, a tower faced with glass, and cables consisting of 82 to 156 strands. Tower width (length in the longitudinal direction of the bridge) is generally used as an argument against saddles. According to Montens (1998), the minimum radius of saddles presents a drawback as this dictates a minimum tower width. Figg (2004) claims that without the cradle system to pass strands over saddles individually, the tower width of the Maumee River Bridge would have had to be 3 m wider. As constructed, the tower is 4.1 m wide and varies in length from 8.8 to 6.3 m (Bonzon 2008). For most extradosed bridges however, the tower width is unlikely to be governed by the saddle radius since the total angle break in the cables is small compared to that in a cable-stayed bridge. Most towers that are detailed with anchorages require internal chambers for stressing and inspection, but towers of only 3 m width and 2 m depth have been designed with internal stressing chambers (Reis et al. 1999) for main spans of less than 100 m. If the anchorages cross through the tower and anchor on opposite sides, as is the case with the Chandoline Bridge (Menn 1991) and many cable-stayed bridges with spans less than 100 m (Walther et al. 1999), the tower dimensions can be kept to a minimum, but this requires twin cables on one side to avoid torsional forces in the tower. The minimal dimensions of saddles allow for a cable configuration closest to a perfect fan, as is the case in many of the extradosed bridges constructed in Japan, where the vertical spacing of saddles is as small as 300 mm (Ogawa et al. 1998). The use of anchorages in the towers requires a larger vertical spacing, although the Shin-Meisei Bridge has anchorages spaced at 600 mm intervals (Kasuga 2006). Based on experience to date with towers for cable-stayed bridges, it may seem clear that saddles would lead to a more economical cable system than anchorages, but anchorages are more commonly used. If the live load stress range permits, conventional anchorages could be used to anchor the cables instead of stay cable anchorages, but extradosed bridges constructed to date have mostly used stay cable anchorages. For the five extradosed bridges (the Odawara, Tshukuhara, Ibi River, Shin-Meisei, and Himi bridges) designed 1.
Japan Prestressed Concrete Engineering Association (November 2000). Specifications for Design and Construction of Prestressed Concrete Cable-Stayed Bridges and Extradosed Bridges (in Japanese). Kasuga (2006) explains the parts of the code relevant to the design of extradosed bridges.
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by Kasuga, the first two were detailed with saddles, while the latter three with anchorages in the towers, even though four of the five bridges have a stress range in the extradosed cables of less less than 50 MPa. Kasuga (2006) claims that the steel boxes for the anchorages allow for inspection of the stay cables from inside the tower the during maintenance, but gives no other reason for this design choice. A detailed cost comparison between conventional anchorages and saddles in the towers, carried out for the North Arm Bridge in Vancouver (Griezic et al. 2006), found anchorages to be more cost-effective. According to Griezic et al., cable anchorages simplify cable installation and provide greater redundancy than saddles against cable loss, since the cable loss does not propagate to the other side of the tower. However, anchorages in the tower lead to larger bending moments in the event of cable rupture and during cable replacement. Cable saddles eliminate the need for internal inspection access and therefore reduce the tower dimensions, but require more strand to keep the total stress (the sum of axial and flexural stresses in the cables over the saddles) below the allowable stress. As constructed, each cable of the North Arm Bridge has 58 strands, but two additional strands would have been required had saddles been used. In summary, the use of saddles may lead to cost savings but presents some challenges in design. It is not apparent why saddles are not preferred in all circumstances. One possible explanation lies in the prototype testing required by the aforementioned recommendations. Since it is the design engineer’s responsibility to detail the saddle, as it is “an integral element of the pylon conceptual design and the geometric layout of the towerhead” (PTI 2001), engineers who are unfamiliar with the technology may shy away from the system.
3.4.5
Equivalent Modulus of Elasticity for Stay Cables
Cable sag effects are one of the sources of geometric nonlinearity in cable-stayed bridges. Cable sag is usually accounted for by considering a straight chord member with an equivalent modulus of elasticity. An equivalent modulus, first suggested by Ernst1 in 1965 and described in Figure 3-15, is based on the assumption that the catenary of a cable with low sag/chord length can be modeled as a parabola (Walther et al. 1999). For extradosed bridges, the horizontal projected lengths of the cables are generally short, and the dead load stress in the cables typically account for 70% or more of the total stress at SLS. For a cable made up of strand, this corresponds to a permanent stress of at least 550 MPa (0.30 fpu), for which there is very little sag. For a cable length of 70 metres and a permanent stress of 500 MPa, as would be longest cable in an extradosed bridge with a 150 m main span, the equivalent modulus of elasticity of the cable is only 0.5% lower than the effective modulus of elasticity. Thus in the modelling of an extradosed bridge, the cables can adequately and safely be modelled as linear elements without considering cable sag effects.
1.
Ernst, J.H. (1965). “Der E-Modul von Seilen unter Berücksichtigung des Durchhanges.” Der Bauingenieur, 40(2), 52-55. Ernst presented the equivalent modulus plotted as a function of stress with seperate curves for different cable lengths. Leonhardt & Zellner (1970) plotted the equivalent modulus as a function of cable length as in Figure 3-15, which is more useful since all cables in the bridge generally have a similar loaded tensile stress.
65 1.0
E E eq = ---------------------------2 ( L0 γ ) E 1 + --------------3 12f
f = 600 f = 500 f = 400
Eeq / E
0.9 f = 300
0.8
0.7
f = 200
0.6 0
50
100
150
200
Horizontal Projected Length, m
where Eeq is the equivalent elastic modulus of inclined cables; E is the cable effective elastic modulus; L0 is horizontal projected length of the cable; γ is the weight per unit volume of cable (87 kN/m³ for strand inside HDPE sheathing injected with wax), and f is the cable tensile stress (MPa).
Figure 3-15. Ratio of equivalent to initial modulus of elasticity showing the influence of a cable’s sag on its stiffness (plot adapted from Leonhardt & Zellner 1970).
3.4.6
Preliminary Design of Stay Cables at Serviceability Limit States
The stress range in a stay cable due to live load is an important consideration for the design of the cables against failure due to fatigue. This can either be addressed explicitly by considering a fatigue limit state (FLS), or implicity by designing based on an allowable stress in the cable at SLS. The first method is consistent with a limit states approach to design. At the fatigue limit state, the stress range due to the fatigue load must be less than the constant amplitude fatigue threshold, in order to ensure a 75 year design life of the cables, with some safety factors to account for loading uncertainties. While this is perhaps the most thorough treatment of cable fatigue, it is difficult to apply in practice because a representative spatial model of the entire bridge must be available in order to determine the maximum stress range in each individual cable due to live load at FLS. For this reason, the second method is preferable for preliminary design of the stays, since the cross-section of both the cables and the girder will change as the design is refined. Both the SETRA Recommendations (SETRA 2001), and Japan Prestressed Concrete Engineering Association’s Specifications1, present rational approaches to designing stays based on SLS loads, that consist of limiting the cable’s allowable axial stress at SLS based on the maximum axial stress range (the difference between the highest and lowest stress) in the cable due to SLS live load, as shown in Figure 3-16. The axial stress in each cable at SLS can be determined from a plane frame model. The SETRA Recommendations (SETRA 2001) limit the allowable stress of a stay cable fa to between 0.46 and 0.60 of the guaranteed ultimate tensile strength fpu, for a maximum axial stress range due to live load at SLS ΔσL between 140 MPa and 50 MPa: Δσ L – 0.25 × f pu ≤ 0.6f pu f a ≤ 0.46 ⎛ ----------⎞ ⎝ 140 ⎠ The Recommendations require one value of the allowable stress to be set for the entire cable system, based on the maximum stress range due to live load in any cable in the structure. For a girder that is simply supported on the piers, the highest stress range under SLS live load will generally occur in the backstay cable, whereas for a girder fixed at the piers, it will occur in a main span cable. These SLS requirements 1.
See Footnote 1, page 63.
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were defined to cover, but not to substitute, ULS and FLS verification (Lecinq 2001) and they offer the advantage of working with a realistic maximum stress to be expected in the cable at SLS. The Japan Prestressed Concrete Engineering Association’s Specifications transition the allowable stress from 0.40 to 0.60 fpu for a stress range due to live load at SLS ΔσL between 100 MPa and 70 MPa for a strand system: ⎧ 0.6f pu ⎪ f a = ⎨ ( 1.067 – 0.00667Δσ L )f pu ⎪ ⎩ 0.4f pu
Δσ L ≤ 70MPa 70MPa ≤ Δσ L ≤ 100MPa Δσ L ≥ 100MPa
and between 130 MPa to 100 MPa for a prefabricated wire system: ⎧ 0.6f pu ⎪ f a = ⎨ ( 1.267 – 0.00667Δσ L )f pu ⎪ ⎩ 0.4f pu
Δσ L ≤ 100MPa 100MPa ≤ Δσ L ≤ 130MPa Δσ L ≥ 130MPa
These values have been established to ensure bridges previously designed for FLS (Kasuga 2006) would have been conservatively designed with the SLS requirements. The Specifications permit an allowable stress to be determined for each stay individually, related to the stress caused by live load in that particular stay. 0.65 0.60
F SLS /F pu
0.55 0.50 SETRA
0.45
Japan Strand
Japan Wire
0.40 0.35
40
60
80 100 120 140 Live Load Stress Range, MPa
160
Figure 3-16. Allowable stress in cable stays as a function of the stress range due to live load at SLS
In cable-stayed bridges, the allowable stress at SLS has traditionally been limited to 0.45 fpu based on a safety factor of 2.2 against rupture (Gimsing 1997), which takes into account secondary flexural stresses that are not considered in the analysis (Menn 1990). There does however seem to be a practical limitation to the allowable stress in the stay imposed by relaxation. Relaxation in low-relaxation strand is negligeable when the permanent stress in the cable is less than 50% fpu, but accelerates beyond this threshold (Gimsing 1997). With respect to cable-stayed bridges, relaxation results in a decrease in the initial prestrain of the cables which causes the girder to deflect downwards and carry more dead load, both undesirable actions. Walther (1994) suggests that an allowable stress limit of 0.55 fpy (0.50 fpu) is justifiable given the following reasons: many of the secondary stresses can now be accounted for by advanced analysis, cables have failed due to corrosion but never to fatigue alone, and modern stay cable systems have multiple layers of corrosion protection with anchorages that are designed to limit flexural stresses in the strands. This view has been reflected in specifications by SETRA (2001) and fib (2005),
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which allow the use of an allowable stress of 0.50 fpu at SLS provided the cables are tested dynamically with flexural stresses (introduced through shims at the anchorages - cables must resist 0.95 fpu after 2 million cycles of 200 MPa dynamic load at an upper stress of 0.45 fpu). SETRA allows the stress in an individual strand to be as high as 0.60 fpu, as sometimes required for strand by strand stressing, provided it falls below 0.55 fpu within a few hours (Lecinq et al. 1999). According to the CEB-FIP (1993) Model Code, relaxation in low relaxation prestressing strand at 1000 hours is 1% when initially stressed to 0.60 fpu or less, 2% at 0.70 fpu, and approximately 5% at 0.80 fpu. Given that the relaxation at 50 years is taken as 3 times the initial value, an allowable stress of 0.60 fpu for an extradosed cable is a reasonable upper limit.
3.4.7
Verification of Stay Cables at the Fatigue Limit State
The PTI Recommendations (PTI 2001) requires that the cable forces comply with a Strength Limit State and a Fatigue Limit State. For a theoretically infinite fatigue life, the maximum stress range in each cable must be less than the constant amplitude fatigue threshold stress. The fatigue load consists of a single design truck, in a single lane, and the load effect is then increased by a Dynamic Load Allowance of 15% and by a factor of 1.4 to account for longer spans of cable-stayed bridges than those for which the fatigue provisions of the AASHTO LRFD (AASHTO 2004) were originally designed. The factored load effect γ(ΔF) must be less than half the constant amplitude fatigue threshold (ΔF)TH as described by the following equations. γ ( ΔF ) ≤ ( ΔF ) η 1 γ ( ΔF ) ≤ --- ( ΔF ) TH 2 where (ΔF) is the stress range due to the passage of the fatigue load (ΔF)TH is the constant amplitude fatigue threshold (taken as 110 MPa for parallel strands) γ is the load factor of 0.75 1 ∴0.75 ( ΔF ) ≤ --- ( 110 ) 2 ( ΔF ) ≤ 73 MPa The CHBDC (CSA 2006a) live load at FLS consists of one CL-625 Truck, positioned in the centre of one travelled lane, and amplified by a dynamic load allowance of 1.25. The CHBDC does not specify a constant amplitude fatigue threshold stress, but states that the fatigue stress range for cable stays that are not readily inspectable or replaceable shall not exceed 0.75 of the fatigue stress range established by test, which is consistent with the PTI Recommendations. For both the AASHTO LRFD and the CHBDC, the stress range due to the fatigue load (including all amplification factors) must be less than 73 MPa. Eurocode 1-2 (CEN 2002b) prescribes a FLS that is similiar to AASHTO LRFD, but with different loads and load factors. There are 5 different fatigue load models that can be selected depending on the bridge. The fatigue load model 2 is most comparable to the North American practice and consists of one of several trucks, in a single design lane. The fatigue load model 3, which consists of 4 axles of 120 kN, spaced at 1.2, 6, and 1.2 m is often used instead for a simplified fatigue verification. The truck axle loads
68
include a dynamic load allowance. The safety factor γMf is made up of a partial safety factor for the steel cable material, a partial safety factor for qualification testing, and a partial safety factor for execution imperfections and bending stresses in cables caused by cable vibration (Lecing et al. 2001). Δσ c γ Ff Δσ E, 2 ≤ --------γ Mf where γFf is the partial safety factor applied to load models (taken as 1.0) ; ΔσE,2 is the equivalent stress range for 2 million cycles; Δσc is the reference value of the fatigue strength, taken as 0.52 of the tested value ΔσTEST , which is typically required to a minimum of 200 MPa, and γMf = 1.5 is the partial safety factor for the fatigue strength. 0.52 × 200 ∴1.0Δσ E, 2 ≤ ------------------------1.5 Δσ E, 2 ≤ 69 MPa The SETRA Recommendations (SETRA 2001) provide a rational reference value for the fatigue strength of the cable steel instead of using a partial safety factor. The reference value of fatigue strength is taken as the endurance limit of the stay cable at 100 million cycles, which is approximately 0.52 of the dynamic stress range at which the cable is tested for 2 million load cycles (SETRA 2001), assuming a 6 dB decay between these two points. The resulting stress range due to the fatigue load must be less than 69 MPa, which is comparable to North American codes. For cable-stayed bridges designed with an allowable stress of 0.45 fpu for the cables, fatigue loading governs for only a few cables under highway loading, even in structures with light girders (Gimsing 1997). If an extradosed bridge is designed for the allowable stress as recommended in the previous section, it is unlikely that any cables will be governed by fatigue when a verification is made at FLS.
3.4.8
Verification of Stay Cables at Ultimate Limit States
According to the SETRA Recommendations (SETRA 2001), the material resistance factor for extradosed cables is 0.75 if the cables have been mechanically tested to ensure ultimate and fatigue strength according to SETRA requirements, 0.67 if they have not been tested. The resistance factor for cables in cable-stayed bridges is 0.70 if they have been tested. The PTI Recommendations (PTI 2001) define a resistance factor of φ=0.65 for stay cables at strength limit states (ULS). The CHBDC specifies a resistance φTC=0.55 for stay cables. The value of φTC=0.55 was established from a calibration of existing bridges (with a known distribution of dead and live loads) designed with an allowable stress design approach (CSA 2006b). As many of the bridges considered were suspension bridges, and most of the cable-stayed bridges were steel or composite, the value of φTC is too low for castin-place concrete cable-stayed bridges, which would have to be designed with an allowable stress of around 0.40 fpu to meet the ULS requirement. This is a different approach than other limit states codes, as mentioned in 3.4.6, which impose an SLS limit instead, and define the cable strength at ULS based on
69
partial safety factors for the steel (after undergoing fatigue testing) and for the flexural stresses at the anchorages. Since an extradosed cable will have a maximum stress at SLS between that of a stay cable (0.45 to 0.50 fpu depending on specifications) and that of a prestressing tendon (0.70 fpu), the resistance factor at ULS should also be somewhere between those of the stay cable (φTC=0.55) and prestressing tendon (φP=0.95). For a maximum stress at SLS of 0.60fpu in an extradosed cable, linear interpolation between these two values gives a resistance factor of φ=0.79 for the extradosed cable at ULS.
3.4.9
Stay Cable Tensioning
The analysis of extradosed bridges is similar to cable-stayed bridges. They are highly indeterminate structures with the stiffness of the deck, the cables, and the piers affecting the distribution of the forces in the structure. If only the structure’s initial geometry is modelled and the dead load is applied, the stays extend and the structure deforms elastically, resulting in high girder and pylon moments and low cable forces. By pretensioning the cables, the cables are shortened at the anchorage to counteract the elastic strains and return the girder to its original undeformed profile (Walther et al. 1999). Since cable-stayed bridges, and most extradosed bridges, are constructed progressively in cantilever, it is possible to end up with any distribution of dead load moments. However, for concrete girders it is desirable to select a moment distribution close to that of a continuous girder on simple supports, also referred to as the natural moment distribution of the continuous beam (Mondorf 2006), to limit the effects of creep. For the purposes of modeling, the pretensioning of the cables is applied as a load case consisting of a strain in each cable which is added to the permanent loads. All methods to determine the cables’ initial tensions are either displacement based, or force based. Displacement based methods seek to reduce the displacements due to dead load on the cable-stayed system to a specified value. Troitsky (1988) describes a procedure to calculate the forces to reduce these displacements directly. First, the forces and displacements in the entire bridge are calculated for a unit force substituted for each cable of the bridge individually. Then, a system of equations is established requiring that the sum of the displacements, due to the unknown stressing forces in the cables, be equal in magnitude and opposite in direction to the dead load displacements at each of the locations where displacement is reduced, which corresponds to the number of unknown cable stressing forces. The cable stressing forces are determined by solving the system of equations. The reduction of displacements is described in more detail in Lazar et al. (1972) and a Fortran computer implementation based on the flexibility method of analysis is described in Troitsky (1988). Wang et al. (1993) present an iterative version of the same basic procedure to account for geometrical nonlinearities which they call the shape finding procedure. Walther et al. (1999) suggest determining the shortening of stays through a series of successive approximations to bring the girder back to its undeformed position, a method that is known as the zero displacement method. In each stage of the iterative procedure, the cable is shortened by a length equivalent to the elongation in the previous iteration, dead load is applied, and a new set of elongations
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result. Convergence is achieved when the change in bending moment in the deck is sufficiently small, which can require 10 or more iterations. Because the vertical displacements of the deck are influenced by bending in the towers, Walther suggests a two-stage process to achieve zero displacements in a cablestayed bridge. In the first stage, the towers are fixed, and the cable strains are adjusted in the centre span to obtain zero deflections. In the second stage, the towers are released and the cable strains in the side spans are adjusted to obtain zero deflections across the entire girder. This approach assumes that the towers can be set to their undeformed geometry through a backstay cable. Figure 3-17 shows the cable prestrains and maximum moments in the girder (of the stiff girder extradosed bridge in Chapter 4 released in rotation at the piers) for 25 iterations of the zero displacement method. While the girder moments stabilise after 5 iterations, the prestrain in some cables has not yet stabilised after 25 iterations. In order to reach the undeformed geometry, the cables in the side spans would have to be further released to lower the main span. In the case of this extradosed bridge with side span lengths of 0.6 of the main span, a ‘zero displacement’ moment distribution is not achievable in the side spans unless a very high tensile force is permitted in the backstay cable.
3
80
89 10
60
Cable 1
40
2
20 1
Midspan
0
0
-20 0
5
10
15 Iteration
20
25
100
4
Cable Strain, mm/m
4 3 2
5 6 7
Girder Moment, MNm
Cable Strain, mm/m
4
b) Towers Released
100
80 60
3
40
2 Side Span 1
20
Midspan
0
0
Girder Moment, MNm
a) Towers Fixed
-20 0
5
10
15 Iteration
20
25
Figure 3-17. Cable pre-strain and maximum moments for 25 iterations of the zero displacement method applied in two staged process: a) towers fixed, main span cable strains adjusted and b) towers released and side span cable strains adjusted. Cable 1 is anchored closest to the pier.
Force based methods seek to reduce the bending moments due to dead load on the cable-stayed system, or to match them to a desired bending moment distribution. One of the first force based methods, the reduction of maximum bending moment, is described as load balancing for cable-stayed bridges (Lazar et al. 1972). The procedure reduces the maximum bending moment in the girder by a specified coefficient, then finds identical unit stresses in all cables to achieve that reduction. Since the force due to dead load and stressing forces is the same in each cable, the cables can all be stressed to their allowable stress, which was considered to yield an optimal design of the cables. With the adaptability provided by strand based cables, which are now standard and where the size and prestressing can be individually varied, achieving a desired dead load moment distribution in the girder is considered much more important than matching the force across all cables. For concrete girders, the dead load moments should be close to those of a continuous beam on simple supports, as discussed in Section 3.2. The force equilibrium method (Chen 1999) searches for cable forces that will give rise to a desirable bending moment distribution in the structure. The method considers only the equilibrium of forces, and assumes that displacements can be controlled by precamber of the girder. For a concrete cable-stayed
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bridge, the target bending moments to be achieved in the final structure {M0} are usually obtained from a model of the girder as a continuous beam on simple supports, to which the dead loads and the prestressing added during construction are applied. The target moments are usually taken at the deck anchorage locations, but other control sections can be chosen. First, a model of the cable-stayed bridge, with all cables removed, is used to obtain the bending moments at every control section for a unit load applied individually to each cable. A matrix [m] of approximate influence coefficients mij contains the bending moment at the ith control section due to a unit force in the jth cable. An initial estimate of the cable forces {T0} can be solved from the following equation, where {Md} contains the bending moments at the control sections due to dead load and construction prestressing on the bridge with cables removed. o
–1
0
d
{T } = [m] ({M } – {M }) The interaction between tower, cables, and girder is taken into account by updating girder bending moments {Mn}, at each iteration n, with the dead load, construction prestressing, and the previously calculated cable forces. The following equations describe the first two iterations of the method. 1
–1
1
0
{ ΔT } = [ m ] ( { M } – { M } ) 1
0
1
{ T } = { T } + { ΔT } 2
–1
2
0
{ ΔT } = [ m ] ( { M } – { M } ) 2
0
1
2
{ T } = { T } + { ΔT } + { ΔT } Since the force equilibrium method does not consider the stiffness of the cables, nonlinearity from cable sag can be considered seperately. The force in a backstay cable anchored directly to the ground, when present in the system, can be paired with a control section in the tower to reduce or eliminate bending in the tower (Chen 1999). The notion of superimposing individual prestress bending moment diagrams was first suggested by Smith (1967). The unit load method (Bruer et al. 1999; Janjic et al., 2003) is similar to the force equilibrium method, except that the entire cable-stayed structure (tower, girder and cables) is used to obtain the bending moments at control sections for a unit load case applied to each degree of freedom (chosen as cable tensioning or jacking points). The expanded unit load method is implemented in RM2006 (TDV 2006) to include staged construction analysis, nonlinearity due to creep and shrinkage, and nonlinarity due to cable sag. In each stage of construction, unit loading cases are applied to different structural systems, and nonlinear effects are taken as linear over any single time interval (Bruer et al. 1999). Loads are accumulated at each stage of construction, and section forces are taken into account as initial displacements. At each time step, an approximation of the tension forces in the cables from a linear analysis is used as a starting point for an iterative procedure to include nonlinearities (Janjic et al., 2003). Gimsing (1997) states that the distribution of dead load moments in the cable-stayed bridge is fully described by the moments at the cable anchor points, Mg,i, and the girder dead load, gi,i+1, between each anchor point, as shown in Figure 3-18. The cable forces, Ti, can be found from the dead load moments and conversely the dead load moment distribution can be found from cable forces through the following equation.
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Mi – 1 – Mi Mi + 1 – Mi 1 1 T i = ⎛ --- ( g i – 1 ,i λ i – 1 ,i + g i, i + 1 λ i, i + 1 ) + ------------------------- + -------------------------⎞ ------------⎝2 λ i – 1 ,i λ i, i + 1 ⎠ sin ϕ i The equation is derived from an approximate method of analysis, which considers the stiffening girder as a continuous beam of elastic supports (Troitsky 1988). If shortening of both the tower and girder is neglected, the elastic support spring constant K (force per unit displacement), is as follows: Ec Ac 2 K i = T i sin ϕ i = ------------ sin ϕ i Li The solution to the beam on elastic supports analogy forms the basis of the second part of the equation suggested by Gimsing (1997). The first part of the equation is the tension required to counteract the elastic extension of the cables. The moment envelope of the girder due to live load is characterised by higher
λi,i+1 gi,i+1
λi-1,i Ti+1
gi-1,i
positive moments away from the piers and higher negative moments closer to
Ti
the piers. For steel girder cross-sections, it is usually preferable to have larger
Ti-1 φ
positive than negative moments in regions of high compressive force. Mi-1
Mi
Mi+1
Therefore, a more favourable moment distribution under dead and live load
Figure 3-18. Cable force can be achieved by altering the dead load distribution to yield a higher corresponding to dead load moment distribution (adapted positive moment demand across the girder, instead of minimizing the bending from Gimsing 1997). in the dead load condition (Gimsing 1997). This stressing strategy, however,
is not appropriate for concrete girders where creep causes a significant redistribution of dead load bending moment unless the distribution resembles that of a continuous beam on simple supports. If the desired dead load moment distribution is that of a continuous beam on simple supports, and the cable anchor is thought of as a node that must be held vertically, the pretensioning of the cable must exactly counteract all elastic deformation in the system. As dead load g is applied, the girder deflects downwards and the cable must be prestrained to raise the girder to its initial position. The prestrain of the cable will in turn cause an axial shortening of the girder, which will cause the initial anchor point to displace towards the tower. Similarly, the cable prestrain causes an axial shortening of the tower, which will cause a downwards displacement of the tower. Therefore, the cable must be shortened by an amount equal to those displacements. If the geometry and sectional properties of the cables, tower, and girder are known, this elastic deformation can be calculated explicitly. Assuming a constant girder area, the axial shortening in the girder Δgi at point i, will be caused by the force in all cables from i+1 to the outermost cable n and is Δ gi
L i cos ϕ i = ------------------Eg Ag
n
∑ j = i+1
g j – 1 ,j λ j – 1 ,j --------------------------tan ϕ j
Likewise, the axial shortening in the tower will be caused by the vertical component of force from the cables and from the self-weight of the tower above. Neglecting the self-weight of the tower, the tower shortening will be
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Δ ti
L i sin ϕ i = -----------------Et At
n
∑
g j – 1 ,j λ j – 1 ,j
j = i+1
Therefore, the total prestrain in cable i will be Δ gi Δ ti 1 g i – 1 ,i λ i – 1 ,i + g i, i + 1 λ i, i + 1 ε ci = ------------ ⎛ ----------------------------------------------------------------⎞ + ------------------- + -----------------⎝ ⎠ Ec Ac 2 sin ϕ i L i cos ϕ i L i sin ϕ i ⎛ n ⎞ ⎛ n ⎞ g j – 1 ,j λ j – 1 ,j⎟ 1 ⎛ g i – 1 ,i λ i – 1 ,i + g i, i + 1 λ i, i + 1⎞ 1 ⎜ 1 ⎜ --------------------------- + ---------g j – 1 ,j λ j – 1 ,j⎟ ε ci = ------------ ---------------------------------------------------------------- + -----------⎠ Eg Ag ⎜ ⎟ Ec Ac ⎝ 2 sin ϕ i tan ϕ j ⎟ E t A t ⎜ ⎝j = i + 1 ⎠ ⎝j = i + 1 ⎠
∑
∑
For an extradosed bridge with harp cable configuration and constant section properties, this simplifies to gλ ( n – i ) gλ ( n – i ) gλ ε ci = ----------------------- + ------------------------ + ---------------------Et At E c A c sin ϕ E g A g tan ϕ For the extradosed bridges described in Chapter 4, girder bending moments similar to those of a continuous girder on simple supports were obtained directly by applying a prestrain to each cable, calculated from the above equation. This is also a convenient form to estimate long-term strains in the girder and tower due to creep and shrinkage. Cable forces will change as the erection proceeds. Initial cable forces to be jacked in during erection must be determined in order to produce the final permanent load condition. Once the permament load condition is established, the cable tensions at each stage can be determined by a dismantling procedure, also referred to as a backwards analysis, which involves the same steps as the erection but in the opposite direction (Gimsing 1997). Felber et al. (1999) claim that backwards analysis may be valid for cable-stayed bridges with only steel elements, but is an “oversimplification” for composite and concrete bridges. The backwards analysis has a major disadvantage: it cannot account for time-dependent effects such as creep and shrinkage of concrete. Figure 3-19a shows the cable tensions at each step in a backwards analysis, while Figure 3-19b shows the cable tensions from a staged construction analysis that includes timedependent effects (shown for the stiff girder extradosed bridge of Chapter 4). Each stage is assumed to last 7 days. The final cable tensions from a staged construction are up to 8% lower than the desired tensions, which are the initial (leftmost) values of the backwards analysis. 5.3
a) Backwards Analysis
b) Staged Construction
5.2
5.2
5.1
5.1
5.0
5.0
4.9
4.9
4.8
4.8
4.7
4.7
4.6
Tension, MN
Tension, MN
5.3
4.6
10
5 Construction Stage
1
1
5
10 Construction Stage
Figure 3-19. Tensions of main span cables, at each stage of construction up to midspan closure, resulting from a) backwards analysis and b) Staged construction including time-dependent effects (form traveller not considered).
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The cable’s neutral (unstressed) length and its pre-deformation are intrinsic variables which do not change after initial stressing, unless the cable is restressed (Virlogeux 1994). Therefore, the forces in the structure at any stage can be determined by applying the cable pre-deformations and the self-weight, neglecting the creep of the concrete in previous stages. Introducing the cable pre-deformations as the cable is installed in a staged construction produces permanent forces which are very similar to the case where the pre-dormations are applied at once to the entire structure. Determining the initial predeformations by means a displacement method is convenient for input into analysis software since it works with prestrains directly.
3.5
Towers and Piers
The design of the towers offers great opportunity for creativity and structural expression. A significant decision in the design for the towers is that of whether to use a single mast or two lateral supports. This decision have to be made together with the selection of the cross-section, and the cable arrangement. A harp cable configuration causes significant bending in the tower which necessitates a minimum section width. A single mast above the deck will be more economical than two pylons, but the advantage also carries through to the substructure. The single mast allows for a single pier of relatively narrow width and supported by a single foundation, as compared with two pylons which are usually extended down to ground level and anchored in a single large foundation. Some designers have opted to transition lateral pylons into a single pier column by means of a deep transverse beam which results in a configuration that resembles a tuning fork. Figure 3-20 shows the variety of pier and tower forms that have been used in extradosed bridges. Central support - single mast
Cantilever or wall-type piers
Lateral support - two towers
Twin pier legs
Cantilever or wall-type piers
Hybrid configurations
Multiple pier legs
Figure 3-20. Tower and pier configurations. From left to right: Barton Creek Bridge, North Arm Bridge (LRT), Kiso and Ibi Bridges, Sunniberg Bridge, Odawara Blueway Bridge, Tsukuhara Bridge, Shin-Karato Bridge, Hozu Birdge, Miyakodagawa Bridge and Domovinski Bridge (LRT and road). See Table 2-1 for drawing sources.
The design of the pier is more important for the structural behaviour of the extradosed bridge than the tower itself. The main decision with respect to the pier is whether to keep the superstructure (girder, cables and tower) simply supported on the substructure, which keeps bending in the piers to a minimum, or to embed (fix in rotation) the superstructure on the piers, which is preferable to reduce the live load stress in
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the cables and allows for a more slender girder but increases the bending moments in the piers. A stiff girder extradosed bridge can be designed with the girder either embedded or simply supported on the piers, but a stiff tower extradosed bridge requires moment transfer between the tower and the pier. Depending on the height of the piers, the bending moment due to temperature range, long-term shrinkage of the girder, and live load, may be too large to allow a single pier column to be detailed with adequate bending strength. If a single pier column cannot be used, twin pier legs can be used to provide the desired rotational restraint and longitudinal flexibility for displacement. In seismic regions, twin pier legs may be preferable as they can be designed to be more flexible than a single column or wall-type pier. Transversely, lateral pier columns form a multiple column bent that is subject to higher response modification factors, resulting in lower seismic force demand since the system is more ductile. In summary, the design of towers and piers in an extradosed bridge depends on: 1. central or lateral suspension of the cross-section; 2. the cable configuration; 3. the superstructure fixity with the substructure, and 4. the magnitude of bending due temperature range effects, long-term shrinkage, and live load. The design of towers and piers for an extradosed bridge does not differ significantly from their design in an conventional bridge.
3.6
The Girder Cross-Section
The selection of the cross-section will depend primarily on the roadway width, whether it is supported by one or two planes of cables, and its depth, as determined by span to depth ratios discussed in Section 3.3.3 or otherwise. Aesthetics should play a significant role in the selection and shaping of the cross-section. For cantilever constructed girder bridges, closed single cell box girder cross-sections are used unvariably. For cable-stayed bridges, there are two types of deck cross-section that are generally used: the closed box and the slab. The box section can be either laterally or centrally supported, while the slab cross-section requires lateral support. The slab cross-section is either unstiffened, or stiffened by longitudinal edge girders, located inline with the cables or tucked in. For transverse spacing between cables of 12 m up to 20 m (Leonhardt et al. 1991; Menn 1994), a variable depth solid concrete slab can span between cables. Above this range, it is more economical to provide transverse ribs at 5 to 7 m spacing (T-beams) to reduce self-weight. The precompression in the longitudinal direction increases the moment resistance of the slab allowing it to span between cross-beams with minimal reinforcement. The ideal extradosed bridge cross-section lies somewhere between the closed box and the slab. There is an apparent contradiction between the closed box girder section, which is ideal for longitudinal bending with the webs located at the quarter points, and the stiffened slab which allows for direct support of the longitudinal edge girders from the cables, and is efficient transversely. However, the aesthetics of the slab cross-section suffer when the girders are deeper than 2 m, which is rarely a requirement for cable-stayed bridges. If the deck does not cantilever beyond the girders, the edge girders emphasize the visual depth of
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the superstructure. Sidewalks should always be positioned on deck cantilever overhangs outside the cables and main girders, both for transverse structural behaviour of the cross-section and for aesthetics. Designers have come up with different solutions for extradosed cross-sections. These have been classified in four categories: centrally supported box girders, laterally supported single cell box girders, multiple cell box girders, and laterally supported slabs. Box cross-sections have been used in over 36 of 50 extradosed bridges considered in the Chapter 2 study, while only 13 these were centrally supported. Centrally supported cross-sections can be easily supported by single piers, which is a major advantage when the piers are tall, but central suspension is appropriate only for bridges a divided roadway with two or more lanes in each direction.
3.6.1
Centrally Supported Box Girder Cross-Section
The centrally supported box girder cross-sections consist of either two webs, with steel or prestressed concrete ties linking the cable anchorages to the lower corners of the box, or of two nearly vertical closely spaced interior webs with cables anchored in a diaphragm between them. Long deck cantilever overhangs are supported by struts, inclined webs, transverse ribs or a combination of these. For bridges with a median barrier seperating traffic in each direction, central support is a logical choice, especially in light of the inherent girder depth and torsional stiffness in most extradosed cross-sections. Inclined webs facilitate the removal of formwork, can reduce the width of pier heads, and produce an external formed surface of better quality than a vertical surface (SETRA 2007). An analysis of bid results of 14 cable-stayed bridges that were tendered as both concrete and steel design alternatives in North America, conducted by Figg Engineering Inc. (Goni et al. 1999), found that concrete cable-stayed bridges with central vertical towers, saddles, and a single plane of cables have always been the low bid compared to steel alternatives. These precast segmental concrete designs have won over steel composite designs featuring H-type pylons with two planes of cables supporting a crosssection of edge girders with transverse floorbeams. Additionally, for other bids where the concrete designs had the same configuration as the steel composite designs, the concrete alternatives were always more expensive. However, the competitiveness of precast box girders with central suspension may be limited to wider bridges (with four or more traffic lanes). Figures 3-21 through 3-26 show examples of centrally supported box girder cross-sections from extradosed bridges proposed or constructed.
Figure 3-21. Arrêt-Darré Viaduct, France (concept 1982-83): main span 100 m, span to depth ratio 27, cantilever construction with precast segments with voided webs (Mathivat 1988).
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Figure 3-22. Barton Creek Bridge, United States (completed 1987): main span 103.6 m, span to depth ratio 27 at midspan, cantilever construction, with the fin poured progressively after completion of 3 segments (Gee 1991).
Figure 3-23. Kiso and Ibi River Bridges, Japan (completed 2001): 275 m maximum spans, span to depth ratio 39 at piers and 69 at midspan, cantilever construction with precast segments lifted with 600 tonne barge mounted cranes, and central 95 to 105 m steel sections strand-lifted from barges and made continuous (Casteleyn 1999, Kasuga 2006).
Figure 3-24. Shin-Meisei Bridge, Japan (completed Figure 3-25. North Arm Bridge, Vancouver, Canada 2004): 122.3 m main span, span to depth ratio 35, cast-in- (completed 2008): 180 m main span, span to depth ratio place cantilever construction (Kasuga 2006). 53, cantilever construction with precast segments (Griezic et al. 2006).
Figure 3-26. Trois Bassins Viaduct, Reunion (completed 2008): three main spans of 126 - 104.4 - 75.6 m, with cables overlapping through the middle span, effective span to depth ratio 30 at tallest pier and 50 at midspan, cantilever construction of central box, and construction of deck cantilevers and struts with mobile carriages (Frappart 2005).
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3.6.2
Laterally Supported Single Cell Box Girder Cross-Section
In some two lane bridges, the designers have used a single-cell box girder cross-section, where the box is almost as wide as the cross-section width. Since the vertical component of the cable force is not that large, it is sometimes possible to anchor the cables in short deck cantilever overhangs without transverse diaphragms at anchorage points, as would be required in a cable-stayed bridge. The vertical component of the cable force is transferred in bending to the girder webs, the deck slab thickness is kept to a minimum with transverse prestressing, and the girder webs are inclined inwards to reduce the width of the bottom slab. A wide single cell box cross-section was used for the Ganter Bridge (Vogel & Marti 1997) and for some of the first extradosed bridges such as the Odawara Blueway Bridge (Kasuga 2006), the Tsukuhara Bridge (Kasuga 2006), and the Korror-Babeldoap Bridge (Oshimi et al. 2002). To provide additional protection for the cables from the elements, some of the cross sections have a fascia that lowers from the cantilever overhang to cover the anchorage block outs. When sidewalks are required, they are almost always located on the deck cantilever overhangs outside of the cables. Figures 3-27 through 3-28 show examples of wide single cell, laterally supported box girder crosssections from extradosed bridges constructed to date.
Figure 3-27. Tsukuhara Bridge, Japan (completed 1997): Figure 3-28. Himi Bridge, Japan (completed 2004): main main span of 180 m, span to depth ratio of 33 at piers and span of 180 m, span to depth ratio of 45, cantilever construction (Kasuga 2006). 60 at midspan, cantilever construction in 6 m long segments, transverse tendons in deck (Kasuga 2006).
Figure 3-29. Korror-Babeldoap (Japan-Palau Friendship) Bridge, Palau (completed 2002): main span of 247 m, span to depth ratio of 35 at the piers and 70 at midspan, cantilever construction of concrete portions of spans, and central 82 m steel section lifted from barges and made continuous (Oshimi et al. 2002).
3.6.3
Multiple Cell Box Girder Cross-Section
Some bridges with wide decks feature box cross-sections with multiple webs, however this is not recommended for several reasons. Firstly, cantilever construction of multiple box-sections requires form
79
travellers with additional bays and formwork cores which add cost and complexity to the construction (SETRA 2007). Secondly, flexural strength and stiffness of the cross-section in the longitudinal direction is related to the radius of gyration I ⁄ A which is increased by locating more material towards the flanges (Menn 1990) and eliminating unnecessary webs. Thirdly, the internal webs are not directly supported by the cables unless transverse diaphragms are used to make the section more rigid, which implies that the inner webs are subjected to higher bending moments than the outer webs. As well, it is analytically much more complex to predict the distribution of forces between all webs. For the Ibi River Bridge cross-section shown in Figure 3-23, the most important design consideration was how to efficiently transmit the cable force to the main girder without providing diaphragms each cable anchorage (Casteleyn 1999). A combination of deck ribs and web ribs in each segment, along with three diaphragms in each cantilever were required to ensure adequate transverse rigidity for the 33 m wide section (Kasuga 2006). Referring to Figure 2-7, it is apparent that the multiple box girder cross-sections have the largest effective depths of concrete for their span. It is almost certain that for a similar cost, a more aesthetically pleasing structure can be built with struts or transverse ribs. The only advantage that multiple webs present is the potential to maintain a spacing of around 4 m between webs often found in multiple girder highway bridges, which is efficient for the transverse bending of the deck slab. The deck slab can be kept to a nominal dimension of around 225 mm with minimal reinforcement of 15M at 300 mm spacing in each face and in each direction designed with the CHBDC (CSA 2002a Clause 8.18.4). Figures 3-30 through 3-35 show examples of multiple cell box girder cross-sections from extradosed bridges constructed to date.
Figure 3-30. Odawara Bridge, Japan (completed 1994): main span of 122.3 m, span to depth ratio of 35 at the piers and 55 at midspan, cantilever construction (Kasuga 2006).
Figure 3-31. Shin-Karato Bridge, Japan (completed 1998): main span 140 m, span to depth ratio of 40 at piers and 56 at midspan, cantilever construction (Tomita et al. 1999).
Figure 3-32. Domovinski Bridge, Croatia (completed 2006): 840 m total length, spans of 60 m typical with a main extradosed span of 120 m, span to depth ratio of 30, cantilever construction in 4 m segments (Balić & Veverka 1999).
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Figure 3-33. Rittoh Bridge, Japan (completed 2006): main span of 170 m, effective span to depth ratio of 37 at pier and 61 at midspan, cantilever construction (Yasukawa 2002).
Figure 3-34. Pyung-Yeo Bridge, South Korea (completed 2007): main span of 120 m, span to depth ratio of 34 at piers and 30 at midspan, cantilever construction with one pair of travellers (Masterson 2006).
Figure 3-35. Pearl Harbor Memorial Bridge, United States (under construction): main span of 157 m, span to depth ratio of 31 at piers and 45 at midspan (Stroh et al. 2003).
3.6.4
Laterally Supported Stiffened Slab Cross-Section
A slab supported by cross-beams between longitudinal edge girders forms an efficient system that can span any width required and is increasingly used for cable-stayed bridges (Walther et al. 1999). Stiffened slabs are common for composite cable-stayed bridges, with spans of constructed bridges ranging from 105 m to 600 m (Svensson 1999), and have been used in some concrete cable-stayed bridges such as the East Huntington Bridge (Walther et al. 1999), the Dame Point Bridge (Gimsing 1997) and the Sidney-Lanier Bridge (Callicutt & West 1999). The use of stiffened slabs for extradosed bridges has been limited to short spans or shallow cross-sections. Figures 3-36 through 3-38 show examples of laterally supported edge girder cross-sections from extradosed bridges constructed to date.
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Figure 3-36. Saint-Rémy-de-Maurienne Bridge, France (completed 1996): spans of 52.4 and 48.5 m, effective span to depth ratio of 35, cast-in-place on falsework (Grison & Tonello 1997).
Figure 3-37. Sunniberg Bridge, Switzerland (completed 1998): main spans 128, 140, and 134 m, span to depth ratio of 127, cantilever construction in 6 m stages (Figi et al. 1997).
Figure 3-38. Third Bridge over Rio Branco, Brasil (completed 2006): main span of 90 m, span to depth ratio of 36 at piers and 45 at midspan, cantilever construction (Ishii 2006).
A preliminary design undertaken by the author for Tsable River Bridge near Nanaimo, BC used this concept of longitudinal edge girders along the outside edges of the deck slab. This resulted in an efficient use of reinforcing steel in the deck slab. The main spans of the bridge are 130 m long, with cross-beams spanning 20 m and spaced at 6.0 m, supporting a 250 mm thick deck slab. For 70% of the spans, only 15M reinforcement top and bottom at 300 mm spacing is required.
3.6.5
Composite Cross-Section
Composite cross-sections consisting of a steel girder with concrete have been considered for extradosed bridges. An alternative for the Pearl Harbor Memorial Bridge (Stroh et al. 1999) has been detailed with a composite cross-section, and the Golden Ears Bridge (Bergman et al. 2007) will be the first composite extradosed bridge constructed. The problem with a composite cross-section is that the part of the axial force in the girder, initially resisted by the concrete deck slab, creeps from the concrete into the steel girders over the long-term. The extent of this creep depends on the initial age of concrete deck slab at loading and on the relative stiffness of the two components. In cable-stayed bridges, loss of axial force in the deck slab is mitigated by providing the least amount of structural steel in the longitudinal edge beams, using high strength steel to provide strength without adding stiffness, and by using precast concrete panels that are cast three to six months prior to installation to allow some shrinkage to occur before the they are made composite (Taylor 1994). Compared with a cable-stayed bridge, the girder of an extradosed bridge of equivalent span is subjected to larger bending moments, which requires more steel in girders, and to an axial force of two to three times the magnitude that can be resisted by the same concrete slab dimensioned for transverse behaviour. Thus, the axial force is larger and the stiffness ratio of concrete to overall
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stiffness would be lower for an extradosed bridge, causing much of the compression initially in the concrete to creep into the steel.
Figure 3-39. Golden Ears Bridge, Canada (completion 2009): main span of 242 m, span to depth ratio of 54 at piers and 80 at midspan, cantilever construction with precast deck panels (Bergman et al. 2007).
Takami and Hamada (2001) studied the long-term behavior of a composite extradosed bridge that was designed with the same width and span lengths as the Odawara Blueway Bridge (Kasuga 2006). Their study set out to evaluate if a twin girder composite bridge, an economical system for construction, could be efficiently used for an extradosed design. They were concerned that the composite bridge would have excessive long-term deflections due to creep, leading to tensile stresses in the deck slab at the supports. For a bridge with spans of 74 - 122 - 74 m, they applied an instantaneous dead load to the the deck and compared the deflections and stresses at the initial time and 10000 days. The vertical deflection decreased up to 40 mm in the side spans and increased up to 40 mm in the main span. The compressive stress in the deck slab increased considerably to around 3.5 MPa in side spans, and decreased by less than 0.5 MPa to around 1.5 MPa over the supports. The stress in the steel top flange increased noticeably across the entire bridge, up to 60 MPa, while the steel bottom flange stress increased by up to 50 MPa at the supports. Cable forces increased an average of 12% in the main span. Although Takami and Hamada’s (2001) construction sequence is unknown, their main span to web depth ratio is 30.5, which is not particularly slender. The West Virginia Approach Spans to the Bridge across the Ohio River and Blennerhassed Island have spans of up to 122.2 m with a web depth of 3.05 m, resulting in a main span to web depth ratio of 40 (Wollman 2008). For short spans, it would be possible to erect light steel girders first, pour the deck slab, then install and stress the cables. The benefits of this system are similar to those of prestressing composite girders with external tendons: it limits tension stresses, increases yield load, increases ultimate strength, reduces structural steel weight, and reduces fatigue stresses in the steel (Tong et al. 1992). For short spans, this construction sequence could be more economical than cantilevered construction.
3.7
Tendon Layout
In Mathivat’s (1988) concept of the extradosed bridge, the internal cantilever tendons of the box girder are replaced by the extradosed tendons. The only tendons that are housed within the cross-section are external tendons, draped between pier diaphragms and deviators in the span, which is reasonable for precast construction. However, in many extradosed bridges constructed to date, internal cantilever and continuity tendons have been incorporated into the designs. Thus, there are four types of tendons that can be used in an extradosed bridge, as shown in Figure 3-40 and described as follows.
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Extradosed Tendons (Stays)
Internal Cantilever Tendons
External Continuity Tendons
Internal Continuity Tendons
Figure 3-40. Possible types of tendons in an Extradosed Bridge
Extradosed tendons are anchored in the deck segments and are either anchored in the towers or deviated through the towers by means of saddles. They resist dead load during cantilevering, and resist dead and live load in the final condition. As they are located above the deck surface, they require protection against the elements. Cantilever tendons are internal tendons installed as construction proceeds and anchored at the face of the segment. They resist negative moments during construction and in the final condition, and are efficient since the total prestressing force changes at each section and is built up gradually as cantilevering proceeds. In an extradosed bridge, typically cantilever tendons are only used for cantilevering out to the first extradosed tendons. Internal continuity tendons are installed after span closure and are anchored in block-outs as close to the bottom slab and webs as allowed given the clearance requirements for jacking. They resist positive moments that occur during construction due to temporary construction loads, thermal gradient, and concrete deformations. If external continuity tendons are not used, the internal continuity tendons also resist positive moment from live load and creep. External continuity tendons are installed after the structure is continuous and complement the other tendons. They resist negative moment around the piers and positive moment at midspan under superimposed dead loads, live load and creep. They are anchored in cross-beams at the piers and are deviated around the quarter points of the span by smaller cross-beams. These tendons are installed and stressed from anchorages at the piers.
3.8
Erection
Extradosed bridges are almost always constructed in cantilever, except for short spans. Conventional form travellers, as used for the construction of cantilevered girder bridges and cable-stayed bridges, can also be used for extradosed bridges. The form traveller can produce significant negative bending moments in the deck, depending on the type used. The first concrete cable-stayed bridges, such as the Brotonne Bridge (Mathivat 1983), were constructed with conventional form travellers of the same type used for cantilevered girder bridges. For the Brotonne Bridge, a cable was anchored every two segments, with each segment around 3 m in length. Three segments had to cantilever from the last installed cable before the next cable could be stressed, which produced negative bending moments extending up to 5 cables back and required prestressing tendons to balance these forces. Due to the early age of the concrete when the negative tendons were
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stressed, the long-term bending moments due to creep of the prestressing forces resulted in problematic long-term deflections of the girder (Virlogeux 1994). In subsequent cable-stayed bridges, form travellers were designed to allow stressing of the cables at the face of the segment before advancing the traveller. For extradosed bridges, form travellers have been designed to cast segments of 5 to 7 m in length, to match the cable spacing and speed up construction. Since extradosed cables are often designed not to require restressing, large multistrand stressing jacks can be used for stressing since they can be mounted on the travellers, as was done for the construction of the Tsukuhara Bridge (Ogawa et al. 1998). Typical box girders are stiff, and the erection geometry can be established from the final geometry (Virlogeux 1994). Flexible decks are subject to uncertainties arising from local longitudinal deflections, transverse deflections, and thermal effects of concrete hardening (Virlogeux 1994). With flexible decks, it is more important to balance the permanent loads at all stages of construction, as the final geometry is more sensitive to creep deformations. The cable-stayed form traveller is the best solution to limit the bending moments in the bridge during construction, and is now the preferred method for cast-in-place construction (Virlogeux 1994). The traveler is supported at its nose either by temporary cables or permanent cables which can be decoupled from the traveler after the the concrete is set. Each stage is typically between 5 and 7 m in length, and a cable (or pair of cables) is installed at every stage. This type of traveler was used for the construction of the Diepoldsau Bridge completed in 1985 (Walther et al. 1999), the Dames Point Bridge in 1989 (Abraham et al. 1998), the Burgundy Bridge in 1992 (Virlogeux et al. 1994b), the Sunniberg Bridge in 1997 (Figi et al. 1998) and the Sidney Lanier Bridge in 2003 (McNary 1999). In this system, the cables are stressed to balance the weight of the traveler after advancement, and in three or four increments through the segment pour. For the construction of the Diepoldsau Bridge over the Rhine in Switzerland, precast concrete edge beams under the main slab cross-section, were installed outwards from the previous stage (Walther et al. 1999). Stay cables were then installed and stressed before placement of concrete. The load of the cast-inplace concrete was supported by the stays at either end of the given stage. In the Dames Point Bridge (Abraham et al. 1998), a special coupler joined the form traveller to the stay cables through precast anchorage blocks, which allowed the permanent cables to support the wet concrete during the segment pour. After the concrete achieved strength, the stays were uncoupled from the traveller, thus transferring the cable force to the concrete. This method resulted in a traveller of only 60% of the weight that would otherwise have been required, and a cycle of 4 to 6 days per stage (Abraham et al. 1998). Upon advancing to the next stage, the formwork supports folded downwards and the forms lowered to allow the formwork to clear under the cross beams. In the Burgundy Bridge (Virlogeux et al. 1994b), temporary stays were anchored on the nose of the traveler, ahead of the given segment. The permanent cables were stressed simulatenously with the destressing of the temporary cables. The superstructure of the Sunniberg Bridge (Baumann & Däniker 1999) was constructed in stages of 6 m in length with an unconventional form traveller that extended over two segments. In each stage, the
85 Concrete Edge Beam
Support Rail HEB 700
Section through Edge Beams
Ballast
Support Frame Support Rail HEB 700
Section through Deck Centreline Support Frame Reaction Point
Plan View of form traveller
Work Platform
Outline of Work Platform on form traveller
Figure 3-41. Sunniberg Bridge form traveller (adapted from Figi et al. 1998).
leading edge beams and the trailing deck slab are poured simultaneously, as shown in Figure 3-41. With this configuration, the form traveller is better balanced: casting of the deck slab is offset by 1.5 m with the edge beams, thereby perfectly balancing their dead loads on the stay cable in question. The segment casting cycle was: 1. Pour edge and deck slab, with simultaneous decrease in ballast; 2. Reposition the support rail of the traveller for the next stage, mount jacks to stay cables for initial stressing; 3. Stress stay cables in 4 to 6 steps with simulaneous increase in ballast;l 4. Lower and advance the traveller, and 5. Place reinforcement in edge beams and deck slab and install stay cables. The use of ballast allows for better control and less risk during the deck pour, when problems with stressing of stay cables could have severe consequences. As well, all measurements of deformations and surveying can be completed ahead of the deck pour. This sequence resulted in a one-week construction per pair of segments. The entire bridge deck of 526 m length was constructed with a single pair of form travellers.
3.9
Verification at the Ultimate Limit States
According to the CHBDC (Clause 5.10.2), cable-stayed bridges must be investigated for non-linear effects from cable sag, deformation of the deck superstructure, and material non-linearity at the ultimate limit states. As well, the integrity of the structure must be ensured for the effect of the loss of any stay cable. In an extradosed bridge, cable sag can safely be neglected as explained in Section 3.4.5. The effects of non-linear geometry are small and need only be considered in combination with material non-linearity unless the deck is flexible.
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Walther et al. (1999) suggest using the static method of the theory of plasticity, using elastic force fields, to verify the capacity of the girder at the ultimate limit states. While this should be adequate to prove the resistance at most sections of the girder, the negative moment over the pier from an elastic analysis will generally exceed the section’s resistance. Therefore, a certain level of nonlinear analysis is necessary to verify that the yielding of the girder section causes an increase in the cable forces that can be resisted by the cables. The cable tensions, and corresponding girder moments, can be treated in two different ways at ULS. In a first method, the forces resulting from the pre-deformations are distinguished from those resulting from the dead loads. As with internal prestressing, the cable tensions due to the predeformations are factored differently than those due to dead load, which must be factored by the same coefficient as the dead loads (Virlogeux 1988). Where D is the total dead load force, and CP is the force due to cable predeformations, the force at ULS is: ULSD = αDD + 1.0CP
In a second method, the prestress contributes to the strength only (Walther et al. 1999), and the forces due to dead load and due to cable predeformations can be factored by the same coefficient: ULSD = αD(D + CP)
Menn (1990) explains that this strategy leads to girder moments at ULS that are consistent (of the same sign) with those at SLS, and thus reinforcement required at ULS will also contribute to SLS behaviour. The difference between this second method and the first method, (αD - 1.0)CP, can be thought of as a redistribution of moments in the girder at ULS. The second method appears to be reasonable for an extradosed bridge with a flexible girder, where the girder would exhibit nonlinear behaviour and redistribute load to the cables. The primary system, the tension in the cables and compression in the deck slab, resists collapse and the towers ensure stability. The first method, however, is better suited for extradosed bridges with stiff girders, where the extradosed cables sometimes yield after the formation of a plastic hinge in the girder close to the piers. This is explained schematically by Kasuga (2003).
3.10
Concluding Remarks
This chapter reviewed the loading, discussed design concepts related to distribution of live load between the primary axial load resisting system of the cables and the secondary flexural load resisting system of the girder, and the influence of rotational fixity at the piers. This was followed by discussions on proportions of the girder and tower, cable configuration, cable anchorages in the girder and piers, and cable design at SLS, FLS and ULS. A thorough summary of methods for cable pretensioning was presented to explain a topic that is discussed only briefly in most sources. For cantilever construction, it is only practical to pretension the cables at installation to balance the girder’s self-weight. To balance superimposed dead load, if desired for a stiff girder and necessary for a flexible girder, the cables have to be retensioned after closure of the cantilevers.
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The design of towers and piers will depend mainly on the type of suspension (lateral or central) and the type of connection with the girder which affects the magnitude of forces in the piers. All types of girder cross-sections have been used in extradosed bridges, but a single cell box girder is the preferred crosssection for efficiency in both longitudinal and transverse bending and materials usage for extradosed bridges with stiff girders. A single cell box girder centrally suspended and supported on a central pier appears to result in the most economical and constructable designs. However, central suspension is not practical for a bridge with only two traffic lanes. In the next chapter, the insight gained from the comprehensive literature review and demonstrations of this chapter will be applied to the design of two extradosed bridges: one with a stiff girder to resist live load through the secondary flexural system (and as has been the design strategy used for most extradosed bridges constructed to date), and one with a flexible girder to resist live load through the primary axial system.
4
DESIGN OF CANTILEVER CONSTRUCTED GIRDER BRIDGE, EXTRADOSED BRIDGE WITH STIFF GIRDER, AND EXTRADOSED BRIDGE WITH STIFF TOWER
In this chapter, three different bridges will be designed for the same crossing: a cantilever constructed girder bridge, a stiff girder extradosed bridge, and a stiff tower extradosed bridge. Design assumptions are common for all three bridges, and will be stated up front. The desing procedure varies for each bridge and is described in separate sections along with the structural behaviour and dimensioning of each bridge. Drawings of the three bridges can be found in the Drawings section. A fictional crossing has been assumed which requires a three span bridge with a main span of 140 m length. Cantilever-constructed concrete bridges are known to be competitive at this span length, since this is at the upper limit for composite steel plate girder bridges (Dubois 2004), often considered to be the main competition against concrete bridges for shorter spans, due to the size of flange plates. While other steel bridge types are feasible, such as composite box girder and tied arch bridges, these are usually more expensive options. This span is however shorter than is thought to be economical for cable-stayed bridge construction (Troitsky 1988). This main span is slightly less than the 150 m average main span of extradosed bridges in the Chapter 2 study. The side spans were chosen as 84 m (0.6 of main span) for the girder bridge and stiff girder extradosed bridge to keep the positive moment in the side spans similar that that in the main span, as discussed in Section 3.3.2, and to avoid uplift at the abutments. The side spans were chosen to be 72 m (0.51 of main span) for the stiff tower extradosed bridge to keep the side span positive moments in the girder under permanent loads to a minimum. A stiff tower design was attempted with 84 m side spans, but this entailed an unsupported length of 17 m between the last cable and the abutment, and the resulting side span girder moments greatly exceeded the its capacity. For this crossing, it is possible to add a short approach span at either side of the bridge, but for comparison between the three bridges, the main bridge was kept to three spans. The bridges are continuous, symmetrical about the centre of the main span, with expansion joints at each abutment. The roadway cross-section consists of one lane of 3.75 m width in each direction, shoulders of 2.0 m width, and a starndard MTO PL-2 concrete barrier with a steel railing (Ontario 2001), shown in Figure 4-1. PL-2 refers to
Figure 4-1. Roadway cross-section.
performance level of the barrier required in accordance with the CHBDC (CSA 2006a), based on an exposure index appropriate for this type of bridge if it had an AADT (average annual daily traffic) of 2000 vehicles. Since there are only two traffic lanes, lateral suspension will be used for the two extradosed bridge designs, which adds additional deck width beyond the barrier walls. According to the CHBDC (CSA 2002a), the 11.5 m wide roadway has 3 design lanes.
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89
4.1
Design Assumptions
4.1.1
Material Properties and Detailing
Commonly available materials with typical characteristics were chosen for the designs in this section, as summarised in Table 4-1. In Ontario, new bridges under the jurisdiction of the Ministry of Transportation are constructed with High Performance Concrete with a minimum compressive strength of 50 MPa, unless it is not available at the site’s location (Ontario 2001). This concrete has been adopted for these designs. Table 4-1. Material Characteristics assumed for design. Material High Performance Concrete
Strength Compressive strength: f’c = 50 MPa Cracking strength:
Modulus of Elasticity Ec = 28 100 MPa ( 3000 f' c + 6900 ) ( γ c § 2300 )
1.5
fcr = 0.4 f' c = 2.83 MPa Reinforcement - Grade 400R
fy = 400 MPa
Es = 200 000 MPa
Prestressing Steel: Seven-wire high-strength, low-relaxation strand (CEB-FIP Class 2), Size 15 (Astrand =140 mm2)
Specified tensile strength: fpu = 1860 MPa Yield strength: fpy = 0.90fpu = 1674 MPa
Ep = 200 000 MPa
The concrete deck is overlaid with a 90 mm asphalt and waterproofing system. The concrete deck surface is conservatively considered to be exposed to de-icing chemicals or surface runoff containing deicing chemicals. The periphery of the cross-section (the soffit of the deck cantilever overhangs, the external surface of the webs, and the soffit of the bottom slab) is also considered to be exposed. Minimum concrete covers and placing tolerances are adopted from the CHBDC (CSA 2006a) for the given exposure of the surface and are summarised in Table 4-2. Table 4-2. Concrete Covers and Tolerances specified in the CHBDC (CSA 2006a). Component Top Surface of Deck Slab Soffit of Deck Slab Cantilever Interior Soffit of Deck Slab External Surface of Web Internal Surface of Web Top Surface of Bottom Slab Soffit of Bottom Slab
Longitudinal Prestressing 130 ± 15 70 ± 10 (≤ 300 mm) 80 ± 10 (> 300 mm) 60 ± 10 90 ± 10 80 ± 10 60 ± 10 70 ± 10
Transverse Prestressing 90 ± 15 60 ± 10 -
Mild Reinforcement 70 ± 20 50 ± 10 (≤ 300 mm) 60 ± 10 (> 300 mm) 40 ± 10 70 ± 10 60 ± 10 40 ± 10 50 ± 10
Prestressing steel for internal, external and extradosed tendons is high strength seven wire lowrelaxation strand, and both tendons and extradosed stay cables are assumed to have the basic modulus of elasticity given in the CHBDC (CSA 2006a). Ducts for internal tendons are assumed to be rigid steel, which have a higher curvature friction coefficient, but lower wobble friction coefficient than plastic ducts. Plastic ducts are more durable during construction and therefore well adapted to segmental construction, can be coupled easily by means of half-shell clamps or shrink-wrap couplers, but are more expensive than steel ducts. External ducts are assumed to be polyethylene with rigid steel pipe deviators. Wobble and curvature friction coefficients are adopted from the CHBDC and defined in Table 4-3.
90 Table 4-3. Friction Coefficients (per metre length of prestressing tendon) Sheath Internal ducts: Rigid steel External ducts: Straight plastic Rigid steel pipe deviators
4.1.2
Wobble friction, K
Curvature friction, µ
0.002
0.18
0.000 0.002
0 0.25
Analysis and Limit States Verification
Frame models of the bridges were developed and analysed with the program SAP2000 (CSI 2005). Tendons were modelled as tendon elements in order to explicitly account for long-term effects. For elastic analyses, the external tendons were modeled as internal tendons, which is valid for small curvatures. Relaxation of tendons was included in all analyses of long-term effects according to values given in the CEB-FIP (1993) MC90 for class 2 relaxation: “improved relaxation characteristics for wires and strands”. Accordingly, the long-term relaxation after 50 years is assumed to be three times the relaxation at 1000 hours. The ability of SAP2000 to model tendon relaxation was validated using a model of a 1 m long tendon, subjected to different stress levels for 1000 hours and 50 years. Relaxation at 1000 hours is 1.1% for stress levels under 0.6 fpu, 2% for a stress of 0.7 fpu, and 5% for a stress of 0.8 fpu. At 50 years, relaxation is three times the relaxation at 1000 hours. For the the extradosed bridges, there is a difference between the girder area for structural response and the effective girder area for calculating its self-weight, due to the presence of transverse ribs. Since the internal functionality of SAP2000 was used to calculate the self-weight of the girder, the structural area of the girder was specified as its section property, and the unit weight of the girder concrete was increased to include the additional weight of the transverse ribs. The SETRA formula, given in Section 3.4.6, was used to determine the allowable stress at SLS for the extradosed cables in the stiff girder extradosed design. At SLS I, the CHBDC (CSA 2006a) requires the load case K for all strains, deformation, and displacements to be factored by 0.8 (see Table 3-1). With a structure built in stages, long-term effects (relaxation, creep and shrinkage) are calculated explicitly for each element starting from the time they are introduced into the model. It is difficult to isolate the long-term effects into a single load case because a portion of the effects will have already occurred before the structure is completely assembled. In lieu of factoring the portion of load case K due to long-term effects by 0.8, long-term effects were accounted for by verification at SLS1 of the forces in the structure at 50 years, an age at which most losses have occurred. It is assumed that the bridge will undergo maintenance work within the first 50 years of its service life, at which time the actual deflections of the structure would be taken into account, and additional prestressing could be added if necessary to correct the deflections and/or increase capacity. The structure is also verified at SLS1 for the forces in the structure immediately after construction. It is difficult to calculate the bending moments at ULS in a structure built in phases. One approach is to take the permanent moments from the SLS load case, and add the difference in loads between SLS and
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ULS, as taken from the continuous structure. The following summation of load cases at SLS and ULS illustrates this approach. SLS1perm,0 is the combination of permanent moments following the construction sequence equivalent to: SLS1perm,0 = 1.0SW + 1.0B + 1.0A + 1.0CP + 1.0P SLS1perm,50 is the combination of permanent moments after 50 years, effectively: SLS1perm,50 = 1.0SW + 1.0B + 1.0A + 1.0CP + 1.0P + 1.0K SLS1
= SLS1perm,0 / SLS1perm,50 + MAX/MIN(0.9L, 0.5TG + 0.5L, 1.0TG)
ULS1
= 1.2SW + 1.2B + 1.5A + 1.0CP + 1.0P + 1.7L = SLS1perm,0 + 0.2 SW + 0.2B + 0.5A + 1.7L
ULS2
= 1.2SW + 1.2B + 1.5A + 1.0CP + 1.0P + 1.6L + 1.15K = SLS1perm,0 + 0.2 SW + 0.2B + 0.5A + 1.6L + 1.15(SLS1perm,50 - SLS1perm,0)
where
SW is the self-weight of the girder (cable load is lumped with girder SW); B is the barrier load; A is the asphalt load; CP is the cable pretensioning (prestrain) load; P is the secondary prestress effect; L is the live load; TG is the temperature gradient; K is the load effect of relaxation of prestressing, concrete shrinkage and creep
The load cases added to the SLS1 load combination are calculated from the continuous structure. This will result in a conservative design, since more load is added to the positive moment regions where there is less reserve in bending capacity of the girder than in the negative moment regions at the piers.
4.1.3
Temperature Gradient
The AASHTO (2004) LRFD nonlinear thermal gradient for Zone 3 was adopted for design since the CHBDC (CSA 2006a) linear gradient simplistic and overly conservative for girder depths above 2 metres, as explained in Section 3.1.2. The procedure used to model the nonlinear temperature gradient at each girder section is as follows. •
Determine temperature forces in a restrained system subjected to strains due to the nonlinear temperature gradient (in a spreadsheet);
•
Calculate a curvature corresponding to the temperature forces of the restrained system (i.e. restraint is released and the temperature forces are applied to the system);
•
Calculate a linear temperature gradient corresponding to the above curvature and apply to the computer model. The total stress due temperature gradient is the summation of the primary or self-equilibrating stress
(made up of stresses from the restrained system and stresses from the released system) and the secondary or continuity stresses. The resulting stress from the computer model only accounts for the released stress and the continuity stress. At SLS, the temperature gradient is considered as a load case, and factored by 0.5 when considered simultaneously with live load, and by 1.0 when considered alone, as specified in the AASHTO (2004)
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LRFD code. When the temperature gradient is combined with live load, the live load is factored by 0.5. The temperature gradient has not been considered at ULS.
4.1.4
Construction Sequence and Segment Construction Cycle
The construction sequence was used in preliminary design and later in refined nonlinear analysis. A standard construction sequence was adopted, and assumed to proceed symmetrically about the centreline of the bridge. The major steps are summarised in Figure 4-2. The construction sequence for the extradosed bridges is the same as for the girder bridge with cantilever tendons replaced by extradosed tendons. Figure 4-2. Construction sequence. Construction of piers and pier tables.
Cantilever construction of segments. Tendons are stressed two per segment. Construction of cast-in-place portion of side spans between the cantilevers and abutments, stressing of side span continuity tendons. Closure pour at midspan segment linking the two cantilevers, stressing of main span continuity tendons, and removal of form travellers. Installation and stressing of additional continuity tendons or external tendons, followed by asphalt paving and construction of barrier walls.
A construction cycle of seven days per segment pair (5 working days and 2 weekend days for curing) has been used for staged anlaysis. At the beginning of the cycle, the traveler is advanced into position, the reinforcement is placed, and at the end of the week the concrete is poured. The concrete is left to gain strength over the weekend allowing two to three days for the concrete to achieve the required strength (Mondorf 2006). Then, the tendons are stressed and the traveller is advanced to the next segment position, and the cycle is repeated. Normally, one crew will be sufficient to handle all operations in both travellers, leading to a rate of construction of approximately 1.0 m per day per pair of travellers. For the extradosed bridge, a cycle of 7 days is maintained for each 6 m segment. Longer segments leads to a faster rate of construction, but would require a larger crew or preassembly of reinforcement cages. In the analysis, 28 days is provided between pouring of the pier tables and construction of the first pair of segments with the travellers. The pier tables normally take a long time to construct because they are completed in several pours to limit pour volumes and allow reuse or removal of formwork.
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4.2
Cantilever Constructed Girder Bridge
4.2.1
Layout and Cross-Section
A single cell box girder cross-section was chosen which is consistent with standard practice for a bridge of this width. The two webs intersect the deck slab beyond the quarter points, resulting in 2.5 m deck cantilever overhangs. Inclined webs were chosen for aesthetic reasons. The depth of the cross-section varies parabolically from 8.0 m at the pier to 3.2 m at the centre of the main span, which corresponds to span to depth ratios of 17.5 and 43.8 respectively, based on recommendations from various sources as discussed in Section 3.3. The girder is simply supported on bearings at the piers. The segment length was chosen as close to 3.5 m as possible. Given the desire to maintain a 2 m closure segment between cantilevers, and a pier table projecting approximately 4 m beyond the pier centreline, the cantilever length was divided into 19 segments of 3.42 m length. The approximate size of cantilever tendons was determined from an initial estimate of cantilever moment, and used to detail the deck slab haunches for adequate spacing and cover of prestressing ducts. The bottom slab has a minimum thickness of 250 mm to accommodate internal tendons given cover requirements. The girder cross-section in shown in Figure 4-3, and the general arrangement of the bridge can be found in Drawings, CANT-S1.
Figure 4-3. Girder cross-section of cantilever constructed girder bridge.
4.2.2
Longitudinal Prestressing
Two prestressing schemes were designed for this bridge. The first scheme uses cantilever tendons, bottom slab continuity tendons, and external tendons. The second scheme uses only cantilever and bottom slab continuity tendons. The two prestressing strategies will be compared on the basis of net moment in the girder and material quantity. Starting in the 1980s, it has become standard practice in France and some other European countries to use external tendons, in combination with internal tendons, as a replacement for internal tendons
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previously draped inside the webs (SETRA 2007). The use of external tendons for cantilever constructed girder bridges presents several advantages such as reduced cross-sectional dimensions, ease of installation, controlled stressing, future replaceability of external tendons, reduced shear demand in webs, and fewer blockouts for bottom continuity tendons. In North America, external tendons are not commonly used for cast-in-place construction. After appropriate cross-section dimensions were established, a preliminary design was used to size the remaining tendons, following the construction sequence. The main steps are outlined below, and follow the steps outlined by SETRA (2007). 1. Cantilever tendons were determined to keep the section over the pier uncracked in the determinate cantilever system during construction, for the moment caused by casting of the final segment. A form traveller self-weight of 400 kN was assumed, which is between 35 and 60% of the maximum and minimum segment weights. 2. The pier section was checked to ensure it remains uncracked at casting of the midspan closure segment (weight is added but continuity is not yet acheived). This does not normally control the design since the closure pour is small and the weight is distributed between the two cantilevers. 3. Continuity prestressing for the midspan closure segment was determined in order for it to remain uncracked for positive moment due to the temperature gradient. The form traveller and wet concrete load of the of the final segment are added to the determinate system, and the weight of the form traveller is removed from the continuous system, resulting in a negative moment (this is done first so that the secondary prestress moments from the midspan closure are known when sizing the side span continuity tendons). 4. Continuity prestressing for the side span closures was determined to keep the construction joint uncracked a) for positive moment due to the dead load of the cast-in-place girder after falsework removal and b) for the additional positive moment due to temperature gradient after the structure is fully continuous between the piers. 5. The critical moments were found at the critical design sections, as listed in Table 4-4, for the continuous bridge subjected to dead load, superimposed dead load (asphalt and barrier walls), temperature gradient (as before), and live load (envelope). Additional continuity prestressing or external prestressing was added as required to keep the girder uncracked at the side span section of maximum positive moment, and at midspan. The top stresses at the pier, and just beyond the pier table in the side spans were checked to ensure the concrete top surface remains uncracked. Table 4-4. Critical sections for design and corresponding load cases to produce the maximum load effect. Section Side span closure
Design
Midspan closure
Critical Moment Max Min
9
9 9
Side span max. moment Over pier
Check
9
9 9
9
9 9
DL
SDL
9 9 9 9
9 9 9 9
Load Case to Consider LLmin LLmax Temp
Creep
9 9 9
9
9
9
9
9
9
95
The above preliminary design resulted in the tendon arrangement in Table 4-5. A detailed model was then developed with the tendons from the preliminary design. The tendon layout and anchorage locations were modified to give a reasonable distribution of net bending moment and to keep the bottom concrete fibre uncracked at all sections. The net bending moments, immediately after construction and after 50 years, are shown for tendon schemes in Figure 4-4, along with the moments due to temperature gradient and live load. Calculation of stresses from the results of the detailed model found one pair of main span external tendons to be unnecessary in the case of the mixed tendon arrangement, and one pair of main span internal continuity tendons was added to the internal tendon design. The final tendon arrangement can be found in Drawings, CANT-A-S2 and CANT-A-S3 for the mixed tendon scheme, and CANT-B-S2 and CANT-B-S3 for the internal tendon scheme. Mixed Tendons Staged Construction Permanet Loads + PT (at End of Construction)
Internal Tendons Staged Construction Permanet Loads + PT (at End of Construction)
Mixed Tendons Staged Construction Permanet Loads + PT (at 50 years)
Internal Tendons Staged Construction Permanet Loads + PT (at 50 years)
AASHTO Temperature Gradient
Live Load Moment Envelope (all lanes loaded)
Critical SLS Moment = Permanent Loads + PT + Max (1.0 TG, 0.9 Live Load, 0.5 TG + 0.5 Live Load)
Figure 4-4. Bending moment in cantilever girder bridge (SAP2000 diagrams at the same relative scale). Table 4-5. Preliminary and final tendons for cantilever constructed girder bridge. Cantilever prestressing Continuity prestressing in main span
External prestressing in main span Continuity prestressing in side spans External prestressing in side spans
Internal and external tendons 19 - 2x15-15 tendons 1 - 2x19-15 top tendon 5 - 2x19-15 bottom tendons
8(6) - 27-15 draped tendons installed after structure is continous and all concrete has reached 28 day strength. 2 - 2x19-15
Internal tendons only 19 - 2x19-15 tendons 1 - 2x19-15 top tendon 4 - 2x22-15 bottom tendons at closure 6(7) - 2x22-15 bottom tendons after concrete has reached 28 day strength. None
3 - 2x22-15 bottom tendons at closure
2 - 27-15 draped tendons installed after struc- None ture is continous and all concrete has reached 28 day strength. Note: Numbers in paranthese (X) are final tendons used after checking stresses with a detailed model.
96
4.2.3
Verification at SLS and ULS
The net moment and the top and bottom stresses in the concrete, at the critical sections in the bridge, are shown in Table 4-6 for the mixed tendon design, and in Table 4-7 for the internal tendon design. Only the stresses after 50 years are shown since these controlled the design at SLS. The temperature gradient load case was found to govern the design of the continuity prestressing in the main span at SLS, while live load governed in the side spans. Detailed results can be found in Appendix B. For the calculation of section resistance at ULS, the stress in the external tendons is taken as the stress at 50 years at SLS1. Any increase in stress that might occur due to elongation of the tendon at ULS is neglected. Section resistance was calculated with Response 2000, with axial compression and primary moment of the external tendons included as an external load.. Table 4-6. SLS Forces and Maximum Stresses in the Girder - Internal and External Tendons Side Span
Main Span
Critical Sections Closure Max Mlive Pier CL 0.3 of span 0.4 of span SLS Forces after 50 years (Forces include Mp, units are MNm, MN and MPa) SLS Mmin (t=50 years) -10.28 -0.55 -84.0 14.95 -5.0 SLS Mmax (t=50 years) 18.25 48.6 43.0 86.2 64.2 SLS Axial Force -22.2 -38.7 -138.0 -61.8 -59.0 Top Stress - Mmin -1.7 -4.5 -7.7 -8.2 -6.7 - Mmax -4.8 -9.0 -11.6 -13.9 -13.7 Bottom Stress - Mmin -4.4 -4.6 -12.5 -5.5 -7.9 - Mmax 0.1 1.3 -8.8 1.8 1.6 Forces at ULS1 (αDD + 1P + 1.7L over pier) or ULS2 (αDD + 1P + 1.6L + 1.15K in positive moment regions) Mf Mr Mf / Mr
81.7 80.1 1.02
65.5 69.4 0.95
-702 -1125 0.62
137.3 137.l 1.00
CL span -10.2 57.2 -62.0 -6.7 -14.1 -9.4 1.2 181.3 180.4 1.00
The moment resistance at the side span closure section is inadequate, but can be increased sufficiently by increasing the bottom slab reinforcement from 15M to 20M. Table 4-7. SLS Forces and Maximum Stresses in the Girder - Internal Tendons Side Span Main Span Critical Sections Closure Max Mlive Pier CL 0.3 of span 0.4 of span SLS Forces after 50 years (Forces include Mp, units are MNm, MN and MPa) SLS Mmin (t=50 years) 2.0 -5.6 -179.0 13.7 3.4 SLS Mmax (t=50 years) 30.6 43.9 -53.0 86.5 72.6 SLS Axial Force -17.3 -42.4 -105.0 -60.6 -61.0 Top Stress - Mmin -2.4 -4.5 -2.4 -7.9 -7.8 - Mmax -5.3 -9.0 -6.2 -13.8 -14.8 Bottom Stress - Mmin -1.9 -5.7 -13.7 -5.5 -7.0 - Mmax 2.6 0.3 -9.5 1.9 2.5 Forces at ULS1 (αDD + 1P + 1.7L over pier) or ULS2 (αDD + 1P + 1.6L + 1.15K in positive moment regions) Mf 88.8 64.9 -714 154.8 Mr 109.4 129.3 -1226 282 Mf / Mr 0.81 0.50 0.61 0.55
CL span -2.3 63.9 -66.0 -8.1 -15.4 -8.7 1.7 203 351 0.58
For both designs, the deck surface remains precompressed from the end of construction to 50 years at SLS, and the bottom surface remains uncracked. The bridges have adequate bending resistance at ULS.
97
Based on an elastic analysis, the design with internal tendons appears to have a larger reserve capacity than that with external tendons.
4.3
Extradosed Bridge with Stiff Deck
The objective of this design was to make the girder as slender as possible, but stiff enough relative to the cables to keep the stress range due to live load in the cables below 50 MPa, the limit at which SETRA allows the extradosed cables to stressed to 0.60 fpu in a conventional prestressing anchorage. In keeping with the notion of extradosed tendons as prestressing the deck, retensioning of extradosed cables is avoided, and the girder is designed to remain fully prestressed (uncracked) at SLS. Extradosed cables are anchored in the towers.
4.3.1
Layout and Cross-Section
A harp cable configuration was chosen for the reasons discussed in Section 3.4.1, with a span to tower height ratio of 10. This results in a cable slope of 0.2. Each cantilever consists of 10 segments of 6 m length, with a 6 m closure pour between the two cantilevers and an initial pier table segment projecting 7 m beyond the pier centreline. Several alternatives were considered in the selection of a cross-section for the bridge; each alternative is shown in Table 4-8 with its advantages and disadvantages. Cross-section depths of 4 m and 2 m were considered, to gain insight into how each alternative would work in a variable-depth configuration as well for a constant depth. The dimensions of the deck slab, webs and bottom slab are determined based on cover requirements, standard practice for detailing, and reference cross-sections from Section 3.6. The deck slab is haunched at the webs to allow adequate cover for internal prestressing ducts and anchorages. Some cross-sections feature details such as strut connections and ribs intersecting cables which require careful detailing but which have been previously constructed and are considered feasible. The chosen cross-section consists of a box girder with vertical webs located at the quarter points of the section. There are transverse ribs spanning the full width of the cross-section between the anchorages and spaced at 6.0 m to correspond with the segments. The depth of the cross-section was chosen as a constant 2.8 m to give a span to depth ratio of 50. This is higher than recommended by some studies mentioned in Section 3.3, but was chosen on the basis of aesthetics: it is desirable to keep the girder as slender as possible. To use the extradosed cables most efficiently however, the stress range in the cables due to live load had to be kept under 50 MPa, low enough that there would be no reduction in allowable stress in the cables at SLS. Initial plane frame models indicated that if the girder was simply supported at the piers, the maximum stress range in the backstay cable at SLS would be 114 MPa, far greater than the 50 MPa limit. If however, the girder was on fixed supports, the maximum stress range in the cables at the quarter points of the main span would be 58 MPa, for which there is only a minor reduction in allowable stress at SLS. Therefore, the girder span to depth ratio of 50 was accepted and the girder was embedded on the piers. This resulted in the cross-section in Figure 4-5, and the general arrangement for the bridge can be found in Drawings, EXTG-1.
98
Figure 4-5. Girder cross-section of stiff girder extradosed bridge.
At segments where extradosed cables are anchored, 2 transverse 19-15 mm diametre strand draped tendons are required in the ribs. At other segments, there is a single 12-15 mm diametre strand tendon.
4.3.2
Longitudinal Prestressing
After appropriate cross-section dimensions were established, a preliminary design was used to size the remaining tendons, following the construction sequence, and assuming that the prestressing moments were not affected by the presence of the cables. The main steps are outlined below, and follow a similar methodology to those of the cantilever girder bridge. The form traveller is assumed to weigh half of the self-weight of one segment, in this case 600 kN. 1. Extradosed tendons were determined to balance the dead load of one segment with an allowable stress of 0.6 fpu. This resulted in 19-15mm diametre strand tendons. Several pretensioning strategies were investigated of those discussed in Section 3.4.9, but simply pretensioning to 0.6 fpu to balance the selfweight of a 6 m section of the girder was found to produce minimal net moments in the girder. 2. Internal cantilever tendons were dimensioned, additional to extradosed tendons, as required for the section over the pier to remain uncracked in the determinate cantilever system, for the moment caused by casting of the first cantilevered segment, before the extradosed cable is installed. A single pair of 19-15mm diametre strand tendons was required across the pier table. At this stage, a simple plane frame model was used to obtain the critical moments at the critical design sections for the continuous system subjected to dead load, superimposed dead load (asphalt and barrier walls), temperature gradient, and live load (envelope). 3. Continuity prestressing for the midspan closure segment was dimensioned to keep it uncracked for positive moment due to the temperature gradient. The form traveller and wet concrete load of the of the final segment are added to the determinate system, and the weight of the form traveller is removed from the continuous system, resulting in a negative moment. Three pairs of 19-15mm diametre strand bottom tendons and a single pair of top tendons were required across the midspan joint. The shortest pair is anchored across 5 segments (30 m length) in order to better balance the midspan moment that results from superimposed loads. 4. Continuity prestressing for the side span closures was dimensioned to ensure the closure joint remains uncracked a) for positive moment due to the dead load of the cast-in-place girder after falsework removal and b) for the additional positive moment due to temperature gradient after the structure is fully continuous between the piers. Two pairs of 19-15mm diametre strand tendons were required.
99 Table 4-8. Evaluation of alternative cross-sections for lateral support by planes of cables. Cross-Section Wide box girder
Advantages
Disadvantages
• Cable force transferred directly to all webs. • High out-of-plane bending of webs. • Consistent internal cross-section. • High moments around corners of box. • High stiffness for given section depth. • Viewed in profile,box girder looks heavy without deck cantilever overhangs.
ρ = 0.66, 0.63 Wide box girder with interior struts
ρ = 0.66, 0.63
Multiple cell box girder
• Cable force transferred directly to all webs. • Struts decrease transverse span of deck slab. • Struts are precast or of steel and easy to incorporate into forms. • Cable force transferred directly to all webs.
• In variable depth section, each strut connection is different. • External tendons are difficult to • Decrease in dead load can be minimal of box girderd components are governed by other factors (cover, PT ducts, anchorages). • Viewed in profile, box girder looks heavy without deck cantilever overhangs. • Struts are not visible to the users.
• Additional web decreases transverse span of deck slab, reduced thickness and reinforcement. • Transverse PT not required.
• Additional web adds weight, additional line of PT, and vertical reinforcement. • Cross-section is less efficient. • High transverse deformability. • Distribution of force between webs is difficult to predict. • Additional bay and internal core form required in form traveller.
• Narrow flange is more efficient at midspan, raises the centre of gravity of the concrete. • Ribs allow for a larger deck width without an additional web. • Ribs ensure uniform sectional behaviour. • Structural behaviour is exposed. • Viewed from below, ribs add visual interest and rhythm to plain concrete box girder.
• Ribs restrict access through box girder. • Ribs restrict height of external tendons at the pier. • Rythm from the ribs could conflict with the cables. • Ribs complicate construction: the soffit formwork must be lowered beneath ribs to advance the traveller.
• Narrow flange is more efficient at midspan, raises the centre of gravity of the concrete. • Struts allow for a larger deck width with minimal additional weight and without an additional web. • Struts transfer vertical cable force directly to webs. • Unobstructed access through box girder. • Structural behaviour is exposed. • Viewed from below, add visual interest and rhythm to plain concrete box girder.
• In variable depth section, each strut connection is different. • If struts are anchored at the bottom flange, transverse prestressing must be added which increases depth of flange or adds a cross-rib. • Struts complicate construction: some disassembly of forming surface is required in order to advance the traveller.
• Efficient at midspan, raises the centre of gravity of the concrete. • Appropriate for depths less than 2 m. • Viewed from below, add visuals interest and rhythm to the girder if transverse ribs are used.
• Poor negative bending resistance. • Form traveller must be cable-stayed or split across two segments to limit the bending in the girder.
ρ = 0.66, 0.64
Typical box girder with transverse ribs
ρ = 0.62, 0.61 Box girder with transverse struts
ρ = 0.66, 0.62
Double T section or U section
ρ = 0.39, 0.33
5. Additional continuity prestressing or external prestressing was added as required to keep the girder uncracked at the side span section of maximum positive moment, and at midspan. Four 19-15mm
100
diametre strand tendons were provided at the side spans, and 8 were provided across the main span. The side span and main span external continuity tendons overlap across the pier table. Since the cross-section features transverse ribs, the external tendons cannot be raised into the top slab at the pier, but reach their highest point at 1.0 m below the deck surface. Since the girder centroid is only 1.3 m below the surface, the external tendons are ineffective in reducing the negative moment demand. While pockets could be detailed to pass the external tendons through the ribs, this would have required special detailing of reinforcing steel and transverse tendons for 16 of the ribs, which was considered to be undesirable. The prestressing from this preliminary design is summarised in Table 4-9. Table 4-9. Preliminary and final tendons for extradosed bridge. Preliminary Design Extradosed prestressing Cantilever prestressing Continuity prestressing in main span External prestressing in main span Continuity prestressing in side spans External prestressing in side spans
10 - 2x19-15 tendons 1-2x19-15 tendons 1 - 2x19-15 top tendons 3 - 2x19-15 bottom tendons 8 - 19-15 draped tendons installed after structure is continous and all concrete has reached 28 day strength. 2 - 2x19-15
Final Design 10 - 2x19-15 tendons 1 - 2x19-15 top tendons 6 - 2x19-15 bottom tendons 6 - 27-15 draped tendons installed after structure is continous and all concrete has reached 28 day strength. 4 - 2x27-15
4 - 19-15 draped tendons installed after struc- 2 - 27-15 draped tendons installed after structure is continous and all concrete has reached ture is continous and all concrete has reached 28 day strength. 28 day strength.
Note: Numbers in paranthese (X) are final tendons used after checking stresses with a detailed model.
4.3.3
Comparison with Bending Moments in a Girder Bridge
To assess the feasibility of the chosen cross-section, it is desirable to have an idea of the bending moment capacity of that section. For this purpose, the bending moment of the extradosed cross-section was compared with that in a girder bridge with a box girder cross-section of similar depth. Although most of the superimposed dead load moment in the extradosed girder will be balanced by internal prestressing, it is useful to get a sense of whether these bending moments could be accommodated in the initial crosssection. Five girder bridges were analysed for the purpose of determining maximum bending moments, both in negative and positive bending, likely to occur in a box girder section of given depth at SLS and ULS. They have maximum spans of 50, 75, 100, 150, and 200 m. The first two have constant depth cross-sections with side spans of 0.8 of the main span, while the others are variable depth with side spans of 0.6 of the main span. Cross-sections were given typical dimension based on references and the recommendations that were summarised in Section 3.3. They have the same roadway width, asphalt thickness and barriers as in the cantilevered post-tensioned bridge in Section 4.2. The maximum moments are summarised in Table 4-10.
101 Table 4-10. Maximum Bending Moments in a Typical Box Girder Bridge Span (m)
50
3.60
Area (m2)
3.1 Inertia (m ) Load Case Moments (MNm) Self-weight 18.62 Asphalt 2.62 Barriers 1.40 Live Load 11.70 SLS Moment 33.2 ULS Moment 47.9 Mself / Msls 0.56 αMself / Muls 0.47
75
3.74
100
3.98
3.98
-35.38 -4.98 -2.66 -9.83 -51.9 -69.8 0.68 0.61
46.34 5.90 3.15 21.04 74.3 104.0 0.62 0.53
-88.00 -11.20 -5.99 -18.79 -122.1 -161.5 0.72 0.65
5.38
3.95
6.76
4.41
8.09
3.3
27
6.4
51
10
139
58.3 7.4 3.9 20.9 88.4 121.3 0.66 0.58
-192.1 -23.0 -12.3 -39.5 -263 -347 0.73 0.66
152.6 16.7 9.0 39.6 214 286 0.71 0.64
-507.0 -51.7 -27.6 -76.6 -655 -849 0.77 0.72
260.0 25.0 13.4 56.0 349 461 0.75 0.68
-1093.0 -96.6 -51.6 -134.0 -1362 -1746 0.80 0.75
500 ULS
1600
Maximum Moment, MNm
Maximum Moment, MNm
1800 ULS
1400 1200 1000
SLS -ve
800 600
ULS
400 200 0 40
200
3.57
6.8
4
150
400 SLS +ve 300 ULS 200 SLS -ve 100
SLS +ve 0
60
80
100 120 140 160 Length of Main Span, m
180
200
2
3
4 Depth of Box Girder, m
5
6
Figure 4-6. Maximum bending moment in a typical girder bridge of 12.4 m width: a) as a function of longest span, and b) as a function of girder depth.
The maximum moments in the five girder bridges considered are shown in Figure 4-6a as a function of their maximum span lengths. As the span length increases, the negative moment increases more rapidly than the positive moment. However, when the moments are shown in Figure 4-6b as a function of the girder depth, the positive moments are typically twice as large as the negative moments. This is not surprising since the centroid in a box girder is located closer to the deck surface than to the soffit, making the box section more favourable for positive bending. The extradosed bridge in this section has a maximum bending moment at SLS of around 110 MNm over the pier and at midspan. From Figure 4-6, the maximum SLS moment in a girder bridge with girder depth of 2.8 m is around 75 MNm in negative bending and 130 MNm in positive bending. Since the precompression in the girder will be higher for the extradosed section of 2.8 m depth, the moments in the extradosed girder are reasonable.
102
4.3.4
Detailed Model
A detailed plane frame model was developed for the extradosed bridge to properly model the staged construction, secondary effects, and long-term effects. The staged construction includes P-delta effects, although these were found to be small for this extradosed bridge. The form traveller was included in the staged construction since it imposes a significant negative moment on the girder when it is advanced. The weight of the form traveller was assumed to be centred 2 m beyond the previous segment. Self Weight (SW)
Staged Construction = Permanent Loads + PT (at End of Construction)
Cable Prestrain (CP)
Staged Construction (after 50 Years)
Net Self Weight (SW+CP)
AASHTO Temperature Gradient
Permanent Loads (SW+CP+SDL+PT)
Live Load Envelope
Figure 4-7. Bending moment in extradosed bridge (SAP2000 diagrams at the same relative scale).
The prestressing described in Section 4.3.2 was used as a starting point for the detailed model. Top and bottom stresses at SLS were checked at critical sections, both at the end of construction and after 50 years, to ensure they remained below the cracking stress of the concrete. Due to the presence of the extradosed cables, the prestressing is not 100% effective in offsetting the moment demand in the girder, and the girder stresses were considerably higher than anticipated. Additional tendons were added to oppose the permanent moments. The final tendon arrangement can be found in Drawings, EXT-S2 and EXT-S3. Moment diagrams for the extradosed bridge are shown in Figure 4-7.
4.3.5
Verification at SLS and ULS
The bending moments and stresses at SLS are summarised in Table 4-11. The moments in the girder under permanent loads, at the end of construction and after 50 years, are of particular interest. There is a very
103
large shift in bending moment in the girder over the long-term, which results from relaxation of the cables and prestressing combined with shrinkage and creep of the girder. At midspan, this change in forces results in a 7.7 MPa decrease in the bottom stress of the girder and downwards displacement of 93 mm. The live load moment range causes stress of 7.8 MPa in the bottom slab. To keep the girder uncracked at 50 years, an effective prestress after losses of 10 MPa was provided at midspan. . Table 4-11. SLS Forces and Maximum Stresses in the Girder Side Span Main Span Critical Sections At closure Max Mlive beyond PT Pier CL 0.4 of span SLS Forces at the end of construction (Forces include Mp, units are MNm, MN and MPa) Mmin (end of const.) -26.5 -22.8 12.2 -58.4 11.62 Mmax (end of const.) 2.97 16.9 47.6 -1.95 47.7 Axial Force (end of const.) -50.2 -63.0 -62.9 -95.0 -93.6 Top Stress - Mmin -4.0 -6.2 -10.3 -3.5 -14.5 - Mmax -7.4 -10.8 -14.4 -9.8 -18.7 Bottom Stress - Mmin -12.9 -13.9 -6.2 -17.5 -10.6 - Mmax -6.4 -5.2 1.6 -10.3 -2.7 SLS Forces after 50 years (Forces include Mp, units are MNm, MN and MPa) SLS Mmin (t=50 years) -9.95 -7.42 7.31 -99.5 25.1 SLS Mmax (t=50 years) 19.57 32.2 42.8 -43.1 61.2 SLS Axial Force -33.2 -50.2 -47.1 -82.0 -71.0 Top Stress - Mmin -3.5 -6.2 -7.5 2.5 -12.9 - Mmax -7.0 -10.8 -11.6 -3.8 -17.1 Bottom Stress - Mmin -6.9 -8.7 -5.0 -21.5 -4.47 - Mmax -0.4 0 2.8 -14.2 2.8
CL span -2.61 35.0 -89.2 -12.2 -16.6 -13.1 -4.9 21.2 58.8 -68.6 -12.1 -16.5 -5.0 2.8
In a first verification at ULS, the demand was calculated by starting with the bending moments at SLS at the end of construction, and adding additional bending moments, calculated from the loads applied to the final system, to produce the full ULS load combination, as explained in Section 4.1.2. The axial force in the girder is also increased to correspond with the external loads. The difference between the state of stress at 50 years and the state of stress at the end of construction was considered as the load case K to group all long term effects due to relaxation of prestressing, shrinkage and creep of concrete, and redistribution of the structure from the as-constructed state to the final state. As the long-term effects are considerable, ULS2 (1.6 Live + 1.15 Long-term effects) is more significant than ULS1 at all sections. Table 4-12. ULS Forces in the Girder Critical Sections At closure Forces at ULS2 (αDD + 1P + 1.6L + 1.15K) Mf Axial Force to calculate Mr Mr Mf / Mr
107.3 -8.65 152.8 0.70
Side Span Max Mlive 125.1 -24.1 164.6 0.76
beyond PT 58.1 -49.4 58.8 0.99
Pier CL -212.4 -80.3 -156.9 1.35
Main Span 0.4 of span
CL span
210.4 -41.2 210.5 1.00
227.5 -30.7 229.0 0.99
The utilisation ratios of the girder at ULS are summarised in Table 4-12. At some sections, the resistance is either just adequate to cover the demand, or is insufficient in the case of section over the pier.
104
However, as the load increases and girder loses stiffness due to inelastic response (cracking of concrete followed by yielding of reinforcement), the cables will take up increasingly more load. The moment in the pier will not reach the value predicted by a purely elastic model. Meanwhile, the forces in the cables cannot exceed their factored resistance. A nonlinear analysis is required to get more realistic values of the moment in the girder and the forces in the cables. The extradosed cable tensions, both immediately after construction and after 50 years, are shown in Figure 4-8. b) After 50 years
5
5
4
4 3
2009
2007
2005
2003
2001
1002
2009
2007
2005
2003
2001
1002
0 1004
0 1006
1
1008
1
1004
2
1006
2
Live Load Positive Permanent Loads
1008
3
allowable = 0.58 A fpu = 5.7 MN
1010
Cable Force (MN)
6
1010
Cable Force (MN)
a) After construction 6
Figure 4-8. Tension in cables of stiff girder extradosed bridge.
The pier capacity was also investigated for critical loading (maximum moment and axial force) both during construction and in the final condition. During construction, the maximum moment in the piers occurs at ULS4 (maximum wind) where the final segment is constructed at the end of one cantilever, and wind acts upwards on the opposite cantilever. Temporary prestressing installed between the foundation and the pier table would be required for strength during construction. The piers were jacked apart by 110 mm, corresponding to a force of 2000 kN, to displace the piers outwards by approximately half the value of the displacement due to shrinkage and creep of the girder after 50 years, thus reducing the long-term moments in the piers by half.
4.4
Extradosed Bridge with Stiff Tower
For this design, the objective was to make the girder as slender as possible so that live load is transferred directly to the piers, as an axial force couple between the cables and girder, similar to a cable-stayed bridge. An upper limit of 200 MPa has been set for the stress range in the cables due to live load at SLS, which corresponds to the tested value of strand-based stay cable anchorages (for an upper stress of 0.45 fpu at 2 million load cycles). The girder is designed to be as light as possible in order to achieve economy by reducing material quantities in the girder. Stay cables are anchored in the towers.
4.4.1
Layout and Cross-Section
The same harp cable configuration and tower height to span ratio of 10 as the Stiff Girder Extradosed Bridge were adopted for the Stiff Tower Extradosed Bridge. The segment lengths were also maintained:
105
each cantilever consists of 10 segments of 6 m length, with a 6 m closure pour between the two cantilevers and an initial pier table segment projecting 7 m beyond the pier centreline. Preliminary Investigations Instead of starting with a cross-section as is usually done in the design of a bridge, the maximum stress range that would be permissible in a stay cable was used as the criteria to find a flexible cross-section that would keep the stress range in the cables due to live load to below 200 MPa. To accomplish this, a model of the main span girder, shown in Figure 4-9, was used to obtain forces in the cables due to the CHBDC live load. The girder is fixed in rotation at its ends to simulate the stiff piers, and is supported by springs representing the cables. Each spring coefficient was given a stiffness Ki based on the cable size and length.
Ec Ac 2 K i = T i sin ϕ i = ------------ sin ϕ i Li
Ki
Figure 4-9. Simplified model of main span used to obtain the maximum live load stress range in the cables.
As a starting point, the girder moment of inertia was assumed to be 0.10 m4 per single plane of 1916mm diametre strand cables. The resulting maximum stress range in the cables was 319 MPa, much larger than the 200 MPa limit. Thus for the same girder stiffness, a cable of 319/200 × 19 = 30 strand would be required. The next anchorage size up, for 31-16mm diametre strand cables, was selected. Based on the given anchorage size and an allowable stress of 0.45 fpu at SLS, the maximum live load was subtracted from the working load of the cable to give the force available for permanent loads. Since the barriers and asphalt are known loads, they too were subtracted leaving the force available to resist the girder’s self-weight, which corresponded to a girder cross-sectional area of around 7 m2. There is no structural advantage to selecting a girder that is lighter than this. A
A = 3.15 m2 (structural) A = 3.45 m2 (for sef-weight) I = 0.47 m4
B
A = 3.15 m2 (structural) A = 3.45 m2 (for sef-weight) I = 0.20 m4
Figure 4-10. Two basic girder cross-sections considered for the stiff tower extradosed bridge.
Two cross-sections, both with the same basic shape, were considered for the design. Each crosssection has a different depth, but the width of the edge beams was modified in order to keep the crosssectional area approximately the same, as shown in Figure 4-10. Both cross-sections were introduced into the model of the girder supported by springs, to see if there was any advantage to using the slightly stiffer cross-section (moment of inertia twice as large). For cross-section B, the stress range due to live load at SLS is 200 MPa in the most heavily loaded cable, and deflects a maximum of 198 mm (L/710). For cross-section A, the stress range is 170 MPa and the maximum deflection is 170 mm, 85% of the values for cross-section B. The maximum moment due to live load at SLS in girder A is 47% higher than B, which is approximately equal to the increase in lever arm. An equal concentric prestress would be required in either girder to oppose the stress caused by moment due to live load (-P/A + ML/Sb = 0). Based on these findings, there appears to be no advantage to
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using any stiffer cross-section than the minimum permitted by the limit on the stress range in the cables. A more detailed comparison was made with full frame models of each bridge, but led to the same conclusion. Final Girder Cross-Section The chosen girder cross-section consists of a slab, stiffened with edge beams at the interior of the cable anchorages. There are transverse ribs spanning between the edge beams spaced at 6.0 m to correspond with the segments. The depth of the cross-section is 1.0 m, which corresponds to a span to depth ratio of 140. This resulted in the cross-section in Figure 4-11, and the general arrangement for the bridge can be found in Drawings, EXTT-1.
Figure 4-11. Girder cross-section of stiff tower extradosed bridge.
At segments where extradosed cables are anchored, a single transverse 19-15 mm diametre strand tendon is draped through the rib between the edge beams. The rib is designed as a reinforced concrete beam during construction (under self-weight of the deck only) and a partially prestressed beam in the final condition. Thus, the tendons can be stressed from the fascia off the critical path.
4.4.2
Detailed Model
A plane frame model was developed for the extradosed bridge to model staged construction, secondary effects, and long-term effects. The full cross-section has been used for the calculation of gross section properties. The moments are assumed to be distributed uniformly between both edge beams, which will not in fact be the case for live load. The form traveller was included in the staged construction since it imposes a significant negative moment on the girder when it is advanced. The weight of the traveller was assumed to act 2 m beyond the previous segment and was modeled as a force couple at the two previous cable anchorages. The weight of the cast segment was assumed to be carried by the cable anchored at the end of it, as would be achieved through a cable-stayed traveller, or a traveller supported at its nose by the final cables, as is the standard for construction of flexible girders as discussed in Section 3.8. In a first attempt at staged construction, the cable prestrains were selected to balance all permanent forces at SLS. During construction, however, this resulted in positive moments as high as 4200 kNm one segment back from the cable being stressed, which would cause wide cracks in the soffit of the edge beams. The girder at the same location has a factored resistance of 5800 kNm. P-delta effects amplify the moments in the girder during construction by 5% at one segment behind the stressed cable, and by up to 30% at 30 m from the piers in the final cantilevered stage. Since each cable is prestrained too much at
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installation, the girder curls upwards, and at each stage positive moment is accummulated in the girder. These forces are locked in at closure and result in high positive moments in the permanent condition. In a second attempt at staged construction, the cables were prestrained during cantilevering to balance the self-weight only, and after closure, the cables are prestrained to balance the superimposed dead loads added after. This retensioning was sufficient to give a moment distribution similar to that of a continuous beam on simple supports at the end of construction. Similar to the case of the stiff girder extradosed bridge, the cantilevers were jacked apart 70 mm, corresponding to a force of 700 kN, to counteract the shrinkage and creep of the girder in the main span. Self Weight (SW)
Cable Prestrain (CP)
Net Permanent Loads (SW+CP+SDL)
Net Self Weight (SW+CP) After 50 years
Temperature Gradient (AASHTO)
Live Load Envelope
Temperature Differential (17˚C Stays)
Temperature Range (-46˚C)
Figure 4-12. Bending moment in stiff tower extradosed bridge (SAP2000 diagrams at the same relative scale bending moments in the tower and rigid links have not been shown for clarity in all diagrams except temperature).
Since it is known that the desired moment distribution in the girder at the end of construction is that of a continuous beam on simple supports, permanent moments at SLS were determined from the load cases applied to the continuous structure. To account for long-term effects, a nonlinear load case was created to model the permanent loads at 50 years, starting from the above permanent load condition. Long-term moments due to creep and shrinkage of the girder are small compared to live load moments. The moments induced by temperature gradient, temperature differential between stays and girder (17˚ C), and a constant temperature decrease (41˚ C) from the effective construction temperature are small in comparison to the live load and have not been included in this design. Moment diagrams for the extradosed bridge are shown in Figure 4-12.
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4.4.3
Verification at SLS and ULS
The stresses at the top and bottom of the girder were calculated under SLS forces. Although the cables provide an axial prestress to the girder that increases linearly from 0 MPa at abutments and midspan to 10 MPa at the piers, this is only sufficient to keep the girder uncracked over 20 m at either side of the pier. Full prestressing of the girder is not feasible, as revealed in the following simple calculation. Since only concentric prestressing can be used locally in the girder (eccentric prestressing causes a secondary moment as large as the primary moment), an effective prestress force after losses of 100 000 kN would be required at midspan to keep the girder uncracked. This would require 24×27-15 mm diametre strand tendons, approximately double the quantity in the stiff girder extradosed bridge at midspan. Partial prestressing was thus chosen to limit crack widths to 0.2 mm under SLS loads, as required by the CHBDC (CSA 2006a Clause 8.12.3), and consistent with the design of girder prestressing for cablestayed bridges as explained in Section 3.2.3. Envelopes of the bending moments in the girder between each pair of cables were calculated at SLS and ULS based on elastic analysis. Prestressing and reinforcing steel were provided to meet crack width limitations and provide the required bending resistance, respectively. This resulted in a maximum of 6-2×19-15 mm diametre strand tendons at midspan. The final tendon arrangement can be found in Drawings, EXTT-S2 and EXTT-S3. The longitudinal reinforcement in the edge beams is also shown since it is considerable. The cable tensions immediately after construction and after 50 years are shown in Figure 4-13. b) After 50 years
a) After construction
7
6
6
2009
2007
2005
2009
2007
2005
2003
2001
1002
0 1004
0 1006
1 1008
1
2003
2
2001
2
3
1002
3
4
1004
4
5
1006
5
1008
Cable Tension (MN)
7
1010
allowable = 0.45 A fpu = 7.8 MN
8
1010
Permanent Loads
8
Cable Tension (MN)
Live Load Positive Live Load Negative
Figure 4-13. Tension in cables of stiff tower extradosed bridge.
With the chosen span configuration and construction sequence, there is no uplift at the abutments at SLS, but a large abutment diaphragm provides additional ballast which prevents uplift at ULS.
4.5
Design Comparison
4.5.1
Girder Cross-Section
The dimensions of the cantilever constructed girder bridge cross-section are mostly governed by durability requirements, those of meeting minimum cover of reinforcement and prestressing ducts, and therefore it cannot be made much lighter.
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The extradosed girder cross-sections, on the other hand, could be made lighter. The area of stiff girder bridge cross-section could be reduced by around 10%, while that of the stiff tower bridge could be reduced by 15%. However, this would not offer any advantage with respect to structural behaviour of the cables or girder. The same size cables would still be required to meet limits on the stress range in the cables, and the magnitude of the moments in the girder would not change substantially. The bending in the girder, in the case of the stiff girder bridge, is due mostly to superimposed dead load and live load, while in the case of the stiff tower extradosed bridge, it is entirely due to live load. At some span range, there is a high premium for the dead load of the box girder in the girder bridge, which implies an economical transition to the extradosed form.
4.5.2
Material Quantities
The material quantities in the girder, for the three bridges designed in this Chapter, are summarised in Table 4-13. The reinforcing steel ratios have been calculated based on typical sections, but reinforcing steel that varies significantly by section (shear reinforcement in girder bridges and longitudinal bottom reinforcement in stiff tower extradosed bridge) was calculated separately and smeared into a typical section. A detailed breakdown of prestressing tendon lengths and sizes can be found in Appendix C. Table 4-13. Material quantities in girder and cables. Girder Bridge Cantilever Bridge Stiff Girder Stiff Tower Materials Internal Prestressing Mixed Prestressing Extradosed Bridge Extradosed Bridge* 2985 1.00 2985 1.00 2570 0.86 1825 0.66 Concrete (m3) Prestressing Steel (Mg) 161.7 1.00 146.6 0.91 178.4 1.10 159.4 1.06 Internal PT 49.1 161.7 109.8 31.1 External PT 36.3 36.8 Stays/Extradosed Cables 68.2 119.1 Transverse PT 24.9 9.2 212 Anchorages 224 432 492 1184 916 Duct Couplers 204 240 Length of Duct / Sheath 7486 7726 373 1.00 269 0.72 231 0.62 219 0.63 Reinforcing Steel (Mg) 120 90 90 120 Ratio (kg/m3) * Relative factors for the stiff tower extradosed bridge are multiplied by 1.078 to account for its shorter total length.
The mixed tendon arrangement for the cantilever constructed girder bridge uses around 90% of the prestressing and 70% of the reinforcing steel of the same bridge designed with internal tendons only. The total number of anchorages is approximately equal. The stiff girder extradosed bridge requires 10% more prestressing steel, but has only 85% of the concrete and 60% of the reinforcing steel of the girder bridge with internal tendons. The stiff tower extradosed bridge requires 6% more prestressing steel, but has only 66% of the concrete and 60% of the reinforcing steel of the girder bridge with internal tendons. The stiff girder and stiff tower extradosed
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bridges require approximately the same prestressing and reinforcing steel, but the stiff tower bridge has only 70% of the concrete of the stiff girder bridge. Table 4-14. Average material quantities in girder and cables. Materials Deck Surface Area (m²) Average Concrete thickness (m) Prestressing Steel per volume of concrete (kg/m³) per deck surface area (kg/m²) Reinforcing Steel (kg/m³)
Girder Bridge Cantilever Bridge Stiff Girder Stiff Tower Internal Prestressing Mixed Prestressing Extradosed Bridge Extradosed Bridge 3822 3822 4338 4001 0.78 0.78 0.59 0.46 54 42 120
49 38 90
59 39 90
80 41 120
The average material quantities are summarised in Table 4-14, for comparison with bridges in the Chapter 2 study. Figure 4-14 shows that these designs have relatively low average concrete depths compared with those of the study. The ratios of prestressing (per volume of concrete) in the cantilever bridges correspond very closely with Menn’s (1990) estimate of 49 kg/m³ for a 140 m span. Based on material quantities alone, the cantilever girder bridge with mixed tendon arrangement uses the least amount of prestressing, while the stiff tower extradosed bridge uses the least amount of concrete. Material quantities reflect design efficiency, but a cost estimate is needed to determine if lower material quantities translate to lower costs. Menn's Estimate
Cantilevered Regression
Average depth of concrete, m 3/m2
1.1
SETRA Estimate
Cantilever Constructed Girder Extradosed Cable-Stayed
1.0 0.9 Extradosed Regression
0.8 0.7
Cable-Stayed Regression
0.6 0.5 0.4 0.3 0
100
200
300 Longest Span, m
400
500
Figure 4-14. Average girder concrete thickness of Chapter 4 bridge designs compared with Chapter 2 study bridges.
4.5.3
Cost Comparison
The combined cost of cables and prestressing is expected to account for a considerable portion of the total cost of the structure, up to 25% of the total cost of each bridge based on published cost breakdowns of similar sized structures. Similarly, the total cost of the prestressing, girder concrete and reinforcing steel could amount to 60% of the total cost of the structure. Given that the total quantities of prestressing steel do not vary significantly between the designs, the costs have been broken down according to tendon type. The assumed costs per unit of concrete, prestressing, and reinforcing steel are given in Table 4-15, and were determined based on costs of recent
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cast-in-place concrete structures built in Ontario. At first glance, it seems unusual that the extradosed cables would cost more than the stay cables, but there is a reason for this. The cost of the stay/extradosed cables was calculated based on estimates of individual components of the system, and the supply and delivery, engineering, and installation are assumed to add an additional 40 to 45% of the cost of the materials alone. While the cost of strand is proportional to the total weight, the cost of the ducts and connections are mostly influenced by cable length. Cost estimates for each bridge are given in Table 4-16. Table 4-15. In-place costs of materials in girder and cables. Materials Concrete Stay Cables Extradosed Cables Internal Longitudinal Tendons External Longitudinal Tendons Internal Transverse Tendons Reinforcing Steel
Unit m³ Mg Mg Mg Mg Mg Mg
Unit Cost ($) 1000 13100 13800 6500 8700 10000 4000
Table 4-16. Cost estimate of bridges (costs in $1000). Component
Girder Bridge Cantilever Bridge Stiff Girder Stiff Tower Internal Prestressing Mixed Prestressing Extradosed Bridge Extradosed Bridge* Concrete 2985 2985 2571 1825 Stay/Extradosed Cables 941 1557 Internal Longitudinal Tendons 1051 713 319 202 External Longitudinal Tendons 320 316 Internal Transverse Tendons 249 92 Reinforcing Steel 1493 1075 926 876 Total 5530 5090 5010 4550 Cost per m² of Effective Deck* 1450 1330 1310 1285 Relative Cost 1.13 1.04 1.02 1.00 * Effective deck area is calculated based on the travelled lanes and barriers.
Based on this simple estimate, both extradosed girders appear to have a lower cost than the girder bridges. This estimate is most sensitive to the cost of the concrete. If the cost of concrete increases by 50% to $1500/m³, the girder is 10% more expensive than the siff tower extradosed bridge. If on the other hand the cost of concrete decreases by 100% to $500/m³, then the stiff girder extradosed bridge is 4% more expensive than the girder bridge. In all cases, the girder bridge with internal tendons is the most expensive. These estimates do not take the cost of the tower and piers into consideration. Twin piers are more expensive than single piers, and although the cost of short towers is not considerable, these two factors are likely to make the girder bridges more economical than the extradosed bridges.
5
CONCLUSIONS
This chapter summarises the conclusions of the preceeding chapters, highlights key factors which must be considered in the design of an extradosed bridge, and suggests future studies.
5.1
Review of Extradosed Bridges
The study of existing extradosed bridges in Chapter 2 reveals that extradosed bridges have been built for spans from 66 to 275 m, and there is a large variability in the proportions adopted. Span to depth ratios of between 30 and 35 are most common at the piers, while span to depth ratios of up to 60 are common at midspan. As the main span length increases, designers have opted for variable depth cross-sections with larger span to depth ratios, and a larger ratio of pier depth to midspan depth. It is clear from the litterature that the term ‘extradosed bridge’ is used to describe a cable-stayed bridge with a stiff girder that carries live load through flexural behaviour. Bridges have been designed with low towers but flexible girders and stiff piers, and they rely on the axial force couple between stays and girder to carry live load. In this thesis, they are referred to as stiff tower extradosed bridges.
5.2
Design Considerations
Loads that must be considered for the design of an extradosed bridge, as determined from Chapter 3, are live load, temperature gradient in the girder, and temperature differential between girder and stays. Stays should be of light colour to minimize temperature differential. Central suspension is preferable as it leads to a lower total live load in the cables. Extradosed bridges, like cable stayed bridges, should be designed with an allowable cable stress at SLS, and then verified at FLS and ULS. The live load stress range in an extradosed cable is the primary factor used to set the allowable stress range at SLS that can be used to produce a design that will satisfy FLS and ULS requirements. The stress range due to live load in the cables is affected by the girder stiffness and the fixity of the support on the piers. When the girder is stiff, the stress range in the cables due to live load will be small in comparison with permanent loads. To reduce the magnitude of this stress range, the girder should be fixed at the piers unless foundations or thermal movements dictate otherwise. Span to depth ratios should be chosen to limit the stress range to a value that is acceptable for the type of anchorage chosen. Recommendations for span to depth ratio can only be considered rough guidelines, as they are specific to the amount of live load for which they have been developed. As the stress range due to live load in the cables is the primary concearn with extradosed bridges, the live load model has a significant effect on the design. Cable pretensions can be determined by one of the commonly used methods for cable-stayed bridges, to achieve a moment distribution close to that of a continuous girder on simple supports at cable locations. Retensioning to adjust the moment distribution before or after superimposed dead load is applied can be avoided for a stiff girder extradosed bridge, but not for a stiff tower extradosed bridge.
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5.3
Comparison between Extradosed and Cantilever Constructed Girder Bridges
A design comparison in Chapter 4 of a stiff girder extradosed bridge, a stiff tower extradosed bridge, and two cantilever constructed girder bridges (one with internal tendons and one with internal and external tendons) found that extradosed bridges require a comparable amount of prestressing as in a girder bridge, but a reduced quantity of concrete. A cost estimate found that neglecting the costs of towers, the superstructure cost of an extradosed bridge is on par or less than that of a girder bridge. Since concrete accounts for a significant portion of the superstructure cost, the extradosed bridges are at an advantage as the span increases. Already for a span of 140m, an extradosed bridge is a competitive bridge form. If side spans of longer than 50% of the main span area required, a stiff tower extradosed bridge cannot be used. In the case of the stiff girder extradosed bridge, creep and shrinkage in the concrete and relaxation in the extradosed cables cause long term moments in the girder at midspan of similar magnitude to the moment due to live load. Overall, the stiff girder form of extradosed bridge does not offer any huge advantage over the stiff tower form, other than its ability to span multiple piers on simple supports. There are different approaches to designing extradosed bridges. The design may start with a specific cross-section that is already in use elsewhere on the project, in which case the designer would like to find an optimal extradosed solution for a given girder stiffness. If, however, the design is not constrained, the choice of girder stiffness should be made for other reasons than that of economy alone, since an extradosed bridge will be competitive. The cost of the cables may be largely determined by the stress range, as this will determine the type of anchorages required.
6
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Iida, J., Nakayama, H., Wakasa, T., Kasuga, A. and Mizuno, K. (2002). “Design and Construction of ShinMeisei Bridge.” Proceedings of the 1st fib Congress: Concrete Structures for the 21st Century, Osaka, Japan. Ishii, M. (2006). “Sistemas estruturais de pontes extradorso” thesis, presented to Departamento de Engenharia de Estructuras e Geotécnica - Escola Politécnica da Universidade de São Paulo in partial fulfillment of the requirements a Master degree. Janjic, D., Pircher, M., Pircher, H. (2003). “Optimization of Cable Tensioning in Cable-Stayed Bridges.” Journal of Bridge Engineering, 131. Jaques, J.A.L. and Fort, J.R. (2005). “Puente sobre el río Deba (Autopista Vitoria - Eibar).” Revista de Obras Públicas, 3459, 71-74. Jordet, E. and Svensson, H. (1994). “Helgeland Bridge, A Slender Concrete Cable-Stayed Bridge Located in A Severe Environment.” International conference A.I.P.C.--F.I.P. : ponts suspendus et à haubans : cable-stayed and suspension bridges. Deauville, October 12-15, 1994. Bagneux, France. Kasuga, A., Shirono, Y., Nishibe, G. and Okamoto, H. (1994). “Design and construction of the Odawara Port Bridge - the first extradosed prestressed bridge.” FIP - XIIth International Conference, May 29th - June 2, Washington. Kasuga, A. (2003). “Extradosed Bridges in Japan.” Role of Concrete Bridges In Sustainable Development, Proceedings of the International Symposium dedicated to Professor Jiri Strasky, University of Dundee, Scotland, 17-30. Kasuga, A. (2006). “Extradosed bridges in Japan.” Structural Concrete, 7(3), 91-103. Kato, T., Norio, T., Fukunaga, Y. and Uehira, Y. (2001). “Design and Construction of Extradosed Bridge Miyakodagawa Bridge - in the New Tomei Expressway.” Cable-Supported Bridges - Challenging Technical Limits, IABSE Conference, Seoul 2001. Korean Group of IABSE and Korea Highway Corporation. Kikuchi, H., Monichiro, T. and Nakamura, T. (2002). “Construction of the Lao-Nippon Bridge over the Mekong in Laos.” Proceedings of the 1st fib Congress: Concrete Structures for the 21st Century, Osaka, Japan. Komiya, M. (1999). “Characteristics and Design of PC Bridges with Large Eccentric Cables.” Extradosed Bridge Technology in Japan and the New Pearl Harbor Memorial Bridge. Federal Highway Administration / U.S. Department of Transportation and The Connecticut Department of Transportation, Washington, DC. 55-80. Kumarasena, Sena (2005). “A Comparison of Cable-Stayed vs. Extradosed Bridges.” Sixth International Bridge Engineering Conference: Reliability, Security, and Sustainability in Bridge Engineering. July 17-20, 2005. Kutsuna, Y., Kasuga, A., Morohashi, A. (2002). “Non-linear Beahior of the Ibi River Bridge under Ultimate Loads.” Proceedings of the 1st fib Congress: Concrete Structures for the 21st Century, Osaka, Japan. Lacaze, J-J. (2002). “Autoroute A90 - Le viaduc du Pays de Tulle.” Travaux, 782.
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DRAWINGS
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130
131
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133
APPENDIX A Chapter 2 Supplementary Information
134
135 Table A-1. Extradosed Bridges in Chapter 2 Study. Name and Location
Operational -Girder Depth h x Width w Date -Span Lengths
Longest Tower L:H L:h Span Height mid
L:h pier
Ac pier
Ac mid
Ixx mid
Ixx Pier Box Central Dwg in pier Fix. Gird. Susp. Fig 2-1
Brief Description
Reason for Selection of Extradosed Bridge Type
Wide single cell concrete box girder, cable-panel stayed.
Visual strength, curved roadway.
174 14.9 11.7 69.6 34.8 7.17 11.6 7.29
45
Vogel & Marti 1997, Kasuga 2006
Single cell concrete box girder with voided webs and struts supporting deck cantilevers.
Economy in materials.
100
12 16.1
26
Mathivat 1986, Mathivat 1988, Virlogeux 2002a
9.7 7.17 11.6 8.15
82
Gee 1990
30.3 30.3 12.3 12.3 19.7
20
Reis & Pereira 1994
26
Taniyama et al. 1994, Kasuga et al. 1994
Sources
1
Ganter Bridge, Switzerland
2
Arrêt-Darré Viaduct, France
3
Barton Creek Bridge, Austin, USA
1987 3.7 - 10.7 x 17.7 47.6 + 103.6 + 57.9
Single cell concrete box girder with webs inclined inwards into a central fin above the deck level, and transverse struts supporting the deck slab.
Visual signficance from road marking entrance into Estates.
4
Socorridos Bridge, Madeira, Portugal
Single cell concrete box girder, cable-panel stayed.
Tall piers.
5
Odawara Blueway Bridge, Japan
1993 3.5 x 20 54 + 85 + 106 + 86 1994 2.2 - 3.5 x 13 73.3 + 122.3 + 73.3
Wide double cell concrete box girder.
Navigational clearance, height restriction from airport.
6
Saint-Rémy-de-Maurienne Bridge, Savoie, France
1996 2.2 x 13.4 52.4 + 48.5
U shaped concrete deck with transverse ribs between edge beams.
Shallow clearance over roadway.
7
Tsukuhara Bridge, Japan
Wide single cell concrete box girder.
Fit with adjacent CS pedestrian bridge.
8
Kanisawa Bridge, Japan
9
Shin-Karato Bridge, Kobe, Japan
1997 3 - 5.5 x 12.8 65.4 + 180 + 76.4 1998 3.3 - 5.6 x 17.5 99.3 + 180.0 + 99.3 1998 2.5 - 3.5 x 11.5 74.1 + 140.0 + 69.1
Two and three cell concrete box girder.
Shallow depth girder spans over unstable slope.
140
12 11.7
10 Sunniberg Bridge, Switzerland
1998 1.1 x 12.375 59.0 + 128.0 + 140.0 + 134.0 + 65.0
Concrete slab with edge stiffening beams.
Tall piers, emphasis on aesthetics.
140
15
11 Santanigawa (Mitanigawa) Bridge, Japan
1999 2.5 - 6.5 x 20.4 57.9 + 92.9 1999 3.3 - 5.1 x 18 111.5 + 185.0 + 111.5
Double cell concrete box girder.
2000 3.5 - 6 x 11.3 109.3 + 89.3 2000 3 - 6.5 x 13.8 52.0 + 123.0 + 143.0 + 91.5 + 34.5 2000 2.1 - 3.2 x 11 60.8 + 105.0 + 57.5 2000 3 - 6 x 23 94 + 3 x 140 + 94 2000 2.8 - 5 x 9.2 84.82 2000 15 x 23.4 180 + 312 + 180
Single cell concrete box girder.
19 Yukisawa-Ohashi Bridge, Japan
2000 2 - 3.5 x 15.8 70.3 + 71.0 + 34.4
Two cell concrete box girder with wide sidewalks on deck cantilever overhangs outside of cable planes.
107 11.5
9.3 53.5 30.6 10.7 17.1 6.09
31
Nunoshita et al. 2002, Kasuga 2006
20 Hozu Bridge, Japan
2001 2.8 x 15.3 33 + 50 + 76 + 100 + 76 + 31 2001 4.3 - 7.3 x 33 154 + 4 x 271.5 + 157
Single cell concrete box girder.
100
10
10 35.7 35.7 11.6 14.6 12.2
17
Sumida et al. 2002, Kasuga 2006
Hybrid cross section: four cell concrete box girder near piers and steel Economy, heavy prefabrication. box girder in central 100 m with moment and shear connection.
271.5
30
22 Kiso River Bridge, Japan
2001 4.3 - 7.3 x 33 160 + 3 x 275 + 160
Hybrid cross section: four cell concrete box girder near piers and steel Economy, heavy prefabrication. box girder in central 100 m with moment and shear connection.
275
30
23 Miyakodagawa Bridge, Japan
2001 4 - 6.5 x 19.9 134 + 134
Parallel double cell box concrete box girders.
214
24 Nakanoike Bridge, Japan
2001 2.5 - 4 x 21.4 60.6 + 60.6 2002 2.5 - 3 x 13.7 62.1 + 90.0 + 66 + 45.0 + 29.1 2002 3.5 - 7 x 11.6 82 + 247 + 82
12 Second Mandaue - Mactan (Marcelo Fernan) Bridge, Mactan, Philippines
13 Matakina Bridge, Nago, Japan 14 Pakse (Lao-Nippon) Bridge, Laos 15 Sajiki Bridge, Japan 16 Shikari Bridge, Japan 17 Surikamigawa Bridge, Japan 18 Wuhu Yangtze River Bridge, Wuhan, China
21 Ibi River Bridge, Japan
25 Fukaura Bridge, Japan 26 Korror Babeldoap Bridge, Palau
1980 2.5 - 5 x 10 127 + 174 + 127 Proposed 3.75 x 20.5 60 + 4 x 100 + 52
Units of m
Concrete box girder.
Three cell concrete box girder.
Single cell concrete box girder.
8 12.5 26.7 26.7
103.6
28
106
122.3 10.7 11.4 55.6 34.9 80 180
5.9 13.6 36.4 36.4 16 11.2
180 22.1
115 12.8 Height restriction from airport.
Long navigational span of long viaduct.
Concrete box girder.
Navigation clearance, height restriction from airport.
Landmark structure with good seismic resistance.
8.7 2.49 2.5
60 32.7 7.87 17.7 10.4
73
40 8.57 13.8 7.57
9.3 127 127 6.14 9
Ogawa et al. 1998 Kasuga 2006
8.1 54.5 32.1 56
Grison & Tonello 1997, Kasuga 2006
23
8.3 0.25 0.5
46 17.7
38.8 180
Tomita et al. 1999 Figi et al. 1997, Figi et al. 1998, Menn 1998, Baumann & Däniker 1999 Nishimura et al. 2002, Stroh et al. 2003
160 26.4
6.1 45.7 26.7
Kasuga 2006
143
9.5 47.7
15
22 7.74 12.8 10.4
90
Nakamura 2001, Kikuchi et al. 2002, Kasuga 2006 Kasuga 2006
105 12.3
8.5
140
14 46.7 23.3
Stroh et al. 2003
5.2 30.4
Kasuga 2006
312
90 Navigational clearance, height restriction from airport.
8.7
6.4
Kasuga 2006
10
33
247
50 32.8
17
Fang 2004
9.5 20.8 20.8
32 49.9 243
Hirano et al. 1999, Casteleyn 1999, Kutsuna et al. 2002, Kasuga 2006
64 37.7 23.1
32 49.9 243
Hirano et al. 1999, Casteleyn 1999, Kasuga 2006
20 10.7 53.5 32.9 16.3
32 39.8 193
Kato et al. 2001, Terada et al. 2002
97 11.8
Hybrid cross section: wide single concrete box girder near piers and steel box girder in central 82 m.
9 15.4
185 18.3 10.1 56.1 36.3
85 16.5 Double-decker steel truss with composite deck slab on top roadway, two rail lines on bottom level.
9
9 63.1 37.2 23.1 9.2
8.5 10.6 27
Kasuga 2006
8.2 38.8 24.2 36
Kasuga 2006
30
9.1 70.6 35.3 10.5 16.6
20 122
Oshimi et al. 2002, Ewert 2003
136 Table A-1. Extradosed Bridges in Chapter 2 Study (continued). Name and Location
Operational -Girder Depth h x Width w Date -Span Lengths
27 Sashikubo Bridge, Japan 28 Shinkawa (Tobiuo) Bridge, Hamamatsu, Japan
29 Deba River Bridge, Gipuzkoa, Spain 30 Himi Bridge, Japan 31 Korong Bridge, Budapest, Hungary
32 Shin-Meisei Bridge, Japan 33 Tatekoshi Bridge, Japan 34 Sannohe-Boukyo Bridge, Aomori, Japan 35 Domovinski Bridge over the River Sava, Croatia
36 Kack-Hwa First Bridge, Gwangju, South
38 Rittoh Bridge, Japan 39 Tagami Bridge, Japan 40 Third Bridge over Rio Branco, Brasil
Brief Description
Reason for Selection of Extradosed Bridge Type
Longest Tower L:H L:h Span Height mid
L:h pier
Ac mid
Ac pier
Ixx mid
2002 3.2 - 6.5 x 11.3 114 + 114 2002 2.4 - 4 x 25.8 38.5 + 45.0 + 90.0 + 130.0 + 80.5
Concrete box girder.
185
22
8.4 57.8 28.5
Three cell concrete box girder.
130
13
10 54.2 32.5 16.9 21.7 12.5
2003 2.7 x 13.9 42 - 66 - 42 2004 4 x 12.45 91.8 + 180.0 + 91.8 2004 2.5 x 15.85 52.26 + 61.98
U shaped concrete deck with transverse ribs between edge beams.
2004 3.5 x 19 89.6 + 122.3 + 82.4 2004 1.8 - 2.9 x 19.14 56.3 + 55.3 2005 3.5 - 6.5 x 13.45 99.9 + 200.0 + 99.9
Three cell concrete trapezoidal box girder.
2006
- x 31.1 55 + 115 + 100
2006 2.6 - 3.5 x 20.55 68.1 + 110.0 + 68.1 2006 4.5 - 7.5 x 19.6 140 + 170 + 115 + 70 (Tokyo bound) 2006 3 - 4.5 x 17.8 80.2 + 80.2 2006 2 - 2.5 x 17.4 54 + 90 + 54
Clearance under bridge.
Single cell doubly composite box girder with corrugated steel webs. Three cell concrete box girder stiffened with transverse ribs, on 40 deg skew. Barriers form structural edge beams, deck is supported by 3 cables deviated on girder, anchored at abutments.
66
Concrete box girder.
45
90 9.45
9.5
36
36 12.4 12.4 5.19 5.2
200
25
Long span of viaduct to cross the river.
120
12
8.6 12.4 24.1
7.4 34.9 34.9 12.3 16.4 17.2
Deck slab with L shaped edge beams (appears as single box girder with incomplete bottom slab) that taper to I beams at midspan.
Kasuga 2006
10 33.8 33.8 27.2 27.2 41.1
11
50
30 11.5 20.1
45 200
7.4
128 14.5
8.8 42.7 28.4
Kasuga 2006 Ishii 2006
9.2
55 33.8
210
24
8.8 52.5 32.3
Wide single cell concrete box girder.
110
15
7.3 46.8 32.8 7.78 12.5 7.55
Mulitple cell concrete box girder.
125 8.85 14.1
2007 3.5 - 4 x 23.5 65 + 120 + 65
Four cell concrete box girder.
120 10.5 11.4 34.3
2007 3.5 x 28.6 55 + 7 x 110 + 55
Wide single cell trapezoidal box girder with internal struts (Bang Na cross section).
2008
- x 14 70 + 3 x 130 + 70 2008 3.4 x 10.31 139 + 180 + 139
Multiple cell concrete box girder.
130
Single cell concrete box girder for LRT, precast segmental construction. Navigational clearance, height restriction from airport.
180
22
49 Trois Bassins Viaduct, Reunion, France
2008 4 - 7 x 22 18.6 - 126.0 - 104.4 - 75.6 - 43.2
Single cell concrete box girder with steel struts supporting long deck cantilevers.
Tall piers, access from one side of gorge only.
200
19 10.5
50 Golden Ears Bridge, Canada
2009 2.7 - 4.5 x 31.5 121 + 3 x 242 + 121
Steel box girders at edge of deck with transverse floor beams composite with precast concrete deck.
Most economical for span length.
242
40
51 Pearl Harbor Memorial (Quinnipiac)
2012 3.5 - 5 x 33.7 75.9 + 157.0 + 75.9
Parallel five cell concrete box girders with inclined exterior webs.
Wide navigational clearance, height restriction from airport.
157 22.6
45 Pyung-Yeo 2 Bridge, Yeosu, South Korea
46 Second Vivekananda Bridge over the Hooghly River, Calcutta, India
47 Cho-Rack Bridge, Dangjin, South Korea 48 North Arm Bridge (Canada Line Extradosed Transit Bridge), Canada
Bridge, New Haven, USA
Navigational clearance, height less than nearby temple.
Wilcox et al. 2002, Yasukawa et al. 2002, Masterson 2004
225 30.5
24
Chungcheongnam-do, South Korea
Balić & Veverka 1999
Kasuga 2006
10 42.3 31.4
220
44 Gum-Ga Grand Bridge,
41
Structurae
45
43 Brazil-Peru Integration Bridge, Brazil
Iida et al. 2002, Kasuga 2006
8 57.1 30.8
7.5
42 Yanagawa Bridge, Japan
Becze & Barta 2006
Kasuga 2006
50
12
2006 4 - 6.5 x 17.4 139.7 + 220.0 + 139.7 2006 4 - 6.5 x 17.4 130.7 + 130.7 2007 2.35 - 3.35 x 16.8 65 - 110 -65 2007 - x 23 85.4 + 5 x 125.0 + 85.3
26
Kasuga 2006
31
8.6
90
41 Tokuyama Bridge, Japan
34
115
Gateway structure to reflect cultural context of Kansai District.
Kasuga 2006
Jaques 2005
45
110 Three cell doubly composite box girder with corrugated steel webs. Towers are shaped to reflect a japanese crane in flight.
51
9.1
Span of 200 m to cross protected River and train line.
Sources Kasuga 2006
180 19.8
122.3 16.5
Multiple cell concrete box girder.
Ixx Pier Box Central Dwg in pier Fix. Gird. Susp. Fig 2-1
24.4 24.4
90 10.5
Five cell concrete box girder supports light rail between cable planes. 2006 3.55 x 34 48 + 6x60 + 72 + 120 + 72 + 2x60 + 48
Korea
37 Nanchiku Bridge, Japan
Units of m
110
14
36 6.25 9.85 2.34 9.2
Stroh et al. 2003 Kasuga, 2006
23
Structurae Structurae
30 19.5
29 36.4
63
Masterson 2006
Binns 2005
7.9 31.4 31.4
Structurae
8.2 52.9 52.9
6.3
8.1 10.6
15
50 28.6 12.4 18.6 26.7 137
Frappart 2005, Boudot et al. 2007 Trimbath 2006, Bergman et al. 2007
6 89.6 53.8 6.9 44.9 31.4 24.5 27.5 42.3
Griezic 2006
95
Stroh et al. 2003
137 Table A-2. Contilever Constructed Girder Bridges in Chapter 2 Graphs. Name and Location
Operational -Girder Depth h x Width w (m) Date -Span Lengths (m)
Girder Description
Tendon Description
Single cell box girder with wide deck cantilevers partially prestressed.
1
Felsenau Bridge
1975 3 - 8 x 26.2 38 + 5x48 + 94 + 12 + 144 + 12 + 144 + 12 + 94 + 6x48 + 38
2
Chinon Bridge, Indre-et-Loire
1984 2.8 - 4.5 x 10.9 48 + 70 + 48
3
Nantura Viaduct (A40)
Single cell box girder.
internal PT using 12T15 tendons
4
Poncin Viaduct (A40)
1986 3 - 6.65 x 12.15 124.3 + 113.5 + 113.5 + 104 + 93.7 + 93.1 + 91.7 + 90.1 + 88.4 + 54.9 1986 4 - 10 x 19.6 40 + 70 + 79 + 117 + 155 + 105
Single cell box girder.
5
Tacon Viaduct
6
Beaumont sur Oise bridge
1986 3 - 6 x 19.5 25 + 40 + 55 + 90 + 80 + 30 1988 3 - 6.6 x 10.7 30 + 120 + 30
7
Bridge over the Loch d'Auray (Kerplouz Bridge)
1989 2.5 - 5.2 x 20.4 46 + 84 + 51 + 39 + 39 + 34.8
8
Champ du Comte Viaduct (RN90)
1989 2.9 - 5.8 x 19.1 60 + 5x100 + 60 + 60 + 3x100 + 60
Single cell box girder, parallel structures.
9
Doubling of the General Audibert Bridge over the Loire, Nantes
1989 1.8 - 3.8 x 18 51 + 67 + 45
Parallel single cell box girders connected to 18 12T13 tendons m total width.
Total Spans
Cant Spans
Cant Longest Length Span (m)
17
4
512
144
3
3
166
10
10
cantilever PT: 2x27 - 19T15 tendons continuity PT: 2x7 - 12T15 tendons external PT: 2x10 - 19T15 tendons
6
Single cell box girder, parallel structures.
internal PT using 12T15 tendons
Single cell box girder.
L:h mid
hpier: Q Conc Q PT Q Q PT Sources hmid (m³) Long (t) Trans (t) Reinf (t) Menn 1990
18
2.7
10130
0
0
0
70
25 15.6
1.6
1385
50
0
205
SETRA 2007
967
113.5
37.8 17.1
2.2
9800
430
0
1250
SETRA 2007
3
365
155
38.8 15.5
2.5
11270
516
77
1267
SETRA 2007
6
2
170
90
15
2
4820
185
0
554
SETRA 2007
9T15 and 12T15 internal tendons and 19T15 external tendons. Short end spans counterweighted with 9 m long solid concrete.
3
3
180
120
40 18.2
2.2
1793
73
0
226
SETRA 2007
cantilever PT: 12T15 tendons continuity PT: 2x6 - 12T15 tendons external PT: 2x2 - 12T15 tendons
6
3
0
84
33.6 16.2
2.1
0
0
0
0
SETRA 2007
12
12
1040
100
34.5 17.2
2
17000
850
0
3800
SETRA 2007
3
3
163
67
37.2 17.6
2.1
2150
85
0
338
SETRA 2007
cantilever PT: 2x9 - 12T15 tendons continuity PT: 2x4 - 12T15 tendons external PT: 2x7 - 12T15 tendons
cantilever PT: 2x13 - 12T15 and 2 x 1 -19T15 tendons continuity PT: 2x5 - 12T15 tendons external PT: 2x3 - 19T15 and 2 x - 12T15
48
L:h pier
30
10 Bourran Viaduct at Rodez
1991 3 - 6 x 12 44 + 75 + 100 + 74 + 29
cantilever PT: 2x16 - 12T15 tendons continuity PT: 2x3 - 12T15 tendons external PT: 2x4 - 19T15 tendons
5
3
0
100
33.3 16.7
2
3000
119
0
365
SETRA 2007
11 Bridge over the Saint-Denis River,
1991 2.5 - 4.5 x 16.8 50 + 76 + 76 + 50
cantilever PT: 2x14 - 12T15 tendons continuity PT: 2x2 - 12T15 tendons external PT: 2x6 - 19T15 tendons
4
4
254
76
30.4 16.9
1.8
3100
166
0
437
SETRA 2007
5
5
568
169
48.3 18.8
2.6
13900
969
0
1711
SETRA 2007
cantilever PT: 20 to 30 - 12T15 tendons continuity PT: 2 to 14 - 12T15 tendons external PT: 8 to 12 - 19T15 tendons
7
7
0
90
17
2
4680
222
0
769
SETRA 2007
cantilever PT: 12T15 tendons continuity PT: 12T15 tendons external PT: 19T15 tendons
8
3
290
136
42.5 20.9
2
10300
585
0
1520
SETRA 2007
cantilever PT: 2x35 - 19T15 tendons continuity PT: 2x7 side; 2x2 main - 19T15 tendons external PT: 19T15 tendons
3
3
308
144
48
18
2.7
362
5
10350
SETRA 2007
Reunion
12 Bridge over the Seine, Gennevilliers 13 Limay Viaduct, Yvelines
14 Auxonne and Maillys Viaducts, Cote d'Or
15 Bridge over Truyère at Garabit, Garabit
1992 3.5 - 9 x 18.06 110 + 169 + 96 + 169 + 114 1992 2.7 - 5.3 x 14.2 50 + 90 + 75.2 + 59.5 + 75.2 + 90 + 58
Single cell box girder.
1993 3.2 - 6.5 x 24 52 + 77 + 136 + 77 + 55 + 55 + 55 + 45
Parallel single cell box girders.
1993 3 - 8 x 20.5 82 + 144 + 82
33.3
16 Piou Viaduct
1994 5 x 21 45 + 81 + 90 + 84 + 72 + 42
Single cell box girder with strutted deck cantilevers.
6
6
0
90
18
18
1
6500
228
92
819
SETRA 2007
17 Corniche Bridge, Dole
1995 2.5 - 5.5 x 14.5 45 + 5 x 80 + 48
Single cell box girder with corrugated steel webs.
7
7
496
80
32 14.5
2.2
4100
190
0
0
SETRA 2007
18 Rivoire Viaduct (A51), Isère
1996
- x 20 64 + 113 +70 1997 4.5 - 12 x 19 50 + 70 + 130 + 190 + 132
Concentric twin single cell box girder.
3
3
0
113
4000
174
0
690
SETRA 2007
5
3
0
190
42.2 15.8
2.7
650
0
3900
SETRA 2007
2002 4.5 - 9 x 14.75 121 + 205 + 131 2002 4 - 10 x 19.3 68.3 + 123.7 + 180 + 10.8 + 180 + 10.8 + 180 + 100.1
Single cell box girder.
3
3
457
205
45.6 22.8
2
7750
598
0
913
SETRA 2007
Single box girder with transverse ribs at 3.533 m spacing.
6
5
786
190.8
47.7 19.1
2.5
17500
930
70
2500
Lacaze 2002
19 Viaur Valley Viaduct
20 Bridge over the Rhine, Strasbourg 21 Tulle Viaduct, Ville de Tulle, A89, Correze
cantilever PT: 19T15 tendons continuity PT: 19T15 tendons external PT: 27T15 tendons
138 Table A-3. Cable-Stayed Bridges in Chapter 2 Graphs. Name and Location
Operational -Girder Depth h x Width w (m) Date -Span Lengths (m)
Units of m
Girder Description
Stayed Spans
Stayed Span
Total Length
L:h mid
Ac mid
Ac pier
Ixx mid
Ixx pier
Q Conc Effective (t) Depth
Sources
1
Brotonne Bridge, France
1977 3.8 x 19.2 58.5 + 58.5 + 143.5 + 320 + 143.5 + 70 + 55.5 + 30
Single cell box girder, centrally suspended with prestressed diagonal ties to webs.
3
320
880
84
9.46 12.61 20.8 31.1
8534
0.505
Mathivat 1983
2
Ed Hendler Birdge (PascoKennewick), Washington
1978 2.15 x 24.33 39.93 + 123.9 + 299 + 123.9 + 45.2 + 45.2 + 45.2
Slab supported on transverse ribs between triangular edge box girders, lateral suspension.
3
299
722
139
11.3 11.65 2.55 2.55
8180
0.465
Mondorf 2006
3
Diepoldsau Bridge, Switzerland
1985 0.55 x 14.5 15 + 18 + 19.5 + 40.5 + 97 + 40.5 + 19.5
Solid slab laterally suspended.
3
97
250
176
6.33
0.13
1583
0.437
Walther et al. 1999
4
East Huntington Bridge, West Virginia
1985 1.53 x 12.6 274.3 + 185.3
Slab supported on transverse steel ribs between concrete edge beams.
2
460
460
301
6.2
1.41
2850
0.492
Walther et al. 1999
5
Sunshine Skyway, St. Petersburg, Florida
1987 4.3 x 28.85 164.6 + 365.8 + 164.6
Single cell box girder, centrally suspended with prestressed diagonal ties to webs.
3
366
695
85
16.1
38.8
11190
0.558
Gimsing 1997
6
Akkar Bridge, Sikkim, India
1988 0.8 x 11.1 77 + 77
Slab supported on transverse ribs between concrete edge beams.
2
154
154
192
3.74
0.08
576
0.337
Schlaich 1991
7
Dame Point Bridge, Jacksonville, Florida
1989 1.5 x 32.3 198 + 396 + 198
Slab supported on transverse ribs between concrete edge beams.
3
396
792
264
15.23
2.29
12062
0.472
Gimsing 1997
8
Chandoline Bridge, Sion, Switzerland
1990 2.5 x 27 72 + 140 + 72
Single cell box girder, centrally suspended with prestressed diagonal ties to webs and struts supporting wide cantilever overhangs.
3
140
284
56
11.3
8
3209
0.419
Menn 1990
9
Skarnsundet Bridge, Norway
1991 1.2 x 13 20 + 3 x 27 + 190 + 530 + 190 + 3 x 27 + 4 x 20
Two cell triangular box girder, laterally suspended.
3
530
530
442
8.53
16.1
4521
0.656
Hansvold 1994
10 Helgeland Bridge, Norway
1991 2.1 x 11.95 177.5 + 425 + 177.5
Slab supported on transverse ribs between concrete edge beams.
3
425
780
202
7.98
0.99
6224
0.668
Jordet & Svensson 1994
11 Burgundy Bridge, Chalon-sur-Soane,
1992 1 x 15.54 17 + 27.6 + 44.85 + 151.8 + 44.85 + 27.6 + 21.1 + 16.7
Slab supported on transverse ribs between concrete edge beams, with sidewalks on cantilever overhangs from beam soffit.
3
152
352
152
6.28
0.58
2207
0.404
Virlogeux et al. 1994b
12 Evripos Bridge, Euboea, Greece
1992 0.45 x 14.1 90 + 215 + 90
Solid slab of constant depth laterally supported.
3
215
395
478
6.35
0.11
2508
0.450
Bergermann & Strathopulos 1988
13 Quetzalapa Bridge, Mexico
1993 1.6 x 21.4 11 + 94.5 + 213 + 94.5 + 11
Slab supported on transverse ribs between precast concrete edge beams.
3
213
424
133
11.04
2.31
4681
0.516
Revelo et al. 1994
14 Cheasapeake & Delaware Canal
1995 3.65 x 38.8 45.72 + 45.72 + 45.72 + 228.6 + 45.72 + 45.72 + 45.72
Parallel precast box girders supported on centrally suspended delta frame.
3
229
503
63
19
30
9555
0.490
Goni et al. 1999
France
Bridge, Delaware
Girder Cross-Section
APPENDIX B Chapter 4 Supporting Calculations
139
140 Cantilever Constructed Girder Bridge Preliminary PT Design A - Mixed Tendons Side Span Length Half Span Length Segment Length Closure Segment PT Cover Long PT Trans PT L&T Reinf
84 70 3.42 2
m m m m
Deck width
12.41 m
Top of Top Bot of Top Ext side Int side Top of Bot Bot of Bot 130 60 60 60 90 60 70 40 70 60 40 40 Comp at transfer fci = Comp limit 0.6fci = Tensile limit 0.5fcri =
30 MPa 18 MPa 1.10 MPa
Comp comp at SLS f'c = Comp limit 0.6f'c = Tensile limit fcr =
50 MPa 30 MPa 2.83 MPa
Section Properties Variable h g Ag yt yb I St Sb ρ
For PT design For +ve M Check midspan at pier side span max LL 3.2 8 3.68 m 0.25 1 m 2 7.935 13.459 8.65 m 1.312 3.719 1.6035 m 2.003 4.281 2.0765 m 4 11.9 127.2 18 m 3 9.07 34.20 11.23 m 3 5.94 29.71 8.67 m 0.571 0.594 0.625
Description height of cross-section thickness of bottom slab Area distance from cgc to top distance from cgc to bottom Moment of Inertia
geometric output
Moment on pier during the casting of the final segment Weight Lever Arm Mg = 13414.1 x 34.5 = 3112.8 x 23 = Mtraveler = 400 x 67.29 = Mconst = 428.1 x 34.5 = Mself cantilever =
Moment 462787 71594 26916 14771 576068
kNm kNm kNm kNm kNm
Description Constant area Variable area Assumed to be 40 Mg 500 Pa construction load
Cantilever Tendons Assume the segments over the pier have reached the 28 day strength by the time the last segment is cast. Segments Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e 18 2 15 140 0.6 1860 105 20 etop = 3.557 m = yt - cover - duct/2 - 25mm duct eccentricity P≥ 78607 kN P required to limit top concrete stress below cracking stress OK Pprovided = 84370 kN ft (MPa) = 1.80 MPa Pe = 300060 kNm = 0.52 of Mcant Stress Check at casting of the closure segment Mself = 13608.5 x 35 3157.9 x 23.333333 Mtraveler = 200 x 69 Mconst = 434.4 x 34.5 Total Cantilever Moment P= 89057 kN -P/A +Mcant/S -Mp/s ft (MPa)= -6.6 16.9 -9.3 fb (MPa)= -6.6 -19.5 10.7
= = = = =
476298 73684 13800 kNm 14985 kNm 578767 kNm
Constant area Variable area Assumed to be 40 Mg 500 Pa construction LL
Total 1.0 -15.4
≤ ≥
2.83 OK -30 OK
141 Continuity Prestressing for the Closure Segment The continuity tendons must take up the forces resulting from the removal of the form traveler and the effects of the thermal gradient. Assume the side spans are already closed and continuous. Mtraveler = Mtempgradient = Mmid,max =
-6082 kNm +ve 67800 kNm +ve 61718 kNm +ve Tendons
Strand 2
P Top pair Pair 1 Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 Pair 7 P=
35623 -34777 -P/A
ft (MPa)= fb (MPa)=
19 e
5937 5937 5937 5937 5937 5937
Mp + Mps =
Removel of traveller - From SAP model of continuous structure, two point loads upwards Positive moment due to nonlinear temperature gradient - CHBDC Gradient
Strand Ap Effective fpu (MPa) duct diam duct e 140 0.6 1860 105 Mp
1.110 -1.871 -1.871 -1.871 -1.871 -1.871 -1.871 -1.871
lc 6587 -11105 -11105 -11105 -11105 -11105 0 0 -48940
+Mtrav/S -Mp/S Total 0.7 3.8 0.0 -1.0 -5.9 -11.4 +Mmid/S -Mp/S Total -4.5 -6.8 3.8 -7.5 -4.5 10.4 -5.9 0.0
-4.5 -4.5 -P/A
ft (MPa)= fb (MPa)=
Mps 15.68 22.52 29.36 36.2 43.04 56.72 63.56 15.68
20
Mptot -738 1786 2329 2872 3414 4499 0 0 14162
5849 -9319 -8776 -8234 -7691 -6606 0 0
Mps = -Mp* Ltendon/Lspan Mptot = Mp + Mps
≤ ≥
1.10 OK -18 OK
≥ ≤
-18 OK 1.10 OK
Continuity Prestressing at Side Span Closures The continuity prestressing must take up the forces from the removal of the falsework CIP section 15 m Rpier = 260 kN Dead load 194.4 kN/m Rabut = 2656 kN -1 x ratio x M 0 2000 0 0 0 4000 0.2 3 7092 6000 8000 0.4 6 12435 10000 0.6 9 16028 12000 0.8 12 17872 14000 16000 0.9 13.5 18137 18000 1 15 17965 20000 Mself = 18137 kNm
3
5
Strand Strand Ap Effective fpu (MPa) duct diam 2 19 140 0.6 1860 125 ebot = -1.856 m P≥ 1860 kN Pprovided = 11874 kN -22033 kNm Pe = Mp cont side = After falsework removal of end spans - stresses at side span maximum moment section -P/A +Mself/S -Mp/S Total ft (MPa)= -1.5 -2.0 2.4 -1.1 ≤ 1.10 fb (MPa)= -1.5 3.1 -3.7 -2.2 ≥ -18
duct e
x (m) 7
9
11
Moment (kNm)
1
Pairs provided
Tendons
2
25
OK
OK OK
After midspan closure - add stresses from midspan continuity tendons and temperature gradient Mself = 18137 MTgradient = 12107 Mtotal = 30244 kNm Continuity PT (from above) Mp cont side = -22033 Mp cont mid = 2529 = Mps cont mid x (Lcip / Lside) Mp cont = -19504 -P/A +Mtotal/S -Mp/S Total ft (MPa)= -1.5 -3.3 2.2 -2.7 ≤ -18 OK fb (MPa)= -1.5 5.1 -3.3 0.3 ≥ 1.10 OK Creep As an initial estimate, reserve a margin of 2 Mpa for the stress in the lower axis over the pier, find corresponding moment. Mcreep = 59425 kNm Mcreep = 2 MPa * Sb
13
15
142 External Prestressing Design Moments at critical sections due to external loads and temperature gradient Side Span Side Span Location Closure Max LL +ve 4.02 m Pier Midspan x= -140 -119 -74.02 -70 0 Mself cant 15990 -33000 -512803 -543800 0 Mtraveler 1290 3220 7350 7720 -6080 Mself cont 34020 12700 -379000 -434300 108800 Mbarriers 2020 647 -20069 -23161 6252 Masphalt 4090 1310 -40640 -46901 12660 Mtemp AASHTO 11300 28250 59745 67800 67800 Mlive, max 21000 33174 12000 12000 33160 Mlive, min -10010 -25200 -65600 -72100 -10300 Mcreep 9904 24761 56581 59425 59425 Mmin 14381 -50503 -625203 -671032 3562 Mmax 52194 27649 -449836 -478917 140057 Mmin = Mself cant + Mtravrem + Mbarriers + Masphalt + Ml,min Mmax = Mself cant + Mtravrem + Mbarriers + Masphalt + Mtemp + Mcreep + Ml,max Prestressing at Midspan Continuity PT (from above) Pcont = 35623 kN Mp cont = -34777 kNm External PT Deviator to Pier = 44.6 m Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e Pier offset Mid offset 10 19 140 0.6 1860 125 25 250 300 Primary PT Moment Secondary PT Moment Pext = 29686 kN x= 30.89 m x = Me1 / (Me1 + Me2) * Deviator epier = e2 = 3.3815 m 2 * S1 = 3101000 emid = e1 = -1.5005 m 2 * S2 = -610597 Me2 = 100382 kNm S3 = -2262797 Me1 = -44543 kNm Mps = 1626 kNm Mps = (2S1 + 2S2 +S3)/Lmain Mp ext , midspan = -42917 kNm =Me2 + Mps Combined Continuity and External PT Pcont+Pext = 65308 Mp,cont+Mp,ext = -77695 Stresses at Midspan - Minimum and maximum moment conditions -P/A +Mmin/S -Mp/S Total ft (MPa)= -8.2 -0.4 8.6 -0.1 ≤ 2.83 OK fb (MPa)= -8.2 0.6 -13.1 -20.7 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -8.2 -15.4 8.6 -15.1 ≥ -30 OK fb (MPa)= -8.2 23.6 -13.1 2.3 ≤ 2.83 OK Prestressing at Side Span Closure Continuity PT (from above) Pcont = 11874 kN Mp,cont = -22033 kNm Secondary PT Moment from Midspan Tendons Mps,cont = 2529 kNm =Mps,cont,pier * Lclosure / Lside Mps,ext = 290 kNm =Mps,ext,pier * Lclosure / Lside Mps,midspan = 2819 kNm External PT Deviator to Pier = 44.6 m Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e Pier offset Side offset 2 19 140 0.6 1860 125 25 250 300 Primary PT Moment Secondary PT Moment Pext = 5937 kN x= 13.71 m x = Me1 / (Me1 + Me2) * Deviator e1 = e2,mid = -1.501 m S1 = 310100 e2 = e1,pier = 3.382 m S2 = -61060 Me1 = -8909 kNm S3 = -217371 Me2 = 20076 kNm S4 = -66815 Mps,pier = 837 kNm Mps = (2S1 - 2S2 - S3)/Lmain Mps,side closure = 149 kNm Mps,side = Mps,pier * Lclosure / Lside Mp ext,side = -8759 kNm =Me1 + Mps, side closure Combined Side Span Continuity, Side Span External PT, Midspan Cont and Ext. Secondary PT Moment Pcont+Pext = 17811 kN Mp,cont+Mp,ext = -27973 kNm Stresses at Side Span Closure - Minimum and maximum moment conditions -P/A +Mmin/S -Mp/S Total ft (MPa)= -2.2 -1.6 3.1 -0.7 ≤ 2.83 OK fb (MPa)= -2.2 2.4 -4.7 -4.5 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -2.2 -5.8 3.1 -4.9 ≥ -30 OK fb (MPa)= -2.2 8.8 -4.7 1.8 ≤ 2.83 OK
143 Check Stresses at Max Live Load Moment in Side Span Cantilever PT Pcant = 25311 kN =P * x / Lhalf Mp,cant = 36473 kNm =P * x / Lside * e Main Span Continuity PT Mps,cont = 5901 kNm =Mps,cont,pier * x / Lside Side Span Continuity PT Pcont = 11874 kN Mp,cont = -22033 kNm Main Span External PT to the section under consideration Mps,ext = 677 kNm =Mps,ext,pier * x / Lside Side Span External PT Pext = 5937 kNm eo -1.042 m Meo = -6184 kNm Mps = 349 kNm Mp,ext = -5835 kNm Combined Cantilever PT, Main Span Continuity PT, Main Span External PT and Side Span External PT Ptot = 43122 kN Mptot = 15184 kNm Stresses at Side Span Section of Maximum Live Load Moment -P/A +Mmin/S -Mp/S Total ft (MPa)= -5.0 4.5 -1.4 -1.8 ≤ 2.83 OK fb (MPa)= -5.0 -5.8 1.8 -9.1 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -5.0 -2.5 -1.4 -8.8 ≥ -30 OK fb (MPa)= -5.0 3.2 1.8 0.0 ≤ 2.83 OK Check Stresses over Pier in Final Condition Cantilever PT Pcant = 84370 kN Mp,cant = 300060 kNm Main Span Continuity PT Mps,cont = 14162 Main Span External PT Pext= 29686 Mp,ext = 100382 kNm =Mp, ext at pier Mps,ext = 1626 kNm =Mps,ext,pier Mps,midspan = 102008 kNm Side Span External PT Pext= 5937 Mp,ext = 20076 kNm =Mp, ext at pier Mps,ext = 837 kNm =Mps,ext at pier Mps,ext = 20913 kNm Combined Cantilever PT, Main Span Continuity PT, Main Span External PT and Side Span External PT Ptot = 119992 kN Mptot = 437144 kNm Neglect Contribution from Midspan External Tendons (ie section just beyond) Stresses at Pier Section of Minimum Moment (highest negative moment) -P/A +Mmin/S -Mp/S Total ft (MPa)= -8.9 19.6 -12.8 -2.1 ≤ 2.83 OK fb (MPa)= -8.9 -22.6 14.7 -16.8 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -8.9 14.0 -12.8 -7.7 ≥ -30 OK fb (MPa)= -8.9 -16.1 14.7 -10.3 ≤ 2.83 OK Check Stresses 4.02 m from Pier in Final Condition Combined Cantilever PT, Main Span Continuity PT, and Side Span External PT Ptot = 90307 kN Mptot = 336762 kNm Neglect Contribution from Midspan External Tendons (ie section just beyond) Stresses beyond Pier -P/A +Mmin/S -Mp/S Total ft (MPa)= -6.7 18.3 -9.8 1.7 ≤ 2.83 OK fb (MPa)= -6.7 -21.0 11.3 -16.4 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -6.7 13.2 -9.8 -3.4 ≥ -30 OK fb (MPa)= -6.7 -15.1 11.3 -10.5 ≤ 2.83 OK
144 Cantilever Constructed Girder Bridge PT Design A - Detailed Model SLS Stress Checks Joint x from CL Location Section Moments Mself+Msdl+Mp Staged @ end construction Staged @t=50 years Mthermal Mlive min Mlive max Msls min Staged @ end construction Staged @t=50 years Msls max Staged @ end construction Staged @t=50 years PT after Transfer - final Pi Pf
199 -154 Abutment Mid
120 -139 S Closure Mid
Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
Top - Min Top - Max Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
212 -28 End PT 0.3 main
216 -15 0.4 Main
299 0 CL Main Mid
-3300 -2000 12150 -9200 22500
15700 18800 27600 -21500 32000
-101000 -116700 67800 -54000 9000
-32000 -33000 76000 -56700 8800
11000 22150 64000 -8000 37200
-12700 2200 62000 -8000 37200
-24600 -4300 61500 -6600 39500
kNm kNm
-23100 -21100
-11580 -10280
-3650 -550
-149600 -165300
-83030 -84030
3800 14950
-19900 -5000
-30540 -10240
kNm kNm
-23100 -21100
16950 18250
45500 48600
-33200 -48900
44000 43000
75000 86150
49300 64200
36900 57200
kN kN
-24000 -21900
-24500 -22200
-42500 -38700
-111000 -103400
-142000 -131800
-68500 -61800
-65600 -59000
-68300 -62000
7.940 11.970 1.323 1.877 9.048 6.377
7.940 11.970 1.323 1.877 9.048 6.377
8.460 17.320 1.573 2.081 11.011 8.323
12.844 107.936 3.567 3.898 30.262 27.690
13.460 127.218 3.832 4.168 33.197 30.524
8.850 21.990 1.755 2.233 12.530 9.848
8.180 14.310 1.440 1.971 9.938 7.260
7.940 11.970 1.323 1.877 9.048 6.377
-3.09 1.28 -1.87 -1.81 -4.96 -1.82 2.66 -4.90 -0.43
-5.02 0.33 -4.13 -4.69 -9.16 -0.44 5.47 -5.46 0.44
-8.64 4.94 1.10 -3.70 -7.54 -5.40 -1.20 -14.04 -9.84
-10.55 2.50 -1.33 -8.05 -11.88 -2.72 1.44 -13.27 -9.11
-7.74 -0.30 -5.99 -8.04 -13.73 0.39 7.62 -7.35 -0.12
-8.02 2.00 -4.96 -6.02 -12.98 -2.74 6.79 -10.76 -1.23
-8.60 3.38 -4.08 -5.23 -12.68 -4.79 5.79 -13.39 -2.82
-2.80 1.14 -2.02 -1.66 -4.81 -1.61 2.86 -4.41 0.07
-4.57 0.05 -4.41 -4.52 -8.99 -0.07 5.84 -4.64 1.26
-8.05 5.46 1.62 -2.59 -6.43 -5.97 -1.77 -14.02 -9.82
-9.79 2.53 -1.30 -7.26 -11.09 -2.75 1.41 -12.54 -8.38
-6.98 -1.19 -6.88 -8.18 -13.86 1.52 8.75 -5.46 1.77
-7.21 0.50 -6.46 -6.71 -13.67 -0.69 8.84 -7.90 1.63
-7.81 1.13 -6.32 -6.68 -14.13 -1.61 8.97 -9.41 1.16
MPa MPa MPa MPa MPa MPa MPa MPa
2.55 2.55 -0.47 -0.47 -3.62 -3.62 -6.64 -6.64
Stresses - Staged Construction Model at t=30000 days (80 years) Pf/A MPa -2.76 -Mmin/St -Mmax/St
200 -70 Pier CL Pier
-23100 -21100 0 0 0
Stresses - Staged Construction Model after Construction Pf/A MPa -3.02
Top - Min Top - Max
101 -74 S Pier Table Pier
kNm kNm kNm kNm kNm
Section Properties (gross concrete section) A m2 I m4 yt m yb m St m3 Sb m3
-Mmin/St -Mmax/St
114 -119 S Max LL Mid
MPa MPa MPa MPa MPa MPa MPa MPa
2.33 2.33 -0.43 -0.43 -3.31 -3.31 -6.07 -6.07
145 Cantilever Constructed Girder Bridge PT Design A - Detailed Model ULS Moment Capacity Check Joint x from CL Location Section Moments and Axial Forces Mself Masphalt Mbarriers Mlive min Mlive max M @ End of Construction P @ End of Construction Long-term Moment Shift Long-term Axial Force Shift ULS1 - α D D + 1 P + 1.7 L Mmin Mmax P ULS2 - α D D + 1 P + 1.6 L + 1.15 K Mmin Mmax P ULS9 - 1.35 D + 1 P M P
199 -154 Abutment Mid
114 -119 S Max LL Mid
101 -74 S Pier Table Pier
200 -70 Pier CL Pier
212 -28 End PT 0.3 main
216 -15 0.4 Main
299 0 CL Main Mid
kNm kNm kNm kNm kNm kNm kN kNm kN
0 0 0 0 0 -23100 -24000 2000 2100
33800 4146 2218 -9200 22500 -3300 -24500 1300 2300
10600 1280 685 -21500 32000 15700 -42500 3100 3800
-336700 -37040 -19816 -54000 9000 -101000 -111000 -15700 7600
-393000 -42700 -22844 -56700 8800 -32000 -142000 -1000 10200
30800 3215 1720 -8000 37200 11000 -68500 11150 6700
85470 9730 5205 -8000 37200 -12700 -65600 14900 6600
104360 12020 6430 -6600 39500 -24600 -68300 20300 6300
kNm kNm kN
-23100 -23100 -24000
-9663 44227 -24500
-17953 72997 -42500
-282623 -175523 -111000
-232909 -121559 -142000
5512 82352 -68500
-3300 73540 -65600
-7652 70718 -68300
kNm kNm kN
-20800 -20800 -21585
-7248 45722 -21855
-12238 76562 -38130
-295278 -193578 -102260
-228389 -122709 -130270
19134 95174 -60795
14635 90675 -58010
16353 94063 -61055
kNm kN
-23100 -24000
10757 -24500
20098 -42500
-238745 -111000
-192490 -142000
23507 -68500
22442 -65600
18384 -68300
2.51 1.00 0.98
1.58 1.00 0.99
3.96 1.00 0.99
-2.85 1.00 0.99
1.10 1.00 0.98
1.41 1.00 0.98
1.64 1.00 0.98
-22200 -9800 -1.487 -14573 -12400 0 76 0.63 -21415 81709 80100 1.02
-38700 -9800 -1.237 -12123 -28900 180 38 0.51 23035 65649 69400 0.95
-103400 -9800 3.067 30053 -93600 570 0 0.63 315120 -640452 -1032000 0.62
-131800 -38200 3.492 133402 -93600 570 0 0.63 339974 -701765 -1125000 0.62
-61800 -28400 -0.945 -26838 -33400
-59000 -28400 -1.370 -38908 -30600 120 114 0.50 -7688 137271 137100 1.00
-62000 -28400 -1.487 -42231 -33600 30 190 0.59 -44969 181263 180400 1.00
Increase in Demand over SLS Moment M ULS / M SLS P ULS1 / P SLS (End Const) P ULS2 / P SLS (50 years) Moment Capacity P total P external e external Mp external P internal Top Strand internal Bot Strand internal fpe / fpu internal Mpint = Pf*e Mf = M ULS - Mp ext - Mp int Mr (from Response 2000) Mf / Mr
120 -139 S Closure Mid
kN kN m kNm kN no. no. kNm kNm kNm
-21900 -9800 -1.487 -14573 -12100
Change bottom reinf to 20M
146 Cantilever Constructed Girder Bridge Preliminary PT Design B - Internal Tendons Only Side Span Length Half Span Length Segment Length Closure Segment PT Cover Long PT Trans PT L&T Reinf
84 70 3.42 2
m m m m
Deck width
12.41 m
Top of Top Bot of Top Ext side Int side Top of Bot Bot of Bot 130 60 60 60 90 60 70 40 70 60 40 40 Comp at transfer fci = Comp limit 0.6fci = Tensile limit 0.5fcri =
30 MPa 18 MPa 1.10 MPa
Comp comp at SLS f'c = Comp limit 0.6f'c = Tensile limit fcr =
50 MPa 30 MPa 2.83 MPa
Section Properties Variable h g Ag yt yb I St Sb ρ
For PT design For +ve M Check midspan at pier side span max LL 3.2 8 3.68 m 0.25 1 m 2 7.935 13.459 8.65 m 1.312 3.719 1.6035 m 2.003 4.281 2.0765 m 4 11.9 127.2 18 m 3 9.07 34.20 11.23 m 3 5.94 29.71 8.67 m 0.571 0.594 0.625
Description height of cross-section thickness of bottom slab Area distance from cgc to top distance from cgc to bottom Moment of Inertia
geometric output
Moment on pier during the casting of the final segment Weight Lever Arm Mg = 13414.1 x 34.5 = 3112.8 x 23 = Mtraveler = 400 x 67.29 = Mconst = 428.1 x 34.5 = Mself cantilever =
Moment 462787 71594 26916 14771 576068
kNm kNm kNm kNm kNm
Description Constant area Variable area Assumed to be 40 Mg 500 Pa construction load
Cantilever Tendons Assume the segments over the pier have reached the 28 day strength by the time the last segment is cast. Segments Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e 18 2 19 140 0.6 1860 110 20 etop = 3.554 m = yt - cover - duct/2 - 25mm duct eccentricity P≥ 78640 kN P required to limit top concrete stress below cracking stress OK Pprovided = 106868 kN ft (MPa) = -2.20 MPa Pe = 379809 kNm = 0.66 of Mcant Stress Check at casting of the closure segment Mself = 13608.5 x 35 3157.9 x 23.333333 Mtraveler = 200 x 69 Mconst = 434.4 x 34.5 Total Cantilever Moment P= 112805 kN -P/A +Mcant/S -Mp/s ft (MPa)= -8.4 16.9 -11.7 fb (MPa)= -8.4 -19.5 13.5
= = = = =
476298 73684 13800 kNm 14985 kNm 578767 kNm
Constant area Variable area Assumed to be 40 Mg 500 Pa construction LL
Total -3.2 -14.4
≤ ≥
2.83 OK -30 OK
147 Continuity Prestressing for the Closure Segment The continuity tendons must take up the forces resulting from the removal of the form traveler and the effects of the thermal gradient. Assume the side spans are already closed and continuous. Mtraveler = Mtempgradient = Mmid,max =
-6082 kNm +ve 67800 kNm +ve 61718 kNm +ve Tendons
Strand 2
P Top pair Pair 1 Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 Pair 7 Pair 8 Pair 9 Pair 10 Pair 11
6875 6875 6875 6875 6875 6875 6875 6875 6875 6875 74683 -73224
P= -P/A ft (MPa)= fb (MPa)=
22 e
5937
Mp + Mps =
Removel of traveller - From SAP model of continuous structure, two point loads upwards Positive moment due to nonlinear temperature gradient - CHBDC Gradient
Strand Ap Effective fpu (MPa) duct diam duct e 140 0.6 1860 125 Mp
1.095 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856 -1.856
lc 6498 0 -12756 -12756 -12756 -12756 -12756 -12756 -12756 -12756 -12756 -12756 -121059
+Mtrav/S -Mp/S Total 0.7 8.1 -0.7 -1.0 -12.3 -22.8 +Mmid/S -Mp/S Total -9.4 -6.8 8.1 -8.1 -9.4 10.4 -12.3 -11.3
-9.4 -9.4 -P/A
ft (MPa)= fb (MPa)=
Mps 15.68 15.68 22.52 29.36 36.2 43.04 49.88 56.72 63.56 70.4 77.24 84.08
25
Mptot -728 0 2052 2675 3298 3921 4545 5168 5791 6414 7038 7661 47835
5770 0 -10704 -10081 -9457 -8834 -8211 -7588 -6965 -6341 -5718 -5095
Mps = -Mp* Ltendon/Lspan Mptot = Mp + Mps
≤ ≥
Install top tendon pair and 3 pairs for closure 1.10 OK -18 NO GOOD Install remaining bottom tendons after concrete
≥ ≤
-18 OK 1.10 OK
has reached 28 d strength
Continuity Prestressing at Side Span Closures The continuity prestressing must take up the forces from the removal of the falsework CIP section 15 m Rpier = 260 kN Dead load 194.4 kN/m Rabut = 2656 kN -1 x ratio x M 0 2000 0 0 0 4000 0.2 3 7092 6000 8000 0.4 6 12435 10000 0.6 9 16028 12000 0.8 12 17872 14000 16000 0.9 13.5 18137 18000 1 15 17965 20000 Mself = 18137 kNm
3
5
Strand Strand Ap Effective fpu (MPa) duct diam 2 22 140 0.6 1860 125 ebot = -1.856 m P≥ 1860 kN Pprovided = 20624 kN -38267 kNm Pe = Mp cont side = After falsework removal of end spans - stresses at side span maximum moment section -P/A +Mself/S -Mp/S Total ft (MPa)= -2.6 -2.0 4.2 -0.4 ≤ 1.10 fb (MPa)= -2.6 3.1 -6.4 -6.0 ≥ -18.00
duct e
x (m) 7
9
11
13
15
Moment (kNm)
1
Pairs provided
Tendons
3
25
OK
OK OK
Install 3 tendon pairs at side span closure at midspan
After midspan closure - add stresses from midspan continuity tendons and temperature gradient Mself = 18137 MTgradient = 12107 Mtotal = 30244 kNm Continuity PT (from above) Mp cont side = -38267 Mp cont mid = 8542 = Mps cont mid x (Lcip / Lside) Mp cont = -29725 -P/A +Mtotal/S -Mp/S Total ft (MPa)= -2.6 -3.3 3.3 -2.7 ≤ 2.83 OK fb (MPa)= -2.6 5.1 -5.0 -2.5 ≥ -30 OK Creep As an initial estimate, reserve a margin of 2 Mpa for the stress in the lower axis over the pier, find corresponding moment. Mcreep = 59425 kNm Mcreep = 2 MPa * Sb
148 External Prestressing Design Moments at critical sections due to external loads and temperature gradient Side Span Side Span Location Closure Max LL +ve Pier Midspan x= -140 -119 -70 0 Mself cant 15990 -33000 -543800 0 Mtraveler 1290 3220 7720 -6080 Mself cont 34020 12700 -434300 108800 Mbarriers 2020 647 -23161 6252 Masphalt 4090 1310 -46901 12660 Mtemp AASHTO 11300 28250 67800 67800 Mlive, max 21000 33174 12000 33160 Mlive, min -10010 -25200 -72100 -10300 Mcreep 9904 24761 59425 59425 Mmin 14381 -50503 -671032 3562 Mmax 52194 27649 -478917 140057 Mmin = Mself cant + Mtravrem + Mbarriers + Masphalt + Ml,min Mmax = Mself cant + Mtravrem + Mbarriers + Masphalt + Mtemp + Mcreep + Ml,max Prestressing at Midspan Continuity PT (from above) Pcont = 74683 kN Mp cont = -73224 kNm External PT Deviator to Pier = 49 m Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e Pier offset Mid offset 0.000001 27 140 0.6 1860 125 25 250 340 Primary PT Moment Secondary PT Moment Pext = 0 kN x= 34.22 m x = Me1 / (Me1 + Me2) * Deviator epier = e1 = 3.3815 m 2 * S1 = 0 emid = e2 = -1.4605 m 2 * S2 = 0 Me1 = 0 kNm S3 = -0.301893 Me2 = 0 kNm Mps = 0 kNm Mps = -(2S1 + 2S2 +S3)/Lmain Mp ext , midspan = 0 kNm =Me2 + Mps Combined Continuity and External PT Pcont+Pext = 74683 Mp,cont+Mp,ext = -73224 Stresses at Midspan - Minimum and maximum moment conditions -P/A +Mmin/S -Mp/S Total ft (MPa)= -9.4 -0.4 8.1 -1.7 ≤ 2.83 OK fb (MPa)= -9.4 0.6 -12.3 -21.1 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -9.4 -15.4 8.1 -16.8 ≥ -30 OK fb (MPa)= -9.4 23.6 -12.3 1.8 ≤ 2.83 OK Prestressing at Side Span Closure Continuity PT (from above) Pcont = 20624 kN Mp,cont = -38267 kNm Secondary PT Moment from Midspan Tendons Mps,cont = 8542 kNm =Mps,cont,pier * Lclosure / Lside Mps,ext = 0 kNm =Mps,ext,pier * Lclosure / Lside Mps,midspan = 8542 kNm External PT Deviator to Pier = 49 m Tendons Strand Strand Ap Effective fpu (MPa) duct diam duct e Pier offset Side offset 0.000001 27 140 0.6 1860 125 25 250 340 Primary PT Moment Secondary PT Moment Pext = 0 kN x= 14.78 m x = Me1 / (Me1 + Me2) * Deviator e1 = e2,mid = -1.461 m S1 = 0 e2 = e1,pier = 3.382 m S2 = 0 Me1 = 0 kNm S3 = 0 Me2 = 0 kNm S4 = 0 Mps,pier = 0 kNm Mps = (2S1 - 2S2 - S3)/Lmain Mps,side closure = 0 kNm Mps,side = Mps,pier * Lclosure / Lside Mp ext,side = 0 kNm =Me1 + Mps, side closure Combined Side Span Continuity, Side Span External PT, Midspan Cont and Ext. Secondary PT Moment Pcont+Pext = 20624 kN Mp,cont+Mp,ext = -29725 kNm Stresses at Side Span Closure - Minimum and maximum moment conditions -P/A +Mmin/S -Mp/S Total ft (MPa)= -2.6 -1.6 3.3 -0.9 ≤ 2.83 OK fb (MPa)= -2.6 2.4 -5.0 -5.2 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -2.6 -5.8 3.3 -5.1 ≥ -30 OK fb (MPa)= -2.6 8.8 -5.0 1.2 ≤ 2.83 OK
149 Check Stresses at Max Live Load Moment in Side Span Cantilever PT Pcant = 32060 kN =P * x / Lside Mp,cant = 46119 kNm =P * x / Lside * e Main Span Continuity PT Mps,cont = 19931 kNm =Mps,cont,pier * x / Lside Side Span Continuity PT Pcont = 20624 kN Mp,cont = -38267 kNm Main Span External PT Mps,ext = 0 kNm =Mps,ext,pier * x / Lside to the section under consideration Side Span External PT Pext = 0 kNm eo -1.042 m Meo = 0 kNm Mps = 0 kNm Mp,ext = 0 kNm Combined Cantilever PT, Side Span Continuity PT, Side Span External PT, Midspan Cont PT and Ext. Secondary PT Moment Ptot = 52684 kN Mptot = 27783 kNm Stresses at Side Span Section of Maximum Live Load Moment -P/A +Mmin/S -Mp/S Total ft (MPa)= -6.1 4.5 -2.5 -4.1 ≤ 2.83 OK fb (MPa)= -6.1 -5.8 3.2 -8.7 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -6.1 -2.5 -2.5 -11.0 ≥ -30 OK fb (MPa)= -6.1 3.2 3.2 0.3 ≤ 2.83 OK Check Stresses over Pier in Final Condition Cantilever PT Pcant = 106868 kN Mp,cant = 379809 kNm Main Span Continuity PT Mps,cont = 47835 Main Span External PT Pext= 0 Mp,ext = 0 kNm =Mp, ext at pier Mps,ext = 0 kNm =Mps,ext,pier Mps,midspan = 0 kNm Side Span External PT Pext= 0 Mp,ext = 0 kNm =Mp, ext at pier Mps,ext = 0 kNm =Mps,ext at pier Mps,ext = 0 kNm Combined Cantilever PT, Main Span Continuity PT, Main Span External PT and Side Span External PT Ptot = 106868 kN Mptot = 427645 kNm Neglect Contribution from Midspan External Tendons (ie section just beyond) Stresses at Pier Section of Minimum Moment (highest negative moment) -P/A +Mmin/S -Mp/S Total ft (MPa)= -7.9 19.6 -12.5 -0.8 ≤ 2.83 OK fb (MPa)= -7.9 -22.6 14.4 -16.1 ≥ -30 OK -P/A +Mmax/S -Mp/S Total ft (MPa)= -7.9 14.0 -12.5 -6.4 ≥ -30 OK fb (MPa)= -7.9 -16.1 14.4 -9.7 ≤ 2.83 OK
150 Cantilever Constructed Girder Bridge PT Design B - Detailed Model SLS Stress Checks Joint x from CL Location Section Moments Mself+Msdl+Mp Staged @ end construction Staged @t=50 years Mthermal Mlive min Mlive max Msls min Staged @ end construction Staged @t=50 years Msls max Staged @ end construction Staged @t=50 years PT after Transfer - final Pi Pf
199 -154 Abutment Mid
120 -139 S Closure Mid
kNm kNm kNm kNm kNm
-35900 -28850 0 0 0
1650 10300 12600 -9200 22500
12000 13800 28150 -21500 32000
36800 21000 39000 -29400 30600
-53000 -86500 67800 -54000 9000
-68000 -128000 75000 -56700 8800
47400 26700 65200 -17900 15700
19600 23000 63500 -10300 24500
-8100 10600 62000 -8000 37200
-28400 3600 60300 -6600 39500
kNm kNm
-35900 -28850
-6630 2020
-7350 -5550
10340 -5460
-101600 -135100
-119030 -179030
31290 10590
10330 13730
-15300 3400
-34340 -2340
kNm kNm
-35900 -28850
21900 30550
42075 43875
75800 60000
14800 -18700
7000 -53000
112600 91900
83100 86500
53900 72600
31900 63900
kN kN
-20800 -16700
-22700 -17300
-51700 -42200
-61000 -50300
-116700 -98500
-124000 -105000
-74000 -63300
-76500 -60600
-78200 -61000
-83200 -66000
7.940 11.970 1.323 1.877 9.048 6.377
7.940 11.970 1.323 1.877 9.048 6.377
8.460 17.320 1.573 2.081 11.011 8.323
9.336 28.866 1.985 2.429 14.542 11.885
12.844 107.936 3.567 3.898 30.262 27.690
13.460 127.218 3.832 4.168 33.197 30.524
9.930 38.590 2.262 2.669 17.060 14.459
8.850 21.990 1.755 2.233 12.530 9.848
8.180 14.310 1.440 1.971 9.938 7.260
7.940 11.970 1.323 1.877 9.048 6.377
-2.86 0.73 -2.42 -2.13 -5.28 -1.04 3.43 -3.90 0.58
-6.11 0.67 -3.82 -5.44 -9.93 -0.88 5.06 -6.99 -1.06
-6.53 -0.71 -5.21 -7.25 -11.75 0.87 6.38 -5.66 -0.16
-9.09 3.36 -0.49 -5.73 -9.57 -3.67 0.53 -12.76 -8.55
-9.21 3.59 -0.21 -5.63 -9.42 -3.90 0.23 -13.11 -8.98
-7.45 -1.83 -6.60 -9.29 -14.05 2.16 7.79 -5.29 0.34
-8.64 -0.82 -6.63 -9.47 -15.28 1.05 8.44 -7.60 -0.21
-9.56 1.54 -5.42 -8.02 -14.98 -2.11 7.42 -11.67 -2.14
-10.48 3.80 -3.53 -6.68 -14.00 -5.38 5.00 -15.86 -5.48
-4.99 0.50 -3.98 -4.48 -8.97 -0.67 5.27 -5.66 0.28
-5.39 0.38 -4.13 -5.01 -9.51 -0.46 5.05 -5.85 -0.34
-7.67 4.46 0.62 -3.20 -7.05 -4.88 -0.68 -12.55 -8.34
-7.80 5.39 1.60 -2.41 -6.20 -5.87 -1.74 -13.67 -9.54
-6.37 -0.62 -5.39 -7.00 -11.76 0.73 6.36 -5.64 -0.02
-6.85 -1.10 -6.90 -7.94 -13.75 1.39 8.78 -5.45 1.94
-7.46 -0.34 -7.31 -7.80 -14.76 0.47 10.00 -6.99 2.54
-8.31 0.26 -7.06 -8.05 -15.37 -0.37 10.02 -8.68 1.71
Section Properties (gross concrete section) A m2 I m4 yt m yb m St m3 Sb m3
Stresses - Staged Construction Model after Construction Pf/A MPa -2.62 -Mmin/St -Mmax/St
Top - Min Top - Max Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
MPa MPa MPa MPa MPa MPa MPa MPa
3.97 3.97 1.35 1.35 -5.63 -5.63 -8.25 -8.25
Stresses - Staged Construction Model at t=30000 days (80 years) Pf/A MPa -2.10 -2.18 -Mmin/St -Mmax/St
Top - Min Top - Max Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
MPa MPa MPa MPa MPa MPa MPa MPa
3.19 3.19 1.09 1.09 -4.52 -4.52 -6.63 -6.63
-0.22 -3.38 -2.40 -5.56 0.32 4.79 -1.86 2.61
114 -119 S Max LL
110 101 -105 -77 S End PT S Pier Table Pier
200 -70 Pier CL Pier
208 -42 M End PT
212 -28 0.3 Main
216 -15 0.4 Main
299 0 CL Main Mid
151 Cantilever Constructed Girder Bridge PT Design B - Detailed Model ULS Moment Capacity Check Joint x from CL Location Section Moments and Axial Forces Mself Masphalt Mbarriers Mlive min Mlive max M @ End of Construction P @ End of Construction Long-term Moment Shift Long-term Axial Force Shift ULS1 - α D D + 1 P + 1.7 L Mmin Mmax P ULS2 - α D D + 1 P + 1.6 L + 1.15 K Mmin Mmax P ULS9 - 1.35 D + 1 P M P
199 -154 Abutment Mid
120 -139 S Closure Mid
kNm kNm kNm kNm kNm kNm kN kNm kN
0 0 0 0 0 -35900 -20800 7050 4100
33300 4054 2169 -9200 22500 1650 -22700 8650 5400
12060 1411 755 -21500 32000 12000 -51700 1800 9500
-50000 -5862 -3136 -29400 30600 36800 -61000 -15800 10700
-339300 -37027 -19809 -54000 9000 -53000 -116700 -33500 18200
-396000 -42872 -22936 -56700 8800 -68000 -124000 -60000 19000
-62600 -7526 -4026 -17900 15700 47400 -74000 -20700 10700
kNm kNm kN
-35900 -35900 -20800
-4869 49021 -22700
-21282 69669 -51700
-26738 75262 -61000
-235135 -128035 -116700
-269613 -158263 -124000
kNm kNm kN
-27793 -27793 -16085
5998 58968 -16490
-17062 71739 -40775
-41968 57092 -48695
-268260 -166560 -95770
kNm kN
-35900 -20800
15483 -22700
16979 -51700
16151 -61000
1.93 1.00 0.95
1.64 1.00 0.97
-16700 0
-17300 0
0 -16700
0 -17300 0 132 0.50 -29877 88845 109400 0.81
Increase in Demand over SLS Moment M ULS / M SLS P ULS1 / P SLS (End Const) P ULS2 / P SLS (50 years) Moment Capacity P total P external e external Mp external P internal Top Strand internal Bot Strand internal fpe / fpu internal Mpint = Pf*e Mf = M ULS - Mp ext - Mp int Mr (from Response 2000) Mf / Mr
kN kN m kNm kN no. no. kNm kNm kNm
114 -119 S Max LL
110 101 -105 -77 S End PT S Pier Table Pier
200 -70 Pier CL Pier
208 -42 M End PT
212 -28 0.3 Main
216 -15 0.4 Main
299 0 CL Main Mid
-10300 24500 19600 -76500 3400 15900
87090 9886 5289 -8000 37200 -8100 -78200 18700 17200
102080 11710 6265 -6600 39500 -28400 -83200 32000 17200
-118 57002 -74000
2090 61250 -76500
1719 78559 -78200
-12096 66274 -83200
-332943 -227263 -102150
-22133 33197 -61695
7030 65160 -58215
24024 100064 -58420
25364 103074 -63420
-191648 -116700
-229633 -124000
21447 -74000
19600 -76500
27693 -78200
13619 -83200
0.99 1.00 0.97
8.91 1.00 0.97
4.29 1.00 0.97
0.36 1.00 0.97
0.75 1.00 0.96
1.38 1.00 0.96
1.61 1.00 0.96
-42200 0
-50300 0
-98500 0
-105000 0
-63300 0
-60600 0
-61000 0
-66000 0
0 -42200 228 132 0.45 6817 64922 129300 0.50
0 -50300
0 -98500
0 -105000 722 0 0.56 381381 -714324 -1226000 0.58
0 -63300
0 -60600
0 -61000 152 352 0.46 -54768 154832 282000 0.55
0 -66000 38 484 0.49 -100289 203363 350800 0.58
152 Stiff Girder Extradosed Bridge PT Design - Detailed Model SLS Stress Checks x from CL Location Section Moments Mself+Msdl+Mp Staged @ end construction Staged @t=50 years Mthermal Mlive min Mlive max Msls min Staged @ end construction Staged @t=50 years Msls max Staged @ end construction Staged @t=50 years PT after Transfer - final Pi Pf
-154 Abutment Mid
Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
Top - Min Top - Max Mmin/Sb Mmax/Sb
Bottom - Min Bottom - Max
-77 S Pier Table Pier
-70 Pier CL Pier
-15 0.4 Main Mid
0 CL Main Mid
-22900 -6300 6970 -4050 22550
-17300 -1930 10600 -6100 28550
19450 14600 10600 -8100 21900
-2600 -28000 25660 -17700 1500
-25200 -66300 25000 -36860 3610
14500 28000 27000 -3200 36900
0 23800 25000 -2900 38900
kNm kNm
-51900 -40600
-26545 -9945
-22790 -7420
12160 7310
-18530 -43930
-58374 -99474
11620 25120
-2610 21190
kNm kNm
-51900 -40600
2971 19571
16875 32245
47640 42790
19278 -6122
-1951 -43051
47710 61210
35010 58810
kN kN
-49900 -32600
-50200 -33200
-63000 -50200
-62900 -47100
-67900 -59000
-95000 -82000
-93640 -71000
-89200 -68600
7.115 8.340 0.972 1.828 8.580 4.562
7.115 8.340 0.972 1.828 8.580 4.562
7.115 8.340 0.972 1.828 8.580 4.562
7.115 8.340 0.972 1.828 8.580 4.562
8.620 11.230 1.252 1.548 8.970 7.255
9.500 11.640 1.300 1.500 8.954 7.760
7.115 8.340 0.972 1.828 8.580 4.562
7.115 8.340 0.972 1.828 8.580 4.562
-7.06 3.09 -0.35 -3.96 -7.40 -5.82 0.65 -12.87 -6.40
-8.85 2.66 -1.97 -6.20 -10.82 -5.00 3.70 -13.85 -5.16
-8.84 -1.42 -5.55 -10.26 -14.39 2.67 10.44 -6.18 1.60
-7.88 2.07 -2.15 -5.81 -10.03 -2.55 2.66 -10.43 -5.22
-10.00 6.52 0.22 -3.48 -9.78 -7.52 -0.25 -17.52 -10.25
-13.16 -1.35 -5.56 -14.52 -18.72 2.55 10.46 -10.61 -2.70
-12.54 0.30 -4.08 -12.23 -16.62 -0.57 7.67 -13.11 -4.86
-7.06 0.86 -3.76 -6.19 -10.81 -1.63 7.07 -8.68 0.01
-6.62 -0.85 -4.99 -7.47 -11.61 1.60 9.38 -5.02 2.76
-6.84 4.90 0.68 -1.95 -6.16 -6.06 -0.84 -12.90 -7.69
-8.63 11.11 4.81 2.48 -3.82 -12.82 -5.55 -21.45 -14.18
-9.98 -2.93 -7.13 -12.91 -17.11 5.51 13.42 -4.47 2.86
-9.64 -2.47 -6.85 -12.11 -16.50 4.64 12.89 -5.00 2.79
MPa MPa MPa MPa MPa MPa MPa MPa
6.05 6.05 -0.96 -0.96 -11.38 -11.38 -18.39 -18.39
Stresses - Staged Construction Model at t=30000 days (80 years) Pf/A MPa -4.58 -4.67 -Mmin/St -Mmax/St
-101 S End PT Mid
-51900 -40600 0 0 0
Stresses - Staged Construction Model after Construction Pf/A MPa -7.01
Top - Min Top - Max
-119 S Max LL Mid
kNm kNm kNm kNm kNm
Section Properties (gross concrete section) A m2 I m4 yt m yb m St m3 Sb m3
-Mmin/St -Mmax/St
-133 S Closure Mid
MPa MPa MPa MPa MPa MPa MPa MPa
4.73 4.73 0.15 0.15 -8.90 -8.90 -13.48 -13.48
1.16 -2.28 -3.51 -6.95 -2.18 4.29 -6.85 -0.38
153 Stiff Girder Extradosed Bridge PT Design - Detailed Model ULS Moment Capacity Check x from CL Location Section Moments and Axial Forces Mself Pself Masphalt Pasphalt Mbarriers Pbarriers Mlive min Mlive max M @ End of Construction P @ End of Construction Long-term Moment Shift Long-term Axial Force Shift ULS1 - α D D + 1 P + 1.7 L Mmin Mmax P ULS2 - α D D + 1 P + 1.6 L + 1.15 K Mmin Mmax P ULS9 - 1.35 D + 1 P M P
-154 Abutment Mid
-119 S Max LL Mid
-101 S End PT Mid
-77 S Pier Table Pier
-70 Pier CL Pier
-15 0.4 Main Mid
0 CL Main Mid
kNm kN kNm kN kNm kN kNm kNm kNm kN kNm kN
0 0 0 0 0 0 0 0 -51900 -49900 11300 17300
53660 -2060 6490 -390 3472 -209 -4050 22550 -22900 -50200 16600 17000
66020 -4910 7950 -588 4253 -315 -6100 28550 -17300 -63000 15370 12800
-3560 -8340 -380 -1015 -203 -543 -8100 21900 19450 -62900 -4850 15800
-115500 -11600 -13860 -1385 -7415 -741 -17700 1500 -2600 -67900 -25400 8900
-235800 -19870 -27900 -2360 -14926 -1263 -36860 3610 -25200 -95000 -41100 13000
104450 -15200 12400 -1800 6634 -963 -3200 36900 14500 -93640 13500 22640
117800 -12784 14000 -1500 7490 -802 -2900 38900 0 -89200 23800 20600
kNm kNm kN
-51900 -51900 -49900
-15114 30106 -50849
-9640 49265 -64339
4737 55737 -65184
-64203 -31563 -71061
-151957 -83158 -100407
37477 105647 -97773
27128 98188 -92667
kNm kNm kN
-38905 -38905 -30005
4381 46941 -31299
8645 64085 -49619
-30 47970 -47014
-91643 -60923 -60826
-195536 -130784 -85457
53322 117482 -71737
54788 121668 -68977
kNm kN
-51900 -49900
-632 -51131
10078 -65035
18000 -66364
-50471 -72704
-122719 -103223
57719 -99927
48752 -94480
2.40 1.01 0.94
1.99 1.02 0.99
1.17 1.04 1.00
2.09 1.05 1.03
1.97 1.06 1.04
1.92 1.04 1.01
2.07 1.04 1.01
-33200 -8000 -1.468 -11744 -27410 0 216 0.49 -46405 0 -8649 105091 152800 0.69
-50200 -8000 -1.468 -11744 -27410 0 216 0.49 -46405 -14790 -24129 122234 164600 0.74
-47100 -8000 -0.988 -7904 0 0 0.0001 0.00 0 -39100 -49384 55874 58800 0.95
-59000 -8000 -0.042 -336 0 0 0.0001 0.00 0 -51000 -62161 -91307 -100800 0.91
-84800 -29700 0.200 5940 -9950 76 0 0.50 10945 -45150 -80257 -212421 -156900 1.35
-71000 -21700 -1.468 -31856 -33900 0 270 0.48 -57393 -15400 -41233 206730 210500 0.98
-68600 -21700 -1.468 -31856 -48000 54 324 0.49 -64361 1100 -24067 217885 229000 0.95
Increase in Demand over SLS Moment M ULS / M SLS P ULS1 / P SLS (End Const) P ULS2 / P SLS (50 years) Moment Capacity P total P external e external Mp external P internal Top Strand internal Bot Strand internal fpe / fpu internal Mp int = Pf*e P extradosed N axial force to calculate Mr * Mf = M ULS - Mp ext - Mp int Mr (from Response 2000) Mf / Mr
-133 S Closure Mid
kN kN m kNm kN no. no. kNm kN kN kNm kNm
-32600 -8000 0.172 1376 -27410
* to account for the increase in axial force in the girder due to increase in permanent loads
-169600 for 2x2x27 -196500 for 3x2x27 -220560 for 4x2x27 25M in top slab will raise Mr by 20000 kNm
154
Stiff Girder Extradosed Bridge PT Design - Forces in Cables 7000 6000 5000 4000 3000
SLS Prelim Max kN
2000
SLS End of Const Max kN
1000
SLS Final Max kN
ACTUAL PIER Cable Distance from CL Cable length Cable Area
x Lo A
m m m2
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
0
1010 1009 1008 1007 1006 1005 1004 1003 1002 1001 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 -137 -131 -125 -119 -113 -107 -101 -95 -89 -83 -57 -51 -45 -39 -33 -27 -21 -15 -9 -3 68.327 62.208 56.089 49.970 43.852 37.733 31.614 25.495 19.376 13.257 13.257 19.376 25.495 31.614 37.733 43.852 49.970 56.089 62.208 68.327 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 0.00532 899
800
701
602
503
404
305
206
107
8
998
1097
1196
1295
1394
1493
1592
1691
1790
1889
996 119 64 6 5006 230 -81 5229 4628
1102 132 70 -9 4900 293 -44 5149 4563
1196 143 76 -21 4804 276 -26 5084 4514
1270 152 81 -30 4724 256 -13 5037 4483
1319 158 84 -35 4667 222 -7 5013 4474
1335 160 86 -38 4640 205 -3 5010 4481
1312 157 84 -38 4649 200 -1 5020 4493
1243 149 80 -34 4701 194 -1 5036 4497
1126 135 72 -29 4804 181 -2 5035 4469
964 115 62 -22 4966 156 -2 5033 4421
1162 138 74 7 4735 201 -12 5117 4564
1390 166 89 6 4497 240 -18 5116 4617
1562 186 100 4 4330 267 -24 5118 4652
1674 200 107 1 4228 286 -29 5098 4643
1724 206 110 -2 4188 293 -34 5071 4607
1718 205 110 -5 4205 293 -39 5043 4554
1663 198 106 -8 4272 284 -43 5018 4491
1565 187 100 -12 4382 271 -47 4987 4411
1431 171 91 -15 4529 253 -51 4962 4327
1268 152 81 -18 4706 232 -56 4936 4233
Serviceability Limit States SLS Prelim Permanent * kN 6184 6204 6219 6228 6229 6220 kN SLS Prelim Max 6391 6467 6468 6458 6428 6404 kN SLS End of Const Max 5436 5413 5333 5267 5213 5194 SLS Final Max kN 4835 4826 4762 4713 4674 4665 kN 280 303 272 242 205 187 Live Load Force Range MPa 53 57 51 46 39 35 Live Load Stress Range SETRA Allowable 0.58 0.59 0.58 0.59 0.60 0.60 0.60 Actual fp/fpu 0.55 0.55 0.54 0.53 0.53 0.52 Actual / Allowable 0.95 0.95 0.94 0.92 0.91 0.91 * Large difference from Prelim Force and End of Construction force is due to internal and exteral PT.
6201 6381 5200 4673 181 34 0.60 0.53 0.91
6172 6347 5211 4672 176 33 0.60 0.53 0.91
6137 6299 5198 4631 164 31 0.60 0.53 0.91
6107 6247 5173 4561 142 27 0.60 0.52 0.91
6109 6291 5298 4745 192 36 0.60 0.54 0.93
6141 6357 5332 4833 232 44 0.60 0.54 0.94
6178 6419 5359 4893 262 49 0.60 0.54 0.94
6208 6465 5355 4900 283 53 0.59 0.54 0.94
6228 6492 5335 4872 295 55 0.58 0.54 0.94
6238 6501 5306 4818 298 56 0.58 0.54 0.93
6239 6495 5273 4747 294 55 0.58 0.53 0.93
6234 6477 5231 4655 286 54 0.58 0.53 0.92
6222 6450 5190 4555 274 51 0.59 0.52 0.91
6207 6416 5145 4442 259 49 0.60 0.52 0.90
262 79 17 340 5718 5564 0.58
249 74 16 330 5705 5551 0.58
225 67 14 307 5650 5502 0.57
193 58 12 265 5561 5432 0.56
232 69 15 342 5776 5598 0.58
278 83 18 408 5902 5691 0.60
312 93 20 455 5998 5765 0.61
335 100 21 485 6039 5791 0.61
345 103 22 499 6039 5785 0.61
344 103 22 497 6008 5754 0.61
333 99 21 483 5953 5706 0.60
313 93 20 460 5874 5635 0.59
286 85 18 430 5782 5555 0.58
254 76 16 395 5677 5461 0.57
Load Cases from SAP SELF ASPHALT BARRIERS TGAASHTO CABLES LIVEL, Max LIVEL, Min NLCANTCON NL50YEARS
Ultimate Limit States ULS Self ULS Asphalt ULS Barriers ULS Live Load Max ULS1 - End of Construction ULS9 - End of Construction ULS1 Actual fp/fpu
0 kN 11 kN 22 kN 33 kN 44 kN 77 kN 88 kN 55 kN 66 kN
0.2 0.5 0.2 1.7 0.35
199 59 13 392 5892 5641 0.60
220 66 14 497 5947 5606 0.60
239 71 15 470 5880 5580 0.59
254 76 16 435 5818 5563 0.59
264 79 17 377 5749 5560 0.58
267 80 17 348 5722 5563 0.58
155 Stiff Tower Extradosed Bridge PT Design - Detailed Model SLS Stress Checks x from CL Location Section Moments Mself+Msdl+Mp Staged @ end construction Staged @t=50 years Mthermal Mlive min Mlive max Msls min Staged @ end construction Staged @t=50 years Msls max Staged @ end construction Staged @t=50 years Axial Compression in Girder P from permanent loads P corresp to Max LiveL P from PT initial P from PT final P after construction P final after losses
-137 C10 Mid
kNm kNm kNm kNm kNm
kNm kNm kNm kNm
kN kN kN
-131 C9 Mid
-125 C8 Mid
Mmin/Sb Mmax/Sb
Bottom - Permanent Loads Bottom - Min Bottom - Max
-107 C5 Mid
-101 C4 Mid
-95 -73 -67 -45 C3 Pier CLPier CL C3 Mid Pier Pier Mid
-39 C4 Mid
-33 C5 Mid
-27 C6 Mid
-21 C7 Mid
-15 C8 Mid
-9 C9 Mid
-3 0 C10 CLMid Mid Mid
853
919
830
660
473
330
291
427
-2301
-2300
243
39
22
138
433
595
840
1031
3285
3485
3300
2855
2263
1622
1008
-5023
-4819
780
942
1170
1453
1776
2102
2381
2588
2662
316
686
1012
1277
1474
1610
1720
1722
1676
1585
1593
1598
1603
1609
1615
1622
1628
1632
1632
-1115
-1820
-2050
-1951
-1655
-1260
-1260
-1926
-7165
-7034
-2960
-2930
-2900
-2890
-2805
-2590
-2193
-1684
7635
5448
8816
9575
9020
7970
6742
5850
5105
555
460
5050
5636
6140
6625
7070
7650
7700
7716
7634
1106
-1306
-8750
-8631
-2421
-2598
-2588
-2463
-2092
-1736
-1134
-485
7978
-725 -11472 -11150
-1884
-1695
-1440
-1148
-749
-229
407
1072
9534
8778
9072
9281
9282
-151
-719
-1015
-1096
-1017
-804
-843
1610
1647
1640
1544
1366
1129
488
6009
9402 10257
9800
8825
7686
6932
6399
-461
-618
6062
6390
6830
7388
8088
7769 11768 12912 12440 11207
9619
8263
6980
-3183
-3137
6599
7293
7978
8703
9431 10285 10613 10838 10838
-6095 -12250 -18460 -24715 -31014 -37350 -43730 -50160 -63259 -63590 -50510 -44080 -37690 -31315 -24800 -18400 -12160 -532
-1014
-1606
-6595
-7847
-41560 -41560 -41560 -34633 -27707 -20780 -13853
-6927
0
0
-35623 -35623 -35623 -29686 -23748 -17811 -11874
-5937
0
0
-2675
-3923
-5260
-9778 -11770 -10200
-6795
-5484
-4161
-2859
-5900
230
-1556
-410
-9166
-8020
0
0
0
-6927 -13853 -20780 -27707 -34633 -41560
0
0
0
-5937 -11874 -17811 -23748 -29686 -35623
-47655 -53810 -60020 -59348 -58721 -58130 -57583 -57087 -63259 -63590 -50510 -44080 -37690 -38242 -38653 -39180 -39867 -40533 -41330 -41718 -47873 -54083 -54401 -54762 -55161 -55604 -56097 -63259 -63590 -50510 -44080 -37690 -37252 -36674 -36211 -35908 -35586 -35393
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
6.300
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
0.700
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
1.333
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
0.571
-9.06 -10.04 -10.09
-8.02
-7.00
-5.98
-6.07
-6.14
-6.22
-6.33
-6.43
-6.56
1.82
1.95
1.94
1.85
1.57
1.30
0.85
0.36
-5.98
Stresses - Staged Construction Model after Construction Pf/A MPa -7.56 -8.54 -9.53
Top - Permanent Loads Top - Min Top - Max
-113 C6 Mid
2613
Section Properties (gross concrete section) A m2 6.300 I m4 0.400 yt m 0.300 yb m 0.700 St m3 1.333 Sb m3 0.571
-Mmin/St -Mmax/St
-119 C7 Mid
MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa
-9.42
-9.32
-9.23
-9.14
0.11
0.54
0.76
0.82
0.76
0.60
0.63
0.98
6.56
6.47
-4.51
-7.05
-7.69
-7.35
-6.62
-5.76
-5.20
-4.80
0.35
0.46
-4.55
-4.79
-5.12
-5.54
-6.07
-6.58
-6.80
-6.96
-6.96
-6.92
-7.85
-8.90
-8.93
-8.97
-8.98
-8.92
-8.74 -11.77 -11.82
-7.84
-6.97
-5.97
-5.97
-5.81
-5.77
-5.70
-5.66
-5.73
-7.45
-8.00
-8.77
-8.60
-8.56
-8.62
-8.51
-8.08
-3.48
-3.62
-6.20
-5.05
-4.04
-4.22
-4.57
-4.92
-5.48
-6.07 -12.54
-12.07 -15.59 -17.22 -16.77 -15.94 -14.99 -14.34 -13.86
-9.70
-9.63 -12.56 -11.79 -11.11 -11.61 -12.20 -12.80 -13.13 -13.39 -13.52
-0.26
-1.26
-1.78
-1.92
-1.78
-1.41
-1.48
-2.29 -15.31 -15.10
-4.24
-4.55
-4.53
-4.31
-3.66
-3.04
-1.98
-0.85
13.96
10.52
16.45
17.95
17.15
15.44
13.45
12.13
11.20
-1.08
10.61
11.18
11.95
12.93
14.15
15.36
15.88
16.24
16.24
-6.07
-6.93
-8.07
-8.27
-8.49
-8.65
-8.63
-8.31 -14.07 -14.12
-7.59
-6.93
-5.94
-5.83
-5.38
-5.18
-4.86
-4.63
-4.62
-7.83
-9.80 -11.30 -11.34 -11.10 -10.63 -10.62 -11.35 -25.35 -25.20 -12.25 -11.54 -10.51 -10.38
-9.80
-9.26
-8.31
-7.28
7.40
9.55
9.81
9.68
6.12
4.22
2.99
2.14 -10.85 -11.18
2.59
4.19
5.97
6.86
8.02
9.14
Stresses - Staged Construction Model at t=30000 days (80 years) Pf/A MPa -6.62 -7.60 -8.58 -8.64 -8.69
-8.90 -10.04 -10.09
-8.02
-7.00
-5.98
-5.91
-5.82
-5.75
-5.70
-5.65
-5.62
1.41
1.27
1.08
0.86
0.56
0.17
-0.31
-0.80
-7.15
-Mmin/St -Mmax/St
Top - Permanent Loads Top - Min Top - Max Mmin/Sb Mmax/Sb
Bottom - Permanent Loads Bottom - Min Bottom - Max
MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa
2.95
7.91
8.42
7.73
-0.81
-8.76
-8.83
-1.21
-1.24
-1.23
-1.16
-1.02
-0.85
-0.37
0.54
8.60
8.36
-5.83
-8.83
-9.68
-9.33
-8.41
-7.21
-6.20
-5.24
2.39
2.35
-4.95
-5.47
-5.98
-6.53
-7.07
-7.71
-7.96
-8.13
-8.13
-4.66
-5.14
-5.97
-6.16
-6.55
-7.06
-7.61
-8.15 -13.81 -13.71
-7.43
-6.29
-5.11
-4.82
-4.49
-4.17
-3.91
-3.71
-3.62
-7.83
-8.83
-9.81
-9.79
-9.72
-9.60
-9.19
-8.36
-1.44
-1.73
-6.60
-5.73
-4.90
-5.05
-5.26
-5.58
-6.01
-6.45 -12.77
-12.45 -16.42 -18.27 -17.96 -17.10 -15.97 -15.02 -14.14
-7.65
-7.74 -12.97 -12.47 -11.97 -12.44 -12.89 -13.46 -13.66 -13.78 -13.75
2.82
2.88
2.87
2.70
2.39
1.98
0.85
-1.27 -20.08 -19.51
13.60
20.59
22.60
21.77
19.61
16.83
-3.30
-2.97
-2.52
-2.01
-1.31
-0.40
0.71
1.88
16.68
14.46
12.22
-5.49
11.55
12.76
13.96
15.23
16.50
18.00
18.57
18.97
-2.05
-1.85
-2.49
-2.86
-3.70
18.97
-4.80
-5.99
-7.14 -18.83 -18.53
-6.65
-5.35
-3.94
-3.37
-2.71
-2.07
-1.53
-1.12
-3.81
-4.72
-5.71
-5.93
-0.96
-6.30
-6.78
-7.97 -10.17 -30.12 -29.61 -11.31
-9.96
-8.50
-7.92
-7.13
-6.15
-4.99
-3.77
11.07
6.97
13.00
14.01
13.13
10.92
8.08
5.25
7.47
8.80
10.17
11.74
12.36
12.80
12.83
5.63
-5.57
3.31 -15.61 -15.58
3.02
156 Stiff Tower Extradosed Bridge PT Design - Detailed Model SLS & ULS Capacity Check x from CL Location Section Moments and Axial Forces Mself Pself Masphalt Pasphalt Mbarriers Pbarriers Mlive min Mlive max M @ End of Construction P @ End of Construction Long-term Moment Shift Long-term Axial Force Shift
-137 C10 Mid kNm kN kNm kN kNm kN kNm kNm kNm kN kNm kN
ULS1 - α D D + 1 P + 1.7 L Mmin kNm Mmax kNm P kN ULS2 - α D D + 1 P + 1.6 L + 1.15 K Mmin kNm Mmax kNm P kN ULS9 - 1.35 D + 1 P M kNm P kN Increase in Demand over SLS Moment M ULS / M SLS P ULS1 / P SLS (End Const) P ULS2 / P SLS (50 years) SLS Moment M50% Mmax Pcorresponding
kNm kNm kN
ULS Moment Mmax Pcorresponding *
kNm kN
Bot Reinf Centered PT (19-15 units) Mcr M0.2 cracks Mr Msls / M0.2 Mf / Mr
-131 C9 Mid
-125 C8 Mid
-119 C7 Mid
-113 C6 Mid
7746 13043 15054 14600 12570 -1466
-3850
1083
1826
-205 579
-107 C5 Mid 9818
-101 C4 Mid 6655
-95 -73 -67 -45 C3 Pier CLPier CL C3 Mid Pier Pier Mid 3210 -20500 -20105
-9 C9 Mid
-3 0 C10 CLMid Mid Mid
-4869
-487 1425
948
464
-540
-994
-1550
-2183
-2873
-3556
-4228
977
1130
1099
947
743
507
248
-2910
5714
-15 C8 Mid
8860
1389
4118
-21 C7 Mid
-9660
1771
2600
-27 C6 Mid
7350
2054
1050
-33 C5 Mid
-7085 -11037 -15527 -20343 -25253 -30000 -37763 -47300 -40090 -35485 -30500 -25330 -19950 -14673 2112
-803
-39 C4 Mid
9945 10374
-2852
-90
171
385
590
803
1020
1220
1365
-5329
-6649
-5623
-4971
-4271
-3543
-2788
-2049
-1348
-679
-67
-1557
-1526
-48
91
206
316
430
546
653
730
762
-110
-289
-532
-829
-1168
-1537
-1902
-2262
-2851
-3557
-3008
-2659
-2285
-1895
-1492
-1096
-721
-363
-36
-1115
-1820
-2050
-1951
-1655
-1260
-1260
-1926
-7165
-7034
-2960
-2930
-2900
-2890
-2805
-2590
-2193
-1684
7635
5448
8816
9575
9020
7970
6742
5850
5105
555
460
5050
5636
6140
6625
7070
7650
7700
7716
7634
853
919
830
660
473
330
291
427
-2301
-2300
243
39
22
138
433
595
840
1031
1106
-6095 -12250 -18460 -24715 -31014 -37350 -43730 -50160 -63259 -63590 -50510 -44080 -37690 -31315 -24800 -18400 -12160
-5900
230
1557
1556
1760
2366
2655
2640
2382
1933
1331
1164
1542
1638
1510
1248
995
55
581
-2519
537
903
1148
1315
1343
1507
1541
-1924 -20348 -20010
-5004
-4628
-4154
-3593
-2705
-1719
-376
12321 19623 21400 20161 17611 14598 12142 10029
-2722
-7224
-7270
8613
-6513 -13348 -20480 -27863 -35445 -43163 -50939 -58726 -74046 -77086 -61941 -54194 -46383 -38532 -30482 -22578 -14910
3300
4445
4896
4741
4153
3344
1712
-1063 -22762 -22203
13800 21463 23496 22295 19553 16147 13088 10187 -10410 -10213
-4091
986 17025
9934 11214 12582 14082 15689 16443 16966 17024
-3297
-2544
-1792
-880
273
1616
-7286
92
2945 18051
8725 10409 11920 13432 14920 16657 17445 17985 18050
-6513 -13348 -20480 -27863 -35445 -43163 -50939 -58726 -74046 -77086 -61941 -54194 -46383 -38532 -30482 -22578 -14910
-7286
92
4597
5245
5502
-6718 -13888 -21474 -29411 -37621 -46014 -54479 -62932 -79339 -83717 -67562 -59170 -50660 -42084 -33281 -24636 -16265
-7969
24
4146
6465
7234
6874
5824
4513
3130
1800 -11039 -10869
-86
498
1139
1896
2864
3716
1.78
1.82
1.82
1.79
1.74
1.68
1.58
1.46
3.27
3.26
1.32
1.43
1.49
1.54
1.58
1.62
1.64
1.66
1.67
1.07
1.09
1.11
1.13
1.14
1.16
1.16
1.17
1.17
1.21
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
0.40
1.07
1.09
1.11
1.13
1.14
1.16
1.16
1.17
1.17
1.21
1.23
1.23
1.23
1.23
1.23
1.23
1.23
1.23
0.40
3305
4886
5139
4719
4060
3364
2924
2724
-2051
-2093
2516
2575
2785
3119
3615
4038
4305
4503
4541
6009
9402 10257
9800
8825
7686
6932
6399 -11472 -11150
6062
6390
6830
7388
8088
8778
9072
9281
9282
-6574 -13163 -19905 -27123 -34545 -42084 -49666 -57222 -72059 -74183 -59690 -52329 -44908 -37431 -29736 -22145 -14733
-7300
-139
13800 21463 23496 22295 19553 16147 13088 10187 -10410 -10213
8725 10409 11920 13432 14920 16657 17445 17985 18050
-6513 -13348 -20480 -27863 -35445 -43163 -50939 -58726 -74046 -77086 -61941 -54194 -46383 -38532 -30482 -22578 -14910
-7286
92
20-25M40-30M44-30M40-30M22-30M10-25M10-25M10-25M10-25M10-25M10-25M10-25M10-25M10-25M20-25M30-25M34-25M40-25M40-25M 12 12 12 10 8 6 4 2 0 0 0 0 0 2 4 6 8 10 12
-19300 -19300 6700 6470 8700 11000 11333 11022 10667 10333 10889 11111 -23667 -23667 12644 11600 8778 8556 9778 10327 10222 10256 9778 15300 22100 23600 22900 19800 17100 17300 17400 -21300 -21300 17600 16385 13800 14100 15600 16900 17400 18200 18300 0.69 0.85 0.91 0.89 0.83 0.74 0.64 0.58 0.48 0.47 0.48 0.55 0.78 0.86 0.83 0.85 0.89 0.90 0.95 0.90 0.97 1.00 0.97 0.99 0.94 0.76 0.59 0.49 0.48 0.50 0.64 0.86 0.95 0.96 0.99 1.00 0.99 0.99 * to account for the increase in axial force in the girder due to increase in permanent loads At SLS Moment, bot stress is up to 30 Mpa.
157
Stiff Girder Extradosed Bridge PT Design - Forces in Cables 8000 7000 6000 5000
77 kN 88 kN 55 kN 66 kN
Serviceability Limit States SLS Prelim Permanent SLS Prelim Max SLS End of Const Max SLS Final Max Permanent fp/fpu Live Load Force Range Live Load Stress Range SETRA Allowable Actual fp/fpu Actual / Allowable
Ultimate Limit States ULS Self ULS Asphalt ULS Barriers ULS Live Load Max ULS1 - End of Construction ULS9 - End of Construction ULS1 Actual fp/fpu
kN kN kN kN kN MPa
0.45
0.2 0.5 0.2 1.7 0.35
2003
2002
2001
1001
1002
1003
1004
1005
1006
2010
44 kN
2009
33 kN
2008
22 kN
2007
0 kN 11 kN
2006
m m m2
2005
Load Cases from SAP SELF ASPHALT BARRIERS TGAASHTO CABLES LIVEL, Max LIVEL, Min NLCANTCON NL50YEARS
x Lo A
2004
ACTUAL PIER Cable Distance from CL Cable length Cable Area
1007
SLS Final Max kN 1008
SLS End of Const Max kN
1000 0 1009
SLS Prelim Max kN
2000
1010
4000 3000
1010 -137 68.327 0.0093
1009 -131 62.208 0.0093
1008 -125 56.089 0.0093
1007 -119 49.970 0.0093
1006 -113 43.852 0.0093
1005 -107 37.733 0.0093
1004 -101 31.614 0.0093
1003 -95 25.495 0.0093
1002 -89 19.376 0.0093
1001 -83 13.257 0.0093
2001 -57 13.257 0.0093
2002 -51 19.376 0.0093
2003 -45 25.495 0.0093
2004 -39 31.614 0.0093
2005 -33 37.733 0.0093
2006 -27 43.852 0.0093
2007 -21 49.970 0.0093
2008 -15 56.089 0.0093
2009 -9 62.208 0.0093
2010 -3 68.327 0.0093
899
800
701
602
503
404
305
206
107
8
998
1097
1196
1295
1394
1493
1592
1691
1790
1889
1490 209 112 -9 3645 543 -239 6057 5521
2426 341 182 -37 2753 566 -69 6141 5785
3295 464 248 -52 1923 886 -17 6224 6035
4027 567 303 -57 1224 1156 -7 6300 6252
4575 645 345 -54 706 1353 -3 6368 6425
4909 693 371 -47 402 1474 -1 6428 6545
5005 707 378 -38 338 1535 -26 6485 6611
4839 685 366 -27 546 1523 -49 6541 6620
4373 620 332 -17 1070 1420 -56 6610 6590
3542 503 269 -8 1982 1194 -48 6700 6530
3254 462 247 0 2263 1162 -83 6635 6398
4118 584 312 0 1321 1379 -77 6540 6419
4697 664 355 1 692 1485 -62 6478 6432
5068 715 382 1 289 1522 -41 6452 6455
5281 743 397 1 58 1520 -24 6450 6482
5360 753 403 1 -34 1498 -43 6459 6503
5322 746 399 0 -6 1460 -71 6456 6491
5170 723 387 -1 132 1410 -104 6437 6446
4909 686 367 -2 372 1357 -144 6410 6376
4544 635 340 -3 708 1310 -192 6363 6270
5456 5944 6546 6010 0.35 703 76 0.54 0.38 0.84
5702 6212 6651 6295 0.36 572 62 0.56 0.38 0.85
5929 6726 7021 6832 0.36 812 87 0.52 0.41 0.90
6121 7161 7340 7293 0.36 1046 113 0.49 0.42 0.94
6271 7489 7585 7642 0.37 1220 131 0.47 0.44 0.97
6374 7701 7755 7872 0.37 1327 143 0.46 0.45 1.00
6429 7811 7867 7993 0.37 1405 151 0.45 0.45 1.01
6437 7808 7912 7991 0.38 1415 152 0.45 0.46 1.02
6394 7672 7888 7867 0.38 1328 143 0.46 0.46 1.01
6296 7371 7775 7605 0.39 1118 120 0.48 0.45 1.00
6227 7273 7681 7444 0.38 1121 121 0.48 0.44 0.99
6335 7576 7780 7660 0.38 1310 141 0.46 0.45 1.00
6408 7745 7815 7769 0.37 1392 150 0.45 0.45 1.00
6454 7824 7821 7824 0.37 1406 151 0.45 0.45 1.00
6479 7847 7817 7849 0.37 1390 149 0.45 0.45 1.00
6482 7830 7807 7851 0.37 1387 149 0.45 0.45 1.00
6460 7774 7770 7806 0.37 1378 148 0.45 0.45 1.00
6413 7682 7706 7715 0.37 1363 147 0.45 0.45 0.99
6335 7556 7632 7598 0.37 1351 145 0.46 0.44 0.98
6227 7406 7542 7449 0.37 1352 145 0.46 0.44 0.97
298 105 22 923 7405 6691 0.43
485 171 36 963 7796 7173 0.45
659 232 50 1505 8669 7626 0.50
805 284 61 1965 9414 8014 0.54
915 323 69 2300 9974 8316 0.58
982 346 74 2507 10337 8519 0.60
1001 354 76 2610 10526 8617 0.61
968 342 73 2590 10514 8603 0.61
875 310 66 2413 10274 8474 0.59
708 251 54 2030 9744 8210 0.56
651 231 49 1976 9543 8022 0.55
824 292 62 2344 10061 8294 0.58
939 332 71 2525 10345 8479 0.60
1014 357 76 2587 10486 8610 0.61
1056 371 79 2583 10540 8697 0.61
1072 376 81 2546 10534 8739 0.61
1064 373 80 2483 10456 8719 0.60
1034 362 77 2397 10308 8636 0.60
982 343 73 2308 10116 8497 0.58
909 318 68 2227 9884 8294 0.57
APPENDIX C Chapter 4 Quantities
158
159
C.1 Breakdown of Prestressing Quantities of Chapter 4 Bridges The prestressing quantities, by tendon type, for bridges designed in Chapter 4 are summarised in Table C1, Table C-2, Table C-3, Table C-4. Table C-1. Prestressing quantities in cantilever-constructed girder bridge with internal prestressing. Tendon Unit Internal 19-15mm dia. strand Internal 22-15mm dia. strand
Cable Length (m) 5840 1646
Anchorages Duct Couplers 156 768 68 416 Total longitudinal prestressing - 162 tonnes
Table C-2. Prestressing quantities in cantilever-constructed girder bridge with internal and external prestressing. Tendon Unit Internal 15-15mm dia. strand Internal 19-15mm dia. strand External 27-15mm dia. strand
Cable Length (m) 5840 646 1240
Anchorages Duct Couplers 156 768 36 148 20 Total longitudinal prestressing - 146 tonnes
Table C-3. Prestressing quantities in stiff girder extradosed bridge. Tendon Unit Extradosed* 19-15mm dia. strand† Internal 19-15mm dia. strand Internal 27-15mm dia. strand External 27-15mm dia. strand Transverse 19-15mm dia. strand Transverse 12-15mm dia. strand
Cable Length (m) 3264 224 1496 1224 1120 112
Anchorages Duct Couplers 160‡ 20 12 56 192 20 160 16 Total longitudinal prestressing - 154 tonnes Total transverse prestressing - 24.9 tonnes
*Extradosed tendons do not require retensioning. †Strand for extradosed cables shall be galvanized and invidually sheathed. ‡Extradosed anchorage: external multistrand anchorage with stay cable protection details.
Table C-4. Prestressing quantities in stiff tower extradosed bridge. Tendon Unit Stay Cables* 31-16mm dia. strand† Internal 19-15mm dia. strand Transverse 15-15mm dia. strand
Cable Length (m) 3264 1488 560
*Stay cables require retensioning with gradient jack or strand by strand. †Strand for stay cables shall be galvanized and invidually sheathed.
Anchorages Duct Couplers 160 72 240 160 Total longitudinal prestressing - 150 tonnes Total transverse prestressing - 9.2 tonnes
APPENDIX D Presentation Handout
160
Behaviour and Design of Extradosed Bridges
Average Thickness of Concrete Girder 1.17
Kris Mermigas - MASc Thesis Presentation - 28 August 2008
Girder Regression
1.1
Cantilever Constructed Girder Extradosed Cable-Stayed KM Design
Average Thickness of Concrete Girder in Extradosed Bridges 0.91 0.9
0.9 Extradosed Regression
0.7 0.6
Cable-Stayed Regression
0.5
Average depth of concrete, m 3 /m2
Average depth of concrete, m3 /m2
1.0
0.8
161
Miyakodagawa Pyung-Yeo Korong
0.8
Hozu YukisawaOhashi
0.7
Regression
Shin-Karato Domovinski Odawara
0.6
Tsukuhara Pearl Harbor
Shinkawa Shin-Meisei
Saint-Remy
Ganter
Ibi Kiso
Himi
Pakse Rittoh
North Arm
Socorridos
Trois Bassins
0.5
Barton
Brazil-Peru
Sunniberg
Arret-Darre
0.4 0.34 0.3 0
0.4 0.39
100
200
300 Longest Span, m
400
0.3
500
50
100
150
200
250
275
Longest Span, m
Moment of Inertia of Deck at Midspan
Span to Girder Depth Ratios
140
25
120 100 Span : Depth
Extradosed Cable-Stayed
20
Constant Depth Simple Supports Constant Depth Embedded Variable Depth Simple Supports Variable Depth Embedded
at midspan at pier
80 60
55
40
30 Mathivat
20
15
0
50
66
100
150
200
250
275
Longest Span, m 10
Span to Tower Height Ratios
15 Mathivat
14 5
12
Sunniberg
0
0
100
200
300 Longest Span, m
400
500
530
Span : Height of Tower
Midspan I per 10 m width, m 4
Rio Branco
10
8
8 6
5 Cable-Stayed Typical
4 2 0
50
66
100
150
200 Longest Span, m
250
275
Cantilever Constructed Girder Bridge Permanent Loads (at end of construction)
Stiff Girder Extradosed Bridge Live Load Envelope -1100 kNm
Live Load Envelope -24800 kNm Permanent Loads (at end of construction) 33000 kNm Temperature Gradient
Permanent Loads (after 50 years)
Stiff Tower Extradosed Bridge Live Load Envelope
25600 kNm
29700 kNm
Permanent Loads (at end of construction)
2500 kNm
9670 kNm
Temperature Gradient
Temperature Gradient
61300 kNm
Permanent Loads (after 50 years)
21400 kNm
25200 kNm
Permanent Loads (after 50 years)
3000 kNm
1630 kNm
Span : Depth = 44
Span : Depth = 17.5
Span : Depth = 140
Span : Depth = 50
A = 7.9 m2 I = 11.9 m4
A = 7.1 m2 I = 8.3 m4
A = 13.5 m2 I = 127 m4
Quantity (m³ or Mg)
Average quantities
1.00
0.78 m
2985
780
146.6
0.91
49 kg/m³ 38 kg/m²
1033 713 320
270
0.72
90 kg/m³
1075 5090
280 1330
109.8 36.8
A = 6.1 m2 I = 0.41 m4
Cost Cost per m² ($×1000) roadway ($)
2985
269
7630 kNm
38800 kNm
-4330 kNm
Concrete (m3) Prestressing Steel Stays/Extradosed Cable Internal PT External PT Transverse PT Reinforcing Steel Total
162
Quantity (m³ or Mg)
1.00
Concrete (m3) Prestressing Steel Extradosed Cables Internal PT External PT Transverse PT Reinforcing Steel Total
Average quantities
Quantity (m³ or Mg)
Cost Cost per m² ($×1000) roadway ($)
2570
0.86
0.59 m
2570
670
178.4 68.2 49.1 36.3 24.9 231
1.10
59 kg/m³ 39 kg/m²
470
0.62
90 kg/m³
1810 926 319 316 249 926 5010
170 1310
0.98
Average quantities
Cost Cost per m² ($×1000) roadway ($)
1825 0.66* 0.46 m 1825 510 Concrete (m3) Prestressing Steel 159.4 1.06* 80 kg/m³ 1871 530 Stays 31.1 41 kg/m² 1557 Internal PT 202 External PT 119.1 Transverse PT 9.2 92 Reinforcing Steel 219 0.63* 120 kg/m³ 876 250 Total 4550 1285 * Relative factors are multiplied by 1.078 to account for the bridge’s shorter total length.
0.97
