OPTIMIZATION OF SPAN-TO-DEPTH RATIOS IN HIGH-STRENGTH CONCRETE GIRDER BRIDGES
by
Sandy Shuk-Yan Poon
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 Sandy Shuk-Yan Poon (2009)
Optimization of Span-to-Depth Ratios in High-Strength Concrete Girder Bridges
Sandy Shuk-Yan Poon Master of Applied Science Graduate Department of Civil Engineering University of Toronto 2009
ABSTRACT Span-to-depth ratio is an important bridge design parameter that affects structural behaviour, construction costs and aesthetics. A study of 86 constant-depth girders indicates that conventional ratios have not changed significantly since 1958. These conventional ratios are now questionable, because recently developed high-strength concrete has enhanced mechanical properties that allow for slenderer sections. Based on material consumption, cost, and aesthetics comparisons, the thesis determines optimal ratios of an 8-span highway viaduct constructed with high-strength concrete. Three bridge types are investigated: cast-in-place on falsework box-girder and solid slabs, and precast segmental span-byspan box-girder. Results demonstrate that total construction cost is relatively insensitive to span-todepth ratio over the following ranges of ratios: 10-35, 30-45, and 15-25 for the three bridge types respectively. This finding leads to greater freedom for aesthe tic expressions because, compared to conventional values (i.e. 18-23, 22-39, and 16-19), higher ranges of ratios can now be selected without significant cost premiums.
ii
Optimization of Span-to-Depth Ratios in High-Strength Concrete Girder Bridges
Sandy Shuk-Yan Poon Master of Applied Science Graduate Department of Civil Engineering University of Toronto 2009
ABSTRACT Span-to-depth ratio is an important bridge design parameter that affects structural behaviour, construction costs and aesthetics. A study of 86 constant-depth girders indicates that conventional ratios have not changed significantly since 1958. These conventional ratios are now questionable, because recently developed high-strength concrete has enhanced mechanical properties that allow for slenderer sections. Based on material consumption, cost, and aesthetics comparisons, the thesis determines optimal ratios of an 8-span highway viaduct constructed with high-strength concrete. Three bridge types are investigated: cast-in-place on falsework box-girder and solid slabs, and precast segmental span-byspan box-girder. Results demonstrate that total construction cost is relatively insensitive to span-todepth ratio over the following ranges of ratios: 10-35, 30-45, and 15-25 for the three bridge types respectively. This finding leads to greater freedom for aesthe tic expressions because, compared to conventional values (i.e. 18-23, 22-39, and 16-19), higher ranges of ratios can now be selected without significant cost premiums.
ii
ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my supervisor, Professor Paul Gauvreau, whose encouragement, guidance, and support enabled me to complete this thesis. I am also indebted to my research colleagues for their insightful advice and assistance throughout my graduate studies: Cathy Chen, Billy Cheung, Davis Doan, Negar Elhami Khorasani, Eileen Li, Kris Mermigas, Jason Salonga, Jimmy Susetyo, Brent Visscher, and Ivan Wu. Lastly, I would like to thank my family f amily for their support and encouragement over these past two years.
iii
TABLE OF CONTENTS Abstract.............................................................................................................................................. ii Acknowledgements........................................................................................................................... iii Table of Contents ............................................................................................................................. iv List of Figures ................................................................................................................................. viii List of Tables .................................................................................................................................... xi List of Symbols ............................................................................................................................... xiii 1
2
3
Introduction............................................................................................................................... 1
1.1
The Significance of Optimizing Span-to-Depth Ratio ........................................................ 1
1.2
Objectives and Scope .......................................................................................................... 5
1.3
Thesis Structure ................................................................................................................... 6
Typical Span-to-Depth Ratios of Existing Bridges................................................................. 7
2.1
Cast-in-Place Box-Girder .................................................................................................... 7
2.2
Cast-in-Place Slab ............................................................................................................. 12
2.3
Precast Segmental Box-Girder .......................................................................................... 16
2.4
Concluding Remarks ......................................................................................................... 18
Analysis Overview ................................................................................................................... 19
3.1
Analysis Model ................................................................................................................. 19
3.2
Materials ............................................................................................................................ 21
3.2.1
Prestressing Tendons ................................................................................................. 21
3.2.2
Concrete Covers ........................................................................................................ 22
3.3
Loads ................................................................................................................................. 22
3.3.1
Load Combinations and Load Factors ...................................................................... 22
3.3.2
Live Loads ................................................................................................................. 23
3.4
Design Requirements ........................................................................................................ 24
3.4.1
Ultimate Limit States Design Requirements ............................................................. 24
3.4.1.1
Flexural Strength ................................................................................................... 24
3.4.1.2
Shear Strength ....................................................................................................... 25
3.4.2
Serviceability Limit States Design Requirements ..................................................... 26
3.4.2.1
Stress ..................................................................................................................... 26
3.4.2.2
Vibration ............................................................................................................... 26
3.4.2.3
Deflection .............................................................................................................. 27 iv
3.5 4
Other Preliminary Analysis Assumptions ......................................................................... 27
Analysis of Cast-in-Place on Falsework Bridges .................................................................. 28
4.1
Cast-in-Place on Falsework Construction ......................................................................... 28
4.2
Cast-in-Place on Falsework Box-Girder ........................................................................... 28
4.2.1 4.2.1.1
Cross-Section ........................................................................................................ 29
4.2.1.2
Prestressing Tendon Layout .................................................................................. 30
4.2.2
4.3
Analysis Results ........................................................................................................ 31
4.2.2.1
Structural Behaviour and Dimensioning ............................................................... 31
4.2.2.2
Vibration Limits .................................................................................................... 32
4.2.2.3
Deflections ............................................................................................................ 33
4.2.2.4
Material Consumption ........................................................................................... 34
4.2.2.5
Limiting Factors of Span-to-Depth Ratios ............................................................ 36
Cast-in-Place on Falsework Solid Slab ............................................................................. 38
4.3.1
5
Model ........................................................................................................................ 28
Model ........................................................................................................................ 39
4.3.1.1
Cross-Section ........................................................................................................ 39
4.3.1.2
Prestressing Tendon Layout .................................................................................. 39
4.3.2
Strip Method versus Beam Model ............................................................................. 40
4.3.3
Analysis Results ........................................................................................................ 42
4.3.3.1
Structural Behaviour and Dimensioning ............................................................... 42
4.3.3.2
Maximum Reinforcement Criterion ...................................................................... 43
4.3.3.3
Vibration Limits .................................................................................................... 44
4.3.3.4
Deflections ............................................................................................................ 45
4.3.3.5
Material Consumption ........................................................................................... 46
4.3.3.6
Limiting Factors of Span-to-Depth Ratios ............................................................ 47
Analysis of Precast Segmental Span-by-Span Box-Girder.................................................. 48
5.1
Precast Segmental Span-by-Span Construction ................................................................ 48
5.2
Model ................................................................................................................................ 49
5.2.1
Cross-Section ............................................................................................................ 50
5.2.2
Elevation and Prestressing Tendon Layout ............................................................... 50
5.3
Longitudinal Bending Moments ....................................................................................... 51
5.3.1
Construction Moments .............................................................................................. 51
5.3.2
Moments due to Thermal Gradient ........................................................................... 54 v
5.4
6
Loss of Prestress ................................................................................................................ 57
5.4.1
Friction Losses .......................................................................................................... 57
5.4.2
Creep and Shrinkage Losses ..................................................................................... 58
5.4.3
Losses due to Relaxation of Prestressing Steel ......................................................... 59
5.4.4
Total Prestress Losses ............................................................................................... 59
5.5
Behaviour of Unbonded Tendons at Ultimate Limit States .............................................. 60
5.6
Analysis Results ................................................................................................................ 61
5.6.1
Structural Behaviour and Dimensioning ................................................................... 61
5.6.2
Vibration Limits ........................................................................................................ 61
5.6.3
Deflections ................................................................................................................ 62
5.6.4
Material Consumption ............................................................................................... 63
5.6.5
Limiting Factors of Span-to-Depth Ratios ................................................................ 64
Cost Comparisons ................................................................................................................... 65
6.1
Material Costs ................................................................................................................... 65
6.1.1 6.1.1.1
Concrete Material Unit Price ................................................................................ 65
6.1.1.2
Cast-in-Place versus Precast Concrete .................................................................. 66
6.1.1.3
Falsework versus Erection Truss........................................................................... 67
6.1.1.4
Formwork .............................................................................................................. 67
6.1.1.5
Prestressing Tendons ............................................................................................. 67
6.1.2
6.2
Material Unit Prices .................................................................................................. 65
Material Cost Comparisons ....................................................................................... 67
6.1.2.1
Concrete Cost Comparison ................................................................................... 68
6.1.2.2
Prestressing Cost Comparison ............................................................................... 69
6.1.2.3
Reinforcing Steel Cost Comparison ...................................................................... 70
6.1.2.4
Total Superstructure Cost ...................................................................................... 73
Overall Construction Costs ............................................................................................... 76
6.2.1
Construction Cost Breakdown .................................................................................. 76
6.2.2
Total Construction Cost Comparison ........................................................................ 77
6.3
Other Cost Factors ............................................................................................................ 78
6.4
Sensitivity Analysis ........................................................................................................... 79
6.4.1
Sensitivity with Respect to Changes in Material Unit Prices .................................... 79
6.4.2
Sensitivity with Respect to Changes in Construction Cost Breakdown .................... 83
6.5
Concluding Remarks ......................................................................................................... 85 vi
7
Aesthetics Comparisons.......................................................................................................... 86
7.1
8
Visual Impact of Span-to-Depth Ratio .............................................................................. 86
7.1.1
Effects of Viewing Points ......................................................................................... 92
7.1.2
Other Factors that Affect Visual Slenderness ........................................................... 94
7.2
Evolution of the Visually Optimal Span-to-Depth Ratio .................................................. 97
7.3
Concluding Remarks ....................................................................................................... 102
Conclusions ............................................................................................................................ 103
8.1
Conventional Span-to-Depth Ratios ............................................................................... 103
8.2
Maximum Span-to-Depth Ratios .................................................................................... 103
8.3
Material Consumption Comparisons............................................................................... 104
8.4
Total Construction Cost Comparisons ............................................................................ 104
8.5
Aesthetic Comparisons.................................................................................................... 105
8.6
Optimal Span-to-Depth Ratios ........................................................................................ 105
Reference........................................................................................................................................ 107 Appendix A: Chapter 2 Supplementary Information ................................................................ 111
A.1 Cast-in-Place on Falsework Box-Girder ............................................................................ 112 A.2 Cast-in-Place on Falsework Solid Slab .............................................................................. 116 A.3 Precast Segmental Span-by-Span Box-Girder.................................................................... 118 Appendix B: Supporting Calculations ........................................................................................ 119
B.1 Flexural Strength for Bonded Tendons at ULS .................................................................. 119 B.2
Shear Strength at ULS ....................................................................................................... 120
B.3 Thermal Gradient Moments................................................................................................ 122 B.4 External Tendon Force ....................................................................................................... 124 B.5 Total Construction Cost...................................................................................................... 125 Appendix C: Summary of Results ............................................................................................... 126
C.1 Cast-in-Place on Falsework Box-Girder............................................................................. 127 C.2 Cast-in-Place on Falsework Solid Slab .............................................................................. 128 C.3 Precast Segmental Span-by-Span Box-Girder .................................................................... 129 C.4 Sensitivity with Respect to Changes in Construction Cost Breakdown ............................. 130
vii
LIST OF FIGURES Figure 1-1. Recommended ratios for cast-in-place box-girder ......................................................... 2 Figure 1-2. Recommended ratios for cast-in-place slab.................................................................... 2 Figure 1-3. Recommended ratios for precast segmental box-girder ................................................. 2 Figure 2-1. Span-to-depth ratios of cast-in-place box-girders ........................................................ 10 Figure 2-2. Span-to-depth ratios of cast-in-place box-girders ........................................................ 11 Figure 2-3. Span-to-depth ratios of cast-in-place slabs ................................................................... 13 Figure 2-4. Span-to-depth ratios of cast-in-place slabs ................................................................... 14 Figure 2-5. Span-to-depth ratios of precast segmental box-girders ................................................ 17 Figure 2-6. Span-to-depth ratios of precast segmental box-girders ................................................ 17 Figure 2-7. Span-to-depth ratios for all bridge types ...................................................................... 18 Figure 3-1. Typical plan and elevation ........................................................................................... 19 Figure 3-2. Typical deck arrangement ............................................................................................ 19 Figure 3-3. Summary of analysis cases ........................................................................................... 20 Figure 3-4. Live loads: CL-625 truck load (top); CL-625 lane load (bottom) ................................ 23 Figure 3-5. Flexural resistance: a) cross-section, b) concrete stains, c) equivalent concrete stresses, d) concrete forces .............................................................................................................................. 24 Figure 3-6. Construction cost economy from increasing the number of stirrup spacing ................ 25 Figure 3-7. Deflection limits for highway bridge superstructure vibration (CHBDC 2006) .......... 26 Figure 4-1. Moment comparison of bridges with constant and reduced end span length ............... 29 Figure 4-2. Typical cross-section for cast-in-place on falsework box-girder ................................. 29 Figure 4-3. Typical reinforcing steel layout .................................................................................... 30 Figure 4-4. Typical tendon profile .................................................................................................. 30 Figure 4-5. Changes in sectional modulus and cross-sectional depth ............................................. 32 Figure 4-6. Deflection for superstructure vibration limitation ........................................................ 33 Figure 4-7. Deflections: a) dead load, b) long-term, c) short-term ................................................. 33 Figure 4-8. Material consumptions for cast-in-place on falsework box-girder ............................... 35 Figure 4-9. Tendon arrangement that limits further increase in span-to-depth ratio....................... 36 Figure 4-10. Interior box cavity limitation...................................................................................... 37 Figure 4-11. Height of access diminishes as span-to-depth ratio increases .................................... 37 Figure 4-12. Concrete reduction due to increase in L/h ratio for solid slab and box-girder ........... 38 Figure 4-13. Voided slab ................................................................................................................. 38 Figure 4-14. Typical cross-section for cast-in-place on falsework solid slab ................................. 39 viii
Figure 4-15. Typical reinforcing steel layout .................................................................................. 39 Figure 4-16. Transverse distribution of longitudinal bending moment in slabs.............................. 41 Figure 4-17. Maximum reinforcement criterion: a) concrete stains, b) equivalent concrete stresses, c) concrete forces .............................................................................................................................. 44 Figure 4-18. Deflection for superstructure vibration limitation ...................................................... 44 Figure 4-19. Deflections: a) dead load, b) long-term, c) short-term ............................................... 45 Figure 4-20. Material consumptions for cast-in-place on falsework solid slab .............................. 47 Figure 5-1. Precast segmental span-by-span construction method ................................................. 49 Figure 5-2. Span-by-span erection girder: a) overhead truss, b) underslung girder........................ 49 Figure 5-3. Typical cross-section for precast segmental span-by-span box-girder ......................... 50 Figure 5-4. Typical reinforcing steel layout .................................................................................... 50 Figure 5-5. Typical tendon profile .................................................................................................. 51 Figure 5-6. Construction moments for segmental span-by-span method........................................ 52 Figure 5-7. Redistribution of dead load moments due to creep ...................................................... 54 Figure 5-8. Redistribution of dead load and prestress moments due to creep................................. 54 Figure 5-9. Thermal gradient effects ............................................................................................... 55 Figure 5-10. Moments due to thermal gradient ............................................................................... 56 Figure 5-11. Intentional angle changes ........................................................................................... 57 Figure 5-12. Long-term loss of prestress due to relaxation (Menn 1990)....................................... 59 Figure 5-13. Compatibility conditions for bonded and unbonded tendons ..................................... 60 Figure 5-14. Deflection for superstructure vibration limitation ...................................................... 62 Figure 5-15. Deflections: a) dead load, b) long-term, c) short-term ............................................... 62 Figure 5-16. Material consumptions for precast span-by-span box-girder ..................................... 63 Figure 5-17. Access limited by height of interior box cavity.......................................................... 64 Figure 5-18. Access limited by height of interior box cavity and external tendons........................ 64 Figure 6-1. Concrete material unit price ......................................................................................... 66 Figure 6-2. Concrete material cost comparison .............................................................................. 68 Figure 6-3. Total concrete cost comparison .................................................................................... 68 Figure 6-4. Prestressing tendon cost comparison ............................................................................ 69 Figure 6-5. Cost comparison of stirrups and minimum reinforcing steel ....................................... 71 Figure 6-6. Cost distribution of stirrups and minimum reinforcing steel........................................ 72 Figure 6-7. Total reinforcing steel cost comparison ....................................................................... 73 Figure 6-8. Total superstructure material cost comparison ............................................................. 74 Figure 6-9. Total superstructure cost comparison (including cost of concrete placement)............. 75 ix
Figure 6-10. Total construction cost comparison ............................................................................ 78 Figure 6-11. Total construction cost comparison (+50% concrete unit price) ................................ 80 Figure 6-12. Total construction cost comparison (-50% concrete unit price) ................................. 80 Figure 6-13. Total construction cost comparison (+50% prestressing tendon unit price)............... 81 Figure 6-14. Total construction cost comparison (-50% prestressing tendon unit price) ............... 81 Figure 6-15. Total construction cost comparison (+50% reinforcing steel unit price) ................... 82 Figure 6-16. Total construction cost comparison (-50% reinforcing steel unit price) .................... 82 Figure 6-17. Total construction costs under changes in construction cost breakdown ................... 84 Figure 7-1. Cast-in-place on falsework box-girder with L=50m .................................................... 87 Figure 7-2. Cast-in-place on falsework solid slab with L=30m ...................................................... 88 Figure 7-3. Precast segmental span-by-span box-girder with L=50m ............................................ 89 Figure 7-4. Visual effects of increasing span-to-depth ratios from 10 to 35................................... 90 Figure 7-5. Effect of increasing span length (box-girder with h=2.5m) ......................................... 91 Figure 7-6. Viewed from 300m ....................................................................................................... 92 Figure 7-7. Viewed from 150m ....................................................................................................... 92 Figure 7-8. Viewed from 75m ......................................................................................................... 92 Figure 7-9. Effects of pier width-to-height ratio and span-to-depth ratio ....................................... 93 Figure 7-10. Effect of span-to-depth ratio as viewing angle becomes less oblique ........................ 94 Figure 7-11. Effect of bridge height on perceived superstructure slenderness ............................... 95 Figure 7-12. Effect of pier configuration on perceived superstructure slenderness........................ 95 Figure 7-13. Effect of deck cantilever length on perceived superstructure slenderness ................. 96 Figure 7-14. Glenfinnan Viaduct, 1901 (Cortright 1997) ............................................................... 97 Figure 7-15. Slender bridges by Maillart ........................................................................................ 98 Figure 7-16. Waterloo Bridge over the Thames (Darger 2002) ...................................................... 99 Figure 7-17. Changis-sur-Marne Bridge, 1948 (Mossot 2007) ....................................................... 99 Figure 7-18. Sketches to evaluate aesthetic impact of span-to-depth ratios (O'Connor 1991) ....... 99 Figure 7-19. Neckar Valley Viaduct, 1977 (Leonhardt 1982) ...................................................... 100 Figure 7-20. Kocher Valley Viaduct, 1979 (Leonhardt 1982) ...................................................... 100 Figure 7-21. Pregorda Bridge, 1974 (Menn) ................................................................................. 101 Figure 7-22. Felsenau Bridge, 1974 (Menn) ................................................................................. 101 Figure C-1. Cast-in-place on falsework box-girder with L=50m.................................................. 130 Figure C-2. Cast-in-place on falsework solid slab with L=25m ................................................... 130 Figure C-3. Precast segmental span-by-span box-girder with L=40m.......................................... 130
x
LIST OF TABLES Table 1-1. Description of recommended ratios ................................................................................. 3 Table 2-1. Summary of cast-in-place box-girders............................................................................. 7 Table 2-2. Summary of cast-in-place slabs (continued).................................................................. 13 Table 2-3. Summary of precast segmental box-girders................................................................... 16 Table 3-1. Material properties ......................................................................................................... 21 Table 3-2. Material resistance factors (CSA 2006) ......................................................................... 21 Table 3-3. Prestressing tendon properties (CSA 1982) ................................................................... 21 Table 3-4. Corrugated metal duct properties (DSI 2008)................................................................ 21 Table 3-5. Concrete cover requirements (CSA 2006) ..................................................................... 22 Table 3-6. Load combination .......................................................................................................... 22 Table 3-7. Load factors ................................................................................................................... 22 Table 3-8. DLA factor (CSA 2006) ................................................................................................ 23 Table 3-9. Multi-lane loading modification factor (CSA 2006) ..................................................... 23 Table 4-1. Summary of structural response and dimensioning of cast-in-place on falsework boxgirder ................................................................................................................................................. 31 Table 4-2. Summary of material consumptions for cast-in-place on falsework box-girder ............ 34 Table 4-3. Results from beam model and strip method .................................................................. 42 Table 4-4. Summary of structural response and dimensioning of cast-in-place on falsework solid slab .................................................................................................................................................... 43 Table 4-5. Concrete strengths required to satisfy maximum reinforcement criterion..................... 44 Table 4-6. Summary of material consumption for cast-in-place on falsework solid slab ............... 46 Table 5-1. Prestress losses due to friction ....................................................................................... 58 Table 5-2.
Prestress losses due to anchorage set ............................................................................ 58
Table 5-3. Prestress losses due to creep and shrinkage ................................................................... 59 Table 5-4. Effective prestress after all losses .................................................................................. 59 Table 5-5. Prestress at ULS ............................................................................................................. 61 Table 5-6. Summary of structural response of precast span-by-span box-girder ............................ 61 Table 5-7. Summary of material consumption for precast span-by-span box-girder ...................... 63 Table 6-1. Material unit prices ........................................................................................................ 65 Table 6-2. Concrete material unit price........................................................................................... 66 Table 6-3. Comparison of changes in cross-sectional depth and prestressing demand................... 70 Table 6-4. Total superstructure cost variations ............................................................................... 76 xi
Table 6-5. Construction cost breakdown (Menn 1990)................................................................... 77 Table 6-6. Material unit price changes ............................................................................................ 79 Table 6-7. Summary of material unit price sensitivity analysis ...................................................... 83 Table 6-8. Summary of cost study .................................................................................................. 85 Table C-1. Summary of results of cast-in-place on falsework box-girder analysis ...................... 127 Table C-2. Summary of results of cast-in-place on falsework solid slab analysis ........................ 128 Table C-3. Summary of results of precast segmental span-by-span box-girder analysis.............. 129
xii
LIST OF SYMBOLS A
Gross cross-sectional area
Ac
Area of concrete
Ap
Area of prestressing steel
As
Area of reinforcing steel
Av
Stirrup area
C
Compressive force
c
Depth of compression region
e(x)
Eccentricity of tendon at location x
Ec
Concrete elastic modulus
Ep
Prestressing tendon elastic modulus
Es
Reinforcing steel elastic modulus
f’c
Concrete compressive strength
f cr
Concrete tensile strength
f pu
Prestressing tendon ultimate strength
f py
Prestressing tendon yield stress
f r
Free stress due to temperature gradient
f y
Reinforcing steel yield stress
h
Girder depth
I
Moment of inertia
Ic
Moment of inertia of gross uncracked concrete section
L
Span length
L/h
Span-to-depth ratio
lp
Arc length of tendon between anchors
mop
Moment when qp is applied to prestressing band
mp
Moment when qp is applied to slab
Mr
Flexural resistance
Mr
Restraint moment
MSLS
SLS moment demand
msp
Self-equilibrating moment in strip method
MULS
ULS moment demand
n
Distance from base of cross-section to neutral axis (Section 5.3.2); E p /Ec (Section 5.4.2) xiii
P
Prestressing force
P0
Jacking force
Pr
Axial restraint force
qp
Prestressing deviation force
S
Sectional modulus
s
Stirrup spacing
T
Tensile force
z
Moment lever arm
α(x)
Sum of angle changes of tendon between stressing locations and point x
αc
Thermal coefficient of concrete
αD
Dead load factor
αp
Prestress load factor
αx
Intentional angle change of tendon
∆
Deflection
∆P
Loss of prestress force
∆α
Unintentional angle change of tendon
∆σp,rel
Prestress loss due to relaxation of steel
ε0
Final strain
εc
Concrete strain
εcs (t)
Time-varying shrinkage strain
εcu
Ultimate strain for concrete
εf
Free strain due to temperature gradient
θ(y)
Thermal differential
μ
Coefficient of friction
σp0
Jacking stress
σ p∞
Effective prestress after all losses
φ(t)
Creep coefficient
φc
Concrete resistance factor
φp
Prestressing tendon resistance factor
φs
Reinforcing steel resistance factor
ψ
Final curvature of bending
xiv
1
INTRODUCTION
1.1
The Significance of Optimizing Span-to-Depth Ratio Span-to-depth ratio, also known as slenderness ratio (L/h), is an important bridge design
parameter that relates a bridge’s span length to its girder depth. In the industry, this ratio is usually used to establish the superstructure depth and is chosen during the conceptual design phase before detailed calculations are performed. Selecting the ratio at an early stage of the design process permits approximate dimensional proportioning which is needed for preliminary analysis to evaluate the feasibility, cost-efficiency, and aesthetic merits of the design in comparison with alternative design concepts (ACI-ASCE 1988). The ratio is commonly chosen based on experience and typical values used in previously constructed bridges with satisfactory performance in order to ensure that the design does not deviate drastically from past successful practice. The ratio can also be determined by optimizing the combination of span length and superstructure depth to create a cost-efficient and aesthetically-pleasing structure, but this generally involves an iterative process. Therefore, instead of optimizing the span-to-depth ratio for every design concept, it is more common to select ratios from a range of conventional values. The choice of slenderness ratio is particularly critical in the design of girder-type bridges, because it directly affects the cost of materials and construction of the superstructure. For instance, using a high ratio (i.e. slender girder) reduces the concrete volume, increases the prestressing requirement, and simplifies the construction due to a lighter superstructure. Moreover, slenderness ratio has significant aesthetic impact, because the overall appearance of a girder-type bridge is highly dependent on the proportion of the superstructure (Leonhardt 1982). As stated previously, despite the significance of span-to-depth ratio, the industry has generally relied on the same proven range of ratios over the past decades. Figures 1-1 to1-3 show the recommended ranges of slenderness ratios outlined in different publications for three types of prestressed concrete constant-depth girders: cast-in-place box-girder, cast-in-place slab, and precast segmental box-girder. A brief description of the recommendations from each publication is given in Table 1-1.
1
2 40 ACI-ASCE 1988
30 AASHTO 1994
Span-todepth ratio
Duan et al. 1999
Leonhardt 1979
20
Hewson 2003 Menn 1990
Leonhardt 1979
10
Cohn & Lounis 1994 Multiple-cell box-girder Incremental launching method
0 1975
1980
1985
1990
Year
1995
2000
2005
2010
Figure 1-1. Recommended ratios for cast-in-place box-girder 50
ACI-ASCE 1988
40
AASHTO 1994 Cohn & Lounis 1994
Span-todepth ratio
30 Menn 1990 Cohn & Lounis 1994
20
Hewson 2003
Leonhardt 1979
10
Voided slab Solid slab
0 1975
1980
1985
1990
Year
1995
2000
2005
2010
Figure 1-2. Recommended ratios for cast-in-place slab 25 ACI-ASCE 1988
20 AASHTO-PCI-ASBI 1997
Span-todepth ratio
Gauvreau 2006
15 Duan et al. 1999
10
5
0 1975
1980
1985
1990
Year
1995
Figure 1-3. Recommended ratios for precast segmental box-girder
2000
2005
2010
3 Table 1-1. Description of recommended ratios
Author
Year
Description
Leonhardt
1979
Fritz Leonhardt, a Professor of Civil Engineering at the University of Stuttgart, suggests ratios based on values from previously constructed prestressed concrete bridges with good performance. For cast-in-place single-cell box-girder, a ratio of 21 is recommended. The suggested ratio is lowered to around 12 to 16 when incremental launching method is used due to the large negative construction moments associated with this co nstruction method. For castin-place slab, he suggests values from 18 to 36, with the higher values used for longer spans and for bridges with lighter traffic.
ACI-ASCE
1988
The American Concrete Institute-American Society of Civil Engineers (ACI-ASCE) Committee 343 on Concrete Bridge Design defines span-to-depth ratio recommendations for common bridge types based on typical values. These recommendations are intended to provide general guidelines for preliminary design. For cast-in-place, post-tensioned multiple-cell box-girder, ACI-ASCE recommends ratios from 25 to 33. The recommended ratio for precast multiple-cell continuous box-girder is around 22. These ratios are higher than the ones for single-cell boxgirder, because a multiple-cell box section has more webs to acc ommodate tendons compared to a single-cell section with similar width. The recommended range of ratios i s between 24 and 40 for cast-in-place, post-tensioned slab.
Menn
1990
Christian Menn is a Professor of Structural Engineering at the Institute of Structural Engineering in Zurich. His suggestions are based on existing bridges with satisfactory performance in terms of structural behaviour, aesthetics, and economics. He recommends ratios between 17 and 22 for cast-in-place box-girders, because girders with ratios below 17 would appear too heavy. On the other hand, girders with ratios above 22 have substantial cost increase due to the significantly higher longitudinal prestressing demand. Menn also suggests a maximum practical limit of 25 for solid slab and a maximum cost-effective slab depth of 0.8m.
AASHTO
1994
The American Association of State Highway and Transportation Officials (AASHTO) defines optional criteria for span-to-depth ratios in Cl.2.5.2.6.3 of the LRFD Bridge Design Specifications. These values are based o n traditional maximum ratios of constant-depth continuous highway bridges with adequate vibration and deflection response. To ensure proper vibration and deflection behaviours, the maximum ratios are determined to be 25 for cast-in-place box-girder and 37 for cast-in-place slab.
Cohn & Lounis
1994
M.Z. Cohn is a Professor of Civil Engineering at the University of Waterloo and the span-todepth ratios suggested in this paper are part of the results of a Ph.D. thesis prepared by Z. Lounis. These ratios are established f rom a systematic, multi-level optimization approach that determines the ideal cross-sectional dimensions, span layouts and superstructure system based on cost, material consumption, and aesthetics. For cast-in-place single-cell box-girder, the optimum ratio is found to range from 12 to 20. The ratio increases with span length and decreases with bridge width (e.g. a ratio of 12 corresponds to a span of 20m and a width of 16m while a ratio of 20 corresponds to a span of 50m and a width of 8m). This range of ratios is slightly lower relative to the ones from other publications, because this study investigates a simply-supported system while the ratios from other public ations are mostly based on continuous systems. A simply-supported girder tends to be deeper since it experiences greater moments at midspan compared to a continuous structure. Cohn & Lounis also suggest the range of optimum ratios for voided and solid slabs are 22 to 29 and 28 to 33 respectively.
AASHTOPCI-ASBI
1997
The American Segmental Bridge Institute (ASBI) has established various standard precast sections for segmental construction to enhance uniformity and simplicity for forming and production methods. Using these standard sections generally lead to practical and costeffective solutions. The ranges of span-to-depth ratios obtained from these standard sections are 17 to 19 fo r span-by-span method and 17 to 20 fo r balanced cantilever method.
Duan et al.
1999
Lian Duan is a Senior Bridge Engineer with the California Department of Transportation and a Professor of Structural Engineering at Taiyuan University o f Technology in China. A span-todepth ratio of 25 is recommended for cast-in-place multiple-cell box-girder based on typical values from existing bridges. A range of ratios from 12.5 to 20 is recommended for precast segmental box-girder. This range is based on frequently used standard precast sections from Federal Highway Administration (FHWA).
4 Table 1-1. Description of recommended ratios (continued)
Author
Year
Description
Hewson
2003
Nigel Hewson is a recognized expert in the design and construction of prestressed bridges and is an Associate Lecturer at the University of Surrey on this subject. He suggested a span-todepth ratio of 20 for cast-in-place single-cell box-girder and a maximum ratio of 20 for cast-inplace voided slab. Both of these recommendations are based on typical values.
Gauvreau
2006
A span-to-depth ratio of 17 is recommended for precast segmental span-by-span constructed box-girder. This value corresponds to the lower limit of span length used for this construction method (30m) and the minimum height requirement of a box section to provide sufficient access space within the box (1.8m). The recommended ratio is lower than the one for cast-inplace box-girder, because a larger depth is needed to compensate for the reduced tendon eccentricity due to the use of external unbonded tendons.
As shown in the previous graphs, there has been no significant increase in the recommended span-to-depth ratio since 1979 despite the advancement in material strengths and construction technologies. Recent developments have resulted in high-strength materials which theoretically should lead to more slender structural components and longer span lengths. In particular, highstrength concrete with compressive strength of 40 to 140 MPa has been achieved by lowering the water-to-cement ratio and incorporating chemical admixtures (Kosmatka et al. 2002). Because of their enhanced mechanical properties like higher ultimate strengths and modulus of elasticity, highstrength concrete structures can resist the same level of loads using slenderer sections, resulting in lightweight structures. The reduction in self-weight is especially critical in long-span bridges, because the dead load consumes approximately 75% of the load-bearing capacity in long-span bridges constructed with normal-strength concrete (TRB 1990). High-strength concrete lowers the dead load contribution by using thinner sections and improves the load-bearing capacity by increasing strength, thus slenderer bridges with longer spans can be attained. High-strength concrete has been applied to various types of structures. For instance, concrete with compressive strength of 60 MPa is commonly used for large bridges in Europe (Muller 1999) while the building industry has been using concrete with strengths of over 100 M Pa for years (Hassanain 2002). However, most short- and medium-span bridges are being constructed with concrete strengths of less than 50 MPa, because high-strength concrete is more expensive, especially if the designer still uses the typical span-to-depth ratios as defined decades a go based on normal-strength concrete (Hassanain 2002). For instance, the unit price of concrete rises by about 68% when the compressive strength changes from 30 MPa to 60 MPa (Dufferin Concrete 2009). This indicates a substantial material cost increase if the same guidelines for superstructure proportioning of normal-strength concrete bridges are applied to high-strength concrete bridges, causing the application of high-strength concrete in bridges to be economically unfeasible. Therefore, with the advent of high-strength materials, recommended span-to-depth ratios need to be
5 updated to match the improvement in material strength and stiffness and to provide an economic incentive for the application of these materials in bridges.
1.2
Objectives and Scope The purpose of this thesis is to determine the ideal range of span-to-depth ratios for post-
tensioned girder bridges constructed with current high-strength materials based on aesthetic comparisons and optimization parameters such as material consumption and total construction cost. The three bridge types considered in this study are cast-in-place on falsework box-girder and solid slab, and precast segmental span-by-span box-girder. The objectives of this study are summarized as follows:
Provide a study on the evolution of span-to-depth ratios in concrete girder bridges constructed over the past 50 years and establish a range of conventional ratios.
Determine the amount of prestressing and the concrete strength needed to satisfy safety and serviceability requirements as a function of span-to-depth ratio for the three types of bridge considered.
Compare the material consumptions and total construction costs for bridges with different slenderness ratios and determine the most cost-effective ratios.
Investigate the sensitivity of the construction cost results with respect to changes in material unit cost and construction cost breakdown.
Examine the visual impact of different span-to-depth ratios and especially evaluate the aesthetic influence of using the cost-effective ratios instead of conventional ones.
Update the recommendations for span-to-depth ratios based on economic and aesthetic considerations.
The results of this research are expected either to confirm that the conventional ratios are already optimal for new high-strength materials or to demonstrate that more slender sections can be attained. The study focuses on the superstructure only while the prestressing and concrete strength demands for the substructure are not explicitly accounted for. Also, only bridges with typical span lengths are analyzed in this study: 35m to 75m for cast-in-place box-girder, 20m to 35m for cast-inplace solid slab, and 30m to 50m for precast segmental box-girder.
6
1.3
Thesis Structure The thesis is organized in eight chapters: Chapter 1 provides the background and motivation of optimizing span-to-depth ratio. Chapter 2 examines the span-to-depth ratios of existing bridges and discusses their changes over
the past 50 years. This information along with the span-to-depth ratio recommendations described in Chapter 1 leads to values for conventional slenderness ratios. These conventional ratios serve as a basis for cost and aesthetic comparisons in the later chapters. Chapter 3 outlines the general analysis model and method used for all three bridge types. It also provides a breakdown on all the analysis cases that need to be considered and discusses specific design criteria that must be satisfied. The specific analysis models and analysis results for cast-in-place on falsework box-girder and solid slab, and precast segmental span-by-span box-girder are described in Chapter 4 and 5 respectively. Analysis results include structural responses, material consumptions, and factors that limit further increase in slenderness ratio. The construction method and design issues unique to each bridge type are also discussed. Chapter 6 compares the material costs and total construction costs for bridges with varying span-to-depth ratios for the three bridge types. Optimal ratios with the lowest costs are determined and in particular, cost savings associated with using the optimal ratios instead of conventional ones are examined. Furthermore, a sensitivity analysis is performed to demonstrate the effects of changing unit costs and total construction cost breakdown on the analysis results. Chapter 7 explores the aesthetic impacts of varying span-to-depth ratios and discusses the public perception on visually optimal ratios. Chapter 8 provides a conclusion for this study by summarizing the optimal span-to-depth ratios for the three bridge types as well as their improvement over conventional ratios in terms of material consumptions, construction costs, and aesthetics. These optimal ratios lead to updated span-to-depth ratio recommendations for bridges constructed with current high-strength materials.
2
TYPICAL SPAN-TO-DEPTH RATIOS OF EXISTING BRIDGES This chapter describes a study of 86 existing constant-depth girder bridges and presents a
compilation of their span-to-depth ratios. Specifically, the study determines the range of ratios typically used in the industry and examines its variations over the past 50 years. Three bridge types are considered: cast-in-place box-girder, cast-in-place slab, and precast box-girder. A majority of these bridges has span-to-depth ratios within the suggested ranges discussed in Chapter 1, indicating that a representative sample of bridges has been used.
2.1
Cast-in-Place Box-Girder First, the study investigates 44 constant-depth cast-in-place box-girders. Table 2-1 provides the
basic information as well as a cross-sectional drawing for each bridge. Additional information, including the span arrangement, girder dimensions, designer and references, is given in Appendix A.1. Figure 2-1 shows the span-to-depth ratios with respect to the span lengths and compares these ratios with the recommended values described in Se ction 1.1. Figure 2-2 plots the ratios with respect to the completion years in order to illustrate the trend in slenderness ratio over time. Table 2-1. Summary of cast-in-place box-girders Bridge no.
Name
Location
Span-todepth ratio
Construction method
1
Grenz Bridge at Basel
Switzerland
17.7
N/A
2
Sart Canal-Bridge
Belgium
12.0
Incremental launching
3
Weyermannshaus Bridge
Switzerland
18.9
N/A
4
Eastbound Walnut Viaduct
U.S.A.
23.0
CIP on falsework
5&6
Taiwan High Speed Rail (1) & (2)
Taiwan
11.4
Span-by-span
7
Pregorda Bridge
Switzerland
22.2
Span-by-span on falsework
8&9
Almese Viaduct & Condove Viaduct
Italy
18.2
Balanced cantilever
10
Gravio Viaduct
Italy
18.2
Balanced cantilever
Legend: N/A = no data CIP = cast-in-place
Split cross-section:
7
Cross-section
N/A
8 Table 2-1. Summary of cast-in-place box-girders (continued) Bridge no.
Name
Location
Span-todepth ratio
Construction method
11
Borgone Viaduct
Italy
18.2
Balanced cantilever
12
Quadinei Bridge
Switzerland
20.0
Span-by-span on falsework
13
Altstetter Viaduct
Switzerland
21.6
N/A
14
Reuss Bridge
Switzerland
17.7
CIP on falsework
15
Cerchiara Viaduct
Italy
18.5
Balanced cantilever
16
Castello Viaduct
Italy
18.5
Balanced cantilever
17
Costacole Viaduct
Italy
18.5
Balanced cantilever
18
Ferroviario Overpass at Bolzano
Italy
28.1
N/A
19
Krebsbachtal Bridge
Germany
12.9
Incremental launching
20
Shatt Al Arab Bridge
Iraq
12.8
Incremental launching
21
Ancona Viaduct
Italy
20.7
Segmental
22
Felsenau Bridge (approaches)
Switzerland
16.0
Span-by-span on falsework
23
La Molletta Viaduct
Italy
20.8
Segmental
24
Fosso Capaldo Viaduct
Italy
20.8
Segmental
25
Sihlhochstrasse Bridge
Switzerland
29.5
N/A
26 to 28
Grosotto Viaduct, Grosio Viaduct, Tiolo Viadut
Italy
20.0
Balanced cantilever
29
Denny Creek Viaduct
U.S.A.
20.9
N/A
30
Woronora River Bridge
Australia
14.7
Incremental launching
Cross-section
N/A