Title

Performance of in-situ concrete stitches in precast concrete segmentalbridges

Advisor(s)

Au, FTK

Author(s)

Leung, Chun-yu, Cliff.; 梁鎮宇.

Citation

Issued Date

URL

Rights

Leung, C. C. [梁鎮宇]. (2012). Performance of in-situ concrete stitches in precast concrete segmental bridges. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4961775 2012

http://hdl.handle.net/10722/180955

The author retains all proprietary rights, (such as patent rights) and the right to use in future works.

PERFORMANCE OF IN-SITU CONCRETE STITCHES IN PRECAST CONCRETE SEGMENTAL BRIDGES

LEUNG CHUN YU CLIFF 梁鎮宇

Ph.D. THESIS

THE UNIVERSITY OF HONG KONG

2012

Abstract of thesis entitled

Performance of In-Situ Concrete Stitches in Precast Concrete Segmental Bridges Submitted by

LEUNG Chun Yu Cliff for the degree of Doctor of Philosophy at The University of Hong Kong in December 2012

Multi-span precast concrete segmental bridges are commonly constructed using the balanced cantilever method, which essentially involves sequentially extending precast segments outwards from each pier in a balanced manner. A gap of 100 to 200 mm wide is usually provided around the mid-span location between the last two approaching segments to facilitate erection. In-situ concrete is then cast to ‘stitch’ the segments together, thus making the bridge deck continuous.

In the current practice, the in-situ concrete stitches are usually

designed to be capable of sustaining considerable sagging moment but only minimal hogging moment.

Failure of stitches may occur under exceptional

circumstances that may potentially trigger a progressive collapse. However, relatively little research in this area has been carried out. In view of this, the author is motivated to undertake an extensive study of the behaviour of in-situ concrete stitches and the effects of their performance on the robustness of typical segmental bridges. Experimental study is carried out to examine the behaviour of in-situ stitches under different combinations of internal forces.

Series of stitch specimens of different configurations are tested.

Subsequent parametric studies are conducted numerically to examine the effects

of various parameters on the load-displacement characteristics of the stitches. Formulae for strength estimation are proposed based on the results. A study of robustness involves analyzing the collapse behaviour of a structure in an extreme event and the analysis should be carried out up to and then well beyond the state of peak strength of structural members. A finite element programme for post-peak analysis is therefore developed for the present study. As the ability of a member section to sustain large inelastic deformation can ultimately affect the robustness of a structure, an investigation is conducted to examine the effects of steel content, yield strength and prestressing level on the ductility and deformability of prestressed concrete sections. Using the programme developed, the formation of collapsing mechanisms of a multi-span segmental bridge deck in an extreme event is examined. A typical bridge deck is subject to prescribed accidental load on its span in order to analyze the sequence of failure. Substantial redistribution of internal forces along the deck is observed as failures initiate, thus causing subsequent failures of other deck sections even though they have been designed to resist the internal forces at the ultimate limit state. The results indicate that any span of a multispan bridge may become a temporary end-span in the event of collapse of an adjacent span and the strength of the sections must be designed accordingly to prevent progressive failure. As a span becomes a temporary end-span, the in-situ concrete stitches may experience substantial moment and shear, and their failure could potentially trigger progressive collapse of the entire bridge deck. Towards the end of the thesis, important design considerations that can enhance the performance of in-situ concrete stitches and robustness of precast concrete segmental bridges are presented.

(478 words)

To my family

PERFORMANCE OF IN-SITU CONCRETE STITCHES IN PRECAST CONCRETE SEGMENTAL BRIDGES

by

LEUNG Chun Yu Cliff

BASc (Civil), University of Waterloo MSc (Eng), The University of Hong Kong

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at The University of Hong Kong

December 2012

DECLARATION I declare that this thesis represents my own work, except where due acknowledgement is made, and that it has not been previously included in a thesis, dissertation or report submitted to this University or to any other institution for a degree, diploma or other qualification.

Signed _____________________ LEUNG Chun Yu Cliff

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(This page is intentionally left blank)

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ACKNOWLEDGEMENTS The course of the author’s study has not always been a smooth one and there have been times of frustration. Without the supports from the people around the author, completion of this study would not have been possible. The author would now like to express his personal gratitude to them. First and foremost, the author would like to express his deepest gratitude to his supervisor, Professor F.T.K. Au, for his patience, support and guidance in the course of the author’s study. Professor Au has offered much invaluable academic and personal advice to the author from time to time. This thesis would not have been possible without the kind assistance of York Zeng for all his help in carrying out some of the most critical numerical analyses and derivations.

York has also given a great deal of help in the

experiments. For this the author is truly grateful. The author would like to thank Enoch Chan for his guidance during the initial stages of the research. His initial contributions have paved an easier way for the author. The help and dedication from all the technicians of the Department of Civil Engineering is much appreciated. Thanks also go to all the undergraduate students and research assistants who have taken part in this project. The Research Grants Council of the Hong Kong Special Administrative Region, China is gratefully acknowledged for supporting the work described in this thesis. The author would like to thank all his friends, especially H.H. Tsang, Wilson Fung, Fiona Chan, Enoch Chan, Zachary Chan, Vivian Wong, Mickey Chan, Tony Si, Jing Zhang, R.J. Jiang, Dixon Lau, Leo Lau, Eugenia Wu, Janet Yuen, Wallis Chan and K.Y. Leung, for their friendship and support through good and bad times in the course of the author’s study. Throughout these years, someone has often inspired the author with his philosophical insights, life experiences and interesting business ideas. He has often encouraged the author to think differently with a critical mind. The author

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would like to thank Professor A.K.H. Kwan for his encouragement and guidance along the way. The author cannot express enough his gratitude towards the kindness of all his relatives who have taken care of him during his stay in Hong Kong. To the author’s parents and sister, no words can express how thankful he is towards their support and encouragement and how much he appreciates their sacrifices in these years when he is not around. This thesis is dedicated to them all.

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TABLE OF CONTENTS Declaration ............................................................................................................ i Acknowledgements ............................................................................................... iii Table of Contents .................................................................................................. v List of Tables ......................................................................................................... x List of Figures ...................................................................................................... xi List of Abbreviations ........................................................................................ xviii List of Symbols ................................................................................................... xix

CHAPTER 1

INTRODUCTION

1.1

Precast Concrete Segmental Bridge............................................................. 1

1.2

Progressive Collapse and Robustness of Structures .................................... 2

1.3

In-Situ Concrete Stitches, Progressive Collapse and Robustness of Precast Concrete Segmental Bridges ........................................................... 3

1.4

Numerical Methods for Nonlinear Analyses of Prestressed Concrete Members ...................................................................................................... 4

1.5

Present Research and Outline of the Thesis ................................................ 4

CHAPTER 2

2.1

LITERATURE REVIEW

Joints of Precast Concrete Segmental Bridges .......................................... 11 2.1.1 Overview of previous studies ........................................................ 11 2.1.2 Shear strength of joint of precast concrete segmental bridge ........ 12

2.2

Progressive Collapse and Robustness........................................................ 15

2.3

Nonlinear Analysis of Prestressed Concrete Members ............................. 17 2.3.1 Development of numerical methods for nonlinear analysis .......... 17 2.3.2 Numerical methods for prestressed concrete members ................. 18

2.4

Deformability and Ductility of Prestressed Concrete Sections ................. 19

2.5

Conclusions ............................................................................................... 21

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CHAPTER 3

POST-PEAK ANALYSIS OF PRESTRESSED CONCRETE MEMBERS BY FINITE ELEMENT METHOD

3.1

Overview ................................................................................................... 25

3.2

General Scheme ......................................................................................... 25

3.3

Finite Element Formulation ....................................................................... 26

3.4

Section Analysis ........................................................................................ 27 3.4.1 Constitutive models used ............................................................... 28 3.4.2 Performing the analysis ................................................................. 30

3.5

Iteration Process ........................................................................................ 31

3.6

Conclusions ............................................................................................... 33

CHAPTER 4

DUCTILITY AND DEFORMABILITY OF PRESTRESSED CONCRETE SECTIONS

4.1

Overview ................................................................................................... 39

4.2

Method of Analysis ................................................................................... 40

4.3

Parametric Study ....................................................................................... 40

4.4

Moment-Curvatures Curves and Ductility Factors.................................... 41 4.4.1 Moment-curvature curves .............................................................. 42 4.4.2 Ductility factors ............................................................................. 42

4.5

Reinforced Concrete Sections ................................................................... 45

4.6

Prestressed Concrete Sections ................................................................... 46

4.7

Effects of x/d.............................................................................................. 47

4.8

Strength-Ductility-Deformability Performance......................................... 49

4.9

Conclusions ............................................................................................... 51

CHAPTER 5

EXPERIMENTAL INVESTIGATION ON BEAM SPECIMENS WITH IN-SITU CONCRETE STITCHES

5.1

Overview ................................................................................................... 65

5.2

Experimental Programme .......................................................................... 65 5.2.1 Configuration of the specimens ..................................................... 66

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5.2.2 Preparation of the specimens ......................................................... 66 5.2.3 Test setup ....................................................................................... 67 5.3

Experimental Observations........................................................................ 68 5.3.1 M Series ......................................................................................... 68 5.3.2 MV Series ...................................................................................... 69 5.3.3 V Series .......................................................................................... 71

5.4

Parametric Study........................................................................................ 71 5.4.1 Method of analysis ......................................................................... 72 5.4.2 Model calibration ........................................................................... 73 5.4.3 Effects of stitch width .................................................................... 74 5.4.4 Effects of prestressing force .......................................................... 74 5.4.5 Effects of concrete strength ........................................................... 75 5.4.6 Effects of shear key........................................................................ 75 5.4.7 Effects of prestressing force on crack width at construction joints............................................................................................... 77

5.5

Conclusions ............................................................................................... 79

CHAPTER 6

EXPERIMENTAL INVESTIGATION ON SHEAR SPECIMENS WITH IN-SITU CONCRETE STITCHES

6.1

Overview.................................................................................................. 107

6.2

Experimental Programme ........................................................................ 107 6.2.1 Configuration of the specimens ................................................... 107 6.2.2 Preparation of the specimens ....................................................... 109 6.2.3 Test setup ..................................................................................... 109

6.3

Experimental Observations...................................................................... 110 6.3.1 Specimens with plain stitch ......................................................... 110 6.3.2 Specimens with keyed stitch ........................................................ 111 6.3.3 Behaviour of specimens with different initial level of prestress ....................................................................................... 113 6.3.4 Behaviour of specimens with different joint roughness .............. 113

6.4

Parametric Study...................................................................................... 114 6.4.1 Method of analysis ....................................................................... 114 -vii-

6.4.2 Model calibration ......................................................................... 114 6.4.3 Effects of stitch width .................................................................. 115 6.4.4 Effects of initial prestressing level .............................................. 116 6.4.5 Effects of concrete grade ............................................................. 117 6.4.6 Effects of aspect ratio of stitch .................................................... 117 6.5

Proposed Design Formula for Keyed Stitches......................................... 118

6.6

Conclusions ............................................................................................. 122

CHAPTER 7

NUMERICAL SIMULATION OF FULL SCALE INSITU CONCRETE STITCHES

7.1

Overview ................................................................................................. 145

7.2

Description of the North Vernon Bridge ................................................. 145

7.3

Method of Analysis ................................................................................. 146

7.4

Full Range Behaviour of the Bridge ........................................................ 146

7.5

Variation of Stitch Width ........................................................................ 147

7.6

Conclusions ............................................................................................. 149

CHAPTER 8

COLLAPSE MECHANISMS AND ROBUSTNESS DESIGN

8.1

Overview ................................................................................................. 157

8.2

Method of Analysis ................................................................................. 157 8.2.1 Configuration of the bridge analysed .......................................... 157 8.2.2 Nonlinear finite element analysis ................................................ 158 8.2.3 Modelling of the bridge deck....................................................... 158

8.3

Variation of Internal Moments and Formation of Collapse Mechanisms ............................................................................................. 159 8.3.1 Scenario A ................................................................................... 160 8.3.2 Scenario B.................................................................................... 161 8.3.3 Scenario C.................................................................................... 162

8.4

Effects of Providing Top Reinforcement to In-Situ Concrete Stitches ... 164

8.5

Overall Assessment and Recommendation for Design ........................... 165

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CHAPTER 9

DESIGN CONSIDERATIONS

9.1

Overview.................................................................................................. 181

9.2

In-Situ Concrete Stitches ......................................................................... 181 9.2.1 Provision of shear keys ................................................................ 181 9.2.2 Dimension on in-situ concrete stitch ........................................... 181 9.2.3 Initial prestressing level ............................................................... 182 9.2.4 Importance of surface roughness ................................................. 182 9.2.5 Estimation of shear strength ........................................................ 182

9.3

Robustness Design of Multi-Span Precast Concrete Segmental Bridges ..................................................................................................... 183

CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 10.1 Conclusions ............................................................................................. 189 10.2 Recommendations for Future Work ........................................................ 191 10.2.1 Behaviour of in-situ concrete stitches reinforced by FRP ........... 191 10.2.2 Shear keys of PCSB ..................................................................... 191 10.2.3 Robustness of multi-span bridges ................................................ 191

REFERENCES ................................................................................................ 193

List of Publications by the Author..................................................................... 203

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LIST OF TABLES Table 5.1

List of beam specimens tested. ....................................................... 80

Table 6.1

List of shear specimens tested....................................................... 123

Table 8.1

Tendon arrangements. ................................................................... 168

Table 8.2

Details of finite element mesh. ..................................................... 168

Table 8.3

Amount of top prestressing strands in a stitch. ............................. 168

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LIST OF FIGURES Figure 1.1

Balanced cantilever construction (not to scale). ............................. 7

Figure 1.2

Construction of a PCSB using the balanced cantilever method (Hewson, 2012). ................................................................ 7

Figure 1.3

Formwork for casting of the in-situ concrete stitch (Hewson, 2012). .............................................................................................. 8

Figure 1.4

In-situ concrete stitch in precast concrete segmental bridge. ......... 8

Figure 1.5

An example of initiation of progressive failure along a bridge deck. .................................................................................... 9

Figure 1.6

Logical flow of thesis chapters. .................................................... 10

Figure 2.1

Typical test setups used in the studies belonging to Categories (a) and (b) (not to scale). ............................................ 22

Figure 2.2

Definition of Ak and Asm

Figure 2.3

Indented construction joint as specified in Eurocode 2. ............... 23

Figure 2.4

Potential numerical instability when analysis proceeds to the post-peak state as a result of adopting tangent stiffness in the formulatioin. ....................................................................... 24

Figure 3.1

Beam element used in the analysis. .............................................. 34

Figure 3.2

Constitutive model of the beam element used. ............................. 34

Figure 3.3

Constitutive model of concrete. .................................................... 35

Figure 3.4

Constitutive model of non-prestressed steel. ................................ 35

Figure 3.5

Constitutive model of prestressing steel. ...................................... 35

Figure 3.6

The iteration process: (a) moment and curvature of a typical element increasing from Points I to J; (b) iterations of moment and curvature from Points I to J. .................................... 36

Figure 3.7

Flowchart of the nonlinear finite element analysis....................... 37

Figure 4.1

Sections analyzed: (a) reinforced concrete section; and (b) prestressed concrete section. ........................................................ 53

Figure 4.2

Moment-curvature curves of (a) reinforced concrete sections; and (b) prestressed concrete sections. ........................... 54

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Figure 4.3

Definitions of yield curvature and ultimate curvature for (a) reinforced concrete section; and (b) prestressed concrete section........................................................................................... 55

Figure 4.4

Reinforced concrete sections: variations of (a) flexural strength; and (b) ductility factor. .................................................. 56

Figure 4.5

Reinforced concrete sections: variations of (a) yield curvature; and (b) ultimate curvature. .......................................... 57

Figure 4.6

Prestressed concrete sections: variations of (a) flexural strength; and (b) ductility factor. .................................................. 58

Figure 4.7

Prestressed concrete sections: variations of (a) yield curvature; and (b) ultimate curvature. .......................................... 59

Figure 4.8

Effects of x/d ratio of reinforced concrete (RC) sections and prestressed concrete (PC) sections on (a) flexural strength; and (b) ductility factor. ................................................................. 60

Figure 4.9

Effects of x/d ratio of reinforced concrete (RC) sections and prestressed concrete (PC) sections on (a) yield curvature; and (b) ultimate curvature. ........................................................... 61

Figure 4.10

Flexural strength-ductility performance of (a) reinforced concrete sections; and (b) prestressed concrete sections. ............. 62

Figure 4.11

Flexural

strength-deformability

performance

of

(a)

reinforced concrete sections; and (b) prestressed concrete sections. ........................................................................................ 63 Figure 5.1

The beam specimens tested. ......................................................... 81

Figure 5.2

Formwork and reinforcement layout for beam specimens of the M Series .................................................................................. 82

Figure 5.3

Scraping the surface of the construction joint using a needle gun. ............................................................................................... 82

Figure 5.4

Surface of the construction joint after treatment. ......................... 82

Figure 5.5

Anchorage at the dead end of the prestressing tendon. ................ 83

Figure 5.6

Prestressing equipment at the live end of the prestressing tendon. .......................................................................................... 83

Figure 5.7

The Avery testing frame: (a) the frame; and (b) specimen placed in the frame. ...................................................................... 84 -xii-

Figure 5.8

Setup of loading tests.................................................................... 85

Figure 5.9

Typical failure of specimens of (a) M Series; (b) MV Series; and (c) V Series. ........................................................................... 86

Figure 5.10

Stages of crack propagation for specimens of the M Series. ........ 87

Figure 5.11

Experimental load-deflection response for specimens of (a) M Series; (b) MV Series; and (c) V Series. .................................. 88

Figure 5.12

Stages of crack propagation for specimens of the MV Series. ..... 89

Figure 5.13

Configuration of the shear keys in the beam specimen. ............... 90

Figure 5.14

Bond-slip model adopted .............................................................. 90

Figure 5.15

Comparison

between

load-displacement

relationships

obtained experimentally and numerically .................................... 91 Figure 5.16

Parametric study - effects of stitch width for M Series specimens with (a) plain stitch and (b) keyed stitch..................... 92

Figure 5.17

Parametric study - effects of stitch width for MV Series specimens with (a) plain stitch and (b) keyed stitch..................... 93

Figure 5.18

Parametric study - effect of stitch width for V Series specimens with (a) plain stitch and (b) keyed stitch..................... 94

Figure 5.19

Parametric study - effects of stitch width on peak strength.......... 95

Figure 5.20

Parametric study - effects of prestressing force for M Series specimens with (a) plain stitch and (b) keyed stitch..................... 96

Figure 5.21

Parametric study - effects of prestressing force for MV Series specimens with (a) plain stitch and (b) keyed stitch. ......... 97

Figure 5.22

Parametric study - effects of prestressing force for V Series specimens with (a) plain stitch and (b) keyed stitch..................... 98

Figure 5.23

Parametric study - effects of prestressing force on peak strength. ........................................................................................ 99

Figure 5.24

Parametric study - effects of concrete strength for M Series specimens with (a) plain stitch and (b) keyed stitch................... 100

Figure 5.25

Parametric study - effects of concrete strength for MV Series specimens with (a) plain stitch and (b) keyed stitch. ....... 101

Figure 5.26

Parametric study - effects of concrete strength for V Series specimens with (a) plain stitch and (b) keyed stitch................... 102

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Figure 5.27

Parametric study - effects of concrete strength on peak strength. ...................................................................................... 103

Figure 5.28

Parametric study - effects of shear keys for specimens of various series. ............................................................................. 104

Figure 5.29

Parametric study - effects of prestressing force on maximum crack width at the construction joint. ......................................... 105

Figure 6.1

A shear specimen comprising two precast units connected by an in-situ concrete stitch. ....................................................... 124

Figure 6.2

Shear key configurations. ........................................................... 124

Figure 6.3

Examples of shear specimen configuration. ............................... 125

Figure 6.4

Formwork and reinforcement arrangement for the precast unit. ............................................................................................. 126

Figure 6.5

A chiselled construction joint surface and device for measuring surface roughness. .................................................... 126

Figure 6.6

Arrangement of linear variable displacement transducers (LVDT) for the shear specimens. ............................................... 127

Figure 6.7

Modes of failure of the shear specimens. ................................... 128

Figure 6.8

Experimental load-displacement relationships of specimens with (a) plain stitch; (b) stitch with single key; and (c) stitch with multiple keys. ..................................................................... 129

Figure 6.9

Experimental load-displacement relationships of specimens with different shear key configurations. ..................................... 130

Figure 6.10

Experimental load-displacement relationships of specimens with different prestressing force. ................................................ 130

Figure 6.11

Experimental load-displacement relationships of specimens with different construction joint roughness. ............................... 131

Figure 6.12

The effect of roughness on shear dilation in experiments. ......... 131

Figure 6.13

Experimental

and

numerical

load-displacement

relationships of externally prestressed specimens. ..................... 132 Figure 6.14

Experimental

and

numerical

load-displacement

relationships of internally prestressed specimens. ..................... 133 Figure 6.15

Parametric study - effects of stitch width for externally prestressed specimens. ............................................................... 134 -xiv-

Figure 6.16

Parametric study - effects of stitch width for internally prestressed specimens. ................................................................ 135

Figure 6.17

Parametric study - effects of prestressing for externally prestressed specimens. ................................................................ 136

Figure 6.18

Parametric study - effects of prestressing for internally prestressed specimens. ................................................................ 137

Figure 6.19

Parametric study - effects of concrete strength for externally prestressed specimens. ................................................................ 138

Figure 6.20

Parametric study - effects of concrete strength for internally prestressed specimens. ................................................................ 139

Figure 6.21

Parametric study - effects of aspect ratio on shear strength. ...... 140

Figure 6.22

Comparison between predicted values and FEA results of shear capacity. ............................................................................ 141

Figure 6.23

Maximum principal strain field and stress vectors in specimen E-K(M)-200(600)-60-1 when (a) inclined cracks formed; and (b) shear keys are sheared. ..................................... 141

Figure 6.24

Definition of Ak and Asm. ............................................................ 142

Figure 6.25

Formation of shear cracks in a keyed stitch (shear keys are not shown). ................................................................................. 142

Figure 6.26

Comparison between predictions from Equation (6.9) and FEA results of shear capacity. .................................................... 143

Figure 7.1

Configuration of the North Vernon Bridge (not to scale). ......... 150

Figure 7.2

Full-range behaviour of North Vernon Bridge subject to imposed displacement. ............................................................... 151

Figure 7.3

Development of maximum principal strain fields and principal stress vectors in stitch (1.6 m width) of North Vernon Bridge. ........................................................................... 152

Figure 7.4

Effects of stitch width on structural response of North Vernon Bridge. ........................................................................... 153

Figure 7.5

Development of maximum principal strain fields (left) and principal stress vectors (right) of stitch (200 mm width) in North Vernon Bridge at different stages: (a) when diagonal cracking across the stitch initiates; (b) when cracking -xv-

between the flange and web progresses; and (c) to (e) subsequent progressive failure of shear keys along the right construction joint.. ...................................................................... 154 Figure 7.6

Variations of internal forces at stitch section of North Vernon Bridge ignoring material nonlinearity. .......................... 155

Figure 7.7

Comparison between predicted load-carrying capacity based on shear strength of stitch and overall numerical results of North Vernon Bridge. ................................................................. 156

Figure 8.1

Configuration

of

the

bridge

analysed:

(a)

general

arrangement; and (b) deck section. ............................................ 169 Figure 8.2

Tendon arrangements: (a) interior span; and (b) end span. ........ 170

Figure 8.3

Finite element mesh for analysis. ............................................... 170

Figure 8.4

Examples of accidental loads on the bridge deck: (a) blasting as a result of vehicle collision; and (b) impact by a large boulder during a mudslide. ................................................ 171

Figure 8.5

Variation of deck moments in Scenario A. ................................ 172

Figure 8.6

Variation of deck displacements in Scenario A: (a) failure at M3; (b) after failure at M3; and (c) failure at Pier Sections B and E.. ......................................................................................... 173

Figure 8.7

Variation of deck moments in Scenario B. ................................ 174

Figure 8.8

Variation of deck displacements in Scenario B: (a) prior to failure at Pier Sections C and D; and (b) collapse of Span 3. .... 175

Figure 8.9

Variation of deck moments in Scenario C. ................................ 176

Figure 8.10

Variation of deck displacements in Scenario C: (a) failure at Pier Sections C and D; and (b) failure at Pier Sections B and E. ................................................................................................ 177

Figure 8.11

Top reinforcement in the form of prestressing strands that are anchored to the blisters at the soffit of the deck. .................. 178

Figure 8.12

Deck displacements for stitches with and without top reinforcement. ............................................................................ 178

Figure 8.13

Variations of moments in Stitches 2 and 3 with imposed deflection. ................................................................................... 179

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Figure 8.14

Summary of load-deflection relationship at M3 for various scenarios. .................................................................................... 179

Figure 9.1

Definition of key angle θkey. ....................................................... 185

Figure 9.2

Schematic diagram illustrating the provision of top reinforcement in the form of strands or bars across in-situ stitch (not to scale)...................................................................... 186

Figure 9.3

Schematic diagram illustraing the provision of top reinforcement in the form of fibre reinforced polymer (FRP) sheets across in-situ stitch (not to scale)..................................... 187

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LIST OF ABBREVIATIONS AASHTO

American Association of State Highway and Transportation Officials

ACI

American Concrete Institute

CSA

Canadian Standards Association

DOF

Degree of freedom

FEA

Finite element analysis

LVDT

Linear variable displacement transducer

M3

Mid-span of Span 3 of the typical bridge deck analysed

PC

Prestressed concrete

PCI

Prestressed Concrete Institute

PCSB

Precast concrete segmental bridge

PVC

Polyvinyl chloride

RC

Reinforced concrete

RSUF

Residual strength utilization factor

ULS

Ultimate limit state

-xviii-

LIST OF SYMBOLS A

Parameter in the constitutive model of concrete (definition used in Chapter 3); cross-sectional area (definition used in Chapter 6)

a

Node a of the beam element used

Ai

Area of joint

Ak

Area of the base of all shear keys

Aps

Area of prestressing steel

As

Area of steel reinforcement

Asm

Area of contact between smooth surfaces on the failure plane (definition used in Chapter 2); area of contact between smooth surfaces as defined in Figure 6.24 (definition used in Chapters 5 and 9).

Ast

Area of non-prestressed tension steel

B

Parameter in the constitutive model of concrete

b

Node b of the beam element used (definition used in Chapter 3); width of the section analysed (definition used in Chapter 4)

B

Strain matrix

c

Empirical factor that depend on the roughness of an interface

d

Effective depth

dmid

Deflection at the mid-span of the central span

dp

Effective depth of prestressing steel

dt

Effective depth of non-prestressed tension steel

Ec

Modulus of elasticity of concrete

Eps

Modulus of elasticity of prestressing steel

Es

modulus of elasticity of non-prestressed steel

EI

Flexural rigidity

(EI)a

Flexural rigidity at Node a of the beam element used

(EI)b

Flexural rigidity at Node b of the beam element used

Fs

Shear force at the location of the in-situ stitch

FL

Applied force acting on the bridge deck

f

Nodal force vector

f ′c

Cylinder strength of concrete

-xix-

fcd

Compressive strength of concrete (used by Eurocode 2)

fci

Concrete stress corresponding to the inflection point of the descending branch of the constitutive model of concrete

fck

Characteristic compressive strength of concrete

fco

Peak stress of concrete

fctd

Tensile strength of concrete (used by Eurocode 2)

fpc

Compressive stress in concrete

fpe

Effective prestressing stress

fpy

Yield strength of prestressing steel

fpu

Ultimate strength of prestressing steel

ft

Tensile strength of concrete

fy

Yield strength of non-prestressed steel

fyt

Yield strength of non-prstressed tension steel

Gc

Pre-crack shear modulus of concrete

Gc,cr

Post-crack shear modulus of concrete

h

Overall depth of the section analysed (definition used in Chapter 4); depth of in-situ stitch (defintion used in Chapter 6)

i

ith load step

K

Empirical coefficient used in the constitutive model of prestressing steel (definition used in Chapter 3); a parameter in the shear strength formula from the AASHTO guide

Ke

Element stiffness matrix

Kg

Global stiffness matrix

k

A parameter in the shear strength formula from Eurocode 2

L

Length of the beam element

lp

Plastic hinge length

M

Bending moment

Mo

Bending moment at which the stress at the extreme tension fibre is zero at transfer of prestress

Mp

Peak resisting moment of a section

m

Bending moment calculated after an iteration step

N

Empirical coefficient used in the constitutive model of prestressing steel

n

nth iteration step -xx-

Na

Interpolation function at Node a

Nb

Interpolation function at Node b

Q

Empirical coefficients used in the constitutive model of prestressing steel

q

Bond stress of between concrete and bonded tendon

s

Slip between concrete and bonded tendon

Tu

Ultimate force acting normal to an interface exerted by steel reinforcement

u

Empirical factor that depend on the roughness of the interface

u1

Vertical translational degree of freedom at Node a of the beam element used

u2

Rotational degree of freedom of Node a of the beam element used

u3

Vertical translational degree of freedom of Node b of the beam element used

u4

Rotational degree of freedom of Node b of the beam element used

V

Shear force

Vnj

Shear strength of joint of precast concrete segmental bridge

Vu

Ultimate shear capacity in the shear friction model

vRdi

Shear resistance between concrete that are cast at different times

w

Width of in-situ stitch

x

Abscissa of the beam element (definition used in Chapter 3); neutral axis depth at ultimate state (definition used in Chapter 4)

z

Distance between the critical section and the point of contraflexure

αdw

Coefficient in the modified AASHTO shear strength formula to account for the effect of aspect ratio of stitch

αsc

Coefficient in the modified AASHTO shear strength formula to account for the effect of stress concentration

β

Shear retention factor

δ

Nodal displacement vector

εc

Strain in concrete

εci

Concrete strain corresponding to the inflection point of the descending branch of the constitutive model of concrete

εco

Strain in concrete at peak stress -xxi-

εp

Residual strain in non-prestressed steel

εps

Strain in prestressing steel

εpu

Ultimate strain in prestressing steel

εs

Strain in non-prestressed steel

θ

Angle of shear crack

θkey

Shear key angle

φ

Section curvature

φo

Curvature at which the stress at the extreme tension fibre is zero at transfer of prestress

φr

Residual curvature

φra

Residual curvature at Node a of the beam element used

φrb

Residual curvature at Node b of the beam element used

φy

Yield curvature

φu

Ultimate curvature

φ

Friction angle

ψ

Acute angle between steel reinforcement and the interface

µ

Coefficient of friction (definition used in Chapter 2); ductility factor (definition used in Chapter 4)

ρ

Ratio of steel area to joint area

ρps

Prestressing steel ratio

ρst

Non-prestressed tension steel ratio

σ

Normal stress of infinitesimal element

σc

Stress in concrete

σn

Stress per unit area caused by the normal force across the interface

σps

Stress in prestressing steel

σs

Stress in non-prestressed steel

τ

Shear strength of joint (definition used in Chapter 2); shear stress of an infinitesimal element (definition used in Chapter 6)

τmax

Maximum shear stress in a section

υ

Strength reduction factor

-xxii-

CHAPTER 1 INTRODUCTION

1.1

Precast Concrete Segmental Bridges Precast concrete segmental construction of bridges is commonly adopted

nowadays due to its versatility in coping with difficult terrains and construction constraints with the absence of formwork and falsework. The superstructure of a precast concrete segmental bridge (PCSB) is essentially an assemblage of precast deck segments of 2 to 3 m in length with subsequent prestressing to tie the segments together. One of the popular methods of constructing the PCSB is the balanced cantilever method in which the bridge piers are initially constructed and precast deck segments are extended outwards from the piers in a balanced manner (Figure 1.1(a)). In Figure 1.2, the construction of a PCSB using the balanced cantilever method is illustrated. A gap of 100 to 200 mm wide is usually provided around the mid-span location between the last two approaching segments to facilitate erection (Figure 1.1(b)). In-situ concrete is then cast to ‘stitch’ the segments together, thus making the bridge deck continuous. The formwork for casting the stitch is shown in Figure 1.3. The in-situ concrete stitch is also known as the closure pour or closure joint. Since in-situ concrete stitch found in typical PCSB tends to be narrow in width (Figure 1.4), provision of longitudinal reinforcement to the stitch is usually avoided. Benaim (2008) mentioned that for narrow stitches up to 250 mm wide, non-prestressed longitudinal reinforcement might not be necessary. Therefore most in-situ stitches are only reinforced by the continuity tendons running through the bottom region of the stitch so that the stitch is capable of sustaining considerable sagging moment while the hogging moment capacity is only nominal.

Failure of the in-situ stitches can only occur under exceptional

circumstances when hogging moment is high.

For example, an exceptional

scenario might result from the loss of a bridge pier caused by a serious collision, in which case the cantilevering span could induce substantial hogging moment at -1-

the in-situ stitch of the adjacent span. Hogging moment could also be induced by vertical earthquake loading or even rupturing of sections resulting from terrorist attack or blasting due to vehicle collision (Figure 1.5). In any case, if it really occurred, the subsequent redistribution of internal forces might result in the formation of additional plastic hinges at the in-situ stitches elsewhere along the bridge deck and collapsing mechanisms might eventually form. Thus the in-situ concrete stitches are location of potential weakness that can trigger progressive collapsing mechanism under exceptional circumstances.

1.2

Progressive Collapse and Robustness of Structures Progressive collapse is the sequential spreading of failure and collapse

that is initiated by structural failure in a localized area. Some of the infamous cases include the Ronan Point incident in Newham, United Kingdom in the 1968 and the attack on the World Trade Centre in New York, United States in the 2001. Catastrophic failures have also occurred to bridges and examples include the recent fatal incident of the I-35W Bridge in Minneapolis, United States, which resulted in the collapse of the entire bridge due to the failure of a gusset plate; as well as the incident of the Haeng-Ju Grand Bridge in Seoul, Korea in 1992, which has eleven spans of 800 m length in total collapsing upon the failure of a temporary pier during construction (Starossek, 2006). To prevent progressive collapse, a structure should be designed to have adequate robustness. Structural robustness can be defined as the ability of a structure to guard against disproportionate collapse in the event of a localized failure. The design of most existing structures is in line with the limit state principle with certain factors of safety imposed. However this conventional practice cannot guarantee the robustness of a structure. Although a structure has been designed for the standard load cases at the ultimate limit state (ULS), it may still be subjected to rare extreme loading. Such extreme loading may be so much higher than the standard design loading at ULS that designing for the extreme loading with standard safety factors may be totally unwarranted. Knoll and Vogell (2009) commented that the circumstances of a structural accident had little to do with the numerical safety margins which were used in the design of

-2-

the failed structural system, but had to be attributed to something which was not anticipated altogether at the time and in the context of design and analysis. Therefore, a mere amplification of safety margins might not have prevented the accident. In other words, adequate robustness cannot be simply achieved by raising the safety factors used. Studies in the past have actually identified that structural robustness depends on a number of factors including the redundancy of the structure, the load-carrying capacity, ductility and deformability of the structural elements. Although the importance of robustness design is well recognized, detailed guidance of implementation is rarely found in most design codes around the world.

In the review of the Pentagon crash in 2001, Mlakar et al. (2005)

recommended that there should be a focused effort to review research and practical experience in the area of structural robustness and prepare an authoritative guide that would be useful to the design community. However, up to now, such guide does not seem to exist.

1.3

In-Situ Concrete Stitches, Progressive Collapse, and Robustness of Segmental Bridges Multi-span

PCSBs

are

susceptible

to

progressive

collapse,

as

aforementioned, that may be triggered by the failure of one of the in-situ concrete stitches in an exceptional scenario. Whether the bridge deck will suffer localized damage only or experience progressive collapse depends on how the internal moments redistribute themselves and how well bridge sections can cope with this change in moment that may not have been anticipated. The amount of moment redistribution that a bridge deck can undergo is strongly dependent upon the post-peak inelastic deformability of the structure at the plastic hinge locations. In other words, the post-peak behaviour of a structural element does have marked influence on its ability to redistribute moments and, more importantly, it does affect the robustness of a structure. The joints of precast segmental bridges have been extensively studied by various researchers done in the past. However, the joints studied are either dry or epoxy joints that exist between precast segments, but not the in-situ concrete

-3-

stitching joints. In fact, there has been little, if any, research that has been carried out on in-situ concrete stitches. Therefore, the understanding of the behaviour of in-situ concrete stitches is limited and the effects of the performance of the stitches on the global behaviour of the entire bridge deck, namely redistribution of internal moment and formation of collapse mechanism, are not well understood.

Therefore, comprehensive analyses of prestressed

concrete members with in-situ stitches and an appropriate numerical method for carrying out such analyses are required.

1.4

Numerical Methods for Nonlinear Analyses of Prestressed Concrete Members Although numerical methods for nonlinear analyses of prestressed

concrete members exist, the methods can either predict behaviour up to the peakload-carrying capacity only (Priestley and Park, 1972; Scordelis, 1984), or are too generalized with a substantial number of effects, such as creep, shrinkage, steel relaxation, tension-stiffening, dynamics, etc., taken into account (Cohn and Krzywiecki, 1987; Cruz et al., 1998; Marí, 2000) which may not be necessary for the problem at hand. Since the post-peak behaviour of a structural element has marked influence on its ability to redistribute moments, a method for the fullrange nonlinear analysis of prestressed concrete structures is necessary. The method need not be sophisticated but should be computationally efficient and numerically stable such that the behaviour of the structure can be adequately captured with only the critical effects taken into account.

1.5

Present Research and Outline of the Thesis The present research involves extensive experimental testing and

numerical simulation with the following aims: (a) to gain better understanding of the behaviour of in-situ concrete stitches subject to different combinations of internal forces; (b) to develop a numerical method for full-range nonlinear analysis of prestressed concrete members that can simulate the behaviour of PCSB in an extreme event;

-4-

(c) to examine the effects of the performance of in-situ concrete stitches and various critical structural parameters on the distribution of internal forces and formation of collapse mechanisms along the deck of a PCSB in an extreme event; and (d) to provide guidelines for the design of in-situ concrete stitches that ensure adequate robustness of multi-span PCSB. In Chapter 2, a review of the previous studies related to the present research, namely the joints of the precast concrete segmental bridge, progressive collapse and robustness, methods for nonlinear analyses of prestressed concrete members, and ductility and deformability of concrete sections is presented. The drawbacks of these investigations are discussed. In Chapter 3, the development of the finite element programme for nonlinear post-peak analysis of prestressed concrete members is elaborated. The formulation of elements, the constitutive models assumed and the scheme to obtain admissible solutions are discussed. As the inelastic deformability of member sections can ultimately affect the robustness of a structure, an investigation is conducted alongside with the present study to examine the effects of steel content, yield strength, and prestressing level on ductility and deformability of prestressed concrete sections. The results are presented in Chapter 4. Chapters 5 and 6 then focus on examining the performance of structural members with in-situ concrete stitches. The results of the experimental and numerical studies and the effects of various parameters on the behaviour of beam and shear specimens with in-situ concrete stitches are presented respectively. Numerical study on full-scale in-situ concrete stitches that are found in typical segmental bridges are also conducted apart from those done on the experimental specimens. In Chapter 7, the results of the numerical simulation of full scale in-situ concrete stitches are presented. In Chapter 8, the variation of internal forces and formation of collapse mechanisms of a typical multi-span segmental bridge deck in an extreme event is examined and an overall assessment of robustness is conducted. In Chapter 9, design guidelines for in-situ concrete stitches and robustness of multi-span segmental bridges are provided. -5-

Final conclusions are drawn in Chapter 10 and the possible work in related topics that can be further carried out are recommended. A comprehensive flow chart illustrating the logical flow of the chapters of the thesis is provided in Figure 1.6.

-6-

lifting boom

precast segment Pier

Pier

Pier

(a) extension of precast segments in a balanced manner in-situ concrete stitch

gap is provided to facilitate erection (b) erecting the final segment Figure 1.1. Balanced cantilever construction (not to scale).

Figure 1.2. Construction of a PCSB using the balanced cantilever method (Hewson, 2012).

-7-

Figure1.3. Formwork for casting of the in-situ concrete stitch (Hewson, 2012).

In-situ concrete stitch

Figure 1.4. In-situ concrete stitch in precast concrete segmental bridge.

-8-

Moment diagram

Vehicle collision Stitch

Stitch

(a) vehicle collision resulting explosion at one of the in-situ stitches, which causes immediate rupture Redistribution of moment

Ruptured stitch

Stitch now ruptures due to substantial hogging moment

(b) redistribution of moment occurs as soon as stitch ruptures, resulting in substantial hogging moment at the adjacent stitch Figure 1.5. An example of initiation of progressive failure along a bridge deck.

-9-

Present Research Performance of in-situ concrete stitches in precast concrete segmental bridges

Prong 1: To gain a better understanding of the behaviour of in-situ concrete stitches under different combinations of internal forces

Prong 2: To examine the effects of the performance of in-situ stitches on the global behaviour and robustness of PCSB

Chapter 2: Literature review on relevant topics was carried out. Chapters 5 and 6: Experimental and numerical studies on stitch specimens were carried out.

Chapter 3: A programme for post-peak nonlinear analysis for prestressed concrete member was developed for this investigation.

Chapter 7: Numerical analysis on fullscale stitches was conducted.

Chapter 8: Using the programme developed, the effects of the performance of in-situ on robustness of PCSB was examined.

Chapter 4: Using the programme developed, a side-study on ductility and deformability of RC and PC sections was carried out.

Chapter 9: Based on the results of the studies, various design considerations were proposed.

Figure 1.6. Logical flow of thesis chapters.

-10-

CHAPTER 2 LITERATURE REVIEW

2.1

Joints of Precast Concrete Segmental Bridges

2.1.1

Overview of previous studies A number of studies on the joints of precast concrete segmental bridges

(PCSB) have been conducted previously. In the review by Buyukozturk et al. (1990), these studies are divided into two categories, namely (a) tests performed on models of segmental bridges; and (b) tests on the shear behaviour of the joints. Typical test setup used in the studies of each category is illustrated in Figure 2.1. Up to the time when the review was done, the studies in the first category conducted included those by Moustafa (1974), Finsterwalder et al. (1974), Kashima and Breen (1975), Kupfer et al. (1982), Specht and Vielhaber (1986), Abdel-Halim et al. (1987), and Sowlat and Rabbat (1987); while the studies in the second category conducted included those by Jones (1959), Gaston and Kriz (1964), Sims and Woodhead (1968), Diaz (1975) and Koseki and Breen (1983). The studies of the first category have essentially concluded that the behaviour and ultimate strength of PCSB having epoxied joints is similar to the behaviour of beams that are monolithically cast. The studies of the second category have identified that the main parameters that affect the shear behaviour of the joints are prestressing level, epoxy thickness, shape of the key, concrete strength, surface preparation, and the contact area of the joint. Also, it was found that dry joints (i.e. joints without adhesive agent) generally had strength lower than that of epoxied joints. Apart from those studies reported by Buyukozturk et al. (1990), there are some in the first category that have been conducted recently. Takebayashi et al. (1994) have performed a full-scale destructive test on a PCSB with dry joints and external tendons to improve the understanding of the deformation characteristics of PCSB. Hindi et al. (1995) have also done full-scale tests to compare the behaviour of segmental bridges with dry and epoxied joints. Megally et al. -11-

(2003) have investigated the seismic performance of joints of PCSB by performing cyclic load tests of specimens comprising four full-scale precast segments assembled by prestress. As for the recent studies of the second category, Hewson (1992) examined the performance and the benefits of using dry joints. Zhou et al. (2005) conducted shear tests of joint specimens that aimed to validate the existing AASHTO shear strength formulae (AASHTO, 1999) and it was found that the strength of dry multiple-keyed joints tended to be greatly overestimated, while the strength of epoxied single- and multiple-keyed joints was underestimated by up to 40% of the tested values. There are also those studies done by Turmo et al. (2006) examining the shear strength of dry joints made with concrete with and without steel fibres; and by Issa and Abdalla (2007) to evaluate the performance of the epoxy-jointed match-cast single shear keys in terms of shear, fatigue, ultimate strength, and water tightness for different types of epoxy and curing conditions.

2.1.2

Shear strength of joint of precast concrete segmental bridge The joints of PCSB is essentially a kind of precast concrete connection

and research in precast concrete connection can be dated back to as early as 1960's (Birkeland and Birkeland, 1966; Mast, 1968). Birkeland and Birkeland (1966) introduced the concept of shear friction to model the shear behaviour of precast connections.

The shear friction model predicts the ultimate shear

capacity Vu across a construction joint or interface by Vu = Tu tanφ

(2.1a)

where Tu is the ultimate clamping force acting normal to the interface that is exerted by the steel reinforcement across the interface as shear takes place; and tanφ is essentially the coefficient of friction of the interface surface. Tu is given by Tu = Asfy

(2.1b)

where As is the area of the steel reinforcement and fy is the yield strength of the reinforcement.

Subsequent research on shear friction has been extensively

-12-

carried out by Mattock and his collaborators (Hofbeck et al., 1969; Mattock et al., 1975; Mattock, 1981; Mattock, 2001). Using the shear friction concept and under the guidance of the work by Hofbeck et al. (1969), a formula to predict the shear strength of dry joints of PCSB has been derived and presented in the guidelines by AASHTO for the design of PCSB (AASHTO, 1989; AASHTO, 1999) (referred to as the AASHTO guide hereafter), namely Vnj = Ak

f c′ (12 + 0.017 f pc ) + 0.6 Asm f pc

(2.2)

where Vnj is the shear strength in kips; Ak is the area of the base of all keys in the failure plane (Figure 2.2) in in2; f′c is the compressive cylinder strength of concrete in psi; fpc is the compressive stress in concrete in psi after allowance for all prestress losses determined at the centroid of the cross-section; and Asm is the area of contact between smooth surfaces on the failure plane (Figure 2.2) in in2. At around the same time when Equation (2.2) was first presented in the 1989 version of the AASHTO guide, Buyukozturk et al. (1990) carried out extensive experimental testing on joints of PCSB. In the tests, dry and epoxied joints with and without shear keys were examined. The effects of confining force and thickness of epoxy were studied. Based on the regression analysis of test results, formulae for shear strength τ (psi) were proposed: for flat dry joint:

τ = µσ c ;

(2.3)

for flat epoxied joint:

τ = 10.25 f c′ + 0.98σ c ;

(2.4)

for keyed dry joint:

τ = 7.8 f c′ + 1.36σ c ; and

(2.5)

for keyed epoxied joint: τ = 11.1 f c′ + 1.2σ c

(2.6)

where µ is the coefficient of friction; σc is the confining pressure across the joint in psi; and f′c is the compressive cylinder strength of concrete in psi. The proposed formulae were, however, incorporated in neither the 1989 nor the 1999 version of the AASHTO guide. The formulae for shear strength prediction as given by Equations (2.2) to (2.6) are derived for joints between precast segments that are narrow in width, i.e.

-13-

1 to 3 mm for typical epoxied joints and even less for dry joints. For the in-situ concrete stitching joints that are of concern to the present study, the width is typically much wider (i.e. 100 to 200 mm) and it is doubtful whether or not these formulae can be readily extended to predict the shear strength of the in-situ stitch. In the latest Eurocode for the design of concrete structures (EN1992-2), the shear resistance vRdi between concrete that are cast at different times is estimated as vRdi = c fctd + u σn + ρfy (µ sinψ + cosψ) ≤ 0.5 υ fcd

(2.7)

where c and u are factors that depend on the roughness of the interface; fctd is the tensile strength of concrete; σn is the stress per unit area caused by the minimum external normal force across the interface that can act simultaneously with the shear force, positive for compression, such that σn < 0.6fcd, and negative for tension; ρ is expressed in terms of is the area of steel reinforcement crossing the interface As and is the area of the joint Ai as As/Ai; ψ is the acute angle between the steel reinforcement and the interface; fcd is the concrete compressive strength; and υ is the strength reduction factor. For indented construction joint complying with Figure 2.3, c and u are taken as 0.5 and 0.9 respectively.

If no

reinforcement is provided, the shear resistance of an indented joint as estimated by Equation (2.7) should be vRdi = 0.5 fctd + 0.9 σn

(2.8)

It is stated in the code provisions that Equation (2.7) can be applied to calculate the shear resistance of grouted joints between slab and wall elements, which are similar to the in-situ concrete stitching joints in PCSB. Therefore, it seems plausible that Equations (2.7) and (2.8) can also be applied to estimate the shear strength of in-situ concrete stitches.

However, there is one essential

difference between the two types of joints: the grouted joints between slab and wall elements are mainly designed for shear only, while the in-situ concrete stitches, especially those located near the mid-span, experience substantial bending moment and shear. The interaction between moment and shear may affect the applicability of Equations (2.7) and (2.8) to estimate the shear resistance of in-situ concrete stitches.

-14-

2.2

Progressive Collapse and Robustness The incident of the Ronan Point Apartment in 1968 has raised the

concern for incorporating robustness into structural design and more research that focuses on the robustness of buildings has been conducted, which have ultimately led to new design principles and changes to the building codes (Ellis and Currie, 1998; Beeby, 1999; Alexander, 2004; Pearson and Delatte, 2005; Ellingwood, 2005).

Various studies to derive quantitative evaluation of

vulnerability and damage tolerance of structures have been carried out ever since (Lind, 1995). Following the tragedy of the World Trade Centre in 2001, the call for structural robustness has become more pressing and various approaches to analyse structural robustness have been proposed (Marjanishvili, 2004; Maes et al., 2006; Baker et al., 2008) and more assessment methods based on probability and reliability theory that attempt to quantify structural robustness have also been put forward (Agarwal et al., 2006; Canisius et al., 2007; Starossek and Haberland, 2008a). An extensive summary of different types of progressive collapse has been provided by Starossek (2007) while the existing approaches for the quantification of robustness have also been reviewed by Starossek and Haberland (2008b). Extensive study of progressive collapse and structural robustness has been carried out by a research team at the Imperial College London, UK. A comprehensive framework for the assessment of progressive collapse has been developed and presented by Izzuddin et al. (2007), Izzuddin et al. (2008) and Gudmundsson and Izzuddin (2010). The assessment involves examining the potential for progressive collapse initiated by the sudden loss of a column. An example of its implementation on a typical seven-storey steel-framed building has been demonstrated by Vlassis et al. (2008). The research team has also demonstrated the use of this framework to assess the progressive collapse of a steel-composite building subject to localized fire (Fang et al., 2011; Fang et al., 2012) as well as subject to impact load from failed floor (Vlassis et al., 2009). The framework and examples illustrated have, however, focused mainly on steel structures rather than concrete structures, nor multi-span bridges.

-15-

There have been various studies on the design considerations for robustness of bridges in general (Starossek, 2009) and bridges of particular types, including highway overpasses (Stempfle and Vogel, 2006), cable-stayed bridges (Wolff and Starossek, 2008), railway bridges (Wisniewski et al., 2006), while studies on the robustness of bridges based on probabilistic or risk analysis have also been conducted (Canisius et al., 2007; Baker et al., 2008; Starossek and Haberland, 2008b). Large-scale experimental investigation on the vulnerability of bridge components subject to blasting have been conducted by Seible et al. (2008) in a mitigation programme against progressive collapse of bridge structures. Robustness of bridges during construction has also been examined. Rosignoli (2007) has carried out a study on the robustness and stability of launching gantries and shuttering systems for bridges in construction. Marjanishvili (2004) proposed qualitative procedures to assess structures for progressive collapse. Most of the studies mentioned above either present general qualitative guidelines for robustness design or attempt to derive quantitative indicators that measure robustness of bridges. However, there has been a lack of systematic examination of how the properties of various structural components, the strength of in-situ stitches for instance, affect structural integrity, a view which is also shared by Lee and Sternberg (2008). Moreover, robustness indicators are useful only if accurate prediction of collapse behaviour is available. Numerical methods specifically catering for analyzing structural collapses and predicting collapse loads have been developed in the past. Guralnick and Yala (1998) have developed an energy method to analyse the formation of collapse mechanisms of reinforced concrete frames by examining the sequence of plastic hinge formation and the associated ultimate collapse loads. However, the method and the constitutive models used are for reinforced concrete structures only. It may not be directly applicable to prestressed concrete structures.

Ghali and Tadros (1997) have presented a method for dynamic

collapse analysis of multi-span bridges. The displacement of the structure as it collapses and upon impacting the sea surface and seabed can be determined. However, it is believed that the progressive collapse of typical bridges can be adequately analysed by static analysis. Dynamic analysis can be avoided since -16-

its implementation is less straightforward as the static counterpart and dynamic failure analysis is extremely sensitive to any minor variations of material properties and the time history of the failure process.

2.3

Nonlinear Analysis of Prestressed Concrete Members

2.3.1

Development of numerical methods for nonlinear analysis The solution to a nonlinear analysis usually involves an iterative process

that requires substantial computational resources and for this reason it has not been popular until the 1970’s when computers became more advanced and widely used. Ngo and Scordelis (1967) are among the pioneers to apply the finite element method to the analysis of reinforced concrete structures. Although their method is very restrictive, which assumes linearly elastic material behaviour and the locations of cracks must be predefined prior to the analysis, the concept has nevertheless paved the way for subsequent development of nonlinear analysis of concrete structures.

In the years to follow, nonlinear

analysis methods for concrete structures emerged and such analysis was performed on various kinds of structures.

Full-range nonlinear analysis of

reinforced concrete (RC) frames was performed by Lazaro and Richards (1973) to examine the formation of plastic hinges and mechanisms as well as the collapse load. Similar analysis on RC frames and RC beams was done by Darvall and Mendis (1985) and Kim and Lee (1992) taking into account the hardening of steel and softening of concrete, while Cope and Vasudeva Rao (1977) devised a nonlinear analysis procedure using the constant stiffness iteration approach to analyse the post-cracking range of RC slab, which at that time was not widely studied. Substantial work on the development of numerical methods for prestressed concrete (PC) members has also been conducted from the 1970's and up to recently (Priestley and Park, 1972; Kang and Scordelis, 1980; Scordelis, 1984; Warner and Yeo, 1986; Cohn and Krzywiecki, 1987; Campbell and Kodur, 1990; Cruz et al., 1998; Marí, 2000), which will be discussed in the next section. By the 1990's, methods of nonlinear analysis have become more sophisticated. Hu and Schnobrich (1990) developed a constitutive model of

-17-

concrete specifically catering for the cracked behaviour and taking into account the effects of tension stiffening, stress degrading, reduction in shear stiffness after the cracking of concrete. Riva and Cohn (1990) developed a generalized finite element programme for nonlinear analysis of concrete structures using realistic material models and accounting for the full-range behaviour.

2.3.2

Numerical methods for prestressed concrete members Priestley and Park (1972) were among the earliest to develop numerical

methods for nonlinear analysis of such members in their study of moment redistribution in continuous PC beams. Their methods essentially idealized the beam as a series of beam segments whose behaviour was given by the relationship between moment and average curvature of the section at the centre of the segment. Compatibility conditions between segments were applied to determine the bending moment distribution and, eventually, the ultimate load of the beam. This method of analysis can predict the behaviour of the members up to the peak capacity only. Kang and Scordelis (1980) and Scordelis (1984) presented numerical models for nonlinear analysis of both reinforced and prestressed concrete structures based on the finite element method.

Geometric and material

nonlinearities, as well as time-dependent effects, have been taken into account. The models presented in Scordelis (1984) can be made up of combinations of frame, plane stress, shell, and solid elements so that the method is applicable to the analysis of frames, slabs, panels and thin shells.

In formulating the

equilibrium relationship, tangent stiffness is used and this can result in numerical difficulties when the behaviour of the structure advances to the softening state (i.e. the post-peak state) since the tangent stiffness may either has a value of zero or being negative (Figure 2.4). Therefore, it is doubtful whether this method can adequately capture the post-peak behaviour. A numerical method of full-range nonlinear analysis has been developed by Warner and Yeo (1986) in their investigation of ductility requirement for prestressed concrete members. The method is similar to that by Priestley and Park (1972), which idealizes the member as a series of beam elements whose behaviour is governed by the moment-curvature relationship at the centre of the -18-

element. The secant stiffness is adopted in the method of analysis and this should allow greater numerical stability for analysis at the post-peak state. Therefore, this method offers a promising way for analyzing prestressed concrete members with limited ductility in which softening of plastic hinges would occur. Campbell and Kodur (1990) has implemented the method by Warner and Yeo (1986) in their computer programme. Cohn and Krzywiecki (1987) developed a sophisticated computer programme STRUPL-1C to analyze various structural plasticity problems in prestressed concrete, such as the formation of plastic hinges and collapse mechanisms. A structure is idealized by one-dimensional prismatic elements between nodes of the structural members. Lumped plasticity model rather than plastic hinge with finite length is adopted for modelling the plastic behaviour of the element. Cruz et al. (1998) presented a method for nonlinear time-dependent analysis of segmentally constructed structures. Unlike the methods as previously mentioned, concrete and tendons are discretely modelled by beam and tendon elements respectively.

Both instantaneous and long-term behaviour are

considered since the method is specifically caters for the behaviour of the structure from the beginning of construction to the end of its service life. The method was implemented by Marí (2000) and an example was presented to demonstrate the capability of the model to reproduce the effects of a complex construction process in which cracking, creep and shrinkage of concrete and relaxation of prestressing steel took place.

2.4

Ductility and Deformability of Prestressed Concrete Sections Numerical and experimental studies of the ductility of reinforced concrete

sections and members have been carried out by various researchers (Desayi et al., 1974; Park and Dai, 1988; Pam et al., 2001; Ashour, 2002; Kwan et al., 2002; Bernardo and Lopes, 2004; Whitehead and Ibell, 2004; Rao et al., 2008; Bai and Au, 2009) but there have been relatively few studies on the ductility of prestressed concrete sections. Thompson and Park (1980) examined the effects of the prestressing steel content and distribution on the ductility of prestressed

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concrete sections and, based on their theoretical and experimental findings, recommended that a limit on the prestressing steel content should be imposed. They have, however, not considered the possible effect of the prestressing force. Naaman et al. (1986) studied both theoretically and experimentally the effects of the non-prestressed and prestressing steel contents, prestressing force, prestressing steel grade, concrete strength and concrete confinement. From each moment-curvature curve, they derived the yield curvature as the curvature at the intersection point between the initial linear portion and the final linearized portion, and the ultimate curvature as the curvature at maximum moment. They found that decreasing the prestressing force had an unfavourable effect on the ductility, or, in other words, increasing the prestressing force would increase the ductility. Cohn and Riva (1991) studied by numerical analysis the effects of various parameters, including the sectional shape, reinforcement index and ratio of prestressing steel to total steel. They defined the yield curvature as the curvature at which the strain increment in the reinforcing or prestressing steel reached a value of 0.2% and the ultimate curvature as the curvature at maximum moment. Based on the numerical results they have come up with, the ductility increases with the ratio of prestressing steel to total steel and therefore prestressing has a positive effect on the ductility. However, the degree of prestressing (i.e. a dimensionless parameter directly proportional to the prestressing force applied) was kept constant and not considered in the study. Zou (2003) conducted a state-of-the-art review on the existing ductility indices and addressed their drawbacks in measuring the ductility of beams prestressed by fibre-reinforced polymer (FRP) tendons. A new index was proposed and it was verified by correlating the values of the index to the experimental failure modes of beams prestressed by FRP and steel tendons. However, it was not the focus then to examine the effect of prestressing on the ductility of prestressed concrete sections. Du et al. (2008) and Au et al. (2009) investigated the ductility of prestressed concrete members with unbonded tendons by numerical analysis. The ductility of a prestressed member with unbonded tendons has been found to be quite different from that of a similar prestressed member with bonded tendons. -20-

Hence, the bonding of prestressing tendons has significant effects on the ductility and separate studies are needed for prestressed members with bonded and unbonded tendons. Moreover, extensive parametric studies show that, for prestressed members with unbonded tendons, the ductility decreases as the prestressing steel content increases but increases as the prestressing force increases.

2.5

Conclusions In this chapter, an extensive review on various relevant topics to the

present research, namely the joints of PCSB, progressive collapse, robustness, and methods for nonlinear analysis of PC members, has been conducted. Various drawbacks and limitations of the methodologies adopted in the previous studies have been identified and this has motivated new approaches to be adopted in the present research.

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Load from actuator End block

Segment-to-segment joint Prestressing tendon Strong floor

Precast deck segments (a) Category (a)

Load from actuator Precast units Prestressing rod Joint with shear key

(b) Category (b)

Figure 2.1. Test setups typically used in the studies belonging to Categories (a) and (b) (not to scale).

-22-

Ak (sum of all these individual areas)

Asm (sum of these individual areas)

Base areas of shear keys Figure 2.2. Definition of Ak and Asm.

45° ≤ ψ ≤ 90° Reinforcement ≤ 10r ψ

≤ 30°

new concrete

Shear r ≥ 5 mm

≤ 10r

old concrete

Figure 2.3. Indented construction joint as specified in Eurocode 2.

-23-

Tangent stiffness of zero

M

Tangent stiffness being negative

Potential numerical instability

φ

Figure 2.4. Potential numerical instability when analysis proceeds to the postpeak state as a result of adopting tangent stiffness in the formulation.

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CHAPTER 3 POST-PEAK ANALYSIS OF PRESTRESSED CONCRETE MEMBERS BY FINITE ELEMENT METHOD

3.1

Overview In this chapter, the development of a numerical method for nonlinear

post-peak analysis of prestressed concrete members is elaborated. The general scheme of the method is introduced, followed by discussions on the formulation of the element used, the constitutive models adopted, and the iteration scheme employed for determining admissible solution.

3.2

General Scheme The finite element method is adopted for the numerical programme

developed, which caters mainly for the analysis of typical precast concrete segmental bridge deck that is provided with simple supports that do not restrain axial deformation.

With this configuration, the bridge deck is essentially a

prestressed continuous beam that can be idealized as a series of beam elements having rotational and vertical translational degrees of freedom at each node with the axial degree of freedom neglected. Such a beam element is well suited for analysing the global behaviour of the deck and is computationally efficient for nonlinear analysis. Assuming that the properties at intermediate sections are related to those at the nodes by suitable rules of interpolation, the constitutive behaviour of the beam element is governed by the corresponding momentcurvature relationship of sections at the nodes of the element.

Further

elaboration on the constitutive behaviour of the element will be given in due course. The beam element used is illustrated in Figure 3.1, where u1 and u3 are the vertical translational degrees of freedom (DOFs) at Nodes a and b respectively; u2 and u4 are the rotational DOFs at Nodes a and b respectively; (EI)a and (EI)b are the flexural rigidities at Nodes a and b respectively; φra and

-25-

φrb are the residual curvatures at Nodes a and b respectively; and L and x are the length and abscissa of the element respectively. With the bridge deck discretized, incremental load or displacement is then applied on the structure, upon which a series of iterations are performed to obtain the admissible nodal displacements that satisfy the constitutive behaviour of each element. In summary, the method of analysis essentially involves three steps, namely (a) discretising the bridge deck by a series of beam elements; (b) performing section analysis on the nodal sections of each element to obtain the moment-curvature relationship; and (c) performing iterations to obtain the admissible nodal forces and displacements for each imposed load or displacement increment. The present method of analysis differs from that adopted by similar programmes previously developed mainly in three aspects namely (i) the behaviour of element is governed by the moment-curvature relationship of sections at the element nodes rather than the section at the middle of the element; (ii) initial stiffness rather than tangent stiffness is used in element formulation; and (iii) the iteration process involves updating the residual curvature instead of the flexural rigidity (EI). The advantage of (i) is that abrupt change in curvature across element boundary can be minimize, especially when elements are located within regions with rapid change in curvature. The use of tangent stiffness can present numerical difficulties when the behaviour of the structure advances to the softening state, and the advantage of (ii) is that such problem can be avoided. Since initial stiffness is used, the iteration process will involve iterating for the residual curvature rather than the flexural rigidity.

3.3

Finite Element Formulation The constitutive behaviour of a section, which includes loading and

unloading, is governed by the moment-curvature relationship (Figure 3.2) M = EI (φ − φ r )

(3.1)

where M is the bending moment, EI is the flexural rigidity that is taken as the slope of the initial elastic branch of the moment-curvature curve, and φ and φr are -26-

the section curvature and the residual curvature respectively.

The moment-

curvature relationship is obtained by section analysis (Au and Leung, 2011). The inelastic behaviour of the section is thus taken into account by the residual curvature. The use of such M-φ curves helps to account for the limited ductility of plastic hinges on the overall structural behaviour.

Assuming the tendon

eccentricity and other sectional properties to vary linearly along the element, the values of EI and φr within the element can be interpolated from those of Nodes a and b at the ends. The deck is assumed to have sufficient shear reinforcement so that shear failure can be ruled out. Derivation of the force-displacement relationship for each element using the potential energy approach gives the load vector f as f = K e δ - ∫ B T [N a (EI )a + N b (EI )b ]( N aφ ra + N bφ rb ) B dx

(3.2)

where the element stiffness matrix Ke, the strain matrix B, and the interpolation functions Na and Nb are given respectively as K e = ∫ B T [N a (EI )a + N b (EI )b ] B dx

(3.3)

 6 12 x 4 6 x 6 12 x 2 6 x  B =  − 2 + 3 ;− + 2 ; 2 − 3 ;− + 2  L L L L L  L L  L

(3.4)

x  N a = 1 −  L 

(3.5)

x Nb =   L

(3.6)

δ is the displacement vector; (EI)a and (EI)b are the flexural rigidities at Nodes a and b respectively, φra and φrb are the residual curvatures at Nodes a and b respectively; x is the x-coordinate in the element axial direction; and L is the length of element as defined earlier in Figure 3.1.

3.4

Section Analysis To obtain the moment-curvature curve as illustrated in Figure 3.2, section

analysis, which is also referred to as moment-curvature analysis, is performed for

-27-

the section at each node of the element. Such analysis is done numerically by a computer programme developed based on the approach of Ho et al. (2003), which is intended for analysing reinforced concrete sections. Modifications have been made such that fully and partially prestressed sections can be analysed.

3.4.1

Constitutive models used

Constitutive model for concrete The constitutive model of concrete under uniaxial compression proposed by Attard and Setunge (1996), which is applicable to concrete strength ranging from 20 to 130 MPa, is adopted. This model is comprehensive and has taken into account the various effects on concrete under uniaxial action, such as the effect of confinement. In this model, the relationship between concrete stress σc and strain εc is

σc f co

A(ε c ε co ) + B (ε c ε co ) = 2 1 + ( A − 2 )(ε c ε co ) + (B + 1)(ε c ε co ) 2

(3.7)

where fco and εco are the uniaxial compressive strength and the strain at peak stress, respectively. The formulae for determining the values of A and B are as follows. (a) For the ascending branch of the stress-strain curve:

A = Ec ε co f co ;

(3.8a)

( A − 1)2

(3.8b)

B=

0.55

− 1 ; and

(b) For the descending branch of the stress-strain curve: f ci (ε ci − ε co ) ; ε co ε ci ( f co − f ci ) 2

A=

(3.9a)

B=0.

(3.9b)

The parameters Ec, εco, fci and εci in the above formulae can be determined using the following equations: Ec = 4370 (fco)0.52 ;

(3.10a)

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εco = 4.11 (fco)0.75 / Ec ;

(3.10b)

fci/fco = 1.41 – 0.17 ln(fco) ; and

(3.10c)

εci/εco = 2.50 – 0.30 ln(fco).

(3.10d)

When the concrete strain decreases during strain reversal at the post-peak state or during unloading, the stress-strain path is assumed to follow a straight line having a gradient equal to the initial tangent modulus. The constitutive model for concrete accounting for such stress-path dependence is shown in Figure 3.3. It is assumed that concrete cannot take tension and the effect of tension stiffening is not considered. Such assumptions are reasonable for the present analysis which primarily focuses on the ultimate state of the structure.

Constitutive model for non-prestressed steel The constitutive behaviour of non-prestressed steel is assumed to be linearly elastic - perfectly plastic. To allow for strain reversal, namely the decrease of strain despite monotonic increase of curvature, stress-path dependence is taken into account by incorporating an unloading path having a gradient equal to the initial tangent modulus. The stress-strain relationship can be described by: (a) Elastic stage:

σs = Es εs

(3.11a)

(b) Beyond yielding:

σs = fy

(3.11b)

(c) On unloading after yielding: σs = Es (εs – εp)

(3.11c)

where σs, εs, Es and fy are the stress, strain, elastic modulus and yield strength of the non-prestressed steel, respectively, and εp is the residual strain. The residual strain εp is also the permanent strain at zero stress, which can be evaluated from the stress and strain values at the previous loading step as εp = εs – σs / Es

(3.11d)

The constitutive model for the non-prestressed steel is shown in Figure 3.4.

-29-

Constitutive model for prestressing steel For the prestressing steel, the constitutive model proposed by Menegotto and Pinto (1973) is adopted. Naaman (1983) showed that this model can adequately simulate the stress-strain behaviour of prestressing steel. In this model, the stress is related to the strain by the following equations:



σ ps = E ps ε ps Q + 

[1 + (E

1− Q

 1N  

N ps ε ps ( Kf py ) )]

Q = ( f pu − Kf py ) (E ps ε pu − Kf py )

(3.12a)

(3.12b)

where σps, εps, Eps, fpy, fpu and εpu are the stress, strain, elastic modulus, yield stress, ultimate stress and ultimate strain, respectively, and N, K and Q are empirical coefficients with values depending on the type of tendon used. Naaman (1985) recommended that for 7-wire strands with an ultimate stress fpu of 1863 MPa, the values of N and K may be taken as 7.344 and 1.0618 respectively, the value of fpy may be taken as 85% of fpu and the value of εpu may be taken as 0.069. His recommended values are adopted in the present study. The constitutive model for prestressing steel is shown in Figure 3.5.

3.4.2

Performing the analysis An iterative process with the prescribed curvature applied incrementally

is adopted. At each iteration step, the strain variation is determined assuming that plane sections remain plane after bending, and the stresses in the concrete and steel are evaluated from their respective constitutive models. Axial equilibrium is used to determine the position of neutral axis after which the resisting moment is calculated. This iterative process is repeated until sufficient length of the fullrange moment-curvature curve has been obtained. The moment-curvature curves obtained from the section analyses are then input into the computer programme for global structural analyses for assignment to each element.

-30-

3.5

Iteration Process The analysis starts by forming the element and global stiffness matrices

Ke and Kg respectively. As mentioned above, the iteration scheme of the present technique adopts the initial stiffnesses, and hence the global stiffness matrix remains unchanged throughout the entire analysis. The value of flexural rigidity EI required for computing the element stiffness matrix Ke is taken as the slope of the elastic region of the moment-curvature curve corresponding to each element, as shown in Figure 3.2. The cumulative incremental load or displacement is applied at the specified location, upon which iterations are performed and the residual curvature of each element is updated until a set of admissible displacements and forces at all nodes is obtained. The procedure of the iteration process at any load step i is explained as follows. Step 1. A set of nodal displacements and forces is determined by solving Equation (3.2). The curvature φin at the representative section of each element can be calculated from the nodal displacement vector δn of that element by

φin = B δ n

(3.13)

where the subscript i and superscript n refer to the ith load step and nth iteration step respectively.

The moment min corresponding to curvature φin is then

calculated using Equation (3.1). The residual curvature φ r in Equation (3.1) is taken as the residual curvature determined from the previous load step or iteration step. Step 2.

For each element, the maximum permissible moment M in

corresponding to the calculated curvature φin is obtained from the momentcurvature curve.

The moment-curvature curve is treated effectively as an

envelope in the sense that the calculated moment cannot exceed the moment given by the curve at a certain curvature. In other words, M in is the moment on the moment-curvature curve at the calculated curvature φin . Step 3. The calculated moment min is checked against the maximum permissible moment M in . If min is greater than M in by a certain tolerance, the

-31-

maximum moment M in corresponding to the calculated curvature is adopted and the residual curvature is updated accordingly as

(φ )

n +1 r i

M in =φ − EI n i

(3.14)

which is inferred from the re-arranged form of Equation (3.1), and (φr )i

n +1

is the

updated residual curvature to be used in the next iteration step. On the contrary, if min is less than M in , then min will be taken as the moment that the section is subject to and the residual curvature will not be updated. Once the residual curvatures of all elements have been determined, Steps 1 to 3 are repeated until the calculated moments and curvatures of all elements are sufficiently close to the moment-curvature curve of the corresponding element. The iteration process can also be demonstrated graphically using Figure 3.6. Figure 3.6(a) shows that the moment and curvature of a typical element increase from Points I to J. An enlarged view of the moment-curvature curve between Points I and J is shown in Figure 3.6(b) to illustrate the iteration process. Suppose that the initial moment and curvature of the element at the beginning of load step i lie on Point I, and the element has residual curvature (φr )i . Referring to Figure 3.6(b), the calculated moment and curvature after the first iteration step are mi1 and φi1 respectively, giving Point 1 with moment exceeding the maximum permissible moment M i1 as denoted by Point 2. The residual curvature is thus updated with a new value (φr )i . The computer programme then proceeds to the 1

second iteration step, which gives the moment and curvature corresponding to Point 3. As the moment of Point 3 does not go beyond the moment-curvature curve, the residual curvature is kept unchanged. Subsequent iterations yield Point 4 and so forth. The iteration cycles will be terminated as the values of moment and curvature converge to those at Point J. Suppose that upon reaching Point J, the section in this element undergoes unloading (Figure 3.6(a)) because some other elements of the structure have reached their peak moments and gone to the post-peak range. The moment and curvature of the unloading element will follow the unloading path of the

-32-

moment-curvature curve from Point J, which is assumed to be parallel to the elastic slope. Neither Warner and Yeo (1986) nor Campbell and Kodur (1990) have clearly explained their treatment of the moment-curvature relationship as a section undergoes unloading. The flowchart of the entire iteration process is presented in Figure 3.7. Although the nonlinear M-φ relationship inclusive of the post-peak behaviour has been taken into account, the geometric nonlinearity of large deflection is not accounted for. However this is considered sufficient to evaluate the essential behaviour relevant to robustness of this type of bridges.

3.6

Conclusions The development of a method for nonlinear post-peak analysis of

prestressed concrete members have been presented in this chapter. The member is idealized as a series of beam elements whose behaviour is governed by the moment-curvature relationship at the element nodes. The method of analysis essentially involves three steps, namely (i) discretization; (ii) performing section analysis on nodal sections to obtain the moment-curvature relationship; and (iii) performing iterations. The present method has adopted formulation and iteration strategies different from those used by programmes previously developed that improve numerical stability of the analylsis especially at the post-peak stage.

-33-

u1 u2

(EI)b

(EI)a

φrb

φra a

u3 u4 b

x L

Figure 3.1. Beam element used in the analysis.

M i

Mi

EI

EI 1

1

φri φ i

φ

Figure 3.2. Constitutive model of the beam element used.

-34-

σc Loading curve Unloading curve

Ec 1

Ec

1

εc

Figure 3.3. Constitutive model of concrete. σs σs = fy fy

Es

Es

1

1 εs

εp

Figure 3.4. Constitutive model of non-prestressed steel. σps fpu Kfpy

tan-1(QEps)

fpy

Eps 1

Eps 1

εpy

εpu

εps

Figure 3.5. Constitutive model of prestressing steel.

-35-

M J I Loading EI 1 Unloading

EI

φ

(φr)j (φr)i

(a) M m

1

1 i

4 J

M i1

I

2 3

φ i1

(φr )i

( )

φ r 1i

φj

φ

(b) Figure 3.6. The iteration process: (a) moment and curvature of a typical element increasing from Points I to J; (b) iterations of moment and curvature from Points I to J.

-36-

Begin Input all necessary geometric data; and generate mesh Perform section analysis; and obtain and store moment-curvature (M-φ) curves of the section at each node Form global stiffness matrix K For load step i

i=i+1

Apply non-incremental and incremental force or displacement

For equilibrium iteration step n Solve for displacement δ from the force-displacement relationship f = Kδ - ∫ B T [N a (EI )a + N b (EI )b ](N a φ ra + N b φ rb ) B dx

n

Obtain bending moment mi and calculate the corresponding n curvature φi at the nodes of each element by φin = min / EI + (φr )in Determine the moment M i on the M-φ curve corresponding to φin for each element n

n

Check convergence n || M mi || < tolerance for all elements Yes n i -

No

Check if i = last load step

No

n=n+1

Yes End Figure 3.7. Flowchart of the nonlinear finite element analysis. -37-

n

For element with || M i - mi || > tolerance, update residual curvature by n +1 (φr )i = φin − M in / EI

(This page is intentionally left blank)

-38-

CHAPTER 4 DUCTILITY AND DEFORMABILITY OF PRESTRESSED CONCRETE SECTIONS

4.1

Overview Catastrophic collapse of a structure is often caused not by the design

loads at the ultimate limit state (ULS) but by extreme events, such as high energy impact, strong earthquake or terrorist attack. When an extreme event occurs, the loads acting on the structure could far exceed the design loads at ULS and some members of the structure might have reached the post-peak state at which a member has already exhausted its peak load carrying capacity. At the post-peak state, the ductility or the ability to sustain inelastic deformation without excessive reduction in load carrying capacity would become the most important parameter governing the safety of the structure. Hence, for safety beyond the ULS, the provision of sufficient ductility is at least as important as the provision of adequate strength. It is well known that the flexural ductility of a reinforced concrete section depends mainly on whether the section is under-reinforced or over-reinforced. If the section is under-reinforced such that the tension steel yields before the concrete fails in compression, the section would fail in a ductile manner. Conversely, if the section is over-reinforced such that the tension steel does not yield even when the concrete fails in compression, the section would fail in a brittle manner. Therefore, the amount and yield strength of the tension steel, which together determine whether the section is under-reinforced or overreinforced, are the major factors affecting the flexural ductility of a reinforced concrete section. For a prestressed concrete section, however, the flexural ductility is much more complicated. As for a reinforced concrete section, the flexural ductility of a prestressed concrete section depends on whether the section is under-reinforced or over-reinforced, but it is not so clearly defined. Since prestressing has great effects on the flexural behaviour of the section, it may be envisaged that the -39-

prestressing force and steel content should also have some effects on the flexural ductility. Hence, quite obviously, there are more structural parameters affecting the flexural ductility of a prestressed concrete section than a reinforced concrete section. Actually the usual practice of measuring the flexural ductility in terms of the ductility factor defined as the ratio of ultimate curvature to yield curvature could be misleading, because a reduction in the yield curvature without any increase in the ultimate curvature could produce an apparent increase in ductility. Careful study of the previous research findings leading to the conclusion that increasing the prestressing force would increase the ductility, such as those by Naaman et al. (1986), revealed that the apparent increase in ductility was due solely to the reduction in yield curvature rather than any increase in ultimate curvature. Therefore the common belief that prestressing increases ductility has to be critically re-examined. To provide justification for this argument and investigate how the ability of a concrete section to withstand inelastic deformation could be better measured, a parametric study on the effects of various parameters on the full-range moment-curvature behaviour and flexural ductility of reinforced and prestressed concrete sections is carried out.

4.2

Method of Analysis Theoretical moment-curvature analysis of reinforced and prestressed

concrete sections is performed using the method as presented in Section 3.3. The constitutive models of concrete, non-prestressed, and prestressing steel used in the analysis are those given in Section 3.3.1.

4.3

Parametric Study In the light of the study of size effect on full-range analyses by Bai (2006),

a normalisation approach is adopted with suitable dimensionless parameters so that the findings can be applied to cases of the same material properties and reinforcement arrangement but of different dimensions. The ratios of dimensions in the vertical direction are found to be essential parameters that govern structural behaviour. A parametric study is carried out to examine the effects of

-40-

various parameters on the flexural strength, yield curvature, ultimate curvature and curvature ductility factor of both reinforced and prestressed concrete sections. Figure 4.1 shows the sections analyzed, which are all rectangular with an overall depth h of 1,400 mm and a width b of 700 mm. In order to focus on the effects of prestressing, the uniaxial compressive strength of the concrete fco is taken as 60 MPa. Since the elastic moduli of non-prestressed and prestressing steel Es and Eps respectively seldom vary, they are just taken as 200 GPa. For the reinforced concrete sections containing only non-prestressed steel, the effective depth to the tension steel dt is taken as 0.9 h. The yield strength of tension steel fyt varies from 460 to 620 MPa, whereas the tension steel ratio ρst = Ast/bdt varies from 1 to 4%, in which Ast is the area of tension steel. For the prestressed concrete sections containing only prestressing steel, the effective depth to the prestressing steel dp is also taken as 0.9 h and the ultimate strength of the prestressing steel fpu is 1860 MPa. The effective prestressing stress fpe is so varied that the prestressing stress ratio fpe/fpu ranges from 0.3 to 0.7. The prestressing steel ratio ρps = Aps/bdp varies from 0.2 to 1.4%, in which Aps is the area of prestressing steel. These ranges of parameters are so chosen that the maximum compressive stress of the concrete section at transfer would not exceed 0.6 times the concrete strength at transfer and no tensile stress occurs in the concrete section, taking into account the bending moment induced by dead load. These limitations are commonly adopted in various codes of practice, such as Eurocode 2 (European Committee for Standardization, 2004), ACI 318 (ACI Committee 318, 2005) and CSA A23.3 (CSA Technical Committee on Reinforced Concrete Design, 1994).

4.4

Moment-Curvatures Curves and Ductility Factors From the above analysis, full-range moment-curvature curves of

reinforced and prestressed concrete sections, each comprising of a pre-peak branch and a post-peak branch, are generated. Based on these curves, the yield curvature and ultimate curvature were determined for detailed study. Sagging moments and curvatures are taken as positive for convenience of subsequent discussions.

-41-

4.4.1

Moment-curvature curves Figure 4.2(a) shows the moment-curvature curves of reinforced concrete

sections with tension steel yield strength fyt = 580 MPa and tension steel ratio ρst = 1, 2, 3 or 4%. It can be seen from these curves that as ρst increases, the flexural strength increases but the ductility decreases. For tension steel ratio ρst not exceeding 3%, the section is under-reinforced and a distinct yield point can be identified in the moment-curvature curve. For tension steel ratio at 4%, the section becomes over-reinforced and there is no distinct yield point in the moment-curvature curve. Hence, the tension steel ratio has great effects on the flexural strength and ductility of reinforced concrete sections. Figure 4.2(b) shows the moment-curvature curves of the prestressed concrete sections with prestressing steel ratio ρps = 1% and prestressing stress ratio fpe/fpu = 0.3, 0.4, 0.5, 0.6 or 0.7. Unlike those of the reinforced sections, the moment-curvature curves of the prestressed sections do not start with zero moment at zero curvature, but extend into the negative curvature region because of the prestress. Furthermore, the moment-curvature curves of the prestressed sections do not exhibit any distinct yield points. It is also seen that as fpe/fpu increases, the pre-peak branch of the curve gives not only a higher resisting moment but also a more rapid increase in resisting moment with the curvature, whereas the post-peak branch changes very little. Hence, the prestressing stress ratio has more effects at the pre-peak state than at the post-peak state.

4.4.2

Ductility factors From the moment-curvature curves, the curvature ductility of the concrete

sections analyzed may be evaluated in terms of a ductility factor µ, which is usually defined as the ratio of ultimate curvature φu to yield curvature φy, namely

µ=

φu φy

(4.1)

However, different researchers have been using different definitions for φu and φy. Regarding the ultimate curvature φu, some researchers, such as Naaman et al. (1986), defined φu as the curvature at maximum moment with the resisting

-42-

moment of the section at the post-peak state ignored, while others, such as Du et al. (2008), defined φu as the curvature at which the resisting moment has reached the peak and dropped to 85% of the peak resisting moment. In order to take into account the resisting moment at the post-peak state, the definition used by Du et al. (2008) is adopted, as illustrated in Figure 4.3. The major difficulty in the determination of yield curvature φy is that the moment-curvature curve does not always have a distinct yield point. To overcome this, the yield curvature has been arbitrarily taken as the curvature at the point on the moment-curvature curve marking obvious transition from elastic to inelastic deformation. For example, Naaman et al. (1986) defined the yield curvature as the curvature at the intersection point between the initial linear portion and the final linearized portion of the moment-curvature curve. However, the method they used to find the intersection point for prestressed sections is not the same as that for reinforced sections. The definition used by Naaman et al. (1986) is adopted herein, except that a consistent method is employed to find the intersection point for both reinforced and prestressed sections. Prestressed sections behave quite differently from reinforced sections mainly in the existence of positive moment at zero curvature and within the negative curvature region. For a reinforced section, the origin of the moment-curvature curve with zero moment and zero curvature is usually taken as the reference point. However for a prestressed section, there is no such origin that can be taken as the reference point. Hence, the reference point for a prestressed section is arbitrarily chosen as the point at which the stress at the extreme tension fibre is zero at transfer. This is a reasonable assumption for the critical sections of members with eccentric prestressing, which have been properly designed for economy. The moment and curvature at this reference point are denoted by Mo and φo, respectively. For a reinforced section, this reference point is just the origin with Mo = 0 and φo = 0. For a prestressed section, this reference point is somewhere in the negative curvature region, as illustrated in Figure 4.3. Having found the reference point (φo, Mo), the initial linear portion of the moment-curvature curve is constructed by drawing a straight line through the reference point and the point on the pre-peak branch at which the resisting

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moment M corresponds to 75% of the increment necessary to reach the peak resisting moment Mp, namely when M = Mo+0.75(Mp-Mo). On the other hand, the final linearized portion is just taken as the horizontal line passing through the peak of moment-curvature curve. The intersection between these two straight lines is taken as the “yield point” from which the yield curvature φy is determined, as shown in Figure 4.3(b). This method for determination of the yield point is consistent with the methods previously used for both reinforced sections (Kwan et al., 2002) and prestressed members (Park and Falconer, 1983; Du et al., 2008). Similarly the ultimate curvature φu is taken as the curvature of the point on the post-peak branch with resisting moment M corresponding to 85% of the increment from Mo necessary to reach the peak resisting moment Mp, namely when M = Mo+0.85(Mp-Mo). Note that the definition of curvatures φy and φu for reinforced concrete sections as shown in Figure 4.3(a) is a special case of that for prestressed concrete sections as shown in Figure 4.3(b) on setting Mo = 0 and φo = 0. Taking a simply supported prestressed concrete member as example, prestressing creates negative curvatures and camber. If such a beam is load tested to failure in order to determine the ductility, the initial conditions should have included the effects of dead load and prestressing. If the beam has been properly designed for economy, the bottom fibre should have maximum compressive stress while the top fibre should have roughly zero stress, which satisfies the conditions for the reference point. To encompass prestressed concrete sections as well, the ductility factor µ is rewritten in a more general form as

µ=

φu − φ o φ y − φo

(4.2)

which degenerates to Equation (4.1) on noting that φo = 0 for reinforced concrete sections. This approach is also consistent with experimental practice. Imagine that a number of prestressed concrete and reinforced concrete beam specimens are tested by displacement control to failure. All displacement measurements are set zero at the beginning of experiment, which implies taking the initial conditions as reference. The above method to define the ductility factor implies

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that the same procedure applies to both reinforced and prestressed concrete sections.

4.5

Reinforced Concrete Sections The variations of the flexural strength and ductility of the reinforced

concrete sections, expressed in terms of dimensionless parameters Mp/fcobh2 and µ, with the tension steel ratio ρst at different steel yield strength fyt are plotted in Figures 4.4(a) and 4.4(b), respectively. From Figure 4.4(a), it is observed that the flexural strength parameter Mp/fcobh2 increases with increasing tension steel ratio and/or steel yield strength until the section becomes over-reinforced, and then the flexural strength parameter slowly converges to a constant value of around 0.25 despite further increases in tension steel ratio and steel yield strength. This observation implies that the steel yield strength has significant effect on the flexural strength only when the section is under-reinforced and has basically no effect on the flexural strength when the section is over-reinforced. Figure 4.4(b) shows that the flexural ductility µ decreases with increasing tension steel ratio and/or steel yield strength until the section becomes over-reinforced, and then the flexural ductility slowly converges to a constant value of around 1.6 despite further increases in tension steel ratio and steel yield strength. This observation implies that although the use of more and/or higher strength steel increases the flexural strength while the section is still under-reinforced, it also reduces the flexural ductility. The variations of the yield curvature φy and ultimate curvature φu with the tension steel ratio ρst at different steel yield strength fyt are plotted in Figures 4.5(a) and 4.5(b), respectively. From Figure 4.5(a), it is seen that the yield curvature φy increases with increasing tension steel ratio and/or steel yield strength until the section becomes over-reinforced. Then the yield curvature decreases as the tension steel ratio further increases and is no longer dependent on the steel yield strength. Hence, the effects of the tension steel ratio and steel yield strength depend on whether the section is under- or over-reinforced. When the section is under-reinforced, the yield curvature increases as the tension steel ratio or steel yield strength increases because a larger curvature is needed to

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cause yielding of the tension steel at higher tension steel ratio or steel yield strength. When the section is over-reinforced, the yield curvature decreases as the tension steel ratio increases because a higher tension steel ratio leads to a higher initial stiffness and thus an apparent yield point at smaller curvature, and the yield curvature becomes independent of the steel yield strength because the tension steel actually does not yield at all. Figure 4.5(b) shows that the ultimate curvature φu decreases with increasing tension steel ratio and/or steel yield strength until the section becomes over-reinforced, and then the ultimate curvature slowly converges to a constant value of around 0.6×10-5 rad/mm despite further increases in tension steel ratio and steel yield strength. Considering Figures 4.4 and 4.5 together, it is evident that the reduction of ductility factor µ as the steel yield strength fyt increases is due to both increase in the yield curvature φy (the denominator in the definition of µ) and decrease in the ultimate curvature φu (the numerator in the definition of µ).

4.6

Prestressed Concrete Sections To examine the effects of prestressing, the variations of the flexural

strength and ductility of the prestressed concrete sections, expressed in terms of dimensionless parameters Mp/fcobh2 and µ, with the prestressing steel ratio ρps at different prestressing stress ratio fpe/fpu are plotted in Figures 4.6(a) and 4.6(b), respectively. From Figure 4.6(a), it is observed that the flexural strength increases steadily with the prestressing steel ratio ρps. For ρps below 0.8%, the flexural strength is not sensitive to the prestressing stress ratio, but for ρps above 0.8%, the flexural strength is slightly higher at a higher prestressing stress ratio. Figure 4.6(b) shows that the flexural ductility decreases as the prestressing steel ratio ρps increases, but increases as the prestressing stress ratio fpe/fpu increases. Hence, in both reinforced and prestressed sections, an increase in the tension steel ratio or prestressing steel ratio reduces the flexural ductility. However, it appears that an increase in prestressing stress ratio fpe/fpu improves the flexural ductility, which is the same phenomenon observed by Naaman et al. (1986). The above observation of higher flexural ductility at higher prestressing force should be treated with caution. Flexural ductility is often measured in terms

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of the ductility factor µ, which can be increased by an increase in ultimate curvature φu and/or a decrease in yield curvature φy. If the ductility factor µ is increased due to an increase in ultimate curvature φu, one may consider that the flexural ductility has improved in view of the larger amount of energy absorption before failure. However if the ductility factor µ is increased solely by a reduced yield curvature φy, then it is questionable if the flexural ductility has really improved. To illustrate this point, the variations of the yield curvature φy and ultimate curvature φu with the prestressing steel ratio ρps at different prestressing stress ratio fpe/fpu are plotted in Figures 4.7(a) and 4.7(b), respectively. Figure 4.7(a) shows that the yield curvature changes only slightly as the prestressing steel ratio increases but decreases substantially as the prestressing stress ratio increases. This is because, as the prestressing stress increases, the pre-peak branch of the moment-curvature curve is shifted further to the left, leading to substantial decrease in the yield curvature. On the other hand, Figure 4.7(b) shows that the ultimate curvature decreases as the prestressing steel ratio increases but is virtually insensitive to the prestressing stress ratio. Considering Figures 4.6 and 4.7 together, it is evident that the increase in the ductility factor µ as the prestressing stress ratio fpe/fpu increases is due solely to decrease in yield curvature φy. Such apparent increase in the ductility factor without increase in the ultimate curvature should not be construed as any improvement in the flexural ductility at all. For this reason, the common measure of the flexural ductility in terms of the ductility factor should be reviewed.

4.7

Effects of x/d Although the combined effects of the various steel-related parameters are

fairly complicated, it has been found in previous studies (Desayi et al., 1974; Park and Dai, 1988; Pam et al., 2001; Kwan et al., 2002; Bai and Au, 2009) that the combined effects of the tension steel ratio and steel yield strength on the flexural behaviour of reinforced sections may be evaluated in terms of the ratio of neutral axis depth x at peak moment to effective depth d to tension steel (for reinforced or prestressed section). In fact, the maximum allowable x/d ratio is often used in codes of practice to stipulate the minimum ductility required for

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reinforced concrete sections. It is therefore desirable to find out whether the combined effects of the prestressing steel ratio and prestressing stress ratio on the flexural behaviour of prestressed sections may also be evaluated in terms of the x/d ratio. To study the effects of the x/d ratio on the flexural strength and ductility of both reinforced and prestressed sections, the values of Mp/fcobh2 and µ obtained for the representative reinforced and prestressed sections analyzed and reported here are plotted against the x/d ratio in Figures 4.8(a) and 4.8(b), respectively. Figure 4.8(a) shows that all the data points plotted, including those of reinforced and prestressed sections, lie on virtually the same curve. In fact, it can be shown that the data points of the other reinforced and prestressed sections analyzed in the study also lie on this curve, although they have been omitted for clarity. Hence, it may be concluded that for both reinforced and prestressed sections, the flexural strength is governed solely by the x/d ratio, irrespective of whether the section is reinforced or prestressed. However, from Figure 4.8(b), it can be seen that the relationship between the ductility factor µ and the x/d ratio depends on whether the section is reinforced or prestressed, and it varies significantly with the steel yield strength or the prestressing stress ratio. To study the effects of the x/d ratio on the yield and ultimate curvatures of both reinforced and prestressed sections, the values of φy and φu obtained for the reinforced and prestressed sections analyzed and reported here are plotted against the x/d ratio in Figures 4.9(a) and 4.9(b), respectively. From Figure 4.9(a), it can be observed that the relationship between the yield curvature φy and the x/d ratio depends on whether the section is reinforced or prestressed, and the yield curvature varies significantly with the steel yield strength or the prestressing stress ratio. For reinforced sections in particular, the yield curvature at a fixed x/d ratio is larger at higher steel yield strength; whereas for prestressed sections, the yield curvature at a fixed x/d ratio is smaller at higher prestressing stress ratio. Hence, for both reinforced and prestressed sections, the yield curvature cannot be evaluated as a simple function of the x/d ratio. Nevertheless, Figure 4.9(b) shows that all the data points of ultimate curvature for the reinforced sections lie on one curve, whereas all those of the prestressed sections lie on another curve. These two curves, with one for reinforced sections and the other for prestressed sections, -48-

are so close together that for practical applications they may be merged into one single curve. Hence one may conclude that, regardless of whether the section being considered is reinforced or prestressed, the ultimate curvature may be evaluated as a simple function of the x/d ratio. In summary, one may regard that in general the flexural strength and ultimate curvature of a prestressed concrete section are essentially the same as those of a reinforced concrete section having the same x/d ratio. However, the flexural ductility and yield curvature of a prestressed concrete section are highly dependent on the prestressing stress ratio and therefore cannot be directly compared to those of a reinforced concrete section having the same x/d ratio. In fact, the occasional higher flexural ductility of a prestressed concrete section (e.g. curve in Figure 4.8(b) for fpe/fpu =0.7) than a reinforced concrete section having the same x/d ratio is rather misleading; the flexural ductility of the prestressed concrete section appears to be higher only because of the reduction in yield curvature caused by prestressing (e.g. curve in Figure 4.9(a) for fpe/fpu =0.7). To overcome this anomaly, one should avoid any reliance on the yield curvature for ductility evaluation. The ability of a concrete section to sustain inelastic deformation without excessive reduction in load carrying capacity should preferably be evaluated in terms of the ultimate curvature.

4.8

Strength-Ductility-Deformability Performance Since the steel-related parameters affect the flexural strength and ductility

at the same time, the ductility performance of reinforced and prestressed sections should be compared on the equal strength basis. For this purpose, the concurrent flexural strength (in terms of Mp/fcobh2) and flexural ductility (in terms of µ) that can be achieved by the reinforced and prestressed sections analyzed are plotted in Figures 4.10(a) and 4.10(b), respectively. Each curve in Figure 4.10(a) shows the concurrent flexural strength and ductility that can be achieved at different steel yield strength, while each curve in Figure 4.10(b) shows the concurrent flexural strength and ductility that can be achieved at different prestressing stress ratio. From these curves, it is evident that as the flexural strength increases, the flexural ductility decreases, and vice versa. Particularly in the case of reinforced sections,

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a higher steel yield strength at the same flexural strength leads to a lower flexural ductility; and in the case of prestressed sections, a higher prestressing stress ratio at the same flexural strength leads to a higher flexural ductility apparently. However, such observed effects of the steel yield strength and prestressing stress ratio should be interpreted with extreme care as explained below. To overcome the above confusion associated with the equivalent yield point, the concept of deformability is advocated as an alternative measure of the ability of a reinforced or prestressed section to sustain inelastic deformation. To measure the maximum deformation that a section can sustain without excessive reduction in load carrying capacity, one may adopt the ultimate curvature φu, which is the curvature at which the resisting moment has dropped to a point corresponding to 85% of the maximum imposed moment after reaching the peak resisting moment. However, the ultimate curvature is dependent on the depth of section. It is proposed herein to multiply the ultimate curvature φu by the overall depth h and take the product φuh as a dimensionless measure of curvature deformability. In this regard, it is noteworthy that Cohn and Riva (1991) always multiplied the curvature by the depth of section to convert the curvature into a dimensionless value. As before, since the steel-related parameters affect the flexural strength and deformability at the same time, the deformability performance of reinforced and prestressed sections should be compared on equal strength basis. For this purpose, the concurrent flexural strength (in terms of Mp/fcobh2) and flexural deformability (in terms of φuh) that can be achieved by the reinforced and prestressed sections analyzed are plotted in Figures 4.11(a) and 4.11(b), respectively. This time, all the data points in Figure 4.11(a) fall on the same curve, indicating that the steel yield strength has no effect on the concurrent flexural strength and deformability that can be achieved. Likewise, all the data points in Figure 4.11(b) fall on the same curve, indicating that the prestressing stress ratio has no effect on the concurrent flexural strength and deformability that can be achieved. Hence, it may be concluded that on the equal strength basis, the steel yield strength and prestressing stress ratio have no effect on the deformability. Finally, it can be shown that the two curves in Figure 4.11, though plotted separately for clarity, are almost identical to each other, revealing that on -50-

the equal strength basis, reinforced and prestressed sections actually have very similar deformability. The belief that prestressing can improve the ability of a section to withstand inelastic deformation is a misconception. As for reinforced sections, due care should be exercised in the provision of sufficient ductility or deformability in the design of prestressed sections so as to avoid brittle failure.

4.9

Conclusions Full-range moment-curvature analysis is carried out on reinforced

concrete sections with only non-prestressed steel reinforcement and prestressed concrete sections with only bonded prestressing tendons, which takes into account material nonlinearity and stress-path dependence. The effects of nonprestressed steel content, prestressing steel content, steel yield strength and prestressing stress are examined. From each moment-curvature curve, the peak resisting moment, yield curvature and ultimate curvature are determined, from which the dimensionless flexural strength parameter Mp/fcobh2 and the flexural ductility factor µ can be evaluated. As the tension steel ratio and/or steel yield strength of a reinforced section increase, the flexural strength increases but the ductility factor decreases until the section becomes over-reinforced. It is also found that the reduction in ductility factor as the steel yield strength increases is due to both increase in yield curvature and decrease in ultimate curvature. For prestressed sections, it is observed that the flexural strength increases with the prestressing steel ratio but is insensitive to the prestressing stress ratio except at high prestressing steel ratio. The ductility factor decreases as the prestressing steel ratio increases. On the other hand, the ductility factor increases as the prestressing stress ratio increases but such increase is due solely to the decrease in yield curvature and should not be construed as improvement in flexural ductility. Correlation of the flexural strength, ductility factor, yield curvature and ultimate curvature with the neutral axis depth ratio x/d at maximum moment reveals that for both reinforced and prestressed sections, the flexural strength and ultimate curvature are uniquely related to the x/d ratio, but the ductility factor and yield curvature are not. More importantly, the flexural strength and ultimate

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curvature of a prestressed section are virtually the same as those of a reinforced section having the same x/d ratio. Furthermore, comparison of the ductility factors of reinforced and prestressed sections on the equal strength basis reveals that the ductility factor of a prestressed section can be higher than that of a reinforced section, thus giving rise to the impression that prestressing can increase ductility. Since there is actually no increase in ultimate curvature by prestressing, this is just an illusion. To avoid this, it is proposed to measure the ability of a section to withstand inelastic flexural deformation in terms of a dimensionless deformability factor, defined as the ultimate curvature multiplied by the overall depth. Comparing the deformability factors of reinforced and prestressed sections on the equal strength basis, it becomes clear that the deformability of a prestressed section is virtually the same as that of a reinforced section. As for reinforced sections, due care should be exercised to provide sufficient deformability in the design of prestressed sections so as to avoid brittle failure. Since the deformability of prestressed sections are related to the x/d ratio in the same way as reinforced sections, the simplest way of providing minimum deformability for prestressed sections is to follow the current practice for reinforced sections of limiting the x/d ratio in the design codes.

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h

dt h

Prestressing steel tendon of area Aps

dp

Non-prestressed steel of area Ast b

b

(a)

(b)

Figure 4.1. Sections analyzed: (a) reinforced concrete section; and (b) prestressed concrete section.

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25000

fyt = 580 MPa ρst = 1 % ρst = 2 % ρst = 3 % ρst = 4 %

Moment M (kNm)

20000 15000 10000 5000 0 0.0

1.0

2.0

3.0

Curvature φ (×10 rad/mm) -5

(a) reinforced concrete sections

8000

ρps = 1 % fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

Moment M (kNm)

7000 6000 5000 4000 3000 2000 1000 0 -0.5

0.0

0.5

1.0

Curvature φ (×10-5 rad/mm) (b) prestressed concrete sections

Figure 4.2. Moment-curvature curves of (a) reinforced concrete sections; and (b) prestressed concrete sections.

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M

Peak moment

Mp 0.85Mp 0.75Mp

Yield curvature

φu

φy

0

Ultimate curvature

φ

(a) reinforced concrete sections (not to scale) M Peak moment Mp Mo+0.85(Mp-Mo) Mo+0.75(Mp-Mo)

Yield curvature

Mo

φo

0

Ultimate curvature

φu

φy

φ

(b) prestressed concrete sections (not to scale)

Figure 4.3. Definitions of yield curvature and ultimate curvature for (a) reinforced concrete section; and (b) prestressed concrete section.

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0.30

Mp / (fcobh2)

0.25 0.20 0.15 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

0.10 0.05 0.00 0

2

4

6

ρst (%) (a) flexural strength 25 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

20

µ

15 10 5 0 0

2

4

6

ρst (%) (b) ductility factor

Figure 4.4. Reinforced concrete sections: variations of (a) flexural strength; and (b) ductility factor.

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1.0 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

φy (×10-5 rad/mm)

0.8 0.6 0.4 0.2 0.0 0

2

4

6

ρst (%) (a) yield curvature 5 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

φu (×10-5 rad/mm)

4 3 2 1 0 0

2

4

6

ρst (%) (b) ultimate curvature

Figure 4.5. Reinforced concrete sections: variations of (a) yield curvature; and (b) ultimate curvature.

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0.30

Mp / (fcobh2)

0.25 0.20 0.15

fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

0.10 0.05 0.00

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ρps (%) (a) flexural strength 25 fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

20

µ

15 10 5 0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ρps (%) (b) ductility factor

Figure 4.6. Prestressed concrete sections: variations of (a) flexural strength; and (b) ductility factor.

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1.0

φy (×10-5 rad/mm)

0.8

fpe/fpu = 0.3 fpe/fpu = 0.4

fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ρps (%) (a) yield curvature 5 fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

φu (×10-5 rad/mm)

4 3 2 1 0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ρps (%) (b) ultimate curvature

Figure 4.7. Prestressed concrete sections: variations of (a) yield curvature; and (b) ultimate curvature.

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0.30

RC (fyt = 460 MPa) RC (fyt = 540 MPa) RC (fyt = 620 MPa) PC (fpe/fpu = 0.3) PC (fpe/fpu = 0.5) PC (fpe/fpu = 0.7)

Mp / (fcobh2)

0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/d (a) flexural strength 25

RC (fyt = 460 MPa) RC (fyt = 540 MPa) RC (fyt = 620 MPa) PC (fpe/fpu = 0.3) PC (fpe/fpu = 0.5) PC (fpe/fpu = 0.7)

20

µ

15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/d (b) ductility factor

Figure 4.8. Effects of x/d ratio of reinforced concrete (RC) sections and prestressed concrete (PC) sections on (a) flexural strength; and (b) ductility factor.

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1.0

RC (fyt = 460 MPa) RC (fyt = 540 MPa) RC (fyt = 620 MPa) PC (fpe/fpu = 0.3) PC (fpe/fpu = 0.5) PC (fpe/fpu = 0.7)

φy (×10-5 rad/mm)

0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/d (a) yield curvature 5.0

RC (fyt = 460 MPa) RC (fyt = 540 MPa) RC (fyt = 620 MPa) PC (fpe/fpu = 0.3) PC (fpe/fpu = 0.5) PC (fpe/fpu = 0.7)

φu (×10-5 rad/mm)

4.0 3.0 2.0 1.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/d (b) ultimate curvature

Figure 4.9. Effects of x/d ratio of reinforced concrete (RC) sections and prestressed concrete (PC) sections on (a) yield curvature; and (b) ultimate curvature.

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0.30 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

Mp / (fcobh2)

0.25 0.20 0.15 0.10 0.05 0.00 0

5

10

15

20

25

µ (a) reinforced concrete sections 0.30 fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

Mp / (fcobh2)

0.25 0.20 0.15 0.10 0.05 0.00 0

5

10

15

20

25

µ (b) prestressed concrete sections

Figure 4.10. Flexural strength-ductility performance of (a) reinforced concrete sections; and (b) prestressed concrete sections.

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0.30 fyt = 460 MPa fyt = 500 MPa fyt = 540 MPa fyt = 580 MPa fyt = 620 MPa

Mp / (fcobh2)

0.25 0.20 0.15 0.10 0.05 0.00 0

2

4

6

8

φuh (×10 ) -2

(a) reinforced concrete sections 0.30 fpe/fpu = 0.3 fpe/fpu = 0.4 fpe/fpu = 0.5 fpe/fpu = 0.6 fpe/fpu = 0.7

Mp / (fcobh2)

0.25 0.20 0.15 0.10 0.05 0.00 0

2

4

6

8

φuh (×10 ) -2

(b) prestressed concrete sections

Figure 4.11. Flexural strength-deformability performance of (a) reinforced concrete sections; and (b) prestressed concrete sections.

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CHAPTER 5 EXPERIMENTAL INVESTIGATION ON BEAM SPECIMENS WITH IN-SITU CONCRETE STITCHES

5.1

Overview One of the aims of the present research is to investigate the behaviour of

in-situ concrete stitches under different combinations of internal forces. For that purpose, a number of beam specimens of different configurations that are either provided with or without in-situ concrete stitches are fabricated. Loading tests on the specimens are carried out in which different arrangements of supports and points of loading are applied. In so doing, the in-situ concrete stitch in the specimen is subject to either pure bending moment, pure shear force, or a combination of moment and shear, and the behaviour of the stitch is examined. Since only a limited number of specimen configurations can be experimentally tested, finite element analysis is used to carry out parametric studies to examine the effects of various structural parameters on the behaviour of the specimens. Experimental data are used to calibrate the finite element models. In this chapter, the methods and the results of the experimental and parametric studies of the beam specimens are presented.

5.2

Experimental Programme The experimental programme involves the testing of 15 beam specimens

of different configurations. All specimens are either comprised of two precast units joined together by an in-situ concrete stitch of 50 mm or 100 mm in width, or one part is match cast against another precast unit.

All specimens are

internally prestressed by either a bonded tendon or an unbonded tendon, which are referred to as bonded and unbonded specimens respectively hereafter. The specimens are divided into three series, namely (i) M Series, (ii) V Series, and (iii) MV Series of which the in-situ stitch is subject to pure bending moment, pure shear and a combination of moment and shear respectively.

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5.2.1

Configuration of the specimens The specimens tested are illustrated in Figure 5.1. All specimens are

1,400 mm long with a rectangular section of 200 mm depth and 150 mm width. To produce the desired internal force at the stitch, the stitch is placed at the midspan of the specimens of the M and V Series (Figure 5.1(a)), while the stitch is placed at 350 mm from one of the ends for the MV Series (Figure 5.1(b)). The precast units are reinforced by non-prestressed reinforcement comprising of Grade S460 longitudinal deformed bars and Grade S275 stirrups arrange as shown in Figure 5.1. Non-prestressed reinforcement is not provided to the in-situ stitch. Eccentric prestressing is provided to the specimens of the M and MV series with the tendon located at a depth of 133 mm, while the specimens of the V series are concentrically prestressed.

Grade 60 concrete with mean

compressive cube strength at 28 days of at least 60 MPa is used in all specimens. Prestressing is done by post-tensioning of a 12.9 mm diameter 7-wire steel strand. The specimens are identified using this convention: (series)-(stitch width)-(concrete grade)-(prestressing force)-(bonded or unbonded (i.e. B or U)). For example, M-50-60-100-B identifies a specimen belonging to the M Series with a 50 mm wide stitch, made with Grade 60 concrete, prestressed to 100 kN using a bonded tendon. A list of all specimen configurations tested is provided in Table 5.1.

5.2.2

Preparation of the specimens Prior to casting the precast units, the non-prestressed reinforcement cages

are fixed and placed in the formwork along with the polyvinyl chloride (PVC) duct as illustrated in Figure 5.2, which corresponds to the formwork setup for the specimens of the M Series. Mould oil is applied onto the formwork interior and then concrete is cast. The precast units are removed from the formwork one week after casting and the surfaces of the units that are to serve as construction joints are scraped using a needle gun. By doing so, the laitance is removed for better bonding surface. The scraping operation and the treated surface are shown in Figures 5.3 and 5.4 respectively. For specimens with in-situ concrete stitch, the stitch is usually cast on the day after the joint surface is treated.

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For

specimens without in-situ concrete stitch, the remaining part is match-cast against the precast unit the day after the joint surface is treated. For unbonded specimens, prestressing is applied 28 days from the day of casting the in-situ stitch and loading test is subsequently performed on the same day.

For bonded specimens, prestressing and grouting is performed on

confirmation that the stitch has gained sufficient strength two weeks after the casting of stitch.

Loading test is then performed 28 days from the day of

grouting. The prestressing operation is carried out as follows: a 7-wire steel strand is anchored at one end (i.e. the dead end) of the specimen by a wedge and barrel anchorage (Figure 5.5), while at the other end (i.e. the live end), the strand is stressed by a hydraulic jack with the prestressing force monitored by a load cell (Figure 5.6).

Stressing is terminated when the desired prestressing force is

achieved. The spacer (Figure 5.6) is dissembled and the strand is cut at the location of the spacer. The hydraulic jack is then removed. The grout for bonded specimens comprises of water and ordinary Portland cement with a water-to-cement ratio of 0.55. A vertical grout tube is provided at near each end of the specimen with one for injection of grout and other for vacuuming of air from the duct to assist in grouting.

5.2.3

Test setup Monotonic load testing is carried out using the Avery testing frame

(Figure 5.7) with a capacity of 1,000 kN. The arrangements of supports and imposed load are illustrated in Figure 5.8. By loading the specimens as shown in Figures 5.8(a), 5.8(b) and 5.8(c), the stitch or the construction joint is subject to pure bending moment, pure shear, or a combination of moment and shear respectively. The test is carried out using a displacement-controlled scheme at a ram rate of 1 mm/min. The applied load is monitored by a load cell mounted against the actuator of the testing frame. Displacement is monitored at discrete locations along the beam by linear variable displacement transducers (LVDT) and the readings are recorded by a data-logger.

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Variation in prestressing force with applied load is monitored for unbonded specimens. Since tendon stress is assumed uniform along the tendon length, prestressing force can be measured by load cells that are mounted at the anchorages. For bonded specimens, however, tendon stress is not uniform. Such a method to obtain variation of prestressing force is not worthwhile and so the prestressing force variation is not monitored.

5.3

Experimental Observations

5.3.1

M Series The typical flexural failure observed for specimens of the M Series is

illustrated in Figure 5.9(a). The numbers that are written on the specimen are the ram displacements corresponding to the extent of the cracks. All specimens of the series have similar crack propagation as shown in the figure, which can be categorized by four different stages, namely (a) cracking initiates at both construction joints; (b) the cracks propagate vertically along the joints as load increases; (c) the cracks then propagate laterally as they reach the compression zone and the cracks along the joints in the tension zone are opened, with widths of 2 mm to 3 mm are typically observed at the bottom edge of the specimen; and (d) spalling and crushing of concrete at the top surface occur. The four stages of crack propagation are illustrated in Figure 5.10.

It is

worthwhile to note that regardless whether bonded or unbonded tendons are used, no noticeable crack is observed other than those found along the joints and in the compression zone in the vicinity of the stitch. This cracking pattern is different from that typically observed in prestressed beams that are continuously cast and reinforced with loads applied in a similar fashion: for bonded beams, a number of flexural cracks are usually present along the beam, while for unbonded beams, fewer number of relatively wider cracks are usually found (Mattock et al., 1971). The load-displacement relationships of the specimens of the M Series are plotted in Figure 5.11(a) where both bonded and unbonded specimens are presented.

The overall trend of the load-displacement relationships for all

specimens is essentially similar, which consist of an initial elastic branch,

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followed by another linear branch with reduced steepness, and finally a plateau may occur until an abrupt drop occurs corresponding to the failure of the member. The transition from the elastic branch to the second linear branch corresponds to the initiation of cracking along the construction joints, while the plateau is attributed to cracking and spalling of concrete in the compression zone as well as “yielding” of the prestressing tendon at higher load level. From Figure 5.11(a), it can be seen that regardless of the stitch width or whether bonded or unbonded tendons are used, the elastic flexural stiffnesses are similar. The peak strengths of most specimens are more or less the same, which vary from 80 kN to 90 kN, with the exception of Specimen M-100-60-100-B, which has a slightly higher peak strength of 105 kN. The plot shows little correlation between the peak strength and the stitch width, which suggests that the width of the stitch does not have significant effect on the strength of the member within the range of stitch widths tested. Figure 5.11(a) shows that within the pre-peak range, the load-displacement relationships of the bonded specimens are close to each other forming one band while those of the unbonded specimens are also close to one another forming another band. This reveals that the variation of stitch widths has little effect on the load-displacement response.

The figure also shows that the unbonded

specimens tend to have higher strength than the bonded specimens from the elastic limit to the peak. The reason for this may be attributed to the prestressing loss in the bonded specimens since they are tested approximately 28 days after grouting. Numerical analysis shows that decrease in prestressing force tends to have negative effect on the strength of the member.

5.3.2

MV Series The typical failure observed for the specimens of the MV series is

illustrated in Figure 5.9(b).

The stages toward failure can be described as

follows: (a) flexural cracking occurs in the vicinity of the mid-span as load reaches approximately 30% to 40% of the peak load; (b) soon afterwards or simultaneously on occasion, cracking initiates at the construction joint that is closer to the loading point; -69-

(c) as load approaches the peak, flexural cracks in the vicinity of the mid-span continue to propagate more or less towards the loading point, while rapid propagation of the crack at the construction joint takes place vertically along the joint until it reaches the compression zone upon which it propagates laterally towards the loading point; (d) shear slip take place along the construction joint as soon as the crack at the joint propagates laterally; and (e) once the load has already reached its peak, spalling of concrete occurs in the compression zone between the construction joint and the loading point, and the load subsequently drops. These stages of crack propagation are illustrated in Figure 5.12.

Although

cracking along the construction joint is evident, the width of the crack is much narrower than that found in the specimens of the M Series. Maximum width of less than 1 mm is typically observed for the vertical construction joint crack found in the specimens of the MV Series, while maximum crack width of 3 to 4 mm at the joint is common for the specimens of the M Series. The reason for this is that the former is primarily caused by shear, which causes slippage along the interface that results in little widening of the crack, while the latter is primarily caused by flexure, which causes substantial widening of the crack due to concentration of curvature. The load-displacement relationships of the specimens of the MV Series are plotted in Figure 5.11(b). The peak strengths of the unbonded specimens with stitch vary only slightly from 79 kN to 83 kN, while those of the bonded specimens are approximately the same at 70 kN. Similar to the specimens of the M Series, it is observed that the unbonded specimens tend to have slightly higher strength than the bonded specimen.

Similar reasoning for the M Series

specimens may be applied to this case.

From Figure 5.11(b), reduction in

loading can be observed after the peak strength is reached and reduction can be relatively more abrupt as in the case of the bonded specimens. This abrupt drop is due to a relatively large slippage along the construction joint in stage (d) of the failure sequence as mentioned above.

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5.3.3

V Series The typical failure mode for the specimens of the V Series is

characterized by brittle diagonal cracking across the in-situ stitch as illustrated in Figure 5.9(c). Before reaching the peak load, flexural cracking is observed at the points of maximum moment, while the stitch is undamaged and no crack is observed along the construction joints. As soon as the peak load is reached, a major diagonal crack across the stitch suddenly forms and slip along the construction joint is observed. By then, the load on the specimen drops abruptly. The load-displacement relationships of the specimens of the V series are plotted in Figure 5.11(c). For specimens other than the bonded specimen with 50 mm stitch and the unbonded specimen without in-situ stitch, the peak strength ranges from 180 kN to 220 kN and they have failed in a brittle manner when the peak load occurs and the beam has only sustained relatively small deformation. Upon reaching the peak, the load abruptly drops but it is evident that residual strength exists and gradually increases.

This behaviour is attributed to the

resistance provided by the internal tendon, which is holding the cracked specimen together. From Figure 5.11(c), no apparent correlation between the peak strengths and the bonding of the tendon can be observed. The bonding of the tendon appears to have no significant effect on the peak strength, which is to be verified by the numerical results to be presented in due course. For the bonded specimen with 50 mm stitch, and the unbonded specimen without in-situ stitch, neither an apparent peak nor a sudden drop in load is found in the load-displacement relationships.

The reason for this is that diagonal

cracking similar to that found in the other specimens has never occurred and the stitch remains intact throughout the entire loading history, while excessive cracking at the points of maximum loading is evident.

5.4

Parametric Study Parametric study is carried out to further examine the effects of stitch

width, prestressing force, concrete strength, bonding of the prestressing tendon, and the provision of shear keys on the load-deflection relationships of members

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with in-situ concrete stitch.

The effect of prestressing force on the crack

development at the construction joint is also investigated. The effect of providing shear keys to beam specimens is only examined numerically and not experimentally tested. The configuration of shear keys analyzed is that given in the guideline by the American Association of State Highway and Transportation Officials (AASHTO) (AASHTO, 1999) and is illustrated in Figure 5.13. In the parametric study, the models are divided into two groups, namely those having stitches without shear keys (referred to as plain stitches hereafter) and those having stitches with shear keys (referred to as keyed stitch hereafter). In each group, members with bonded and unbonded tendons are analyzed. The ranges of values examine for the other parameters are as follows: stitch width is varied from 50 mm to 150 mm; prestressing force is varied from 40 kN to 120 kN; and concrete grade is varied from 40 MPa to 80 MPa.

5.4.1

Method of analysis The parametric study is carried out numerically using the finite element

method.

The commercial finite element package MIDAS FEA is adopted.

Models of the specimens are constructed using four node plane stress elements for the concrete and truss elements for the prestressing tendons and nonprestressed reinforcement.

Line interface elements are used to model the

construction joints, while bond-slip interface elements are used to model the interface between the concrete and prestressing tendon. The constitutive models proposed by Attard and Setunge (1996) and Hordijk (1991) are adopted for concrete under uniaxial compression and tension respectively. The post-crack shear behaviour of concrete is taken into account by applying a constant shear retention factor β to the pre-crack shear modulus Gc to reduce the shear stiffness after cracking.

Therefore, the post-crack shear

modulus Gc,cr is given as Gc,cr = β Gc

(5.1)

where β is taken as 0.1. This value is within the range of values suggested by Vecchio (2000). It is observed that the analysis tends to be more stable when a -72-

higher value of β is adopted but it may be urealistic if a high value is used since the shear stiffness tends to be much lower once shear crack is formed. The constitutive model proposed by Menegotto and Pinto (1973) is used for prestressing steel, while the non-prestressed steel is assumed to behave in an elasto-plastic manner. The behaviour of the interface element used for the construction joints is governed by the Mohr-Coulomb friction model (MIDAS Information Technology, 2008), which is commonly adopted for interface modelling. The bond-slip relationship as given in the CEB-FIP Model Code (Comité EuroInternational du Béton, 1993) is used to model the behaviour of the interface between concrete and bonded tendon. The model is shown in Figure 5.14 where s is the slip of the tendon in mm and q is the bond stress in MPa. The values of the model parameters suggested by Tabatabai and Dickson (1993) are used and they are also adopted by Ayoub and Filippou (2010) in their finite element analysis of prestressed concrete girders with bond-slip taken into account. For unbonded specimens, a small constant value of bond stress is assumed. Staged nonlinear analysis is adopted taking into account the posttensioning process, in which the tendon is first stressed and imposed displacement is subsequently applied. For bonded specimens, the bond stress of the bond-slip elements is adjusted in the staged analysis to model the process in which the tendon is initially unbonded during stressing and subsequently bonded by grouting.

5.4.2

Model calibration The finite element models are calibrated against the experimental results

for various empirical parameters of the interface element model, namely the normal and tangential stiffnesses, cohesion and internal friction angles. Loaddisplacement relationships obtained experimentally and calculated using the calibrated models are plotted in Figure 5.15 and good agreement is generally observed. Parametric studies are carried out using the calibrated models to study the effects of various parameters as reported in the following sections.

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5.4.3

Effects of stitch width The load-displacement relationships of bonded and unbonded members

having different stitch widths are plotted in Figures 5.16 to 5.18 for members of the M, MV and V Series respectively. To examine the effects of stitch width, the peak load-carrying capacities of members are plotted against stitch widths in Figure 5.19.

Figures 5.16 to 5.18 have been obtained for members having

concrete strength of 60 MPa and prestressing force of 100 kN. From Figures 5.16 to 5.18, it is evident that all members have nearly identical loaddisplacement relationships. Figure 5.19 further reveals that the peak strengths are rather insensitive to the variation in stitch width and whether bonded or unbonded tendons are provided.

The results suggest that, for stitch widths

ranging from 50 mm to 150 mm, the width of the stitch has little effect on the load-displacement behaviour and the peak strength. In Figure 5.16, it appears that bonded specimen have slightly higher strength than unbonded member, however, the trend is less obvious for the members of the MV and V Series as shown in Figures 5.17 and 5.18.

5.4.4

Effects of prestressing force The load-displacement relationships of bonded and unbonded members at

different initial prestressing forces are plotted in Figures 5.20 to 5.22 for members of the M, MV and V Series respectively. All of them are for members having stitch width of 50 mm and concrete strength of 60 MPa. From the figures, four effects are observed as prestressing force increases: (a) the increase in elastic limit; (b) the increase in strength of member at the same displacement beyond the elastic limit; (c) the increase in peak strength; and (d) the decrease in displacement at which peak strength occurs or the decrease in deformability. The plots in Figures 5.20 to 5.22 also reveal that the strength of bonded members is consistently higher than that of the unbonded members beyond the elastic range for all cases of prestressing force applied, except for those members with

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shear keys that are subject to pure shear (V Series) (Figure 5.22(b)), whose loaddeflection relationships are nearly identical whether bonded or unbonded tendon is provided. To conclude, the increase in the prestressing force can increase the strength of members with plain and keyed stitches but deformability simultaneously decreases, which is especially the case for members with keyed stitches.

5.4.5

Effects of concrete strength The load-displacement relationships of bonded and unbonded members

having different concrete strengths are plotted in Figures 5.24 to 5.26 for members of the M, MV and V Series respectively. All of them are for members with stitch width of 50 mm and prestressing force of 100 kN. From the figures, it can be seen that the load-displacement relationships of members of the same series do not deviate much, and the peak strength occurs at larger displacement as the concrete strength increases. The peak strength of the members of each series is plotted against the concrete strength in Figure 5.27, which clearly shows the increase in peak strength as concrete strength increases and whether bonded or unbonded tendon is provided has little influence on the peak strength. Therefore, the results suggest that variation of concrete strength has little effect on the trend of load-displacement response but it will increase the peak strength and the corresponding displacement as concrete strength increases. The increase in concrete strength will not have noticeable benefit if one wishes to improve the strength of the member while providing a certain deformation prior to the failure of the member.

5.4.6

Effects of shear keys Members having stitch width of 50 mm, concrete strength of 60 MPa and

prestressing force of 100 kN are selected to examine the effects of provision of shear keys. The load-displacement relationships of these members with plain and keyed stitches are compared in Figures 5.28 to 5.30 for members of the M Series, MV Series and V Series respectively.

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For members with in-situ stitch subject to pure moment (M Series), it is evident by examining Figure 5.28(a) that the load-displacement relationships for members with plain and keyed stitches are very similar, except for the slight difference in peak strength. For bonded members with plain stitch and keyed stitch, the peak strengths are about 101 kN and 93 kN respectively, while for unbonded members with plain and keyed stitch, the peak strengths are about 96 kN and 92 kN respectively. For members with in-situ stitch subject to shear (MV and V Series), the load-displacement relationships of members with plain and keyed stitches are quite different. It can be seen that the slope of the ascending branch is greater for the members with keyed stitch, which implies that the stiffness of each of these members is larger. However, the peak strengths do not vary substantially among members with plain and keyed stitches. The effects on the peak strengths can be further examined by comparing the curves corresponding to the plain and keyed stitches in Figures 5.19, 5.23 and 5.27. It is evident that the effect of shear key is most pronounced for members of the V Series and comparisons reveal that members with plain stitch tend to have higher strength than those with keyed stitch. This behaviour may seem surprising at first but it is largely attributed to the stress concentration developed in the keyed stitches that can cause premature failure, and further elaboration is provided below. Figure 5.28 reveals that deformability reduces when shear keys are provided. Similar observations can also be made when (a) and (b) of Figures 5.16 to 5.18, Figures 5.20 to 5.22, and Figures 5.24 to 5.26 are compared, which also reveal that provision of shear keys can be influential to the deformability of the member. Deformability decreases with the provision of shear key and it may be due to the introduction of stress concentrations at the corners and roots of the shear keys, which in turn promotes concrete fracturing at these locations. Therefore, the concrete of keyed stitch in the compression zone often fails sooner than that of plain stitch due to stress concentration. It is also observed that the reduction in deformability with the provision of shear keys is more at higher prestressing force and concrete strength. Considering the bonded members for instance, by comparing the plots in Figures 5.20(a) and 5.20(b), the displacement at peak strength reduces from 15.8 mm to -76-

15.5 mm when a relatively low prestressing force of 40 kN is applied, while the displacement at peak strength reduces from 12.2 mm to 10 mm when the prestressing force increases to 120 kN. Comparison between Figures 5.24(a) and 5.24(b) reveals that the provision of shear keys to bonded members has no noticeable change in the displacement at peak strength when relatively low concrete strength of 40 MPa is used, but the displacement reduces considerably from 19.1 mm to 13.4 mm when the concrete strength is increased to 80 MPa. The reason for this behaviour is that, as the prestressing force increases, stress concentration becomes more significant and therefore fracturing of the stitch in the compression zone would occur at a lower displacement. As the concrete strength increases, the concrete becomes more brittle and may be more sensitive to the detrimental effect of stress concentration. To conclude, the effect of providing shear keys to in-situ concrete is most pronounced when the stitch is subject to shear, which will be further examined in the next chapter.

The provision of shear keys has little benefit to strength

improvement when the stitch is subjected to pure moment.

Moreover, the

introduction of stress concentration by the shear keys can reduce the deformability of the members under flexure.

5.4.7

Effects of prestressing force on crack width at construction joints As mentioned in Section 5.3.1, members with in-situ concrete stitches

subject to moment have noticeable cracks at the construction joints only. The concentration of cracking will lead to the concentration of curvature at the joints. Therefore one can envisage that crack widths at the construction joints may become excessive as loading increases. Moreover, concentration of curvature in beams with in-situ stitch may result in cracks that are wider than those found along beams that are continuously cast and reinforced at the same applied load, since beams of the latter category would not have concentration of curvature until a plastic hinge begins to form. Thus beams with in-situ stitches are more prone to violating the crack limits imposed by most codes of practice. Excessive crack width can also lead to the intrusion of water or any other harmful substance that can damage the tendons and causes serviceability problems. One of the prevention strategies is to increase the level of prestress. -77-

Therefore, it is

worthwhile to examine the effects of prestressing force on the development of crack width at the construction joints. Members with in-situ concrete stitches subject to pure moment are further studied for the effects of prestressing force on the crack width at the construction joint. The variations of maximum crack width at the construction joint with applied load at different prestressing forces are plotted in Figure 5.29, where both bonded and unbonded members with plain stitch are included. The maximum crack width is taken as the width of the crack at the bottom edge of the member. The joint remains closed when the value of maximum crack width is zero. The end point of each curve corresponds to the state at which the peak strength of the member is reached. From the figure, it is evident that the load that initiates cracking increases as the prestressing force increases. Once cracking occurs, the width of the crack increases rapidly. Under the same applied load, unbonded members have wider cracks than bonded members. The reason for this may be that the bonded tendon is better in distributing crack widths in the vicinity of the plastic hinge. Codes of practice usually impose limits on the maximum crack width allowable in member design. For instance, Eurocode 2 (European Committee for Standardization, 2004) allows maximum crack width of up to 0.2 mm and 0.4 mm for bonded and unbonded members respectively, while the crack width allowed by ACI Code (ACI Committee 318, 2005) is derived based on the maximum crack with of 0.4 mm. From Figure 5.29, it can be seen that when a bonded member is prestressed to 40 kN, the maximum crack width can reach as high as 4.2 mm when 80 kN of load is applied. If the service load is roughly taken as 1/3 of the ultimate design load and if the ultimate load is 80 kN, then the member has a service load of approximately 27 kN, at which the maximum crack width is approximately 0.5 mm, which has exceeded most code limits. As the prestressing force applied increases, the problem of exceeding the crack limit will therefore be less likely. It is recommended that the prestressing force should be applied in such an amount that will result in no tension at the construction joint under service load, which in turn can ensure that no cracking at the joint would occur.

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5.5

Conclusions The behaviour of in-situ concrete stitches subject to pure bending

moment, pure shear, and a combination of moment and shear has been examined experimentally using beam specimens.

Typical failure modes have been

identified and the progression to failure has been observed. Extensive parametric study has been carried out to study the effects of several parameters, namely the width of stitch, initial prestressing forces, concrete strength, provision of shear keys and bonding of tendon. It is found that the variation of stitch width has little effect on the peak load-carrying capacity for the range of stitch widths examined. Increase in the prestressing force can increase the load-carrying capacity of the member but deformability simultaneously decreases, especially for members with keyed stitches.

Concrete strength is positively correlated with peak load-carrying

capacity but increase in concrete strength will have no noticeable benefit if one wishes to improve the strength of the member while providing a certain deformation prior to the failure of the member. The provision of shear keys has pronounced effects on the stitch when it is subjected to shear. However, when the stitch is subjected to pure moment, the provision of shear keys has little benefit to strength improvement and the introduction of stress concentration by shear keys can reduce the deformability of the member under flexure.

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Table 5.1. List of beam specimens tested. No.

Specimen ID

1 2 3 4 5

M-0-60-100-U M-50-60-100-U M-100-60-100-U M-50-60-100-B M-100-60-100-B

6 7 8 9 10

MV-0-60-100-U MV-50-60-100-U MV-100-60-100-U MV-50-60-100-B MV-100-60-100-B

11 12 13 14 15

V-0-60-100-U V-50-60-100-U V-100-60-100-U V-50-60-100-B V-100-60-100-B

Stitch width (mm) 0 50 100 50 100

Concrete grade (MPa) 60 60 60 60 60

Initial prestressing force (kN) 100 100 100 100 100

Unbonded Unbonded Unbonded Bonded Bonded

MV

0 50 100 50 100

60 60 60 60 60

100 100 100 100 100

Unbonded Unbonded Unbonded Bonded Bonded

V

0 50 100 50 100

60 60 60 60 60

100 100 100 100 100

Unbonded Unbonded Unbonded Bonded Bonded

Series

M

Tendon type

Note: Specimens are identified using this convention: (series) - (stitch width) - (concrete grade) (prestressing force)-(bonded or unbonded (i.e. B or U))

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Precast segment

Prestressing tendon

In-situ stitch

Load cell Anchorage 150

Varies

A 700

A-A 700

M Series: dps =133

(a) M and V Series A Precast segment

V Series: dps = 100

Precast segment

150 2-T10 R8-130 2-T10

200 133

Prestressing tendon

2-T10 R8-130 2-T10

200 dps

A Precast segment

In-situ stitch Varies 300

A 1,100

A-A

(b) MV Series

Figure 5.1. The beam specimens tested (dimensions in mm).

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Reinforcement cage PVC duct for tendon

Precast unit

Stitch

Precast unit

Figure 5.2. Formwork and reinforcement layout for beam specimens of the M Series.

Scraped surface PVC duct

Figure 5.3. Scraping the surface

Figure 5.4. Surface of the

of the construction joint using a

construction joint after treatment.

needle gun.

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Barrel Wedges Strand

Figure 5.5. Anchorage at the dead end of the prestressing tendon.

Hydraulic jack

Load cell

Anchorage

Anchorage

Spacer

Figure 5.6. Prestressing equipment at the live end of the prestressing tendon.

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(b)

(a) Figure 5.7. The Avery testing frame: (a) the frame; and (b) specimen placed in the frame.

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Load from actuator Spreader beam

300

300

300

300

(a) M Series

Load from actuator Spreader beam

300

300

300

300

(b) V Series Load from actuator

600

350

250

(c) MV Series

Figure 5.8. Setup of loading tests (dimensions in mm).

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Point of loading

Point of loading

Stitch (a) M Series (points of support are not shown)

Point of loading

Point of support

Stitch

Point of support

(b) MV Series

Point of loading

Stitch

Point of support

Point of support

(c) V Series (point of loading to the right is not shown)

Figure 5.9. Typical failure of specimens of (a) M Series; (b) MV Series; and (c) V Series. -86-

Load

Load Construction joints Crack initiates

Support

(a) Stage (a)

Support

Propagates upward (b) Stage (b)

(c) Stage (c) Concrete spalling

(d) Stage (d)

Figure 5.10. Stages of crack propagation for specimens of the M Series.

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Applied load (kN)

140

M-50-60-100-B M-100-60-100-B M-50-60-100-U M-100-60-100-U M-0-60-100-U

120 100 80 60 40 20 0 0

5 10 15 20 Mid-span displacement (mm)

25

Applied load (kN)

(a) M Series 140 120 100

MV-50-60-100-B MV-100-60-100-B MV-50-60-100-U MV-100-60-100-U MV-0-60-100-U

80 60 40 20 0 0

5 10 15 20 Mid-span displacement (mm)

25

(b) MV Series

Applied load (kN)

500

V-50-60-100-B V-100-60-100-B V-50-60-100-U V-100-60-100-U V-0-60-100-U

400 300 200 100 0 0

5

10 15 20 Displacement at stitch (mm)

25

(c) V Series

Figure 5.11. Experimental load-deflection response for specimens of (a) M Series; (b) MV Series; and (c) V Series.

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Load Construction joints Support

Cracks

(a) Stage (a)

Support

Crack initiates (b) Stage (b)

(c) Stage (c)

shear slip along joint

(d) Stage (d) Concrete spalling

(e) Stage (e)

Figure 5.12. Stages of crack propagation for specimens of the MV Series.

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Detail ‘A’

Beam with keyed stitch w/2 w/2

30

25 15 30 15 30 15 30 15 25

Stitch centerline Detail ‘A’ Figure 5.13. Configuration of the shear keys in the beam specimen (dimensions in mm).

q q = qmax (s/s1)0.4

qmax

Model parameters qmax = 3.5 MPa qf = 1.38 MPa s1 = 0.15 mm s2 = 16.0 mm s3 = 38.0 mm

qf s1

s2

s3

s

Figure 5.14. Bond-slip model adopted.

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Applied load (kN)

150

M-50-60-100-U (Exp.) M-100-60-100-U (Exp.) M-50-60-100-U (FEA) M-100-60-100-U (FEA)

100

50

0 0

5 10 15 20 Mid-span displacement (mm)

25

(a) M-50-60-100-U and M-100-60-100-U

Applied load (kN)

150

MV-50-60-100-U (Exp.) MV-100-60-100-U (Exp.) MV-50-60-100-U (FEA) MV-100-60-100-U (FEA)

100

50

0 0

2 4 6 8 Mid-span displacement (mm)

10

(b) MV-50-60-100-U and MV-100-60-100-U

Applied load (kN)

250

V-50-60-100-U (Exp.) V-100-60-100-U (Exp.) V-50-60-100-U (FEA) V-100-60-100-U (FEA)

200 150 100 50 0 0

2

4 6 8 Displacement at stitch (mm)

10

(c) V-50-60-100-U and V-100-60-100-U

Figure 5.15. Comparison between load-displacement relationships obtained experimentally and numerically.

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150

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

120

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

90 60

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

30 0 0

5 10 15 Mid-span displacement (mm)

20

(a) plain stitch

150

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

120

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

90 60

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

30 0 0

5 10 15 Mid-span displacement (mm)

20

(b) keyed stitch

Figure 5.16. Parametric study - effects of stitch width for M Series specimens with (a) plain stitch and (b) keyed stitch.

-92-

150

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

120

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

90 60

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(a) plain stitch

150

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

120

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

90 60

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(b) keyed stitch

Figure 5.17. Parametric study - effects of stitch width for MV Series specimens with (a) plain stitch and (b) keyed stitch. -93-

250

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

200

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

150 100

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

50 0 0

1 2 3 Displacement at stitch (mm)

4

(a) plain stitch

250

Fpe = 100 kN fcu = 60 MPa

Applied load (kN)

200

Bonded w = 50mm w = 75mm w = 100mm w = 150mm

150 100

Unbonded w = 50mm w = 75mm w = 100mm w = 150mm

50 0 0

1 2 3 Displacement at stitch (mm)

4

(b) keyed stitch

Figure 5.18. Parametric study - effects of stitch width for V Series specimens with (a) plain stitch and (b) keyed stitch. -94-

Peak strength (kN)

250 M Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 50

75 100 Stitch width (mm)

150

(a) M Series

Peak strength (kN)

250 MV Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 50

75 100 Stitch width (mm)

150

(b) MV Series

Peak strength (kN)

250 V Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 50

75 100 Stitch width (mm)

150

(c) V Series

Figure 5.19. Parametric study - effects of stitch width on peak strength. -95-

150

w = 50 mm fcu = 60 MPa

Applied load (kN)

120

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

90

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

60 30 0 0

5 10 15 Mid-span displacement (mm)

20

(a) plain stitch

150

w = 50 mm fcu = 60 MPa

Applied load (kN)

120

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

90

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

60 30 0 0

5 10 15 Mid-span displacement (mm)

20

(b) keyed stitch

Figure 5.20. Parametric study - effects of prestressing force for M Series specimens with (a) plain stitch and (b) keyed stitch. -96-

150

w = 50 mm fcu = 60 MPa

Applied load (kN)

120

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

90

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

60 30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(a) plain stitch

150

w = 50 mm fcu = 60 MPa

Applied load (kN)

120

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

90

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

60 30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(b) keyed stitch

Figure 5.21. Parametric study - effects of prestressing force for MV Series specimens with (a) plain stitch and (b) keyed stitch.

-97-

250

w = 50 mm fcu = 60 MPa

Applied load (kN)

200

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

150

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

100 50 0 0

1 2 3 Displacement at stitch (mm)

4

(a) plain stitch

250

w = 50 mm fcu = 60 MPa

Applied load (kN)

200

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

150

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

100 50 0 0

1 2 3 Displacement at stitch (mm)

4

(b) keyed stitch

Figure 5.22. Parametric study - effects of prestressing force for V Series specimens with (a) plain stitch and (b) keyed stitch. -98-

Peak strength (kN)

250 M Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Initial prestressing force (kN)

120

(a) M Series

Peak strength (kN)

250 MV Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Initial prestressing force (kN)

120

(b) MV Series

Peak strength (kN)

250 V Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Initial prestressing force (kN)

120

(c) V Series

Figure 5.23. Parametric study - effects of prestressing force on peak strength. -99-

150

w = 50 mm Fpe = 100 kN

Applied load (kN)

120

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

90

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

5 10 15 20 Mid-span displacement (mm)

25

(a) plain stitch

150

w = 50 mm Fpe = 100 kN

Applied load (kN)

120

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

90

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

5 10 15 20 Mid-span displacement (mm)

25

(b) keyed stitch

Figure 5.24. Parametric study - effects of concrete strength for M Series specimens with (a) plain stitch and (b) keyed stitch. -100-

150

w = 50 mm Fpe = 100 kN

Applied load (kN)

120

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

90

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(a) plain stitch

150

w = 50 mm Fpe = 100 kN

Applied load (kN)

120

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

90

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

2 4 6 8 Mid-span displacement (mm)

10

(b) keyed stitch

Figure 5.25. Parametric study - effects of concrete strength for MV Series specimens with (a) plain stitch and (b) keyed stitch. -101-

250

w = 50 mm Fpe = 100 kN

Applied load (kN)

200

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

150

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

100 50 0 0

1 2 3 Displacement at stitch (mm)

4

(a) plain stitch

250

w = 50 mm Fpe = 100 kN

Applied load (kN)

200

Bonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

150

Unbonded fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

100 50 0 0

1 2 3 Displacement at stitch (mm)

4

(b) keyed stitch

Figure 5.26. Parametric study - effects of concrete strength for V Series specimens with (a) plain stitch and (b) keyed stitch. -102-

Peak strength (kN)

250 M Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Concrete strength (MPa)

120

(a) M Series

Peak strength (kN)

250 MV Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Concrete strength (MPa)

120

(b) MV Series

Peak strength (kN)

250 V Series Bonded; plain Unbonded; plain Bonded; keyed Unbonded; keyed

200 150 100 50 0 40

60 80 100 Concrete strength (MPa)

120

(c) V Series

Figure 5.27. Parametric study - effects of concrete strength on peak strength. -103-

Applied load (kN)

150

w = 50 mm Fpe = 100 kN fcu = 60 MPa

100

Bonded plain stitch keyed stitch Unbonded plain stitch keyed stitch

50

0 0

5 10 15 Mid-span displacement (mm)

20

(a) M Series

Applied load (kN)

150

w = 50 mm Fpe = 100 kN fcu = 60 MPa

100

50

Bonded plain stitch keyed stitch

0

Unbonded plain stitch keyed stitch 0

2 4 6 8 Mid-span displacement (mm)

10

(b) MV Series

Applied load (kN)

250

w = 50 mm Fpe = 100 kN fcu = 60 MPa

200 150

Bonded plain stitch keyed stitch

100

Unbonded plain stitch keyed stitch

50 0 0

2 4 6 8 Displacement at stitch (mm)

10

(c) V Series

Figure 5.28. Parametric study - effects of shear keys for specimens of various series.

-104-

4.5

w = 50 mm fcu = 60 MPa

Max. crack width (mm)

4.0 3.5

Bonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

3.0 2.5

Unbonded Fpe = 40 kN Fpe = 80 kN Fpe = 120 kN

2.0 1.5 1.0 0.5 0.0 0

20

40 60 80 Applied loading (kN)

100

120

Figure 5.29. Parametric study - effects of prestressing force on maximum crack width at the construction joint.

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-106-

CHAPTER 6 EXPERIMENTAL INVESTIGATION ON SHEAR SPECIMENS WITH IN-SITU CONCRETE STITCHES

6.1

Overview The previous chapter presents the use of beam specimens to examine the

behaviour of in-situ concrete stitches subject to different combinations of bending moment and shear. Another type of specimen, which is referred to as the shear specimen, is used to examine the behaviour of stitches under direct shear. The configuration of the shear specimen is similar to those used by Hofbeck et al. (1969) and Buyukozturk et al. (1990) in their study on shear friction and shear strength of joints of precast concrete segmental bridges (PCSB), respectively. In fact, many of the existing formulae that predict the strength of PCSB joints and precast concrete connections are actually derived based on the results from tests on specimens of similar configuration. The shear specimens are used to obtain the shear strength of in-situ concrete stitches so that the values can be directly compared to those predicted by the existing formulae for PCSB joints and the validity of the existing formulae can be investigated. Parametric studies on the shear specimens are also carried out in conjunction with the experimental tests to study the effects of various structural parameters. In this chapter, the results of loading tests and parametric studies on the shear specimens are presented.

The shear strengths of the in-situ stitches

obtained from experiments are compared to those predicted by formulae from the AASHTO guide (AASHTO, 1999), ACI code (ACI Committee 318, 2008) and Eurocode 2 (EN1992-2) (European Committee for Standardization, 2004).

6.2

Experimental Programme

6.2.1

Configuration of the specimens Each specimen is assembled from two L-shaped precast units with an in-

situ concrete stitch cast to join the two units together as illustrated in Figure 6.1. -107-

The out-of-plane dimension of the specimen is 250 mm and the depth of the stitch is 200 mm. The precast units and in-situ stitch of each specimen are made with Grade 60 concrete with mean compressive strength at 28 days of at least 60 MPa. The stitch is subject to prestress of either 1, 2 or 5 MPa. Stitch widths of 100 mm and 200 mm are examined. Construction joint without intentional roughening and with roughness of 2 mm, 4 mm and 6 mm are tested. The method of roughening is described in the next section. The specimens are designed in such a way to simulate the shear behaviour of box girders of different web configurations. Various parts of the web of a box girder may or may not be provided with shear keys, and they may or may not have prestressing tendons running through. Therefore, the specimens tested are either internally prestressed by a bonded tendon or externally prestressed by a clamping device, while the stitch is either plain or provided with shear keys. For those specimens with shear keys, either one large key with a depth of 50 mm (Figure 6.2(a)) or two smaller keys with a depth of 30 mm (Figure 6.2(b)) are provided at each interface. Figure 6.3(a) shows an externally prestressed specimen with two shear keys at each interface, while an internally prestressed specimen without shear key is shown in Figure 6.3(b). Where a specimen is internally prestressed, a 7-wire steel strand is used. The shear specimens are identified using the convention: (prestress type) - (joint type)(joint parameter) - (stitch width in mm) - (concrete grade in MPa) (initial prestress level in MPa). The field 'prestress type' is assigned either with 'E' or 'I' for specimens that are externally or internally prestressed respectively. For plain stitch, the field 'joint type' is assigned with 'P'.

The field 'joint

parameter' is the roughness of the joint in mm where the value is 0 for joint that is not intentionally roughened. For keyed stitch, the field 'joint type' is assigned with 'K' and the field 'joint parameter' is either assigned with 'S' for single-keyed stitch or 'M' for multiple-keyed stitch.

As an example, E-P(0)-100-60-1

identifies an externally prestressed specimen with 100 mm wide and 200 mm deep plain stitch that is not intentionally roughened and the specimen is made with Grade 60 concrete with an initial prestressing level of 1 MPa. configurations of all the specimens tested are listed in Table 6.1.

-108-

The

6.2.2

Preparation of the specimens Prior to the casting of the precast units, the non-prestressed reinforcement

cages are fixed and placed in the formwork along with the polyvinyl chloride (PVC) ducts as illustrated in Figure 6.4, which corresponds to the setup for an internally prestressed specimens with multiple shear keys. A tube is protruded vertically from the precast unit (Figure 6.4) for internally prestressed specimens that require grouting.

Similar to the beam specimens, the precast units are

removed from the formwork one week after casting and the surfaces that are to be joined with the in-situ stitch are scraped to remove laitance. The specimens that are to be examined for the effects of construction joint roughness will have the joint surface chiselled after the laitance is removed. A method similar to the one outlined in the technical report on precast concrete products by the European Committee for Standardization (European Committee for Standardization, 2008) is adopted to measure the degree of roughness, which is essentially taken as the maximum depression within a certain surface area. The joint surface is divided into 20 rectangular sectors (Figure 6.5) and a measuring device as shown in Figure 6.5 is used to monitor the degree of roughness of each sector during chiselling. Excessive roughening is avoided by chiselling progressively until the desired degree of roughness is achieved. For the externally prestressed specimens, prestressing is applied 28 days from the day of casting the in-situ stitch and loading test is subsequently performed on the same day.

For the internally prestressed specimens,

prestressing and grouting is performed on the confirmation that the stitch has gained sufficient strength two weeks after the casting of stitch. Loading test is then performed 28 days after the day of grouting. The prestressing and grouting operations are similar to those for the beam specimens as described in Section 5.2.2.

6.2.3

Test setup The test is carried out using the Avery testing frame as described in

Section 5.2.3. The test is carried out using displacement-controlled scheme at a ram rate of 0.5 mm/min. Three linear variable displacement transducers (LVDTs)

-109-

are attached to the precast units to monitor the shear displacement of the in-situ stitch (Figure 6.6), which is taken as the relative vertical movement between the two precast units, as well as the dilation of the stitch under shear. The LVDT readings are recorded by a data-logger. Steel plates are placed above and below the specimen as illustrated in Figures 6.3(a) and 6.3(b). Gypsum packing is provided to achieve even contact between the steel plate and the specimen, as well as between the steel plate and the actuator.

6.3

Experimental Observations

6.3.1

Specimens with plain stitch The typical mode of failure observed from the specimens with plain stitch

is the sudden sliding along the construction joints between the precast unit and the in-situ concrete stitch as illustrated in Figure 6.7(a). The load-displacement relationships of specimens with plain stitch are plotted in Figure 6.8(a). The externally prestressed specimens, i.e. E-P(0)-100-60-1 and E-P(0)-200-60-1, behave largely similarly. The peak strengths of both specimens are between 50 and 60 kN. The drop in strength after reaching the peak corresponds to the sudden sliding initiated at the joints. Upon failure, the residual strength of the stitch is mainly contributed by the sliding friction between the precast unit and in-situ stitch. The long smooth plateau in the post-peak range of their loaddisplacement curves suggests that the residual strength can be modelled by Coulomb friction. The observations above suggest that the strength of plain stitches depends on the cohesive quality and the friction resistance at the joint which accounts for the initial cracking strength and the residual strength respectively. The load-displacement behaviour of the internally prestressed specimens is similar to those of the externally prestressed specimens as shown in Figure 6.8 except that the peak strength of the specimen with stitch of 200 mm width (IP(0)-200-60-1) is not reached until a relatively large displacement has taken place.

Nevertheless, the peak strengths of the two internally prestressed

specimens are similar, namely 55 kN for I-P(0)-100-60-1 and 52 kN for specimen I-P(0)-200-60-1.

In fact, by examing the load-displacement

-110-

relationship of E-P(0)-100-60-1 and E-P(0)-200-60-1, it is evident that the peak strength of the externally prestressed specimen with 200 mm stitch also occurs at a relatively large displacement.

The reason for this is that the additional

displacement caused by the rotation of the stitch before slippage occurs. The rotation of stitch as shear displacement takes place is illustrated in Figure 6.7(c). As the shear displacements of the internally prestressed specimens advance beyond the peak strength, their behaviour become different from that of the externally prestressed specimens where hardening is evident, which is very likely attributed to the dowel resistance from the internal tendon. When the residual strength of the plain stitch is mainly contributed by friction, one can envisage that the residual strength should be significantly influenced by the normal force induced by prestressing. If the residual strength is to be maintained, then sufficient prestressing force should be provided.

6.3.2

Specimens with keyed stitch The typical failure mode observed from specimens with keyed stitch is

the sudden formation of a diagonal crack across the stitch as shown in Figure 6.7(b). The structural behaviour up to failure is described as follows. Prior to failure, cracks develop along the construction joints. Once cracking initiates, slippage along the cracks and widening of the cracks are evident. Without any cracking elsewhere in the specimen, a diagonal crack suddenly forms across the stitch as the critical shear displacement is reached. The load-displacement relationships of the specimens with single-keyed stitch and multiple-keyed stitch are plotted in Figures 6.8(b) and 6.8(c) respectively. For most specimens except those that are internally prestressed having 200 mm stitch, there is an abrupt reduction in strength when the peak strength is reached, which corresponds to the formation of a diagonal crack across the stitch as shown in Figures 6.8(b) and 6.8(c). In the figures, the steep descending branch of the load-displacement curves is only shown partially for clarity. There is in fact no residual strength in the specimen when the diagonal crack forms.

-111-

Something unexpected is observed from Figures 6.8(b) and 6.8(c). Unlike the 200 mm keyed stitches that are externally prestressed, those that are internally prestressed have no abrupt drop in strength and no diagonal crack has ever formed across the stitch of these specimens. This phenomenon is possibly attributed to the occasional inability of the stitch to form a critical compressive strut across the stitch due to the significant rotation mentioned above. In the experiment, it is observed that the rotation of the stitch is substantial for 200 mm stitch that is internally prestressed, as explained below. Those stitches that are externally prestressed are done so by four 20 mm threaded rods (Figure 6.3(a)), while a single 7-wire steel strand is used to prestress those internally prestressed stitches (Figure 6.3(b)). Therefore, the strain in the strand should be larger than that in the threaded rods when the same prestressing force is applied. In other words, the confinement against dilation for the internally prestressed stitches is lower than that for the externally prestressed ones. With the dilation movement less confined, the rotation of the stitch is therefore increased and the formation of a critical compressive strut that causes diagonal cracking becomes less likely. This may explain why those specimens that are internally prestressed would have diagonal cracking occuring at a relatively large displacement or not at all. This explanation may also be applied to the unusual behaviour of I-K(M)-200-60-1 (Figure 6.8(c)) where the diagonal crack is formed at a relatively large displacement. Comparing the load-displacement relationships between the specimens with plain stitch (Figure 6.8(a)) and keyed stitch (Figure 6.8(b) and 6.8(c)), it is found that the peak strength of the keyed stitch tends to be higher than that of the plain stitch. The load-displacement relationships of the specimens with plain and keyed stitches of 100 mm width are plotted together in Figure 6.9 for ease of comparison. For example, for the 100 mm stitches that are externally prestressed, the plain stitch can attain a peak strength of 63 kN, while the single-keyed stitch and the multiple-keyed stitch can attain substantially higher strength of 135 kN and 180 kN respectively, which are nearly double and triple the strength of the plain stitch respectively. Therefore, despite the brittle behaviour, the provision of shear keys can nevertheless increase the shear strength of the stitches significantly. -112-

6.3.3

Behaviour of specimens with different initial level of prestress The experimental behaviour of the specimens with different initial levels

of prestress are compared by plotting their load-displacement relationships together in Figure 6.10. The behaviour of the plain stitches is first examined. It is evident that the level of prestressing has marked influence on the shear strength of in-situ stitches. By raising the level of prestressing from 1 MPa to 5 MPa, the strength of the stitch has increased by approximately three times. However the post-peak behaviour becomes less ductile. Figure 6.10 shows that the post-peak behaviour of specimen E-P(0)-10060-5 is not as smooth as that of E-P(0)-100-60-1. Since the prestressing force in specimen E-P(0)-100-60-5 is substantially higher, the resistance against sliding by the exposed aggregates along the construction joint becomes very large. When the applied load is large enough to overcome that peak resistance, sudden slippage along the construction joint occurs and the resistance drops. Subsequent to the first occurrence of slippage, the resistance builds up again with further increase in shear displacement and sudden slippage occurs again later when the second peak resistance is overcome by the applied load. Therefore the post-peak branch of response of specimen E-P(0)-100-60-5 under shear has a 'zig-zag' shape. This observation also implies that the roughness at construction joint should have a strong effect on the post-peak shear strength of plain stitches. For the keyed specimens, it can be seen from Figure 6.9 that the peak strengths of both specimens E-K(M)-100-60-2 and E-K(S)-100-60-2 are higher than that of specimen E-K(M)-100-60-1, which also suggests that the initial prestressing force has favourable effect on shear strength. The peak strengths of E-K(M)-100-60-2 and E-K(S)-100-60-2 are similar to that of E-P(0)-100-60-5. This implies that the provision of shear keys at the in-situ stitch helps to reduce the level of prestressing required to achieve certain shear strength, albeit at the expense of reduction in ductility.

6.3.4

Behaviour of specimens with different joint roughness The behaviour of specimens with intentional roughening of the

construction joint is examined The load-displacement relationships of specimens -113-

E-P(0)-100-60-1, E-P(4)-100-60-1 and E-P(6)-100-60-1, which have joints with no intentional roughening, 4 mm roughness and 6 mm roughness respectively, are plotted in Figure 6.11. Increase in both peak and residual strengths is evident with the increase in joint roughness. This is attributed mainly to two factors: (a) the increase in the coefficient of friction as a result of the increase in aggregate interlocking; and (b) the increase in shear dilation with higher roughness that leads to the increase in the tendon stress, which in turn results in higher normal force acting across the stitch.

The shear dilation is plotted against shear

displacement in Figure 6.12 and increase in shear dilation with displacement is evident. From Figure 6.11, it is worthwhile to note that by roughening the joint to attain a roughness of 6 mm, the peak strength of the stitch substantially increases, which has similar effect of providing shear key to the stitch, and yet the behaviour remains very ductile, which is a quality not exhibited by keyed stitches.

6.4

Parametric Study Parametric study is carried out to further examine the effects of stitch

width, initial prestressing force, concrete strength, bonding of the prestressing tendon, the provision of shear keys and the aspect ratio of the stitch on the loaddisplacement relationships.

6.4.1

Method of analysis The parametric studies are carried out numerically using the commercial

finite element package MIDAS FEA. Details of modelling and the choice of various constitutive models are the same as those provided in Section 5.4.1.

6.4.2

Model calibration The finite element models are calibrated against the experimental results

for various empirical parameters of the interface element model, namely the normal and tangential stiffnesses, cohesion and internal friction angle. Figures 6.13 and 6.14 show the both the experimental and calculated load-displacement relationships for stitches that are externally and internally prestressed -114-

respectively. Good agreement of peak strengths is generally observed, but for those specimens with 200 mm stitch, discrepancies between experimental and numerical values are evident at advanced shear displacement. This might have been due to the difficulty of numerical convergence when the analysis is carried out at relatively large displacement where significant damage has taken place. The load-displacement curves that do not have an abrupt drop at the end of the curve correspond to those models with stitch having no diagonal cracking failure.

6.4.3

Effects of stitch width The load-displacement relationships of externally and internally

prestressed stitches with width ranging from 50 mm to 100 mm are plotted in Figures 6.15 and 6.16 respectively. From Figures 6.15(a) and 6.16(a), it is evident that the width of the in-situ concrete stitch has little effect on the trend of the load-displacement relationships for plain stitches. Approximately the same shear capacities of 65 kN are found for the externally prestressed stitches, while shear capacities of 40 to 60 kN are observed for the internally prestressed stitches. From Figures 6.15(b) and 6.15(c), as well as from Figures 6.16(b) and 6.16(c), it can be seen that the load-displacement behaviour of keyed stitches is quite different from that of the plain stitches. There appears to be no definite correlation between the peak strength and the stitch width. However, the peak strengths of single-keyed and multiple-keyed stitches do not deviate too substantially; the peak strengths of single-keyed and multiple-keyed stitches vary from 130 to 170 kN and from 110 to 140 kN respectively. However, the stitches that are relatively narrow (i.e. width of 50 mm) or relatively wide (i.e width of 200 mm) have different behaviour. From Figures 6.15(c) and 6.16(c), it is evident that the 50 mm multiplekeyed stitch has much higher strength than the other multiple keyed stitch. For stitches that are narrow, the rotation of stitch (Figure 6.7(c)) is not significant and hence, together with the better interlocking provided by the multiple keys, the shear resistance can be mobilized more effectively. This is why the 50 mm stitch tends to have a higher strength. However, from Figures 6.15(b) and 6.16(b), it is found that the shear strengths of the 50 mm single-keyed stitches are less than

-115-

those of the 100 mm and 200 mm single-keyed stitches. Unlike those that are multiple-keyed, stress concentration at the roots of the keys is more pronounced for single-keyed stitch, thus cracking initiates at the roots that results in diagonal cracking at a relatively low load. Hence, the shear capacities are lower for 50 mm single-keyed stitch. For those stitches that are 200 mm wide, numerical simulations have revealed that substantial rotation of the stitch occurs as shear displacement increases. This phenomenon is also observed in the experiments, and failure of the stitch by diagonal cracking does not occur either.

Therefore, the peak

strength of these stitches cannot be determined without having substantial displacement but damage would have occurred elsewhere by then. The observations above suggest that the width of the stitch may have certain influence on the shear strength of keyed stitch. It is suspected that the effects of aspect ratio, which is taken as the ratio of the depth to the width of the stitch, may be of greater significance than the width of the stitch. The effects of aspect ratio are examined in due course.

6.4.4

Effects of initial prestressing level To examine the effects of initial prestressing level, the load-displacement

relationships of stitches with initial prestressing level ranging from 1 to 4 MPa are plotted in Figures 6.17 and 6.18 for externally and internally prestressed stitches respectively. From Figures 6.17(a) and 6.18(a), it can be seen that both peak and residual strengths for plain stitches increase as initial prestressing force increases.

Since the shear strength of plain stitches relies significantly on

frictional resistance, which is proportional to the normal force applied, therefore increase in initial prestressing level would have favourable effects on the strength of plain stitches. Figures 6.17(b) and 6.17(c), as well as 6.18(b) and 6.18(c), also reveal positive correlation between peak strength and initial prestressing level for keyed stitches. However, it is found that the increase in peak strength with initial prestressing level diminishes when prestressing level is increased from 3 MPa to 4 MPa. Unlike plain stitches, the shear strength of keyed stitches is governed by the resistance against diagonal splitting crack across the stitch. In other words, -116-

the strength in the principal direction along the diagonal would govern the resistance. The prestressing force can provide confinement that would increase the strength in the principal direction. However, when the prestressing force is further increased, the gain in the principal strength is no longer proportional to the increase in prestressing force, and thus the resistance to the diagonal cracking will have no obvious increase.

6.4.5

Effects of concrete grade To examine the effects of concrete grade, the load-displacement

relationships of stitches with concrete grade ranging from 40 to 80 MPa are plotted in Figures 6.19 and 6.20 for externally and internally prestressed stitches respectively. For the plain stitches, Figures 6.19(a) and 6.20(a) show that the load-displacement relationship is insensitive to the change in concrete grade. This is a reasonable behaviour since the strength of plain stitch is controlled by cohesive quality and friction at the joint surface, which are somewhat insensitive to the strength of concrete. On the other hand, increase in shear strength with concrete strength of stitch is evident in Figures 6.19(b) and 6.19(c) as well as 6.20(b) and 6.20(c). Since the strength of keyed stitches depends on the resistance to diagonal cracking, the tensile strength of concrete is critical.

The tensile strength

essentially increases with the compressive strength of concrete, and therefore an increase in concrete grade will ultimately increase the shear strength of keyed stitches.

6.4.6

Effects of aspect ratio of stitch The externally prestressed stitches with shear keys are further examined

for the effects of the aspect ratio of stitch on the shear strengths. Models of stitches with depths of 400 mm and 600 mm have also been analysed so that a broader range of aspect ratios can be investigated. In Figures 6.21(a) and 6.21(b), the shear strengths are plotted against the aspect ratios of single keyed and multiple-keyed stitch respectively at different initial prestressing levels. Notice that the specimen identifier has been slightly modified where X is the width of

-117-

the stitch that varies with aspect ratio and it is taken as the depth of the stitch divided by the aspect ratio. The number in the bracket following X indicates the depth of the stitch in mm. In most cases, a mild trend of increase in shear strength with aspect ratio of stitch can be observed. The relatively low values of shear strength with low aspect ratios of stitch is attributed to the ineffective transfer of shear due to substantial rotation of the stitch concrete as previously discussed.

6.5

Proposed Design Formula for Keyed Stitches An infinitesimal element within the keyed stitch is subject to axial and

shear stresses, from which the maximum principal stress can be determined. Failure is assumed to occur when the maximum principal stress reaches the tensile strength of concrete ft (MPa) (Tureyen and Frosch, 2003), so the critical shear stress τ (MPa) is expressed as:

τ=

f t + f tσ 2

(6.1)

where the normal stress σ is taken as positive for compression. Taking the maximum shear stress τmax (MPa) as τmax = 1.5 V/A in terms of the shear force V (N) and the cross sectional area A (mm2), and applying it to Equation (6.1) gives V =

2 2 A f t + f tσ 3

(6.2)

which can be used to predict the shear capacity of the stitch. The tensile strength of concrete is usually expressed as a function of the characteristic compressive strength of concrete fck. Eurocode 2 (EN1992-2) provides expressions of ft in terms of fck.

Other empirical expressions for predicting shear capacity of

concrete under normal stress are provided by various guides and codes, such as AASHTO guide (1999):

V = 0.17K f ck A K = 1+

(6.3)

0.04σ ≤2 f ck

(6.4)

-118-

and ACI code (ACI Committee 318, 2008):

 σ V = 0.171 +  f ck A  14 

(6.5)

The formula from Eurocode 2 (European Committee for Standardization, 2004) also considers the depth of section h (mm):

(

)

V = 0.035k 2 / 3 f ck + 0.15σ A k = 1+

(6.6)

200 ≤2 h

(6.7)

where the influence of imposed deformations on normal stress σ of section can be ignored. Therefore, the input for normal stress σ using initial prestress level after allowance for all prestress loss determined at the centroid of the cross section is acceptable. Comparison of the shear capacities predicted using Equations (6.2), (6.3), (6.5) and (6.6) with those from finite element analysis are summarised in Figure 6.22.

These predicted values are all lower than those obtained from finite

element analysis. Equation (6.3), (6.5) and (6.6) are understood to provide lower bound estimates to ensure safety, among which Equation (6.3) from AASHTO tends to be the most conservative. Although the failure criterion of Equation (6.2) is based on the maximum principal stress, it may apply to a point at a stage only, which is not yet the failure of the keyed stitch. Finite element analysis reveals that the inclined cracks due to excessive tensile stresses do not form at the centroid of section, but they tend to occur at the upper or lower portion of stitch.

Such crack

development can be monitored, for instance, from the strain field of E-K(M)200(600)-60-1 obtained from finite element analysis as depicted in Figure 6.23. Figure 6.23(a) shows the maximum principal strain field on the left and the principal stress vectors on the right when inclined cracks form as the stress reaches the tensile strength of concrete. As the load increases, the inclined cracks propagate until some of them touch the boundary of stitch. After the completion of inclined crack propagation, parts of the stitch become separated and the uncracked regions between the inclined cracks act as struts to transfer -119-

shear through the stitch. The failure of keyed stitch occurs when the shear keys are sheared off as shown in Figure 6.23(b). The value calculated by Equation (6.2) is actually the shear-cracking load shown in Figure 6.23(a), whereas the strength of shear keys that transfer shear through strut action after cracking as shown in Figure 6.23(b) is not taken into account. Although shear cracks occur early and damage the stitch, the shear strength of a keyed stitch is governed by the strength of shear keys located in the compression region, i.e. the ends of strut. In the light of this, the AASHTO (1999) design formula estimating the shear capacity of joints with shear keys in precast prestressed segmental bridge can be utilised, which is expressed in SI unit as:

V = Ak (1 + 0.2σ ) f ck + 0.6 Asmσ

(6.8)

where Ak (mm2) is the area of all base areas of shear keys and Asm (mm2) is the area of contact between smooth surfaces as defined in Figure 6.24. According to Figure 6.23, the formation of inclined cracks has damaged the integrity of keyed stitch, which reduces the number and the base area of shear keys that are effective in transferring shear through strut action. Therefore, the joint areas Ak and Asm in Equation (6.9) due to shear cracking should be reduced accordingly.

In addition, Zhou et al. (2005) showed experimentally that

Equation (6.9) from the AASHTO guide always overestimated the shear capacity of multiple-keyed dry joints, possibly due to increasing stress concentration in joints as the number of keys increases. Zhou et al. (2005) recommended the use of a reduction factor for the number of keys above unity. In the light of this, two coefficients, namely the coefficient αdw, where subscript ‘dw’ denotes the aspect ratio or depth/width ratio, and the coefficient αsc, where the subscript ‘sc’ denotes stress concentration, are introduced to Equation (6.9):

  V = α dwα sc  Ak (1 + 0.2σ ) f ck + 0.6 Asmσ   

(6.9)

The derivation of coefficient αdw is illustrated by reference to the stitch of width w in Figure 6.25, which is subjected to compressive force N, bending

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moment M and shear force V. Since the tensile strength of the construction joint is very low, the tensile stress induced by external loads can only reach the critical value in the compressive zone of stitch, for example the upper portion of stitch in Figure 6.25, where the shear crack forms. The finite element analysis conducted also confirms that the shear cracks always develop at either the upper or lower portion of the keyed stitches. As a result, the angle of shear crack θ can then be calculated assuming that the shear crack forms when the maximum principal stress reaches the tensile strength of concrete, develops perpendicular to the direction of maximum principal stress, and starts from the corner of stitch:

tan 2θ =

τ σ /2

f t + f tσ 2

=

(6.10)

σ /2

As the reduction coefficient αdw should reflect the remaining effective area of shear keys, it is taken as the ratio of the remaining depth (h – w tanθ) to the entire depth h as:

α dw

 w w 1 = 1 − tan θ = 1 − tan arctan h h 2 

 2 f t + f tσ   σ /2  

(6.11)

As for the stress concentration coefficient αsc, Zhou et al. (2005) observed a 62% difference in shear strength of three-keyed dry joints between predictions from Equation (6.9) from AASHTO and experimental results. Based on this, the stress concentration coefficient αsc can be estimated as:

α sc =

1 = 0.62 1 + 62%

(6.12)

The predicted values of shear strength of keyed stitches using Equation (6.9) are plotted in Figure 6.26, which shows that they remain as lower bound predictions but with better correlation with finite element results as compared with the use of Equation (6.2).

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6.6

Conclusions Experiments on shear specimens comprised of two L-shaped precast units

connected by an in-situ concrete stitch have been carried out to examine the behaviour of in-situ stitches under direct shear. The experimental programme has been elaborated in detail. From the experiments, it is evident that plain and keyed stitches typically fail by frictional sliding along the construction joint and diagonal cracking across the stitch respectively.

The effects of several

parameters, namely the width and aspect ratio of stitch, initial prestressing forces, concrete strength, provision of shear keys, and roughness of the construction joint surface, have been observed experimentally and further analysed by finite element method. It is found that there tends to be no definite correlation between stitch width and the peak strength of the stitch. On the other hand, the aspect ratio of stitch, is more influential as the shear capacity increases mildly with increase in aspect ratio. The increase in initial prestressing force has favourable effects on strength but the effect diminishes when prestressing level is increased from 3 MPa to 4 MPa. The peak strength is positively correlated with concrete strength and roughness of the surface at the construction joints. For plain stitches, it is observed that by roughening the joint to attain a roughness of 6 mm, the peak strength of the stitch increases substantially yet the behaviour remains very ductile, which is a quality not exhibited by keyed stitches. A design formula for keyed stitches has been proposed, which is a modified version of the existing AASHTO formula for estimating the shear strength of segment-to-segment joints of PCSB. Two coefficients are introduced to the AASHTO formula to take into account the characteristics of in-situ concrete stitches.

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Table 6.1. List of shear specimens tested. Initial Conc. prestress. grade level (MPa) (MPa)

No.

Specimen ID1

Prestress type2

Stitch type3

Stitch width (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

E-P(0)-100-60-1 E-P(0)-200-60-1 E-P(4)-100-60-1 E-P(4)-200-60-1 E-P(6)-100-60-1 E-P(6)-200-60-1 E-P(0)-100-60-5 E-P(0)-200-60-5 E-K(S)-100-60-1 E-K(S)-200-60-1 E-K(M)-100-60-1 E-K(M)-200-60-1 E-K(S)-100-60-2 E-K(M)-100-60-2 E-K(S)-200-60-2 E-K(M)-200-60-2

Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext Ext

P P P P P P P P K(S) K(S) K(M) K(M) K(S) K(M) K(S) K(M)

100 200 100 200 100 200 100 200 100 200 100 200 100 100 200 200

4 4 6 6 -

60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60

1 1 1 1 1 1 5 5 1 1 1 1 2 2 2 2

17 18 19 20 21 22 23 24 25 26

I-P(0)-100-60-1 I-P(0)-200-60-1 I-P(4)-100-60-1 I-P(4)-200-60-1 I-P(6)-100-60-1 I-P(6)-200-60-1 I-K(S)-100-60-1 I-K(S)-200-60-1 I-K(M)-100-60-1 I-K(M)-200-60-1

Int Int Int Int Int Int Int Int Int Int

P P P P P P K(S) K(S) K(M) K(M)

100 200 100 200 100 200 100 200 100 200

4 4 6 6 -

60 60 60 60 60 60 60 60 60 60

1 1 1 1 1 1 1 1 1 1

Roughness (mm)

4

Notes: 1. Specimens are identified using this convention: (prestress type) - (joint type)(joint parameter) (stitch width in mm) - (concrete grade in MPa). 2. Ext = external prestressing; Int = internal prestressing. 3. P = plain stitch; K(S) = single-keyed stitch; K(M) = multiple-keyed stitch. 4. '-' indicates stitch without intentional roughness applied to mating surface.

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Precast unit

In-situ concrete stitch Precast unit Figure 6.1. A shear specimen comprising two precast units connected by an in-

25

50

15

30

25

15

30

50

15

25

30

15

50

25

situ concrete stitch.

30

50 (a) single shear key

(b) multiple shear key

Figure 6.2. Shear key configurations (dimensions in mm).

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Load from actuator Stitch with two shear keys

Steel plate

Hydraulic jack

700 mm

Width varies

45 mm thick steel plate

Depth = 200 mm

Load cell

20 mm diameter threaded rod

Internal forces primarily acting on the stitch

700 – 800 mm (a) in-situ concrete stitch with shear keys and externally prestressed Load from actuator

Steel plate

Stitch without shear key Internal prestressing tendon Hydraulic jack

700 mm

Width varies Depth = 200 mm

Internal forces primarily acting on the stitch

Load cell

700 – 800 mm (b) in-situ concrete stitch without shear key and internally prestressed

Figure 6.3. Examples of shear specimen configuration. -125-

Reinforcement

Grout tube

Shear key Figure 6.4. Formwork and reinforcement arrangement for the precast unit.

Figure 6.5. A chiselled construction joint surface and device for measuring surface roughness.

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LVDT

Figure 6.6. Arrangement of linear variable displacement transducers (LVDT) for the shear specimens.

-127-

Sliding

(a) sliding along the construction joint

Diagonal cracking

(b) diagonal cracking across in-situ stitch

Rotation

(c) cracking along the construction joint followed by rotation of in-situ stitch

Figure 6.7. Modes of failure of the shear specimens.

-128-

Applied load (kN)

80 E-P(0)-100-60-1 E-P(0)-200-60-1 I-P(0)-100-60-1 I-P(0)-200-60-1

60 40 20 0 0

1

2 3 4 Shear displacement (mm)

5

6

(a) plain stitch

Applied load (kN)

200 E-K(S)-100-60-1 E-K(S)-200-60-1 I-K(S)-100-60-1 I-K(S)-200-60-1

150 100 50 0 0

3

6 9 12 Shear displacement (mm)

15

(b) stitch with single key

Applied load (kN)

250 E-K(M)-100-60-1 E-K(M)-200-60-1 I-K(M)-100-60-1 I-K(M)-200-60-1

200 150 100 50 0 0

3

6 9 12 Shear displacement (mm)

15

(c) stitch with multiple keys

Figure 6.8. Experimental load-displacement relationships of specimens with (a) plain stitch; (b) stitch with single key; and (c) stitch with multiple keys.

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250

External E-P(0)-100-60-1 E-K(S)-100-60-1 E-K(M)-100-60-1

Applied load (kN)

200

Internal I-P(0)-100-60-1 I-K(S)-100-60-1 I-K(M)-100-60-1

150 100 50 0 0

2 4 6 Shear displacement (mm)

8

Figure 6.9. Experimental load-displacement relationships of specimens with different shear key configurations.

250

Plain E-P(0)-100-60-1 E-P(0)-100-60-5

Applied load (kN)

200

Keyed E-K(M)-100-60-1 E-K(M)-100-60-2 E-K(S)-100-60-2

150 100 50 0 0

2 4 6 Shear displacement (mm)

8

Figure 6.10. Experimental load-displacement relationships of specimens with different prestressing force.

-130-

250

External E-P(0)-100-60-1 E-P(4)-100-60-1 E-P(6)-100-60-1

Applied load (kN)

200 150 100 50 0 0

2 4 6 Shear displacement (mm)

8

Figure 6.11. Experimental load-displacement relationships of specimens with different construction joint roughness.

Shear dilation (mm)

1.6

External E-P(0)-100-60-1 E-P(4)-100-60-1 E-P(6)-100-60-1

1.2

0.8

0.4

0.0 0

2 4 6 Shear displacement (mm)

8

Figure 6.12. The effect of roughness on shear dilation in experiments.

-131-

Applied load (kN)

80 E-P(0)-100-60-1 (Exp.) E-P(0)-200-60-1 (Exp.) E-P(0)-100-60-1 (FEA) E-P(0)-200-60-1 (FEA)

60 40 20 0 0

1

2 3 4 Shear displacement (mm)

5

(a) plain stitch

Applied load (kN)

250 E-K(S)-100-60-1 (Exp.) E-K(S)-200-60-1 (Exp.) E-K(S)-100-60-1 (FEA) E-K(S)-200-60-1 (FEA)

200 150 100 50 0 0

2

4 6 8 10 Shear displacement (mm)

12

(b) stitch with single key

Applied load (kN)

250 E-K(M)-100-60-1 (Exp.) E-K(M)-200-60-1 (Exp.) E-K(M)-100-60-1 (FEA) E-K(M)-200-60-1 (FEA)

200 150 100 50 0 0

2

4 6 8 10 Shear displacement (mm)

12

(c) stitch with multiple keys

Figure 6.13. Experimental and numerical load-displacement relationships of externally prestressed specimens.

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Applied load (kN)

80 I-P(0)-100-60-1 (Exp.) I-P(0)-200-60-1 (Exp.) I-P(0)-100-60-1 (FEA) I-P(0)-200-60-1 (FEA)

60 40 20 0 0

1

2 3 4 Shear displacement (mm)

5

(a) plain stitch

Applied load (kN)

150 I-K(S)-100-60-1 (Exp.) I-K(S)-200-60-1 (Exp.) I-K(S)-100-60-1 (FEA) I-K(S)-200-60-1 (FEA)

120 90 60 30 0 0

3 6 9 12 Shear displacement (mm)

15

(b) stitch with single key

Applied load (kN)

150 I-K(M)-100-60-1 (Exp.) I-K(M)-200-60-1 (Exp.) I-K(M)-100-60-1 (FEA) I-K(M)-200-60-1 (FEA)

120 90 60 30 0 0

3 6 9 12 Shear displacement (mm)

15

(c) stitch with multiple keys

Figure 6.14. Experimental and numerical load-displacement relationships of internally prestressed specimens.

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Applied load (kN)

80

External fpi = 1 MPa fcu = 60 MPa

60

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

40 20 0 0

1 2 3 Shear displacement (mm)

4

(a) plain stitch

Applied load (kN)

200

External fpi = 1 MPa fcu = 60 MPa

150

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

100 50 0 0

3 6 9 Shear displacement (mm)

12

(b) stitch with single key

Applied load (kN)

250

External fpi = 1 MPa fcu = 60 MPa

200 150

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

100 50 0 0

1 2 3 4 Shear displacement (mm)

5

(c) stitch with multiple keys

Figure 6.15. Parametric study - effects of stitch width for externally prestressed specimens.

-134-

Applied load (kN)

80

Internal fpi = 1 MPa fcu = 60 MPa

60

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

40 20 0 0

1 2 3 Shear displacement (mm)

4

(a) plain stitch

Applied load (kN)

200

Internal fpi = 1 MPa fcu = 60 MPa

150

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

100 50 0 0

3 6 9 Shear displacement (mm)

12

(b) stitch with single key

Applied load (kN)

250

Internal fpi = 1 MPa fcu = 60 MPa

200 150

w = 50 mm w = 100 mm w = 150 mm w = 200 mm

100 50 0 0

3 6 9 Shear displacement (mm)

12

(c) stitch with multiple keys

Figure 6.16. Parametric study - effects of stitch width for internally prestressed specimens.

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Applied load (kN)

150

External w = 100 mm fcu = 60 MPa

120 90

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

60 30 0 0

1 2 3 Shear displacement (mm)

4

(a) plain stitch

Applied load (kN)

200

External w = 100 mm fcu = 60 MPa

150

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

100 50 0 0.0

0.5 1.0 1.5 2.0 2.5 Shear displacement (mm)

3.0

(b) stitch with single key

Applied load (kN)

250

External w = 100 mm fcu = 60 MPa

200 150

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

100 50 0 0.0

0.5 1.0 1.5 2.0 2.5 Shear displacement (mm)

3.0

(c) stitch with multiple keys

Figure 6.17. Parametric study - effects of prestressing for externally prestressed specimens.

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Applied load (kN)

120

Internal w = 100 mm fcu = 60 MPa

90

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

60 30 0 0.0

0.5 1.0 1.5 2.0 2.5 Shear displacement (mm)

3.0

(a) plain stitch

Applied load (kN)

200

Internal w = 100 mm fcu = 60 MPa

150

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

100 50 0 0.0

0.5 1.0 1.5 2.0 2.5 Shear displacement (mm)

3.0

(b) stitch with single key

Applied load (kN)

200

Internal w = 100 mm fcu = 60 MPa

150

fpi = 1 MPa fpi = 2 MPa fpi = 3 MPa fpi = 4 MPa

100 50 0 0.0

0.5 1.0 1.5 2.0 2.5 Shear displacement (mm)

3.0

(c) stitch with multiple keys

Figure 6.18. Parametric study - effects of prestressing for internally prestressed specimens.

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Applied load (kN)

80

External w = 100 mm fpi = 1 MPa

60

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

40 20 0 0

1 2 3 Shear displacement (mm)

4

(a) plain stitch

Applied load (kN)

200

External w = 100 mm fpi = 1 MPa

150

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

100 50 0 0

1 2 3 Shear displacement (mm)

4

(b) stitch with single key

Applied load (kN)

250

External w = 100 mm fpi = 1 MPa

200 150

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

100 50 0 0

1 2 3 Shear displacement (mm)

4

(c) stitch with multiple keys

Figure 6.19. Parametric study - effects of concrete strength for externally prestressed specimens.

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Applied load (kN)

80

Internal w = 100 mm fpi = 1 MPa

60

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

40 20 0 0.0

0.5 1.0 1.5 Shear displacement (mm)

2.0

(a) plain stitch

Applied load (kN)

150

Internal w = 100 mm fpi = 1 MPa

120 90

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

1 2 3 4 Shear displacement (mm)

5

(b) stitch with single key

Applied load (kN)

150

Internal w = 100 mm fpi = 1 MPa

120 90

fcu = 40 MPa fcu = 60 MPa fcu = 80 MPa

60 30 0 0

1 2 3 4 Shear displacement (mm)

5

(c) stitch with multiple keys

Figure 6.20. Parametric study - effects of concrete strength for internally prestressed specimens.

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900

Depth = 200mm E-K(S)-X(200)-60-1 E-K(S)-X(200)-60-2 E-K(S)-X(200)-60-4

Shear strength (kN)

800 700 600

Depth = 400mm E-K(S)-X(400)-60-1 E-K(S)-X(400)-60-2 E-K(S)-X(400)-60-4

500 400 300

Depth = 600mm E-K(S)-X(600)-60-1 E-K(S)-X(600)-60-2 E-K(S)-X(600)-60-4

200 100 0 0

1

2 3 4 5 Aspect ratio of stitch

6

7

(a) single-keyed stitch 800

Depth = 200mm E-K(M)-X(200)-60-1 E-K(M)-X(200)-60-2 E-K(M)-X(200)-60-4

Shear strength (kN)

700 600 500

Depth = 400mm E-K(M)-X(400)-60-1 E-K(M)-X(400)-60-2 E-K(M)-X(400)-60-4

400 300

Depth = 600mm E-K(M)-X(600)-60-1 E-K(M)-X(600)-60-2 E-K(M)-X(600)-60-4

200 100 0 0

1

2 3 4 5 Aspect ratio of stitch

6

7

(b) multiple-keyed stitch

Figure 6.21. Parametric study - effects of aspect ratio on shear strength.

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1000

Max. principal stress: Eq. (6.2) AASHTO: Eq.(6.3) ACI: Eq.(6.5) Eurocode: Eq. (6.6) Perfect correlation line

Predicted (kN)

800 600 400 200 0 0

200 400 600 800 Finite element results (kN)

1000

Figure 6.22. Comparison between predicted values and finite element results of shear capacity.

principal stress vectors

principal strain field

principal strain field

(a)

principal stress vectors (b)

Figure 6.23. Maximum principal strain field and stress vectors in specimen EK(M)-200(600)-60-1 when (a) inclined cracks formed; and (b) shear keys are sheared. -141-

Ak (sum of all these individual areas)

Asm (sum of these individual areas)

Base areas of shear keys Figure 6.24. Definition of Ak and Asm.

w tanθ

θ

M

V

h – w tanθ

N

h

shear crack

V M + Vw

N

w

Figure 6.25. Formation of shear cracks in a keyed stitch (shear keys are not shown).

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1000 Perfect correlation line

Predicted (kN)

800 600 400 200 0 0

200 400 600 800 Finite element results (kN)

1000

Figure 6.26. Comparison between predictions from Equation (6.9) and finite element results of shear capacity.

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CHAPTER 7 NUMERICAL SIMULATION OF FULL SCALE IN-SITU CONCRETE STITCHES

7.1

Overview In this chapter, the behaviour of full scale in-situ concrete stitches

commonly found in typical segmental bridges is studied by numerical method. The North Vernon Bridge in Indiana, USA, which has been built using the balanced cantilever method with a 1.6 m in-situ concrete stitch at the mid-span of the central span, is used as the typical case for analysis. The full range behaviour of the bridge deck is analysed and the damage propagation within the stitch during the entire load history is monitored. Parametric studies are also carried out to examine the effects of stitch width on the response of the bridge deck. Towards the end of the chapter, the application of the modified AASHTO formula derived in Chapter 6 (i.e. Equation (6.9)) to predict the load-carrying capacity of the deck based on the shear capacity of the stitch is demonstrated.

7.2

Description of the North Vernon Bridge The North Vernon Bridge (Prestressed Concrete Institute, 1978) is a 3-

span prestressed concrete segmental bridge with a total length of 116.04 m and a constant cross section depth of 2,745 mm, as shown in Figure 7.1. It is assumed that all supports can resist uplift as necessary. It is constructed with two end support segments of 1.6 m, two intermediate support segments of 2.74 m, 44 span segments of 2.44 m and an in-situ stitch of 1.6 m at the centre of central span. The bridge was built by the balanced cantilever method, where the stitch of central span was cast in-situ without shear reinforcement after the segmental cantilevers were erected from each pier. Twenty-six cantilever tendons and eight continuity tendons were provided at the cantilever span and at the mid-span, respectively. Each tendon consisted of twelve 13 mm diameter strands with an ultimate load of 2,202 kN. All tendons were initially stressed to 70% of their

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ultimate load.

Design was carried out such that the compressive stress of

concrete was limited to 38 MPa and no tension should occur for any load combinations prescribed by AASHTO design code. Further details can be found from the manual by the Prestressed Concrete Institute (1978).

7.3

Method of Analysis To study the response of an in-situ stitch of 1.6 m length under combined

bending and shear, a two dimensional finite element model of the North Vernon Bridge taking into account different transverse thicknesses of the flanges and webs is established using the parameters verified by experiments and incorporating large-deformation analysis. Since shear behaviour is of primary interest, increasing displacement is imposed on the precast segment immediately adjacent to the in-situ stitch at central span in addition to all permanent loading, which generates the maximum shear force on the stitch. For convenience in numerical modelling of the in-situ stitch, only the construction joints adjacent to the in-situ stitch are simulated by the Coulomb model, whereas the other joints are assumed perfectly rigid.

Material non-linearity is taken into account in

modelling all segments.

7.4

Full Range Behaviour of the Bridge The full-range behaviour of the bridge under imposed displacement

adjacent to the mid-span stitch of the central span is presented in Figure 7.2, clearly showing the three-stage characteristics together with the formation of plastic hinges in the deck. The development of maximum principal strain fields and principal stress vectors in the stitch is shown in Figure 7.3. Initially at Stage 1, the load-deflection relationship is approximately linear. As shown in Figure Figure 7.3(a), even after the formation of a horizontal crack at the interface between the upper flange and web of the stitch, the upper flange can still help in carrying bending moment. With the increase in imposed displacement, Figure 7.3(b) shows that the horizontal crack keeps propagating until it spans the entire stitch.

Some inclined cracks initiated from the left construction joint also

propagate and reach the horizontal crack, separating the upper flange from the

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web. The stitch is adversely affected by excessive cracking, and it gradually transforms to a plastic hinge thereby causing more bending moment at the pier segment closer to the imposed load. The overall stiffness of bridge decreases and the load-deflection curve approaches the straight line corresponding to Stage 2 in Figure 7.2. The transition from Stage 2 to Stage 3 occurs with the flexural failure of pier segment above the right interior pier causing formation of the second plastic hinge. With further increase in imposed displacement, another major diagonal crack develops from the right construction joint to the bottom flange as shown in Figure 7.3(c), and shear transfer relies on the strut formed in the uncracked region between the two major diagonal cracks. Figure 7.3(d) shows that shear failure of the stitch is mainly caused by the shearing off of shear keys in the compression zone, which leads to substantial reduction in loadcarrying capacity at Stage 3 or effectively failure of the entire bridge.

7.5.

Variation of Stitch Width The effects of stitch width on the structural behaviour of the bridge at

Stages 1 and 2 are further investigated. The stitch width is varied from 200 mm to the original design of 1,600 mm. In order to change the stitch width while maintaining the span lengths of the bridge, the segmental cantilever spans in the main span are increased accordingly to compensate. The finite element results are plotted in Figure 7.4, which show that higher aspect ratios generally enhance the peak load-carrying capacity. Cases with stitch widths below 500 mm do not show obvious characteristic of Stage 2 behaviour as in Figure 7.2, while the transition from Stage 1 to Stage 2 becomes less apparent as the aspect ratio of the stitch increases. The strain and stress patterns in Figure 7.5 for a 200 mm wide stitch show that the crack development in thinner stitches is similar to that for the thicker stitches as shown in Figure 7.3. Although a diagonal crack forms early at the lower portion of the stitch web as shown in Figure 7.5(a), the remaining part of the web is largely intact and therefore the shear and moment resistances are not much reduced. Similarly, a horizontal crack also propagates at the interface between the upper flange and web of the stitch due to stress concentration, and

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detaches the upper flange of the stitch. With increase in imposed displacement, the shear keys at construction joint do not fail simultaneously but rather fail sequentially from the upper portion of stitch web. The development of principal stress vectors plotted in Figure 7.5 shows that the number of effective shear keys reduces with the gradual loss of shear keys in the compression zone, which also results in decrease of stiffness in the transition from Stage 1 to Stage 2. When all the shear keys in the compression zone are damaged as shown in Figure 7.5(e), shear transfer by strut action is no longer possible. The shear force is then mostly taken by shear friction on the vertical cracked surface near the damaged shear keys, which accounts for the much reduced shear resistance at the failure of the entire bridge. The enhancement of ultimate load-carrying capacity of the bridge with the increase in aspect ratio of the stitch is mostly attributed to the restraint of shear crack formation. Since the damage due to shear cracking is reduced, the residual moment resistance of stitch after cracking remains high thereby resulting in a higher ultimate load-carrying capacity. However, it should be noted that the governing factor of ultimate load-carrying of the bridge is still the moment capacity of the pier segments above the intermediate supports, which is more critical than the shear strength of the stitch. The variations of internal forces at the stitch section of North Vernon Bridge are shown in Figure 7.6. The zero initial bending moment results from the construction sequence as the stitch has been cast after completion of two segmental cantilevers. The shear force Fs (MN) at the stitch and the loading force FL (MN) resulting from the imposed displacement on the bridge can then be expressed approximately in terms of the mid-span deflection of central span dmid (mm) as Fs = 0.21dmid and FL = 0.6dmid respectively, thereby giving FL = 2.87Fs = 2.87×106V in terms of the shear capacity V (N) of different units. Applying Equation (6.9) derived in Chapter 6, i.e.

  V = α dwα sc  Ak (1 + 0.2σ ) f ck + 0.6 Asmσ   

(7.1)

to calculate the shear capacity V at the stitch, one can estimate the load-carrying capacity FL based on shear capacity. The estimated values of load-carrying

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capacity based on shear capacity of the stitch are compared to finite element results of ultimate load considering all aspects in Figure 7.7. The results further confirm that the load-carrying capacity of North Vernon Bridge is governed by flexural strength at nearby pier segments, whereas the shear failure of the stitch is relatively remote.

7.6

Conclusions The behaviour of in-situ stitches under combined bending and shear is

studied by a full-scale finite element model of North Vernon Bridge, with imposed displacement to induce maximum shear.

The full-range behaviour

shows a three-stage response associated with shear cracking of stitch, flexural failure of pier segment and shear failure of the bridge. A higher aspect ratio of stitch tends to increase the ultimate load-carrying capacity as the damage is confined to a thinner stitch so that a higher residual moment resistance can remain.

However, the ultimate load-carrying capacity of the bridge is still

governed by the flexural strength of pier segments rather than the shear strength of the stitch.

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116.04 29.01

29.01

29.01

29.01

In-situ stitch (a) span arrangement (dimensions in m)

610 915 460 610

1,980

610 460 915

610

230

127 2,515 2,745

356 152 254 2,388 3,048 Cantilever tendons Continuity tendons (depth varies)

Bottom flange thickness: 203 for regular segment; 330 for pier segment; 203-330 for transition segment

(b) deck section (dimensions in mm).

Figure 7.1. Configuration of the North Vernon Bridge (not to scale).

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40

Applied load (MN)

1

2

35

1

30

2

25

3

20 15 3

10 5 0 0

50 100 150 Mid-span displacement (mm)

200

Figure 7.2. Full-range behaviour of North Vernon Bridge subject to imposed displacement.

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(b)

(a)

Principal stress vectors

Principal strain field

Principal stress vectors

Principal strain field

Principal stress vectors

Principal strain field

Principal stress vectors

Principal strain field

(d)

(c)

Figure 7.3 Development of maximum principal strain fields and principal stress vectors in stitch (1.6 m width) of North Vernon Bridge.

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40

w = 200 mm w = 400 mm w = 600 mm w = 1,200 mm w = 1,600 mm

Applied load (MN)

35 30 25 20 15 10 5 0 0

20

40 60 80 Mid-span displacement (mm)

100

Figure 7.4. Effects of stitch width on structural response of North Vernon Bridge.

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(a)

(b)

(c)

(d)

(e)

Figure 7.5. Development of maximum principal strain fields (left) and principal stress vectors (right) of stitch (200 mm width) in North Vernon Bridge at different stages:(a) when diagonal cracking across the stitch initiates; (b) when cracking between the flange and web progresses; and (c) to (e) subsequent progressive failure of shear keys along the right construction joint.

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70

10

60

Force (MN)

8 50

7

40

6 5

30

4

20

3 2

10

Bending moment (MNm)

9

Applied load Shear force Tendon force Bending moment

1

0

0 0

50

100

Mid-span displacement (mm)

Figure 7.6. Variations of internal forces at stitch section of North Vernon Bridge ignoring material nonlinearity.

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45

Prediction

Applied load (MN)

40

FEA

35 30 25 20 15 10 5 0 200

300

400 500 800 1200 1600 Stitch width (mm)

Figure 7.7. Comparison between predicted load-carrying capacity based on shear strength of stitch and overall numerical results of North Vernon Bridge.

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CHAPTER 8 COLLAPSE MECHANISMS AND ROBUSTNESS DESIGN

8.1

Overview Unlike buildings, bridges have fewer load paths to redistribute loads in

case of local failures. Robustness is of concern particularly to multi-span bridges to avoid progressive collapse. If part of a span ruptures, substantial hogging moment is induced in the deck sections over the adjacent piers. Whether or not progressive failure ensues therefore depends largely how well can the deck sections cope with this change in moment that may not have been anticipated. In this chapter, the deck of a typical multi-span segmental bridge built using the balanced cantilever method is analysed for its global behaviour in an extreme event.

The variation of internal forces along the bridge deck is

examined as incremental displacement is imposed on the deck to simulate an extreme loading condition. The effects of the performance of in-situ concrete stitch on the redistribution of internal forces and the possibility of forming collapse mechanisms are investigated. Three scenarios are analysed in which the bridge deck is provided with slightly different moment capacities and an overall assessment reveals that the responses of the deck to the monotonic increasing imposed displacement are rather different in each scenario. The effects of providing reinforcements across the stitch are examined. Based on the results, different categories of robustness and various measures to improve robustness are proposed.

8.2

Method of Analysis

8.2.1

Configuration of the bridge analysed Figure 8.1 shows a typical multi-span segmental bridge that is used in the

analysis. The bridge is constructed using the balanced cantilever method and it has a deck configuration that is largely similar to that of the North Vernon Bridge as presented in the previous chapter. The deck is supported on bearings and is -157-

assumed to have the same cross section except for certain segments with thicker bottom flanges, including the segment above each pier (the pier segment) and those adjacent to the pier segment (the transition segments with linearly varying bottom flange thickness). The bridge has rather low statical indeterminacy and is therefore prone to progressive collapse. The bridge and its variations are to satisfy the ultimate limit states for standard load cases. The cantilever tendons are stressed after erection of each pair of segments. After casting the 200 mm wide in-situ concrete stitch and allowing time for it to gain strength, the continuity tendons of the span are stressed. Each tendon consists of twelve 13 mm diameter strands with cross sectional area per strand of 98.5 mm2, ultimate strength fpu of 1,862 MPa and elastic modulus Eps of 195 GPa, unless otherwise stated. To account for various losses of prestress, the effective tendon stress is taken to be 60% of the ultimate strength (Prestressed Concrete Institute, 1978). All the cantilever tendons are symmetrical about the pier from which construction begins, while all the continuity tendons over interior spans are symmetrical about the mid-span. Tendons anchored to the same segment joints constitute a tendon group.

Additional information on the tendon

arrangements is given in Figure 8.2 and Table 8.1. Each deck segment is also reinforced by non-prestressed steel with yield strength fy of 460 MPa and elastic modulus Es of 200 GPa. The amounts of longitudinal non-prestressed steel for the top flange, bottom flange and web as shown in Figure 8.1(b) are 6,266 mm2, 1,570 mm2 and 2,512 mm2 respectively. Longitudinal non-prestressed steel is not provided to in-situ concrete stitches.

8.2.2

Nonlinear finite element analysis The collapse analysis is carried out using the nonlinear finite element

procedures as described in Chapter 3.

8.2.3

Modelling of the bridge deck A symmetrical finite element mesh is adopted, with details shown in

Figure 8.3 and Table 8.2. Plastic hinges may form around the in-situ stitches and the sections above piers (i.e. pier sections), and hence finer elements are used -158-

there. For example, upon imposition of displacements, plastic hinges may form at certain locations where the post-peak branches of moment-curvature (M-φ) relationship (Figure 3.6(a)) are traced after the moment capacities are reached, while unloading occurs elsewhere. The nonlinear analysis helps to decide if onward loading or unloading occurs. The plastic hinge length lp within which all sections are assumed to proceed along the post-peak branch of the M-φ relationship is quite controversial, and existing formulae for calculating them are found to be largely inconsistent (Mendis, 2001). Formulae for estimation of plastic hinge length lp by Sawyer (1964), Corley (1966) and Mattock (1967) are respectively

l p = 0.25d + 0.075z

(8.1a)

l p = d / 2 + 0.2z / d (d and z in inches)

(8.1b)

l p = 0.5d = 0.05z

(8.1c)

where d is the effective depth, and z is the distance between the critical section and the point of contraflexure. Taking d = 2,595 mm and z = 30,000 mm for extreme scenarios in which certain parts have ruptured, the above estimates of lp are 2,899 mm, 1,892 mm and 2,798 mm respectively. As parametric study shows that the outcome is not too sensitive to the plastic hinge length, it is simply taken as the overall depth, namely 2,745 mm.

8.3

Variation of Internal Moments and Formation of Collapse Mechanisms Displacement is gradually applied to the mid-span of Span 3 (i.e. M3) to

simulate an extreme unforeseen load there, as for example when blasting due to vehicle collision occurs (Figure 8.4(a)) or when a large boulder falls onto a hillside viaduct during a mudslide (Figure 8.4(b)). As the ensuing changes in deck moments and the possible sequence of plastic hinge formation will affect the vulnerability of the bridge to progressive collapse, three different scenarios with some variations of properties from those specified in Section 2 are examined, namely

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(a) Scenario A: moment capacities at Pier Sections C and D are not reached; (b) Scenario B: moment capacities at Pier Sections C and D are reached; and (c) Scenario C: moment capacities at the most critical sections are reached. These cases are further analysed to examine possible outcomes of no collapse, limited collapse and progressive collapse. It is assumed that a plastic hinge is formed once the moment capacity is reached and deformation is continued. The typical M-φ curve in Figure 3.2 also shows that continued deformation along the post-peak branch further reduces the resisting moment.

8.3.1

Scenario A The tendon and reinforcement arrangement is that in Section 8.2.1.

Figure 8.5 shows the variation of deck moments in Scenario A under permanent loading and imposed displacement at M3, where positive values denote hogging moments. Figure 8.6 shows the vertical displacements of deck where positive values are upward. Failure at a section is denoted by a cross. These sign conventions hold hereafter unless otherwise stated.

Figure 8.5 also shows

extremely low hogging moment capacities at in-situ stitches and extremely low sagging moment capacities at pier sections as they are not required by the standard load cases. Figure 8.5(a) shows the deck moments when M3 deflects downwards by 25 mm, where the moments still stay within the moment capacities. When the deflection at M3 reaches 200 mm, Stitches 2 and 4 in the adjacent spans fail under hogging moments. As the deflection at M3 increases to 325 mm, the sagging moment there has reached its capacity, causing Stitch 3 to fail and giving the deck moment in Figure 8.5(b). The deformed shape in Figure 8.6(a) clearly shows the “kinks” of significant curvatures at Stitches 2, 3 and 4. Figure 8.6(b) shows the deformed shape when the deflection at M3 reaches 750 mm, with pronounced hinging at Stitches 2, 3 and 4. When the deflection at M3 reaches 1,250 mm, the sagging moments induced at Pier Sections B and E then cause flexural failure there. The deck moment in Figure 8.5(c) shows that 5 plastic hinges are formed. The sequence of plastic hinge formation is summarised in the bridge elevation in Figure 8.5.

The deformed shape in Figure 8.6(c) is

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characterised by “kinks” not only at Stitches 2, 3 and 4, but also at Pier Sections B and E. The moment at each plastic hinge reduces with further rotation along the post-peak branch of its M-φ curve and hence the moments at the plastic hinges become extremely low. Even if the deflection at M3 continues to increase, the deck moments do not depart much from those shown in Figure 8.5(c). Since none of the spans has collapsed and no mechanism is formed, one may say that the bridge has superb robustness. From Figures 8.5(c) and 8.6(c), one may conclude that a necessary condition for this superb robustness is provision of adequate moment capacities at Pier Sections C and D. Examination of the deck moments over Span 2 in Figures 8.5(c) suggests that the maximum moment at Pier Section C can be estimated from an equivalent cantilever having C as fixity with a length lying between 50% and 100% of Span 2.

8.3.2

Scenario B In Scenario B, the hogging moment capacities in the segments adjacent to

the piers are reduced to approximately 70% of those in Scenario A so as to examine the effects of failure of Pier Sections C and D on robustness. Figures 8.7 and 8.8 show respectively the variations of deck moments and vertical displacements under permanent loading and imposed displacement at M3. When the deflection at M3 increases to 360 mm, flexural failure has occurred at M3 giving the deck moments in Figure 8.7(a). As the deflection at M3 reaches 650 mm, Stitches 2 and 4 fail in hogging moment. Unlike Scenario A, because of the reduced flexural stiffness and strength of the segments adjacent to the piers in Scenario B, sagging moment increases more rapidly at M3 causing failure there before Stitches 2 and 4. When the deflection at M3 reaches 2,905 mm as shown in Figure 8.8(a), the hogging moments at Pier Sections C and D reach their moment capacities (Figure 8.7(b)), which are about to cause flexural failure there. Further increase of the deflection at M3 to 2,915 mm causes not only flexural failure at Pier Sections C and D, but also formation of plastic hinges at Stitches 1 and 5 because of excessive hogging moments. As the deflection at M3 reaches 2,925 mm, the flexural failure at Pier Sections C and D has progressed so much along the post-peak branches of their M-φ curves that the

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moments there have decreased to nearly zero, giving the deck moment in Figure 8.7(c). The three well-developed plastic hinges in Span 3 therefore lead to its collapse as shown in the deck displacement in Figure 8.8(b). One critical issue is whether the collapse of Span 3 will trigger progressive collapse. Immediately prior to the flexural failure at Pier Sections C and D, the deck moments from End A to Stitch 2 and from Stitch 5 to End F are relatively small as shown in Figure 8.7(b), which are predominantly sagging except for the vicinity of Pier Sections B and E.

The corresponding deck

displacements in Figure 8.8(a) show predominantly sagging curvatures in Spans 1 and 5, and hogging curvatures in Spans 2 and 4. With the flexural failure at Pier Sections C and D, substantial changes to the deck moments and displacements ensue. Stitches 2 and 4, which have previously formed plastic hinges under hogging moments, are put in slight sagging moments after the reversal. At the same time, large parts of Spans 1, 2, 4 and 5 are caused to take substantial hogging moments (Figure 8.7(c)), resulting in reversals of deflections (Figure 8.8(b)). Collapse is confined to Span 3 only if the increased hogging moments at Pier Sections B and E do not exceed their corresponding moment capacities, when Spans 2 and 4 act effectively as new end spans after the collapse of Span 3. Since collapse is limited to one single span, one may say that the bridge has adequate robustness. To achieve adequate robustness, in the extreme event of loss of a certain span for whatever reasons, the adjacent spans should have sufficient moment capacities to resist the increased sagging and hogging moments when they behave as the new end spans.

8.3.3

Scenario C In Scenario C, the hogging moment capacities in the segments adjacent to

the piers are reduced to approximately 55% of those in Scenario A so as to examine the effects of formation of multiple plastic hinges at pier sections during the failure process on robustness. Figures 8.9 and 8.10 show respectively the variations of deck moments and vertical displacements under permanent loading and imposed displacement at M3.

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Because of the further reduced flexural stiffness and strength of the segments adjacent to the piers, sagging moment increases rapidly at M3. When the deflection at M3 increases to 385 mm, flexural failure has occurred at M3. As the deflection at M3 reaches 1,505 mm, Stitches 2 and 4 fail in hogging moment. When the deflection at M3 reaches 2,745 mm (Figure 8.10(a)), the hogging moments at Pier Sections C and D reach their moment capacities (Figure 8.9(a)), which are about to cause flexural failure there. At that time, the hogging moments at Pier Sections B and E are still relatively low. Further increase of the deflection at M3 to 2,750 mm causes not only flexural failure at Pier Sections C and D, but also formation of plastic hinges at Stitches 1 and 5 because of excessive hogging moments. As the deflection at M3 reaches 2,765 mm, the hogging moments at Pier Sections B and E have reached their respective moment capacities, and the flexural failure at Pier Sections C and D has progressed so much along the postpeak branches of their M-φ curves that the moments there have decreased to nearly zero, giving the deck moment as shown in Figure 8.9(b). The three welldeveloped plastic hinges in Span 3 therefore lead to its collapse. After this, Spans 2 and 4 effectively act as end spans, leading to reversal from hogging moments to sagging moments in the regions from Stitch 2 to Pier Section C and from Pier Section D to Stitch 4, as well as substantial increase of hogging moments around Pier Sections B and E. Further increase of the deflection at M3 to 2,775 mm causes failures at the Pier Sections B and E, resulting in substantial rearrangement of deck moments (Figure 8.9(c)). With the reduction of moments at Pier Sections B, C, D and E to nearly zero, Spans 1, 2, 4 and 5 then act as if they were simply supported. In particular, the rapid increase in sagging moments in Spans 2 and 4 causes formation of plastic hinges in sagging moments at locations close to the piers, as the prevailing moments there under standard load cases are essentially hogging. These additional plastic hinges therefore trigger the collapse of Spans 2 and 4. The progressive failure of the bridge deck is evident and the deck is said to have inadequate robustness.

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8.4

Effects of Providing Top Reinforcement to In-Situ Concrete Stitches Examination of the previous scenarios shows that the collapse of a bridge

designed for standard load cases under extreme circumstances is often triggered by failure of sections by moments of signs unexpected in regular design. A parametric study is therefore carried out with the bridge described in Section 8.2.1 to examine the effects of providing nominal top reinforcement across insitu concrete stitches on the global bridge behaviour under an extreme event. Table 8.3 shows the 6 models investigated, namely Models A1 to A6, with different numbers of prestressing strands provided to the top of each of the insitu stitches. The strands are anchored to blisters located beneath the soffit of top flange and effectively prestressed to a relatively low stress level of 0.2fpu where fpu is the ultimate tensile strength of prestressing strands. A schematic diagram illustrating how the top reinforcements are provided is shown in Figure 8.11. The deformed shapes for Models A1 and A4 just before flexural failure at M3 are plotted in Figure 8.12. Without the top reinforcement across in-situ stitches in Model A1, Stitches 2 and 4 should have already formed plastic hinges in hogging moments, as characterized by the “kinks” at these stitches, before the flexural failure at M3.

However, by providing a relatively small amount of top

reinforcement across in-situ stitches in Model A4, the integrity of Stitches 2 and 4 is preserved, thereby reducing the chance of forming a mechanism. The effectiveness of providing top reinforcement across stitches is further studied by examining the variations of moments at Stitches 2 and 4 with imposed deflection for various models. Because of symmetry, only Stitch 2 is examined. The development of deck moments at critical sections with reference to the case of permanent loading may be described by the residual strength utilisation factor (RSUF), which is defined by the ratio of the remaining strength utilised to the absolute value of the entire remaining strength. For convenience, RSUF is taken to be positive for sagging moments and negative for hogging moments. In other words, for the case of permanent loading, RSUF is zero, while for the case when the moment capacity is reached, RSUF is 1 or -1. Results of RSUF for Models A1, A2 and A4 in Figure 8.13 show that, while the development of moment at M3 with imposed deflection there is little affected by the provision of top

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reinforcement at in-situ stitches, the development of hogging moments at the insitu stitches in the adjacent spans is substantially affected by such provision of top reinforcement there. For example, if no top reinforcement is provided across the in-situ stitches (i.e. Model A1), Stitches 2 and 4 will fail in hogging moments at imposed deflection at M3 of 200 mm. By providing about 5.8% of the amount of tendons at interior supports in Scenario A across each of the in-situ stitches (i.e. Model A4), Stitches 2 and 4 are prevented from failure in hogging moments before the failure at M3 in sagging moment at an imposed deflection there of around 350 mm. Therefore by providing a nominal amount of top reinforcement across the in-situ stitches, the bridge is maintained intact for a longer period during an extreme event.

8.5

Overall Assessment and Recommendation for Design From the above analyses of precast concrete segmental bridges, the

following classification of robustness is proposed: (a) Superb robustness: No collapse of any span occurs when an unforeseen load is applied. Upon imposition of displacement to simulate the unforeseen load, the structure is able to deform at reasonable resistance in spite of formation of plastic hinges. (b) Adequate robustness: Collapse of at most one span may occur when an unforeseen load is applied there.

Upon imposition of displacement to

simulate the unforeseen load, the resistance drops to almost zero after formation of plastic hinges, which are mostly located in the collapsed span. (c) Inadequate robustness: Progressive collapse of more than one span may occur when an unforeseen load is applied. Upon imposition of displacement to simulate the unforeseen load, the resistance drops to almost zero after formation of plastic hinges spread over a few spans. Figure 8.14 summarises the load-deflection relationship at M3 in various scenarios. In particular, some key points are identified by labels each comprising a letter and a number to denote key stages in each extreme event such as formation of plastic hinges. For convenience, Scenarios A, B and C are still denoted by A, B and C respectively, while AR denotes Scenario A with top

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reinforcement of Model A4 provided across in-situ stitches. The numbers come from the sequence of plastic hinge formation shown previously in Figures 8.5, 8.7 and 8.9 as appropriate. The load-deflection curves share the common property of increasing resistance up to deflections around 300-400 mm, followed by an abrupt drop associated with flexural failure at M3. Thereafter the resistance increases again possibly until something drastic happens. The graph of Scenario A shows that its resistance reaches a plateau after the formation of 5 plastic hinges in 3 stages and maintains its ability to deform further, which explains its superb robustness. The graph of Scenario B shows an abrupt reduction in resistance after the formation of 7 plastic hinges in 4 stages. As it results in collapse of a span without spreading to adjacent spans, it has adequate robustness. The graph of Scenario C shows an abrupt reduction in resistance after the formation of 13 plastic hinges in 6 stages, with the last two stages occurring almost at the same time. As the unforeseen load on Span 3 results in collapse of not only Span 3, but also the adjacent spans, it has inadequate robustness. Compared with Scenario A, providing top reinforcement of Model A4 across in-situ stitches strengthens the bridge up to an imposed deflection of 830 mm, after which the resistance drops to virtually the same value as in Scenario A with ability to deform further. This is also a case of superb robustness. These results reveal that, even though a bridge is designed for the standard load cases of ultimate limit states, different performance in robustness is possible. As performance in robustness is significantly affected by the full-range behaviour of plastic hinges at susceptible locations including the in-situ stitches and pier sections, analyses by imposed displacement at the in-situ stitches are desirable. Various susceptible locations should be examined to explore possible progressive collapse. A comprehensive evaluation of robustness should include not only unforeseen loads on the deck, but also possible uplift at deck ends due to unforeseen loads, loss of supports, etc. To ensure the robustness of a bridge, any span should be designed for the deck moments after the adjacent span has collapsed as if the remaining span has become a new end span. Although analysis of standard load cases may indicate absence of internal force of certain sign (e.g. hogging moment at in-situ stitches), -166-

it is still desirable to provide nominal resistance against such unforeseen internal forces for the sake of robustness. If possible, a bridge should be designed to achieve superb or at least adequate robustness.

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Table 8.1. Tendon arrangements. Tendon group 1 2 3 - 13 1 2 3 4

1

Number of Depth of tendon centroid tendons from top of deck Cantilever tendons (Figure 2(a)) ii 4 Constant at 150mm iii 4 iv – xii 2 each Continuity tendons along interior span (Figure 2(a)) iii 2 iv 2 Constant at 2,595mm v 2 vi 2 Continuity tendons along end span (Figure 2(b)) Varies from 1,002mm at i to i and xvi 2 2,595mm at iv; constant at 2,595mm from iv to xvi

Anchorage points

Table 8.2. Details of finite element mesh. Region a b c d e f g

Length (mm) 8,627 11,814 6,814 13,628 1,272 100 1,372

No. of equal elements 8 6 8 6 4 3 4

Table 8.3. Amount of top prestressing strands in a stitch. Model A1 A2 A3 A4 A5 A6

No. of prestressing strands 0 6 12 18 24 30

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Area of prestressing steel (mm2) 0 592 1,183 1,775 2,366 2,958

A Stitch 1 B Stitch 2 Span 1 10 30

In-situ concrete stitch C Stitch 3 D Stitch 4 M3 Span 3 30 30

Span 2 30 30

Span 4 30 30

E Stitch 5 F Span 5 30 10

(a) general arrangement (dimensions in m)

610 915 460 610

1,980

610 460 915

610

230

127 2,515 2,745

356 152 254 Top flange area Web area Bottom flange area

2,388 3,048

Bottom flange thickness: 203 for regular segment; 330 for pier segment; 203-330 for transition segment

(b) deck section (dimensions in mm)

Figure 8.1. Configuration of the bridge analysed: (a) general arrangement; and (b) deck section (not to scale).

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Pier

Cantilever tendons iv v vi vii

Joint numbers i ii iii

Anchorage

Continuity tendons Mid-span viii ix x xi xii

11 segments @ 2,600

Stitch

(a) interior span Abutment Joint numbers i iv ii iii

v vi

4 segments @ 2,475

Continuity tendons xi viii ix x

vii

xv

Pier xvii

xvi

11 segments Anchorage @ 2,600

Stitch

(b) end span

Figure 8.2. Tendon arrangements: (a) interior span; and (b) end span (dimensions in mm).

A

Stitch CL 10 10 a

20 b

a

Detail ‘A’

f

c

f

Detail ‘A’

30 d

c

d

Detail ‘A’

Pier CL e

Stitch CL

C

30

Detail ‘B’

Stitch CL e

Stitch CL

B

g

g

Detail ‘B’

30 c

c

Detail ‘B’

d

Detail ‘A’

Plastic hinge region Other region CL Centreline See Table 8.2 for a to g

Figure 8.3. Finite element mesh for analysis (dimensions in m).

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Stitch (a)

Stitch (b)

Figure 8.4. Examples of accidental loads on the bridge deck: (a) blasting as a result of vehicle collision; and (b) impact by a large boulder during a mudslide.

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3 A 200

Sequence of plastic hinge formation 1 2 1

B

C

Moment capacity

D

3 E

F

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

F

E

F

E

F

-50 -100

(a) early loading stage (max. deflection = 25 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

-50 -100

(b) after failure at M3 (max. deflection = 325 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

-50 -100

(c) after failure at Pier Sections B and E (max. deflection = 1,250 mm)

Figure 8.5. Variation of deck moments in Scenario A.

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A

B

C

D

E

C

D

E

F

Vertical displacement (m)

1.0 0.5 0.0 -0.5

A

B (c) (b) (a)

(a) (b) (c)

-1.0

F

(a) (b) (c)

-1.5

Figure 8.6. Variation of deck displacements in Scenario A: (a) failure at M3; (b) after failure at M3; and (c) failure at Pier Sections B and E.

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Sequence of plastic hinge formation 2 2 2 3 3

4 A 200

B

C

Moment capacity

D

4 E

F

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

F

E

F

-50 -100

(a) after failure at M3 (max. deflection = 360 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

-50 -100

(b) prior to failure at Pier Sections C and D (max. deflection = 2,905 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

-50 -100

(c) after failure at Pier Sections C and D (max. deflection = 2,925 mm)

Figure 8.7. Variation of deck moments in Scenario B.

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F

A

B

C

D

E

C

D

E

F

Vertical displacement (m)

1.0 0.0 -1.0

A

B

F

(a) (b)

(a) (b)

(a) (b) Collapsed span

-2.0 -3.0 -4.0

Figure 8.8. Variation of deck displacements in Scenario B: (a) prior to failure at Pier Sections C and D; and (b) collapse of Span 3.

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4 A 200

Sequence of plastic hinge formation 1 5 6 2 6 3 3 6 2 B

C

Moment capacity

6 5

D

E

4 F

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

F

-50 -100

(a) prior to failure at Pier Sections C and D (max. deflection = 2,745 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

F

-50 -100

(b) after failure at Pier Sections C and D (max. deflection = 2,765 mm)

200

Moment capacity

Moment due to loading

Moment (MNm)

150 100 50 0 A

B

C

D

E

-50 -100

(c) after failure at Pier Sections B and E (max. deflection = 2,775 mm)

Figure 8.9. Variation of deck moments in Scenario C.

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F

A

B

C

D

B

(a) C

D

E

F

Vertical displacement (m)

2.0 1.0 0.0

A

(a)

(a)

(b) -1.0

(b)

(a)

E (a)

F

(b) Collapsed spans

-2.0 -3.0

Figure 8.10. Variation of deck displacements in Scenario C: (a) failure at Pier Sections C and D; and (b) failure at Pier Sections B and E.

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A

Anchorage

Top reinforcement Blister Precast segments

A

Stitch

Elevation view

Top reinforcement

Blister

Section A-A Figure 8.11. Top reinforcement in the form of prestressing strands that are anchored to the blisters at the soffit of the deck (not to scale).

A

B

C

B

C

D

E

F

Vertical displacement (m)

0.2 0.1 0.0

A

D

E

F

-0.1 -0.2 -0.3

No top reinforcement (Model A1) Top reinforcement provided (Model A4)

-0.4 Figure 8.12. Deck displacements for stitches with and without top reinforcement.

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Residual strength utilization factor

1.0

A2 A2

A1 A1

Stitch 2: Stitch 3:

A4 A4

0.5 Imposed deflection (mm)

0.0 0

50

100

150

200

250

300

350

-0.5

-1.0

Figure 8.13. Variations of moments in Stitches 2 and 3 with imposed deflection.

Scenario B Scenario A Scenario C Scenario AR (with top reinforcement across stitches)

6000

AR1 A2 A1

Load (kN)

5000

B1 C1

4000

AR2 A3

3000

B3

B2 C4

2000 1000

C5 B4

C3

C2

0 0.0

0.5

1.0

1.5 2.0 2.5 Imposed deflection (m)

3.0

3.5

Figure 8.14. Summary of load-deflection relationship at M3 for various scenarios.

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CHAPTER 9 DESIGN CONSIDERATIONS

9.1

Overview In this chapter, various salient points of the design of in-situ concrete

stitches and the robustness of precast concrete segmental bridges (PCSB) that have been compiled from the results of the studies are presented.

9.2

In-Situ Concrete Stitches

9.2.1

Provision of shear keys In most cases, the provision of shear keys has positive effect on the shear

strength of in-situ stitches despite the occasional premature failure possibly as a result of significant stress concentration at the roots and corners of the shear keys. Therefore, the effects of stress concentration should be borne in mind in the design of shear keys for PCSB. One can consider increasing the shear key angle θkey as defined in Figure 9.1 from that found in the conventional design as provided by, for instance, the AASHTO guide (AASHTO, 1999). Large shear keys that are spaced far apart should be avoided to prevent significant stress concentration at the roots and corners of just a few keys. Numerical results have shown that keys that are relatively large are more prone to stress concentration at the roots. The arrangement of shear keys as specified by the AASHTO guide (AASHTO, 1999) is reasonable.

9.2.2

Dimensions of in-situ concrete stitch There is no definite correlation between the width the in-situ stitches and

their load-carrying capacity for stitch width ranging from 100 to 200 mm. In most cases, it is found that the width of stitch has little effect on the load-carrying capacity. The aspect ratio of stitch (i.e. ratio of depth to width), on the other

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hand, is more influential as the shear capacity increases mildly with increase in aspect ratio.

9.2.3

Initial prestressing level The increase in initial prestressing level can increase the shear strength of

the in-situ stitch but it is found that the effect subsides when the prestressing level is increased from 3 MPa to 4 MPa. In-situ stitches that are located in an area of high moment should have sufficient initial prestressing force so that the stitches experience no tension. Results reveal that significant concentration of curvature would occur at the construction joint between the stitch and precast segment once the joint cracks, which would lead to excessive crack width. This measure is to prevent excessive crack width at the location of the construction joint.

9.2.4

Importance of surface roughness The surface roughness at the construction joint has significant effect on

the shear strength of the in-situ stitch. Not only does the increase in roughness contribute to the increase in the frictional resistance, it also increases the lateral dilation that in turn increases the stress in the tendon and hence a larger prestressing force, which can ultimately increase the shear capacity of the stitch.

9.2.5

Estimation of shear strength The shear strength V of keyed stitch can be estimated by the modified

AASHTO formula that is introduced in Section 6.5, namely

  V = α dwα sc  Ak (1 + 0.2σ ) f ck + 0.6 Asmσ   

α dw

 w 1 = 1 − tan arctan h  2 

 2 f t + f tσ   σ / 2  

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(9.1)

(9.2)

where αdw is the coefficient to account for the effects of the aspect ratio of stitch, while the coefficient αsc is taken as 0.62 to account for the effects of stress concentration; Ak is the area (mm2) of all base areas of shear keys and Asm is the area of contact between smooth surfaces of the joint as defined in Figure 6.24; fck and ft are the characteristic compressive strength (MPa) and the tensile strength (MPa) of concrete respectively; w and h are the width (mm) and the entire depth (mm) of the stitch respectively; and σ is the normal stress across the stitch (MPa).

9.3

Robustness Design of Multi-Span Precast Concrete Segmental Bridges Even though a bridge is designed for the standard load cases of ultimate

limit states, different performance in robustness is possible. The robustness for the design of multi-span precast concrete segmental bridges can be evaluated as follows: (a) Superb robustness: No collapse of any span occurs in an extreme event and the structure is able to deform at reasonable resistance in spite of formation of plastic hinges. (b) Adequate robustness: Collapse of at most one span may occur in an extreme event. Formation of plastic hinges is expected to be confined to one span. (c) Inadequate robustness: Progressive collapse of more than one span may occur in an extreme event. Formation of plastic hinges is expected to spread over a few spans. If possible, a bridge should be designed to achieve superb or adequate robustness. To do so, any span should be designed for the deck moments after the adjacent span has collapsed as if the remaining span has become a new end span. Nominal top reinforcement should be provided across in-situ stitches to enhance the robustness of PCSB. By doing so, the integrity of stitches can be preserved when they are subject to hogging moment under unexpected circumstances, thereby reducing the chance of forming a mechanism along the bridge deck. Top reinforcements may be provided in the form of, for example, prestressing strands or threaded bars that are anchored to the blisters at the soffit

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of the deck (Figure 9.2), or fiber reinforced polymer (FRP) sheets that are adhered to the soffit of the deck (Figure 9.3).

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θkey θkey

Figure 9.1. Definition of key angle θkey.

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Anchorage

A

Top reinforcement Blister Precast segments

A

Stitch

Elevation view

Top reinforcement in the form of prestressing strands, threaded bars etc.

Blister

Section A-A Figure 9.2. Schematic diagram illustrating the provision of top reinforcement in the form of strands or bars across in-situ stitch (not to scale).

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Anchorage

A

FRP sheet Bonded Unbonded

A

Stitch

Elevation view

FRP sheets

Section A-A Figure 9.3. Schematic diagram illustrating the provision of top reinforcement in the form of fibre reinforced polymer (FRP) sheets across in-situ stitch (not to scale).

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CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

10.1

Conclusions Extensive experimental and numerical studies have been carried out to

investigate the behaviour of in-situ concrete stitches and the effects of their performance on the global response of multi-span precast concrete segmental bridges (PCSB). A numerical method for nonlinear finite element analysis has been developed for the global analysis. In Chapter 4, the ductility and deformability of reinforced and prestressed concrete sections are examined since they are essential to the redistribution behaviour of internal moment along a continuous member that can ultimately affects the global response of a bridge deck. It has found that the ability of a prestressed concrete section against inelastic deformation measured by the convention curvature ductility factor can be higher than that of a reinforced concrete section on an equal strength basis, thus giving rise to the impression that prestressing can increase ductility. However, this is just an illusion as there is actually no increase in ultimate curvature by prestressing and the apparent increase in ductility factor is due solely to the reduction in yield curvature. Therefore, for prestressed concrete sections, it is more appropriate to adopt the concept of flexural deformability, which is taken as the product of the ultimate curvature and the overall depth of section. In Chapters 5 to 6, the behaviour of plain and keyed in-situ concrete stitches subject to different combinations of internal forces has been examined. The typical failure modes of plain and keyed stitches are frictional sliding along the construction joint and diagonal cracking across the stitch respectively. The effects of several parameters, namely the width and aspect ratio of stitch, initial prestressing forces, concrete strength, provision of shear keys, bonding of the tendon, and roughness of the construction joint surface, have been observed experimentally and analysed using a commercial finite element package. The -189-

variation of stitch width tends to have little effect on the load-carrying capacity in most cases, and the shear capacity tends to increase with the aspect ratio of stitch. The increase in initial prestressing force has favourable effects on the load-carrying capacity, albeit with diminishing return. The concrete strength and roughness of the joint surface are correlated with the load-carrying capacity. Providing shear keys to in-situ stitch is effective against shear, but insignificant improvement on load-carrying capacity is observed when the stitch is subjected primarily to bending moment.

Moreover, due care should be exercised in

providing shear keys to avoid significant stress concentration with sharp key corners. Stress concentration may have detrimental effects that can result in premature failure of stitches. Towards the end of Chapter 6, an upper-bound formula that predicts the shear strength of keyed stitch has been derived based on the numerical results. In Chapter 7, the behaviour of a typical segmental bridge with full scale stitches is discussed. The analysis of full-range behaviour shows a three-stage response associated with shear cracking of stitch, flexural failure of pier segment and shear failure of the bridge. The development of internal stress and strain within the stitch has been examined. The ultimate load-carrying capacity of the bridge is governed by the flexural strength of pier segments rather than the shear strength of the stitch. In Chapter 8, the global response of a typical segmental bridge deck in an extreme event is examined for its robustness. The variation of internal forces and formation of collapse mechanisms is analysed. It is found that, although a bridge is designed for the standard load cases of ultimate limit states, different performance in robustness is possible. Three classes of robustness are proposed, namely superb, adequate and inadequate robustness.

A bridge should have

superb or adequate robustness by ensuring that any span should be designed for the deck moments after its adjacent span has collapsed and the original span has become a new end span. Providing nominal top reinforcement across the stitch has found to be effective in enhancing robustness of the bridge deck.

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10.2

Recommendations for Future Work

10.2.1

Behaviour of in-situ concrete stitches reinforced by FRP Most of the existing PCSBs are designed without reinforcement across

the in-situ stitches. As the present research has demonstrated the effectiveness of the provision of nominal reinforcement, it would be worthwhile to consider the possibility of retrofitting existing PCSB. Fibre-reinforced polymer (FRP) is a popular material for retrofitting nowadays due to the relative ease of its application, especially for large flat surfaces such as bridge decks. The use of FRP for retrofitting in-situ concrete stitches is promising and it should be futher investigated.

10.2.2

Shear keys of PCSB One of the problems with the use of shear keys is the detrimental effects

of stress concentration. The intensity of stress concentration depends on the exact dimensions of the shear keys. The present study has only examined the configuration recommended by AASHTO, which may not be the optimal design. A systematic analysis of shear key configurations on the capacity of stitches should be undertaken.

10.2.3

Robustness of multi-span bridges The topic on robustness of multi-span bridges is a topic worthwhile of

further investigation.

While the present study has been confined to the

robustness of bridge deck subject to accidental load on the deck in an extreme event, the loss of pier is another scenario that can be investigated. robustness of other bridge forms should also be studied.

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The

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Journal

of

Composites

for

PUBLICATIONS BY THE AUTHOR

Refereed Journal Papers: [1]

F.T.K. Au, C.C.Y. Leung and A.K.H. Kwan (2011). Flexural ductility and deformability of reinforced and prestressed concrete sections. Computers and Concrete, 8(4):473-489.

[2]

Y. Zeng, F.T.K. Au and C.C.Y. Leung (2011). Analysis of in-situ stitches in precast concrete segmental bridges. The IES Journal Part A: Civil & Structural Engineering, 5(1):1-15.

[3]

C.C.Y. Leung, F.T.K. Au and A.K.H. Kwan (2012). Nonlinear analysis and moment redistribution of PC members. Proceedings of the Institution of Civil Engineers – Engineering and Computational Mechanics. (in press)

[4]

F.T.K. Au, C.C.Y. Leung and A.K.H. Kwan. Collapse mechanism and robustness design of prestressed concrete segmental bridges, Proceedings of the Institution of Civil Engineers – Bridges. (accepted)

[5]

C.C.Y. Leung and F.T.K. Au. Flexural behaviour of precast concrete members with in-situ concrete stitches. Magazine of Concrete Research. (under preparation)

Refereed Conference Papers: [1]

F.T.K. Au, R.J. Jiang, C.C.Y. Leung, L.Z. Xie, P.K.K. Lee, K.Y. Wong and W.Y. Chan. Development and calibration of finite element models for structural health monitoring of Ting Kau Bridge, Proceedings of the 3rd International Symposium on Environmental Vibrations: Prediction, Monitoring, Mitigation and Evaluation, Taipei, 28-30 November, 2007, pp. 559-566.

[2]

C.C.Y. Leung and F.T.K. Au. Behaviour of in-situ concrete stitches in segmental prestressed concrete bridges, Proceedings of the Second International

Postgraduate

Conference -203-

on

Infrastructure

and

Environment, Hong Kong, 1-2 June, 2010, Vol. 2, pp. 256-262. (awarded the Best Presenter Award of the Infrastructure Stream) [3]

C.C.Y. Leung and F.T.K. Au. Shear behaviour of in-situ concrete stitches of precast concrete segmental bridges, Proceedings of the 21st Australian Conference on the Mechanics of Structures and Materials, Melbourne, Australia, 7-10 December, 2010, pp. 233-237.

[4]

Au F.T.K. and Leung C.C.Y. Full-range analysis of multi-span prestressed concrete segmental bridges. Proceedings of the Twelfth East Asia-Pacific Conference on Structural Engineering and Construction, Hong Kong, 2011, EASEC12-529.

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